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A 


t&TRQNQM 


TREATISES 


PHYSICAL    ASTRONOMY, 


LIGHT    AND    SOUND. 


CONTRIBUTED    TO    THE    ENCYCLOPAEDIA    METROPOLIT  AN  A. 


SIR  JOHN  F.  W.  HERSCHEL,  BART,,  M.A.,  F.R.S. 


ST     JOHN'S   COLLEGK,    CAMBRIDGE. 


LONDON     AND     GLASGOW: 
RICHARD     GRIFFIN     AND     COMPANY 

PUBLISIIEB8  TO  THE  USIVEBSITT  OF  GLASGOW. 


ASTRONOMY  UBRARV 


ASTRONOMY 
U8RARY 


CONTENTS. 


PHYSICAL   ASTRONOMY. 


INTRODUCTION 


Page 
647 


PART  I. 


SKCTHIX      I.  On  the  Circular  and  Elliptic  Motions  of  the 

Ptanets  and  Satellites    ...  .649 

II.  On  the  Attractions  of  Spherical  Bodies     .      Col 

III.  Theory  of  Elliptic  Motion          .         .        .       653 

IV.  On  the  Velocities  of  the  Planetary  Motions, 

and  the  Determination  a  priori  of  the  ele- 
ments of  their  Orbits      ....      658 
V.  Peculiar  Cases  of  Celestial  Motion    .        .      GOO 
VI.  On  the  Determination  of  the  Planetary  Or- 
bits a  posteriori,  or  from  Observation       .       663 

PAUT  II. 

SECTION  I.  Of  the  Perturbation  of  the  Elliptic  Motions 
of  the  Heavenly  Bodies,  arising  from  their 
Mutual  Attraction  ....  673 
Investigation  of  the  Forces  exerted  by  one 
body  to  disturb  the  Orbit  of  another  re- 
volving about  a  common  central  body,  and 
of  the  differential  equations  of  their  Mo- 
tions   673 

II.  General  Theory  of  the  Planetary  Perturba- 
tions depending  on  their  Mutual  Configu- 
rations   679 

III.  Investigation  of  the  Perturbations  neglect- 

ing the  Eccentricities  of  both  Orbits      .      683 


Page 


SECTION  IV.  Of  the  Method  of  taking  into  account  the 
effect  of  the  Eccentricities  of  the  Orbits  on 
the  Planetary  Perturbations,  and  of  the 
origin  of  the  Secular  Equations  of  their 

Motions 688 

V.  Of  the  Inequalities  dependingon  the  Squares 

and  higher  powers  of  the  Eccentricities  .  694 
VI.  Of  the  Variations  of  the  Elements  of  the 
Planetary  Orbits,  and  the  Secular  Equa- 
tionsof  their  Motions.  Theory  of  the  Major 
Axes,  Inclinations,  Nodes,  Eccentricities, 
and  Aphelia, 699 

PART  III.  OF  THE  THEORY  OF  THE  MOON. 

SECTION      I.  Rigorous  Investigation  of  the  Differential 
Equationsof  the  Moon's  Motion,  and  gene- 
ral expression  of  the  disturbing  Forces    . 
II.  Approximate  Integration  of  the  equation  of 
the  Moon's  Orbit 

III.  Expression  of  the  Moon's  MeanLongitudein 

terms  of  the  True,  and  vice  versa;  and  of 
the  variation  and  evectiou  of  the  Moon  . 

IV.  Of  the  Effect  of  the  Inclination  of  the  Plane 

of  the  Moon's  Orbit.  Of  the  Motion  of  the 
Nodes,and  the  Precession  of  the  Equinoxes      725 

Alphabetical  Index 733 


714 
719 


724 


SOUND. 


PART  I.— OF  THE  PROPAGATION  OF  SOUND  IN  GENERAL  . 

SECTION  1.  Of  the  Propagation  and  Velocity  of  Sound  in 

Air 

2.  Mathematical  Theory  of  the  Propagation  of 

Sound  in  Air,  and  other  Elastic  Fluid  Media 

3.  Of  the  Linear  Propagation  of  Sound  in  Gases 

and  Vapours 

4.  Of  the  Propagation  of  Sound  through  Liquids 
."i.  Of  the  Propagation  of  Sound  in  Solids  and  in 

Mixed  Media 

l».  Of  the  Divergence  and  Decay  of  Sound 
7.  Of  the  Reflexion  and  Refraction  of  Sound,  and 

of  Echos 

1' AIM  II. — OF  MUSICAL  SOUNDS 

SKCTION  1.  Of  the  Nature  and  Production  of  Musical 
Sounds  


747 
747 
754 

764 

767 

770 
773 

774 

777 

777 


SECTION  2.  Of  the  Vibrations  ofMechanical  Strings  or  Cords  778 

3.  Of  the  Vibrations  of  a  Column  of  Airof  defin- 

ite Length 785 

4.  Of  Musical  Intervals,  of  Harmony,  and  Tem- 

perament   , 790 

5.  Of  the  Sonorous  Vibrations  of  Bars,  Rods,  and 

Plates 801 

PABTHI 804 

SECTION  1.  Of  the  Communication  of  Vibrations,  and  of 

the  Vibrations  of  Systems          .        .        .  804 

2.  Of  the  Communication  of  Vibrations  from  one 

Vibrating  Body  to  another         .        .        .  807 

3.  Of  the  Voice 815 

Remarks  on  Written  Language       .  817 

Alphabetical  Index 821 


M677225 


CONTEXTS. 


LIGHT. 


PART  I. — OF  UNPOLABIZBD  LIGHT          .... 

SECTION  1.  Introduction 

2.  Of  Photometry  

3.  Of  the  regular  Reflexion  of  Unpolarized  Light 

from  Plane  Surfaces 

4.  Of  Reflexion  from  Curved  Surfaces 

5.  Of  Caustics  by  Reflexion,  or  Catacaustics    . 

OF  THE  REGULAB  REFRACTION  OF  LlGHT 
BY  UNCKYSTALLIZED  MEDIA 

6.  Of  the  Refraction  of  Homogeneous  Light  at 

Plane  Surfaces 

7.  Of  Ordinary  Refraction  at  Curved  Surfaces, 

and  of  Diacaustics,  or  Caustics  by  Refraction 

8.  Of  Caustics  by  Refraction,  or  Diacaustics   . 

9.  Of  the  Foci  of  Spherical  Surfaces  for  Central 

Rays 

10.  Of  the  Aberration  of  a  System  of  Spherical 

Surfaces 

11.  Of  the  Foci  for  Oblique  Rays,  and  of  the 

Formation  of  Images          .... 

12.  Of  the  Structure  of  the  Eye,  and  of  Vision  . 

Of  Optical  Instruments       .        . 

PART  II. — CHROMATICS 

SECTION  1.  Of  the  Dispersion  of  Light  .... 

2.  Of  the  Achromatic  Telescope 

3.  Of  the  Absorption  or  Extinction  of  Light  by 

Uncrystallized  Media          .... 

PART  III. — OF  THE  THEORIES  OF  LIGHT 

SECTION  1 .  Of  the  Newtonian,  or  Corpuscular  Theory  of 
Light  

2.  General  Statement  of  the  Undulatory  Theory 

of  Light 

3.  Of  the  Interference  of  the  Rays  of  Light     . 

4.  Of  the  Colours  of  Thin  Plates      . 

5.  Of  the  Colours  of  Thick  Plates    . 

6.  Of  the  Colours  of  Mixed  Plates  . 

7.  Of  the  Colours  of  Fine  Fibres  and  Striated 

Surfaces 

8.  Of  the  Diffraction  of  Light 


Page 
341 

341 
344 

332 
354 
3GO 


3G7 


367 

375 
377 

379 
385 

392 
396 
401 

405 

405 

421 

430 
439 

439 

449 
456 
463 
473 

477 

478 
479 


I'age 
494 

494 


PART  IV. — OF  THE  AFFECTION!  OF  POLARIZED  LIGHT   . 

SECTION  1.  Of  Double  Refraction 

Of  the  Law  of  Double  Refraction  in 

Crystals  with  one  Optic  Axis  .      495 

Of  the  Polarization  of  Light       .        .      503 
'2.  General  Ideas  of  the  Distinction  between  Po- 
larized and  Unpolarized  Light    .         .         .       503 

3.  Of  the  Polarization  of  Light  by  Reflexion    .      505 

4.  Of  the  Laws  of  Reflexion  of  Polarized  Light  .      509 

5.  Of  the  Polarization  of  Light  by  ordinary  Re- 

fraction, and  of  the  Laws  of  the  Refraction 

of  Polarized  Light 511 

6.  Of  the  Polarization  of  Light  by  Double  Re- 

fraction        513 

7.  Of  the  Colours  exlu'bited  byCrystallizcd  Plates 

when  exposed  to  Polarized  Light,  and  of  the 
Polarized  Rings  wliich  surround  their  Optic 
Axes 515 

8.  On  the  Interferences  of  Polarized  Rays        .      529 

9.  Of  the  Application  of  the  Undulatory  Doctrine 

to  the  Explanation  of  the  Phenomena  of  Po- 
larized Light  and  of  Double  Refraction  .  533 

10.  Of  Circular  Polarization      ....      548 

11.  Of  the  Absorption  of  Light  by  Crystallized 

Media 554 

12.  OntheEffectsofHeatandMechanicalViolence 

in  modifying  the  Action  of  Media  on  Light, 
and  on  the  Application  of  the  Undulatory 
Theory  to  their  Explanation  .  .  .  562 

13.  Of  the  Use  or  Properties  of  Light  in  affording 

Characters  for  determining  and  identifying 
Chemical  and  Mineral  Species,  and  for  in- 
vestigating the  intimate  Constitution  and 
Structure  of  Natural  Bodies       .        .        .568 
Table  of  Refractive  Indices         .        .      569 
Table  of  Intrinsic  Refractive  Powers  .      573 
Table  of  Dispersive  Powers         .        .      574 
Table  of  the  Inclinations  of  the  Optic 
Axes  in  Uniaxial  and  Biaxial  Crystals 
severally 576 

14.  On  the  Colours  of  Natural  Bodies        .        .      579 

15.  Of  the  Calorific  and  Chemical  Rays  of  the  Solar 

Spectrum 


Alphabetical  Index 


581 
583 


ENGRAVINGS. 

PHYSICAL  ASTRONOMY,— Plate  of  Diagrams. 
SOUND, — Plates  1  to  6. 
LIGHT.— Plates  1  to  14. 


PHYSICAL    ASTRONOMY. 


Astronomy  THE  object  of  the  philosopher  in  the  investigation 
— •- \s~~~'  of  nature  is  to  arrange  and  classify  facts  and  pheno- 
mena, with  a  view  to  trace  the  agency  of  their  remote, 
or,  at  least,  their  proximate  causes,  and  ascend  as  high 
as  the  imperfection  of  human  means  of  observation, 
and  the  limited  powers  of  the  human  intellect  will 
allow  us  in  the  scale  of  generalization. 

To  beings  endowed  with  more  perfect  faculties, 
and  more  comprehensive  intelligence  than  ourselves, 
much  of  that  complication  we  observe  in  natural 
phenomena  would  disappear :  many  effects,  which 
seem  to  us  independent  of  each  other,  and  linked  by 
no  natural  connection,  would  be  in  their  eyes  col- 
lateral results  of  one  and  the  same  principle  :  much, 
that  to  us  seems  fortuitous  would  to  them  appear  pre- 
disposed and  regularly  arranged.  The  laws  of  nature 
would  at  once  be  reduced  in  number  and  enlarged 
in  extent ;  and  that  higher  order  of  generalization 
which  would  consist  in  classifying  together  laws  of 
the  same  kind,  and  referring  them  to  others  yet  more 
universal,  would  exercise  their  power  and  constitute 
their  science.  That  such  would  be  the  case  with  more 
perfect  beings,  our  own  experience,  limited  as  it  is, 
amply  shews.  The  man  who  has  learnt  to  regard  the 
fall  of  a  leaf  and  the  precession  of  the  equinoxes  as 
results  equally  certain  and  unavoidable  of  a  law 
capable  of  being  stated  in  three  lines,  and  understood 
by  a  child  of  ten  years  old,  has  made  already  a  con- 
siderable step  in  this  way — the  patient  exercise  of  his 
natural  reason  has  stood  him  in  the  stead  of  a  sharper 
intellect ;  and  secrets  which  an  angel  might  pene- 
trate perhaps  at  a  glance,  become  revealed  to  man 
by  the  slow,  yet  sure,  effects  of  persevering  thought. 

The  progress  of  modern  science  has  done  more 
than  the  keenest  metaphysical  reasoning,  and  has 
given  us  the  most  convincing  proofs  of  the  agency  of 
one  general  and  intelligent  cause  throughout  the 
whole  system  of  nature.  When  we  see  on  all  sides 
phenomena  grouping  themselves  under  laws  in- 
telligible and  simply  expressed,  which  are  them- 
selves subordinate  to  others,  yet  more  simple  and 
extensive;  when  we  see  every  anomaly  which  threat- 
ened destiuction  to  a  theory,  becoming,  in  the  pro- 
gress of  our  knowledge,  its  firmest  support ;  every 
inequality  disappearing  when  viewed  from  a  higher 
level ;  every  exception  proving  a  rule  of  greater 
generality  ;  all,  in  short,  conveying  more  and  more 
towards  order  and  simplicity  the  more  severely  we 
scrutinize  it ;  it  is  impossible  not  to  allow  that  that 
last  great  step,  which  unites  all  the  phenomena  of 
the  universe  under  one  general  head,  and  refers  them 
to  one  all-pervading  agency — however  inconceivably 
remote,  and  surpassing  probably  the  utmost  limits  of 
the  human  intellect  to  comprehend,  if  explained,  would 
Still  be  but  the  continuation  and  final  completion  of  a 
VOL.  in 


chain  of  reasoning  whose  first  links  we  hold  within    physical 
our  grasp — the   consummation  of  a  process  actually  Astronomy. 
begun — the  termination  of  a  career  into  which  we  are  — ~V~~-"' 
fairly  entered. 

It  is  difficult  to  avoid  such  contemplations  at  the 
outset  of  an  essay  on  physical  astronomy  ;  they  crowd 
upon  us ;  and  in  rejecting  them  we  should  reject  the 
noblest  use  of  the  sublimest  of  sciences.  For  scarcely 
in  any  are  the  phenomena  presented  by  nature  more 
various  and  more  complicated  ;  in  none  is  the  gene- 
ralization so  complete,  the  final  result  so  simple,  or 
the  object  more  imposing 

From  what  has  just  been  said,  it  may  be  gathered 
that  the  object  of  physical  astronomy  is  to  reduce, 
under  general  laws,  the  motions  and  phenomena  of 
the  heavenly  bodies,  and  investigate  their  causes  ;  to 
trace  the  history  of  what  has  already  happened  in  our 
own  system,  and  to  ascertain  what  changes  the  causes 
demonstrably  in  action  (unless  interfered  with  by  others 
we  have  no  knowledge  of,)  will  superinduce  in  thecourse 
of  ages,  and  thus  to  appreciate  the  stability  of  the  pre- 
sent order  of  creation.  In  a  more  limited  and  practical 
point  of  view,  the  physical  astronomer  is  called  on  to 
furnish  formulae  deduced  from  theory  for  determining 
the  state  of  the  system  at  any  assigned  instant ;  and 
adapted  for  the  purposes  of  the  observer,  so  as  to 
serve  as  a  basis  for  the  construction  of  tables  ;  and  to 
descend,  by  the  application  of  his  general  principles,  to 
those  more  refined  inequalities  which,  owing  to  their 
minuteness,  or  the  length  of  their  periods,  would 
escape  or  mislead  the  observer  unassisted  by  theory. 

There  is  one  feature  in  physical  astronomy  which 
renders  it  remarkable  among  the  sciences,  and  has 
been  the  chief,  if  not  the  only,  source  of  the  per- 
fection it  has  attained.  It  is  this — that  the  funda- 
mental law  embracing  all  the  minutiae  of  the  pheno- 
mena so  far  as  we  yet  know  them,  presents  itself  at 
once,  on  the  consideration  of  broad  features  and  gene- 
ral facts,  deduced  by  observations  even  of  a  rude  and 
imperfect  kind,  in  such  a  form  as  to  require  no  modifi- 
cation, extension,  or  addition  when  applied  in  minute 
detail.  In  other  sciences,  when  an  induction  of  a 
moderate  extent  has  led  us  to  the  knowledge  of  a  law 
which  we  conceive  to  be  general,  the  further  progress 
of  our  inquiries  frequently  obliges  us  either  to  limit 
its  extent  or  modify  its  expression.  To  those  who  are 
familiar  with  the  history  of  chemistry,  instances  of 
this  will  present  themselves  at  every  turn.  In  physical 
optics,  the  general  representation  of  all  the  series  of 
polarised  tints  and  the  colours  of  natural  bodies  by  a 
certain  universal  scale — the  Cartesian  law  of  refrac- 
tion when  applied  to  the  extraordinary  ray  in  crystal- 
lized media,  and  even  to  the  ordinary,  if  the  reports 
of  some  recent  experiments  are  to  be  relied  on — toge- 
ther with  innumerable  other  laws,  simple,  natural. 
4  v 


648 


PHYSICAL    ASTRONOMY. 


Astronomy,  and  resting  on  extensive  inductions,   have  all   been 
*— — •/"•••'  either  overset,   extended,   or  materially  modified   by 
the  progress  of  the  science. 

In  physical  astronomy,  however,  when  taken  in  that 
limited  acceptation,  which  restricts  it  to  the  explana- 
tion of  the  planetary  motions,  our  first  conclusion  is 
our  last.  The  law  on  which  all  its  phenome  in 
depend,  flows  naturally  and  easily  from  the  simplest 
among  them,  as  presented  by  the  rudest  observation ; 
and,  in  point  of  fact,  such  has  really  been  the  order 
of  investigation  in  this  science.  The  rude  supposition 
of  the  uniform  revolution  of  the  moon  in  a  circle  about 
the  earth  as  a  centre,  led  Newton  at  once  to  the  true 
law  of  gravity,  as  extending  from  the  earth  to  its 
companion.  The  uniform  circular  motions  of  the 
planets  about  the  sun,  in  times  following  the  progres- 
sion assigned  by  observation  in  Kepler's  rule,  con- 
firmed the  law,  and  extended  its  influence  to  the 
boundaries  of  our  system.  Every  thing  more  refined 
than  this — the  elliptic  motions  of  the  planets  and 
satellites  —  their  mutual  perturbations  —  the  slow 
changes  of  their  orbits  and  motions,  denominated 
secular  variations — the  deviation  of  their  figures  from 
the  spherical  form — the  oscillatory  motions  of  their 
axes,  which  produce  nutation  and  the  precession  of 
the  equinoxes — the  theory  of  the  tides,  both  of  the 
ocean  and  the  atmosphere,  have  all  in  succession 
been  so  many  trials  for  life  and  death  in  which  this 
law  has  been,  as  it  were,  pitted  against  nature  ;  trials, 
whose  event  no  human  foresight  could  predict,  and 
where  it  was  impossible  even  to  conjecture  what 
modifications  it  might  be  found  to  need.  Even  at  this 
moment,  if  among  the  innumerable  inequalities  of  the 
lunar  or  planetary  motions  any  one,  however  small, 
should  be  discovered  decidedly  not  explicable  on  the 
hypothesis  of  a  force  varying  as  the  inverse  square  of 
the  distance,  that  hypothesis  must  be  modified  till  it 
accounts  for  it.  It  is  hardly  necessary  to  add,  how- 
ever, that  in  the  present  state  of  science,  this  is  a  case 
not  to  be  contemplated. 

Still,  these  are  refinements.  The  deviations  of  the 
planetary  orbits  from  circles  are  small,  their  devia- 
tions from  ellipses  excessively  minute :  the  lunar  orbit 
alone  presents  results  of  perturbation  so  large  as  to 
strike  us  at  once  with  the  appearance  of  irreconcilable 
anomalies,  but  it  is  only  by  a  refinement  of  calcula- 
tion that  we  can  trace  them  all  to  the  laws  of  gravity; 
but  the  motions  of  comets  put  the  truth  and  generality 
of  this  law  to  a  severe  and  rude  test,  by  giving  it  a 
trial  under  the  greatest  varieties  of  distance,  position, 
and  velocity  of  motion,  and  instancing  its  influence 
on  matter  of  a  rarity  almost  spiritual,  and  differing  so 
utterly  from  that  of  which  our  planet  consists,  as 
scarcely  to  authorize  the  admittance  of  any  property 
in  common. 

The  above  observations  have  been  made  in  con- 
formity with  the  general  language  of  natural  philoso- 
phers, and  the  customary  acceptation  of  the  term 
physical  astronomy,  ani  are,  no  doubt,  strictly  appli- 
cable when  we  confine  ourselves  to  the  celestial  phe- 
nomena of  our  own  immediate  system,  and  the  motions 
of  those  larger  masses  of  matter  of  which  planets  and 
tncir  satellites  consist.  The  cautious  philosopher 
however  will  still  regard  it  as  worthy  inquiry,  whether, 
at  enormous  distances,  like  those  of  the  fixed  stars,  or 
at  such  comparatively  microscopic  intervals  as  those 
we  are  ordinarily  conversant  with  on  the  surface  of 


our  planet,  the  rigorous  law  of  a  force  as  the  inverse  Physical 
square  of  the  distance  may  not  suffer  some  modifica-  Astronomy 
tion.  An  emanation,  like  light,  traversing  in  sue-  ^~~^~~"^ 
cession  every  part  of  space,  may  be  conceived  to  go 
on  without  acceleration  or  retaidation,  without  loss  or 
change,  to  the  remotest  regions  ;  but  an  active  and 
immediate  intercourse  carried  on  between  points 
infinitely  distant,  is  not  only  incapable  of  demonstra- 
tion, but,  could  it  be  proved,  must,  I  suppose,  be 
referred  to  direct  spiritual  agency.  At  the  same  time, 
it  is  worthy  attention  how  strict  and  indissoluble  a 
bond  gravity  establishes  between  remote  objects — to 
see  this  in  its  real  light,  we  must  compare  it  with  the 
most  effectual  of  our  ordinary  means  of  transmitting 
power.  If  the  earth  and  sun  were  connected  by  a  rod 
of  cast  iron,  in  one  piece,  an  impulse  or  pull,  however 
violent  and  sudden,  applied  at  the  sun,  would  not 
begin  to  be  felt  at  the  earth  tiL  after  a  lapse  of 
eighteen  months  from  the  moment  of  its  communi- 
cation, while  a  change  in  the  sun's  attraction,  such  as 
might  arise  from  a  sudden  alteration  of  its  figure  or 
density,  would  demonstrably*  affect  our  planet  in  an 
instant  of  time  many  thousand  times  less  than  the 
least  interval  perceptible  to  our  senses. 

The  subsistence  of  sidereal  clusters,  in  which  the 
compression  or  crowding  of  the  stars  is  carried  to  the 
extent  we  have  instances  of  in  many  parts  of  the 
heavens,  seems  hardly  compatible  with  a  gravitating 
force,  unopposed  by  some  principle  of  conservation, 
unless  we  suppose  them  in  a  state  rapidly  verging  to 
a  catastrophe.  On  the  other  hand,  with  regard  to 
small  distances,  we  have  no  distinct  proof,  that  within 
a  few  inches,  or  even  miles,  from  a  material  point,  the 
law  of  gravity  may  not  begin  to  deviate  appreciably 
from  the  Newtonian  law.  The  experiments  of  Mask- 
elyne  and  Cavendish,  which  may  perhaps  be  adduced 
as  supporting  its  rigorous  application,  are  far  too 
gross,  and  differ  too  widely  in  their  results,  to  be 
cited  in  so  delicate  a  matter,  besides  which,  their 
results,  as  applied  to  such  an  inquiry,  are  affected 
with  an  unknown  element,  the  mean  density  of  the 
earth.  At  much  closer  intervals  we  are  certain  of  the 
existence  of  attractive  and  repulsive  forces  following 
a  widely  different  law;  and  by  what  imperceptible 
gradations  these  shade  into  that  of  gravity,  or  whe- 
ther they  are  to  be  regarded  as  distinct  from  it  in 
their  nature  and  origin,  is  a  point  whose  consideration 
seems  reserved  for  a  much  higher  state  of  science 
than  we  can  boast  of  having  yet  attained. 

But  it  is  quite  sufficient  for  the  purposes  of  physical 
astronomy  to  know,  that  as  far  as  the  motions  of  the 
great  masses  of  matter  connected  with  our  system 
either  in  the  heavens  or  on  our  globe  are  concerned, 
observation  and  theory  present  no  difference  capable 
of  being  made  an  objection  to  the  strict  expression  of 
Newton's  law,  and  we  shall  therefore  wave  all  further 
discussion  of  the  subject,  and  proceed  to  the  object  of 
the  present  essay,  in  which  the  reader  will  be  pre- 
sumed acquainted  with  the  general  facts  of  astronomy, 
with  the  principles  of  mechanical  philosophy,  and  so 
much  of  analysis,  of  the  differential  and  integral  cal- 
culus, plane  and  spherical  trigonometry,  as  shall 
render  it.  unnecessary  for  us  to  interrupt  the  general 
chain  of  our  reasoning  to  demonstrate  such  theorems, 
&c.  as  we  shall  have  occasion  to  call  to  our  aid. 

•  Laplace,  Sytltme  du  Mondt,  p.  28P. 


PHYSICAL    ASTRONOMY 


649 


Astronomy  SECTION  I. 

"— *~\*~*^          ON  THE  CIRCULAR  AND  ELLIPTIC  MOTIONS  OF  THE 
PLANETS  AND  SATELLITES. 

By  observing  the  places  of  the  sun,  moon,  and 
planets  in  the  heavens  at  different  times,  and  mea- 
suring; their  angular  diameters,  we  have  learnt,  that, 
provided  certain  excessively  small  inequalities,  to  be 
hereafter  considered,  are  disregarded,  their  motions 
are  all  compatible  with  the  supposition  of  the  planets 
revolving  about  the  sun  in  elliptic  orbits  of  small 
eccentricity,  and  having  the  sun's  centre  in  one  focus ; 
each  orbit  lying  wholly  in  a  fixed  plane  peculiar  to 
itself,  and  siiehtly  inclined  to  the  ecliptic,  or  plane, 
in  which  the  sun  appears  to  revolve  about  the  earth. 
We  learn,  moreover,  that  if  the  apparent  motion  of 
the  sun  be  transferred  in  a  contrary  direction  to  the 
earth,  and  the  sun  be  supposed  at  rest  in  space,  the 
motion  so  assigned  to  the  earth  will  be  such  as  to 
include  it  in  the  expression  of  the  same  law.  The 
supposition  then,  of  the  sun  at  rest,  and  the  earth  in 
motion,  being  agreeable  to  this  general  analogy,  and 
supported  by  incontrovertible  arguments  drawn  from 
the  great  magnitude  of  the  former  in  comparison' with 
the  latter  body,  is  assumed  as  a  demonstrated  truth. 
Observation  moreover  assisted,  it  is  true,  by  calcula- 
tion, but  independent  of  all  theory,  (i.e.  of  all  reason- 
ing from  causes,)  has  taught  us  the  truth  of  the 
following  remarkable  laws,  in  which  also  the  earth  is 
included  among  the  planets. 

The  areas  described  about  the  suns  centre  by  the  radius 
vector  of  any  one  of  the  planetary  orbits  (or  the  line 
drawn  from  the  sun  to  the  place  of  the  planet,}  are  pro- 
portional to  the  times  of  their  description. 

F'E-  1-  Let  S  be  the  sun,  and  APP'  part  of  the  orbit  of  any 
planet.  Then,  A  being  assumed  as  a  point  of  depar- 
ture, the  area  ASP  is  to  the  area  ASP7  as  the  time  of 
the  planet's  describing  the  arc  AP  of  its  orbit  is  to 
the  time  of  its  describing  AP'. 

The  squares  of  the  periodic  times  of  different  planets, 
Cor  of  the  times  of  a  complete  revolution  of  each  about 
the  sun,}  are  as  the  cubes  of  their  mean  distances  from 
the  sun,  or  of  the  greater  semiaxes  of  their  respective 
ellipses. 

The  periodic  time  of  the  earth  is  365'2564,  and  that 
of  Mars  686'9796.  The  greater  semiaxis  of  the 
earth's  orbit  being  1,  that  of  Mars  is  1'523693  ;  and 
we  may  easily  satisfy  ourselves,  by  executing  the 
computation,  that  (365'2564)2  ;  (686'9796)2  '  '  I3  ' 
( 1 -523693  )3. 

These  three  laws,  viz.  1st.  The  elliptic  motion  of  the 
planets  about  the  sun  as  a  focus  :  2dly.  The  propor- 
tionality of  the  areas  to  the  times  :  and  3dly.  The  law 
of  the  periodic  times,  which  have  immortalized  the 
name  of  Kepler,  and  whose  discovery,  and  the  manner 
of  it,  afford  at  once  matter  of  humiliation  and  triumph 
to  the  human  intellect,  were  all  deduced  immediately 
from  observation,  as  insulated,  and,  for  aught  their 
discoverer  knew,  unaccountable  facts.  It  shall  now 
be  our  business  to  demonstrate  their  mutual  depend- 
ence, and  to  shew  how  the  general  law  of  attraction 
may  be  derived  from  them  most  simply. 

The  analogy  observed  between  the  motions  of  the 
other  planets  and  of  the  earth,  affords  a  reasonable 
presumption  of  tneir  being  masses  of  matter  subject 


to  the  same  mechanical  laws  of  rest,  impulse,  and  Physical 
resistance,  as  that  of  which  our  own  planet  consists.  Astronomy 
Moreover,  from  what  we  know  of  the  constitution  of  "~~v 
our  own  atmosphere,  and  its  rapid  diminution  of 
density  as  we  recede  from  the  earth,  we  have  every 
reason  to  believe,  that  the  immense  space  in  which 
their  revolutions  are  performed,  is  either  completely 
void,  or  at  least  free  from  any  material  substance 
capable  of  sensibly  resisting  or  impeding  their  motions, 
or  preventing  any  external  impulse  they  may  receive 
from  acting  on  them  with  its  full  effect  Setting  out 
with  these  assumptions  (the  strict  truth  of  which  will 
be  best  tried  by  the  conclusions  they  will  lead  to,)  it 
is  obvious  that  as  the  planets,  instead  of  moving  con- 
tinually forward  in  straight  lines,  as  masses  of  inert 
matter  would  do  if  projected  in  space  and  left  to 
themselves,  are,  in  fact,  constantly  deviating  from 
this  rectilinear  progression — they  must  be  under  the 
perpetual  influence  of  some  agency  external  to  them- 
selves, which,  (by  the  second  law  of  motion,)  can  be 
no  other  than  that  of  a  mechanical  force  acting  in  a 
direction  inclined  to  that  in  which  they  move  at  any 
instant. 

The  enormous  distance  at  which  the  planets  are 
from  the  sun,  and  their  own  minuteness,  compared 
with  it,  permit  us  at  present  to  regard  them  as  points  ; 
and  it  will  be  shewn  hereafter  that  this  supposition 
introduced  here  merely  for  simplification,  is  strictly 
legitimate.  Let  us  then  consider  the  motion  of  a 
material  point  perpetually  deflected  from  a  straight  line 
by  the  action  of  an  external  force  ;  and  to  this  end 
let  us  conceive  the  curve  OPQ  described  by  the  planet 
to  be  replaced  by  a  polygon  of  an  infinite  number  of 
sides  OP,  PQ,  '&c.  and  setting  out  from  O,  let  it 
describe  the  chord  OP  in  the  first  instant  of  time  d  t.  Fig.  2, 
In  an  equal  subsequent  instant  it  would,  if  left  to 
itself,  go  on  describing  PR  equal  to  OP,  and  in  the 
same  straight  line.  But  since  we  have  regarded  the 
curve  as  replaced  by  an  elementary  polygon,  we  must 
(on  the  principles  of  the  differential  calculus)  con- 
ceive the  deflecting  force  to  act  by  interrupted  impulses 
at  the  angles  of  that  polygon.  Let  the  first  impulse 
therefore  be  conceived  to  take  place  at  P.  Then, 
since  the  material  point  P,  in  virtue  of  the  motion 
inherent  in  it  at  P,  would  have  described  PR  in  the 
instant  d  t ;  but,  in  virtue  of  that  motion,  combined 
with  the  new  motion  it  receives  at  P,  does  actually 
describe  PQ  in  the  same  time,  that  new  motion  (by 
the  composition  of  motions,)  must  be  such  as,  alone, 
would  carry  it  from  P  over  a  space  Pf  equal  and 
parallel  to  RQ  ;  and  as  the  change  of  motion  takes 
place  in  the  direction  in  which  the  moving  force  acts, 
Pf  must  be  the  direction  of  the  deflectingforce.  Prolong 
Pv  indefinitely,  and  take  in  it  any  point  S  ;  join  OS, 
SQ,  SR.  Then,  since  OP  =  PR,  the  area  OSP  = 
SPR  =  SPQ,  because  QR  is  parallel  to  PS. 

A  force  may  be  conceived  to  tend  to  any  point  in 
the  line  of  its  direction.  We  see,  therefore,  that  any 
point  to  which  the  force  acting  at  P  tends  is  charac- 
terised by  this  remarkable  property,  that  the  areas 
described  about  it  in  equal  evanescent  instants  on 
either  side  of  P  are  equal.  This  property  belongs  to 
every  point  in  the  line  PcS,  but  (as  is  obvious)  to 
no  point  situated  out  of  that  line  ;  and  any  point 
possessing  this  property  may  be  regarded  (at  least  for 
that  moment)  as  a  point  of  tendency,  or  centre  of  the 
force  acting  on  the  body  at  P. 
'4  P  "2 


650 


PHYSICAL    ASTRONOMY. 


Astronomy  Now,  as  we  have  seen,  it  is  matter  of  observation 
that  each  planet  describes  areas  proportional  to  the 
times,  and  consequently  equal  areas  in  equal  infinitely 
small  times  before  and  after  any  given  instant,  about 
a  fixed  point  in  the  system  coincident  with  the  sun's 
centre.  This  point,  therefore,  possesses  at  all  times 
and  in  all  positions  of  the  planets,  the  property  above 
demonstrated  to  belong  to  a  point  situated  in  the 
direction  of  the  deflecting  force  ;  or,  in  other  words, 
the  forces  deflecting  the  planets  in  their  orbits  are 
invariably  directed  to  the  centre  of  the  sun. 

The  moon,  (neglecting  periodical  inequalities,) 
describes  about  the  centre  of  the  earth,  and  the  satel- 
lites of  Jupiter  and  Saturn  about  their  respective  pri- 
maries, areas  proportional  to  the  times  of  their  descrip- 
tion. The  forces,  therefore,  which  deflect  them  in 
their  orbits,  are  directed  (small  causes  of  inequality 
being  neglected)  to  the  centres  of  the  earth,  of  Jupi- 
ter, and  of  Saturn  respectively. 

Having  ascertained  the  directions  of  the  forces 
which  deflect  the  planets  from  their  rectilinear  paths, 
and  retain  them  in  their  orbits,  we  come  now  to  esti- 
mate their  intensity,  and  investigate  the  laws  of  their 
action.  In  order  to  this,  we  shall  find  it  more  simple 
to  abandon  the  supposition  of  the  interrupted  impul- 
sive action  of  the  deflecting  force,  and  consider  the 
body  P  as  deflected  from  the  tangent  PR,  and  describing 
not  the  chord  PQ,  but  the  infinitesimal  arc,  the  force 
being  supposed  to  act  during  the  whole  time  d  '. 
This  time,  however,  being  infinitely  small,  the  forcx 
may  be  regarded  as  constant;  and  since  the  angle 
PSQ  between  its  first  and  last  directions  is  also 
evanescent,  it  must  be  considered  as  acting  con- 
stantly in  a  direction  parallel  to  PS  or  QR.  If  then  we 
take  F  to  represent  the  force  at  P,  and  g  =  32ft'190S 
(or  double  the  space  through  whicn  a  heavy  body 
falls  in  the  first  second  at  the  earth's  surface,)  and 
suppose  unity  to  represent  the  force  of  gravity,  we 
shall  have  by  mechanics, 


versed  sine  = 


(arc)'2 


diameter 

and  since  QR  or  Pt>  is  in  this  case  equal  to  the  versed 
sine  of  PQ,  we  must  have, 


Physical 

Astronomy 


QR=2-R: 

and,  finally,   (since  d  t  =  1") 


F  = 


R 


g 


and  this  is  the  general  expression  for  the  force  perpe- 
tually urging  a  body  to  the  centre  of  a  circle.  To 
reduce  it  to  numbers  in  the  case  of  the  moon,  we  have 
R  =  238783m  =  39165700  x  g  (since  g  =  32ft'1903) 

and  —  =  39165700 

g 

moreover, 

T  =  27d'32167=  2360592". 
Calculating  from  which  data,  we  find 

F  =  0-00028394  = 


If,  therefore,  we  know  by  observation  the  nature  of 
the  orbit,  and  the  velocity  of  the  body  at  any  point  P, 
we  may  thence  calculate  the  magnitude  of  the  deflec- 
tion QR  produced  in  any  very  minute  time  d  t .-  and 
thus  the  intensity  of  the  deflecting  force  will  become 
known.  Let  us,  for  instance,  take  the  case  of  the 
moon  ;  and,  supposing  her  orbit  a  circle,  with  the 
earth  in  the  centre,  let  us  inquire  the  actual  magni- 
tude of  QR,  the  deflection  from  the  tangent  produced 
in  some  extremely  minute  portion  of  time  as  1",  by 
the  force  retaining  it  in  its  orbit. 

Call  the  mean  radius  of  the  moon's  orbit  R,  her 
period  (in  seconds  of  mean  time)  T  j  then  will  her 
velocity  (being  equal  to  the  circumference  of  her 
Dibit,  divided  by  the  number  of  seconds  in  the  time 

of  one  revolution,)  be  represented  by  — — ,      where 

TT  =  3'74159,  &c. ;   and   this   is  the  actual  length  of 

the    arc  described  in    1" '.      Now,   since    (neglecting    intensity  of  gravity  is  diminishtd 

higher  powers  of  the  arc  than  the  square) 

*  QR  here  represents  the  deflection  from  the  tangtnt,  and  is 
only  Iwlf  the  length  of  QR  in  the  last  figure,  which  represents  the 
deflection  from  the  preceding  chord  prolonged. 


So  that  the  force  by  which  the  moon  is  retained  in  its 
orbit,  is  about  3522  times  feebler  than  that  of  gravity 
at  the  earth's  surface. 

When  we  observe  a  tendency  in  all  bodies  at  the 
earth's  surface  to  approach  or  fall  towards  its  centre, 
and  if  hindered  from  approaching,  still  to  press  towards 
that  point,  we  express  these  phenomena,  by  saying, 
that  they  are  attracted  towards  the  earth.  At  all 
moderate  elevations  above  its  surface,  and  in  the  same 
geographical  situation  this  tendency  seems  invariable  ; 
but  at  great  elevations,  the  delicate  indications  of 
modern  instruments  will  detect  a  decrease  in  its 
energy;*  and  indeed  the  gradual  enfeebling  of  attrac- 
tion towards  any  body  by  an  increase  of  distance,  is 
not  only  a  natural  supposition  of  itself,  but  is  borne 
out  by  the  strong  analogy  of  magnetic  and  electrical 
attractions.  At  vast  elevations  then,  like  that  of  the 
moon,  there  is  reasonable  ground  to  expect  a  consi- 
derable diminution  of  attraction  ;  and  if  the  force  by 
which  the  moon  be  retained  in  her  orbit  be  nothing 
more  than  this  same  attraction  modified  by  the  remote- 
ness of  the  two  bodies,  we  see  that  an  increase  of  the 
distance  to  about  60  times  the  earth's  radius  from  its 
centre,  is  sufficient  to  weaken  it  more  than  3500  times. 
As  the  distance  then  from  the  earth's  centre  increases, 
the  attraction  diminishes  in  a  much  more  rapid  pro- 
gression. What  the  exact  nature  of  this  progression 
is,  we  must  satisfy  ourselves  by  other  phenomena  ; 
but  even  from  the  rude  calculation  already  made,  (in 
which  every  correction  has  been  neglected)  we  may 
perceive,  that  a  law  of  decrease  as  the  squares  of  the 
distances  (the  next  in  simplicity  to  the  distances 
themselves,)  has  a  primd  facie  probability.  In  fact, 

6oJ  :  i2  ::  3000  :  i. 

Having  only  one  attendant  satellite,  however,  we 


At  an  elevation  of  a  mile  above  the  surface  of  the  earth,  tue 

;  and  a  pendulum  clock. 

beating  seconds  at  the  level  of  the  sea,  would  Iose21'8!)8  seconds 
a  day  at  this  altitude,  a  quantity  not  to  be  overlooked.  Any 
traveller  having  leisure,  and  the  proper  apparatus,  might  try  the 
experiment  in  the  barrack  on  Mont  Corns,  or  at  the  Hospice  of 
St.  Bernard. 


PHYSICAL   ASTRONOMY. 


651 


Astronomy,  have  no  means  of  obtaining  any  further  verification  of 
*>^>-v-v'  such  a  law,  in  this  way  ;  but  if  we  regard  the  earth 
as  well  as  the  other  planets,  as  so  many  satellites  of 
the  sun,  we  have  here  ample  room  to  satisfy  ourselves, 
having  a  progression  of  no  less  than  eleven  distances, 
from  that  of  Mercury  to  that  of  Uranus,  on  which  to 
ground  the  assumption  of  a  law.  And  here  we  have 
the  advantage  of  dispensing  altogether  with  numerical 
calculation,  and  substituting  in  its  stead  the  third  law 
of  Kepler  :  for  if  we  call  R  and  r  the  radii  of  the 
circles  described  by  any  two  planets  round  the  sun, 
T  and  t  their  periodic  times,  F  and /the  forces  re- 
taining them  in  their  orbits,  we  have  (as  a  result  of 
observation,) 


which,  combined  with  the  equations, 
4^      R  4 

*  TV  J    ~ 


which  give 


we  shall  find 


9 

F_ 

£ 

7 


R 


Thus,  then  we  encounter  the  same  rate  of  diminu- 
tion in  the  attractive  tendencies  of  each  of  the  planets 
towards  the  sun,  which  the  lunar  motions  had  given 
reason  to  surmise  in  the  case  of  the  earth  and  rnoon  ; 
but  in  the  case  now  under  consideration,  the  verifica- 
tion is  much  more  satisfactory,  and  the  numerical 
coincidences,  when  the  calculations  are  gone  through, 
complete  ;  the  third  law  of  Kepler,  on  which  the 
whole  is  founded,  being  almost  rigorously  exact. 

We  may  now,  with  great  confidence,  presume  the 
inverse  proportion  of  the  squares  of  the  distances  to 
be  the  law  of  variation  of  that  force  which  retains  the 
bodies  of  our  system  in  their  orbits  ;  but  previous  to 
assuming  its  generality,  it  will  be  right  to  compare 
the  force  retaining  the  moon  in  its  path  round  the 
earth,  and  that  deflecting  the  earth  in  its  orbit  about 
the  sun.  Calling  R,  r,  the  respective  radii  of  the 
earth's  and  moon's  orbits,  and  T,  t,  their  periodic 
times,  we  have  still 

F       R       /  t  \  « 


Now  we  have 


6023799 


z 

t 


365-25638 


R         23405          t        27-32167 
So  that,  executing  the  calculation, 

F 

-r=2-17399. 

The  sun  then,  although  more  than  380  times  the  dis- 
tance of  the  moon,  exerts  a  force  of  more  than  double 
the  intensity  on  the  earth  compared  with  the  earth's 
attraction  on  the  moon.  At  equal  distances,  then, 
the  forces  exerted  by  the  sun  and  earth  would  be  in 
the  ratio  of 

2-17399  x   (— V  :   1  or  328196   :   1.  und  Am>  ") 

This  enormous  difference  in  the  attractive  energies 


of  the  two  bodies,  must  evidently  be  owing  to  some  Physical 
equally  striking  difference  in  the  bodies  themselves ;  Astron°™y- 
and  when  we  consider  the  immense  magnitude  of  the 
sun  (in  comparison  with  our  planet)  we  shall  not  be 
at  a  loss  to  what  cause  to  assign  it.  Whatever  be  the 
cause  of  attraction,  we  may  fairly  conclude  that,  if  it 
be  the  result  of  a  force  inherent  in  matter,  two  equal 
and  similar  bodies  (i.  e.  each  containing  the  same 
quantity  of  attracting  matter,)  placed  close  together, 
will  each  attract  a  third  placed  at  a  distance,  with 
equal  forces ;  and  both  together,  with  double  the  force 
of  either  separately,  and  pursuing  the  same  idea — that 
328196  such  bodies  as  the  earth,  if  placed  close  toge- 
ther, and  forming  one  mass  in  the  place  of  the  sun, 
would  attract  as  the  sun  actually  does  :  in  other 
words,  that  the  sun  only  attracts  other  bodies  with 
more  energy  than  the  earth,  by  reason  of  its  being  a 
greater  boiiy,  and  containing  a  greater  quantity  of 
attracting  «j.  gravitating  matter. 

By  such  icasonings  we  are  led  to  assume,  as  a 
general  law,  that  similar  and  equal  particles  of  matter, 
however  situated  in  space,  attract,  or  tend  to  each 
other  with  a  force  directly  proportional  to  their  masses 
or  quantities  of  gravitating  substance,  and  inversely 
proportional  to  the  squares  of  their  distances  from 
each  other  ;  and  having  arrived  at  this  law  by  the 
steps  described,  we  must  now  proceed  to  verify  its 
rigorous  exactness,  by  applying  it  in  succession  to  the 
phenomena  as  presented  by  nature  in  our  system, 
which  will  be  the  object  of  the  following  sections. 

SSCTION  II. 
On  the  attractions  of  spherical  bodies 

The  earth,  sun,  and  planets,  as  well  as  their  satel- 
lites, being  shewn  by  observation  to  be  spherical  bodies 
of  great  magnitude,  it  becomes  necessary  to  examine, 
in  limine,  whether  the  law  of  attraction  above  stated 
be  compatible  with  this  fact — in  other  words,  whether 
from  a  knowledge  that  the  gross  attractions  of  the 
whole  masses  follow  that  law  of  decrease,  we  can 
argue  that  the  attraction  of  each  elementary  molecule 
follows  the  same. 

Let  BDCE  be  the  attracting  sphere,  m  the  body  Fig.  3. 
attracted,  which  at  present  we  will  suppose  to  be  a 
single  particle,  taking  its  mass  as  unity.  Suppose 
mBAC  the  axis  of  the  sphere  passing  through  the 
molecule  m,  and  let  M,  M'  be  two  equal  and  similar 
molecules,  similarly,  but  oppositely  situated  with 
respect  to  the  axis.  Each  of  these  molecules  will 

attract  m  with  a  force  represented  by  ,  but  the 

directions  of  their  attractions  not  coinciding,  we  must 
resolve  them  into  others,  whose  effects  may  directly 
assist  or  counteract  each  other.  Draw  MPM'  (which 
will  of  course  be  perpendicular  to  mAC),  and  if  we 

M 

take  Mm  =/,  and  MP  =  />,  we  shall  have  — •    to    re- 
present  the   force    of  M  on  m,   which,    reduced   to 
the  directions  m  C  and   PM,  give  the  partial  forces 
M       mP       jM       MP 
—  X  — —  and  -^  x  —^77,  that  is,   (if  we  call  AP,  x, 


M  (a  —  x)         ,    M  '  p 

— and   • — — — , 


65-.' 


PHYSICAL    ASTRONOMY. 


A-'tronomy.  The  partial  forces  of  M'  are  represented  by  the  same 
— ~~--,"~/  quantities,  but  the  latter  of  them,  acting  in  the  direc- 
tion PM'  destroys  the  partial  force  of  M  in  the  oppo- 
site direction  PM,  while  the  former  conspires  with 
the  corresponding  force  of  M,  and  doubles  its  effect,  so 
that  we  have 

2  (a  —  x)  •  M 

~fr~ 

for  the  attraction  of  this  pair  of  molecules  in  the  direc- 
tion m  A  ;  and  the  sum  of  all  such  pairs  throughout 
the  sphere  being  found  by  the  ordinary  rules  of  the 
integral  calculus  will  express  the  whole  attraction  of 
the  sphere.  The  simplest  way  will  be  to  regard  the 
molecule  M  as  a  parallelepiped  included,  first,  between 
two  consecutive  positions  of  the  plane  DE  perpendi- 
cular to  the  axis  A  m,  separated  from  each  other  by 
the  interval  d  x  ;  2dly.  Between  two  cylindrical  sur- 
faces, having  for  their  bases  the  circle  MM',  whose 
radius  is  PM  (=/>)  and  the  same  circle  in  its  conse- 
cutive position,  when  its  radius  varies  from  p  to  p+dp; 
and  3dly.  Between  two  consecutive  positions  of  the 
plane  PM  m,  assumed  during  its  rotation  about  P  m 
as  an  axis.  In  virtue  of  this,  if  we  put  the  angle 
FPM  =  0  the  dimensions  of  the  molecule  in  a  direc- 
tion perpendicular  to  PM  will  be  p  d  0,  and  its  dimen- 
sions in  other  two  directions  being  respectively  dx 
and  dp,  we  have 

M  =  pdp  .  dO  .  dx; 

so  that  the  whole  attraction  (A)  will  be  expressed  by 
the  triple  integral 

>2   (a  —  x)   .  pdp  .  d  6  .  dx 


The  variables  /),  0,  and  x,  are  here  independent ;  and  it 
is  therefore  indifferent  with  which  we  begin,  we  will 
commence  with  0,  because / being  ==  V(a  —  x)*  +  p* 
is  independent  on  0.  Thus  we  have 


A  = 


-J]' 


2  (a  —  x)  pdp  .  dx 


(0  +  Const.) 


This  integral  must  be  extended  only  from  0  =  o  to 
0  =  TT,  or  over  only  half  the  circumference  of  the 
circle  MM',  otherwise  the  attraction  of  each  molecule 
M,  M'  (having  been  grouped  in  pairs,)  would  be  re- 
peated twice  over.  Then  we  have 

/V»  2  IT  (a  —  x)  dx  .  pdp 
JJ         {(a -*)•+/,«}+      ' 

If  we  now  perform  the  integration  relative  to  p  re- 
garding x  as  constant,  we  get 

A  =  A  ,  (a-x)dx  .  f  Const.  -  =] 

J  (  V  (a  —  x)3  +  p3) 

But  the  integral  in  this  case  is  to  be  extended  from 
P  =  o  to  p  =  PD  =  Vr*  —  xs,  r  being  the  radius  of 
the  sphere,  so  that  it  becomes 


represents   the   mass   of    the    sphere, 


(a  —  x 
(«-*)  d  .  (a  -  x) 


.=yw(a-a 

"Ml/1  (—')'•(—') 

«y      V2  a  (a  —  x)  —  (a4  —  r8) 

=Const.  +2 7rz  +  -  *\  2  a2—  r1— a*  [ ' 
31  J 


1 


This  integral  must  be  extended  from  x  =  AC  =  —  r 
to  x  =  AB  =  +  r,  when  it  finally  becomes,  after  all 
reductions, 


Now,  —  n-r5 

21 

which  being  called  S,  we  have 

S 

-rf' 

an  equation  which  shews,  that  the  attraction  of  the 
sphere  is  expressed  by  the  whole  mass,  divided  by  the  square 
of  the  distance  of  its  centre  from  the  attracted  molecule, 
and  is  therefore  precisely  the  same  as  if  the  whole  sphere 
were  condensed  into  its  centre. 

The  hypothesis,  then,  which  refers  the  observed 
attractions  of  the  great  masses  composing  our  system 
to  the  effect  of  the  mutual  attraction  of  their  ultimate 
molecules,  varying  according  to  the  same  law,  has 
nothing  in  it  incompatible  with  mathematical  reason- 
ing; but  it  is  a  very  remarkable  coincidence  that  this 
should  so  happen,  as  the  only  mathematical  laws  of 
attraction  which  would  lead  to  a  similar  conclusion, 
are  that  of  nature,  and  that  in  which  the  force  is 
directly  proportional  to  the  distance,  or  one  resulting 
from  the  combination  of  these  two  laws. 

Let  us  next  examine  the  case  when  the  attracted 
body  is  also  a  sphere  of  sensible  magnitude.  S  and  s 
being  the  two  spheres,  and  a  the  distance  of  their 
centres,  it  has  been  shewn  that  S  will  attract  every 
molecule  of  s  with  the  same  force  as  if  it  were  con- 
densed into  its  centre.  Now  the  mass  of  a  single 
molecule  having  been  regarded  as  unity,  S,  the  mass 
of  the  first  sphere  will  be  proportional  to,  and  repre- 
sent, the  number  of  molecules  it  consists  of.  The 
attraction  then  of  s  on  S  will  be  the  same  as  if  the 
latter  sphere  were  removed,  and  in  its  centre  a  single 
molecule  placed,  endowed  with  an  attractive  energy  S 
times  as  great  as  that  of  any  molecule,  such  as  S 
actually  consists  of.  But  the  attraction  of  5  on  one 
molecule  of  the  last  named  kind  has  been  shewn  to  be 

—  ,  therefore   its  attraction  on  S  being  S  times  as 


forcible,  will  therefore  be  represented  by 


S  xs 


This  expression  represents  the  absolute  force  with 
which  the  two  spheres  tend  to  each  other,  or  the  num- 
ber of  pounds,  grains,  or  other  units,  which  must  be 
opposed  to  it  in  order  to  hinder  their  approach.  This 
in  mechanical  language,  is  called  the  moving  force; 
and  we  therefore  see,  that  the  moving  force  with  which 
two  homogeneous  spheres  attract  each  other,  is  as  the 
product  of  their  masses  directly,  and  the  square  of  the 
distances  of  their  centres  inversely. 

The  moving  force  then  is,  of  course,  the  same  on 
each  sphere  ;  and  in  consequence  of  the  equality  of 
action  and  re-action,  (which  always  refers  to  moving 
force,)  it  ought  to  be  so.  Were  the  spheres  allowed 
to  approach  each  other,  however  their  velocities  would 
obviously  be  different,  the  greater  moving  slower  than 
the  less.  In  fact,  the  accelerating  force  on  any  body 
being  equal  to  the  moving  force  divided  by  the  mass 
moved,  we  have 

accelerating  force  on  S  =  — 

c 

accelerating  force  on  s  =  — . 
a* 
Suppose  now  the  two  spheres  at  liberty  in  space,  and 


PHYSICAL    ASTRONOMY. 


653 


Astronomy,  moving,  in  consequence  of  their  mutual   attraction 
v— - -v—  and  any  projectile  force.     If  to  both  of  them  we  apply 

an  accelerating  force  —  equal  to   that   exerted  by  s, 

but  in  an  opposite  direction,  or  towards  S,  the  sphere 
S  will  be  urged  by  forces  destroying  each  other,  and 
will  therefore  either  remain  at  rest,  or  move  uniformly 
in  a  right  line  :  but  the  sphere  s  will  now  be  urged 

by  an  accelerating  force  equal  to  - — — .      Now,    it  is 

shown  in  mechanics,  that  the  application  of  a  common 
accelerating  force  to  all  the  bodies  of  a  system,  does  not 
alter  their  relative  motions.  Hence,  if  we  refer  the 
motions  of  our  spheres,  not  to  a  fixed  point  in  space, 
but  to  the  centre  of  one  of  them  S  ;  or  take  that 
centre  as  the  origin  of  our  co-ordinates,  we  must 

then  put  — —  for  the  accelerating  force  animating 

the  other.* 

Let  us  now  consider  the  attractions  of  spheres  not 
homogeneous,  but  composed  of  concentric  strata 
varying  in  density  according  to  any  law  of  the  dis- 
tance from  their  centres.  There  is  every  reason  to 
suppose  this  the  actual  constitution  cf  the  sun  and 
planets ;  and  it  therefore  becomes  necessary  to  exa- 
mine this  case.  Now,  any  stratum  of  infinitesimal 
thickness  rf  r,  may  be  regarded  as  the  difference  of 
two  spheres  s,  and  s  +  d  s,  homogeneous,  and  of  the 
same  density  as  the  stratum,  their  radii  being  r  and 

r  +  d  r.     The  attraction  of  s  on  a  molecule  equal  to 

c 

1  placed  at  a  distance  a,  is  — -,  and  that  of  s  +  d  s  is 

a* 

s  +  ds 
a1 

=  —  ;  and  therefore   the  same  as  if  the 

a2  a*       a2 

stratum  were  collected  in  its  centre.  As  this  is  true 
of  every  stratum  separately,  whatever  be  its  density, 
and  their  attractions  do  not  interfere,  it  will  be  true 
of  all  together ;  so  that,  whether  the  sphere  be  homo- 
geneous, or  composed  of  concentric  layers,  or  strata 
of  different  density,  the  same  property  still  holds  good; 
and  all  we  have  demonstrated  in  the  case  of  homoge- 
neous spheres,  remains  true  in  this. 

SECTION  III. 

Theory  of  elliptic  motion. 

We  are  now  enabled  to  enter  on  the  general  theory  of 
the  planetary  motions,butwe  will  still  confine  ourselves 
to  a  case  of  comparative  simplicity.  The  vast  mass  of 
the  central  body  of  our  system,  compared  with  those 
which  circulate  round  it,  permits  us  to  regard  their 
motions  as  influenced  by  it  alone,  and  to  neglect,  in 
a  first  approximation,  all  the  minute  effects  arising 
from  the  mutual  attractions  of  the  planets  and  satel- 
lites on  each  other.  The  case  then  we  propose  to 
consider  in  this  section  is  that  of  the  sun  and  a  single 
planet,  or  a  primary  and  one  of  its  satellites. 

*  If,  however,  we  suppose  the  sphere  S  forcibly  retained  in  its 
place  by  some  external  agency  not  acting  on  s,  the  case  will  be 
c 

different;  and  —  will  continue  to  represent  the  accelerating 
force  on  S. 


consequently  the  attraction  of  the  stratum  is 
ds 


Let  M  represent  the  mass  of  the  sun  or  central  body,    Physical 
and  m  that  of  the  planet ;  and,  fixing  the  origin  of  the  Astronomy. 
co-ordinates  in  the  centre  of  M,  let  x,  y,  z,  represent  v—"~v~~~"/ 
the  co-ordinates  of  m.     Also  let  T  be  the  radius  vector, 
or  line  joining  the  centres  of  the  two  bodies  ;    so  that 
r2  =  x2  +y*  +  z*,  and  t  the  time  (in  seconds  of  mean 
solar  time,)  elapsed  since  any  fixed  epoch. 

M 

The  accelerating  force  of  M  on  m  is  — ,  and  that  of  Fig.  4. 

m  on  M  — ;  and  since  we  regard  M  as  fixed,  the  latter 

quantity  must  be  added  to  the  force  animating  m,  ac- 
cording to  the  observation  made  in  the  last  section ; 
so  that  the  relative  accelerating  force  acting  on  m  in 

the  direction  of  the  radius  vector  m  M  will  be 

which  being  resolved  into  forces  in  the  directions 
m  P,  PQ,  QM,  of  the  three  co-ordinates,  becomes  mul- 

x     y  z 

tiplied  by  — ,  — ,  and  — ,  respectively,  and  produces 

the  partial  forces 

_    (M  +  m)  x        _  (M  +  OT)  y       _  (M  +  m)  z 

The  effect  of  an  accelerating  force  P  acting,  during  an 
instant  of  time  d  t,  on  a  body  in  a  direction  parallel  to 
any  given  axis,  that  of  the  x,  is  to  produce  a  variation 
in  its  velocity  in  that  direction,  which  is  to  the  varia- 
tion gravity  on  the  earth's  surface  would  produce  in 
the  same  time,  as  the  force  P  is  to  the  accelerating 
force  of  gravity  which  we  will  represent  by  unity. 
Now  gravity  producing  the  variation  g  .  d  t  in  that 
time,  the  variation  produced  by  the  force  P,  will  be 
P  .  g  d  t ;  or,  if  instead  of  taking  one  foot,  as  we  have 
hitherto  done,  for  the  unit  of  linear  measure,  we  take 
g  (=32ft-190S)  for  our  standard  unit,  simply  Pdt. 
But  to  this  the  negative  sign  should  be  prefixed,  as 
the  force  P  tends  to  dimmish  the  co-ordinate  x.  Again, 

the  velocity  in  the  direction  x  being  — ,   its  variation 

is   d  — - ;  we  have  therefore  d  — -  =  —  P  d  t;  or,  sup- 
d  t  rl  * 


— - 
at 


posing 


R  = 


and  writing  for  P  its  value  '. 
d-T 


dt 


similarly, 


and 


(3) 


These  equations  contain  the  whole  theory  of  the 
planetary  motions,  neglecting  their  mutual  perturba- 
tions ;  and  if,  instead  of  supposing  R  = 3 ,  we 

suppose  it  equal  to  — — ,  the  same  equations  will  ex- 
press the  motion  of  a  point  m  about  a  centre  of  force 
M,  attracting  it  with  a  force  represented  by  any  func- 
tion <f>  (r)  of  the  distance  MOT. 

If  we  eliminate   R  from   the   two  first   of    these 


654 


PHYSICAL    ASTRONOMY. 


Astronomy,  equations,  by  multiplying  the  first  by  y,and  the  second 
v—  v—  -^  by  —  x,  and  adding,  we  get 


that  is, 


or 


,  dx          dy 
yd  --  xd  —  -  =  o 
dt  dt 

y  d3  x  —  x  c(2  y  _  (y  d  x  —  x  dy) 

dt  dP 

yd*x  —  xd?y  _  d't 


y dx  — x dy        d  t 

Each  member  of  this  equation  being  a  complete  dif- 
ferential, because  y  d*x  —  I  d2y  =  d  .  (y  dx  —  x  dy), 
we  get  by  integration 

y  dx  —  x  dy  =  hdt ;  (4) 

and  similarly, 

zdy  —  ydz  =  h" d  t;  (6) 

If,  now,  we  multiply  the  first  of  these  three  equations 
(in  which  h,  h',  and  h'',  represent  arbitrary  constant 
quantities)  by  z,  the  second  by  —  y,  and  the  third  by 
x,  and  add  all  together,  we  get 

hz-h'y+h"x  =  o;  (7) 

which  is  the  equation  of  a  plane  passing  through  the 
origin  of  the  co-ordinates.  Consequently,  the  curve 
described  by  the  body  is  one  of  simple  curvature,  and 
its  plane  passes  through  the  centre  of  attraction. 

For  simplicity,  let  us  suppose  the  plane  of  the  orbit 
to  be  coincident  with  that  of  the  x  and  y,  we  have 
then  z  =  o,  and  our  equations  are  reduced  to 

dx 

a  -r-  • 
d  t 


y  dx  —  xdy  =  hdt 

Fig.  5.     The  area  of  the  elementary  sector  M  m  m',  described 
by  the  radius  vector  M  m  in  the  instant  d  t,  is  equal  to 

Mmro'P'-Mm'P' 
=  MmP  — Mro'P'  +  Pmm'P' 

xy  _  ydx  —  xdy 

2  2 

If,  then,  we  call  A  the  area  described  by  the  radius 
vector  since  the  commencement  of  the  time  t,  we 
have  ydx  —  x  dy  =  ZdA.,  and  consequently 

this  equation  expresses  the  proportionality  between 
the  areas  and  times  in  Kepler's  law  ;  and  since  the 
process  by  which  it  is  deduced,  is  independent  of  any 
particular  value  of  R,  R  having  been  eliminated  to 
obtain  it,  the  analytical  demonstration  here  given 
applies  generally  for  all  possible  laws  of  central  force. 
In  order  next  to  investigate  the  nature  of  the  curve 
described,  we  must  eliminate  t,  which  will  be  easiest 
done  after  a  transformation  of  the  co-ordinates.  Let 
then  the  angle  AM  m  described  by  the  radius  vector, 
since  the  origin  of  the  time  t,  be  called  0,  and  we  have 
mft  =  rd0,  and  the  elementary  sector  m  M  TO'  = 

-,  ord  Ai  r-d0,  so  that 


2 


Now,  since  r*  =  x3  +  y*,  we  have 
rdr  =  xdx  +  y  dy 


(9) 


so  that  (putting  d  s  =  m  m'  =  Vd  x"  +  d  y*) 
x  d*x  +  y&y  =riPr+dr'—  ds?. 
But     ds*  —  dr3  =.  m  m'1  —  m'  ^2  =  m  [i?  =  r2  d  01 
whence,         x  d?  x  +  y  tPy  =  r  d?  r  —  r1  d  0*. 
This  premised,  since  our  equations  (1)  and  (2)  giro 

o  =d*x  —  l-^tft  +  Rx-  dP 
d  t 

o  =  d*y-  -jtd*t  +  Ry  •  dp 

if  we  multiply  the  first  of  them  by  x,  and  the  second 
by  y,  and  add,  we  have 

dtt          (10) 


but  if  we  take  the  logarithmic  differential  of  the  equa- 
tion hdt  =  r'ldO,  and  suppose  d  0  constant,  (which 
we  are  at  liberty  to  do,  having  as  yet  taken  ao  dif- 
ferential constant,)  we  get 


So   that,    making    this  substitution  for   — ,  and  for 

x  d2  x  +  y  d2  y,  writing  its  value    TO?-  r  —  r-  d0*,  and 
r  dr  for  x  dx  +  y  dy,  we  get 

o  =  r  d2  r  —  r2  d  0*  —  2  d  r4  +  R  r2  d  f, 

in  which  we  have  put  for  R  its  value  —        -  ,     and 


for  d  t,   —  -  —  ,  it  becomes 


o  =  r    -  r  - 


/r 


Put  r  = — ,  and  since  dr  = -,     and    d2  r   =   — 

a  a2 

iPu       2du2 
— - — r-  — — ,  it  becomes 


d2u 


M  +  m 


This  equation  (being  the  simplest  case  of  an  equa- 
tion of  the  second  order  of  a  linear  form,)  is  imme- 
diately integrable,  and  gives 


u=/cos  (0+g) 

whence, 


,'IS) 


M  +  m 


r  = 


1    + 


i  •»     /* 

rr—  '  cos  (°  +  s) 


f  and  g  being  two  arbitrary  constants. 

In  any  conic  section,  if  we  call  a  the  semiaxis 
major,  a  (I  —  e2)  the  semiparameter,  and  0+g  the 
angle  included  between  the  radius  r  and  the  vertex 
nearest  the  focus  from  which  r  is  supposed  to  take 
its  origin,  we  have 


~  1  +  e  •  cos  (0  +  g) 

Consequently  we  see  that  the  curve  described  by  the 
oody  must  necessarily  be  a  conic  section,  having  the 
body  M  in  the  focus,  and  the  relation  between  the 
arbitrary  constants  f,  h,  and  the  axis  and  semipara- 


PHYSICAL    ASTRONOMY. 


655 


Astronomy  meter  of  this  conic  section  will  be  (if  we  call  p  the 
* /-— '  semiparameter,  or  putp  =  a  (1  —  ee) 


/  = 


<*  (1  - 


(15) 


h  =  Va  (1  -  e2)  (M  +  m)  =  Vp  (M  +  m)  ;  (16) 
If  we  suppose  the  angle  0  to  commence  from  the 
vertex  nearest  the  focus,  or  from  the  nearer  apside,  or 
perihelion  of  the  orbit,  we  have  g  =  o,  and 


1  +  e  .  cos  0  1  +  e  cos  0 ' 
If  the  value  ofe  be  less  than  unity,  the  conic  section 
described  is  an  ellipse,  if  equal  and  a  infinite,  a  para- 
bola, if  greater,  and  a  negative,  an  hyperbola. 

But  to  complete  the  theory  of  the  planetary  motions, 
it  is  necessary  to  know,  not  only  the  nature  of  the 
orbit,  in  general,  but  also  whereabouts  in  it  the  body 
will  be  at  any  moment  assigned.  For  this  purpose 
we  must  obtain  a  finite  equation  involving  t,  and 
either  r  or  0,  or  some  functions  of  them.  Now,  the 
equation  (9)  gives  hd  t,  or 

VaTl  -  ea)  (M  +  m)  x  d  t  =  r2  d  0 
and  substituting  for  r,  its  value  in  equation  (17) 

a4(l  —  e2)4  d0 

dt=     .: =L-  .  TT-. —  (18) 


and 


t 


C  = 


V  M  +  m        (I  +  e  .  cos  0)* 
OT(!— e»)f       /»  d0 


VM  +  TO      J     (1  +  «  .  cos  0)* 
To  integrate  this,  take  another  variable  »,  such,  that 

cos  v  —  e 
cos0  =  - :  (19) 


whence, 


1  -  e 


—  e"  .  sin  v 


d 
d 

1  -  e   . 

cos  v 
(1  - 

*8)  • 

sin  v 

a           d  .  cos  6 

ti- 
de . 

e  .  cos  i')* 

Vl 

-e* 

sin  0 
1  +  e  .  cos  0  =  — 


1  —  e 


cos 


1 


1  —  e  .  cos  v 

and  substituting  these  expressions  in  the  value  of  t 
above  given,  it  will  be  found  to  reduce  itself  to  the 
following  very  simple  form  : 

«7  /' 

t  +  C  =  —  --     /   d  v  (  1  —  e  .  cos  c) 
VM.  +  m*J 


so  that,  making 
of 


1 


(20) 


VM  +  TO  n 
and  taking  the  integral  to  commence,  when  v  =  o,  or 
0  —  o,  that  is,  from  the  instant  of  the  body  leaving 
the  lower  apside, 

nt  =  v  —  e  .  sin  e  ;  (21) 

This  equation  fixes  the  relation  between  t  and  v  ;  but 
that  between  v  and  0  may  be  expressed  more  con- 
veniently for  the  purposes  of  calculation  than  by  equa- 
tion (19),  as  follows.  By  the  equation  last  men- 
tioned, we  get 

(1  +  e)  (1  —  cos  e) 


1  —  cos  0  = 


I  —  e  .  cos  v 


1  +  cos  0  = 


whence, 


1  —  cos  0 


(I  -e)  (1  +  cose) 
1  —  e  .  cos  v 

l+e      1  —  cos  v 


Physical 
Astronomy. 


1  +  cos  6       1  —  e      1  +  cos  v 


or,  tan  i  0  = 


•  tan 


(22) 


Finally,  we  obtain  immediately  the  relation  between 
r  and  v,  by  substituting,  in  equation  (17)  for  cos  6,  its 
value  in  (19),  when  we  find 

r  =  a  (1  -  e  .  cos  «)  ;  (23) 

These  equations  comprise  the  whole  theory  of  the 
motions  of  bodies  in  conic  sections.  Equation  (17) 
exhibits  the  relation  between  r  and  0,  or  the  polar 
equation  to  the  curve;  while  (21),  (22),  and  (23), 
express  relations  between  the  time,  t,  the  true  anomaly 
(as  it  is  called)  0,  and  the  radius  vector,  r,  respectively, 
and  an  auxiliary  angle  v,  to  which  the  name  of  the 
eccentric  anomaly  has  been  given.  The  quantity  n  t, 
when  reduced  into  angular  measure,  by  multiplying 
it  by  180°,  and  dividing  it  by  IT  =  3'14159,  &c.  is 
called  also  the  mean  anomaly.  The  eccentric  anomaly 
may  be  exhibited  geometrically,  as  follows  : 

On  the  major  axis  AB  of  the  ellipse,   let  a  semi-  Fig.  6. 
circle  Am'B  be  described,   and    draw   the   ordinate 
Pmm'  through  m.     Then,    AM  m  will   be    the   true 
anomaly  of  the  body  m,  and  AC  m'  the  eccentric.   For, 
if  we  take  CP  =  x, 

CP  x 

cos  AC  m,  =  — 


cos  AM  m  =  — 


C  m' 
MP 


a 

a  e  +  x 


M  TO  a  +  ex 

because,  by  the  property  of  the  ellipse,  M  m  =  a  +  ex. 
Hence,  we  have 

1  __  cos  AM  TO      (1  +  e)  (a  +  x) 
I  +  cos  AM  TO      (1  —  e)  (a  —  x) 

=£-:*— 


1  +  e  _   1  —  cos  AC  m' 
l—e      1  +  cos  AC  m' 
and,  consequently, 

l+e 
tan  \  AM  TO  =  -  -  x  tan  |  AC  m' 

Let  T  represent  the  time  of  one  entire  revolution 
in  the  ellipse,  or  the  periodic  time,  then,  as  0  in- 
creases from  0  to  360°  (or  2  IT)  t  increases  from  0  to 
360  (2  TT)  also.  Consequently,  by  (21)  we  get 

T       c,  i-      27r 

raT  =  27r  ;         T  =  — 


VOL.  III. 


The  periodic  times,  then,  of  several  bodies  revolv- 
ing about  the  same  central  body,  are,  in  the  sesqui- 
plicate  ratio  of  the  major  axes  of  their  orbits,  (or 
mean  distances  from  the  central  body,)  directly,  and 
in  the  subduplicate  ratio  of  the  sum  of  the  masses  of 
the  revolving  and  central  body  inversely.  The  mags 
of  the  sun  being  enormously  great  compared  with 
4  a 


656 


PHYSICAL    ASTRONOMY. 


Astronomy,  those  of  the  planets,  we  may  neglect  m  in  comparison 
'  with  M  ;  and  M  being  the  same  for  the  whole  system, 
we  have  T  QC  <*T  ;     T*  OC  «3 

which  is  no  other  than  Kepler's  third  law. 

The  periodic  times  of  the  planets  being  very  exactly 
known,  we  might  expect  to  find  in  equation  (24)  the 
means  of  ascertaining  the  masses  of  the  planets,  sup- 
posing that  of  the  sun,  and  any  one  of  them,  known. 
Thus,  if  m,  m',  be  the  masses  of  two  planets,  and  T, 
T7,  their  periodical  times,  we  have 

M  +  m 
X  ^— — ;  (2o) 


On  applying  calculation,  however,  all  we  learn  from 
this  relation  is,  that  the  resulting  masses  are  so  small, 
as  to  be  incapable  of  accurate  determination  by  this 
method ;  their  values,  as  deduced  from  it,  being 
materially  affected  by  the  small  uncertainties  still  pre- 
vailing as  to  the  lengths  of  the  periods,  and  by  the 
mutual  perturbations  of  the  sun  and  planets.  There 
is  a  case,  however,  where  it  may  be  used  with  advan- 
tage, viz.  in  that  of  a  planet  accompanied  by  a  satel- 
lite. If  we  call  M  the  sun,  and  m  the  planet,  and 
neglect  the  mass  of  the  latter  in  comparison  with  the 
former,  and  that  of  the  satellite  in  comparison  with 
the  primary,  we  have  at  once 

M       m       /a'V       /T 


Thus,  in  the  case  of  the  earth,  we  have 

a'  _  60  23799       T  _  365'25638 
T~      23405~~ ;    T7  ~  27-32167 

So  that,  by  executing  the  computations,  we  find 


—  =  0  00000304697   = 
M 


1 


328196 

To  find  v  in  terms  of  t,  or  to  calculate  the  eccentric 
(and  thence  the  true)  anomaly  at  any  given  instant, 
we  must  resolve  the  transcendental  equation, 

n  t  =  v  —  e  .  sin  v 

This  can  only  be  done  in  a  series;  and  fortunately, 
in  the  case  of  the  planets,  e  is  so  small,  that  a  series 
ascending  by  powers  of  c  will  converge  sufficiently. 
Now  we  have 

p  =  n  t  +  e  .  sin  v 

and,  since  e  is  small,  and  sin  v  necessarily  less  than  1, 
n  t  itself  expresses  the  value  of  v  within  a  limit  less 
than  e,  and  is  therefore  a  first  approximation.  Again, 
if  in  sin  «  we  write  its  value  for  v  or  (n  t  +  e  .  sin  r) , 
we  get 

v  =  n  t  +  e  .  sin  (n  t  +  e  .  sin  D) 
So  that, 

v  =  n  t  +  e  .  sin  n  t 

is  an  approximation  carried  one  step  farther,  or  to  the 
first  power  of  e.  Let  this  b°  again  substituted  for  e, 
and  we  have 

c  =  n  t  +  e  .  sin  {  n  t  +  e  .  sin  n t } 
But  (neglecting  the  squares  and  higher  powers  of  e) 
we  have 
sin  ]  n  t  +  e  .  sin  n  t  j   =  sin  n  t  +  e  .  sin  n  t  .  cos  n  t 


=  sin  n  t  -{ sin  2  n  t 


So  that  we  find 


v  =  nt  +  e  .  sin  .•» 1  +  -'—  sin  2  n  t 


(2G) 


tion,  pushed  to  the  third  power  of  e,  and  so  on,  as  far    Physical 
as  we  please.  Astronomy 

For  numerical  calculation,  however,  the  equation  *— • "v"™* 
v  =  n  t  +  e  .  sin  v  furnishes  the  readiest  solution ; 
as  we  have  only  to  reduce  e  into  seconds,  (taking 
57°  17'  44"-8  for  the  arc  equal  to  radius  or  1 ,  or  add- 
ing the  logarithm  5-3144251  to  the  logarithm  of  e, 
which  gives  at  once  that  of  the  number  of  seconds  e 
is  equal  to,)  and  assuming  n  t  (the  given  mean  ano- 
maly) fora  first  approximation,  correct  it  successively, 
as  in  the  following  example. 

Required  Jupiter's  eccentric  anomaly  corresponding 
to  53°  of  mean  anomaly. 

Here  n  t  =  53°  ,  and  in  Jupiter's  orbit  we  have 
e  =  0-046077  log. .  8-6819374 

5-3144251 


e  =  9916'A6  ;  log.  .  3'9963625 

Take  v  =  53° log  sin  53°      9'9023486 


e  .  sin  v  =    2°  12' 0"  =  7919"'8  ;    3-8987111 


v  =  55   12  0  Corrected  value,  with  which  re- 
suming the  process. 

log  sin  v  =9'9144221 
e. .      ..39963625 


2°  15'  43"  =  8143'  -0 log 3  9107846 

53°  =  n  t 


55    15    43.  ...  Second  corrected  value. 
Another  repetition  of  the  very  same  process  gives  v  = 
55°  15'  49"'  1,  which  is  true  within  O"'2. 

But  even  this  process,  simple  as  it  is,  becomes 
tedious  for  the  orbit  of  Mercury,  and  those  of  the  new 
planets,  Pallas  and  Juno,  in  which  the  value  of  e  is  not 
very  small  ;  and  here  we  must  have  recourse  to  the 
well  known  method  of  trial  and  error,  which  may  be 
applied  in  this  case  as  follows  :  —  Having  assumed  by 
estimation  a  value  of  v,  (neglecting  minutes  and 
seconds,)  by  noticing  whether  the  term  e  .  sin  v  is 
additive  or  subtractive,  and  increasing  or  diminishing 
n  t  accordingly  :  calculate  the  value  of  v  —  e  .  sin  v 
for  that  and  the  next  subsequent  degree  ;  and  let  the 
values  so  found  be  called  V,  and  V  :  should  either  of 
these  be  exactly  equal  to  the  proposed  value  of  n  t, 
the  corresponding  value  of  c  will  be  the  truth  ;  but  as 
this  will  probably  never  happen,  we  have  only  to  say, 


v-  v  :  v-  n  t  \  :  -SGOO"  :  x  =  - 


x  (V-« 


which  correction  being  applied  with  its  proper  sign 
to  the  latter  of  the  two  assumed  values  of  v,  will  give 
an  approximate  value.  Let  the  value  of  v  —  e  .  sin  « 
be  again  computed  with  this  value  of  v,  and  call  the 
result  V".  This  will  always  be  found  very  nearly 
equal  to  n  t  ;  but  if  not  exactly  so,  the  correction 


If  we  again  repeat  the  process,  we  get  an  approxima- 


must  be  computed  and  applied  to  the  new  value,  and 
so  on. 

For  instance  :  Let  n  t  —  33'2°  28'  65",  and  e  — 
50600"=  14°  3'  20''.  Here  sin  nt  is  negative;  so 
that  e  .  sin  n  t  is  subtractive,  and  v  must  he  less  than 
n  t.  Take  then  for  the  two  values  of  v,  325°  and 
326°  respectively,  and  compute  as  follows  :  — 


PHYSICAL    ASTRONOMY.  657 

Astronomy.                                   sin  325°  -  97585913  (neg)  sin  326°  -  97475617  (neg)  Physical 

* — v '  loge" 47041513  47041513  Astronomy 

4-4627426  (nee;)  4-4517130  (neg) 

e  .  sin  325°  =  —  29023"'O  =  —  8°  3'  43"'0  e  .  sin  326°  =  —  28295  2  =  -  7°  51'35"'2 

Hence  V  =  333°  3'  43"'0  V  =  333°  51'  35"-2 

V  —  V  =  2S72"-2  V  —  n  t  =  —  4960'2 

Log  3600 3-5563025 

Log  2872-2    3-45821 47 


0'0980878 
Log  4960-2    3-6954992 


Log  6217-1    ........   37935870 

v  =  326°  —  6217"'l  =  324°  16'  23" 

Taking  this  for  a  new  value  of  v,  we  find,  by  another     Finally,  if  we  resolve  the  equation 
repetition  of  the  process,  /\  _j_e 

V"  =  332°  28'  49"'6  ;     V"  -  n  t  =  -  5"'4  tan  a  6  =  V    \IT~e  '  tan  5  » 

36OO  we  eet 

and,  "     x  (V"-«0  =  +  6"768 

G  =  v  +  e  .  sin  e  +  —  .  sin  2  v  +  &c. 
so  that  v  =x  324°  16'  29"768  .  .     4 

4i.    v-      j    .uv.    c  j         A  »i.  •       wherein,  if  we  write  for  «  its  value,  we  shall  obtain 

which  is  true  to  the  hundredth  of  a  second  ;  and  this 

case  is  nearly  the  worst  that  can  occur  in  the  theory  0  =  n  t  +  2  e  .  sin  n  t  +  —  e*  .  sin  2  n  t  +  &c. 

of  the  planets.  4 

If  we  substitute  the  value  of  »  given  in  equation  (26)  To  carry  these  series  to  the  hi£her  Powers  of  e>  and 

in  the  expression  for  r,  r  =  a  (1  -  e  .  cos  v)  ;   (23),  *?  ascertain  their  law   we  must  have  recourse  to  par- 

and  develope  in  powers  of  e,  we  obtain  r  in  a  series  *lcular  theorems  for  facilitating  such  developements, 

of  powers  of  e,  and  cosines  of  n  t,  and  its  multiples,  but  these  will  suffice  for  our  present  purpose  ;  and  the 

s  reader  who  wishes  to  proceed  farther  in  the  mvestiga- 

r  _  a  I  j  _  e    COsn<+—  (1  _  cos2n<)  _  e3x&c  >•  tion,  will  findin  the  second  book  of  the  Mfaanique  Celeste, 

1                           2                                          )  '  arts.  xx.  xxi.  xxii.  every  information  he   can   desire. 
We  shall  content  ourselves  here  with  merely  setting  down  the  formulae,  which  are  as  follows  :  — 

e2 
—  .  2  .  sin  2  n  t 

(27) 


sin 


(28) 


i;  —  •«  t    f  c    .   sin  n  i  T     ~-     • 

.  i  .  siu  x  n  i 

e3 

'} 

1.2.3. 

e4 

f    ^, 

n  t  -  4  .  23 

.  sin  2  n  t  [ 

1.2.3 

.4  .23[       '  S1 

* 

{r4        '     t 

;  n  f           E       o. 

t     Qin      O     «     /         [ 

5  .  4 

-I-&C.  ; 

Sill  O  7*  *  ~f"  • 

1.2 

r                   e*                              e2 

—  =  1  +  —  e  .  cos  n  <  cos  2  n  t 

a                   2                             2 

e*  r 

3f\r\a  ^  *)  #            ^ 

cos  n<> 

2  .  4( 

e4 

-|  42  .  cos  3  n  <  - 
o  I 

-  4  .  2*  .  cos 

2»<  | 

2       4r 

e6 

f 

5 
!  —  5  .  3s  .  cos  3  n  <  +  - 

.  4 

2.4  . 

.  2  ' 

1 

.  cos  n  t  V 
^  J 

-&C.; 

2es  2e* 

0  =  »  +  2  e  .  sin  v  +  —  -  .  sin  2  «  H  --  .  sin  3  v  +  &c.  ;  (29) 

£  3 

where  e  =  —  —  —  and  is,  of  course,  a  fraction  smaller  than  e.     Also,  if  we  neglect  powers  of  e 

i  +  vl  —  ea 
higher  than  the  fifth, 

0  =  nt  +  J2e--i-e3  +  ^es|  .  sin  n  t  +  \—  e2  -  ^-e*}  sin  2  TI  t 

96     J  C4  24     ) 

113    ,       43     )     .  103  1097 

4  5 


658 


PHYSICAL     ASTRONOMY. 


Astronomy. 


SECTION    IV. 


On  the  velocitiet  of  the  planetary  motions,  and  the  de- 
termination a  priori  of  (he  elements  of  their  orbits. 

The  angular  velocity  of  a  body  is  measured  by  the 
angle  which  it  appears  to  describe  in  any  very  small 
time  to  the  eye  of  a  spectator.  In  fact,  if  we  call  0 
the  angle,  and  t  the  time,  we  have 

dO 
Angular  velocity  about  the  sun  =  — 

Now,  by  our  9th  equation,  we  have  r°d9  =  hdt, 
and,  consequently,  —  =  —  ;  that  is  to  say,  the  angular 

velocity  in  any  orbit  is  inversely  as  the  square  of  its 
distance  from  the  centre  ;  and  this  law  is  general  for 
all  central  forces. 

The    paracentric  velocity   is  the  approach   to,   01 

recess   from,  the  centre  ;    and  is   measured   by   —  . 

dr      dr      <I0 

Now,  —  =  —  .  —  ,  but 

dt      dO      dt 


dr 
00'' 


a  (I  —  e°)  .  e  sin  0  e  .  sin  0 


a  (1  —  e*)  .  r" 


(1  +  e  .  cos  O'f 
and,  consequently, 

d  T  _     he  .  sin  6     _       /  M  +  m       e  .  sin  0 
dl~a  (l-es)  r>~  V    a  (T^I8)    '   ~~  ' 

To  complete  our  knowledge  of  the  body's  motion, 
we   must   inquire   its  linear  velocity  at  any  instant. 

To  this  end,  if  we  call  V  the  velocity,  V  =  — ,  ds  be- 
ing the  element  of  the  curve.   Now,  first,  we  must  re- 

mark, that  if  we  write  for  d  t  its  value  - - '• — -, 

h 


this  gives 


But 


y  d  x  —  i  d  y 


y d x  —  xdy' 


(32) 


ds 


expresses  the  length  of  a  perpendi- 
cular dropped,  or  a  tangent  to  the  curve  fram  the  origin 
of  the  co-ordinates.  Thus  we  see,  that  the  velocity  is 
inversely  as  the  perpendicular  so  let  fall,  and  directly  as 
the  quantity  h,  or  the  area  described  in  a  given  time. 
Moreover,  we  have  d  s2  =  d  r4  +  r*  d  ff* ;  so  that, 

writing  for  dr  its  equal -,  or  —  r*  dti, 


Now,  by  reason  of  the  equation  ra  dO  =  h  d  t,  we  have 


Again,  if  we  differential  the  equation  (12),  we  find 


=  /•-/«.  COS   (0+g)« 

ft      (         M  +  "l\  a 

•  I  «  -      —  ^  —  I  ,     (by  equation  12) 

es       /  1  \* 

=  P*~  \     ~    ~p)  '     (b>'efluations  15>  16) 


p"  -  1 


writing  for  u  its  value  — .     Consequently, 


pr 


we  find 


P\r        a 
because  p  =  a  (1  —  es.)     Hence,  we  finally  obtain 

(—}*  —  —  (-  ___  L\ 
\dt)      ~  p   \T      a  / 


Physical 
Astronomy 


or 


V«=(M 


(33) 


On  the  determination  of  the  elements  of  the  plane- 
tary orbits,  a  priori,  there  is  no  occasion  to  enter  into 
any  very  extensive  discussion,  in  a  practical  point  of 
view.  Since,  however,  it  is  a  subject  generally 
touched  upon  in  astronomical  works,  and  is  not  with- 
out its  interest,  when  we  consider  what  changes  in 
our  own  system  may  have  taken  place,  or  may  yet 
take  place  from  the  action  of  violent  causes,  we  shall 
devote  a  part  of  our  space  to  its  consideration. 

Let  its  suppose,  then,  a  body  of  a  given  mass,  to  be 
launched  in  space  from  a  given  point,  with  a  velocity 
and  direction  also  given  ;  and  to  be  attracted  by  ano- 
ther body,  whose  mass  is  also  given  with  a  force  in- 
versely as  the  square  of  the  distance  :  it  is  required 
to  determine  the  form,  magnitude,  and  position  of  the 
conic  section  it  will  describe. 

The  plane  passing  through  the  attracting  body  and 
the  primitive  direction  of  projection  will,  of  course, 
be  that  in  which  the  orbit  will  lie,  there  being  no 
force  to  draw  the  body  out  of  this  plane.  Taking  then 
the  central  body  for  the  origin  of  two  co-ordinates, 
x  and  y,  lying  in  this  plane,  and  retaining  all  the  other 
denominations  of  the  foregoing  pages,  and  consider- 
ing, as  the  unit  of  velocity,  that  with  which  a  body 
would  describe  a  space  equal  to  32'  1908  feet  in  one 
second,  we  shall  have,  by  (29), 


CM  +* 


>(-!.  L) 


and,  consequently, 
1          2 


(31) 


a         T        M  +  m 

Now,  by  hypothesis,  the  masses  of  the  central  and 
revolving  body  are  given  :  and  the  distance  from  the 
centre,  as  well  as  the  velocity  with  which  the  latter  is 
projected.  If,  then,  we  suppose  the  quantities  in  this 
equation  to  correspond  to  the  point  of  projection,  r  is 
this  distance,  and  V,  M,  m,  are  known  ;  so  that  we 
have  at  once  the  value  of  a,  the  semiaxis  major  of  the 
orbit. 

This  result  is  a  remarkable  one.  It  shews  us,  that 
the  major  axis  of  the  orbit  is  independent  of  the  angle 
of  inclination  to  the  radius  at  which  the  original  pro- 
jection takes  place  :  in  other  words,  that  any  number 
of  bodies  (of  equal,  or  of  exceedingly  small  magni- 
tudes compared  with  the  central  one,)  launched  from 
one  point  with  equal  velocities,  but  in  any  different 
directions,  will  all  describe  orbits  having  equal  major 
axes.  Another  result,  not  less  curious,  follows  from 
this — that  they  will  all  describe  conic  sections  of  the 
same  nature,  that  is,  all  ellipses,  or  all  parabolas,  or 
all  hyperbolas— for  the  nature  of  the  conic  section 
depends  only  on  the  algebraic  sign  of  its  major  axis. 


PHYSICAL    ASTRONOMY. 


659 


Astronomy.  Thus,  if  a  be  positive,  or  if  V,  the  velocity  of  projec-    once  having  had  a  common  velocity  about  the  sun.    Physical 

The  sroallness  of  the  ruptured  mass  renders  the  sup-  Astronomy 
the  orbit  will  be  an    position  of  an  explosion  less  revolting;  and  we  know, ' 
at  least  from  observation,  that  the  fragments  (if  such) 
are  extremely  minute. 

If  the   orbit   be   a   circle,   we   have   r  =  a,    and 
211 

• — •  =  — ;  so  that,  for  the  velocity  in  a  circle, 

r         a          r  J 


tion,  be  less  than 


ellipse.      If  equal,  a  will  be  infinite 


(^=o), 


and 


the  orbit  will  be  a  parabola;  but  if  greater,  a  is  nega- 
tive, and  the  orbit  will  be  an  hyperbola 

In  the  two  latter  cases,  the  bodies  will  never  return 
to  their  original  point  of  departure,  but  in  the  former 
they  will  do  so :  and  since  the  periodic  time  depends  only 
on  the  major  axes  and  masses,  if  their  masses  be  either 
all  equal,  or  all  extremely  small,  they  will  all  return 
in  the  same  time,  and  a  collision  will  take  place  ; 
after  which  it  is  impossible  to  say  what  will  happen. 
If,  on  the  other  hand,  there  exists  any  sensible  propor- 
tion between  the  revolving  and  central  bodies,  and 
any  considerable  inequality  in  their  masses,  their 
periods  will  be  unequal,  and  each  may  perform  its 
orbit  undisturbed. 

This  is  supposing  their  mutual  attractions  to  be 
neglected.  In  fact,  however,  at  the  instant  of  their 
departure,  these  may  be  incomparably  greater  than 
that  of  the  central  body,  and  will  then  materially 
change  their  velocities,  unless  the  latter  be  so  great 
as  speedily  to  carry  them  beyond  the  sphere  of  their 
mutual  influence.  If  a  small  portion  of  the  earth,  for 
instance,  were  suddenly  projected  from  its  surface, 
the  attraction  of  the  earth  on  it  would,  at  the  moment 
of  its  departure,  exceed  that  of  the  sun  in  the  ratio  of 
3522  ;  2-17399,  or  upwards  of  1600  ;  1.  So  that,  in 
the  first  instants  of  its  motion,  it  would  move  as  if 
influenced  by  the  earth  alone.  But  this  effect  would 
diminish  rapidly ;  by  the  time  the  projectile  had 
reached  the  distance  of  the  moon,  the  sun's  action 
would  already  have  a  preponderance  (as  we  have  seen 
in  section  I,)  in  the  ratio  of  2' 17399  '.  1,  and  it  would 
depend  entirely  on  the  relative  velocity  of  projec- 
tion, whether  such  a  space  could  be  described  in  a 
time  small  enough  to  escape  the  influence  of  the 
earth  or  not. 

It  has  been  a  matter  of  some  speculation,  whether 
the  small  planets  between  Mars  and  Jupiter  may  not 
have  had  their  origin  in  the  destruction,  by  violence,  of 
some  larger  mass  once  revolving  in  the  situation  they 
now  occupy.  The  very  considerable  approximation 
of  their  periodic  times,  which,  in  the  case  of  Ceres 
and  Pallas  is  singularly  near  (within  -,-,',„•  of  the  whole 
period,)  and  the  equally  remarkable  fact  of  the  mutual 
intersections  of  their  orbits  falling  all  in  the  same  part 
of  the  heavens,  (in  a  general  way,)  have  given  rise  to 
this  surmise  ;  and  it  has  even  been  conjectured,  that 
an  explosive  rupture  of  a  former  planet  may  have 
scattered  its  fragments  far  and  wide  over  our  system, 
and  produced  these  singular  bodies.  There  is  no  limit 
to  conjecture  ;  but  if  any  such  event  have  taken  place, 
we  are  forced  to  conclude,  that  the  mass  of  the  rup- 
tured planet  must  have  been  very  small,  or  the  frag- 
ments must  have  collapsed  by  their  mutual  attraction  ; 
or,  at  least,  their  velocities  would  have  been  so  mate- 
rially modified  by  it,  as  to  obliterate  all  traces  of  their 


M  +  m  ,.      </M  +  m 

we  have  V2  = ,  V  = ^—  ;  that  is  to  say, 

Vr 

the  velocity  in  different  circles  is  in  the  sub-duplicate 
ratio  of  the  sum  of  the  masses,  or  the  absolute  force, 
as  it  is  sometimes  called,  directly,  and  of  the  radii 
inversely.  Moreover,  if  we  denote  by  V  the  velocity 


in  a  circle  of  the  radius  r,  or  V  = 
2         1 


M  +  m 


,  we  have 


r2 
V2 


or  «a  ;  V2  ; :  2  a  —  r   ;  a. 

Now,  if  APM  be  an  ellipse,  S,  H,  the  foci,  AM  = 
2  a  and  SP  =  r,  we  have  HP  =  2  a  —  r ;  so  that 

v  :  v;:  VHP  :  VAR 

by  which  property  the  velocity  in  a  conic  section  may 
be  immediately  compared  with  that  in  a  circle  at  the 
same  distance. 

Hence,  when  SP  =  AC,  or  at  the  extremities  of 
the  conjugate  axis,  the  velocity  is  equal  to  that  in 
a  circle. 

vt  

In  the  parabola,  we  have  —  =  2,  or  »  =  V  .  A/2,  so 

that  in  this  curve  the  velocity  bears  a  constant  ratio 
to  that  in  a  circle  at  the  same  distance,  -v/2  ;  1. 

In  the  hyperbola,  HP  increases  without  limit,  and 
the  velocity  bears  continually  a  greater  and  greater 
ratio  to  that  in  a  circle. 

As  the  velocity  depends  only  on  the  distance  and 
major  axis  of  the  conic  section,  and  not  at  all  on  its 
form,  we  may  conceive  the  conjugate  axis  so  dimi- 
nished that  the  conic  section  shall  pass  into  a  straight 
line.  In  this  case,  the  extremity  of  the  axis  will  coin- 
cide with  the  focus,  and  the  velocity  at  any  distance 
r  will  be  that  acquired  by  falling  from  a  distance  2  a 
from  the  centre  to  the  distance  r.  The  expression  for 
this  velocity  is  therefore  still  the  same  with  that  in  the 
conic  section.  Hence,  "  The  velocity  in  a  conic  sec- 
tion at  any  point  is  that  which  would  be  acquired  by 
falling  freely  towards  the  centre,  from  a  distance 
equal  to  the  longer  axis,  to  that  point."  In  the  para- 
bola the  longer  axis  is  infinite,  and  the  velocity  at  any 
point  is,  therefore,  that  acquired  by  falling  from  an 
infinite  distance.  In  the  hyperbola  the  axis  is  nega- 
tive, and  even  an  infinite  fall  is  not  sufficient  to  give  a 
body  all  the  velocity  requisite  for  the  description  of 
this  curve. 

We  have  then  the  following  expressions  : — 


660 


PHYSICAL    ASTRONOMY. 


Astronomy. 


Velocity  in  a  conic  section,  semiaxis  =  a,  distance  = 
circle,  whose  radius  is  r 


r;  v  =  \/(M+ni)  (—  -— ) 
\  r        a  / 

...V=V/E±» 


-parabola  at  any  distance  r, 


a  (1  —  e2)  =  r  x  (2 )  .  sin  A2. 


and 


But  a  (1  —  e2)  is  the  semiparameter  of  the  conic 
section.  Moreover,  callings  the  velocity  in  the  curve, 
and  V  that  in  a  circle ;  and  denoting  by  n  the  ratio 

(v  \*                                                  /             r  \ 
—  1    we  have  already  seen,  that  f  2 1  =  n2. 

Hence,  we  have 

a  (1  -  es)  =  r  x  n2  .  sin  A2 ;  (36) 

and  therefore  is  given,  when  the  angle  and  distance  of 
projection  are  given,  and  also  the  velocity.  When 
the  angle  of  projection  varies,  other  circumstances  re- 
maining, we  see  hence  that  the  parameter  varies  as  the 
square  of  its  sine. 

The  eccentricity  is  easily  found ;  for  we  have 


)  .  sin  A2 

a 


;    but   since   t)2  =  n*  V* 


XT  „ 

Now  —  =  2  —  — 

a  M  +m 


M  +m 


,  we  have 


TV' 


=  n2,  and  — =  2-ns; 
M  +  m  a 


2 =  n2  .  so' that  we  get  by  substitution 


=  A/1  -  n2  (2  -  na)  .  sin  A8 


(37) 


=  A/COS  A2  +  sin  A2  (1  —  n2)2  ' 

We  see,  therefore,  that  the  ratio  e  of  the  eccentricity 
to  the  semiaxis,  or  the  figure  of  the  ellipse,  or  hyper- 
bola, depends  solely  on  the  angle  of  projection  and 
the  ratio  of  the  velocity  of  projection  to  that  in  a  circle 
at  the  same  distance  ;  and  if  this  latter  ratio  remain 
the  same,  the  distance  may  be  varied  to  any  extent 
without  changing  the  figure  of  the  conic  section. 

It  only  remains  to  determine  its  position,  or  the 
angle  made  by  the  greater  axis  (or  line  of  apsides) 
with  the  distance  SP. 


Physical 
Astronomy. 


(35) 


Let  us  now  consider  the  effect  on  the  form  of  the 
conic  section  and  on  the  position  of  its  major  axis, 
arising  from  a  change  not  in  the  velocity,  but  in  the 
angle  of  projection,  i.  e.  the  angle  made  with  the 
radius  vector  by  the  direction  in  which  the  body  is 
projected. 

APM  being  any  conic  section,  and  S,  H,  its  foci,  SY, 
SZ,  perpendiculars  on  a  tangent  at  P,  the  angle  of  pro- 
jection SPy,  which  we  will  call  A,  is  equal  to  HPZ, 
and  therefore 

SY  =  SP .  sin  A,  and  HZ  =  HP  .  sin  A. 
consequently, 

SY  x  HZ  =  SP  .  HP  .  sin  A2; 

but,  by  the  property  of  the  conic  sections,  SY  x  HZ 
=  CD2 ;  and  therefore  CD2  =  SP  .  HP  .  sin  A*.  Now, 
we  have  SP  =  r,  HP  =  2  a  —  r,  CD2  =  a2  (1  -  e2)  ; 
hence,         a2  ( I  —  e*)  =  r  (2  a  —  r)  .  sin  A* 


2  (M  +  m) 


Now,  if  we  call  ASP,  0,  we  have 

a  (1  —  e2) 
"  1  +  e  .  cos  6 ' 

cos  0  =  —  ; 


in  which,  substituting  for  e  and  a  (1  —  e2),  their  values 
before  found,  we  shall  obtain 

n2  .  sin  A2  —  1 

(38) 


cos  6  = 


Vcos  A2  +  (1  —  n2)*  .  sin  A2 
on  which  value  we  may  make  the  same  remark  as  on 
that  of  e,  and  in  both  which  it  will  be  recollected, 


that  n2  = 


M  +m 


SECTION  V. 


On  certain  peculiar  cases  of  the  celestial  motion  ; 
viz.  when  the  orbit  is  of  very  great,  or  very  small 
eccentricity.  Of  circular  and  parabolic  motion. 

The  circumstances  of  circular  motion  are  too  sim- 
ple to  need  much  consideration.  The  velocity  is 
uniform  and  equal  to  that  which  would  be  acquired 
by  falling  down  half  the  radius  with  the  force  at  the 
circumference  continued  uniformly.  This  is  evident, 
if  we  recollect  that  the  velocity  in  a  circle,  by  what 


has  been  just  proved,  is 


-,  but  that  acquired 


by  the  action  of  a  constant  force  F,  acting  through  a 
space  5  r,  is  given  by  the  equation  i>2  =  2F«  =  Frin 

(M+m)r      M  +  OT 

the  present  case  = = .     The  perio- 
dic time  is  equal  to  the  circumference  divided  by  this 

constant  velocity,  or  to  —  ,  as  we  have  before 

VM  +  m 
shewn  by  a  more  general  reasoning. 

In  a  conic  section  of  so  small  an  eccentricity  that  its 
square  may  be  neglected,  it  may  be  worth  while  to 
recapitulate  some  of  the  chief  formula?,  developed  to 
the  first  power.     We  have  then 
r  =  a  (1  —  e  .  cos  0)     ~\ 

=  a  (1  —  e  .  cos  v)       >  ;  (39) 

=  a  (I  —  e  .  cos  nt)  J 
0  =  v  —  e.sinv  \ 

0=  nt  +  Ze  .sinnt    f  ' 

v  =  n  t  +  e  .  sin  n  t ;  (41) 

n  t  =  0  —  2  e  .  sin  0 ;  (42) 

Let  us  consider  the  motion  in  a  parabola,  and  since 

in  this  case  we  have  a  =  CD  ,  e  =  ],  and/)  =  a  (1  — e2), 

if  we  call  the  perihelion  distance  D,  we  have  D  =  %  p, 

and  p  =  2  D.       Then   will   the  equations   (17)    and 

(18)  become 

2D  D 

(43) 


1  +  cos  0 


/      0 

Ccos  ^ 


PHYSICAL    ASTRONOMY. 


661 


Astronomy. 


dt  = 


(2D)* 


-f-  m 


( 


dO 


the  resolution  of  a  cubic  equation  of  the  form 


+  COS  I 


X3  +  3  X  =  A 


Physical 

Astronomy. 


To  integrate  the  latter,  put  tan  ~  =  x,  and  we  have      where  A  =  — 

Z  T 


VM  +  m  , 

.      This   cubic,  it  is  easily 


0  = 


dO  = 


which  substituted,  give 


1    —  X4 
1    —  X* 

1  +I2 

2d£ 

1  -t-^a 


+  m 


x 


Di 


</M   +  TO 


or, 


This  equation  gives  tan  — , 


V  2 


(44) 


and   consequently    0,    by 


shewn,  has  but  one  real  root  ;  and  since  the  co-effi- 
cient of  x  is  an  absolute  number,  and  therefore  x  a 
function  of  A  only,  its  root  may  be  found  in  any  pro- 
posed case  by  a  table  of  single  entry.  In  fact,  sup- 
pose we  have  formed  such  a  table,  containing  the 
values  of  x  (or,  for  greater  convenience,  of  2  x 
arc  (tan  =  a:)  or  0)  for  every  value  of  t  on  the  sup- 
position of  D  =  1,  then  will  this  table  serve  for  all 
cases,  and  for  every  other  value  of  D  ;  for  we  have 
only  to  multiply  t  by  D—  f-  ;  and  calling  the  product 
T,  look  out  in  the  table  for  the  value  of  0  correspond- 
ing to  the  time  T.  A  comet,  describing  a  parabola, 
whose  perihelion  distance  is  1,  will  describe  90°  of 
anomaly  from  the  perihelion  in  about  109  days. 
Hence,  the  use  of  the  table  of  a  comet  of  109  days  given 
in  works  on  astronomy,  Lalande's  Astron.  2d  edit. 
vol.  iii.  p.  335;  Delambre's  Astronomy,  vol.  iii.  p.  434, 
from  which  we  extract  it  as  subjoined. 


Table  of  a  comet  of  109  days. 


a 

True  anomaly. 

l 

1  jj 

True  anomaly. 

i 
a 

True  anomaly. 

1 
a 

True  anomaly 

a 
\ 

a 

True  anomaly 

r, 

a 

True  anonialy. 

0 

0°  0'  O'O 

3o 

43°58/34//8 

7C 

ri°51'23"2 

10r 

88°20/42/' 

14 

98°  56'  22" 

17 

106°20'14"2 

1 

1  23  37-4 

36 

44  59  57'5 

71 

72  27  4-4 

IOC 

88  42  42- 

14 

99  11  12 

17 

106  3059-4 

c 

2  47  11-9 

37 

46  0  26'6 

72 

73  2  13  J 

107 

89  4  260 

14 

99  25  54- 

17 

106  41  39-4 

< 

4  10  40'4 

38 

47  0  2'2 

73 

73  36  510 

10S 

89  25  53-4 

14 

99  40  26- 

m 

10652  14-1 

L 

5  34  O'l 

39 

47  58  44-8 

74 

74  10  57'8 

10! 

89  47  5- 

14 

99  54  50- 

17£ 

107  243-5 

r 

6  57  80 

40 

48  56  34-5 

75 

74  44  34-3 

110 

90  8  1-3 

14 

100  9  6 

ISC 

107  13  7'7 

6 

8  20  1-3 

41 

49  53  32-0 

76 

75  17  41-2 

a 

90  28  42-4 

146 

10023  13-4 

18 

107  23  26-9 

7 

9  42  37'2 

42 

50  49  37'6 

77 

75  50  18-9 

112 

90  49  87 

147 

10037  12-2 

182 

1O7  33  41-1 

8 

11  4  53-0 

43 

51  44  51-8 

78 

76  22  28-1 

11: 

91   9  20-3 

148 

10051  2'8 

183 

107  43  50-2 

9 

12  26  45-9 

44 

52  39  15-2 

79 

76  54  9-3 

114 

91  29  17-6 

149 

101  445-4 

184 

107  53  54  6 

10 

13  48  13-4 

45 

53  32  48-2 

8C 

77  25  23-1 

115 

91  49  07 

150 

101  1820-1 

18r 

108  354-1 

11 

15  9  13-1 

46 

54  25  3L-6 

81 

77  56  10  1 

116 

92  8  30-0 

151 

1O1  31  47  1 

186 

108  13  48-8 

12 

16  29  42-5 

47 

55  I/  25-8 

82 

78  26  30-6 

117 

92  27  457 

152 

101  45  65 

187 

108  23  38-8 

13 

17  49  39-4 

48 

56  8  31-5 

83 

78  56  25-3 

118 

92  46  48-0 

153 

101  58  18-2 

188 

108  33  24-2 

14 

19  9  1-5 

49 

56  58  49'4 

84 

79  25  54-6 

119 

93  5  37-3 

154 

102  11  226 

189 

108  43  5-0 

15 

20  27  46'8 

50 

57  48  20-1 

85 

79  54  59  1 

120 

93  24  13-6 

155 

102  24  19-6 

190 

1085241-3 

16 

21  45  53-4 

51 

58  37  4-3 

86 

80  23  39'1 

121 

93  42  372 

156 

102  37  9'4 

191 

109  2  13-1 

17 

23  3  19'4 

52 

59  25  27 

87 

80  51  55-4 

122 

94  O  48-4 

157 

102  49  52-1 

192 

109  11  40-4 

18 

24  20  3-1 

53 

60  12  15-9 

88 

81  19  48-1 

123 

94  18  47-3 

158 

103  227-8 

193 

109  21  3-5 

19 

25  36  2-9 

54 

60  58  44-8 

89 

81  47  17'9 

124 

94  36  34-2 

159 

103  1456-6 

194 

109  30  22-2 

20 

26  51  17'3 

55 

61  44  29'9 

90 

82  14  25-2 

125 

94  54  9'2 

160 

103  27  18-5 

195 

109  39  36'7 

21 

28  5  45-1 

56 

62  29  32-1 

91 

82  41  10-3 

126 

95  11  327 

161 

103  39  33-8 

196 

1094846-9 

22 

29  19  25-0 

57 

63  13  52  0 

92 

83  7  33'7 

127 

95  28  44-7 

162 

10351  42-4 

197 

09  57  53'] 

23 

30  32  15-8 

58 

63  57  30-3 

93 

83  33  35-9 

128 

95  45  45-4 

163 

104  344-6 

198 

10  655-1 

24 

31  44  167 

59 

64  40  277 

94 

83  59  17-2 

129 

96  2  35-2 

164 

104  15  40  3 

199 

10  15  53  1 

25 

32  55  26'7 

60 

65  22  45-0 

95 

84  24  38-0 

130 

96  19  14-0 

165 

04  27  29-7 

200 

102447-1 

26 

34  5  45'2 

61 

66  4  22'8 

96 

84  49  387 

131 

96  35  42-2 

166 

04  39  12  8 

27 

35  15  11-4 

62 

66  45  22-0 

97 

85  14  19'8 

132 

96  51  59-8 

167 

04  50  49-8 

28 

36  23  44-8 

63 

67  25  43-1 

98 

85  38  41-6 

133 

97  8  7'1 

168 

05  220-7 

29 

37  31  24-9 

64 

68  5  26-9 

99 

86  2  44-4 

134 

97  24  4-2 

169 

05  13  45-6 

30 

38  38  11-5 

65 

68  44  34-0 

100 

86  26  28-7 

135 

97  39  51-3 

170 

05  25  4'7 

31 

39  44  4-2 

66 

69  23  5-1 

101 

86  49  54-7 

136 

97  55  28-6 

171 

05  36  17-9 

32 

40  49  29 

67 

70  1   1  0 

102 

87  13  2-8 

137 

98  10  562 

172 

05  47  25-4 

33 

41  53  7-6 

68 

70  38  22-1 

103 

87  35  53-3 

138 

98  26  14-3 

173 

05  58  27-2 

34 

42  56  18-2 

69 

71  15  93 

104 

87  58  26-5 

139 

98  41  23-0 

174 

06  9  23-4 

35 

43  58  34'8 

70 

"1  51  232 

105 

88  20  42-9 

140 

98  56  22-5 

175 

06  20  142 

662  PHYSICAL    ASTRONOMY. 

Astronomy      2dly.  Let  us  now  consider  the  circumstances  of  the     sary  to  complete  the  theory  of  such  comets  as  are    Physical 
^•"V™"^  motion  of  a  body  in  an  ellipse  of  great  eccentricity,  or     known,  or  may  be  suspected,  to  revolve  in  very  elon-  Astronomy- 
approaching  very  near  to  a  parabola.    This  is  neces-    gated  ellipses. 

Since  e,  the  ratio  of  the  eccentricity,  approaches  very  near  to  unity,  1  —  e  must  be  a  very  small  quan- 
tity ;  and  if  we  put  ft  = ,  we  shall  have  e  = -,    1  —  e=          ,  1  +  e  =          ,   (1  —  e 2)  = 


now,  if  D  =  the  perihelion  distance,  we  have  D=  a  (1  —  e),  consequently 

D  (1  +  e) 2D D 

~  2  +  e  cos  0~  (1+/3)  +  (1—  j3)  .  cosfl  ~  1  +  cos  0  1  —  cos  0 

2  '         2 

D  _ 
\* 

D 


1    —   COS  6 


and  developing  this  in  powers  of  /3,  we  obtain 

D         (  X        0  \4  /       0  \4  ") 

n—  )    -&c.h  (45) 


which  is  a  very  simple  expression  for  the  radius  vector  in  the  elongated  ellipse. 
To  find  the  time,  we  must  make  the  same  substitution  in  the  expression, 


t  ,  c  = 


O*     /' 

+  m     J 


(l+e.cos 
Now  {a  .  (1  -  e»)  }-f  =  *  and  since 


(1  +  e  .  cos  fi)'  =       +        {  1  +  /J  +  (1  -  /3)  .  cos 
4       (1  +  cosg  1  —  cos  0 

~~          " 


(i  +  PY 

if  we  put  tan—  =T,  we  shall  have  -  2__-  —  -=  d  T  ;  and  the  integral  will  become  —  -  —   .  /   j—,     —  =^rt 

^  /  V  \  *»/\*TP«iJ 


whence  we  get  after  all  substitution, 


/27TTM      ni       /»dT(l  + 
-  V     M  +  m  *   J     (1  +  /3. 


.T»)« 
and  the  part  of  this  under  the  integral  sign  being  developed  in  powers  of  ft,  and  the  whole  integrated,  we  get 

'  +  "-»  •N/^'K'  +  D-T'G'  +  D  +  TTl'G  +  r)-*')        <«> 

By  the  aid  of  this  series,  we  may  express  very  readily  the  difference  between  the  true  anomaly  in  a  very 
eccentric  ellipse,  and  that  in  a  parabola  of  equal  perihelion  distance,  for,  T  still  expressing  the  tangent  of 
half  the  former  true  anomaly,  and  T  that  of  the  latter,  if  we  develope  according  to  powers  of  /3,  pursuing  the 
process  only  as  far  as  /?*,  we  get 


, 
process  only  as  far  as  /?*,  we  get 

-c  =71=1     + 


DrA/2       f  T») 

Now,  the  time  being  the  same  both  in  the  parabola  and  ellipse,  this  must  equal      -  ^  T  +  T~  f  •   and 

VM  +  m  (. 

therefore  the  series  within  the  brackets  on  both  sides  must  be  equal. 

,     T>  r         T>        2rs  T         ST>        2  T*         3  T' 

Let  T  +  —  =  a  ;  -----  =  6  •    -----  ---  1  --  =  c 

3'225  88  5  7 

T»  T      TJ      2T5  T       ST5   .   «T5      3T7 


PHYSICAL    ASTRONOMY 


663 


Astronomy  tnen  we  have  A+B/3  +  C/34  =  a; 

1— "~V"™^  and   the  first  member  of  this  equation,   though  apparently  a  function  of  ft,  cannot  be  really   so,  as   the 
second  does  not  contain  it.     Hence,  if  we  call  A  the  difference  between  the  elliptic  and  parabolic  anomaly, 

6  +  A  e 

or    suppose    the    former    anomaly  0  +  A    the    latter  0,   since  T  =  tan  — ,  and  ^  =  tan  — ,  we  may 


express  A,  B,  and  C,  thus  : — 


A  =  a 


d_a      A   ,    rf*a 
d0  '    1        d ffi 


1  .8 


&c. 


Physical 
s  ™°omT 


B  =  b  +  —  .  — H 

^  d0       1        dff* 


1  .-2 


P i  _        t±    i   _ ~        *->• 

d0  '    I        dff*    '1.2 


+  &c. 


+  &c. 


but   A  itself  is  required  to  be  expressed  in  terms  of  /3  and  its  powers  ;   and,  consequently  we  may  suppose 
A  =  P/3  +  Q/3*  +  &c. 
If  then  we  make  these  substitutions,  our  equation  becomes 


+   b  ft    +    P  YQ     •    P*    +    &C- 
+   C  ft3    +    &C. 

whence  we  obtain  by  equating  the  co-efficients  of  /3  to  zero, 

„  da  _  da       Pa       d*  a  db 


Now 
(1 

we 

+ 

da 

o  — 
da 

rd^                 "dO* 

dr 

2    '    d 
T'     da 

d-r        I 

+  T'                  da 

d0 

~  d-r 
therefc 

d  & 
irp    P  — 

3'   dr 

•  +  T'  -J- 

L    -1-    T^    . 

4 

du 

-} 

2                       dO 

2 
Again, 

(1     +   T*)'       ( 

d8  a         d       da       n    ,. 

5 

d6>d6>|2         2 
whence,  after  all  reductions,  we  obtain 


r» 

5( 


18 


Q  = 


-     ,   ,    167     7    ,   24  „ 

T T3     -4 T5    H T7    -^ T9    -(-       ,,„     T 

4  4  20  140  35  175 


and,  finally, 


A  = 


_    T   +    T3    +  » 

5 


20 


167 
140 


35 


18 
175 


(48) 


SECTION  VI. 

On  the  determination  of  the  planetary  orbits  a  pos- 
teriori, or  from  observation. 

The  determination  of  the  elements  of  the  orbit  of 
a  planet,  or  comet,  is  justly  regarded  as  one  of  the 
most  difficult  problems  of  astronomy.  By  the  elements 
of  the  orbit  are  meant  those  constant  data  which 
determine  the  position,  form,  and  magnitude  of  the 
conic  section  described,  and  the  body's  place  in  it  at 
a  given  epoch,  and  are  six  in  number,  viz. 

The  inclination  of  its  plane  to  the  ecliptic, 

The  longitude  of  its  ascending  node, 

The  position  of  its  axis,  or  the  longitude  of  the 

perihelion, 
The  length  of  its  axis, 

VOL.  III. 


The  eccentricity,  or  the  length  of  the  semipara 
meter,  or  the  perihelion  distance, 

The   moment  of  passing  the  perihelion,  or  the 

heliocentric  longitude  at  any  assigned  time, 
which  correspond  to,  and  are  functions  of  the  six 
arbitrary  constants  introduced  by  the  integration  of 
the  equations  (1),  (2),  and  (3).  If  we  knew,  and 
could  subject  to  calculation,  the  causes  which  origi- 
nally determined  the  motions  of  the  heavenly  bodies, 
we  could  assign,  a  priori,  by  the  means  already  pointed 
out,  the  values  of  these  constants  from  the  conditions 
which  subsisted  at  the  commencement  of  their 
motions;  but  as  these  are,  and  must  ever  remain, 
unknown,  we  can  only  discover  the  values  in  question, 
a  posteriori,  by  observation  of  the  apparent  places  of 
each  body  at  different  times. 
4  B 


PHYSICAL    ASTRONOMY. 


Astronomy.      In  this  research,  the  great  difficulties  arise  from  our 

— ~v.-»'  not  observing  from  a  fixed  station  in  the  centre  of 

our  system.     To  an  observer  stationed  in  the  sun,  the 

elements    would  offer  themselves  with    comparative 

ease. 

But  the  motion  and  eccentric  position  of  the  earth 

render  the  whole  affair  infinitely  more  complicated ; 

and  to  analyse  the  problem,  it  becomes  necessary  to 

take    into    consideration    the    relations   between    the 

heliocentric  and  geocentric  places  of  a  heavenly  body. 

To  this  effect,  let  S  be  the  sun,  E  the  earth,  P  the 

planet,  or  comet,  projected  into  p  upon  the  ecliptic  by 

the  perpendicular  Pp.     Let  S  7  be  the  line  of  equi- 

Fi^.  8.     noxes,  parallel  to  which  draw  E  7.     Then  will 

L  =  7  S  E  represent  the  earth's  heliocentric  longitude. 

/  =  -/Sp the  planet's  heliocentric  longitude. 

X  =  7  Ep the  planet's  geocentric  longitude. 

h  =  PSf> the  planet's  heliocentric  latitude. 

ft  =  PEp its  geocentric  latitude. 

R  =  the  earth's  distance  from  the  sun. 
r  =  the  planet's  distance  from  the  sun. 
l>  =  the  planet's  distance  from  the  earth. 
d,5=  the  respective  curtate,  or  projected  distances  of 

the  planet  from  the  sun  and  earth. 
It  is  obvious,  then,  that  we  shall  have  the  following 
relations  : — 

1st.  Pp  =  r  .  sin  b,  also  the  same  Pp=  p  .  sin  ft, 
so  that, 

r  .  sin  b  =  p  .  sin  ft;  (49) 

=  r.cos6  (50) 

=  p.cosft  (51) 

3dly.  Since  the  angle  PSE  of  the  triangle  PSE  is 
equal  to  I  —  L,  and  its  sides  are  R,  d,  and  5 

««  =  Ra  +  d5  —  2  R  d  cos  (I  —  L) ;  (52) 

Lastly  producing  SE  to  C,  the  angle  p  EC  =  p  E  7 
—  CS  i'=  \  —  L,  but  the  angle  pE  c  =  180°  —  p  E  S, 
consequently  the  equation 

<F  =  11=  +  S«  -  2  R  5  .  cos  p  E  S 
gives 

d«  =  R*  +  rf*  +  2  R  S  .  cos  (X  —  L)  :  (53) 

Thus  then  we  have  five  distinct  and  independent 
relations  among  the  ten  quantities  L,  I,  \,  &c.  Now, 
it  is  evident,  that  if  R  and  L  be  given,  (or  the  place 
of  E  fixed)  the  geocentric  latitude  and  longitude,  ft  and 
X,  will  be  determined  when  the  place  of  P  is,  but  this 
is  not  determined  unless  I,  d,  and  b,  are  so.  But  when 
all  the  five  quantities  L,  R,  /,  d,  b,  are  given,  the  five 
equations  above  deduced  suffice  to  determine  the 
others,  and  therefore  express  all  the  relations  subsist- 
ing between  the  heliocentric  and  geocentric  places. 
The  last  may,  however,  be  simplified  Sy  substituting 
for  c2  its  value  given  in  (52),  when  the  whole  becomes 
divisible  by  2  R,  and  gives 

o  =  R  +  «  .  cos  (X  -  L)  -  d  .  cos  (I  —  L)  ;  (54) 
Another  equation  (included,  of  course,  in  the  forego- 
ing,) may  also  be  obtained  it'  we  consider,  that  in  the 
triangle  PSE,  we  have 

r3  =  R«  +  p*  -  2R/>  .cosPES 
but  cos  PES  =  cospES  x  cos  PEp  by  spherical  tri- 
gonometry, the  plane  PEp  being  at  right  angles  to 

pES.     Hence,  we  get  (writing  — ^-— ,)  or/  .  Sec  ft  for 

COS  p 

P,  and  —  cos  (X  —  L)  for  cos  p  ES 

r*  =  R'  +  ««  .  Sec  /3«  +  2  R  o  ,  cos  (X  -  L)  ;       (55) 


In  like  manner,  eliminating  p  from  (49),  it  becomes 

r  .  sin  6  =  &  .  tan  ft  ;  (56) 

and  putting  in  (54)  for  d,  its  value  r  .  cos  b, 
o=R  +  e  .  cos(X— L)— r  .  cos  b  .  cos  (I— L)  (57) 

This  equation  may  also  be  derived  independently, 
if  we  drop  the  perpendicular  p  q,  for  we  then  have 
S  q  =  SP  .  cos  PS  p  .  cos  p  S  9 
=  r  .  cos  b  .  cos  (I  —  L) 
and  also 


Physical 
Astronomy. 


=  R  +  S  .  cos  (X  —  L) 

which  equated,  give  the  equation  in  question.  We 
may  also  derive  another,  which  will  be  useful  in  the 
sequel,  by  equating  two  values  of  the  perpendicular 
itself. 

pq  =  Sp  .sinpS9 
which  equated,  give 

r  .  cos  b  .  sin  (I  —  L)  =  S  .  sin  (X  —  L) ;  (58) 

Thus  we   have  reduced  the   five  equations  to  four, 

55,  56,  57,  58,  which   contain  only  the  radius  vector 
of  the    earth,    that    of  the    planet,  and    its    curtate 
distance  from  the  earth,  and  the  heliocentric  and  geo- 
centric latitudes  and  longitudes  of  the  two  bodies. 

The  problem  of  determining  the  elements  of  P's 
orbit  from  observation  requires  these  relations  to  be 
combined  with  the  conditions  of  P's  motion  and  of  the 
earth's.  Now,  if  we  call  8  the  longitude  of  P's 
ascending  node,  and  i  the  inclination  of  its  orbit,  we 
have,  by  spherical  trigonometry,  for  the  heliocentric 
latitude. 

tan  b  —  tan  i .  sin  (I  —  8)  (59) 

Moreover,  if  we  call  ir  the  longitude  of  the  perihelion 
on  the  orbit,  0  the  true  anomaly,  and  Y'  the  angle 
£3  SA,  or  the  distance  of  the  node  from  the  perihelion 
on  the  orbit,  (see  fig.  9)  we  shall  have  Fig.  9. 

cos  A  S  a  =  tan  8  S  a  .  cot  a  SA 
or          tan  (IT  —  Q)  =  cos  i  .  tan  ty  (60) 

and  also 

sin  b  =  sin  i  .  sin  (^  +  0)  (61) 

these   seven    equations,  of  which  the  four   first,  55. 

56,  57,  58,  express  the  relations  between  the  helio- 
centric   and   geocentric    places    of  a  body  any  how 
situated,  the  other  three,59,6O,61,  merely  express  the 
condition  of  every  point  in  the  orbit  lying  in  a  plane 
passing  through  the  sun.     These  must  be  combined 
with  the  dynamical  results  peculiar  to  the  planetary 
motions,  viz.  1st.  The  proportionality  of  the  areas  to 
the  times  ;  and,  2dly,  The  nature  of  the  orbit,   re- 
garded as  an  ellipse  or  parabola  of  unknown  position, 
form,  and  magnitude  ;  all  which  are  included  in  the 
three  equations,   21,  22,  23,  or  others  equivalent,  to 
them.     Thus  we  have  a   system   of  nine  equations, 
representing,  in   general,   the   relations   between   the 
time   elapsed  since  a  given  epoch,  and  the  observed 
geocentric  latitude  and  longitude. 

These  equations  involve,  as  constant  quantities,  the 
elements  of  the  orbit,  t,  8,  ir,  a,  e,  E,  while  the  variable 
ones,  being  either  given  by  observation,  or  expressed 
in  functions  of  others  that  are  so,  the  whole  system 
of  equations  may  be  regarded  as  expressing  a  known 
(though  a  very  complicated)  relation  among  the  ele- 
ments ;  and  as  each  observation  affords  a  similar 


PHYSICAL    ASTRONOMY. 


665 


\stronomy.  relation,  it  is  evident  that  the  complete  determination 

s-"-v~''  of  the  latter  cannot  require  more  than  six  observations. 

In  fact,  however,  from  the  peculiar  nature  of  some  of 

the  equations,  fewer  will   suffice,   half  that  number 

only  being  necessary. 

The  complication  of  the  relations  in  question  pre- 
cludes all  idea  of  a  direct  solution  of  the  problem, 
and  to  apply  them  to  any  particular  case  indirect  and 
approximative  ones  have  been  invented,  but  with 
every  assistance  from  such  simplification  ;  and,  after 
all  the  force  of  analysis  has  been  exercised  upon  it,  it 
still  remains  a  very  difficult  problenij  and  one  which 
our  limits  will  by  no  means  permit  us  to  enter  upon 
in  its  full  extent.  In  fact,  the  case  of  an  elliptical 
orbit  to  be  determined  is  one  of  very  rare  occurrence  ; 
and  we  shall  therefore  content  ourselves  with  pointing 
out  the  course  to  be  pursued  to  obtain  the  readiest 
approximation  to  the  elements  of  parabolic  motion. 
The  frequent  visits  of  comets  to  our  system  rendtr 
this  an  important  case.  These  singular  bodies,  with 
one  or  two  exceptions,  have  always  been  found  to 
describe  parabolas,  or  ellipses  of  such  extreme  eccen- 
tricity, as  to  be  undistinguishable  from  parabolas 
within  considerable  distances  on  either  side  their 
perihelion. 

Preparatory  to  this  research,  we  must  premise  the 
following  remarkable  theorem. 

In  a  parabola,  if  r,  r  denote  any  two  distances  from 

the  focus,   k  the  chord  of  the  arc  they  include,  and  t 

the  time  of  a  body  describing  that  arc,  and  g  a  certain 

constant  the  same  for  all  parabolas,   we  shall  have 

g  t  =  (r  +  /  4-  fc)i  —  (r  +  r1  —  fc)*  ;  (62) 

To  shew  this,  let  0  and  0'  be  the  true  anomalies  cor- 
responding to  the  radii  r,  r',  and  t  being  the  time  of 
describing  the  intermediate  arc,  and  e  the  time  elapsed 
since  the  perihelion  passage  to  the  commencement  of 
the  time  t,  we  have,  by  (44) 
0 


(          0'         1   / 

t  +  £  =  <7  ;   tan 1 1  tan 

^  **         **   \ 

where  9  =  —  —  Consequently,  we  have 

A/.M   -j-  TO 

f  /      e'  0 . 

t  =  q  .  I  (tan—  -  tan—  j 

1  //        ff \^       i        0  \3\) 

+  Y((tanT)-(tanT))} 

Now,  by  the  property  of  the  parabola,  we  have  also 
D  D 


(cosir)* 


. 
r  — 


0f 0 

And  if  we  put  —     —  =  0,  we  shall  have 
8 

_  cos  j  0'  _     cos  (1  0+0) 

r'  A/777    ~~  COS  i  0    ~  COS  i  0 

or,  —    =:  cos  0  —  tan  i  0  .  sin  0 

Vrr 

whence  we  get 

0                                               r       ) 
tan  —  =  cosec.  <z>  <  cos  0 .  V  (63) 


f  r       1 

=  cosec.  0  <  cos  0 ,  > 

I  <ST7 1 


Similarly,  since^  0  =  i  0'  —  0,  we  have, 

L/l  or       /       =  cos  <*  S'  ~  ^ 

V      r         Vrr 


Physical 
Astronomy, 


and  therefore 


cos 


& 


—I 

77) 


Vrr') 

but  k  being  the  side  of  a  triangle  opposite  to  the  angle 
(0'—  0)  or  2  0  and  r  and  r1  the  sides  adjacent,  we 
have,  by  trigonometry, 


COS  0  = 


V(r  +  r)*  —  k* 


sin  0  = 


A/  A;2  —  (r   —  r)2 


or,  if  we  put 


A/ 


—  A'2  =  R  and 


R  =  2  Vr  r'  .  cos  0  and  S  =  2  Vr  r'  .  sin  0 

R  S 

cos  0  =  -  —       sin  0  =  -  — 

2  Vrr' 
0'  R  -  2  r' 


-  — 
2  Vrr' 

Hence,  we  obtain 

0       R-2r 


;  tan¥= 


6' 


tan__ta.n_  = 


0   _  2  (r  +  r1  —  R) 


also 


1  + 


(R2  +  S2)  -  4  r  R  +  4  r2 

but'R2  +  S2  =  (r  +  r'Y  -  (/  -  r)2  =  4  r  r',  so 
that  we  get 

/  0\8        1 

I  COS    .  I      =  . 

V  2  /         4r 

and,  in  consequence, 


r  +  r'  -  R 


D  =  r.   (cos—) 


0_\* 
2 


4        r  +  r'  —  R 
Now  (r  +  /-  R)  (r  +  r'+  R)  =  (r  +  r')8  -  R2  =  *«  j 


so  that 


D  =  ~  .  (r  +  r'  +  R) 


{0'  0  ~)        S 

tan tan  —  /  =  — 
2  2  j        2 

consequently 

1! 


m.A/2f         ff  01       S2  .  A/2(r+/  +  R) 

tan tan  —  >  — i-==: — -  • 

•  TO  I  4  k  .  A/M  +  TO 

0  0/ 

Now,  if  we  put  T  and  T'  for  tan  —  and  tan  • — •,  we 

have      T/3  —  T3  =  (T'  -  T)  (T'2  +  TT  +  T4 } 
and  therefore 

fpa    i    'j"jv    i    'p'a'i 


«= 


i  + 


+  m  "  '  I  3  | 

The  part    without  the  brackets  we  have    already 
considered.    Let  us  next  examine  that  within.    Now, 
if  in  T2  +  XT'  +  T'2,  we  substitute  for  T  and  T',  their 
4  K  2 


666 


PHYSICAL    ASTRONOMY. 


Ajtronomy.  R  —  2r        ,    —  R  +  2  r'    . 

v^. ~^~.  values ; and   -    — -,  it  will  become 

R3  -  2  R  (r  -f-  /)  +  4  (r2  -  r  r'  +  /*) 

S* 

and  substituting  this,  the  quantity  within  the  brackets 
becomes 

3  s«  +  Ra  +  4  (r*  —  r  /  +  /»)  —  2  R  (r  +  rQ 
3S* 

But  Rs  +  S2  =  4  r  /,  therefore  3  S2  +  R2  =  12  r  r' 
—  2  R4  =  2  A2  —  2  )•-  +  8  r  r'  —  2  r  a.  So  that  our 
expression  reduces  itself  to 


3Sa 

2 


and  therefore  we  have,  finally, 


6  fc  .  v  M  +  m 


but 


-v/r  +  r    +  R  .  (r  +  r>  -  R)  = 
=  <v/(r  +  r'  +  R)  .  (r  +  r'  -  R)^ 


=  fc.  Vr  +  /  -  R 

because  (r  +  r'  +  R)  (r  +  r'  —  R)  =  fc».      So  that 
substituting,  we  get 

_  k  .  Vr  +  /  +  R  +  (r  +  rQ  .  ^r  +  /  —  R       .— 


6  . 


+ 


Now  R  =  A/(r  +  /)*  —  k2,  so  that  the  two  radi- 
cals in  the  above  expression  are  of  the  form 

\fa  ±  */a*  —  k* 

putting  a  for  r  +  r/.  Hence  the  usual  rule  for 
extracting  the  root  of  a  binomial  surd  applies  to 
them,  the  root  being,  as  it  is  shewn  in  all  books  of 
algebra,  _  _ 

Vn  +  k         */a  —  k 


first  and  second  ;  secondly,  from  the  first  and  third  ; 
the  three  following 

o  =  x(y"z  -y1  z')  +  x'  (y  2"  - 
o  =  y  (z"xf  -  z'x')  +  y'  (zx"  - 
o  =  z(x"y'  -  x'y")  +  z'  (x  y"  - 

Now  z  y'  —  z'  y  represents  the  double  area  of  the 
projection  of  the  plane  triangle  included  between  the 
radii,  r,  r'  drawn  to  the  place  of  the  comet  at  the  first 
and  second  observations,  and  the  chord  k  of  the  para- 
bolic arc  joining  their  extremities  on  the  plane  of  the 
z  and  y  ;  similarly,  2  y"  —  z"  y  represents  double  the 
triangle  formed  by  the  projections  of  the  radii  r,  /' 
and  the  chord  k'  joining  their  extremities  on  the  same 
plane  ;  and  if  we  denote  by  k"  the  chord  joining  the 
extremities  of  the  radii  T  ,  r",  the  double  of  the  pro- 
jection of  the  triangle  included  between  r',  r",  and  k" 
will  be  represented  by  z'  y''  —  z"  y'.  If,  therefore, 
we  call  the  surfaces  of  these  triangles  respectively 

S,  S',  S" 

Since  the  projection  of  each  is  equal  to  the  surface 
multiplied  by  the  cosine  of  its  inclination  to  the  plane 
of  the  z,  y,  which  is  the  same  for  all,  we  shall  have  the 


Consequently,  we  have 

1 


(k  (Vr  +rf+  k  +  Vr  +  /  -  k) 


)  ) 

r  —  k) 


Physical 
Astronomy 


6  VM+mX  t=(r  +  /+  k)l  —  (r  +  r/—  k)i  ;    (62) 
which  agrees  with  the  proposition  as  announced,  if 

we  take  g  =  6  -v^M  +  m. 

The  relation  just  proved  to  subsist  between  the 
time  of  describing  any  parabolic  arc,  and  the  rectili- 
near distances  of  its  two  extremities  from  each  other, 
and  from  the  sun,  is  very  useful,  as  it  enables  us  from 
any  three  of  these  quantities  given,  to  find  the  fourth 
without  knowing  either  the  perihelion  distance,  or 
the  position  of  the  perihelion  with  respect  to  the  arc 
described.  In  an  analytical  point  of  view,  it  is  cer- 
tainly remarkable  for  the  length  and  complexity  of 
the  transformations  by  which  it  is  obtained  from  the 
general  principles  of  parabolic  motion.  It  appears  to 
admit  of  no  simple  demonstration,  and  that  above 
given,  tedious  as  it  may  seem,  can  hardly  be  replaced 
by  one  materially  shorter. 

We  shall  now  proceed  to  show  how  this  relation 
may  be  rendered  available  in  computing  the  elements 
of  a  comet's  orbit. 

Let  x,  y,  z,  be  the  co-ordinates  of  the  comet's  place 
at  the  epoch  of  a  first  observation,  x',  y',  z,  their  values 
after  any  time  t  and  x'',  y",  z'',  their  values  after  any 
other  times  t'  has  elapsed.  Let  X,  Y,  Z,  X',  Y',  Z', 
and  X",  Y",  Z'',  be  the  corresponding  co-ordinates 
of  the  earth's  place.  (If  the  plane  of  l;he  ecliptic  be 
chosen  for  that  of  the  X  and  Y,  we  shall  have  Z  =  o, 
Z'  =  o,  Z"  =  o.)  Also  let 

t'  -  t  =  t" 

then  will  /"  be  the  time  elapsed  between  the  second 
and  third  observations, 

Since  the  comet  moves  in  a  plane  passing  through 
the  sun,  we  have 

z  =  a  x  +  b  y  (64,  1) 

z'  =  ax'  +  by'  (64,2) 

z''  =  ai"  +  by't  (64,3) 

from  which,  if  we  eliminate  a  and  b  ;  first,  from  the 

and   lastly,  from  the  second   and  third  ;  we  shall  get 


y"z)  +  xfl  (y'z  -yz') 
xz'')  +  y''  (xz'  -zx') 
t"y)  +  z"  (x'y-xy') 
following  equation  instead  of  (65,1), 

xS"  -iSS'  +  x"S  =  o 
and  similarly, 

y  S"  -  /  S'  +  y"  S  =  o 
z  S"  •     z'S'  +  z"S  =  o 


(65,  1) 
(65,2) 
(65,3) 

(66,1) 

(66,  2) 
(66,  3 


Now,  if  we  retain  the  denominations  of  the  fore- 
going pages,  and  assume  the  line  of  the  equinoxes  S  y 
for  the  axis  of  the  x,  we  have 


x    =w  = 


=  R.cosL  +  S.cosX; 
=  R'  .  cos  L'  +  «'  .  cos  X'  ; 
=  R"  .  cos  L"  +  «"  .  cos  X''  ; 


j> 

x' 

also 

y    =  E  e  -f  p  q  =  R  .  sin  L  -j-  S  .  sin  X  ; 

yf  =  R'  .  sin  L'  +  6'  .  sin  X  ; 


y"  = 


R"  .  sin  L"  +  c"  .  sin  X''  ; 


(67,1) 
(67,  2) 
(67,  3) 

(67,  4) 
(67,  5) 
(67,  6) 


PHYSICAL    ASTRONOMY. 


667 


\stronomy.  we  have  also 


z    =  a  .  tan  ft 

z'  =&' .  tan  ft' 

r"  =  S"  .  tan  ft 

and  by  the  substitution  of  these  values  we  shall  obtain 
equations,  in  which,  instead  of  the  co-ordinates  x,  y,  z, 


(67,  7)     quantities  given  by  the  solar  tables,  or  by  observation    Physical 

(07,  8)     !lt  tne  three  assigned  instants,  so  that  the  only  un-  Astronomy. 

(6"  9}     known  quantities  contained  in  these  equations  are  the  v-~~v"~"'' 

three  curtate  distances,  S,  S',  S",  and   the  areas  S,  S', 

S"  ;   and  by  elimination,  any  one  of  the  three  former 

may  be  expressed  in  terms  of  the  latter.     Let  this 


&c.  the  curtate  distances  S,  S',  S'',  will  be  involved,  viz.     process"be  executed  then  in  succession  for  each  of  the 


(68,  1) 


(68,  2) 


o  =  S"  (S  .  cos  X  +  R  .  cos  L) 

-  S'  (&'  .  cos  X'  +  R'  .  cos  L') 
+  S  (a"  .  cos  X"  +  R"  .  cos  L") 

o  =  S"  (S  .  sin  X  +  R  .  sin  L) 

-  S'  («' .  sin  X'  +  R' .  sin  I/) 
+  S  («"  .  sin  X"  +  R"  .  sin  L") 

o  =  S"  .  S.  tan/3— S'.  S' .  tan/3'+S 

R,  L,  X,  and  ft,  and  their  accented  values,   are  all 

n  =  tan  ft    .  sin  (Xv  — 
-  tan  ft'  .  sin  (X 


}, 

J 
~l 

J 


quantities  S,  &',  S",  and  we  shall  have  the  following 
results  : 


(  a.  «  +  AR).S"-A'R'.S'  +  A''R".S  =  o;  (69,1) 
BR  .  S"  —  (  a  .  (,'  +  BTl')  S'  +  B''R''S  =  o ;      (69, 2) 
CR  .  S"  —  C'R' .  S'  +  (a  .  «"— C"R")  S  =  o  ;     (69, 3) 
.  tan/3" ;  (68,3)     where,  for  brevity's  sake,  we  have  made  the  following 


substitutions  — 


"-V)    -v 

-  *)      X-| 

-  X)    J 


+  tan  ft"  .  sin  (X'  —  X) 

A  =  tan  (3  .  sin  (L  -  X")  -  tan  ft"  .  sin  (L  -  V  )  : 
A'  =  tan  ft'  .  sin  (L'  -  X")  —  tan  ft''  .  sin  (L'  —  X' )  ; 
A"  =  tan  ft'  sin  (L"  -  X")  -  tan  ft"  .  sin  (I/'  -  X' ) ; 


B  =  tan  /3"  .  sin  (L  —  X  )  —  tan  ft 
B'  =  tan  ft"  .  sin  (I/  -  X  )  -  tan  ft 
B"  =  tan  ft"  .  sin  (L"  —  X  )  -  tan  ft 


sin  (L  —  X'')  ; 
sin  (L'  -  X")  ; 
sin  (L"  -  X")  ; 


C  =  tan  ft  .  sin  (L  —  X'  )  —  tan  ft'  .  sin  (L  —  X  )  ; 
C'  =  tan  ft  .  sin  (I/  -  X' )  —  tan  ft'  .  sin  (L'  —  X  )  ; 
C"  =  tan  ft  .  sin  (L"  —  X' )  -  tan  , 


sin  (L"  -  X  )  ; 


(70,1) 

(70,  2) 
(70,  3) 
(70,  4) 

(70,  5> 
(70,  6) 
(70,  7) 

(70,  8) 

(70,  9) 

(70,  10) 


In  these  equations,  the  right  hand  members  are  composed  entirely  of  quantities  given  either  by  observation, 
or  by  the  solar  tables  ;  and  their  calculation,  though  long,  is  greatly  facilitated  by  the  regularity  of  their  com- 
position. The  values  of  a,  A,  B,  C,  &c.  obtained,  we  have,  by  (69,  1)  (69,  2) 

ARS"  -  A'R'S'  +  A"R"S 


whence  we  obtain 


BRS"-  B'R'S'  +  B''R"S 

a  S' 
-i  ¥-       ¥. 

t,'   '          S"  '  B' 
(B'A  -  BA')  .  RS"  +  (B'A"  -  B"A')  R"S 


(71) 


B'  { BRS"  -  B'R'S'  +  B"R"S } 

So  far,  we  have  nothing  but  rigorous  equations,  and  it  does  not  immediately  appear  how  these  can  become 
serviceable  in  the  question  before  us  ;  but  if  we  consider  attentively  the  composition  of  the  numerator  of  the 

fraction  which  forms  the  second  term  of  —  here  statec.,  >ve  shall  find  that  the  equation  last  arrived  at,  affords 

tt 

a  means  of  obtaining  a  first  approximation.  Let  us  consider,  first,  the  term  AB'  —  BA' :.  If  we  substitute 
in  this  the  values  of  A,  B,  A',  B',  in  equations  (70,  2,  3,  5,  6,)  execute  all  the  multiplications,  strike  out  such 
terms  as  mutually  destroy  each  other,  and  then  reduce  as  much  as  possible,  we  shall  find  for  its  value 

tan  ft   .  {sin  (L  -  X')  .  sin  (L'  -  X")  -  sin  (I/  -  X')  .  sin  (L  -  X")  |^j 
+  tan  ft'  .  { sin  (L  —  X")  .  sin  (L'  -  X)  -  sin  (L'  —  X")  .  sin  (L  -  L)  j     > 
.+  tan  ft'',  [sin  (L  -  X)  .  sin  (L'  —  X')  -  sin  (L'  -  X)  .  sin  (L  -  X')  ]      .) 
Now,  by  trigonometry,  we  have,   (since  sin  A  .  sin  B  =  £  {  cos  (A  —  B)  —  cos  (A  +  B)  ] 

sin  (L  -  X)  .  sin  (L'  -  X')  =  i  cos  { (L  -  L')  -  (X  -  X') }  -  i  cos  { (L  +  I/)  -  (X  +  X') } 
sin  (L'  -  X)  .  sin  (L  -  X')  =  1  cos  { (L  -  I/>  +  (X  -  X') }  -  |  cos  { (L  +L')  -  (X  +  X')  } 
and  subtracting,  we  find 

sin  (L  -  X)  .  sin  (L'  -  X')  -  sin  (I/  -  X)  .  sin  (L  —  X')  = 

=  §  cos  { (L  -  L')  -  (X  -  X') }  -  |  cos  { (L  -  L')  +  (X  -  X')  } 

which  another  application  of  the  same  trigonometrical  formula  already  used  converts  hack  again  into 

sin  (L  -  L')  .  sin  (X  -  X') 


muuay    es 

{ 


668 


PHYSICAL    ASTRONOMY. 


Astronomy,  a  similar  transformation  applies  to  the  co-efficients  of  tan  ft  and  tan  ft ;  so  that  our  expression  becomes 
11 — v — '  r      sin  (L  —  L')  .  sin   (\f   —  \")  .  tan  ft 

AB'  —  BA'  =  tan/3"  X    <  +  sin  (L  —  L')  .  sin  (V  —  X  )  .  tan  ft' 
I  +  sin  (L  -  I/)  .  sin  (X    —  X' )  .  tan 

C      tan  ft    .  sin  (V  — 

or  AB'  —  BA'  =  sin  (L  —  L')  .  tan  ft".   J  +  tan  ft'  .  .sin  (V 

1+  tan  ft"  .  sin  (X 
or,  if  we  put  7  =  tan  ft  .  sin  (X'  —X")  ^ 

+  tan  ft'  .  sin  (X"-  X  )  V  ;     (72) 

+  tan/3",  sin  (X    —  X')J 

AB'  -  BA'  =  7  .  sin  (L  -  I/)  .  tan  ft"  (73,  1) 

Now,  it  is  evident  that  -y  is  a  symmetrical  function,  and  is  not  changed  by  putting  at  the  same  time  ft'  for  ft, 
and  X'  for  X,  and  reciprocally  ;  so  that,  by  pursuing  a  process  of  reduction  exactly  similar,  we  should  obtain 


Physical 
Astronomy* 


V'-X")>| 
V"-  X  )  J. 
X-X')J 


Fig.  10. 


AB"  -  BA"  =  7  .  sin  (L   -  L")  .  tan  ft" 
A'B"  -  B'A"  =  7  .  sin  (I/  -  L")  .  tan  ft" 
and  were  we  to  pursue  similar  processes,  combining  A  and  C,  B  and  C,  we  should  get 


and 


CA' 

-AC' 

=  Tf 

.  sin 

(L 

-L') 

.  tan 

ft 

CA" 

—  AC" 

=  7 

.  sin 

(L 

-L") 

.  tan 

P 

C"A' 

—  A"C' 

=  7 

.  sin 

(I/ 

-L") 

.  tan 

ft' 

BC' 

—  CB' 

=  7 

.  sin 

(L 

-L') 

.  tan 

ft 

BC" 

-  CB" 

=  7 

.  sin 

(L 

-L") 

.  tan 

ft 

B'C" 

-  C'B'' 

=  7 

.  sin 

(L' 

-L") 

.  tan 

ft 

(73,  2) 
(73,  3) 

(73,  4) 
(73,  5) 
(73,  6) 

(73,  7) 
(73,  8) 
(73,  9) 


Thus  the  numerator  of  the  fraction  in  —  above  referred  to,  becomes 


(74) 

rigorously  zero;  and  in  all  cases  we  see  that  it  must 
be  a  very  small  quantity  of  the  second  order  at  least, 
so  that  the  whole  expression  (74)  must  be  a  very 
small  quantity  of  the  third  order ;  and  that,  whether 
the  intervals  between  the  consecutive  observations 
approach  to  equality  or  not.* 

Let  us  next  consider  the  denominator  of  the  frac- 

0"       t"  -  f 
tion,  and  retaining  the  substitution  n  =  — 


a  -  t 

it  becomes,  on  the  same  hypotheses  as  before 
B'  .  RS  {n B  -  B'  (1  +  n)  +  B"} 

because  S'  =  S  +  S"  nearly.  We  have,  therefore, 
first  to  consider  the  values  of  B"  —  B'  and  of  B  —  B'. 
Now,  if  in  B"  —  B'  we  substitute  the  values  of  B',  B", 
and  reduce  as  far  as  possible,  we  shall  find 


7  •  tan  ft"  .  {US"  .  sin  (L  -  L')  -  R"S  .  sin  (L'  -  L")  }  ; 

Now,  let  us  suppose  the  three  observations  made  at 
small  intervals  of  time.  The  supposition  almost 
always  holds  good  in  the  case  of  comets,  which  usually 
excite  sufficient  interest  about  the  time  of  their  first 
appearance,  to  induce  astronomers  to  observe  them  as 
frequently  as  possible  ;  so  that  observations  separated 
by  an  interval  of  a  few  days  at  most  may  generally 
be  obtained  for  calculation.  On  this  supposition,  then, 
the  variations  X  —  X',  X'  —  X"  and  X"  —  X  of  the 
observed  geocentric  longitudes  may  be  regarded  as 
very  small  quantities  of  the  first  order  ;  so  that  7  is 
also  a  very  small  quantity  of  the  first  order,  especially 
as  ft  (the  observed  geocentric  latitude)  is  usually 
below  45°:  so  that  tan  ft,  tan  p,  &c.  are  less  than 
unity.  Moreover,  L  —  L',  L'  —  L",  &c.  which  re- 
present the  sun's  motion  in  the  intervals  between  the 
observations,  are  also  very  small  quantities  of  the  first 
order ;  so  that  the  expression  (74)  is,  on  these  ac- 
counts,  a  very  small  quantity  of  the  second  order  *  Mr-  Littrow,  from  whose  excellent  and  most  useful  work  on 

But  besides  this,  since  S,  S"  represent  the  rectili-     theoretieal  and  ptac&cal  astronomy  (Tktorefocb  und  Practise/,? 

npar  friano-loa    «P1V  -wry/    j  Astronomic,  von  J..F.  Littrow,  director  der  Sternwarte  und  pro- 

descnbed    about    the  fess0rder  astron.  an  der  K.  K.  universitat  in  Wien.— Wien,  1821, 

sun  in   the    times  V  —  t   and   if'  —  t'    (which   for  the  2  vols.  8vo.)     I  have  taken  this  expose  of  the  determination  of  a 

moment  we  will  call  0  and  n  B)  they  will   be  nearlv  comet's  orbit  and  the  example  which  follows,  has  merely  remarked 

proportional  to  those  times,   being  extremely  near  to  that  the  quantity  '   is  equal  to  -  £  x  ^   +  a  remainder,  and 
equality  with  the  sectors  SP  n  F,  SF  n'  P",  so  that  we  *  B 

have  very  nearly  S"  =  n     S      Aeain   the  earth's  nrhit  "  this  rcmainder  is  of  the  order  AB'  -  A'B>  and  therefore  may  be 

hpinfr  nparlv  a      ir^l»          A<\  neglected  in  a  first  approximation,   when    the  intervals  are  small 

irly  a  circle,  and  its  motion  in  it  nearly  uni-     a*d  near!y  equa,-.  (v'0f.  H.  p.  129.)  This  is  not,  however,  a  satisfac- 

lorm,  we  nave  torv  view  of  the  subject>  an(j  ;s>  jn  fact(  slurring  over  a  very  consi- 

R  =  R"  and  (L'   —  L")  =  n     (L  L'^  derable  difficulty  and  evading  one  of  the  most  difficult  and  delicate 

i  /r         T  /\   u  •  points  in  the  theory  of  comets.     I  have  therefore  judged  it  better 

~T,  '  Tf   mg  VCry  Sma11'  sin  (L/  —  L")  =      to  enter-  thou?h  somewhat  more  at  length  than  is  absolutely  con- 
If  these  suppositions  were  rigorously     sonant  with  the  nature  of  this  work  into  this  part  of  the  subject, 

1 rather  than  leave  a  doubt  on  the  mind  of  the  student,  or  a  mystery 

attached  to  a  fundamental  proposition.      Meanwhile  I  am  happy 
to  have  to  call  the  attention  of  the  lovers  of  astronomy  to  tht 

=  R"S   .  sin  (L'  L")  interesting  work  alluded  to,  of  which  a  translation  by  any  compe- 

»nH  tVi»  tcnt  Dand  would  be  a  real  accession  to  our  stock  of  elemenUry 

and  the  quantity  within  the  brackets  of  (74)  would  be    works. 


n  .  sin  (L  • 

exact,  we  should  have 

R  .  S"  .  sin  (L  -  L')  =  nRS  .  sin  (L   -  L' ) 


PHYSICAL    ASTRONOMY. 


669 


and  the  quantity  within  the  brackets  being  the  difference  between  two  values  of  the  same  function  at  a  small 

T  /  _  T// 

interval  of  time  is  of  the  first  order,  as  is  also  sin  —  -  —  ,  so  that  B"—B'  is  of  the  second  order.    Similarly,  we  have 

+  L 


B  -  B'  =  -  2  .  tan  ft  .  tan  ft"  sin  L  ~  L' 


.  j 


cot  ft  .  cos 


T/    _    T  "  f      __   T  / 

Now,  sin  -  -  -  =  n  .  sin  —  -  —  ,  and  therefore  we  have 


-  X      _  cot  /3"  .  cos 


i  _  \»\  I 


"-BO=«ii.tai»0  tan/3".sin 


T        I      Q   L'   _|_    T"  xT   _  T  />* 

Let  this  be  reducei.  as  ;..uch  as  possible  ;  and,  putting  M  for  -  ,  and  (1  +  n)  .  sin(  -  ) 

4  \      4      / 


for  sin 


-  T       __     T  /'    v 

in   (-        -  J  it  wi 


will  become 

R  .  S  x  2  n  (1  +  n)*  .  tan  /3«  .  tan  /3"»  .  sin  L  ~  L/  .  sjn  L  ~  U  x 

2  4 

X   [sin  (L'  —  X)  .  cot  /3  —  sin  (L'  —  X")  .  cot  y3"}   x 
X   {  sin  (M  -  X)  .  cot  ft  —  sin  (M  -  X")  .  cot  /3"  j 

This  is  evidently  a  quantity  of  the  fifth  order,  the  two  last  factors  of  it  being  each  the  difference  of  con- 
secutive values. 


Granting  then  the  hypotheses  above  employed,  viz. 
—  1.  The  strict  circularity  and  uniformity  of  the  earth's 
motion.  —  2.  The  proportionality  of  the  rectilinear, 
instead  of  the  parabolic,  sectors  to  the  times  we  s-r 
that  the  numerator  of  the  fraction  vanishes  absolutely, 
while  the  denominator  on  the  same  hypotheses  does 
not  strictly  vanish,  but  only  reduces  itself  to  a  quun- 
tity  of  the  fifth  order.  Hence,  we  are  entitled  (at 
least  in  making  a  first  approximation  to  the  elements) 
to  neglect  the  traction,  and  suppose 
s  „,  A, 

(75) 


S' 
for  —  ;  when  we  obtain, 


and  similarly 


I/ 


S 


"  ~  "  ' 


B' 


S 
"g7 


It  is  true,  from  the  minuteness  of  the  denominator  of 
the  fraction  in  (71)  the  deviation  from  truth  of  the 
hypotheses  assumed  becomes  magnified.  But  these 
hypotheses  themselves  are  so  very  nearly  correct, 
that  unless  in  extremely  unfavourable  cases  the 
equations  (75)  and  (76)  may  safely  be  employed  :  and 
it  will  be  observed  too,  that  independent  of  both  hypo- 
theses,  the  numerator  would  be  of  the  fourth  order  at 
least,  for  if  we  throw  it  into  the  form 


consecutive  values,  and  therefore  a  very  small  quan- 
tity,  —  the  first  is  also. 

To  apply  these  equations  to  the  determination  of  a  Fig.  10. 
comet's  elements,  we  will  first  suppose  the   sectors 
P  n  SF,  &c.  equal  to  the  plane  triangles  SPP',  &c.  or 
at  least  in  the  same  ratio  with  them.     This  amounts 

t?  —  t        & 
to  substituting  ^  _  ^  or 

instea(,  of  the  e     ations  (69j  ^  2,  3,) 

(a  8  +  AR)  x  0"  -  AH'  .  ff  +  A."R"6  =  o  (78,1) 
BR  0"  -  (a  V  +  B'R')  tf  +  B"R"  0  =  o  (78,2) 
CR  *"  -  C'R'  #  +  («  «"  -  C'R")  0  =  o  (78,3) 

which  give  the  values  of  S,  B',  S". 
Now  we  get,  by  the  equation  (55) 

r'°-  —  R'2  +  S"*  •  sec  /3/s  +  2  R'S'  .  cos  (I/-X')  ;     (79) 

whence  the  value  of  r  is  obtained. 

,    ^  «">«  "btained,  the  values  of  I,  S',  &c  and  /  may 

l'e  correcte''  b-v  the  Rowing  process,  let  T  represent 

the.  tlmf  .  between    t.he,  second   °bservat,on  and  any 

as8.!gned  'nSt"nt;    "  K         !•'  V'  ''  fSfff*^   T 
ord»late8  ilt  the  first  observabon,  and  ^,  y",  z",  at  the 

'   WC       VC 

_   ,_  &         ^^     ,      ^        (P  if 
~  "      '  ' 


-y  .  tan  / 


.. 

.  SS"  . 


=y  ~      ' 


i 


<J 


,  / 


-  &c' 


-  fcc. 


(80'3) 

(80,  4) 
_  T  .  T 

and  similar  expressions  for  z,  z".  Now,  assuming  t 
the  second  term  is  evidently  so  ;  and  since  the  factor  as  the  independent  variable,  or  dr  constant,  vve  have, 
of  the  first  within  the  brackets  is  the  difference  of  by  (1),  (2),  (3), 


670 


PHYSICAL    ASTRONOMY. 


Astronomy, 


dr* 


dr" 


dr* 


=  -  (M  +  m)  . 


=  -  (M 


(81,  1,2,3) 


The  numerical  value  of  M  +  m  is  easily  found, 
since  it  is  the  same  for  the  whole  solar  system  except 
in  so  far  as  the  masses  of  the  planets  and  comets 
denoted  by  TO  differ,  and  these  bear  so  minute  a  pro- 
portion to  the  mass  M  of  the  sun,  that  they  may  be 
neglected  :  let  T  denote  the  period  of  any  one  plane, 
as  the  earth,  and  o  the  semi-major  axis  of  its  orbit, 
then  we  shall  have,  by  (24) 


a3       M  +  m 
so  that  —  is  a  quantity  very  nearly  constant  for  the 

whole  planetary  system  ;  and  the  period  of  the  earth 
being  365d.2563S4,  and  the  semiaxis  of  its  orbit  1,  its 

.     (S65-256384)8 
value  for  the  earth  is — ,  and  the  same 

quantity  expresses  its  value  for  all  the  other  planets 
and  comets.     Hence  we  have 


M  +  m  = 


4  X   (3-141592)4 


=  0-0002959122  ; 


(365-256384)* 
and  log  (M  +  w)  =  6-4711628 

To  correct  the  values  of  £,  If,  &c.  we  must  correct 
the  supposition  on  which  they  were  obtained,  of  the 
proportionality  of  the  rectilinear  triangles  instead  of 
the  parabolic  sectors  to  the  times,  or  the  values  of 
S,  S',  S".  Now,  if  c  be  the  inclination  of  the  plane  of 
the  orbit  to  the  plane  of  the  x  and  y,  we  have 

2  S    .  cos  c  =  y  jf  —  x  yf 

2  S'  .  cos  c  =  yV  -  x'y" 

2  S" .  cos  c  =  y  x"  -  x  y" 

In  the  first  of  these,  for  y  and  x  write  their  values  as 
given  in  (80,  1)  and  (80,  3),  and  we  find 


2S.COS 
But  if  the 

6          y'dx'  -x'dy' 

r 

1 

-    1     '                   dr 
0*        y'd^x1  -  x'rPy 

1.2'               d  r2 
0>           y'cPx'  —   t'd 

V 

1.2.3'              d  T3 

.  d4  x'        ,   d'  t/ 

vn  UPS  nr  inri          J      Vi 

-  &c. 


written  in 
-r  T 

the  second  term  of  this,  it  will  vanish.  Again,  if 
we  differentiate  the  equations  (81,  1)  and  (81,  2) 
we  get 


d-r 


so  that,  by  substitution,  we  obtain 
y'd'i'  -s'dy  M  +  TO 


dr 

d/ 

~dT 


) 


y'dx'  -x'dy' 


d  ^  /3  d-r 

If  we    neglect   the   higher  powers   of  0   than 


the 


third,  and  put,  for  the  present,  p  for 
we  shall  have 
2  S  .  cos  c  =  p 


—  y'dx'    Physical 
"; Astronomy 

T   T  ' 


and  similarly, 
2S'.  cose  = 


M  +  TO 


2  S".  cos  c 


=  p  0"  {  1  - 


-— 


(82,  2) 
(82,  3) 


If  we  now  employ  the  value  of  /  as  above  approxi- 
mately determined,  and  substitute  it  in  the  right  hand 
members  of  these  equations,  we  shall  obtain,  dividing 
one  by  the  other  the  radios  of  S,  S',  S",  much  more 
accurately  than  on  the  original  hypothesis,  for  we 
have 


—  =  —  y 

c//  off    " 


£.__£_ 

s"  ~~  &'  x 


i  — 


i  — 


i  — 


+ 


(83,  1) 


(83,  2) 


Now,  in  the  equations  for  determining  S,  e',  «cc.  (69, 
1,  2,  3)  it  is  only  the  ratio  of  S,  S',  S",  that  are  re- 
quired, and  thus  we  are  enabled  to  correct  the  values 
of  their  quantities,  and  thence  again  that  of  /.  To 
carry  the  process  to  a  greater  degree  of  precision, 
however,  the  series  expressing  the  value  of  2  S  .  cos  r 
must  be  continued  further.  Without,  however,  going 
through  the  process  as  here  set  down  we  may  content 
ourselves,  after  computing  first  values  of  tf  and  /  by 
the  equations  (78,  2)  and  (79)  with  calculating  the 
quantity 


-BR.  £  (**._««)}. 


(84) 

which  must  be  applied   as  a  correction  to  the  first 
found  value  of  S'  with  its  proper  sign. 

The  values  of  r,  /,  r",  B,  6',  S",  so  obtained,  the 
remaining  unknown  quantities  may  be  found  as  follows : 

The  equation  (58) 

r  .  cos  b  .  sin  (/  —  L)  =  S  .  sin  (X  —  L) 
divided  by  the  equation  (38) 

r  .  cos  b  .  cos  (/  —  L)  =  R  +  &  .  cos  (X  —  L) 
member  for  member,  gives 


tan  (I  -  L)  — 


11  +  S  .  cos  (X  —  L) 
the  right  hand  member  consists  only  of  known  quan- 
tities, so  that  the  value  of  I  —  L  is  easily  found,  and 
thus  the  heliocentric  longitude  becomes  known  at 
each  of  the  three  observations.  Again,  we  have  by 
the  equation  (56) 

r  .  sin  b  =  S  .  tan  ft 
and  by  (58) 


cos  b 


sin  (X  -  L) 
sin  (I  —  L) 


PHYSICAL    ASTRONOMY. 


671 


Astronomy,  which,  divided  member  for  member,  give 

' '  sin  (I  -  L) 

tan  b  =  tan  p 


(86) 


sin  (X  —  L) 

and  thus  the  heliocentric  latitudes  may  be  computed. 
The  values   of  I  and  4  known,  the  same  equations 
afford  a  value  of  r,  viz. 

r  =  *.^ 
sm  4 

whence  r,  /,  i"  may  be  found  ;  and  if  these  values 
agree  with  those  before  determined,  it  will  be  a  proof 
of  the  correctness  of  the  previous  work.  If  V  and  I" 
be  greater  than  I,  the  comet's  motion  is  direct ;  if  less, 
retrograde. 

The  heliocentric  longitudes  and  latitudes  thus 
obtained  at  each  of  the  three  observations,  we  easily 
obtain  the  inclination  and  place  of  the  node.  In  fact, 
(taking  the  first  and  last  observations,  to  embrace  a 
greater  arc  of  the  great  circle)  our  equation  (40) 
gives 

tan  b    =  tan  i  .  sin  (I  —  a) 

tan  b"  =  tan  i  .  sin  (I"  —  a) 
so  that  we  have 

sin  (I   —  63)  tan  b 

sin(/"—  a)  ~  tan  b" 
which  gives 

tan  4"  (sin  /  .  cos  a 


=  tan  4  (sin  I" 
or         sin  a  { tan  4 

=  cos  a  {sin  / 
and  consequently 

tan  4  .  cos  I"  —  tan  4"  .  cos  I 


—  cos  I .  sin  63) 
cos  a  —  cos  I",  sin  63) 
cos  r  —  cos  I  .  tan  b"  \  = 
tan  4"  —  sin  I" .  tan  b   } 


tan  a  — 


and 


tan 


sin  L  .  tan  b"  —  sin  If'  .  tan  b ' 
tan  b 


(87) 


sin  (/  -  £3; 

If  the  motion  of  the  comet  be  direct,  the  value  of  a 
given  by  the  above  equation,  will  be  the  longitude  of 
the  ascending  node ;  but  if  retrograde,  it  is  evidently 
that  of  the  descending  ;  and  to  get  that  of  the  ascend- 
ing node,  ISO0  must  be  added.  The  equation  (61) 
or  the  corresponding  one, 

tan  (I  —  a)  =  cos  i  .  tan  (^  +  0)  (89) 

gives  the  values  of  Y"  +  0  the  longitudes  on  the  orbit 
reckoned  from  the  node,  and  these  being  obtained, 
(Y-  +  0,  Y-  +  ff,  and  Y-  +  °")*  the  angle  G"  —  0 
between  the  first  and  last  places  of  the  comet  at  the 
sun  becomes  known. 

If  we  denote  this  by  0,  we  have  at  once,  by  the 
equation 

0  (  /T) 

tan  —  =  cosec  0  4  cos  0  —  \/  —  >  (90) 

demonstrated  in  the  foregoing  pages  (p.  665)  the  value 
of  0,  or  the  distance  (on  the  orbit)  of  the  perihelion 
from  the  place  of  the  first  observation  ;  so  that,  sub- 
tracting the  value  found,  from  ty  +  0  before  deter- 
mined, we  get  Y-,  and  thence  TT,  the  longitude  of  the 
perihelion  on  the  ecliptic,  by  the  equation  (60) 


*  The  reader  will  not  confound  8  here  with  the  same  letter  before 
nsed  to  denote  the  time. 


It  only  remains  to  find  the  perihelion  distance  and    Physical 
time   of  perihelion   passage.      The   equation   of  the  Astronomy. 
parabola 


gives  the  former  immediately,  r  and  0  being  known, 
and  the  latter  is  found  as  follows.  In  the  table  of  a 
comet  of  109  days  subjoined,  seek  the  number  of  days 
(by  interpolation)  corresponding  to  the  anomaly  0, 
and  call  it  M,  then,  by  equation  (44)  we  have,  (the 
perihelion  distance  of  such  a  comet  being  unity) 

0          1 


and  consequently,  if  T  be  the  time  of  the  perihelion 
passage  before  the  first  observation, 

T  =  N  x  D* 

If  we  institute  a  similar  operation  for  the  time  T" 
of  the  perihelion  passage  before  the  third  observation, 
we  shall  find 

T"  =  N"  .  D* 

Now,  we  ought  to  have  T"  —  T  =  the  interval 
between  the  first  and  last  observations.  If  this  be 
found  on  trial  to  be  the  case,  it  is  a  proof  of  the  cor- 
rectness of  the  work.  The  perihelion  passage  will  be 
before  or  after  the  first  observation  (in  the  case  of 
direct  motion)  according  as  the  longitude  JT  of  the 
perihelion  is  less  or  greater  than  the  heliocentric  longi- 
tude I  at  the  first  observation  ;  the  reverse  is  the  case 
in  retrograde  motion. 

The  chief  difficulty  in  the  solution  of  the  problem 
of  a  comet's  orbit,  consists  in  the  determination  of  the 
distances  from  the  sun  and  earth,  at  the  moments  of 
the  several  observations.  Our  equations  above  derived 
afford  a  great  variety  of  means  for  accomplishing  this, 
among  which  we  shall  only  select  one,  proposed  by 
Dr.  Olbers,  which  in  many  cases  has  particular  advan- 
tages. It  depends  on  the  property  already  demon- 
strated of  the  chords  of  the  parabolic  arcs. 

Let  k  represent  the  chord  of  the  arc  included  be- 
tween the  two  extreme  places,  then  we  have 

or          k*  =  r4  +  r"2  —  2  (xx"  +  yy"  +  zz") 

In  this  equation,  let  the  values  of  x,  y,  z,  x",  y",  z", 
in  (67,  1,  2,  3,  4,  &c.)  in  terms  of  R  and  B,  &c.  be 
substituted ;  and  if  we  put 

Cf_       0^ 

so  that,   by  (76)         c"  =  m  S 
we  shall  have 

k°~  =  r2  +  r"3 

—  2  m  e2  .  { cos  (X  —  X")  +  tan  ft  .  tan  ft"} 

—  2  m  R  K  .  cos  (X"  —  L) 

-  2  R"  a  .  cos  (X  -  L")  (91) 

-  2  RR"  .  cos  (L  -  L") 

This  equation  combined  with  the  equation  (63) 
6  .  VM  +  m  .  tf  =  (r  +  r"  +  k)l  -  (r  +  r"  -  A:)T 
and  the  two  equations 

r*  =  R2  +  a*  .  sec  /32  +  2  R  S  .  cos  (X  —  L) 

r"3  =  R''2  +  r»2  a*  .  sec  ft''*  +  2  m  R"  «  .  cos  (X"— L) 

gives  four  equations  for  determining  the  four  unknown 

quantities.     To  resolve  them,  we  must  use  the  well 

known   rule  of  false,  one  of  the   simplest  and  most 

4  s 


672 


PHYSICAL    ASTRONOMY. 


A*tronoa.y.  widely   useful    rules    mathematical    science    affords. 

*— '~\—~~/  Assume  a  value  for  S,  as  near  the  truth  as  can  be  con- 
jectured, compute  m  and  thence  m  S,  or  &'' ,  and  having 
used  these  to  obtain  k,  r,  and  i",  substitute  these  in 
the  expression  for  gO'.  Call  the  result  A.  Assume 
then  a  second  value  for  S,  which  call  A>  and  execut- 
ing the  same  process,  call  the  result  B.  Then  we 
shall  have 

B  —  A  :  gt  —  A  ; :  A  —  s  :  x  —  s 

where  x  is  the  corrected  value  of  S,  and  hence  we  get 

Mean  Time,  Paris.  h        ,      „ 

1779.     Aug.  30 11     9  42, 

Sep.     2 10  36     8, 

Sep.     4 10    7  51, 


=  «  +  (A  -«) 


B-A 

and  substituting  this  again,  and  employing  that  one  of 
the  former  values  of  S  which  gave  a  result  nearest  to 
gt  for  a  corresponding  value,  we  shall  obtain  a  yet 
nearer  approximation,  and  so  on. 

It  remains  to  exemplify  the  method  of  computation 
by  an  instance.  Let  us  therefore  take  the  following 
observations  of  the  comet  of  1779,  taken  at  Paris  with 
a  transit  and  a  repeating  circle. 


Physical 
Astronomy, 


X  =  125  48  39-3 
X'  =  132  53  48-5 
\"  =  138  56  31-2 


ft  =  41  53  52-2 
ft'  —  45  54  48-1 
F'  =  48  32  27-8 


L    =  337  29     8-7  R    =  1-0087218 

I/  =  340  22  26'9  R'  =  1  0079991 

L"  =  342  17  47'8  R"=  1-0074854 

The  intervals  therefore  between  the  first  and  second,  first  and  third,  and  second  and  third  observations  are 
respectively*  0  =  2  976690  days,  0'  =  4d'957049,  6"  =  ld-9S0350 

Consequently,  we  have  by  the  equations  expressing  the  values  of  A'  and  C'  reduced  into  numbers 

~  =  1-18408  ;     m  =  ~  =  0-787752 

A  •  6 

and  substituting  this  value  of  m,  and  the  values  of  X,  A",  ft,  ft'',  R,  R",  we  get  the  following  equations,  noticing 
that  g  =  O'  103212 

ra  =  1-01752  -  1-71693  .  S  +  1-80493  .  8*  ;  r"*  =  1-OIM3  —  1-45724  .  S  +  1-41564  .  «s 

(r  +  r"  +  k\±       /r  +  r"  — 
-  2  -  /     ~  \  -  2 
These  equations  are  very  nearly  satisfied  by  taking  S  =  071469,  whence  we  find 

r  =  0-8440;     r"  =  0  8346  ;     It  =  0-1317  ;     o"  =  m  S  =  G'562998 

whence  we  next  deduce  the  heliocentric  longitudes  and  latitudes  at  the  extreme  observations  from  equations 
(85)  and  (86) 

I    =  20°  37'  40"8     and  6    =  49°  26'  23"'l 
I''  =    6    45    39  '7  b''  =  49     46    46  '9 

Since  I"  is  less  than  /,  the  comet's  motion  is  retrograde.     Moreover,  we  have  by  equations  (87)  and  (88) 

&  =  100°  51'  53"-4  ;     i  =  49°  51'  7"'9 

we  obtain  also,  by  following  the  process  pointed  out  in  equations  (89)  and  (90)  0  =  12°  39'  35'  '6,  JT  =  4° 
32'  8"'2  for  the  longitude  of  the  perihelion,  and  D  =  0-830761  for  the  perihelion  distance.  Consequently 
the  time  between  the  first  observation  and  the  perihelion  passage  is 

n       +         tan3      =  +  6-97118  = 


+  m 

=  6d  23h  18'  30" 
30th  Aug.   11       9    42 


10    28    12    Sept.  6,  1799  =  time  of  perihelion  passage. 


Of  the  two  solutions  given  above,  the  latter  only 
supposes  the  orbit  to  be  determined  a  parabola  ;  the 
former  may  be  applied  to  the  determination  of  elliptic 
elements,  but  the  latter  part  of  the  process  from 
equation  (70)  must  be  varied  accordingly  ;  but  as  the 
various  methods  of  determining  and  correcting  elliptic 
af  well  as  parabolic  elements  would  lead  us  a  great 
deal  beyond  our  proposed  limits,  we  must  content 
ourselves  with  referring  to  Mr.  Gauss's  Theoria  Motus 
Corporum  Celestium,  in  which  the  whole  subject  is 
treated  with  the  utmost  generality.  The  reader  may 


also  consult  Sir  H.  Englefield,  On  Comets;  the  Mr'ca- 
nique  Celeste  of  Laplace,  vol.  I.  c.  iv.  ;  Lagrange, 
Mfc.  Analytique  (the  new  edition.)  The  German 
scholar  will  find  it  well  worth  his  while  to  consult  the 
paper  of  the  same  eminent  writer  in  the  Berlin  Ephe- 
meris  for  1785  ;  and  the  volume  of  the  same  work  for 
1789,  p.  197  ;  the  papers  of  Gauss  and  Olbers,  in  the 
Monatliche  Correspondenz ;  as  well  as  the  treatise  of 
the  latter,  Ueber  die  leichteste  und  bequemste  Methode  den 
Bahn  tines  Cometen  zu  berechnen. 


*  The  reader  is  requested  to  take  the  numbers  in  this  example  upon  Mr.  Littrow's  authority,  from  whose  work  the  example  is  taken. 


P  H  Y  S  I  C  A  L     A  S  T  R  O  N  O  M  Y.  6/3 


Physical 
Astronomy. 


PART  II.       SECTION  I. 

Or  THB    PERTURBATION    OF    THE  ELLIPTIC  MOTIONS  OP    THE  HEAVENLY  BODIES,  ARISING   FROM  THEIR  MUTUAL 

ATTRACTION 

IH  the  preceding  investigations  we  have  supposed  only  two  attracting  bodies  to  exist  in  space  ;  and  on 
this  supposition  have  succeeded  in  representing  all  the  phenomena  of  their  motions  in  finite  equations,  by  the 
aid  of  which  their  relative  situations  may  be  assigned  at  any  past  or  future  moment.  The  resulting  formulae 
reduced  into  tables,  and  compared  with  actual  observation,  manifest  an  agreement  sufficiently  complete  to 
leave  no  doubt  of  the  correctness  of  the  principles  from  which  they  have  been  deduced,  at  least,  when 
observations  separated  only  by  moderately  long  intervals  of  time  are  compared  together.  Yet,  on  descend- 
ing to  a  more  rigorous  nicety,  and  especially  on  comparing  together  observations  embracing  a  very  long 
series  of  years,  it.  is  found  that  the  results  of  the  calculations,  founded  on  the  assumption  of  elliptic  motion 
according  to  the  laws  above  demonstrated,  do  not  represent  the  observations  perfectly.  Minute  irregu- 
larities are  still  detected,  and  the  planets  are  found  sometimes  a  little  in  advance,  and  sometimes  a  little 
falling  short,  sometimes  a  little  to  the  right  or  left,  above  or  below,  their  calculated  places  in  their  orbits. 
Moreover,  if  the  elements  of  the  orbits  themselves,  as  deduced  from  modern  observations,  be  compared 
with  those  similarly  deduced  from  ancient  ones,  they  will  be  found  not  to  correspond  exactly,  but  to  differ 
by  small  variations,  which  are  however  greater,  as  the  observations  themselves  compared  are  more  distant  in 
their  dates.  In  a  word,  though  the  elliptic  theory  agrees  very  nearly  with  observation  ;  yet,  to  make  it  tally 
rigorously  with  them,  it  is  necessary  to  introduce  modifications  of  this  kind. 

1st.  The  ellipse  in  which  each  planet  moves,  must  be  conceived  to  change  its  eccentricity  and  position  in 
space,  by  exceedingly  slow  gradations — so  slow  indeed,  as  to  be  insensible  in  a  single  revolution  of  the  planet, 
and  only  discoverable  by  a  comparison,  such  as  we  have  described,  ot  its  nature  in  past  and  present  ages. 

2dly .  The  planet  itself  is  not  always  found  exactly  in  its  place  in  the  ellipse  even  when  so  varied ;  nor  indeed 
is  it  always  found  in  the  exact  periphery  of  the  ellipse  at  all.  But  if  we  conceive  an  imaginary  point  to 
describe  this  ellipse  according  to  the  rigorous  laws  of  elliptic  motion,  the  real  planet  will  never  be  very  far 
distant  from  it,  but  will  oscillate  or  revolve  round  it  like  a  secondary  about  its  primary  in  an  orbit  of  extremely 
small  dimensions,  yet  according  to  laws  of  a  very  complicated  nature,  too  complicated  indeed  for  observation 
alone  to  unravel. 

These  variations  are  much  more  sensible  in  the  orbit  and  motions  of  the  moon,  where,  in  fact,  they  amount 
to  very  considerable  quantities.  If  the  moon,  for  instance,  at  its  greatest  northern  latitude  be  observed  to  pass 
over  a  certain  star,  it  should  continue  to  do  so  each  revolution,  if  its  motion  were  strictly  conformable  to  the 
elliptic  hypothesis.  So  far,  however,  is  this  from  being  the  case,  that  it  will  be  observed  to  deviate  visibly 
southward  every  lunation ;  and  after  a  lapse  of  nine  years  and  a  half,  its  path,  in  the  same  part  of  the  heavens 
as  to  longitude,  will  pass  not  less  than  10°  south  of  the  star  ;  after  which  it  will  again  advance  northward,  and 
in  nine  years  and  a  half  more  will  once  more  pass  over,  or  at  least  very  near  the  star.  Deviations  like  these 
must  have  some  positive  and  decided  cause  ;  as  much  so  as  the  elliptic  motion  itself;  and  if  they  are  to  be 
accounted  for  on  the  theory  of  universal  attraction,  it  is  manifest  that  the  same  mechanical  principles  applied 
in  the  same  way  to  the  case  of  several  bodies  abandoned  in  free  space  to  their  mutual  attractions,  ought  10 
lead  us  to  their  explanation. 

But  the  mathematical  difficulties  to  be  encountered  in  this  research,  are  of  a  much  higher  order  than  in  the 
case  of  two  bodies  only.  There  we  had  no  difficulty  in  integrating  the  differential  equations  of  the  problem. 
Here,  on  the  other  hand,  the  equations  are  too  complicated  to  allow  of  their  integrals  being  exhibited  other- 
wise than  in  series ;  and  even  then  we  have  no  means  of  ascertaining  their  laws.  Fortunately,  however, 
ihis  is  not  necessary  ;  the  enormous  preponderance  of  the  sun's  mass  in  our  system  being  such,  that  the 
fractions,  representing  those  of  each  of  the  other  planets,  are  small  enough  to  allow  of  their  squares  and  products 
being  neglected  without  any  fear  of  inducing  appreciable  errors. 

It  is  a  well  known  theorem,  which  extends  to  all  the  applications  of  mathematical  reasoning  to  natural 
phenomena,  that  when  several  causes  of  motion  act  together,  if  their  effects  taken  singly  are  of  such  an  order 
of  minuteness,  that  their  squares  and  products  may  be  disregarded,  then  their  joint  effect  will  be  the  sum  of 
the  effects  which  would  be  produced  by  each  acting  alone.  This  principle  allows  us  to  regard  the  disturbance 
or  perturbation  of  any  one  planet  produced  by  the  joint  action  of  all  the  others,  as  the  sum  of  the  effects  each 
would  produce  separately;  so  that  we  may  simplify  the  investigation  by  leaving  out  of  consideration  all  but 
a  single  disturbing  planet,  and  applying  the  analytical  formulae  so  investigated  to  each  in  succession,  we  shall 
thus  obtain  expressions  for  the  perturbations  produced  by  each  ;  which,  added  together,  give  the  total 
perturbation. 

Investigation  of  the  forces  exerted  by  one  body  to  disturb  the  orbit  of  another  revolving  about  a  common  central  body, 

and  of  the  differential  equations  of  their  motions. 

Taking  unity  for  the  mass  of  the  central  body,  and  its  centre  for  the  origin  of  the  co-ordinates,  let  m  and 
m'  represent  the  masses  of  two  revolving  bodies,  and  suppose  x,  y,  z,  r  to  denote  the  three  co-ordinates  and 
radius  vector  of  the  body  TO.  whose  perturbations  we  would  investigate  ;  and  x  ,  y  ,  z  ,  r',  the  corresponding 
quantities  relative  to  the  disturbing  body  m.  Moreover,  let  \  represent  the  distance  of  the  two  bodies,  m,  m, 

4  s  2 


674 


PHYSICAL    ASTRONOMY. 


Astronomy,  on  each  other.     Then  will  the  several  attractive  forces  exerted  by  the  bodies  of  the  system  on  each  other  be    Physical 
as  follows  :—  Astronomy. 


1st.  The  sun  (or  central  body)  attracts  TO  and  m'  with  forces  represented  respectively  by  —  and  -jj. 
2dly.  TO  attracts  the  sun  with  a  force  — ,  and  it  attracts  TO'  with  a  force  — . 

7"*  X* 

Sdly.  m' attracts  the  sun  with  the  force  -^,  and  it  attracts  m  with  the  force- • 

Fig.  11          These  are  the  forces  exerted  in  the  directions  of  the  lines  joining  the  several  bodies  in  the  system  ;  but  to 
estimate  their  effects,  they  must  be  reduced  to  the  directions  AS,  BS,  CS,  of  the  three  co-ordinates.     This 

done,  we  shall  find  ,,     ,u    c  *• 

For  the  forces  acting  on  the  sun, 


Force  of  m  =  — 


Force  of  m'  —  — 


m'x* 


in  the  direction  AS 


and  similarly, 


m  y  m'  u' 

and nr-  m  the  direction  JBS 


r'3 

m  2                  TO'  2'   . 
— —  and —  m  the  direction  CS. 


So  that  the  three  forces  acting  on  the  sun,  are  respectively 

'  m  x        TO'  i 


Again,  the  forces  acting  on  m  are 

In  the  direction  AS. 


-( 
-( 


r3 

TO  z 


70 

wV 


Force  of  the  sun  =  + 
Force  of  TO'  .      .  =  — 


r* 

m'  (x'  —  x) 


In  the  direction  BS 


Force  of  the  sun  = 


— 


Force  of  TO'....  =  - 


In  the  direction  CS 


Force  of  the  sun  =  H — - 
r3 


Force  of  TO' 


So  that  the  aggregate  forces  acting  on  TO  in  these  directions  are 

+  7T  ~  m    '       x3 

L.JL    „'  y'  -y 

r3   '  X3 

z  2'  -  z 


TO'  (z'  -  -•) 

53 


r  r3  X3 

and  in  the  same  manner,  the  forces  acting  on  m'  will  be  found  to  be  respectively 

x'  x  -  x1 


+  —  - 


z  -  zf 
X3 


PHYSICAL    ASTRONOMY. 


675 


Astronomy.     From  these  three  sets  of  forces  we  might  determine  separately  the  motions  of  the  sun,  of  TO.  and  of  m! ;    Physical 
^— -V-—  but  as  we  have  chosen  to  suppose  the  sun  at  rest,  and  fix  the  origin  of  our  co-ordinates  in  its  centre,  we  must  Astronomy 

transfer  to  both  m  and  m'  in  a  contrary  direction  the  forces  which  act  upon  it.  This  done,  the  forces  animating  ^•~~-s~~> 

m  in  the  respective  directions  AS,  BS,  CS,  will  become 

(1  +  m)  .x  .  (x1        if 


(1  + 


(1  +  m)  .  z 


f/     y'-y\ 

\r'>  X*     / 


-y 


and  if,  in  these  expressions,  we  exchange  the  accented  letters  for  the  corresponding  unaccented  ones,  and 
vice  versd,  (\  only  excepted)  we  shall  evidently  have  the  forces  acting  on  m'. 

The  first  terms  of  these  expressions,  as  will  be  observed,  are  the  same  as  if  the  mass  m'  of  the  disturbing 
planet  were  nothing,  or  as  if  there  were  no  such  planet ;  the  other  parts,  those  multiplied  by  m',  express  the 
disturbing  forces,  or  the  effect  of  the  attraction  of  m' to  derange  the  orbit  of  TO  from  its  elliptical  form.  These, 
it  will  also  be  observed,  consist  each  of  two  parts;  the  one  expressing  the  action  of  m'  on  the  sun,  and  the 
other  its  action  on  m  with  an  opposite  sign.  It  is  only  therefore  in  virtue  of  the  difference  of  the  attractions 
of  m'  on  the  sun,  and  on  m  that  the  orbit  of  the  latter  is  deranged. 

These  forces  are  susceptible  of  a  very  simple  mode  of  expression  ;  and  the  equations  of  m's  motion  in 
consequence  admit  of  a  very  compendious  form,  if  we  consider,  that  owing  to  the  form  of  the  function  X, 


(92) 


which  is  equal  to  V(x'  —  x)~ 

+  (/  -  y)'2  +  (* 

x1  -  x   _    d 
X'         =  dx 

/-y  ..  d 

1 

'  T 
i 

z)2,  we  have 

Xs              dy 

z'  -  z    _  _    d 
X3         ~  dz 

x 
i 
'  T 

Xlf 

1 

V          dU 

y'  . 

rf  U        zf          dU 

so  that  if  we  assume 

r"          dx   ' 

1 

r» 

dy    '     T*           dz 

_   xx   +  yy'  +  zz' 

(i7*  +  /«  +  z's)f 

the  disturbing  forces  will  be  represented  by 

dQ 


(93) 


m  . 


dfl 
'    dy    ' 


dx    '  •    dy    '  '    dz 

and  the  equations  representing  the  motion  of  m,  will  be  (if  we  put  1  +  m  =  fi)  and  suppose  d  t  the  element 
of  the  time  invariable. 

d*x  x  -"">       - 

0  =  -V-T-  +  /*.  —  +«'. 


d«* 
d2  y 


dt* 


+  p.  .  1L  +  m'  . 
+  /»  •  — r  +  m'  • 


dx 
dQ 

~dj 
dQ 

d  z 


(94) 


Similarly,  if  we  put  U'  =  -        f  y  J    +  Z  ^   and  take  /  =  1  +  m',  and 

Q'  =  U'  -   -1 
X 
the  motion  of  m  will  be  represented  by  the  equations 

o  = 


o  = 


o  = 


d**' 

'     ^    +m 

dO'       > 

^  ,  !,' 

.  —   -4-  vn  , 

dx1 
dQ' 

d*2' 
j.  I,* 

r'3 

dy' 

djy 

(95) 


(96) 


676  PHYSICAL    ASTRONOMY. 

Astronomy.      The  joint  integration  of  these  six  equations  would  determine  the  motions  of  both  TO  and  TO',  but  their  rigorous    Physics! 
•^  integration  being  impracticable  in  the  present  state  of  analysis,  we  are  driven  to  have  recourse  to  methods  of  Astronomr 
approximation,  of  which  the  principle  may  be  explained  as  follows  :  —  ^—  ~v~—  ' 

Suppose  A  =  o  to  be  any  equation,  or  system  of  equations,  susceptible  of  rigorous  integration,  and  let  the 
process  to  be  followed  for  this  purpose,  consist  in  deducing  in  succession  from  A  =  o,  by  any  analytical  artifices, 
or  transformations,  the  equations,  or  systems  of  equations, 

B  =  o,     C  =  o,  ........  K  =  o 

the  last  of  which,  K  =  o,  is  integrable  by  known  methods.     Now,  suppose,  instead  of  A  =  o,  we  had  pro- 

posed the  equation  A  +  m'  .  a  —  o 

where  m'  .  a  is  a  minute  term  of  the  order  of  the  disturbing  forces,  m'  being  an  extremely  small  constant 

quantity,  and  a  any  assigned  function  of  the  co-ordinates.     If  then  we  pursue  with  this  equation  the  very  same 

process  by  which  B  =  o  was  derived  from  A  —  o,  it  is  clear  that  we  shall  obtain  an  equation  which  can  differ 

only  from  B  =  o  by  a  quantity  which  vanishes  when  m'  —  o,  and  which  therefore  must  be  of  the  form  TO'  x  b. 

So  that  instead  of  B  =  o,  we  have 

B   +   TO'    X    b  =   o 

where  b  is  some  explicit  function   of  the  co-ordinates,  their  differential  co-efficients,   and  of  m',  and  either 

developed,  or  at  least  developable  in  a  series  of  ascending  powers  of  m'.     In  like  manner  we  may  deduce 

equations  C  +  m'.c  =  o;  ......  K  -f  m'  .  k  =  o 

in  the  place  of  the  equations  C  =  o,  ......................  K  =  o 

In  order  then  to  obtain  a  final  equation  for  the  solution  of  the  problem  of  three  bodies,  we  have  only  to 

follow  out  any  system  of  processes  and  transformations  which,  in  the  case  of  two  only,  would  prove  successful 

in  reducing  the  differential  equations  to  an  integrable  form. 

This  may  be  accomplished  in  a  great  variety  of  ways,  as  the  equations  of  undisturbed  motion  are  integrable 
by  a  great  many  different  artifices,  besides  those  which  we  have  employed  in  the  former  part  of  this  essay  ;  and 
it  is  therefore  necessary  to  select  among  them  such  as  lead  to  final  equations  best  adapted  to  the  nature  of  the 

case  under  consideration.  Now  two  courses  have  been  adopted  by  geometers  ;  the  former  adapted  to  the 
theory  of  the  planetary,  and  the  latter  to  that  of  the  lunar  perturbations. 

In  the  theory  of  the  planets,  the  disturbing  force  is  so  extremely  minute,  that  its  square  and  higher  powers 
may  be  neglected  with  safety.  This  simplification  being  permitted,  it  becomes  practicable  (as  we  shall  pre- 
sently see)  so  to  conduct  the  investigation  as  to  make  the  time,  or  mean  longitude  of  the  disturbed  planet  our 
independent  variable,  and  thus  to  express  the  perturbations  at  once  in  functions  of  the  time,  or  of  angles  pro- 
portional to  it,  a  simplification  of  the  utmost  moment  in  the  construction  of  tables.  In  the  more  complicated 
theory  of  the  moon,  in  which  the  part  of  the  perturbations  depending  on  higher  powers  of  the  disturbing  force 
than  the  first  is  very  conspicuous,  it  is  no  longer  permitted  to  neglect  them,  and  the  necessity  of  preserving,  as 
far  as  possible,  the  rigorous  expressions  of  the  forces,  &c.  at  least  in  the  difftrcntial  equations,  obliges  us  to 
employ,  not  the  mean,  but  the  true  longitude  of  the  moon  for  our  independent  variable,  as  by  this  means  we 
are  enabled  to  arrive  at  final  equations  perfectly  rigorous,  and  can  thus  estimate  the  influence  which  the  neglect 
of  small  quantities  is  capable  of  producing  in  their  integration,  with  less  likelihood  of  being  misled. 

The  theory  of  the  moon  then  differs  entirely  from  that  of  the  planets  in  its  treatment.  The  general  prin- 
ciples of  approximation,  however,  are  the  same  in  both.  Both  theories,  as  we  shall  see,  lead  to  final  equations 
of  the  form  d*  u 

-j^-  +  n^u  +  m'  .  k  —  o  (97) 

where  na  is  constant,  and  k  an  explicit  function  of  the  several  co-ordinates,  distances,  and  angles,  of  the  pro- 
blem, and  m'  a  very  small  quantity  (which,  in  the  lunar  theory,  also  enters  into  the  composition  of  k,)  of  the 
order  of  the  disturbing  forces.  To  approximate  then  to  the  value  of  u,  we  first  suppose  m  =  o,  and  we  get 
a  value  of  u  corresponding  to  ro'  =  o,  and  which  we  call  its  elliptic  value. 

2dly.^  We  deduce  from  this  the  elliptic  values  of  all  the  variables  which  enter  into  the  composition  of  the 
term  m'  k,  in  terms  of  the  independent  variable  t,  whether  t  represent  the  mean,  or  the  true  longitude.  These 
being  substituted  in  the  last  term,  it  will  become  an  explicit  function  of  the  independent  variable;. 

3dly.  If  we  now  again  integrate  the  differential  equation  so  prepared,  the  value  of  u  will  consist  of  two 
parts  ;  the  first  will  be  the  same  as  before,  viz.  the  elliptic  value,  and  the  second  will  be  u  correction  which 
must  be  of  the  order  of  disturbing  forces,  and  will  express  the  perturbation  of  u  with  a  degree  of  precision 
corresponding  to  their  rirst  power. 

If  the  same  process  of  substitution  be  repeated,  and  the  equation  again  integrated,  another  set  of  terms  will 
be  added  to  the  value  of  «,  which  carry  the  approximation  a  step  farther,  or  to  the  squares  of  the  disturbing 
forces  :  and  were  the  same  process  continued  to  infinity,  the  series  of  terms  so  obtained  would  be  a  rigorous 
analytical  expression  for  u. 

The  final  equation  (97)  is  of  the  form  so  well  known  in  analysis,  under  the  name  of  linear  differential  equa- 
tions, and  (as  we  have  observed)  almost  all  the  equations  on  which  the  planetary  motions  depend  being  of  this 
nature,  it  will  be  right  to  premise  some  few  points  relative  to  their  theory,  to  which  to  refer  hereafter. 

It  is  demonstrated  in  all  works  in  the  differential  and  integral  calculus,  and  will  be  so  in  our  article  on  that 
subject,  that  any  linear  differential  equation  of  the  second  order  is  integrable,  provided  we  can  find  two,  or 
even  one  particular  value  capable  of  satisfying  it  when  deprived  of  its  last  term.  Thus,  if 

M  '      +  x  "  +  n  =  °  (98) 


' 


PHYSICAL    ASTRONOMY.  677 

Astronomy,  be  any  such  equation,  M  and  N  being  functions  of  t,  and  if  a'  and  u'  be  any  two  functions  of  t  which  satisfy    Physical 

^— -v— ^  the  equation  Astronomy 

(P  u                d  u  v— ' V — ' 

T7F  +  M  '  Tt  +  N  "  =  ° 


then  the  complete  integral  will  be 

cr  uf  +  c"u"  -  uf  fd  -.-  r  n  4  ft, 

J        u'J        ,  .   u' 


u'  d  Z,  <"> 


u 
Now,  the  last  term  of  this,  beipg  integrated  by  parts,  becomes 

,  \u"   /»  n  d  <2  /»  n  d  p  x  « 

--      ~ 


« 
Consequently,   the  complete  integral  of  the  equation  (98)  will  be 

u  =  C'  u'  +  C"  u"  -   I  u'    /'      nd'3 


(100) 

M '  to    "" 

Suppose  we  have  the  equation  d2  u 

-r-r  +  naa  +  n  =  o  HOI) 

a  i^ 

d2  u 
then  «'  and  d",  the  particular  integrals  of  -— -  +  n-  u  =  o  offer  themselves  readily,   being  no  other  than 

sin  n  t  and  cos  n  t ;  and  if  these  be  substituted  in  the  general  expression  above  given,  we  get  at  once 
u  =  C'  .  cos  n  t  +  C"  .  sin  n  t 

•f-  -        cos  n  t  I   n  d  <  .  sin  n  t  —  sin  n  t  I  n  d  t  cos  n  <    >  ;  (102) 

and  if  n  =  1,  or  in  the  case  of  d2  M 


dl* 
the  integral  is  u  =  C'  .  cos  t  +  C"  .  sin  t 


+  u  +  n  =  o  (103) 


+   cos  t  .  I  n  d  t  .  sin  <  —  sin  t   I  n  d  t  .  cos  * 


(104) 


These  values  of  u  are  rigorous,  whatever  be  the  value  of  n,  and  independent  of  any  approximation,  as 
it  is  easy  for  the  reader  to  satisfy  himself  by  substituting  them  in  the  differential  equations  from  which  they 
were  deduced,  when  the  whole  will  be  found  to  vanish,  independent  of  any  particular  value  of  II,  or  any  sup- 
position made  as  to  its  magnitude.     The  equation  (1O2)  may  however  be  obtained  perhaps  easier  as  follows. 
(P-u 

ijet    £tl-  +  n4  u  =  —  n  be  multiplied  by  d  f  .  cos  n  t  and  it  becomes 

d4  u  .  eos  n  t  +  jpudt*.  cos  n  t  =  —  I    Udf  .  cos  nt 
and  integrating, 

d  u  .  cos  n  t  +  n  u  d  t  .  sin  n  t  =  —  JH  d  t*  .  cos  n  t 

If  we  again  multiply  this  by  --  we  eet 

cos  n  f 


cos  n  t 


Annt  dt          /» 

n  u  d  t  .  -  =  —  -  -  rr    /   II  d  i  .  cos  n  t 
(cos  n  i)4  (cos  n  t)*J 

d  .  tan  n  t    {* 
=  —  -    /   IT  d  t  .  cos  n  t 

n        J 
and  again  integrating, 

u  1      /»  /i 

-  =  —  —    Id.  tan  n  t   I  II  d  t  .  cos  n  t 
cos  nt  n  J  J 

=  —  j    —  tan  n  t  I  n  d  t  .  cos  n  t  +  I  n  d  t  .  sin  n  1  1 
by  integrating  by  parts.     Hence  we  have 

u  =  -    |   cos  n  t  I  n  d  t  .  sin  n  t  —  sin  n  t  Al  d  t  .  cos  n  t  i  1105) 


G78  PHYSICAL     ASTRONOMY 

Astronomy,  or,  exjrressing  the  arbitrary  constants  by  C'  and  C"  Physical 

— -*.—• '  u  =  C'  .  cos  n  t  +  C"  .  sin  n  t  Astr'onomy 


H —  |   cos  n  t    tndt.sinnt  —  sinnt/ndt.  cos  n  t  j> 


as  before. 

If  n  be  an  explicit  function  of  t,  this  value  of  «  is  always  assignable,  at  least  by  quadratures,  and,  when- 
ever the  integrations  can  be  executed,  in  finite  terms.  In  the  lunar  and  planetary  theories,  n  is  always  reducible 

to  a  series  of  sines  or  cosines  of  the  form          (At  +  B).     Let  us  therefore  consider  this  case  more  closely. 

sin 

Now  any  term  of  H,  such  as  a  x  sin  (A  t  +  B)  will  introduce  into  u  the  term 

cos  n  t  I   sin  n  t  .  sin  (A  t  +  B)  d  t  —  sin  n  t  I  cos  n  t .  sin  (A  t  +  B)  d  1 1 

But  we  have 

sin  n  t  .  sin  (A  t  +  B)  =  f  {  cos  (A  —  n  .  t  +  B)  —  cos  (A  +  n  .  t  +  B)  \ 
whence 


.  .     ,.  ,    .    „.         sin  (A  —  n  .  t  +  B)          sin  (A  +  n  .  t  +  B) 

d  t  .  sin  n  t  .  sin  (A  t  +  B)  = 


2  (A  —  n)  2  .  (A  +  n) 

and  similarly, 


•     /A  *    .    i^  cos  (A  +  n  .  t  +  B)         cos  (A  -  n  .  t  +  B) 

d  t  .  cos  n  t  .  sin  (A  t  +  B)  = - — i — — • - 

2  .  (A  +  »)  2  (A  —  n) 

So  that  the  term  introduced  into  u  will  become 


cos  n  t  .  sin  (A  —  n  .  t  +  B)  +  sin  n  t  .  cos  (A  —  n  .  t  +  B) 
~~2  n  (A  -  n) 


cos  n  <  .  sin  (A  +  «  •  *  +  B)   —  sin  n  t  .  cos  (A  +  n  .  t  +  B) 
—  a . 


2  n  (A  +  n) 
=  a  .  sin  (At  +  E)\- 


_  a  .  sin  (A  t  +  B) 

A2-  n3 

Similarly,  if  a  .  cos  (A  t  +  B;  were  any  term  of  IT,  the  corresponding  term  in  u  would  be  — 

A    ~~  ?( 

II  therefore  consisting  of  a  series  of  terms,  such  as 

we  shall  have 

a  cos 

u  =  .          ( 

A*  —  na       sin  v 

-f  C  .  cos  n  t  +  C'  .  sin  n  t 


n  =  a  .  °  >S  (A  t  +  B)  +  ct  .  C!)S  (A' t  +  B')  +  &c.  (106) 

sin  sin 

«  =          °     .    .  C°S  (At  +  B)  +         °'     .    .  C°S  (Aft  +  B')  +  &c. 
A*  —  na       sin  v  As  —  n»      sm  v 


C  and  C'  being  two  arbitrary  constants. 

cos 
Terms  of  the  form  a  .    .    (A  t  +  B)  being  of  perpetual  occurrence  in  physical  astronomy,  it  is  necessary  to 

designate  them  and  their  several  parts  by  names.  The  whole  term  is  called  an  equation,  or  inequality  :  the  part 
(A  t  +  B)  within  the  sign  sin  or  cos  is  called  the  argument ;  and  the  co-efficient  a,  the  maximum.  The  period 
of  the  inequality,  or  the  time  (in  units  of  time  such  as  t  consists  of)  which  k  occupies  in  passing  through  all  its 

tiff) 

gradations  of  magnitude  and  sign,  is  equal  to  —  — ,  A  being  expressed  in  degrees.     Hence,  the  period  of  an 

A 

inequality  is  longer  or  shorter,  according  as  the  co-efficient  of  the  time  in  its  argument  is  less  or  greater. 

The  arguments  of  all  the  inequalities  in  «  then  are  the  same  as  those  in  n,  one  remarkable  case  only 
excepted,  in  which  A  =  n ;  for,  in  this  case,  As  —  n2  =  o  ;  and  the  term  having  (A  t  +  B)  for  its  argument, 
changes  its  form.  In  fact,  since  the  constants  C  and  C'  are  arbitrary,  we  may  change  them  into  C  — 

'  _  —  and  C'  —  ^     r  respectively,  in  which  case  C  .  cos  n  t  +  C'  .  sin  n  t  will  be  changed  to 

C  .  cos  n  t  +  C'  .  sin  n  t    -  — g  .  sin  (A  t  +  B).  Thus  u  will  contain  the  terms 

A.  °_  „,    •  sin  (A  <  +  B)  -    Ae  a_  B,    •  sin  (n  t+  B) 


PHYSICALASTRONOMY.  670 

Astronomy.  sin  (A  t  +  B)  —  sin   (n  t  +  B)  o  Physical 

_,-., -^      But  when  A  =  n,  a  . i —  =  —  and  differentiating  numerator  and  denomi-  Astronomy 

A8  —  n*  o  ,^_    ^j7, 

nator  with  respect  to  A,  it  becomes  simply  (on  making  A  =  n) 

—  .  cos  (nt  +  B)  (108) 

Similarly,  it'll  contained  the  term  cos  (n  t  +  B),  this  would  introduce  into  u  the  term 

-~  .  sin  (nt  +  B)  (108) 

Terms  of  this  kind  form  an  exception  to  the  law  of  periodicity  observed  by  all  the  rest,  having  t  disengaged 
from  the  sign  sin  or  cos.  If  t  represent  either  the  time,  or  the  mean  or  true  longitude,  of  the  disturbed  body, 
they  will  represent  inequalities,  whose  maxima  go  on  continually  increasing  without  limit.  Such  inequalities, 
if  thev  really  had  an  existence  in  our  system,  must  end  in  its  destruction,  or  at  least  in  the  total  subversion  of 
its  present  state  ;  but  we  shall  see  hereafter,  that  when  they  do  occur,  they  have  their  origin,  not  in  the  nature 
of  the  differential  equations,  but  in  the  imperfection  of  our  analysis,  and  in  the  inadequate  representation  of  the 
perturbations,  and  are  to  be  got  rid  of,  or  rather  included  in  more  general  expressions,  of  a  periodical  nature, 
by  a  more  refined  investigation  than  that  which  led  us  to  them.  The  nature  of  this  difficulty  will  be  easily 
understood  from  the  following  reasoning.  Suppose  that  a  term,  such  as  a  sin  (A  t  +  B)  should  exist  in  the 
Viilue  of  u,  in  which  A  being  extremely  minute,  the  period  of  the  inequality  denoted  by  it  would  be  of  great 
length  ;  then,  whatever  might  be  the  value  of  the  co-efficient  a,  the  inequality  would  still  be  always  confined 
within  certain  limits,  and  after  many  ages  would  return  to  its  former  state.  Suppose  now  that  our  peculiar 
mode  of  arriving  at  the  value  of  u,  led  us  to  this  term,  not  in  its  real  analytical  form  a  .  sin  (A  t  +  B),  but  by 
the  way  of  its  developement  in  powers  of  *,  a  +  /3  t  +  -y  <a  +  &c,  ;  and  that,  not  at  once,  but  piecemeal,  as 
it  were;  a  first  approximation  giving  us  only  the  term  a,  a  second  adding  the  term  y3  t,  and  so  on.  If  we 
stopped  here,  it  is  obvious  that  we  should  mistake  the  nature  of  this  inequality,  and  that  a  really  periodical 
function,  from  the  effect  of  an  imperfect  approximation,  would  appear  under  the  form  of  one  not  periodical. 

This  is  what  actually  takes  place  in  the  theory  of  the  problem  of  three  bodies.  These  terms  in  the  value 
of  u,  when  they  occur,  are  not  superfluous  ;  they  are  essential  to  its  expression,  but  they  lead  us  to  erroneous 
conclusions  as  to  the  stability  of  our  system  and  the  general  laws  of  its  perturbations,  unless  we  keep  in  view 
that  they  are  only  parts  of  series  ;  the  principal  parts,  it  is  true,  when  we  confine  ourselves  to  intervals  of 
moderate  length,  but  which  cease  to  be  so  after  the  lapse  of  very  long  times,  the  rest  of  the  series  acquiring 
ultimately  the  preponderance,  and  compensating  the  want  of  periodicity  of  its  first  terms. 


SECTION  II. 
General  theory  if  the  planetary  perturbations  depending  on  their  mutual  configurations. 

OUR  first  object  in  the  theory  of  the  planets  is  to  transform  the  differential  equations  of  the  disturbed  orbit, 
so  as  to  obtain  final  equations  in  which  the  radius  vector,  and  true  longitude,  or  those  parts  of  them  arising 
from  the  action  of  the  disturbing  forces,  shall  be  expressed  in  terms  of  the  time  ;  and  to  reduce  them  to  the 
general  form  of  the  linear  equation  of  the  second  order,  whose  theory  we  have  just  considered.  To  this  end, 

Let  the  equations  (94)  of  m's  motion  be  respectively  multiplied  by  x,  y,  z,  and  added  together,  and  we  get 
d  x  <P  x  +  d  y  </'2  y  +  d  z  d'2  z  x  dx  +  y  dy  +  z  dz 

(  ,}  O  ,1  Q  d  Q  ) 

+  ro    .      - — dx  +  ——dy  +  — —  dz  \ 

(  d  x  dy  uz  ) 

The  portion  within  the  brackets  of  the  last  term  is  the  differential  of  Q  taken  on  a  supposition  of  the  co-ordi- 
nates of  nt  only  varying.     Let  this  be  represented  by  the  Roman  character  d,  so  that 

d n  dQ  dQ 

dQ  =  -—dx  +  —;—dy  +  ——dz 
dx  dy  dz 

bearing  this  in  mind,  and  that  d  Q  is  only  an  abbreviated  expression  for  this  function,  we  have  by  integration 
dx*  +  dy*  +  dza       Zfi        u,  /» 

__  --  +  -•     •*/ 

in  which  we  must  be  careful  not  to  confound     I    d  Q  with   /  d  Q  or  Q,  d  Q    being    only   an  incomplete 

differential,  and  the  characteristic    /   denoting  an  integration  relative   to   t,   supposes  the   variation  of  the 

±/ 
co-ordinates  of  the  disturbing  as  well  as  the  disturbed  body. 

Again,  if  we  multiply  the  same  equations  (94)  by  x,  y,  z,  respectively,  we  shall  get 

zd*x  +  yd*y  +  zd*z        fi  ,(        dQ  dQ  dQ) 

o  =  — + r-  in   \    x  —7 —   T-  y  — I-  z  - 

dP  r  \         d  jc  dy  dz) 

VOL.   Ill  4    T 


680  PHYSICAL    ASTRONOMY. 

Astronomy  If  we  add  this  to  the  former,  observing  that  Physical 

*»—  v—  -**  Jt      ,t  Astronomy. 

dr4  +  dy»  +  dz*  +  xa*x  +  y  d*y  +  z  d3  z  =  %  d9  (j*  +  y*  +  z«)  = 


we  get 

dQ 


jja  '  1  i  l       if  i  I       *  •  ( 

d  <9  r  a  J  (        dx  dy  dz  $ 

Now,  if  we  put  x  =  p  .  cos  0,  y  =  p  .  sin  0  and  z  =  /i  .  s,  or  suppose  p  =  the  projected  radius  vector  r, 
(see  fig.  11.)  Q  the  angle  it  makes  with  the  axis  of  the  x,  and  s  =  the  tangent  of  the  latitude  of  m,  we  have 
T  dp  1 


and  similarly,  .  —  _     . 

or  or 

Consequently,  substituting  for  x,  y,  z,  these  values 

dQ  dQ  dQ  f  dQ          dx          dQ          dy          dQ          dz  )  dQ 

i -f  y h  2 =:  r   i •  H • — r-   .  }    =    r  .  • 

dx  dy  d  z  (  d  x          dr  dy          dr  dz          d  r   )  dr 

and  it  will    be  observed,  that  this  property  is  altogether  independent  of  the   nature   of  the  function  Q,  and 
belongs  to  every  possible  function  of  the  co-ordinates  x,  y,  z,  x',  y',  z'. 
Thus  we  see,  that 

,    /»  ,  (        da  da  f/Q  )  (        /•»  . 

2m'/    d  Q  +  m'  <  x  -7—  +  y  TSTT  +  *  -7— ^    =  m'  .  \1   I   d   Q 


/»  ,  (        da  da  dQ  ) 

n'J  dQ  +  m>{x  —  +  y  —  +  z  —  J    = 

Hence,  if  we  put  Q  =  3  f  d  Q  +  r  -£  ,  (1  10) 

our  differential  equation  becomes 


Let  us  now  suppose  that  r  represents  only  the  elliptic  value  of  r,  and  x,  y,  z,  Q,  the  elliptic  values  of  the 
co-ordinates,  and  the  value  of  the  function  Q,  which  would  arise  from  writing  the  elliptic  values  of  x,  y,  z,  x', 
if,  z  ,  in  their  expressions  ;  and  let  r  +  in'  S  r,  x  +  m'  S  x,  y  +  m'  S  y,  z  +  m'  £  z,  a  +  m'  S  Q,  &c.  represent 
the  disturbed  values  of  these  quantities  ;  then,  if  we  neglect  m'2,  we  shall  get  by  substitution 


.    d4  .  r-         u          u,  ,      (d*  .r&r        n  .  r  &r  ) 

'  =  i-dF-  V7+T.+  "  •}-^-  +  --7^-  +  Qj 

But  since  r  represents  the  elliptic  value  of  r,  the  first  part  of  this  equation  vanishes  of  itself,  and  to  determine 
f>  r  or  r  S  r,  we  have  the  differential  equation 

'  fr  i       f\ 


This  equation  being  linear,  and  of  the  second  order,  is  immediately  integrable  by  our  general  formula,  equa- 
tion (100)  provided  we  can  find  the  two  particular  integrals  «'  and  u"  of  the  equation 

d9 «          w 

-| «  =  o 

d  t*          r3 

but  since  r,  on  the  supposition  of  the  term  Q  bearing  zero,  may  be  taken  for  the  radius  vector  on  the  hypothesis 
of  elliptic  motion,  it  is  obvious  that  the  elliptic  values  of  either  x,  or  y,  or  z,  will  satisfy  this  equation,  because 
these  values  are,  in  fact,  no  other  than  what  are  derived  from  the  integration  of  equations  precisely  similar,  viz. 

d4  z  fi 

-r^r    +  -7*  =  o 


df          r' 
d*  : 


-r  -v  y  = 


dl* 


_  ,  ,  ,,    d  if          y  d  x  —  .T  d  y       h  d  t  . 

Consequently,  we  may  take  u   =  x  and  r  —  y,  whence  we  get  u    —  -„-  —  -  -  =  -  because 

on  the  hypothesis  of  elliptic  motion  y  dx  —  x  dy  =  hdt,  and  it  is  of  the  elliptic  values  of  x  and  y  that  we  are 

.,«"       x  d  y  —  y  d  x  h  d  t  e  /•,f^n\     • 

now  speaking.      Similarly,  a  d  —  =  -  =  -  y-  -  =  --  ,  and  the  formula  (100)  gives 


PHYSICAL    ASTRONOMY.  681 

Astronomy,  for  the  cjmplete  value  of  u  or  r  S  r.     The  constants  then  being  included  under  the  integral  sign,  we  have          Physical 
*  _,-       m_ J  ~  ,t  Astronomy 

y  I    Q  xd  t  —  x  I  Qy  d  t  *~^^—*' 

rST=^L —j^- *  (113> 

Such  is  the  value  of  r  S  r  when  we  consider  only  the  first  power  of  the  disturbing  force.  It  would  be  abso- 
lutely exact,  but  that  x  and  y  are  only  particular  values  of  u  on  the  hypothesis  of  r  having  its  elliptic  value. 
If  this  supposition  were  not  made,  we  should  have  if  =  x+  m'  S  x  and  u"  =  y  +  m'  8  y  ;  and  substituting 
these  values,  we  should  obtain  terms  in  the  expression  of  m'  £  r  depending  on  the  square  of  the  disturbing 
force,  but  with  these  we  have  no  concern. 

The  perturbation  in  longitude  (m'  S  0)  is  easily  obtained  when  the  value  of  S  r  is  found.     In  fact,  we  have 

d  x9  +  d  y*  +  d  z2  =  d  r2  +  r2  d  d* 
whence  we  get 

r*d02  +  dr2          2  a  u.  ,   /», 

o  = ^ -£  +  £  +  2  m'J  d  O  (114) 

and  if  we  subtract  this  from  the  equation  (111),  we  find 

r2  d  0a         r  d4  r         fi  dtl 


If  in   this  we  substitute  r  +  m'  &  r  for  r  and  6  -f-  m'  8  0  for  0  we  get,    (after  obliterating  the   terms  which 
destroy  each  other  by  reason  of  the  properties  of  elliptic  motion,  and  those  which  contain  m'*) 
Ir^dedSe        SrSrde9        r  d8  $  r         dtr.Kr         firSr  dQ 

-**—+—  I*  ---  Jifi  ---  d^-  +  -^~  -r-d7-> 

From  this,  let  the  term  multiplied    by  -  —  be  eliminated  by  means  of  equation  (115)  and  we  get 

U    t 

2  ra  d  6  d  S  0        r  d2  a  r  —  I  r  .  d3  r         3  pr  K  r  dfi 

~~  ~' 


dP  df  ~ 

And  if  in  this  we  substitute  for  -  its  value  given  by  the   equation   (112)   we  obtain,  (restoring  the 
value  of  Q) 

df  dr&r  +  2rdSr}    +    (  3   fdQ  +  2  r  -—  ^  .  d  t* 

<•--  --  '—±£  -  '-4  --  ('••> 

but  r*d0  =  hdt,  elliptic  values  only  being  considered  in  the  second  member  of  this  equation,   and  conse- 
quently integrating,  we  have 


Now,  we  have  h  =  -v//t  a  (1  —  e5),  and  if  \ve  put  nt  for  the  mean  motion  of  the  disturbed  planet  m,  we  have 
n  =    A/  —  so  that  h  =  n  a2  .  V\  —  e'- ;  and  —  =  — 

jr»3 


h         ft  V  I  -  e" 
Consequently  we  get,  for  the  perturbation  of  the  radius  vector, 

m'  a  •!  cos  6  I  r  .  sin  6  .  Q  n  d  t  —  sin  0   I  t  .  cos  0  .  Q  n  d  /  !• 
*»r-—          -J-  -^-  (120) 

fi  VI  —  e2 
and  the  formula  expressing  the  perturbation  in  longitude,  will  become 

'  x  a  —  m>  (^r  d  ^  r 

no?  V\  —  e*\dt  dt 

am'        (       /•     d  Q  /•/»  ) 

H A  2  /  r .ndt  +  3   II  A  Q  .ndt   \  ;  (121) 

p.  V 1  —  es  I    J         dr  JJ 

It  only  remains  to  determine  the  amount  of  the  perturbation  in  latitude,  or  the  value  of  z  or  of  £s,  if  we 
put  z  =  r  S  s,  in  which  case  S  s  represents  the  tangent  of  the  heliocentric  latitude  of  m  in  its  disturbed  orbit 
above  the  plane  of  its  elliptic  motion.  Now,  if  we  treat  the  equation 

d4z         fiz          .   dQ 

o  =  — —   + 1-  m   

d  t*          r3  d  z 

in  the  same  manner  as  the  equation   by  which  the  value  of  r  S  r  was  found,  viz.  regarding  x  and  y  as  two 
particular  integrals  of  the  equation  — — •  +  — -  =  o,  and  then  completing  the  integration  by  the  general 

4  T  2 


682  PHYSICALASTRONOMY. 

Astronomy  formula  (100)*  we  shall  find  (putting  r  .  cos  0  and  r  .  sin  0,  for  x  and  y)  Physical 

•  -  v  -  /  ,  /»      dQ  /•      dO  )  Astronomy 

mf  a  <  cos  0  I  y  —  —  .  n  d  t  —  sin  0  .  /  x  —  —  .  n  d  t  }  v—  V  —  ' 

m'  a  .  =  -  1  -  J        "  -  =  -  S^—lf  --  L  (122) 

p  V\  —  e* 

This  is  the  latitude  of  TO  above  its  primitive  orbit;  and  if  we  denote  by  s  its  undisturbed  latitude  above  any 
fixed  plane  (as  that  of  the  ecliptic)  slightly  inclined  to  this  orbit,  s  +  S  s  will  be  its  latitude  when  subjected 
to  the  action  of  the  disturbing  forces. 

The  equation  (121)  gives  the  perturbation  in  longitude  when  that  of  the  radius  vector  is  known,  and  the 
latter  may  be  computed  from  the  expression  (12O)  which  is  general  ;  and  considering  the  complication 
of  the  subject,  as  simple  as  can  be  expected.  Its  form  enables  us  to  compute  the  amount  of  perturbation  even 
in  the  most  difficult  cases,  as  in  that  of  a  comet,  by  the  application  of  the  method  of  quadratures.  Meanwhile, 
in  the  theory  of  the  planets,  where  it  is  required  to  develope  the  value  of  S  r  in  series  of  sines  and  cosines  of 
arcs  depending  on  the  configurations  of  the  disturbed  and  disturbing  planet,  it  will  be  found  much  simpler  to 
set  out  immediately  from  the  differential  equations  for  the  disturbed  radius,  and  proceed  in  the  manner  now  to 
be  explained. 

(P  it 

Since  the  form  of  this  equation  is  not  precisely  that  of  the  equation        a     +  «s  u  +  n  =s  o,  the  co-effi- 

u, 

cient  of  the  second  term  instead  of  being  constant,  being  —   a  variable  quantity,   we  must   first  endeavour 

to  transform  it  by  substitution  into  one  of  this  form.     Assuming  then  that  u  is  such  a  quantity  that  its 

d4  u 
elliptic  value  shall  satisfy  the  equation  +  n9  u  =  o,  and  its  disturbed  value  (or  u  +  m'  S  u)  the  equa- 

(i   t 

lion  —  -  -  —  -  -  +  n2  (u  +  m'  c  u)  +  m'  n  =  o,  which  gives  —  —  —   +  n2  S  u  +  U  =  o  we  must  inquire 
d  t*  d  i* 

first,  the  relation  between  r  and  u  ;  and  secondly,  the  value  of  n. 

(Pu 
A  satisfactory  relation  between  r  and  u  is  easily  found.     In  fact,  since  u  is  to  satisfy        a     +  n4  u  =  o,    it 

must  be  of  the  form  u  =  const,  cos  (  n  t  +  const.).  Now  the  developement  of  r  in  terms  of  the  mean 
longitude  gives,  putting  £  for  the  longitude  at  the  epoch  of  the  commencement  of  the  time  t,  and  ir  for 
the  longitude  of  the  perihelion, 

r  —  a  I  1  —  e  .  cos  (n  /  +  e  —  v)  +  ea  sin*  (n  t  +  e  —  ir)  +  &c.  } 

So  that,  if  we  take  u  =  e  .  cos  (itt+e  —  TT),  we  shall  have 

r  =  a  {  (1  +  e2)  -  u  (1  -  i  e2)  -  u*  +  &c.  }  (123) 

This  gives  at  once  8r  =  —  a  6  u  $  1  +2u+e4  X  &c.  } 

=  —  a««(l+2e.  cos  (n  t  +  e  —  ir)  +  e*  .  &c.  }  ;  (124) 

by  which,  when  o  u  is  found,  S  r  may  be  had  at  once. 

It  only  remains  to  discover  n.     Now,  since  our  equation  (111),  if  we  put  r1  =  »  and  m'  <2  /  d  Q  +  r  — 
=s  m'  Q,  becomes 


d  f  *          •/  »  a 

if  we  multiply  by  2  d  t>,  and  integrate,  we  shall  find 


=  8  p-ST-  -  4  „,'       Q  d  r  (126) 

a 


but  because  u  is  a  function  of  r,  and  therefore  of  r*  or  r.  we  have 

>l  11  d  u         dv 

dt       '    d  c          d  t 


1  u  d4  u    /  d  o  \*          d  a 

f^"  ~~    du'     \  d  t  )        '  TV 


dp 

fo  that  d*  a          «  d*  u  (  , —      4  fi  v  t 

d  t?  dv'\  a  „ 

+   ^L(l4-^^-2«'Q)  +n'u 

d  v   (  V  v  a  ) 

Now  u  is  a  certain  function  of  w  (or  of  r)  whose  form  is  determined  by  the  reversion  of  the  series  in  (123) 

d1  u 
and  is  independent  of  the  disturbing  forces.     But  were  tnese  forces  zero,  we  should  have  -~r:i~  +  n?  u  =•  o. 

*  This  formula,  which  perhaps  is  new,  and  which  has  stood  us  in  some  stead  in  the  explanation  of  that  chapter  of  the  M&anique 
Colette,  (cap.  vi.  liv.  2.)  which  I  have  adopted  for  the  groundwork  of  this  part  of  the  present  essay,  may  be  deduced  at  once  from  tht 
general  theory  of  linear  equations,  in  my  paper  O»  various  points  of  Analysis. — Phil.  Trans.  1814. 


PHYSICAL   ASTRONOMY.  683 

Astronomy.  Hence  the  portion  of  the  right  hand  member  of  the  equation  just  deduced,  which  does  not  depend  on  the    Physical 
s—  ~v~~*'  disturbing  forces,  must  be  identically  zero,  in  virtue  of  the  relation  between  u  and  v  ;  and  that  it  is  so,   we  Astronomy 

may  assure  ourselves  by  actual   substitution.     Consequently  we  must  have,   when  the  disturbing  forces  are  v-"-v~" 

regarded, 

<P  u 


that  is  (putting  u  +  m'  S  u  for  u,  and  disregarding  terms  depending  on  m'4) 

tPdu  ,    du  (Pu    /•_ 

—  —  +  n*du  =  -  2  Q  -  --  4  —  —  /  Q  d  v 
dt*  dv  dtfj 

Comparing  this  with  the  equation  +n*8u  +  Il  =  owe  have 


. 

d  v  d 

It  is  desirable  to  express  this  in  terms  of  v.  Now,  as  u  is  a  function  of  v,  v  is  reciprocally  a  function  of  a,  and 

#v 
du  1          <Pu  ~du? 


d  v  d  v    '     d  c-  /  d  v 


du 

and  d3  e 


(•2.)' 

\du  ) 


rr»    *   a-4 

d  v 


d  u  \  d  u 

but  since  v  =  r-,  if  for  r  we  put  its  value  in  terms  of  u  (123),  we  get,  neglecting  higher  powers  of  u  than 
the  first,  »  =  a2  (1  —  2a  +  &c.)  and  substituting  this  in  II,  and  after  the  differentiations  writing  for  u  its 
value  e  .  cos  (nt  +  e  —  TT)  we  get 

Q  /                                                           \            1 
n  = (  1  —  e  .  COS  («  t  +  e  —  ir)     I «2  .  COS  (2  n  t  +  2  e  -  2  TT) 

2  e     /  * 

—  /  Q  n  rl  t .  sin  (n  t  +  e  -  a-)  { 1  +  e  .  cos  (n  t  +  c  —  w)  }  (128) 

and  this  being  the  value  of  II,  we  find  F  u  from  the  equation 

rl*  f  II 

—  +  n*8  u  +  n  =  o.  (129) 


SECTION  III. 
Reduction  of  the  perturbative  function  Q  or  2    /d  Q  +  T—  —  to  a  series  of  sines  and  cosines,  and  investigation 

+_/  "  ^" 

of  the  perturbations,  neglecting  the  eccentricities  ami  inclinations  of  both  orbits. 

WE  have  now  reduced  the  investigation  of  the  perturbation  to  the  integration  of  the  linear  equation  (129) 
and  we  have  before  seen  that  this  is  accomplished  without  difficulty,  when  II,  the  last  term,  is  reducible  to 
sines  and  cosines  of  the  independent  variable  and  its  multiples.  All  then  that  remains  to  be  done  to  get  their 
actual  expressions,  is  to  execute  this  reduction.  This,  however,  is  by  no  means  a  simple  process  ;  and  in  an 
essay  like  the  present  it  is  not  possible  to  pursue  it  into  all  its  details  :  we  shall  therefore  only  carry  it  to  a 
certain  extent  necessary  for  our  future  reference,  and  point  out  the  principles  by  which  it  may  be,  if  required, 
carried  farther. 

Let  us  consider  then  the  value  of  the  function  O.  If  we  write  in  it  r  .  cos  0,  r  .  sin  0,  /  .  cos  ff  and  /  .  sin  &  for 
*,  y,  ^,  y',  and  neglect  zz',  z?*,  (z  —  z')s,  which  are  either  of  the  order  of  the  squares  of  the  disturbing  forces, 
or  of  the  products  of  these  forces  by  the  mutual  inclination  of  the  orbits,  and  put  0  —  ff  =  w,  we  get 

x  x'  +  y  y'  +  ZT?  =  rr1  .  cos  w 


*•  =  A/(*  -  ^  +  (y-  y')*  +  (z  -  zO«  =   vV*  -  2  r  r1  .  cos  w   f  " 
and  r 


=  -7-  .  cos  W  — 


ra  —  2  r  r"  .  cos  w  +  /* 

Let  us  conceive  this  function  developed  in  a  series  of  cosines  of  w,  and  its  positive  and  negative  multiples  to 
infinity  ;  then,  since  the  cosines  of  the  negative  are  equal  to  those  of  the  positive  multiples,  we  may  represent 
fl  as  follows  : 

0  =  R  -f  R'  .  cos  ic  -f  W  .  COB  «  w  +  R'"  .  cos  3  ic  +  &c.  ;  (13O\ 


684  PHYSICAL    ASTRONOMY 

Astronomy,  where  R,  R',  R",  &c.  are  certain  given,  explicit  functions  of  r  and  r1  and  of  these  alone,  depending  solely  on    physical 
•  ~~v~~s  the  peculiar  form  of  Q,  let  A,  A',  A",  &c.  represent  the  same  functions  of  a  and  of.     Then,  if  the  eccentri  •  Astronomy 
cities  of  the  orbits  were  nothing,  we  should  have 


Q  =  — -  .  cos  to 
a"1 


\ 

1  —  2  a  a!  .  cos  to  +  a"  f  i (131) 

V" .  cos  2  to  4-  &c. 


=  A  +  A' .  cos  to  +  A" 

Were  the  eccentricities  nothing,  the  orbits  would  be  circles,  and  the  motion  in  them  uniform.  We  should 
therefore  have  0  =  n  t  and  ff  =  n' t,  whence  w  =  0  —  ff  =  (»  —  n')  t,  so  that  Q  in  this  case  would  be 
expressed  in  the  very  simple  series 

Q  =  A  +  A' .  cos  (n  -  n')  t  +  A"  .  cos  2  (n  —  n')  t  +  &c.  (132) 

Moreover,  since  d  Q  represents  the  differential  of  Q  taken  on  the  hypothesis  that  only  the  disturbed  body 
moves,  we  should  then  have 

dQ                 dQ                 dO                            dQ 
d  Q  =  — —  dr  4 d  0  =  d0  =  ndt  . 

because,  in  the  case  of  circular  orbits  d  r  =  o.     Thus  we  should  have 

d  Q  =  —  n  d  t .  { A'  .  sin  (n  —  n')  t  +  2 .  A" .  sin  (2  n  —  2  n')  t  +  &c. } 
and  integrating  relative  to  t 

2  =  —  4-  -     — ;  { A' .  cos  (n  —  n')  t  +  A"  .  cos  2  (n  —  n')  t  +  &c. }  ;  (133) 

—  being  an  arbitrary  constant. 

Again,  since  R,  R',  R",  &c.  are  explicit  functions  of  r,  /,  and  Q  is  only  so  far  a  function  of  r,  as  this  symbol 
is  contained  in  them,  we  must  have 

dQ  dR  dR' 

r  — —  =  r  — — .  4-  r  — —  .  cos  to  4-  &c.  (134) 

dr  dr  dr 

and  in  the  case  of  circular  orbits,  denoting  by  — — ,  &c.  the  same  functions  of  a,  a'  that  — ; —  denote  of  r,  r', 

da  mt 

dQ  dA  dA' 

T  — —  =  a  — —  +  a  — —  .  cos  to  4-  &c. 
d  r  da  da 

d  A     dA' 

The  values  of  — — ,  — — ,  &c.  are  easily  had  in  functions  of  a,  cf,  when  those  of  A,  A',  &c.  are  found.   Now, 
da      da 

to  obtain  these,  we  may  proceed  as  follows  : 


Take  c  =  cos  to  +  V  —  1 .  sin  to.  Then  (by  trigonometry)  we  shall  have  —  =  cos  to  —  V  —  I  .  sin  to ; 
now,  let  us  consider  the  function  (cf  —  2  a  of  .  cos  to  +  a'*)  ~*  which  agrees  with  the  second  term  of  Q  if 
«  =  — .  This  equals  a  —  *'  (  1  —  <2  —  .  cos  to  +  (  —  )  I  or  a  -*  '  (1  —  2  a  .  cos  to  +  a')  -  s  putting 


—     But  we  have 
a 


1  —  2  o  .  COS  to  +  a4  =  (  —  a 


because  c  +  —  =  2  .  cos  w.     Hence 
c 

(a"  4-  2  a  a' .  cos  to  4-  a")-'  =  a  -  *  '  .  (1  —  o  c)-'.  (   1  —  — 


.  .. 


+  &C. 

hut  c  +  --  =  2  cos  to  ,•  c*  4-  -j  =  2  .  cos  2  to,  &c.  (by  trigonometry)  ;  consequently  we  have 


PHYSICAL    ASTRONOMY. 

(a1  -  2  aa' .  cos  w  +  a")-*  =  o-**  j   1  +   £-y-)    .  a*  +  &c.  j 

!o                        c^o_l_1\  5  ") 

_  a  -j-   — : 1   ,  . — .  a>  -)-  flee.  >•  .  COS  to 

.  cos  2  «;  +  &c. 


(     1.2 
In  the  case  before  us,  where  5  =  --,  this  gives  at  once 

IB 


A  =  -    l  +       *  +          -  +  &c]  (136'  1} 

*"'+ 


&c. 


otc. 

If  o  be  less  than  unity,  these  series  are  convergent ;  but  if  greater,  we  have  only  to  throw  the  expression  into 

the  form  a'-2'(  1  —  2-^-.  cos  to  +   (-f\    )        previous  to  developement,  and  taking  a  =  — ,   instead  of  — , 
\a\a/x  a  o 

a  will  now  be  less  than  1,  and  we  shall  have  A  =  —  (   1  +  (•4-J  "2  +  &c.  j  &c. 

In  the  former  case,  when  -7  is  less  than  1,  or  the   orbit  of  the  disturbed  planet  is  interior  to  that  of  the 
disturbing,  we  have 

2.^  +  &c-;  (137-  0 


' 

and  so  on  ;  and  similar  expressions  are  also  readily  obtained  in  the  case  when  the  orbit  of  the  disturbed 
planet  is  the  exterior.  Thus,  when  the  mean  distances  of  the  two  planets  are  given,  the  values  of  A,  A',  &c. 

and  their  differential  co-efficients  — — ,  &c.  are  reducible  to  numerical  evaluation,  and  may  therefore  be 

d  a 

regarded  as  known  quantities.  The  properties  of  the  series  on  which  they  depend,  afford  many  resources  for 
facilitating  their  evaluation,  and  rules  for  deriving  one  of  these  quantities  from  another,  but  these  we  shall  not 
stay  to  explain.  The  reader  will  find  them  with  every  developement  in  the  second  book  of  Laplace's 
Mecanique  Celeste,  art.  49. 

These  values  once  determined,  we  have  Q,  or  2  /d  n  +  r          ,  expressed  as  follows  : 

*J  d  i" 

2g  d  A         f       d  A'          2  n  A'  )  „  .  "} 

Q=_l  +  a         -  +  \  a  ——  +  -      —;'( ".ns  (n  -n')t  I 

da         l        da         "-"'  V (138,1) 

(       dA"        2nA")  ,.   .       ,       ( 

+  J.  a 1 A  cos  2  (n  —  n')  t+  &c.  \ 

(dan  —  n  )  J 

Such  is  the  value  of  the  perturbative  function  when  the  eccentricities  and  inclinations  of  the  orbits  are  neg- 
lected. Let  us,  for  the  present,  confine  ourselves  to  this  case ;  and,  writing  M,  M',  M",  &c.  for  the  successive 
co-efficients,  we  have 

Q  =  M  +  M'  .  cos  (n  -  n')  t  +  M"  .  cos  2  (»  -  n')  t  +  &c.  (138,  2) 

Now  the  equation  (128)  gives,  when  e  =  o, 

Q  /  1  \ 

n= =  —  n4  a  Q   I  because  n4  =  —  I  or, 

<za  V  a  / 

n  =  —  n2  a  { M  +  M' .  cos  (n  -  n')  t  +  &c. } 
*o  that  the  differential  equation  in  Ku  becomes 

o  -    d~r>"    +  ,tt£  u  _  „!  a  [M  +  M' .  cos  (n  -  i')  t  +  &c.}  (139) 


686  PHYSICAL    ASTRONOMY. 

Astronomy,  and  integrating,  Miysii «l 

x— •~v— •"'  n2  a  M'  n4  a  M"  Astromm.v. 

•     (n  _  ^  _  nt  .  cos  (n-n')t-  4(n_^t_nt™*(»  -  »')  <  -  &c. 

Consequently,  since  by  equation  (124)  6  r  =  —  a  &  u  when  e  =  o, 

n4  a2  M"  «*  a2  M" 

-  «*  -M  +    (n  _  „,).  _  „.   •  cos  <»  -  n  )  t  -  _____  .  Cos  2  (n  -„')<  +  &c.        (140) 

The  value  of  S  r  thus  obtained,  S  0  is  easily  got  from  (131 );  for  in  this  case  —  =  o, 

d  t 


where  2  is  used  to  express  the  sum  of  all  similar  terms  from  i  =  1  to  i  =  TO  inclusive,  and  M '  represents 
the  i*  in  order  of  the  co-efficients  M',  M",  W",  &c.     Moreover 

/'dQ                       dA                        an         dM 
J  r  ~ndt  =  a~.nt  +  V- -__.smtu, 


i  (n  —  n')    da 


i 


2  A  ' 

dt  I  dQ  =  -2-.nt  +  _:_2  —  sin 


. 
a  (n  — 

Uniting  therefore  these  several  parts,  we  get 


xnt 


f  3  a  »»  A'  2  a*  n       rf  A'          2  n  «(«  —  »')  i  1 

">"  Z    )  T-  -  TT,  +  —  -  7-    —  ;  --  —  -  7—  -  -  M'    !•  SID  i  10 

(t(»  —  n)»       t  (n  —  «')     d  a         t2  (n  —  n')2  —  n2          | 

N«w,  first,  since  n  t  represents  the  mean  loni^'.tude  of  »»  as  deduced  from  observation,  the  quantity  n  is 
already  affected  with  the  whole  influence  of  the  planetary  perturbation,  and  consequently  the  part  multiplied 
by  n  t  in  this  expression  of  the  perturbation  in  longitude,  and  which,  if  allowed  to  remain,  would  express  an 
additional  perturbation,  is  superfluous.  This  famishes  the  condition 

2    „  dA 


which  determines  the  constant  g.     Moreover,  in  the  latter  part  of  this  expression,  if  we  write  for  M'  its  value 

rfA»         2nA' 
M'  =  a  .  —^  --  h 


r?  n         n  —  n' 
it  will  admit  reductions,  and  the  value  of  S  0  will  at  length  be  found  as  follows  : 

2  n'  a'      dA' 
_.  d  a  (,'}  H2  +  i2  (»  -  «')*)  .  n2  a  A' 

2 


If  these  expressions  of  S  r  and  S  0  be  each  multiplied  by  mf,  we  have  the  values  of  rri  S  r  and  TO'  5  0  the 
perturbations  of  the  radius  vector  and  the  longitude,  i.  e.  those  parts  of  them  which  are  independent  of  the 
eccentricities  of  the  orbits.  These  expressions  give  room  for  some  remarks.  The  perturbation  in  longitude 
as  we  observe  is  wholly  periodical  and  dependent  on  n  single  angle  w  and  its  multiples.  In  forming  then 
a  table  of  the  values  of  m'  S  Q,  the  numerical  co-efficients  being  computed,  and  the  value  of  ra'  S  0 
thus  reduced  to  the  form  p  .  sin  w+  <?.sin2«>  +&c.  we  may  include  the  whole  of  this  in  one  column, 
entered  under  the  general  argument  w,  instead  of  regarding  it  as  consisting  of  an  infinite  number  of  separate 
inequalities. 

The  same  remark  extends  to  the  periodical  part  of  m'  o  r,  its  arguments  are  the  same  as  in  the  formula 

for  m'  S  6  ;  but  besides  this,  S  T  includes,  as  we  have  seen,  a  constant  part  —  a2  .  M  or  —  Zga  •     a3  .  -- 

rf  a 
which  becomes  by  substituting  forg-  its  value  in  (141) 

1  dA 

const,  part  of  n  r  =  —  -  a3  .  —  —  , 

3  da 

and  therefore 

a3       d  A  1  /       d  A*         "2  n  A'  \ 

•'  «TT  •  -3  ---  |-n2aa.  2  —  -  -  •  -  I    a  --  1  --       I  cos  iw  ;  (143) 

3        da  i2  (n  —  n')*  —  n2\        d  a          n  —  n'J 

In  the  formation  of  a  table  of  m'  8  r  this  constant  part  is  of  course  included  with  the  variable  one,  but  the 
effect  is  remarkable.  It  appears  that  the  action  of  the  disturbing  planet  alters  the  mean  distance  from  the 
sun  of  the  disturbed,  and,  of  course,  its  mean  motion  and  periodical  time  from  what  they  would  have  been  had 
the  disturbing  planet  no  existence.  At  the  same  time,  it  will  be  demonstrated  in  the  following  pages,  that 


PHYSICAL     ASTRONOMY.  687 

Astroaomy  these    alterations  once  produced,   are  permanent  and  unchangeable   in    their    quantity  by    the   subsequent    Physical 
<— -v^»  actions  of  the  bodies  composing  the  system.  Astronomy 

The  angle  w  is  the  difference  of  longitudes  of  the  two  planets,  or  their  heliocentric  elongation  from  each 
other.  If  we  call  e  and  e'  their  epochs,  or  their  actual  longitudes  at  the  commencement  of  the  time  t,  we  have 

w  =  n  t  +  c  —    («'  t  +  e') 

=  n  t  —  n' t  -if  t  —  c' 

and  this,  in  fact,  is  the  argument  of  the  perturbations  when  we  neglect  the  eccentricities  and  inclinations  of 
the  orbits. 

Let  us  next  examine  the  perturbation  in  latitude,  we  have 

dQ     _  z'        z'  —  2 
dz    ~7*  ~    ~  X3 

Hence,  it  is  evident,  that  if  we  neglect  the  inclinations  and  eccentricities,  we  have =  o,  and  the  plane 

d  z 

of  the  disturbed  orbit  does  not  change. 

We  have  thus  determined  the  effect  of  the  action  of  a  third  body  on  the  orbit  and  motion  of  in,  on  the  sim- 
plest supposition,  and  our  results  (to  recapitulate  them)  amount  to  this. 

1st.  That  the  radius  vector  undergoes  a  permanent  change  in  its  mean  value,  and,  of  course,  that  the  period 
and  mean  motion  of  m  are  permanently  altered. 

2d.  That  the  elliptic  value  of  the  radius  vector  receives  an  accession  of  terms,  of  the  form 

p  +  q  .  cos  w  +  r  .  cos  2  w  +  s  .  cos  3  w  +  &c. 
and  that  of  the  true  longitude,  a  series  of  terms  of  the  form 

q' .  sin  w  +  r'  .  sin  2  w  +  /  .  sin  3  w  +  &c. 
w  being  the  difference  of  longitudes,  or  mutual  elongation  of  the  planets,  from  each  other. 

3dly.  That  to  express  the  several  co-efficients  of  these  formulae  in  numbers,  nothing  more  is  required 
than  a  knowledge  of  the  mass  of  the  disturbing  planet,  and  the  mean  distances  and  mean  motions  of  both. 

In  the  cases  then  where  the  disturbing  planet  has  satellites,  as  in  those  of  Jupiter,  Saturn,  and  Uranus,  the 
mass  is  known,  and  the  reduction  of  the  formulae  to  numbers  is  complete.  It  is  fortunate  that  these  are  by 
far  the  most  considerable  bodies  of  our  system,  but  proximity  to  a  certain  extent  supplies  the  place  of  intrinsic 
energy  ;  and,  in  the  case  of  the  perturbations  of  the  earth,  our  uncertainty  of  the  masses  of  Mars  and  Venus 
leaves  us  in  some  degree  at  a  loss.  But  physical  astronomy  furnishes  us  in  this  dilemma  with  considerable 
aid.  Regarding  the  masses  of  these  planets  as  unknown  quantities,  we  may  calculate  in  general  terms  their 
effect,  either  on  the  places  of  the  other  planets  at  assigned  instants,  or,  on  the  elements  of  their  orbits  them- 
selves, which  are  susceptible  of  much  more  accurate  determination,  by  bringing  a  long  series  of  observations 
to  bear  on  them,  and  comparing  the  variations  in  their  values  after  long  intervals,  as  computed  by  theory, 
and  as  given  by  observation,  we  obtain  data  for  the  determination  of  these  delicate  quantities,  so  much  the 
more  precise  as  the  variations  observed  in  the  elements  are  greater,  or,  in  other  words,  as  the  interval  of  time 
in  which  they  are  observed  is  longer.  It  is  thus  that  the  lapse  of  ages  is  necessary  to  give  precision  to  the 
numerical  data  of  our  system,  and  that  continual  and  patient  observation  must  ultimately  lead  us  to  the 
knowledge  of  points  which  elude  the  direct  cognizance  of  our  senses,  and  defy  any  investigation  but  those  in 
which  successive  generations  of  mankind  bear  a  part. 

In  fact,  if  we  regard  the  masses  of  all  the  planets  as  unknown  quantities,  but  their  mean  distances  and 
periodic  times  as  known  ones  ; — the  latter  afford  us  the  means  of  computing  the  values  of  S  r  and  $  0  in  any 
assigned  case,  independent  of  the  value  of  m  ,  which  does  not  enter  into  their  expressions.  Let  us  therefore 
represent  by  m'  o  r  and  m'  £'  0,  the  perturbations  of  the  radius  vector  and  longitude  produced  by  the  planet  m' ; 
by  m"  i"  r  and  m"  S"  0  those  produced  by  m",  and  so  on.  Then  will  the  true  values  of  these  quantities  at  any 
assigned  instant  be 

r  +  m'  I'  r  +  m"  e''  r  +  m"  c"  r  +  &c. 
&  4-  m'  S'  0  +  m"  S"  0  +  m'"  S'"  0  +  &c. 

in  which  r,  £' r,  S" 'r,  &c.  and  0,  I1  0,  S"  0,  &c.  are  quantities  susceptible  of  calculation  from  theory.  Sup- 
pose now  that  we  construct  tables  of  the  values  of  0,  d'  0,  c,"  0,  &c.  (or,  as  we  will  for  a  moment  write  these 
latter  quantities,  0,  0,  ^,  &c.)  then,  at  any  assigned  instant,  we  have  only  to  take  out  of  these  tables  the  values 
of  0,  0,  Yo  &c. ;  and  the  true  longitude  of  m  will  be 

0  +  TO'  0  +  m"  ^  -r  &c.  =  L 

Suppose  now  we  compare  this  formula  with  a  great  multitude  of  observations,  and  thus  obtain  a  series  of 
equations, 

m' .  0,  4   m"  .  YTI  +  m''' .  x,   +  &c.  =  L,  —  0, 
m-  .  03  +  m" .  Y",  +  m'"  .  Xs  +  &c.  =  L,  -  0, 

&c.  =  &c. 

The  only  unknown  quantities  in  these  will  be  the  masses  of  the  disturbing  planets  m',  m",  &c.  and  by  resolving 
these,  we  may  determine  their  values,  and  thus  estimate  the  masses  of  the  planets  by  the  perturbations  they 
produce. 

In  this,  as  in  almost  all  such  delicate  inquiries,  where  the  quantities  to  be  determined  are  exceedingly  small, 
and  the  observations  from  which  they  are  to  be  discovered  liable  to  inaccuracies,  bearing  a  sensible  proportion 
VOL.  in.  4  ti 


688  PHYSICAL    ASTRONOMY. 

Astronomy,  to  the  thing  observed,   (which  in  this  case  is  L  —  0,  or  the  total  perturbation  arising  from  the  united  action    Physical 
s— ~v— -'  of  all  the  planets,)  we  are  obliged  to  employ  a  great  many  more  observations  than  would  be,  mathematically  Astronom; 
speaking,  sufficient,  if  each  were  perfect,  with  a  view  to  destroy  the  errors  of  observation  in  the  mean  result.  N-~~v^ 
The  number  of  disturbing  planets  in  our  system  at  present  known  does  not  exceed  ten;  and  it  would  therefore 
appear  that  ten  observations  of  the  longitude  of  one  disturbed  planet  would  enable  us  to  determine  the  masses 
of  all  the  rest ;  and  so  they  would,   were  the  observations  mathematically  exact,  the  elements  of  the  orbits 
exactly  known,  and  the  theory  by  which  the  values  of  0,  ty,  %,  &c.  are  computed,  complete.     But  each  of 
these  conditions  is  far  from  being  fulfilled  in  the  present  state  of  astronomy ;  and  if  we  would  use  this  method 
to  determine  the  masses  of  the  planets,  we  must  accumulate  many  hundreds  of  observations  made,  not  on  one, 
but  oil  all  of  them,  especially  on  those  subject  to  the  greatest  perturbations. 

The  method  of  treating  a  series  of  equations  more  numerous  than  the  unknown  quantities  they  contain,  so 
as  to  give  them  all  their  proper  influence  on  the  result,  and  obtain  from  them  a  set  of  values  which,  though 
satisfying  neither  of  them  separately,  yet  when  substituted  in  all  of  them,  shall,  on  the  whole,  give  more 
satisfactory  results  than  any  other,  depends  on  the  theory  of  probabilities  and  may  be  found  in  Laplace's 
Theorie  Analytique  des  Probabihtes. 

If  the  mass  of  any  one  or  more  of  the  planets  (m')  for  instance,  be  regarded  as  sufficiently  known  from  other 
methods,  we  need  only  employ  this  mode  for  the  rest,  and  regarding  the  perturbation  m'  6'  0  produced  by  it 
as  known,  place  it  on  the  known  side  of  the  equation,  which  will  thus  become 

m"  &"  0  +  m'"  f,'"  0  +  &c.  =  L  -  0  —  m'  &'  0 

Thus  we  may  determine  for  instance,  the  masses  of  Mars  and  Venus,  by  means  of  an  extensive  series  of  obser- 
vations of  the  sun's  longitude,  or  (which  is  the  same  thing)  by  employing  to  that  end  the  perturbations  they 
produce  on  the  earth.  For  the  masses  of  Jupiter,  Saturn,  and  Uranus,  being  known  from  the  periods  of  their 
satellites  ;  and  those  of  Mercury,  and  the  four  new  planets — Ceres,  Pallas,  Juno,  and  Vesta,  being  so  small,  as 
to  have  little  or  no  influence,  we  have  only  two  unknown  quantities  (in',  m")  to  determine. 

This  method  is  laborious,  certainly  ;  but  considering  the  perfection  of  modern  observations,  the  great  mul- 
titude of  them  which  may  be  brought  to  bear  upon  this  point,  and  the  considerable  degree  of  exactness  which 
the  theory  of  the  planetary  perturbations  has  now  attained,  it  is  not  impossible  that  it  may  one  day  be  made 
to  render  the  best  service  in  determining  the  masses  even  of  those  planets  which  have  satellites.  At  all  events, 
it  is  highly  desirable  that  it  should  be  applied  for  that  purpose,  as  its  results  would  lead  us  to  judge  how  far 
the  latter  method  can  be  depended  on  in  cases  like  that  of  Jupiter  and  Saturn,  where  the  great  deviation  from 

2   77"     X     Q'T 

sphericity  of  the  central  body  renders  the  application  of  the  formula  t  —  — ===  somewhat  liable  to  error. 

v  M  +  m 

In  fact,  this  formula  is  derived  on  the  hypothesis  of  a  force  represented  by  — — — —  but  in  the  case  of  spherical 

bodies  only  does  the  total  attraction  vary  precisely  in  that  ratio*.  This  alone,  however,  will  not  account  for 
the  great  difference  which  Mr.  Gauss  has  lately  found  between  the  mass  of  Jupiter,  as  obtained  from  obser- 
vations of  its  satellites,  and  that  deduced  from  the  perturbations  of  the  small  planets  intermediate  between 
Jupiter  and  Mars,  so  that  the  subject  must  be  regarded  as  open  to  further  investigation,  should  the  calcula- 
tions of  the  last  named  eminent  geometer  be  found  to  coincide  with  a  more  extensive  series  of  observations 
of  those  interesting  bodies  than  the  shortness  of  the  time  they  have  been  known  has  hitherto  allowed. 

SECTION  IV. 

Of  the  method  of  taking  into  account  the  effect  of  the  eccentricities  of  the  orbits  on  the  planetary  perturbations,  and  of 

the  origin  of  the  secular  equations  of  their  motions. 

WHKN  we  regard  the  orbits  as  elliptical,  the  whole  of  the  foregoing  investigations  require  modification,  the 
value  of  the  perturbative  function,  and,  of  course,  of  the  perturbations  themselves,  receiving  accessions  of 
terms  depending  on  the  powers  and  products  of  the  eccentricities.  We  will  here  endeavour  to  explain  the 
manner  in  which  these  terms  originate  ;  and  to  a  certain  (though  limited)  extent,  the  course  pursued  by 
geometers  in  determining  their  form  and  value. 

The  functions  fi,  /  d  Q,  f  — — ,  are  explicitly  given  in  terms  of  r,  r',  and  w  or  0  —  0',   and  contain  no 
J  a  r 

other  symbols.  Hence  it  arose,  that  when  r,  r'  were  supposed  constant,  the  only  cause  of  the  variation  of 
these  functions  consisted  in  that  of  w  ;  and  0  and  0'  being  in  this  case  each  expressed  by  an  arc  proportional 
to  the  time,  it  was  sufficient  to  develope  them  in  cosines  and  sines  of  w,  to  have  at  once  the  expression  of  the 
function  n  in  such  a  form  as  we  required  for  integrating  our  equations.  When,  however,  the  eccentricities 
are  introduced,  all  these  facilities  are  at  an  end;  r,r',  and  w,  are  no  longer  constant  quantities  and  simple 
functions  of  the  time,  but  each  of  them  branches  out  into  a  series  of  powers  of  the  eccentricities,  and  sines 
and  cosines  of  variable  arcs. 

*  Laplace  (Tkeorle  dts  Satellites  de  Jupiter,  p.  102.)  makes  the  deviation  of  the  attraction  of  the  first,  compared  with  the  fourth 
satellite  from  the  law  of  the  inverse  squares  of  the  distances,  only  ~7nre  of.the  whole  attraction  of  the  former,  supposing  Jupiter 
homogeneous.  In  Saturn,  the  attraction  of  the  ring  must  cause  a  much  more  considerable  deviation  from  tlmPJaw. 


PHYSICAL   ASTRONOMY.  689 

Astronomy.      Our  object  being  to  reduce  Q,  &c.  to   sines  and  cosines  of  arcs  proportional  to  the  time,  or  of  the  form     Physical 
'  A  t  +  B,  it  is  evident  that  we  must  substitute  for  r,  r',  and  w,  their  values  so  expressed,  and  then  develope  Astronomy, 
each  term  of  Q  to  the  extent  we  wish.    At  present  we  will  confine  ourselves  to  the  first  powers  of  the  eccen-     ~~v~"~ 
tricities. 

Now  we  have  Q  =  R  +  R' .  cos  w  +  R"  .  cos  2  w  +  &c. 

in  which  R,  R',  R",  &c.  and  w  are  explicit  functions  of  r,  r',  &c.  and  w  =  0  —  Q'  putting  0  and  O1  for  the  true 
longitudes  of  the  two  planets.  Now,  if  we  call  e,  and  e',  the  longitudes  at  the  commencement  of  the  time  t, 
n  I  +  e  and  n' t  +  <•'  will  be  their  mean  longitudes  after  the  lapse  of  that  time,  and  calling  IT  and  ttt  the  longi- 
tudes of  the  perihelion,  the  mean  anomalies  will  be  n  t  -f-  e  —  TT  and  n'  t  +  c'  —  T/.  Hence,  the  true 
anomalies  will  be  (by  equation  30), 

n  t  +  e   —  ir   +  2  e  .  sin  (n  t  +  e   —  ir  )  +  e"1  x  &c. 

n' t  +  e'  —  IT  +  2  tf  .  sin  (n' t  +  e'  —  ?/)  +  e4  x  &c. 
and  the  true  longitudes  of  course  are 

(»  t  +  e )  +  2  e  .sin  (n  t  4-  e  —  IT)  +  e*  x  &c. 

(»'<  +  «')  +  2  e1 .  sin  (n' t  +  e'  —  T/)  +  es  X  &c. 

Hence,  if  we  neglect  the  eccentricities,  we  have  simply  w  =  (n  t  —  »'  t  +  «  —  «')  and  as  R,  R',  &c.  in  this 
case  assume  their  circular  values  A,  A',  &c.,  the  terms  of  Q  not  depending  on  the  eccentricities  will  remain 
as  before, 

A  +  A' .  cos  (n  t  —  n'  t  +  e  —  e')  +  A" .  cos  2  (n  t  —  n' t  +  e  —  e')  +  &c. 

On  the  other  hand,  the  terms  depending  on  the  eccentricities  have  their  origin, 

1st.  In  the  developement  of  the  functions  R,  R',  &c.  5  when,  instead  of  r,  /,  we  put  their  elliptic  values, 

r  =  a  +  A  r,  and  /  =  of  +  A  r' 
denoting  by  A  r  and  A  /  the  parts  of  r,  /  arising  from  the  eccentricities. 

2d.  In  the  substitution  of  W  +  A  w  for  w  in  cos  w,  cos  2  w,  &c.  W  being  the  part  of  to  independent  of  the 
eccentricities,  or 

VV  =  (n  —  n')  .  t  +  (e  —  e') 
and  A  w  being  the  part  depending  on  them,  or 

A  w  =  2  e  .  sin  (n  t  +  e  —  jr)  —  2  e!  .  sin  («'  t  +  e'  —  TT')  +  ea  X  &c.  &c. 

3rd.  In  the  multiplication  of  these  terms  together. 

Now,  if  we  still  continue  to  denote  by  A  that  variation  in  Q  r,  r1 ',  &c.  which  arises  from  the  eccentricities,  we  have 

dQ  dQ    t  dQ 

A  Q   =   A  r  +   T-  A  /  +   — —  A  w 

d  r  d  r  d  w 

in  which  the  differential   co-efficients  , r>  • >  are   to  have  their  circular  values.      If  we  would 

dr      dr        dw 

d2  Q      (A  r)2 
pursue  the  investigation  further,  we  must  add  to  A  Q  the  terms         g  .  — - — — ,  &c. 

Now,  in  general,  we  have 

dQ          dR          dR'  dR" 

-3 —  =  — r-  — : —  .  cos  w  +  — —  .  cos  2  w  +  &c. 

d  r  d  r  d  r  d  r 

d  Q  d  R  d  R'  d  R" 


.    .     —        ,     ,     T^        - — 7-  .  uua  w   -|          - — 7- 

d/  d/  dr'  dr' 

the  circular  values  of  which  are  respectively 


cos  2  w  +  &t. 


d  A'     cos  W  +  ~  .  cos  2  W  +  &c.  (144, 1) 


da  da'  da 

47;t.W-.«rw  +  4£—'w  +  ta;  <U4-2' 

and,  Jh  like  manner,  the  circular  value  of is 

d  w 

—  {  1 .  A' .  sin  W  +  2  .  A"  .  sin  2  W  +  3  .  A'"  .  sin  3  W  +  &c.  }  (144,  3) 

The  values  of  A  r  and  A  /  are  given  by  equation  (28)  if  we  substitute  merely  for  n  t  the  expressions  n  t  + 
e  —  jr,  and  n' t  +  e'  —  T/  ;  which  in  this  case  are  the  mean  anomalies,  because  the  mean  longitudes  n  t,  n' t, 
and  the  constants  e,  ef,  TT,  !/,  are  reckoned,  not  from  the  perihelion  of  the  orbits,  as  in  that  equation,  but  from 
the  line  of  the  equinoxes.  If  then  we  put  V  =  n  t  +  «  —  T,  V  =  n' t  +  t'  —  ir',  we  have 

A  r   =  —  a  e  .  cos  V   +  es  x  &c.  ~J 

A/  =  —  a'e'.cos  V  +  (P  X  &c.    >  j (145) 

and  A  w  =  2  e  .  sin  V  —  2  e' .  sin  V  +  e*  x  &c.  J 

Substituting  therefore  in  A  Q  for  the  differential  co-efficients,  their  circular  values  in  (144,  1,  2,  3,)  and  for 
A  r,  A  /,  and  A  w,  the  values  just  now  found,  it  will  become 

4  u2 


PHYSICAL    ASTRONOMY. 

,r  f  d  A          d  A'  d  A"  |  Physical 

A  Q  =  -  a  e  .  cos  V  J  —  -  +  -j—  .  cos  VV  +  - —  .  cos  2  W  +  &c.  \  Astronomy. 

[  a  a  a  a  da  }  ^  _ 

-  a'e' .  cos  V'l-44-  +  4^T  •  cos  VV  +  4^T  .  cos  2  W  +  &c.  ]•  (146) 

(  d  a'         d  a'  da'  J 

—  (2  e  .  sin  V  —  2  e' .  sin  V)  { A' .  sin  VV  +   2  A"    sm  2  VV  +  &c. } 
+  e-  x   &c. 

New  these  series  are  not  precisely  in  the  form  we  want  them  ;  the  cosines  and  sines  of  V  and  W  being  multi- 
plied together  ;  and  to  disengage  them,  we  must  employ  the  well  known  trigonometrical  formula 

cos  A  .  cos  B  =  i  }  cos  (A  +  B)  +  cos  (A  —  B) }  (A) 

—  sin  A .  sin  B  =  ±  \  pos  (A  +  B)  —  cos  (A  —  B)  J 
By  the  aid  of  this,  we  get 

cos  V  =  cos  V  cos  V  =  cos  V 

cos  V.  cos    W  =  i{cos(V  +    W)+cos(V-    W)  ]  ;  cos  V  .  cos    W'  =  i{cos(V'  +    W')+cos(V—  W')} 
cos  V .  cos2  VV  =  i{  cos  (V  +  2  W)  +  cos  (V-2  W)  j ;  cos  V  .  cos2  VV  =  i[  cos  (V  +  2W;  +cos  (V-  2VV)  j 
&c.  =  &c.  &c.  =  &c. 

sinV.sin  W  =  1  [  cos  ( V  -    VV)-cos(V+    W) } ;     sinV'.fin    W  =  |{cos(V-     W)  -  cos  (V  +    VV)  ] 
sinV.sin2VV=l|cos(V-2W)-cos(V-|-2\V)}  ;     sin  V  .  sin2  VV  =  i[cos  (V-2VV)  -  cos  (V  +  2  VV)  } 
&c.  =  &c.  &c.  =  &c. 

Thus  we  see  that  A  Q  (if  we  confine  ourselves  to  the  first  powers  of  the  eccentricities)  is  reducible  to  a 
series  of  sines  and  cosines,  whose  arguments  are  all  comprehended  in  the  forms  V  +  t  VV,  and  V  +  i  VV ; 
or,  (since  cos  —  A  =  —  cos  A)  in  the  forms  i  VV  +  V  and  i  W  +  V.  That  is,  (since  V  =nt  +  t  —  IT,  and 
W  =  n  t  —  n' t  +  e  —  e')  in  the  forms 

t  (n  t  —  n' t  +  e  —  e')  +  (n  t  +  e  —  ir) 

i  (n  t  —  n' t  +  e  —  e')   +  (n't  +  e'  —  IT') 

If  we  actually  execute  the  substitutions  (still,  for  brevity,  preserving  the  denominations  VV  and  V)  we  shall 
obtain  for  the  value  of  A  Q  the  following  expression — 

A  A 

(147) 


a 

dA 

cos  V 
A  }      INK  1  W 

-.-  V) 

-a' 
—  e' 

-  e' 

-  e' 
j 

,     d 

A 

cos  V  ; 

+  *'} 

i'1 

cos  (W  • 

f  V') 

e\ 

da  ' 
dA' 

L   '    d 

K 

a' 
dA' 

2 
a 

da 
dA.' 

(2 

K 

da' 
dA' 

I 

-t 

-  &c. 
But  our  object  is 

'2 

0 

¥ 

a 

da 
dA" 

9  A7/  '        r*r»c    (Q  \V 

+  V) 
-  V) 

of  Q  the 

fa' 

da' 
dA" 

Vj 

+  2A"J 

0  A"  I 

cos  (2VV 
cos  (2W 

2/dn 

+  V) 
-V) 

dn 

4r  r  - 

da 
dA" 

+  2  A"]  .cos  (2VV 
i  the  developement 

K 

da' 
dA" 

2      da 
to  obtai 

perturbativc 

da'           "  J 
function,  or 

r  d 

r 

The 

part  of  this  independent  of  the  eccentricities  is  already  found,  and  we  have  now  only  to  consider  that 
depending  on  them ;  which,  in  the  notation  above  adopted,  will  be  expressed  by  A  Q,  or 


(148) 


Let  us  first  consider  the  value  of  the  first  part  of  this  expression  2  /  d  A  Q,  and  let  any  term  of  A  Q  be 

represented  by 

M  .  cos  (i  W  +  k  V  +  I V) 

in  which  i  may  be  any  integer  from  o  to  infinity,  and  k  and  I  either  +  1  or  o,  an  expression  which  obviously 
comprehenos  all  the  terms  of  which  A  Q  consists.  This  premised,  it  is  obvious  that  the  co-efficient  M  being 
constant,  the  variation  of  A  Q  can  only  arise  from  the  variation  of  the  angle  iW  +  fcV-f-ZV;  and  as  this 
angle  is  supposed  only  to  vary  by  the  motion  of  m,  we  are  to  suppose  n  t  only  to  vary,  and  «'  t  to  remain 
constant.  So  that  we  have,  for  that  part  of  the  variation  of  A  fi  which  arises  from  the  term  in  question, 

—  M  (i  d  VV  +  k  A  V)     *\n  (i  VV  +  k  V  +  I  V) 
because  d  V  =  o,  since  V  =  n'  t  +  «'  —  -a'.     Now  d  VV  =  n  d  t  and  d  V  =  n  d  t  also.     Consequently,  this 


PHYSICAL    ASTRONOMY  691 

\stronomv.  becomes  —  (i  +  k)  M  .  n  d  t  .  sin  (i  W  +  k  V  +  I  V)  Physical 

'_j—       ,_-  Astronomy. 

and  the  part,  of  the  expression  2  /  d  A  Q  arising  from  this  term,  will  therefore  be  ^- 

•  M  cos  (•  W  +  *  V  +  <  V)  (149) 


-  -  L  ,, 

i  (n  —  n')  +  k  n  +  I  n 

We  see  therefore  by  the  foregoing  reasoning,  that  in   order  to  obtain  that  part  of  the  perturodtive  function 
which  originates  in  the  term  2   /  d  Q,  and  depends  on  the  eccentricities,  we  have  only  to  take  the  terms  of 

the  expression  for  A  Q  (14")  in  their  order,  and  with  their  proper  signs,  and  multiply  each  of  them  respec- 
tively by  that  value  of  the  fraction 

2  (i  +  k)  .  n  _ 
i  («  —  »')  +  k  n  +  I  n' 
which  corresponds  to  the  values  of  i,  k,  I,  in  its  argument,  represented  by  i  W  +  k  V  +  I  V.     For  instance, 

2  n 
the  term  which  has  simply  V  for  its  argument,  must  be  multiplied  by  -  --  =  2,  that  which  has  V  by  o.  — 

Again,  the  terms,  whose  respective  arguments  are  W  +  V,  W  —  V,  W  +  V,  W  —  V,  are  to  be  multiplied, 

4  n  2  71 

according  to  this  rule,  by  the  respective  co-efficients  —  -  T,  o,  2,  and  -  7.      Similarly,  the  terms 

2  n  —  n  n  —  2  n 

which  have  <2  W  +  V,  2  W  -  V,  2  W  +  V,  2  W  —  V,    for  their  arguments,    acquire    the   co-  efficients 

6n  2  ra  4n  4  n 

and  —  -  -  —  r  ;  and  so  on. 


3  n  —  2  n'  '     n  —  2  «"     2  n  —  rc' '  2  n  —  3  »i' 

The  co-efficients  thus  acquired  by  integration  depend  solely  on  the  ratios  of  the  mean  motion.-,,  or  periodic 
times,  of  the  disturbed  and  disturbing  planet,  and  are  of  very  great  importance  in  the  planetary  theory. 
Were  it  not  for  these,  the  theory  of  the  planetary  perturbations  would  be  very  simple,  as  the  same  treat- 
ment, or  nearly  so,  could  be  applied  in  every  case,  and  the  magnitudes  of  the  several  inequalities  would  go  on 
diminishing  with  more  or  less  rapidity,  as  the  arguments  contained  higher  multiples  of  the  mean  motions. 
But  these  co-efficients  prevent  this  regular  progression  of  magnitude  from  taking  place,  and  render  it 
difficult  to  foresee  without  computation  whether  any  assigned  inequality  may  be  neglected  or  not,  and  im- 
possible to  argue  from  its  known  minuteness  in  the  case  of  one  pair  of  planets  to  its  probable  smalness  in 
that  of  another.  Thus  an  inequality,  whose  maximum  we  know  to  be  small  in  the  case  of  Venus  disturbed  by 
the  earth,  may  have  a  considerable  magnitude  in  that  of  Jupiter  disturbed  by  Saturn,  merely  from  a  relation 
subsisting  between  the  periodic  times  in  the  latter  case  which  does  not  in  the  former.  In  fact,  it  is  evident 
that  if  the  periods  should  happen  to  be  so  related  as  to  render  the  denominator  of  any  of  the  foregoing  or 
similar  fractions  very  small  compared  to  the  numerator,  the  inequality  into  which  it  enters  as  a  co-efficient 
will,  on  this  account  alone,  acquire  a  very  great  preponderance.  Thus,  if  the  period  of  the  disturbed  planet 

were  very  nearly  half  or  double  that  of  the  disturbing ;  the  terms,  multiplied  by  -          — j  or  by r 

£  HI    ~~    Jl  'fl    ~~~    &  ft 

would  become  very  large,  and  the  length  of  the  period  of  the  inequality  represented  by  it  would  be  propor- 
tionally increased,  and  in  the  case  of  exact  commensurability  would  actually  become  infinite  ;  that  is  to  say,  the 
disturbance  would  go  on  to  such  an  extent,  as  to  make  a  total  subversion  of  the  original  conditions  of  the 
problem.  The  physical  reason  of  this  is  equally  obvious.  In  the  case  just  instanced,  the  two  planets,  at  every 
revolution  of  the  exterior,  would  be  placed  in  exactly  the  same  circumstances — the  same  disturbing  forces 
would  act  in  the  same  manner  for  a  series  of  ages,  and  their  effects,  instead  of  compensating  each  other  by 
mutual  opposition,  would  go  on  accumulating  in  the  same  direction,  till  their  orbits  at  length  became  entirely 
changed,  and  the  commensurability  of  their  periods  at  length  ceased  to  subsist.  In  fact,  an  equation  thus 
limited  by  no  period,  and  affecting  both  the  longitude  and  radius  vector  of  each  orbit  always  the  same  way, 
is  equivalent  to  an  alteration  of  the  mean  motion  and  mean  distance  ;  and  as  this  would  take  place  in  opposite 
directions  on  the  two  planets,  shortening  the  period  of  one,  and  lengthening  that  of  the  other,  their  periods 
would  continually  recede  from  commensurability ;  the  magnitude  of  the  inequality  in  question,  as  well  as  the 
length  of  its  period,  would  both  acquire  finite  values,  which  even  then  would  continually  diminish,  till  reduced 
within  limits  consistent  with  the  stability  of  the  system.  It  is  probably  by  some  such  gradations(if  we  may 
hazard  a  conjecture  on  a  part  of  the  planetary  theory  so  far  beyond  the  reach  of  analysis  or  exact  reasoning,) 
that  our  system  has  attained,  in  the  course  of  almost  indefinite  ages,  its  present  admirable  state  of  equilibrium, 
in  which  no  inequality  of  enormous  magnitude  exists  ;  and  those  which  have  any  notable  value,  can  be  proved 
to  be  confined  within  comparatively  narrow  bounds. 

Let  us  next  consider  the  part  of  the  developement  of  Q,  which  arises  from  the  term  A    (   r  ) .     Now, 

if  we  regard  only  the  first  powers  of  the  eccentricities,  and  consequently  neglect  the  squares  of  A  r,  &c.  we 
have  ./  rf  Q  \  rf  Q  d  Q 

d  r  d  T 

d  Q  d*  Q  t/2  O  ,/;  Q 

=  —. —  A  r  +  r  .  -——  A  r  +  r —  A  r'  +  r —  A  to 

d  r  d  rs  dr  d  r  d  r  d  ;i- 


692 


PHYSICAL    ASTRONOMY. 


Astronomy.  d  Q  Physical 

*-_r-    -«_-  in  which  the  differential  co-efficients  are  to  have  their  circular  values,  which  we  may  represent  by  — — ,    &c.  Astronomy. 


Now  these  are, 


da 


dA 
da 


d  a 


da 


da* 


d2  A           d2  A'                          rf2  A" 
=  a  -T— -  +  a  -— —  .  cos  W  +  a .  cos  2  W  +  &c. 

n  ft*  ft   /i*  ft  n° 


d  ad  a 


tPA 
d  a  da 


7  +  a 


I  .a 


da  da' 
dA' 


.  cos  W 


rp  \" 
a —  .  cos  2  W  +  &c. 


.  sin  W  +  2  .  a 


da  da' 
dA" 


.  sin  2  W  +  &c. 


.(150) 


dadW  da  da 

Let  these  be  substituted  in  the  above  expression  for  A  (r  — —  J  and  the  values  of  A  r,  A  /  and  A  w,  given 

in  (145)  be  put  for  them,  and  we  shall  have,  by  a  process  exactly  similar  to  that  gone  through  for  A  Q, 

/       d  Q  \  ( (  d  A  d2  A  \        /  d  A'  d*  A'\  1 

A  (  r  — —  )  =  -  a  e  .  cos  V  .  N  — —  +  a  —  —  )  +  (  -j—  +  a  — -  )  cos  W  +  &c.  \ 

\        d  r  /  (  \  d  a  daV\da  da4/  J 

-aV.cosV'.l  a  ,^A  .+  a  ^^  .  .  cos  W  +  &c.  i  (151) 

(dad  a'  dad  a'  j 

—  (2  e  .  sin  V  —  2  e' .  sin  V')(l  a  — —  .  sin  W  +  2  .  &c.l 

(da  j 

which,  by  the  use  of  the  same  trigonometrical  formulae,  and  by  properly  arranging  the  terms,  becomes 
di 


'/cos  V 


d  aa  da 

**_«.*£}co.(W+V)- 

^  A/      3a^A_]Cos(  W 


d2  A 
—  ti'.aa'-  -  r—  ;  cos  V 


da  da' 


da* 


daj 


cos  (  W  +  V) 

V)  —  —  la  of — •  —  2  a—, —  [cos  (  W  +  V) 

2(         da  da'  da) 


da 


da 


''l 

- 


,  „„ /uus  is  •»  —  *  i — s<*  «' —.  —  4  a  — — [cos  (2W  —  V) 

2(da*  daj  2(        dado7  rial 

-  &c.  -  &c.  ;   (152) 

We  are  now  in  a  condition  to  express  the  whole  value  of  A  Q,  by  combining  this  with  the  expressions 
(147)  and  (149)  and  it  is  evident  that  our  result  will  be  of  the  following  form  : — 

A  Q  =  (Ne  cos  V  +  N'  J  cos  V)  +  O  e  cos  (W  +  V)  +  OV  cos  (W  +  V)  (153) 

+  P  e  cos  (W  —  V)  +  P'  tf  cos  (W  -  V)  +  Q  e  cos  (2  W  +  V)  +  &c. 
and  the  co-efficients  of  the  several  arguments  will  be 

(       d3A  d  A 

N  =  —  I  a* 1-   3  a  

(da1  da 

O  = 1  a* 1 7  a  — ; A'  , 

2(       d  as        2  n  —  n'        da         2  n  —  n'        ) 


N'  =  -   a  a'  . 


d3A 
da  da' 

d»A' 
'  d  a  d  a' 


da 


da' 


(154) 


P=-l(a(,.J^-3al^+        2»    /a^__i^A't 

2(  dado'  da          n  —  2  n          do  n  —  2  n  jj 

The  value  of  A  Q  being  thus  obtained,  that  of  II  is  had  by  mere  substitution.  The  part  of  n,  independent 
of  the  eccentricities,  will  remain  as  in  the  preceding  section,  changing  only  in  the  several  arguments  to  into 
W ;  and  if  we  denote  this  by  n,  and  by  A  n  the  part  of  II  which  depends  on  the  eccentricities,  the  equation 
(128)  will  give 


A'tronomv 


PHYSICAL    ASTRONOMY.  093 

e  (  /'  )          A   Q  Physical 

A  II  =  —  \  Q  .  COS  V  -  2  /  Q  n  d  t  .  Sin  V —  Astronomy 

a*  |  ,/  J  o- 


in  which  Q  and  A  Q  denote,  as  we  have  all  along  supposed,  the  parts  of  Q  respectively  independent,  and 
dependent,  on  the  eccentricities. 

A  Q 
The  value  of  A  n  therefore  consists  of  two  parts  ;   the  latter —   is  immediately  obtained    from    the 

expression  of  A  Q  above  found,  and  is  equal  to 

e  (  1 

—  4j  N  .  cos  V   +  O  .  cos  (W  +  V)   +  P   .  cos  (W  -  V  )  +  &c.  !• 

_  ^IJN' .  cos  V  +  O'.  cos  (W  +  V)  +    P' .  cos  (W  -  V)  +  &c.| 
a2  (  ) 

The  former  depends  on  Q,  and  must  be  determined  by  substituting  for  Q  its  value 

Q  =  M  +  M' .  cos  W  +  M" .  cos  2  W  +  &c. 

where  M,  M',  &c.  are  co-efficients,  whose  values  are  assigned  in  equation  (138,  1).  This  substitution  made, 
we  find 

M'  M' 

Q  cos  V  =  M  .  cos  V  +  --  .  cos  (W  +  V)  +  —  .  cos  (W  -  V)  +  &c. 

£  2 

Q  n  d  t  .  sin  V  =  -  M .  cos  V ,  M   "    ',.   cos  (W  +  V)  -  ^-£  cos  (W    -  VI  -  &c. 

2  (2  n  —  n)  2  n 

So  that  the  part  of  A  IT  now  in  question  becomes 

—[3  M  .  cos  V  +     4n~"      M'  .  cos  (W  +  V)  +    2  "  + ,  "    M'  .  cos  (W  —  V)  +  &c.j 
aa  I  4  n  —  2  »'  2  n  ) 

and  the  whole  value  of  A  n  will  be  as  follows  :, — 

AH  =  4{(3M  -  N)  cos  V  +  (44  ".Jg"/  M'  -  o)  cos  (W  +  V)  +  ^^/"/M/  -  P)  cos  (W-V)  +  &c.  t 

-  —  JN' .  cos  V  +  O' .  cos  (W  +  V)  +  P'  .  cos  (W  -  V)  +  &c.)  ;  (155) 

«   I  J 

This  found,  the  integral  of  the  equation        g    +  n2  u  +  II  =  o  will  be  obtained  by  the  expressions  106, 

107,  and  108.  The  parts  of  o  n,  S  r,  and  S  0,  independent  of  the  eccentricities,  have  already  been  found  ;  and 
calling  therefore  A  S  u,  A  8  r,  and  A  S  0,  those  parts  of  these  respective  quantities  which  depend  on  the 
eccentricities,  we  shall  have  (since  V  =  n  t  +  e  —  TT,  W  =  (n  —  n')  t  +  (e  —  e')  and  V  =  n'  t  +  e'  —  T/) 


-  ?{-7^r  "' v/  t  T  '  T  °"  "r  +  V'>  +  (.  -  ^).  -  ...  "" (W  '  V'>  •*• &c]       <156) 

The  perturbation  of  the  radius  vector  is  now  easily  found  ;   for,  as  we  have  by  equation  124, 

£  r  =  —  a3M{l+2e.cosV  +  &c.  J 

this  gives  A8r=   —  a,(At>M-|-2e<5M.  cos  V } 

whence  A  Z  r  is  readily  obtained.  With  regard  to  the  perturbation  in  longitude,  or  mf  S  0,  no  further  diffi- 
culty than  the  length  of  the  substitutions  remains  to  be  encountered  ;  for  the  part  not  depending  on  the 
eccentricities  being  already  obtained,  that  which  depends  on  them  will  be  had  by  merely  substituting  for  B  ri 

r  and  Q,  the  values  of  A  c  r,  A  (r— —  )  and  A  Q  already  obtained  in  the  general  expression  (121). 

d  r  \     d  r  / 

But  as  this  process  of  substitution  and  reduction  presents  no  difficulties  in  principle,  requiring  only  patience 
and  exactness  in  performing  the  numerous  combinations  of  the  terms  which  occur  in  it,  we  shall  not  pursue 
it,  but  content  ourselves  with  observing,  that  the  part  of  it  which  depends  on  the  first  powers  of  the  eccen- 
tricities will,  on  examination,  be  found  to  assume  the  form 

e  .  a  t .  cos  V  +  /  .  6  t  .  cos  (W  +  V)  -| 

-1-  e  {A.  sin  (W  +  V)  +  B  .  sin  (W  -  V )  +  C  .  sin  (2  W  +  V)  +  &c.)     V; (157) 

+  e  { A',  sin  V"  +  B' .  sin  (W  —  V)  +  C'.  sin  (2  W  +  V)  +  &c. }   J 

It  is  here  that  we  first  encounter  the  secular  equations  of  the  planetary  motions,  -in  the  form  of  two  terms 
containing  the  time  t  disengaged  from  the  signs  sin  and  cos,  and  therefore  capable  of  indefinite  increase  and 
diminution.  They  are  multiplied  by  the  eccentricities,  and  therefore  originate  from  the  ellipticity  of  the 


694 


PHYSICAL   ASTRONOMY. 


Astronomy,  planetary  orbits  ;  and  in  the  case  of  strictly  circular  orbits,  would  not  exist.     Similar  terms  occur  of  course    Physical 
<s— y— -^  in  the  value  of  8  r,  being  introduced  by  the  integration  of  the  equation  for  6  u.     But  as  the  discussion  of  these  Astronomy, 
terms  is  one  of  the  most  delicate  and  difficult  points  of  the  planetary  theory,  we  shall  not  enter  upon  it  till  we   v—"v~~" 
have  pointed  out  the  method  of  taking  into  account  the  higher  powers  of  the  eccentricities. 

SECTION  V. 
Of  the  inequalities  depending  on  the  squares  and  higher  powers  of  the  eccentricities. 

IT  is  not  our  intention  to  enter  into  any  detailed  account  of  this  very  complicated  part  of  the  planetary 
theory.  Any  such  attempt  would  lead  us  far  beyond  our  proper  limits  ;  and  the  reader,  who  is  desirous  to 
follow  it  into  its  minutiae,  must  consult  the  original  memoirs  of  Laplace,  Lagrange,  &c.,  the  Mecanique 
Ce"lfste,  and  other  works  of  a  similar  nature.  In  the  foregoing  sections  we  have  however  followed,  as  nearly 
as  possible,  the  course  pursued  in  the  last  named  immortal  work,  supplying  only  such  steps  in  the  analysis 
as  cannot  be  expected  to  be  discovered  by  the  ordinary  student,  (and  they  are  numerous)  and  endeavouring 
throughout  to  place  the  principles  of  the  several  processes  in  as  strong  a  light  as  possible.  In  the  present 
section,  the  explanation  of  the  principles  on  which  the  process  of  approximation  is  to  be  pursued  will  be 
almost  our  sole  object. 

Let  us  resume  the  consideration  of  the  function  Q. 

O  =  R  +  R'  .  cos  w  +  R".  cos  2  w  +  R'" .  cos  3  w  +  &c. 
When  r  -f  A  r,  r1  +  A  /,  and  w  +  A  w,  are  substituted  for  r,  r',  w,  in  this,  Q  becomes  Q  +  A  Q,  and  we  have 


A  Q  = 


dO 


A  r 


dO      A/         d  Q     A  u> 


d 

a 

1 

-  -r 

da'        1 

-  T 

dW 

1 

f 

Q 

(A 

r)2 

i 

Ar 

.A 

r' 

4  

Q 

(A 

rV 

d 

o- 

1  . 

2 

d  a  d  a' 

1 

.  1 

'     d 

a'* 

1 

.2 

d3 

Q 

(A 

r)3 

-L    A-o 

+  &c. 


(158) 


dQ 


1.2.3 

The  differential  co-efficients  of  Q  are  here  supposed  to  have  their  circular  values  denoted  by  — — ,     — -—r, 

a  a        a  a 

dQ 

— — -,   &c. 

In  like  manner,  if  we  consider  the  value  of  A    (  r  I ,  or  the  augmentation  of  r  —  —  produced  by  the 

\      dr  /  d  r 

eccentricities,  we  have  only  to  substitute  for  Q  in  the  expression  (158)  the  circular  value  of  the  function  in 

d  Q 
question,  or  a 


da 
dQ 


,  and  we  get 

Ar         d 


d  Q 


(A  r)a 


d* 


1.2 
&c.  .  . 


d  Q 


A  r  A 


d  Q 


d  fl 


(159) 

in  which  — -,     — .,   &c.  denote  the  differentiation  relative  to  a,  a',  &c.  respectively  of  the  function  to  which 
da      da 

they  are  prefixed,  and  the  division  of  the  resulting  differential  by  d  a,  d  a',  &c.  according  to  the  very  conve- 
nient system  of  notation  explained  in  (Lacroix,  Differential  and  Integral  Calculus,  8vo.  English  translation, 
Appendix.) 

Since  Q  =  A  +  A' .  cos  W  +  A" .  cos  2  W  4-  &c.  we  must  have 
dQ 


cos  *  W  +  &c. 


d  / 

1  da 
dQ\ 

•—  a  —  h  a  —  —  -  .1 
da              da 

(•       ^  ^  \        ^  i 

:os   w   -r  a  —  -  —  .  cos  ^  w 
d  a 

(a   dA^cr  W    1      d  ( 

•f   OCC. 

d  A''  \ 

da  \a 
d  I 

da  / 
dQ\ 

da\       da/       da 
_   d  ,     d  A  \       d 

a            1  cos  w  -t-         1 
\       d  a  /                       da\ 

(adV\  ,  ,-  w   |     d  ^ 

''     da  )• 
•    dA"\ 

da'  V 
d  / 

da  / 
dQ^. 

—  j/la     ,       l''j/ 
aa  \      a  a  /       da 

dA' 

V     da  )  i                   daf\ 
d  A" 

0    -                         clr»    O   \V               JUrc. 

."    da  /  ' 
dA"\ 

dW  \a 
*  (a 

da  ) 
dQ\ 

d  a 
d>  /     dA\       d2 

d  a 

(~                  1    ««D    117"      i                 f 

da«    V  " 
d*       ( 
dadVV  \ 

da  / 
dO\ 

da*V      d  a  S  '   da2 
d  f     d  \'\  _ 

da/                       da"  V 

W"         1  n                  1   lin  2 

a    da/ 
W  —  &c. 

da  / 

da\     da    ' 

da\       da/ 

cos2W  +  &c. 


cos  2  W  +  &c. 


(IhO) 


PHYSICAL    ASTRONOMY. 


695 


Astronomy,  and  so  on.     Now    these  series   may  be  regarded  as    completely  developed ;  for   the    co-efficients  of  their    Physical 
"••"V™" •*  several  terms  cos  W,  sin  W,  &c.  are  quantities  completely  given  in  numbers,  when  a,  a',  A,  A',  A",  &c.  and  Astronomy, 
their  differential  co-efficients  are  known,  by  the  equations  ^-~~V~' 

d      i     d  A  \  _   d  A  d2  A          d      /     d  A'\  _  d  A'  d*A' 

;7~ j  ~ ~dT~  +  a  7u*~  '  T^\a~dTl  -^~a~  +  a~Itf'  &c' 


da 

d 

To7' 

(ft 
da* 


d 
dA 
da 
d  A 
d  a 


d2  A 
da  da' 


da 


d 

~d~7 
d3A 


dA 


a  A'  \ 
"  da  /  ~ 


d*A' 
da  d  a 


-,;  &c. 


da3  ' 


,(161) 


If,  instead  of  r  ,  we  had  any  other  function  to  develope  (such  as  for  instance  r*  . \,  we  might 

d  r  \  d2  r  d  // 

treat  it  exactly  in  the  same  way,  and  should  arrive  at  corresponding  series,  in  which  the  sines  and  cosines  of 
W  and  its  multiples  would  be  combined  with  co-efficients  absolutely  constant,  and  reducible  to  numbers. 

In  the  values  of  A  Q  and  A  (r  -jr7-\  in  (158)  and  (159)  we  see  therefore  that  the  differential  co-efficients 

of  Q  introduce  the  sines  and  cosines  of  W,  and  its  multiples,  combined  only  with  given  quantities,  and  not 
involving  the  eccentricities.  These  latter  arise  from  the  factors  A  r,  A  r',  A  w,  and  their  powers.  Let  us  now 
*xamine  these  more  nearly.  Supposing  then,  as  we  have  done  all  along, 

V=nt   +  e-7r;      V  =  «'  t  +  e'  -  ^  ;      W  =  n  t  -  n'  t  +  £  -  e'  • 
let  us  take  a,  ft,  &c.  as  follows  : — 

a'  =   -  a' .  cos  V 


a  =  —  a  .  cos  V 

?  =•  —  (1  —  cos  2  V) 


7  = ('3  cos  3  V  —  3  cos  V) 

etc. 

p  =  2  sin  V 

9  =  —  sin  2  V 

1  13 

r  = sin  V  +   — -  sin  3  V 

4  12 

fee. 


P'=  —  (1  -cos  2V) 

2 

</  =   -  —  (3  cos  3  V  —  3  cos  V) 

o 

&c. 
p'  =  2  sin  V 


'  =  —  sin  2  V 
4 

•  =  _  _1  sin  V'  + 
4 

&c. 


sin  3  V 


Then  we  shall  have,  by  the  equations  (28)  and  (30) 

Ar'rs  oV  +  /3V*  +  <yV3  +  &c.  V- ; (163) 

Au)  =  (pe+    gea   +   re'-j-  &c.)  —  (p'  d  +  <{  e'*  +  T'  e'3  +  &c.).  J 
whence  we  obtain 

(A  r)*  =  «*e*  +  2a/3e3  +  (/32   +  2  a  7)  e4  +  &c. 

A  r  A  /  =  a  a'  e  e'  +  a  /3'  e  e'2  +  /3  a'  e2  e7  +  &c. 

A  r  A  ic  =  p  <z  e*  +  (p  /3  -f  9  a)  e3  +  &c.  —  p'  a  e  e'  —  &c. 

(A  to*)  =  p*  e2  +  2p  q  e3  +  &c.    +  (p'*  e"  +  2  p'  ^  e'3  +  &c.)   —  2  pp'  e  e7  -  &c. 

and  so  on ;  and  it  only  remains  to  substitute  these  values  in  the  expressions  for  A  Q  and  Air I . 

V     d  r  J 

Confining  ourselves  to  A  Q,  since  it  is  obvious  that  the  process  is  exactly  similar  for  the  other  function,  we 
have,  as  before,  for  the  part  depending  on  the  first  powers  of  the  eccentricities, 

d  Q  d  Q 


which,  developed  into  series  of  sines  and  cosines,  gives  the  result  obtained  in  the  last  section. 

The  part  of  A  Q  depending  on  the  squares  of  the  eccentricities,  consists  of  three  terms  multiplied  respec- 
tively by  e*,  eef,  and  e'2,  which  originate,  1st.  From  the  terms  /3  e*,  /JV3,  q  e8,  and  —  cf  ff1,  in  the  simple 
powers  of  A  r,  A  /,  A  w,  and  which  are  therefore  affected  with  the  differential  co-efficients  of  the  first  order 
only :  2dly.  With  the  terms  a*  e*,  a'*  e'3,  p*  e*,  p'»  e'*,  and  —  2  pp'e  </,  in  (A  r)s,  (A  /)*,  and  (A  u>)»,  and 
•which  are  consequently  affected  with  differential  co-efficients  of  Q  of  the  second  order. 
VOL,  HI.  4  x 


696  P  H  Y  S  I  C  A  L    A  S  T  R  O  N  O  M  Y. 

\stronijtny       Sdly.  With  the  terms  a  p  e9,  a' p' e'-,  a  a' e  a',  a  p'  e  e',  a'  p  e  e',  which  arise  from  combinations  of  A  r  with     Physical 

^— v '   A  «  and  with  A  r'.  Astronomy. 

The  aggregate  of  all  these  terms,  with  their  proper  co-efficients,  is 

Q                d  Q  \ 
+  2  o I   4- 

'  I  I    ITT      I  ' 


da  *  d  W  /         V       da*  '    d  a  d  W        r  d  W2 


d2  Q  ,   d2  Q 

d  a'  d  W    ~  PP    dW* 


,f    ,    d2  n  ,    d2  a 

+  CTa    dTda-'-aXdTdW+a/) 

e™  ( /          dQ  ,  d  n  \        /  .,  d*  Q  d2  Q 

-I <  (2  /3 2  o  1  +  I  of*  • 2  a  o  • — • — — h  » 

+   2   H    Pda'  *dW/^\      75»  Pda'dW+P 


(164) 


d  Wa 

The  co-efficients  of  e3,  e2  e',  e  e"1,  e'3,  and  of  the  higher  powers  and  combinations,  may  in  like  manner  be 
easily  obtained ;  but  the  number  of  terms  of  which  they  consist,  goes  on  increasing  so  rapidly,  that  they  at 
length  become  of  extreme  complexity. 

d  O 

Let  us  now  consider  the  nature  of  the  terms  into  which  the  expressions  for  Q  and  r  — —  resolve  fhem- 

selves  by  the  process  of  developement,  and  the  manner  in  which  they  become  modified  by  the  several  pro- 
cesses of  substitution  and  integration  they  have  to  undergo  in  obtaining  the  values  of  Q,  IT,  S  u,  &  r,  and  S  0. 

It  is  evident,  then,  since  a,  ft,  7,  a',  ft',  &c.  are  all  composed  of  cosines,  and  p,  q,  p',  <?',  &c.  of  sines  of  V, 
V,  and  their  multiples,  without  W,  that  any  product  or  combination  of  these  letters,  (such  as  a2,  ap,  a  a', 
&c.)  is  reducible  by  the  trigonometrical  formulae  so  often  employed  in  the  foregoing  pages  into  the  simple 

1  4-  cos  2  V 
sines  or  cosines  of  arcs,  of  the  form  k  V  +  I  V.     Thus,  o2  or  aa  .  cos  V2  becomes  a*  .  -   — — ,   a  p,  or 

—  2  a  .  sin  V  .  cos  V  becomes  —  a  .  sin  2  V,  a  a'  or  a  a' .  cos  V  .  cos  V  is  reduced  to  — —  (cos  (V  +  V)  + 

cos  (V  —  V')),  ap'  into  a  .  sin  (V  —  V)  —  a  .  sin  (V  4-  V),  and  so  on.  Moreover,  it  is  evident,  that  when- 
ever the  combination  in  question  consists  only  of  the  letters  a,  ft,  a',  ft',  &c.  or  of  these  combined  with  any 
product  of  an  even  dimension,  in  p,  q,  p',  q',  &c.,  that  the  terms  into  which  it  is  resolved  will  consist  entirely 
of  cosines  ;  but  when  a  product  of  an  odd  dimension  in  p,  q,  p',  q',  &c.  occurs,  then  of  sines.  Now,  the 
differential  co-efficient  of  Q  combined  with  any  such  product,  will,  in  the  former  case,  evidently  ve  differen- 
tiated an  even,  and  in  the  latter  an  odd  number  of  times  relatively  to  W  ;  so  that  in  the  former  case  it  will 
represent  a  series  of  cosines,  and  in  the  latter,  of  sines  of  W. 

Every  term  therefore  formed  by  such  combination,  must  be  of  one  or  other  of  the  forms 

cos  i  W  .  cos  (k  V  +  I V)  and  sin  i  W  .  sin  (kV  ±1  V) 
both  which,  being  further  resolved,  produce  terms  comprehended  in  the  form 

cos  (iW  +  kV  ±  IV) 

It  is  thus  demonstrated,  that  the  co-efficients  of  all  the  powers  of  e,  ef,  in  the  developements  of  D  and 
r  • — —  are  generally  reducible  to  series  of  cosines  of  arguments,  of  the  form  z  W  +  k  V  +  I V  ;  but  there  is 

a  connection  between  the  multiples  of  V  anil  V  contained  in  any  argument,  and  the  dimension  of  the  power 
or  product  of  the  eccentricities  to  which  it  belongs  that  we  rmist  now  explain.  In  fact,  it  is  obvious  from 
the  process  above  pursued,  that  if  we  regard  a,  a.  p,  p',  as  quantities  of  one  dimension  ;  ft,  ft',  q,  q', 
as  of  two;  7,  7',  r,  r  of  three,  and  so  on,  the  dimension  of  every  term  multiplied  by  e,  or  </,  wi'l  be 
one;  that  of  the  terms  multiplied  by  e^,e^,  e'2,  will  be  two,  and  soon.  Now,  the  expressions  of  these 
quantities  in  V  and  V  involve,  each,  the  sines  or  cosines  of  multiples  of  V,  V,  as  far  as  the  number  expressing 
its  own  dimension  ;  and  when  these  come  to  be  combined  by  multiplication,  and  then  resolved  by  the  usual 
formula,  it  is  obvious  that  the  resulting  terms  of  the  form  cos  (k  V  +  I  V)  and  sin  (k  V  +  /  V)  can  only 
contain  such  multiples  kV  and  IV  of  V,  V,  as  together  (without  regard  to  their  signs)  do  not  exceed  the 
dimension  of  the  combination  from  which  they  arose. 

Moreover,  since  the  alternate  multiples  of  V,  V  are  absent  in  the  expressions  of  a,  ft,  &c.  the  same  law  will 
hold  good  in  any  combination  of  them  when  developed.  Hence  we  may  state  it  as  a  general  law,  that 

The  co-efficient  of  any  power  or  product  of  the  eccentricities  of  the  dimension  n,  in  the  developement  of  O  or  r  — — 
will  consist  of  a  series  of  cosines,  tlte  form  of  whose  argument  is 

iW+  kV  ±  IV 

in  which  i  may  have  every  possible  value  from  o  to  infinity,  lut  k  and  1  are  restricted  to  certain  particular  values,  vi-e. 
those  which  satisfy  one  or  other  of  the  equations 

ic  +  I  =  n,  k  +  I  =  n  —  2,  k  +  I  =  n  —  4,  &c. 

down  to  k  +  I  —  1 ,  or  k  +  I  =  o,  according  as  n  is  odd  or  even. 

Thus,  as  we  have  already  seen,  the  parts  independent  of  the  eccentricities  consist  of  terms  of  the  form  co.« 


PHYSICAL    ASTRONOMY.  697 

Astronomy,  t  W  only,  and  those  depending  on  the  first  powers  involve  the  arguments  Physical 

' v i  W  +  V,      i  W  -  V,      i  W  +  V,      i  W  -  V  Astronomy. 

and  no  other.  Similarly,  in  the  part  dependent  on  the  squares  and  product  of  the  eccentricities,  the  arguments 
which  can  occur,  are  only 

|W,  iW  +  2  V,  i  W-  2V,  iW  +  V+V,  iW  +  V-  V,  i  W  -  V  +  V,  i  W  -  V  -  V,  iW  +  2V 
and  i  W  —  2  V,  and  so  on. 

Here,  it  will  be  observed,  we  have  again  the  argument  i  W,  which  occurred  in  the  part  independent  on  the 
eccentricities,  and  it  is  easily  seen  to  be  a  general  law,  that  any  particular  argument  which  first  occurs  combined 
with  a  power  or  product  of  the  eccentricities  of  the  dimension  n  will  occur  again,  combined  with  products  of  the  dimensions 
„  .(-  2,  ?i  +  4,  &c.  to  infinity,  but  not  with  n  +  1 ,  n  +  3,  &c.  For  instance,  the  argument  i  W  +  3  V  cannot 
occur  combined  with  any  dimension  of  the  eccentricities  less  than  the  third,  and  will  occur  again  in  the  terms 
multiplied  by  the  5th,  7th,  &c.  dimensions,  but  not  by  the  4th,  6th,  or  any  even  dimensions. 

Since  W  =  n  t  —  n't  +  e  -  e',  V  =  n  t  +  e  —  ir,  and  V  =  n' t  +  e  —  ir",  the  argument  iW  +  kV  +  IV 
is  equal  to  (i  +  k)  .  n  t  —  (i  —  1)  n'  t  +  (i  +  k)  <•  —  (i  —  1)  e'  —  k  IT  -  I  ir1 

If  then  we  would  inquire  in  what  terms  any  proposed  combination  of  n  and  n',  such,  for  instance,  as  (fn  —  g  n')t 
can  originate,  we  have  only  to  put 

i  +  k=f,         i-l  =  g 

which  give  i  =  f  —  k,         i  —  i  -  g  =  (f  —  g)  —  k 

taking  then  in  succession  k  =  o,  k  =  +  1,  k  =  +  2,  &c.  we  get 

i  =f,  t  =/+!,  i  =  /+2,&c. 

l=f-g,      l=f-g  +  l,       Z=/-g  +  2,  &c. 

For  instance,  if  we  would  know  from  what  terms  the  combination  (2  n  —  n')  t  can  originate,  the  corres- 
ponding values  of  i,  k,  I,  are 

,,t.i  =  9,  h  =  o,  I—.,  2dly.{<  =  j:  *Z^lf  \=-\  3dly.{;  =  -  J=  *  ^  \=~\ 
So  that  any  of  the  arguments  comprised  in  the  following  series, 

(      W+V-4V;       2V    -5V; 

~3V>    \3W  _  V  -  2V;     4W-2V-V;     5  W  -  3  V,  &c. 

will  produce  the  combination  in  question.  Now,  the  lowest  sum  of  the  co-efficients  of  V,  V  in  these  argu- 
ments taken  without  regard  to  their  signs,  is  3  :  consequently,  the  combination  2  n  i  —  5  n't  will  first  occur 
among  the  inequalities  multiplied  by  the  cubes  or  products  of  three  dimensions  of  the  eccentricities,  and 
among  them  only  in  such  terms  as  produce  the  arguments 

2  \V  _  3  V,     3  W  -  V  -  2  V,     4  W  -  2  V  -  V,     5  W  -  3  V. 

Let  us  now  examine  the  co-efficients  of  the  several  arguments  as  they  occur  in  the  values  of  Q,  n,  &c. 

d  n 
The  co-efficient  of  any  argument,  such  as  iW  +  k  V  +  IV  in  the  developement  of  Q  or  r  — —  will 

obviously  consist  only  of  combinations  of  a,  a',  A,  A',  A",  &c.  and  their  differential  co-efficients  with  a 
power  or  product  of  e,  e',  and  may  therefore  be  regarded  as  a  given  quantity,  and  its  value,  with  more  or  less 
trouble,  numerically  computed.  Taking  A  for  the  general  representative  of  such  a  combination,  M  will  be  a 
function  of  a,  a,  e,  e',  and  of  these  only,  and 

Mcos  (»W  +  kV  ±  IV) 

will  be  the  general  form  of  any  term  of  Q  or  r— — . 

In  the  value  of  Q,  the  terms  of  r  — —  enter  unchanged ;  but  since 

d  W  =  ndt,     dV  =  ndt,     dV  =  o 
the  term  under  consideration  will  produce  in  d  Q  the  term  —  M  .  (i  +  k)  n  d  t  .  sin  (i  W  +  k  V  +  /  V)   and 

inQ'the  term  ,.2(ir*)n,   .Mcos  (iW  +  kV  +  IV) 

(i  +  k)  n  +  I  n 

so  that  Q  will  contain  two  species  of  terms,  those  whose  co-efficients  are  of  the  form  M,  and  M 


(i  +  k)  n  +  I  n 
The  value  of  Q  substituted  in  II  (equation  128)  will  produce  terms  comprised  in  one  or  other  of  the  forms 

M.cos  (iW  +  fcV  +  IV) 

,.  cos  (iW  +  VV±  IV) 


.f.  , 

(i  +  k)  n  +  In 

MT  ...  M"    a.  /  ,.cos(iW±feV  +  IV) 
(z  +  k)  n  +  In 

M.  -  ('  ±  k]  "*  --  r.cos(iW  +  fV  +  IV) 
\(i±k)n±ln'}    {(i±/0»±J«'} 

4x2 


698  P  H  Y  S  I  C  A  L    A  S  T  R  O  N  O  M  Y. 

\sironoray.in  the  process  of  integration  by  which  £  u  is  derived  from  n  (or  in  the  integration  of  the  equation  (J29))     physical 
^  ~Y    "  '  \  .  \  Astronomy. 

these    terms   again   acquire  factors  of   the   form   -  :  --  or  —  - 

{  (i  ±  k)  n  ±  In'  {»  —  n*          {  (i  +  kj)  n  ±  I  n'  }«  —  n* 

according  as  k  or  IS  occurs  in  the  argument  ;  and  as  these  forms  are  obviously  not  altered  in  the  transition 
from  &  u  to  dr,  the  terms  of  S  r  will  necessarily  be  included  in  one  of  the  forms 

M.-:  --  l-  -  -  -  cos  (i  W  +  fc  V  +  IV) 
{(i  ±k)  n  +  Zn'}2-H2 


M.  -  (l  ±  y)  "  -  -cos  (iW  +  *V  +  IV) 
\'  '"         2 


\(i±k)n±ln'}  {(i±k)n±l  »')"  -  «2 


M.  -  -  -  -  -  cos  (tW  +  kV  +  IV) 
{(i  ±k)n  ±  ln'\  {(i±k)n±ln')*-n*} 

M  .  -  ('  ±  kJ}  "'  -  cos  (i  W  +  k  V  +  I  \', 
{(i±k')n±  In'}  {(i  ±k)n±lnf}  {  (i  ±  k)  n  ±ln')*  -  n*} 

Let  the  several  functions  of  n,  n',  in  the  co-efficients  of  these  terms  be  represented  indiscriminately  by  N, 
then  will  the  general  form  of  the  terms  of  S  r  be  M  .  N  .  cos  (i  W  +  k  V  +  /  V). 

It  remains  only  to  consider  the  nature  of  the  terms  of  which  K  0  consists.     Now  these  will  be, 

1st.  Those  arising  from  the  term  —  6  r,  which  are  of  the  form 

M    V  IL    N^ 

-         *—  sin  (i  W  +  (k  ±  k")  V  +  I  V) 
n  a4  v  1  —  e- 

2dly.  Those  arising  from  r  —  —  ,  whose  form  is 

a  t 

2  M  {  (J  +  *)  n  +  I  n'  '  N 

=-  —  -  —  sin  (i  W  +  (k  ±  k")  V  +  IV) 


n  a2 


/"     d  Q  /V* 

3dly.  Those  arising  from  /  r  — '—  n  ill  and  //  d  Q  .  n  dt  whose  forms  are  respectively 

2aM                        n                                                                  3  r<  M                (i+k)n°-  /-iirL/irj   i»v\ 
-   •    ,.          • —         ,  sin  (i  W  +  fr  V  +  /  V) ,  and .  —        —  — —  sin  (i  W  +  k  V  ±  If  V) 

and  lastly  those  peculiar  terms  containing  /  out  of  the  signs  sin  and  cos  which  we  have  already  noticed  as  giving 
rise  to  the  secular  equations. 

The  complete  enumeration  of  all  the  possible  varieties  of  terms  which  SO  may  contain,  will  therefore  be 
had  by  putting  for  N  each  of  the  four  forms  above  assigned  to  it ;  but  as  those  only  really  differ  importantly 
in  which  the  denominators  of  the  fractions  differ,  we  need  only  enumerate  the  latter  quantities  ;  which  are, 

(i  ±  k}n  ±  In';      {  (i  ±  k)  n  ±  t  n' }a  ; 

{  (i  ±  k)  n  ±  I  n'  [4  _  n'-  ;     ((i  ±  k')  n  ±  I  n')  {((i±k)n  ±1  n')2  -  n* \ 
{(i  ±  V)  n  ±  In'}  {  (i  +  k)  n  ±  In'}  [((i  ±  k)  n  ±  In')*  -  n2} 

These  then  are  the  various  forms  of  the  divisors  with  which  the  processes  of  integration,  &c.,  affect  the 
inequalities  in  longitude.  They  are,  as  we  have  already  remarked,  of  the  highest  importance  in  the  theory 
of  the  planets,  by  reason  of  their  effect  on  the  numerical  values  of  the  maxima  of  the  perturbations  to  which 
they  belong.  Such  is  the  immense  number  of  terms,  or  rather  of  series  of  terms,  branching  out  in  all  direc- 
tions, of  which  the  perturbations  consist,  that  it  is  manifestly  in  vain  to  attempt  to  take  account  of  them 
all.  It  is  therefore  of  the  highest  consequence  to  have  some  guiding  principle  to  direct  us  in  our  choice  of 
the  terms  to  be  retained  or  neglected.  Were  it  not  for  these  divisors,  we  might  safely  rely  on  the  rapid 
convergency  of  the  powers  and  products  of  the  eccentricities;  and  reject,  without  further  examination,  all  in 
which  their  dimension  exceeded  a  certain  limit ;  but  should  there  be  an  approach  to  commensurability  in 
the  periodic  times  of  the  two  planets,  (as,  for  instance,  should  five  times  the  mean  motion  of  the  disturbed 
planet  (5  n'  t)  be  very  nearly  equal  to  twice  that  of  the  disturbing,  (2  n  t))  this  circumstance  will  render 
some  one  of  their  factors  (5  n'  —  -2  n)  very  small.  In  consequence,  all  the  divisors  into  which  this  factor 
enters  will  become  very  small,  and  the  inequalities  affected  by  them  will,  in  consequence,  acquire  from 
this  cause  an  unnatural  magnitude  (if  we  may  use  such  an  expression)  and  must  be  retained,  even  though 
of  such  an  order  as  would  otherwise  authorize  their  rejection. — The  terms  so  affected  too,  will  originate  in 
a  great  variety  of  manners  from  the  developements,  and  may  be  affected  with  various  powers  of  the  eccen- 
tricities ;  so  that  their  number  will  necessarily  be  infinite,  and  for  the  purpose  of  approximation  only  the  most 
prominent  can  be  selected. 

The  equations  of  the  motions  of  Jnpiter  and  Saturn,  known  by  the  name  of  the  great  inequalities  of  these 
planets,  were  long  a  difliculty  in  the  way  of  the  theoretical  astronomer,  and  even  a  stumbling  block  in  the 


PHYSICAL     ASTRONOMY.  699 

Astronomy,  way  of  the  Newtonian  philosophy.  It  was  observed,  on  comparing  very  ancient  observations  of  the  oppositions     Physical 
'  of  these  planets  with  more  modern  ones,  that  their  mean  motions  had  undergone  an  apparent  alteration  ;  that  of  Astronomy. 
Saturn  appearing  to  have  been   retarded,  and  that  of  Jupiter  accelerated.     In  other  words,  that  Saturn  per-  "— — 
petually  lagged  behind,  and  Jupiter  as  constantly  surpassed  the  places,  when  they  ought  to  have  been,  on  the 
hypothesis  of  the  mean  motion,  or  periodic  time,  remaining  invariable. 

We  have  seen  that  every  inequality  of  very  long  period  will  appear,  while  on  the  increase,  to  affect  the 
mean  motion  ;  it  is  obvious  it  must,  if  the  latter,  as  determined  by  observations  comprised  within  the  periods 
of  its  increase,  be  compared  with  the  result  of  similar  observations  made  while  its  value  is  on  the  diminu- 
tion. Now,  the  length  of  the  period  of  any  inequality  depends  on  the  multiples  of  the  mean  motions  found 
in  its  argument ;  and  it  was  not  difficult  for  geometers  to  shew,  that,  so  far  as  the  rirst  powers  or  squares  of  the 
eccentricities  were  concerned,  no  inequalities  of  such  very  long  periods  as  the  case  required,  could  be  found 
in  the  motion  of  either  planet.  The  cubes  and  higher  powers  had  all  along  been  neglected  without  fear  of 
error  ;  but  Laplace  having,  from  other  considerations,  ascertained  that  an  acceleration  in  Jupiter's  motion  being 
supposed,  a  retardation  in  Saturn's  must  follow  of  course,  and  that  in  the  very  proportion  observed  ;  and  that 
therefore  the  phenomenon  was  not  altogether  inconsistent  with  the  laws  of  gravity,  set  himself  to  examine  the 
terms  multiplied  by  the  cubes  of  the  eccentricities.  Here  he  immediately  encountered  the  argument 
(5  n  t  —  2  n't  +  const.)  ;  and  the  mean  motion  of  Jupiter  being  to  that  of  Saturn  nearly  in  the  proportion  of  5 
to  2,  if  we  suppose  n'  to  correspond  to  Jupiter's  and  n  to  Saturn's  motion  the  co-efficient  5  n — 2  n'  is  very 
small,  and  the  corresponding  period  is  found  on  calculation  to  amount  to  918  years.  The  resulting 
inequality  has  also  5  n  —  2  n'  for  its  divisor,  and  its  magnitude  is  thus  increased  as  well  as  its  period  lengthened. 
On  executing  the  calculation,  the  inequalities  of  both  planets  were  found  to  be  such  as  would  completely 
account  for  the  apparent  accelerations  and  retardations  observed. 


SECTION  VI. 

Of  the  variations  of  the  elements  nf  the  planetary  orbits,  and  the  secular  equations  of  their  motions.     Theory  of  the 
major  axes,  inclinations,  nodes,  eccentricities,  ciiul  aphelia. 

WE  have  already  taken  occasion  to  observe,  that  the  motions  of  the  planets  may  be  regarded  as  performed 
in  ellipses,  whose  positions  and  magnitudes  are  continually,  but  very  slowly,  changing,  by  the  effects  of  the 
disturbing  forces.  These  forces  are  so  small,  that  in  a  moderate  period  of  time,  as  for  instance,  in  a  single 
revolution  of  a  planet,  the  change  is  insensible  ;  and,  if  we  allow  for  those  inequalities  which  depend  on  the 
configurations  of  the  disturbed  and  disturbing  planets,  the  theory  of  which  has  been  exposed  in  the  foregoing 
sections,  the  motion,  equated  by  the  application  of  these  corrections,  will  coincide  with  almost  rigorous  exact- 
ness with  the  elliptic  theory.  But  after  the  expiration  of  many  revolutions  of  the  planet,  this  exact  coinci- 
dence will  cease  to  take  place,  even  when  its  place  is  corrected  for  such  periodical  inequalities.  The  place  so 
corrected  is  found,  it  is  true,  in  the  circumference  of  an  ellipse  with  the  sun  in  its  focus,  but  it  is  not  an  ellipse 
of  precisely  the  same  form  and  position  as  before.  Its  elements  have  undergone  a  change,  and  this  change, 
though  imperceptible  in  a  single  revolution,  becomes  gradually  more  and  more  evident,  till  at  length  it  is  too 
remarkable  to  be  overlooked. 

Such  slow  changes  are  what  we  understand  by  the  secular  variations  of  the  elements  of  the  planet's  orbits. 
But  there  is  another  point  of  view  in  which  we  may  consider  the  subject,  which  presents  peculiar  facilities  to 
the  application  of  mathematical  investigation.  It  consists  in  referring  all  the  inequalities  resulting  from 
perturbation,  to  the  variation  of  the  elliptic  elements,  not  merely  those  of  long  periods,  but  those  which  pass 
rapidly  from  their  maxima  to  their  minima,  and  depend  on  the  configurations  of  the  bodies.  We  are  indebted 
to  Lagrange  for  this  view  of  the  subject,  and  shall  endeavour  to  give  an  idea  in  this  section  of  the  luminous 
analysis  of  that  great  geometer. 

If,  at  the  expiration  of  any  instant,  the  disturbing  forces  were  to  cease  acting,  the  planet  would  go  on 
describing  an  exact  ellipse,  of  which  the  infinitesimal  arc  described  in  the  last  instant  would  l>e  an  elementarv 
portion.  The  plane  of  the  ellipse  would  be  that  in  which  this  portion  arid  the  sun's  centre  lie  ;  its  eccentri- 
city, position,  and  magnitude,  would  all  be  determined  from  the  position,  magnitude,  and  curvature  of  this 
element,  and  the  laws  of  elliptic  motion.  In  a  word,  it  would  be  a  real  ellipse  of  curvature  to  the  actual  curve 
described  by  the  planet,  at  that  particular  instant,  subjected  to  the  conditions  of  having  its  focus  in  the  sun, 
and  satisfying  the  other  laws  of  elliptic  motion  during  that  moment.  The  elements  of  the  planet's  orbit  then, 
at  any  moment,  are  no  other  than  the  elements  of  this  ellipse,  and  are  determined  from  three  consecutive 
places  of  the  planet,  infinitely  near  each  other.  Thus  every  inequality  in  its  motion  will  produce  a  corres- 
ponding fluctuation  in  the  elements,  which  will  thus  be  subject  to  as  many  equations,  periodical  or  otherwise, 
as  the  planet's  motion  itself  is  affected  with. 

The  periodical  terms  thus  originating  in  the  elements,  will,  in  the  course  of  many  revolutions,  compensate 
each  other ;  but  if  it  should  happen  that  terms  not  periodical  should  find  their  way  into  their  values,  these 
will  express  secular  changes,  which  it  becomes  of  the  utmost  importance  to  investigate. 

To  determine  the  ellipse  of  curvature  at  any  instant,  is  a  matter  of  no  difficulty  ;  we  have  onlv  to  call  to 
mind  that  any  one  of  the  constants  in  its  equation  may  be  insulated,  and  expressed  in  terms  of  the  co-ordi- 
nates and  their  d'fFerential  co-efficients,  by  the  mere  operations  of  differentiating  and  eliminating;  so  that, 


700  P  H  Y  S  I  C  A  L    A  S  T  R  O  N  O  M  Y. 

Astronomy  supposing  the  co-ordinates  and  their  differential  co-efficients  given  at  any  instant,   any  one  of  the  constants     Physical 
•—  -y-^-  may  have  its  value  ascertained  by  simple  substitution.     But  it  is  not  necessary  to  go  through  these  processes  Astronomy. 
—  we  may  avoid  that  trouble  by  recurring  to  the  origin  of  the  ellipse  itself.     Now  all  we  know  of  it  is,  that  it  ' 
satisfies  the  dynamical  relations  of  the  problem,  on  the  supposition  of  the  disturbing  forces  ceasing  to  act  at 
the  instant  dt.     Consequently  its  equations,  however  transformed,  must  be  such  as  to  satisfy   the  differential 
equations 

d2  x         fix 

TF  +  ^  =  0 


dt*  T3 

<PZ        ,      /'Z    _    - 


d  t* 

and  its  elements  will  be  the  constants  introduced  by  the  integration  of  these,  or  known  functions  of  them. 

For  instance,  let  us  consider  its  major  semiaxis.  If  we  pursue  with  these  the  same  process  of  integration 
by  which  equation  (109)  was  obtained,  viz.  multiply  the  first  by  dx,  the  second  by  Ay,  and  the  third  by  dz 
and  add,  and  integrate,  we  find 

fi  2  fi  d  ,r2  +  (/  ?/2  4-  d  z2 

a    '         r  d  <2 

n  being  the  arbitrary  constant  introduced  by  integration  ;  and  if  we  compare  this  with  (33),  we  shall  see 
that  a  is  the  semiaxis  of  the  ellipse.  Thus  we  know,  that  all  that  is  necessary  to  obtain  the  semiaxis  of 
the  ellipse  of  curvature  at  any  instant  (let  the  body  describe  what  curve  it  will)  is  merely  to  substitute  for 

d  x         d  y         d  z     .      , 
r,  --  ,     —  —,     -  ,  in  the  expression 
d  t          d  t         d  t 


those  values  which,  in  the  case  proposed,  they  actually  have  in  virtue  of  the  real  motion  of  the  body,  such  as 
the  forces  in  action  make  it. 

Now,  in  the  case  of  disturbed  motion,   dx,  dy,  dz,  are  given  by  the  equation  (94)  ;  for  if  we  integrate 
,hese  after  multiplying  them  respectively  by  %  d  x,  2  d  y,  <Zdz,  we  find 

fVxdx  f  d  O 

-d7"* 

d  Q 


/ 

(.- 


n 


dz 
consequently  adding  all  together, 

/d.r\4       /rf'/\a       /dr\2  f'Zxdx  +  'Zydy  +  2zdz  .  /)  d  Q    ,  dQ,  dQ     ,     , 

f 1    +  ( — -  I    =  —  ji  / -, — : — '• 2  m  I  ( o  x  -\ a  y  -\ d  z  ] 

\dt)         \dtJ\dt}  rj  r3  J  \  dx  dy  rl.i          ) 

-  2  m'  /d  Q 
and  substituting,  we  get 

a  =  ?- ,  or    £-  =  2  TO'  A  Q  ;  (165,  1) 

2m'/dQ 

The  same  result  will  be  obtained  as  follows  : — If  we  integrate,  as  in  (109),  and  instead  of  adding  explicitly 

u  f* 

the  arbitrary  quantity  —  to  complete  the  integral,  regard  it  as  included  under  the  sign  /  we  have 


but  if  a  be  the  major  semiaxis  of  the  ellipse  of  curvature, 

jt_        1  n  _  d  xa  +  d  y*  +  d 

~         ~ 


a  dt" 

Hence  we  get 


A  «  8  rf  /d  Q 
a  J 

If  (a)  be  the  major  semiaxis  at  the  commenr-ement  of  the  time  t,  and  /  d  Si  be  taken,  so  as  to  vanish  when 


PHYSICAL    ASTRONOMY.  70! 

Astronomy,  f  =  o,  we  have  p  /*        ,    „  „//*,,  o  lifi-    O\     Ph>"sical 

_r-     -«_-  "r*  =  ~T~;  --  h  »  »»     /   U  U  (lOo.  a>  Astronom/ 

In  this  instance  we  have  had  no  difficulty  in  arriving  at  once  at  the  finite  expression  for  the  varied  element. 
But  it  is  in  other  cases  more  commodious  to  express  its  momentary  variation.  Let  us  therefore  denote  by  the 
characteristic  S,  that  peculiar  variation  by  which  the  ellipse  of  curvature  passes  from  the  form  and  position  it 
had  during  d  t  to  that  which  it  has  in  the  consecutive  instant,  then  S  a,  S  e,  S  TT,  &c.  will  be  the  momentary 
variations  of  its  semiaxis,  eccentricity,  perihelion,  &c.  Moreover,  d  x,  c  y,  o  z,  and  S  r,  will  represent  the 
jxcesses  of  the  values  of  x,  y,  z,  r,  in  the  varied  ellipse  ;  not  over  what  they  were  in  the  former  instant,  but 
.)ver  what  they  would  have  been  had  that  ellipse  remained  unaltered.  But,  both  in  the  one  case  and  the  other, 
he  point  in  the  ellipse  to  which  they  correspond,  coincides  with  the  real  place  of  the  planet.  Hence  the 
ines  x,  y,  z,  r,  are  the  same  in  the  varied  ellipse,  in  the  unvaried,  and  in  the  actual  curve  described  ;  so  that 

d  x         d  y         d  z 

5  r  =  o,  S  y  =  o,  S  z  —  o,  and  Z  r  =  0.     Again,  S  —  -  —  ,  6  —  —  ,  S  —  —  ,    are  the    excesses    of  the   values  of 

d  t          at          d  t 

•  -  ,     —  —,     -  ,  in  the  varied  ellipse  over  what  they  would  have  been  had  the  ellipse  not  varied  —  that 

d  t          d  t          d  t 

s,  had  the  disturbing  force  not  acted,  —  in  the  instant   consecutive   to  d  t.     Their  values  therefore  would 
vanish,   had  the    body   remained    in    its  former  ellipse,  and,  in  general,    will    be  obtained  by   subtracting 

from  the  values  actually  assumed  by  —  -  —  ,  &c.  in  the  consecutive  instant  in  the  curve,  what  would  have  been 

ct  t 

assumed  by  them  had  the  body  continued  in  the  same  ellipse,  or  had  the  disturbing  forces  ceased  to  act  at  the 

d  x  \ 

end  of  d  t.     Now,  in  the  curve  the  consecutive  value  of  —  r  —  is 


//  t 


dt 

d  x  d  x          d  x         (  ft-  x  ,  d  fi 

4-  d = ^ r  m  

dt  dt  dt          1    r3  dx 

because-^-  +  m'— —  being  the  force  in  the  curve,  we  must  have  d  — — -  =  —  \  — \-  m'  —-^->  dt.   On 

r3  dx  d  t  (    r3  d  x  ) 

the  other  hand,  had  the  disturbing  force  ceased  acting,  we  should   have  had  simply  d  -—  =  -    — —  d  t, 

d  x                                              d  x          ft,  x 
so  that  the  consecutive  value  of  — —  would  have  been  merely  -j-j —  d  t. 

Hence  we  have  g  _^ 

d  t  dx    '        1 

,;  „  jo 

(166) 


d  t 
and  similarly  s    dy_ 


d  t 

d  z 
S  - 

dt  dz 

To  explain  how  the  variations  so  obtained  may  be  employed,  let  us  take  aeain  the  case  already  treated. 

P_         2  n        d  x*  +  d  y2  +  d  z3 

a  r  d~F 

If  we  differentiate  this  relative  to  the  characteristic  S,  we  get 
ft  S  a  2  /a,  8  r 


r 


_  Q  f  dx    s    dx          dy         dy          dz         dz  } 
"\dtdt          dt    "    dt    "   dt         dt   } 


in  which,   putting  S  r  =  o,  and  for  S  —  —  &c.,  their  values  above  found,  we  find 

((.   t 

[it  a  ,  (  dfi  rfQ  rfQ 

-=+2m'l     —dx  +  -  —  dy+-  — 
(P  (  dx  dy  d  z 

6  a  is  the  momentary  variation   of  a  in  the  instant  d  t,  so  that  this  equation  may  be  integrated  relative  to  t, 

and  we  get  u.  ,   /• 

•£.**«  m'  /dO 
a  J 

the  same  result  as  before. 

Before  we  proceed  farther,  we  will  stop  to  draw  from  this  expression  of  the  reciprocal  axis,  a  most  impor- 
tant conclusion.  It  is  this  —  that  all  the  variations  to  which  the  major  axes  of  the  planetary  orbits  are  sub- 
jected by  their  mutual  attraction  are  periodical  —  and  that  the  mean  distances,  and  consequently  the  mean 
motions  of  the  planets  are  subject  to  no  secular  variations.  In  fact,  when  we  consider  only  the  first  power  of  the 
disturbing  forces,  we  have  already  proved  that  the  developement  of  Q  is  entirely  composed  of  terms  of  the  form 

A  .cos  (iW  +  kV  +  IV) 
hence  /    d  \V  d  V 

d°  =  A  '  +  k  'sn 


=  A.(i+  /,-)  .  sin  }  (J  +  t)  n  t  —  (J  -  A  n'  t  +  (i  +  k)  c  -  (t  —  I)  c'  -  k  *  —  I  x*  \ 


702  PHYSICAL    ASTRONOMY. 

Agronomy,  which   is  always  periodic  unless  i  +  k  =  o  and  i  —  I  —  o,  when  the  argument  becomes  simply  i  (ir  —  **)  a     Physical 
"— "V"''  constant  quantity ;  but  in  this  case  the  whole  term  vanishes   by  the  disappearance  of  its  co-efficient.     Thus  Astronomy, 

f* 
d  Q  and,  of  course,  /  d  Q  contains  no  term  multiplied  by  t,  and  none  but  what  is  periodic  j  consequently, 

u, 

—  is  periodic  also, 
a 

This  beautiful  result,  the  demonstration  of  which  is  of  almost  elementary  simplicity,  assures  us  of  the 
impossibility  of  any  of  the  bodies  of  our  system  ever  leaving  it  in  consequence  of  the  disturbances  it  may 
experience,  and  secures  the  general  permanence  of  the  whole,  by  keeping  the  mean  distances  and  periodic 
times  perpetually  fluctuating  between  certain  limits  (very  restricted  ones)  which  they  can  neither  exceed  nor 
fall  short  of. 

Let  us  next  consider  the  variation  in  the  position  of  the  plane  of  the  disturbed  orbit.  The  equations 
(4),  (5),  and  (6),  give 

dx  dy  d i  d  z  dy  d  z 

=  ~--        h  = z - - x ->    h = 


dt  dt  d  t  dt  '  '    dt          y    d  t 

These  quantities  in  the  case  of  elliptic  motion  are  constant,  but  in  that  of  disturbed  motion  they  will 
equally  hold  good,  if  h,  h',  h",  be  regarded  as  variable.  Now,  either  on  the  one  or  the  other  supposition,  if 
we  multiply  the  first  by  z,  the  second  by  —  y,  and  the  third  by  x,  and  add,  we  get 

hz  —  h'y  +  h"  x  =  o 

which  is  the  same  with  equation  (7).  But  if  we  mutiply  the  first  by  d  z,  the  second  by  —  d  y,  and  the  third 
by  d  x,  we  shall  also  obtain  by  addition, 

h  d  z  —  h'  d  y  +  h"  d  x  =  o 

Consequently,  even  when  we  regard  h  h'  h"  as  variable,  still  the  equation  h  z  —  h'  y  +  h"  x  =  o  and  its  dif- 
ferential relative  to  x,  y,  z,  subsist  together,  just  as  if  h,  h',  h"  were  constant.  Hence,  it  appears,  that  the 
body  at  the  end  of  the  instant  d  t  is  still  found  in  the  plane  represented  by  h  z  —  h'  y  +  h''  x  =  o,  or,  that 
this  plane  is  the  plane  in  which  the  elementary  arc  described  in  the  instant  d  t,  lies.  If  therefore  we  call  0 
its  inclination  to  that  of  the  x  and  y  taken  as  a  fixed  plane,  and  ia  the  longitude  of  its  ascending  node,  we  have 

tan  0  =  </A2/t+  h'* ;  tan  *>  =  h— ;  p  a  (1  -  e*)  =  h*  +  /»"  +  A"* ;  (167) 

whence  these  elements  (viz.  the  inclination,  the  longitude  of  the  node,  and  the  semiparameter,)  are  expressed 
in  terms  of  h,  h',  h". 

That  the  equation  h  d  z  —  h'  d  y  +  h"  dx  =  o  must  hold  good  at  the  same  time  with  hz  —  h'  y  +  h"  x  =  o 
is  also  evident  from  this  consideration,  that  although  the  ellipse,  it  is  true,  varies  from  one  instant  to 
another,  yet  it  must  be  regarded  as  invariable,  while  the  body  describes  each  of  its  elementary  portions  ; 
because,  by  hypothesis,  it  is  so  adjusted  that  the  body  shall  remain  in  its  circumference  during  the  whole  of 
the  instant  d  t,  and  it  is  not  till  the  consecutive  instant  that  it  is  necessitated  to  change  its  form,  &c.  to 
accommodate  itself  to  the  new  course  taken  by  the  body.  The  same  reasoning  holds  for  any  other  finite 
equation  of  elliptic  motion.  Its  first  differential  may  be  taken,  as  if  the  arbitrary  constants  it  involves  were 
rigorously  such.  The  disturbing  forces  make  no  change  on  x,  y,  z,  r,  their  consecutive  values  remain 
T  +  d  x,  y  +  d  y,  z  +  d  z,  r  +  d  r,  as  before,  and  are  common  both  to  the  curve  and  the  ellipse,  it  is  fo 
the  consecutive  values  of  dx,  d  y,  d  z,  that  they  differ,  these  becoming  d  x  +  d*  x  +  £  dx  for  the  curve,  ana 
dx  +  d*  x  for  the  ellipse. 

This  premised,  we  have  only  to  inquire  the  variations  of  h,  h',  h'' ;   and  to  this  end,  by  the  equations 
(4,  5,  6,)  we  have 

«fc  =  v«Ai_  -.  *y  -  «i'_-» dx     .. dz  .  .!//_.»  rfy      ...  dz 

dt 

In  which,  substituting  for  o 
we  find 


a  L 

A 

t   ' 

0 

dt 

-    J 

c 

;      en    —  z  o  —    11  o  . 
dz    '                          d  t           J        dt 

dx 

,« 

dy 

8 

*», 

eir  values 

^  d  °  .,  ,         _/  d  Q   j  .               _/    d  n  „ 

dt 

dt  ' 

dt'tl 

a  x                      d  y                              d  z 

r 

d  O 

d 

0 

) 

' 

. 

$ 

/, 

—  m' 

j 

d  t  • 

t 

d  y 

y 

d 

X 

J 

ft 

h' 

—  m' 

{ 

d  Q 

d 

Q 

I 

dt; 

>                                                           (168) 

d  z 

d 

X 

( 

d  Q 

tj 

o 

) 

£ 

h" 

dt; 

y  j. 

rl 

r.nd  integrating  relative  to  t, 

4^--y~}dt;  (169,1) 


h'  =  (h')  +  ,„'*          .  _  z          -     d  t ;  (169,  2) 

,/    (       at  d  x  ) 

h"  =  (h")  +  m'fl  y^--z  4^-]  d  t ;  (169,  S) 

J     (        d  z  d  y  ) 


PHYSICAL    ASTRONOMY  703 

Astronomy,  where  (h),  (h'),  and  (h"),  represent  the  primitive  values  of  these  respective  quantities  at  the  commencement    Physical 
v— -v— "'  of  the  time  t.  Astronomy 

As  these  expressions  are  rigorous,  we  may  derive  from  them  all  the  laws  which  regulate  the  motion  of  the  **•• — v~- ' 
nodes  and  the  inclinations  of  the  planes  of  the  orbits;  but  as  it  is  only  the  secular  variations  which  concern 
us  at  present,  we  shall  not  regard  the  periodical  parts  of  the  expressions  within  the  brackets  under  the 
integral  signs.  To  develope  them,  we  must  consider  the  orbits  as  inclined  to  the  plane  of  the  x,  y  ;  but  if  we 
take  the  undisturbed  orbit  of  m  for  this  plane,  the  value  of  z  at  any  time  t  will  be  of  the  order  of  the  dis- 
turbing forces,  and  z'  will  be  a  very  small  quantity ;  so  that  z9,  z  z',  and  z'2,  may  be  neglected.  Hence 

*  (=  Vx*  +  y2  -f  z2)  and  r',  will  represent  with  this  degree  of  approximation,  their  projections  on  the  plane 
of  the  x,  y,  and  we  have  (calling  s  the  tangent  of  m's  latitude,) 

x  =  r  .  cos  0,  y  =  r  .  sin  0,    z  =  r  s ;     x'  =  r' .  cos  0',  y'  =  r' .  sin  0',  z'  =  i*  d 
If  then  we  recur  to  the  expressions  for  the  disturbing  forces  m'  — — ,  &c.  in  Section  I.  Part  II.  we  shall  find 

d°  d°        (170,1) 


d  x  d  y 

d  Q  d  Q 


d  Q  d  Q 

d  y  d  z 

jv"  h' 

Now,  since  h  z  —  h'  y  +  h"  x  =  o,  we  have  z  = j-  *  H — ;-  y 

n  h 

h"  h' 

Suppose  then,  —  =  p  and  —  =  q,  and  let  the  quantities  corresponding  to  p  and  q  in  the  orbit  of »»'  be  p/ 
h  n 

and  q' ;  then,  if0,  0',  be  the  inclinations  of  the  two  orbits  to  the  fixed  plane,  and  w,  a/,  the  longitudes  of 


their  ascending  nodes,  we  shall  have  tan  0  =  A/p*  +  q*,  tan  ui  =  — ,  tan  0'  =  Vp'*  +  <f*,  tan  m'  =  l—^, 

t  9 

and  thus,  when  p  and  q  are  determined,  the  inclinations  and  places  of  the  nodes  are  easily  found.     We  have, 
moreover, 

z  =qy  —  px,      z'  =  q'y'  —  p'  x' 

z  x"  —  x  z'  =  (jf  —  p)  x  x1  +  q  y  x'  —  q1  X  y' 

z  y'  —  y  z'  =  —  (cf  —  q)  y  y'  —  p  x  y'  +  p'  y  x" 

and  if  we  therefore  suppose  for  a  moment  M  =  — , 

r3       A,3 

—  =  m'  M  .  (y  x'  -  x  y') 

dh'  "M       I     rf  r'  'I 

d  t 

dh" 

-^y-  =  m  M.{—  (/  —  q)yyr  —  pxif  +  p'yx} 
Now,  our  design  being  to  eliminate  h,  h',  h",  from  the  formulae,  and  obtain  expressions  involving  only  p  and 

q,  from  which  h,  h',  h",  may  be  deduced  if  wanted,  we  differentiate  the  equations  p  =  —  and  q  =  — ,    when 

n  h 

we  find 

<ip    _  _    _1_  /  d  h"  rf/t^         d  q      _  J_  /  d  h'  dh\ 

dt  h    \  dt         P    dt)          dt    ''  ~  T  \.dt    '  ~  q    dtl 

and  if  we  substitute  in  these,  the  values  of  — ~, ,  and  ,  as  above  found,  we  shall  get 


.  --- 

But  if  we  neglect  the  squares  of  the  disturbing  forces,  and  the  eccentricities  and  inclinations  of  the  orbits,  we 

mf  m'  m' 

nave  — •  =  —  =  — -—,  so  that  these  expressions  become 

V         A/(!  +  m)  (1  —  es)  .a        Va 

dp  (pf  —  p)  xf  —  tqf  —  q)  y' 

—f-    =   jft'M.ii- ^ — M iLJLf  ,(171) 

v  a 
VOL.  in.  4  Y 


PHYSICAL    ASTRONOMY 

rf  n                             (n'  —  T>\  x'  —   ( </  —  O~)  Vf  Physical 

ii-   =  m'  M  .  ^ =^ ^J.  x  (172)  Asironomv. 

3 

Let  us  suppose  the  function  (a9  —  2  a  a'  .  cos  «i  +  a'2)~T  developed  in  a  series  of  cosines  of  w,  and  its 
multiples  S  +  S' .  cos  w  +  S'' .  cos  2  w  +  &c. 

then  we  shall  have,  by  (135) 


s  = 


2/3.5     ,    ,   3".  5.  7    ,  ) 

=  ^tef°  +  2^^°  +&c-} 


and  if  we  neglect  in  the  present  research,  as  is  allowable,  the  eccentricities  and  inclinations,  or  suppose  the 
orbits  circular  and  in  one  plane, 

M  =  -, =  (4-  -  S  }-  S'  .  cos  (6'  -  e)  -  S"  .  cos  2  (Cf  -  0)  -  &c.  (174) 

a '       X3         \<r3  / 

r  =  a,  /  =  a',  0  =  n  t  +  e,  &  =  n'  t  +  t ,  0  —  &  =  n  t  —  n' t  +  e  —  e'  =  W 
2/!(/-p)*'-  (<?'-</)/]  =aa'.2-^-{cOS(e'  +6)  -cos^-G)};  (175) 

+  a  a'.  P  ~?{s'"  (0'  +  0)  -  sin  (0'  -  0) } 
r{  (p'  -  p)  x'  -  (q1  -  q)  y'}  =  aa'.  ^-^- {cos  (tf  +  0)  +  cos  (ff  -  0) }  •  (176) 

-a  a'.  9  ~  9{sin  (0'  +  (?)  +  sin  (0' -  0)} 

Each  of  these  latter  quantities  is  to  be  combined  with  M  by  multiplication,  and  in  resolving  each  of  the  pro- 
ducts of  sines  and  cosines  so  originating  into  simple  sines  and  cosines  of  sums  and  differences,  it  is  obvious  that 
constant  terms  will  arise  whenever  similar  terms  are  combined,  by  reason  of  the  property  cos  A  x  cos  A  —  § 
cos  2  A  +  i.  Now  the  only  argument  common  to  both  factors  is  0"  —  0  ;  and,  of  course,  the  only  terms  in 
(174)  and  (175)  from  whose  combination  a  constant  term  can  originate,  are  —  S' .  cos  (0  —  0')  and  —  a  a'  . 

— *•  .  cos  (&  —  0) .     Consequently,  if  we  reject  all  the  periodical  terms,  and  put  I  =  -    -  S',  we  have 

J^.I.tf-q);  (177,1) 


dt 
and  similarly 


);  (177,  2) 


If  we  go  through  a  process  exactly  analogous,  so  as  to  obtain  differential  equations  for  determining  p'  and 
tf  relative  to  the  orbit  of  mf,  we  shall  find  them  to  be 


These  four  equations,  being  of  the  first  order,  and  with  constant  co-efficients  (for  the  secular  variations  of 

a,  cf,  and  therefore  of  S'  and  of x  S'  or  /,  which  are  symmetrical  functions  of  a,  a',  vanish,)  are  easily 

4 

integrated.     As  they  subsist  simultaneously  among  the  four  variables  p,  p',  q,  q',  we  may  integrate  them  ell 
together,  if  we  assume 

p  =  A  .  sin  (g  t  +  k)        p'  =  A'  .  sin  (g  t  +  k) 

q  =  A  .  cos  (g  t  +  k)         qf  =  A'  .  cos  (g  t  +  k) 

for  if  we  substitute  these  values,  we  find  that  the  variable  part  divides  off,  and  there  remain  the  following 
equations  of  condition  between  the  constants  A,  A',  and  g, 

(A'       A)  ;  (178,  1) 


PHYSICAL    ASTRONOMY.  70S 

Astronomy  m  Physical 

v-^^J.  g  A'  =  —  I  (A  -  A')  j  (178,  2)  Astronomy. 

In  these  one  of  the  constants  A,  A',  g,  remains  indeterminate.     Let  this  be  A,  then  eliminating  A.',  we  get 
for  determining  g, 


'  V  a  V 

,   f—r  >  ......................     (179,  1,2) 

TO  v  a  +  m  v  a 
§•  =  o,  or  g-  =  --  -=7  -  .  / 

v  a  a 

Now  it  is  evident,  that  if  g  and  g'  be  two  values  of  g  which  satisfy  the  equations  of  condition,  then  A  .  sin 
(g  t  +  k)  and  B  .  sin  (g'  t  +  k')  will  each  be  satisfactory  values  of  p,  and  so  for  the  rest  ;  consequently  (the 
equations  being  linear)  their  sum  will  be  so,  and  we  have 

fp   =  A  .  sin  (g  t  +  k)  +  B  .  sin  lc'  (180,  ]) 

\q  —  A  .  cos  (g  t  +  k)  +  B  .  cos  tf  (180,  2) 

(y  =  A'  .  sin  (g  t  +  k)  +  B  .  sin  kf  (180,  3) 

\(f  =  A'  .  cos  (g  t  +  k)  +  B  .  cos  k'  (ISO,  4) 
where  A,  B,  k,  k',  are  four  arbitrary  constants,  and 

m  «/  a  +  m'  V  a'  m  V  a 

g=-I.-  -•       A'  =--—=.  A  5  (181,1,2) 

V  a  of  m  v  a 

From  these  values  of  p  and  q,  p'  and  q'  it  is  easy  to  eliminate  sin  (g  t  +  k)  and  cos  (g  t  +  k)  ;  for  if  we 

multiply  (180,  1)  by  m  V  a,  and  (180,  3)  by  m'  V  a',  and  add,  noticing  that  A  .  m  V  a  +  A'  .  m'  V  a'  =  o 
by  reason  of  the  relation  between  A  and  A'  (181,  2)  we  get 

m  A/  a  .  p  +  mf  V  of  .  p'  =  (m  V  a  +  m'  V  a)  B  .  sin  kf  =  constant  ;  (182) 

and  similarly, 

m  V  a.  q  +  m?  */  a'  .  q'  =  (m  V  a  +  m'  V  a)  B  .  cos  k1  =  const.  ;  (183) 

Moreover,  we  have 

tan  02  =  p*  +  q°-  =  ^A2  +  B2)  +  2  AB  .  cos  (g  t  -r  k  —  k')  ;  (184,  1) 

tan  <p'2  =  />'*  +  q'*-  =  (A.'2  +  B2)  +2  A'B  .  cos  (g  t  +  k  —  V)  ;  (184,  2) 

Consequently, 

m  V7".  tan  03  +  m  V~rf  .  tan  0'2  =  m  \/T(A'2  +  B'2)  +  m'  •/!?  (A's  +  B4)  =  constant  ;  (185) 

The  arbitrary  constants  A,  B,  k,  k',   may  be  determined  in  any  particular  case,  either  by  comparing  the 
general  expressions  for  p,  p',  q,  q',  with  their  actual  values  at  any  assigned  instant  as  derived  from  observa- 

tion, or  from  these  last  derived  equations  ;  for  since  p2  +  9*  =  tan  0a  and  —  =  tan  ui,  we  have 

p  =  tan  0  .  sin  <a  ;     q  =  tan  0  .  cos  ui  ;     p'  =  tan  0'  .  sin  u>'  ;     q'  =  tan  0'  .  cos  <a'  ;  (IStj) 

Now  the  equation  (180)  gives 

(p'  -  p)  =  (A'  -  A)  .  sin  (  g  t  +  k)  ;     q'  -  q  =  (A'  -  A)  .  cos  (gt+k) 

consequently,  if  we  take  t  =  o,  or  if  we  assume  for  our  data,  the  elements  0,  (j)',  u>,  a/,  as  they  were  observed 
at  the  epoch  or  origin  of  the  time  t,  we  find 

»'  —  p         tan  0'  .  sin  n/  —  tan  0  .  sin  ox 

tan  k  =  —.  -  £  =  -  -  ;  (187) 

q  —  q         tan  0'  .  cos  «/  —  tan  0  .  cos  ia 

Hence  A  and  A'  are  found  ;  for,  the  values  of  S,  S',  &c.  being  known  from  equations  (173),  I  =  -  S'  is 
also  known  ;  and  since  by  (181,  2) 

m  V  a  +  m'  V  a'  m'  V  a'  (p'  —  p) 

A'—  A=  --  :——  -  .AwegetA=--  —  -=  -  ^       ^'  -  (188) 

•ft'  vff  (m  v  a  +  m'  v  a')  sin  k 

Again,  if  we  divide  (182)  by  (183),  we  find 

,  .       m  V  a  .  tan  0  .  sin  u>  +  m'  "S  a'  .  tan  0'  .  sin  u>' 

tan  k"  =  -  =  -  -  =  -  -  -  ;  (189) 

TO  A/  a  .  tan  0  .  cos  <a  +  m'  V  a'  .  tan  0*  .  cos  «/ 

m  V  a  .  tan  0  .  sin  <a  +  m'  *S  a'  .  tan  0'  .  sin  to' 

B  =  -  ~  -  •  (190) 

(m  */~a  +  m'  V  a)  .  sin  U 

These  constants  once  computed,  the  laws,  periods,  and  limits  of  the  motions  of  the  planes  of  both  orbits 
are  known.  The  period  in  which  the  inequalities  recur  is  deducible  at  once  from  the  value  of  g.  If  we  express 
the  time  t  in  Julian  years,  n,  n'  represent  the  mean  motions  of  the  planets  m,  m',  in  one  such  year,  in  parts 

4  y  2 


706  PHYSICAL    ASTRONOMY. 


1.X  '          "v  t>U       *       I 

Astronomy.0f  a  whole  circumference,  and  n*  a3  =  1,  that  is,  V  a=  — ,  hence  g  (  =  —  I .  ( — ^  +  _!!i_\  j  w;ii  be  Astror.omy 


found  as  follows  : —  g  =  — >  (m  n  a'  +  m'  n  a)  .  I 

and  if  we  call  T  the  whole  period,   —  g  T  =  1  circumference  =  1  ;  and 

7"  "  (m  n'a'  +  'm  n  a)  .  I  ' 

where  .  _    a  a'    ~, 

—  — ~ — 

If  n,  n'be  expressed  in  seconds,  the  numerator  of  T  instead  of  being  unity,  must  be  360  x  60  x  60"=2196GOO'/. 
The  limits  of  the  variations   of  the  inclinations  are  readily   found  ;  for  it  appears  from  the  equations 
(184,  1,  2)  that  their  maxima  and  minima  occur  when  gt  +  k  —  kf  =  o  and  18O°,  and  have  for  their  cor- 
responding values  A   +  B I  ,    A   —  B 1 

A'  +  BJ     and   A'-B/' 

Now,  it  is  obvious  from  the  values  of  p  and  q,  (180,  1,  2)  that  A  and  B  are  small  quantities  of  the  same  order 
as  p,  q ;  so  that  the  inclinations  can  never  increase  or  diminish  beyond  certain  very  narrow  limits  This 
follows  too  from  the  equation  (185)  ;  for  in  the  present  state  of  our  system,  tan  0  and  tan  0'  being  extremely 
small,  the  sum  m  V  a  ,  tan  0s  +  mf  V  a  .  tan  0'2  is  always  a  very  minute  quantity  ;  and  since  */  a  and  */  of 

must  both  be  taken  positively,  (for  A/  a  =  —  and  n  is  positive  for  both  planets,  the  motions  being  both  in 

the  same  direction)  neither  term  separately  can  exceed  the  value  of  the  constant ;  so  that  0  or  0/  must 
remain  for  ever  confined  to  a  value  not  greatly  different  from  what  it  now  has,  and  the  planes  of  the  planetary 
orbits  must  keep  for  ever  oscillating  within  very  confined  limits  about  their  mean  positions. 

With  regard  to  the  nodes  it  is  different.  These  are  liable  to  great  changes  of  place,  and  may  even  circu- 
late for  ever  in  one  direction  without  returning.  In  fact,  if  we  would  determine  the  maxima  and  minima  of 
their  longitudes,  we  have  only  to  put  d  to  =  o,  d  to'  =  o  ;  the  roots  of  which  equations,  if  real,  will  indicate 
the  stationary  points  ;  and,  if  imaginary,  will  shew  that  such  points  do  not  exist,  or  that  the  nodes  circulate. 
Now  we  have 

d  .  tan  to  d  p          d  q  dp  d  q 

d  o>  =   =  o,  or  d  .  tan  w  =  o,  (.r — —  = — .    or  q  — p  =  o 

1  +  taniu2  p  q    '  '    dt  dt 

in  which  equation,  substituting  for  and  — - —  their  values  in  equations  (177,  1,  2)  we  find 

99'  —  9*  +  pp'  —  p2  =  o,     or  pp'  +  9  q   =  p*  +  q~ ;  (193) 

in  which,  substituting  forp,  p',  q,  qf,  their  values  (180)  we  get  ultimately 

A  +  B  .  cos  (g  t  +  k  —  k')  =o     which  gives     cos  (g  t  +  k  —  k)  =  -     —  ;  (194) 

Hence,  if  B7  A  (no  regard  being  had  to  the  signs)  this  will  correspond  to  a  real  value  of  t,  and  the  node  will 
then  merely  have  a  libratory  motion,  advancing  and  receding  alternately ;  if  B  Z  A,  they  will  circulate  always 
in  one  direction.  In  the  former  case,  if  we  substitute  in  the  value  of  tan  02  (equation  184,  1)  this  value  of 
cos  (gt  +  k  —  k')  we  shall  find  tan  0'2  =  Bs  —  A2 ;  tan  0  =  VB'1  —  A8,  which  gives  the  inclination  cor- 
responding to  the  stationary  points  of  the  node.  These  points  are  attained  when  cos  (gt+k  —  J/)  =  —  — 

while  the  maxima  and  minima  of  the  inclh  -itions  happen  when  cos  (gt  +  k  —  k')  =  +  1.  The  stationary 
positions  of  the  node  therefore  do  not  correspond  either  to  the  maxima  and  minima  of  the  inclinations,  or  to 
the  semi-intervals  between  them. 

If  we  had  considered  more  than  two  bodies  the  results  would  have  been  analogous,  and  we  should  have 
arrived  at  similar  expressions  for  p  and  9  only  containing  more  terms,  and  analogous  equations  to  those  in 
182,  183,  and  185,  viz. 

m  *J  a  .  p  +  m'  "J  n  .  p'  +  m"  V  a"  .  p"  +  &c.  =  const.  ~J 

tn  -/"a".  9  +  mf  V'a'  .  q'  +  m"  J~~a"  .  q"  +  &c.  =  const.  >  , (195) 

m  V  a  .  tan  0s  +  m'  */  a'  .  tan  0A2  +  m"  J  of' .  tan  0"2  +  &c.  =  const.J 

Let  us  apply  this  theory  to  an  example,  and  we  will  take  that  of  the  orbits  of  Jupiter  and  Saturn,  the  two 
principal  planets  of  our  system.  If  we  take  for  our  epoch  the  year  17OO,  we  have,  by  Halley's  tables, 

a>  =  101°     5'  6";     0  =  2=  30'  10";     a  =  9'54007 
a/  =     97°  34'  9'';     0'=  1°  197  10';      a   =  5'20098 

Hence  we  find  the  values  ofp,  pf ',  q,  q',  for  that  epoch,  by  the  equations  (186)  as  follows  : — 
p  =  0-04078,  9  =  —  0-01573  ;     p"  =  OO2283,     9'  =    -  0-O0303 

whence,  having  computed  /.and  assuming  m'  = and  m  =  — — ,  we  shall  find  A,  B,  k,  k'  and  gas  follows  : 

10l>r  3358 


PHYSICAL    ASTRONOMY.  707 

Astronomy  B  =  0'02905  ;      A  =  0'01537  ;      k  =  125°  15'  40//,      kf  =  103°  38'  40''  Physical 

v — v — '  and  A'  =  —  0-00661,  and  finally  g  =  —  25"-5756  Astronomy. 

Hence  we  obtain,  in  the  case  of  Saturn, 


tan  0  =  0-03287  .  Vl  +  0-82665  .  cos  {21°  37'  —  t  X  25"'5756} 
and  for  Jupiter 

tan  0'  =  0-O2980  .  Vl  —  0-43290  .  cos  {21°  37'  —  t  X  25"'5756} 

Also  we  have  B  +  A  =  0-04442  ;  B  —  A  =  0-01368,  so  that  the  maxima  and  minima  of  the  inclinations  of 
Saturn's  orbit  are  2°  32'  40"  and  0°  47',  and  its  greatest  deviation  from  the  mean  state  will  not  exceed  52' 
50".  In  Jupiter's  orbit  the  maximum  is  2°  2'  30",  and  the  minimum  1°  17'  1O",  and  the  greatest  deviation 
from  a  mean  state  0°  22'  40". 

The  longitude  of  the  node,  ui,  has  a  maximum  and  a  minimum  in  both  orbits,  because  B  7  A  ;  and  the 
extent  of  its  librations  will  be,  in  the  case  of  Jupiter's  orbit,  13°  9'  4O",  and  in  that  of  Saturn's,  31°  56'  20" 
on  either  side  of  its  mean  station,  on  the  plane  of  the  ecliptic  supposed  immoveable. 

The  period  in  which  these  changes  take  place,  or  the  whole  time  in  which  the  inclinations  vary  from  their 
greatest  to  their  least  values,  and  the  nodes  from  their  greatest  to  their  least  longitudes,  and  back  again,  is 

equal  to  -  =     .„  ,,„  —  =  50673  Julian  years;  an  immense  period,  and  which  may  serve  to  give   some 

£  *w    *vY  &O 

idea  of  the  extent  to  which  the  Newtonian  theory,  assisted  by  the  refined  methods  of  the  modern  analysis. 
enables  us  to  carry  our  views  of  the  past  and  future  condition  of  our  system  ;  as  this,  though  subject  of  course 
to  some  subordinate  corrections,  is  perhaps  one  of  the  least  uncertain  of  the  results  of  perturbation. 

Let  us  now  consider  the  secular  variations  of  the  eccentricities  and  aphelia.  Our  first  object,  agreeable  to 
the  theory  of  the  variation  of  the  arbitrary  constants  already  exposed,  must  be  to  obtain  such  an  equation  of 
the  ellipse  of  curvature  as  shall  be  adapted  to  our  purpose,  by  containing  these  elements  (or  convenient  func- 
tions of  them)  in  a  state  sufficiently  disengaged  from  the  variables  x,  y,  z,  and  their  differentials.  Now  if  we 

differentiate  the  quantities  —  ,  —  ,  —  ,  noticing  that  ra  =  z2  +  y*  +  z*,  we  get 

x         r  d  x  —  x  d  r  _  r*  dx  —  x  ,r  dr  _  r3  dx  —  %  x  d  (r2) 
d~7~  r2  r3  r3 

_  (r8  -f  y3  +  2°)  d  x  —  (x  d  x  +  y  d  y  +  z  d  z)  .  x 

r» 
_  y  (y  dx  —  x  dy)  +  z  (z  d  x  —  x  d  z) 

r3 
That  is,  substituting  for  y  d  x  —  x  d  y  and  z  d  x  —  x  d  z  their  values  h  d  t,  h'  d  t, 


y>  j    I   ?    I    '•-'•"-  j  *    v. J    ..  j  I  3    \  /         L   r  "     ,    i//  /"  "    ^  j£  ^ 


/  jA  =  -*,+£,  d  ,    I  „  ,  /L\        /      A££_      „,  J 

\r  /  r3  I  I  \r/\  r3  r 


r3 
\ 

I  dt 


,,   ...  ft  x  d*  x        uy  oP  y         ,!>•*•  d*  z 

Now,  in  the  case  of  elliptic  motion,  —  —  =  --  —  -  ;   ^--j-  =  --  —|-  and  -^-  =  --  —  ,  consequently 

substituting  these,  we  obtain 

,  /  z\         ,(        ,    d  y          .  dz  )         ,/y\         ,  f  ,    d  x          ,  d  z  ) 
/id  (  —  )  =  d  {  —  h  ~  --  V  -  };  u.d  (  —  )  =  d  I  h  —  ---  h'"  —  - 
\r  /  {  dt  At  )  \r/>  {        dt  d  t  J 


and  integrating, 


'-:+  "• 


These  equations  will  serve  our  purpose,  as  the  arbitrary  constants/,/',/",  are  completely  disengaged  ;  but 
before  we  proceed  to  employ  them,  we  must  determine  the  values  of  /,/',/",  in  terms  of  the  elements,  and 
uice  versn  •  and  for  this  purpose  must  first  eliminate  the  differential  co-efficients,  which  (from  the  peculiar 
form  of  the  equations)  is  practicable,  by  merely  multiplying  the  first  by  x,  the  second  by  y,  and  the  third  by 


708  PHYSICAL    ASTRONOMY. 

Astronomy.  2  ;  and  adding  the  results,  when  we  get  Physical 


.. 

or,  by  reason  of  the  equations  (4)  (5)  (6), 

,,  r  =  ft  +  fy  +  f"  z  +  (/,«  +  h'*  +  h"*)  j  (197) 

This  equation  expresses  the  general  property  of  the  conic  sections,  in  virtue  of  which  a  line  drawn  from  the 
focus  to  the  circumference  is  always  in  a  given  ratio  to  a  perpendicular  let  fall  from  that  point  on  the  directrix. 
If  we  multiply  (196,  1)  by  h",  (196,  2)  by  —  h',  and  (196,3)  by  h,  and  take  the  sum  of  the  results, 
(observing  that  h"  x  —  h'  y  +  h  z  =  o)  it  will  be  found  that  all  the  variable  terms  will  destroy  each  other, 
leaving  simply  the  equation  of  condition, 

h"f-  h'f  +  hf"  =  o  (198) 

Suppose  X,  Y,  Z,  the  co-ordinates  of  the  perihelion.     At  this  point  dr  =  o,  orxdx  +  ydy  +  zdz=o, 
but  if  for  h,  h',  h",  we  write  their  values  in  (4)  (5)  (6)  we  have 


At  the  perihelion  therefore,  substituting  for  y  d  y  +  z  d  z,  x  d  x  +  z  d  z,  x  d  x  +  y  d  y,  their  equals  —  X  d  X, 
—  Y  d  Y,  and  —  Z  d  Z,  respectively,  these  quantities  become 

d  X'  +  d  Y4  +  d  Z" 


-  X. 

-  Y. 


dt* 

d  X2  +  d  Y*  +  d 


d  X4  +  d  Y4  +  d  Z4 
~Z--         -dp- 

Consequently  we  have,  putting  -  -    =  V2,  X2  +  Y4  +  Z4  =  R4 


/=X  -V';    /'=  Y.£-V«       ;    f"  =  Z  _V«;  (200) 

Hence  it  is  easy  to  obtain  the  following  equation, 

/»  +  /*  +  /"*  =  R«  (^  -  V*)  *  j  (201) 

Now  R  is  the  perihelion  distance,  R  =  a  (1  —  e)  and  V  being  the  velocity  at  the  perihelion,   we  have,  by 

(2          1  \ 
|T  —  —  I     Hence  we  get 

=  R*  (JL  -  AY  =  ^15_-L±)!  =  ^es 

\  a         K  /  a 

(202) 


Moreover  we  have 

/_Y   r_z       .r      __   z 

/--x'  /-x'  -VT^+TF-VX^TY-'' 

But  —  is  the  tangent  of  the  longitude  of  the  perihelion,  or  of  the  angle  which  the  projection  of  the  perihelion 

distance  makes  with  the  axis  of  the  x  :  also,  ?r  is  the  longitude  of  the  perihelion  reckoned  on  the  orbit  ;  and  if 
its  plane  is  but  very  little  inclined  to  that  of  the  x,  y,  this  angle  differs  from  its  projection  on  that  plane  only 
by  a  very  small  quantity  of  the  second  order  ;  so  that  if  we  disregard,  as  we  have  hitherto  done  in  this 
research,  the  squares  of  the  eccentricities,  inclinations,  and  disturbing  forces,  we  have 

tan7r=£;  (204) 

fi  _  __^^^__ 

These  three  equations,  viz.  tan  it  =  •i-,  p.  e  =  */f*  +  /'*  +  /"*,  and  hf  —  h'f  +  h"  f  —  o,    determine 

/,/',/",  in  terms  of  the  elements  of  the  orbit.     The  variations  of  the  eccentricities  and  longitudes  of  the 


PHYSICAL    ASTRONOMY.  709 

Astronomy,  perihelia,  e  and  v,  may  be  immediately  determined  from  those  off  and/',  so  that  it  is  to  these  we  shall  now    Physical 
— -V— ^  turn  our  attention.     Resuming  then  the  equations  (196,  1,  2,3)  let  us  differentiate  them  on  the  supposition  Astronomy. 
°ff>f>f">  being  variable  quantities,   their  variations  being  such  as  to  express  the  effect  of  the  disturbing  ^— •'v ~-/ 
forces  only.     To  express  this  we  have  used  the  characteristic  8,  and  we  shall  continue  to  do  so,  to  keep  the 
principle  on  which  the  process  is  founded  distinctly  in  view.     No  inconvenience  can  arise  from  the  confusion 
of  two  symbols,  S  and  d,  in  the  same  investigation,  when  we  bear  in  mind  that  any  expression  such  as  Sf 
denotes,  strictly,  the  whole  amount  of  the  momentary  change  which  the  quantity/undergoes  while  the  planet 
passes  from  one  elementary  portion  of  its  unknown  curve  to  that  immediately  consecutive  to  it,  while  such  expres- 
sions asdx,dy,  &c.,  denote  the  changes  which  x,  y,  &c.  undergo  while  it  passes  from  one  end  to  the  other  of 
the  same  elementary  portion.     In  both  points  of  view,  the  accumulated  effects  during  a  finite  time  are  obtained 

by  the  same  rules  of  integration  ;  the  whole  variation  off  is  legitimately  expressed  by  I  Sf  just  as  that  of  x 

is  by  Id  x,  and  when  Sfis  expressed  in  terms  of  d  t,  the  integration  must  be  performed  in  the  usual  man- 
ner, as  on  a  function  of  t. 

X  If  Z 

We  have  therefore,  since  S  —  =  o,  S  —  =  o,  £  —  =.  o 


T  T  T 


Substituting  therefore  for  S  h,  o  h'  their  values  in  the  equations  (168)  for  S  —  —  ,    6 

d  t 


—  —        —  —     —   m' 


y  d  x  —  xd  y        .    zd  x  —  x  d  z 


—  —  ,   — 
d  t  d  y 


.  y     x  —  x     y        .    z     x  —  x     z 

—  m  '  —  —  respectively;   and  for  /(,  h  ,  their  values  -  -  —  -  -   and   -  ,  we  get 

(I     Z  (it  ( 


dt 

dO  dO  )  f       dQ 

+   d  z   ' 


dx  +  ('^y-ydz)  d- 


d  x  "  ~  }    \ 

da  >    x  m (205,  1) 

+  (z  d  x  —  x  d  z  )— — 
a  •? 

dQ 


dy  ""•    >    \ 

1    X  m'    (205,  2) 


These  are  the  rigorous  values  of  Sf,  Sf,  and  their  integrals  regarded  as  functions  of  t,  express  the  total 
variations  of  these  elements  produced  in  that  time.  But,  as  we  have  done  in  the  less  complicated  theory  of 
the  nodes  and  inclinations  we  shall  neglect,  in  developing  them,  all  their  periodical  terms,  at  least  such  as 
depend  on  the  configurations  of  the  planets,  as  well  as  all  terms  containing  the  squares  of  the  disturbing 
forces,  eccentricities,  and  inclinations,  and  take  the  primitive  orbit  of  m  for  our  fixed  plane.  This  will  sim- 
plify them  greatly  j  for  in  this  ease  z,  and  — —  are  quantities  of  the  order  of  the  disturbing  forces,  and  there- 
fore, when  multiplied  by  m',  may  be  rejected.  Moreover,  on  this  supposition,  it  is  indifferent  whether  we 
reckon  the  longitudes  on  the  orbit  or  on  the  fixed  plane,  since  the  difference  is  of  the  order  z2 ;  consequently 
representing  that  of  m  by  0,  we  have 

x  =.  r  .  cos  0,  y  =  r  .  sin  0 ; 
whence  we  find 


—  -      -- 
d  0  dy 

Now  we  have  !£  =  £  -  L^JL  =  !!^  ..  "  .  sin  fl*  -  r  .  sin  0 

d  y         /3  X3  /i  \3 

But  since  X  =  V  r*  —  2r  /  .  cos  (0  —  0')  +  r'2 

_   r  .  cos  (0  —  &)          1 

~~/*~         "  T 


.;  f2o6) 

dt 


d  0   _  cos  (0  —  (?)        r  —  r'  .  cos  (0  —  ff) 

dr  ~^~  ~\T~  (207) 


PHYSICAL    ASTRONOMY. 

d  Q              r  .sin  (6  —  0')          r  T1 .  sin  (0  -  tf)  Physical 

•     ,    = pj r  -  — ^3 —  (208)  Astronomy 

dQ         1(     dQ      .    _     dQ         _1  (w) 


__.. 
dy          r  [     dr  d  t 

and  as  we  have  d  y  =  dr.  sin  0  +  r  d  0  .  cos  0,  and  h  d  t  =  r3  d  0, 

e/=  —  {dr.sin  0  +  2rd0.  COS0}—  --  r2  d  0  -    -.s'mO;  (210) 

In  like  manner  we  should  have  found 

«/'=  +  {  d  r.  cos  6  —  2rd0.  sin  0  }^j~    +  r3  d  0  -^-  .  cos0;  (211) 

Now,  if  we  neglect  the  squares  of  the  eccentricities,  we  have 

r  =  a{  1  —  e  .  cos  (0  —  ir)  },  d  r  =  a  e  .  sin  (0  —  ir)  d  0,  r'1  d  0  =  n  a2  d  <, 
whence  we  have 

d  0  =  n  d  «  (1  +  2  e.  .  cos  (0  —  ir)  and  d  r  =  n  a  e  d  t  .  sin  (0  —  ir) 
and  consequently, 

3/  =  —  nadt  .  -—  '-  {(2  —  2e.  cos  (9  —  jr)(l  +2e.cos0—  ^-)  .  cos  0  +  e  .sin0  .  sin  0  —  ir}  —  na*d  t  . 


-—  -        —       .  —  .—   -   .  .          .          —  ir   —  .—  —  j 

06*  or 

which  reduced,  rejecting  e2,  becomes 

«/  =  -  m'  ndt  la  —  —  (<Z  cos  0  +  —  e  .  cos  IT  +  ^-  .  cos  (2  0  —  IT)  )  +  a2  —  —  .  sin  0  1         (212) 

(  ft    (/        \,  £  .4  /(IT 

In  order  to  find  the  secular  part  of  the  variation  off,  we  must   ilevelope  this  expression,  retaining  only  such 
terms  as  are  not  periodical.     Now,  recurring  to  the  notation  of  Sections  3,  4,  5,  Part  II.  we  have 

0=n<  +  e-f2e.sinV=V+7r  +  2e.sinV 
therefore  sin  0  =  sin  (V  +  ir)  +  2  e  .  sin  V  .  cos  (V  +  IT) 

=  sin  (V  +  ir)  +  e  .  sin  (2  V  +  ir)  —  e  .  sin  v 
2  cos  0  =  2  cos  (V  +  ir)  —  4  e  .  sin  V.  sin  (V  +  JT) 

=  2  cos  (V  +  w)  +  2  e    cos  (2  V  +  ?r)  —  2  e  .  cos  IT 

€  €  3  fi  3  6 

—  .  COS  (2  0  —  ?r)   =    -—  .  COS  (2  V  -f  ?r)  ;    —-  .   COS  TT  =    -—  -  .  COS  ?r 
2  2  S  '  * 


neglecting  e2  &c.     Again  we  have 


„ 

+  a2  A 


d  r  da  d  r 


When  this  therefore  is  multiplied  by  sin  0,  the  term  —  e  .  sin  TT  will  combine  with  the  constant  term  a2  —  — 

of  a4  —  —  ,  and  produce  —  as  e  .  -  .  sin  ir,  and  this  is  the  only  constant  term  which  can  arise  from 
da  da 

a2  —  —  ,  because  every  other  term  contains  W,  and  therefore  both  V  and  V  :  while  the  value  of  sin  0  con- 

il  a 

tains  only   V,  and,    of  course,  V    cannot  be  destroyed  by  their   combination.      Again,  with    respect  to 
A  —  —  ,  it  has  been  proved,  that  (regarding  only  the  first  powers  of  the  eccentricities,)  this  is  resolvable 

into  a  series  of  terms  of  the  form  A  .  cos  (t  W  +  V),  A  .  cos  (i  W  +  V).     Now,  since  A  —  —  is  of  the  order 


of  the  eccentricities,  we  may  disregard  the  term  of  sin  0  multiplied  by  e,  and  take  it  simply  equal  to  sin 
(V  +  *•).     If  we  multiply  this  into  the  series  ~  ' 
term  can  produce  a  constant  argument,  viz.  that 


(V  +  T).     If  we  multiply  this  into  the  series  2  A  .  cos  (i  W  +  V)   since  W  =  V  —  V  + •  »•  —  •*',  only  one 

;  into  which  V  does  not  enter,  or  in  which  i  =  o  ;  that  is,  the 


term  multiplied  by  cos  V.     The  co-efficient  in  this  term  in  A  — — is  —  ae  ,  (as  appears  from   (147), 

dr  d  a* 

writing   -    —for  Q,  and  of  course ,  &c.  for  A,  &c.)    Hence  the  term  so  originating  in  a*  — — .  sin  0  will 

AT  do,  a  T 

be  equal  to  the  constant  part  of  —  a?  e .  cos  V  .  sin  (V  +  ir),  that  is,  to --—  .  —  -  .  sin  »•.    Again 

d  <zj  2          d  a2 

the  multiplication  of  sin  (V  +  ir)  with  the  series  2  A  .  cos  (i  W  +  V)  will  produce  a  constant  term  by  the 
combination  of  sin  (V  +  ir)  with  cos  (W  +  V)  =  cos  ( V  -f  ir  —  ir1) .     The  co-efficient  of  cos  (W  +  V)  in 

the  developement  of  A  — —  is (a (-  2  ) .     Hence  tht  constant  Dart  of  a2  — —  .  sin  0  so 

dr  2\dada  da/  dr 


PHYSICAL    ASTRONOMY.  711 

Astronomy  a1  e'    /        d?  A'  d  A'  \  Physical 

-_^    O  originating,  will  be  —   —  —  I  a  -  -  —  -  +  2  —  —  I   X  (the  constant  part  of  sin  (V  +  TT)  .  cos  (  V  +  ir  —  if\    Astronomy. 
•  "yuflud  d  G,  /  ^    _        ^^. 


d2A'   | 

\ 

ad  (/) 


e  (       d  A 

=  —  as  .  —  sin  ?/'  2  — h  <t  , 

da  d 

Let  us  next  consider  the  constant  terms  which  can  originate  from  the  part  multiplied  by  a  — — .        This 

d  & 

dQ       dw  d  O  d  n  dQ 

quantity  is  equal  to  a .  =  a  — —  =  a    ,        +  «  A  — — .     But  we  have 

dwdO  d  w  d\V  d  w 

-j-^-  =  -  A  .  sin  W  -  2  A'  .  sin  2  W  —  &c. 

and  as  all  these  contain  V,  and  this  arc  does  not  enter  into  any  of  the  terms  multiplying  — —  in  the  expres- 
sion of  Sf,  it  is  obvious  that  no  combination  not  containing  V,  can  arise  in  this  way,  i.  e.  no  constant  term 
can  occur  in  the  part  of  the  developement  depending  on  .  With  regard  to  A  — —  being  already  of  the 

order  of  the  eccentricities,  it  need  only  be  combined  with  the  first  term,  2  cos  0,  and  this  term  may  be  regarded 
as  equal  simply  to  2  .  cos  (V  +  IT).  We  have  only  then  to  investigate  the  constant  part  of 

(dQ  d*Q  d8  Q  d2  Q          ) 

2  a  .  cos  (V  +  ir)  1  A  — —  =  A  r  +  A  /   +  — —  A  w  \ 

(_       d  w         dadW  d  a  d  W  d  W2         J 

Now  A  r  =  —  a  t .  sin  V,     A  /  =  —  a!  d .  sin  V,     A  w  =  2  e  .  sin  V  —  2  e'  .  sin  V 

d  A' 

1st.  The  combination     —  2  a2  e  .  cos  V  .  cos  (V  +  ir)  .  2  —  i  sin  i  W 

d  a 

can  contain  no  constant  term,  because  the  terms  sin  V  and  cos  (V  +  ir)  cannot  eliminate  V  from  sin  i  W. 

2dly.  —  2  a  of  e'  .  cos  (V  +  IT)  ,  cos  V  .  2  —  i sin  i  W 

d  a 

will  contain  a  constant  term,  for  taking  i  =  1,  cos  V'  x  sin  W  =  5  sin  (W  +  V)  +  £  sin  (W  —  V),  the 
last  term  of  which  does  not  contain  V,  being  equal  to  +  5  sin  (V  +  ir  —  ir').  The  combination  now  under 
consideration,  will  therefore  contain  the  term 

+  a  a'  e' .  1  .  -—  .  cos  (V  +  ir)  .  sin  (V  +  ir  —  ir') 
which,  developed,  will  produce  a  constant  term, 

—  a  of  .  —  .  T  .  sin  n' 

2        d  J 

Lastly,  the  combination 

4  a  .  cos  (V  +  ir)  { e  .  sin  V  -  e' .  sin  V'}  .  2       i2  .  A' .  cos  i  W 
will  produce  a  constant  term  ;  viz.  that  arising  from 

-  4  a  e  .  cos  (V  -f  ir)  .  sin  V  x   —  I9  A' .  cos  W 

for  sin  V  .  cos  W  produces  a  term  free  from  V,  viz.  |  sin  (V  +  ir  —  IT')  which  again  combined  with  cos 
(V  -(-  ir)  produces  -  —  sin  ir' ;  so  that  the  complete  result  of  this  combination  will  be 

—  a  tf  .  A' .  sin  ir1 
Assembling  therefore  all  the  constant  terms  so  found,  we  get 

Sf  =  m' .  a  n  d  i  .  e  sin  ir  \a  — h  —  -p- r-  f  (213) 

(^      da          2    d  a2  ) 

,f  a    dA'        J    dA'         a  of         d*  A'} 

•4-  m.  andt.er.sm  ir'  {  A'  A • H .  ;  ;• 

(  2da          2do'  4         dada'l 

The  value  of  Sf  may  be  derived  by  a  similar  process,  but  without  executing  the  whole  process,  we  may  obtain 
it  more  readily  by  considering  the  expressions  (210)  and  (211)  we  then  see  that  £/ changes  to  Sf  simply  by 

changing  0  to  0  —  90°,  but  leaving  r,  — —  and  -7—  unchanged.    Now  Q,  — —  and  remain  manifestly 

unaltered,  whatever  direction  we  assume  as  that  from  which  we  reckon  the  longitudes.  Suppose  then  that 
we  shift  this  direction  90°,  then  will  each  of  the  angles  0,  6',  e,  (',  ir,  ir',  be  diminished  9O°,  and  this  wil1. 
produce  no  alteration  in  T,  because  0  —  ir  will  remain  unchanged.  Hence,  the  effect  produced  by  this  change 
on  Sf  will  transform  it  into  Sf.  The  value  of  Sf  therefore  will  be  had  at  once  by  changing  sin  ir  and  sin  r' 
into  —  cos  ir,  and  —  cos  ir'  in  that  of  Sf;  and  this  will  be  found  to  be  confirmed  by  the  direct  process. 

Suppose  now  we  denote  the  constant  co- efficients  within  the  brackets  for  brevity  by  —  /  and  K,  so  as  to  have 
vol..  in.  4  •» 


PHYSICAL    ASTRONOMY. 

d  A         a2      d4  A  Physical 

"  "  Astronomy. 


Then  it  will  be  observed,  that  £  is  obviously  symmetrical  with  respect  to  a,  a',  because  the  co-efficient  A  is 
so  ;  and  the  same  will  be  found  to  hold  good  with  I  ;  for  if  we  write  it  as  follows,  — 


2     d  a    (        da 

and  for  A  substitute  its  value  given  in   (137,  1)  it  will  be  found  that  the  execution  of  the  operations  here 
indicated  will  lead  to  precisely  the  same  result  as  those  denoted  by  -—  -  j  a'2  —j—^  £  :    consequently  /  and  K 

are  both  symmetrical  functions  of  a,  and  a',  and  do  not  alter  when  these  elements  are  mutually  transposed. 

a  a' 
With  regard  to  I,  it  may  be  easily  shewn  to  be  identical  with  the  function  —  —  S'  which  we   have  already 

denoted  by  the  same  symbol.     To  see  this,  we  need  only  write  for  A  and  S'  their  values  in  (136,  1)  and 
(  173,  2),  and  executing  on  the  former  series  the  operations  indicated  by  —  ~  .  -  —  (  a2  -  —  J  ,  (  remembering 

that  a  =  —  ,  —  —  =  --  ;    -  =  -)  --  )   and  multiplying  the  latter  by  -  ,  the  result  will  be  iden- 
a       d  a  a        d  a2  a2  / 

tical.     We  have  then 

Sf  =  —  mf  .  n  a  .  I  d  t  x  e  .  sin  JT  +  TO'  .  n  a  .  K  d  t  x  e'  sin  ir  •>  (z\~\ 

if=  +  m'  .  n  a  .  I  d  I  x  e  cos  it  —  m'  .na.Kdt  x  e'  cos  ir'  -> 
Now,  if  we  recur  to  the  values  off,  f,  f",  we  find 

f  f  h'f  ~  h"f 

J—  =  tan  IT  ;  J  =  sin  JT,    /"  =  —  —  ;  -  - 

f  Jj*  +  f*  h 

but  h'  and  h"  are  of  the  order  of  the  disturbing  forces  when  we  take  the  plane  of  m's  undisturbed  orbit  for  our 

fixed  plane;    consequently,  /''  is  so,  and  therefore    Vf  +  /'*  =    A//*  +  /'2  +/'*  =  /*  e,  neglecting  the 
squares  of  the  disturbing  forces.     Hence  we  have/=  p.  e  .  cos  ir  and/'  =  p.  e  .  sin  IT. 

In  imitation  of  what  we  did  in  the  theory  of  the  nodes  and  inclinations,  let  q  and  p  represent  these  quantities; 


then  d  7  and  d  p  being  the  momentary  variations  of  9  and  p,    will  replace  Sf  and  Sf  ;  and  if  in  like  manner 
we  put  (f  and  p'  for  fif  e  .  cos  •**  and  /t'  e'  .  sin  tr'  ,  the  corresponding  quantities  in  the  orbit  of  m',  we  shall  have 

(putting  —=,  —  =,  for  n  a,  n'  a'  ;  and  supposing  /t  =  1,  and  /*'=!,  which  is  allowable  since  the  squares  of  the 

Va     va 
disturbing  forces  are  neglected,  and  the  quantities  under  consideration  are  all  of  the  first  order  of  these  forces,) 


(218,  1,  2,  3,  4) 


d  t 
dt 

'    dt    ' 

Thus  we  see  that  the  problem  of  the  secular  variations  of  the  eccentricities  and  aphelia,  depends,  exactly  as 
in  the  case  of  the  nodes  and  inclinations,  on  the  simultaneous  integration  of  four  differential  equations  of  the 
first  order  with  constant  co-efficients.  If  we  compare  these,  however,  with  those  of  (177),  we  shall  observe 
that  the  former  research  is  rather  more  complicated  than  the  latter,  by  reason  of  the  two  co-efficients,  I  and  K, 
which  it  involves,  while  the  system  of  equations  on  which  the  nodes  and  inclinations  depend,  involves  only  the 
first.  However,  this  circumstance  does  not  render  the  integration  more  difficult ;  the  same  substitution  suc- 
ceeds, and  the  integrals  have  a  form  almost  exactly  similar ;  we  have  only  to  take 

p  =  A  .  sin  (g  t  +  k)  +  B  .  sin  (h  t.  +  I)  ^ 

q   =  A.cosfef  +  k)  +B:cos(A<  +  0(  . 

p'  =  A',  sin  (gt  +  k)+  B'.  sin  (h  t  +  I)  ( 

q'  =  A'. cos  (g  t  +  k)  +  B'.  cos  (h  t  +  I)} 

If  these  be  substituted,  the  equations  will  be  satisfied,  provided  the  followipg  equations  hold  good  between 
the  constants, 


PHYSICAL    ASTRONOMY.  713 

Astronomy.  m'  ,      ~\  (  ,  ,,  mf  Physical 

g  A  =  —  —  (/A   —  KA.  )  ft  B  =  —  —  (IB  —  A  B  )  Astronomy 

v  a'  V  a 

and 


gA./  =  -=(IA.'-  KA.)  ftB'  =  -        (IK-  KE) 

V  a'  J  V.  A/  a' 

It  is  obvious  that  the  elimination  of  A'  from  the  first  pair,   and  of  B'  from  the  second  pair,  will  lead  to 
equations  of  exactly  the  same  form  for  g  and  h,  which  are  therefore  the  two  roots  of  the  quadratic, 

m  V  a  +  m   V  a'          mm' 

f-lg    -      —==  --  +  —,  ==(!»  -  K*)  =  o  (220) 

V  a  a'  V  a  a' 

and  g  and  ft  being  found,  and  A,  B,  k,  I,  remaining  arbitrary,  A'  and  B'  are  easily  found. 

To  adapt  these  values  to  any  particular  case,  the  general  values  of  p,  q,  p',  </,  at  any  assigned  epoch,  must 
be  made  to  coincide  with  the  observed  values  of  e  .  sin  IT,  e  .  cos  IT,  e'  .  sin  IT  ,  and  e'.  cos  ?/  ;  which  condition 
will  furnish  equations  to  determine  all  the  arbitrary  constants. 

For  example,  in  the  case  of  Jupiter  and  Saturn,  we  shall  find  on  computation  g  =  21"'9905,  h  =  3"-5S51  . 

A  =  0-04877  ;     B  =  0-03532  ;     A'  =  —  0-01715  ;     B'  =  0-04321  -,     k  =  306°  34'  4O"  ;     I  =  210°  16'  40' 
Now,  since  p  =  e  .  sin  ir,  and  q  =  e  .  cos  TT,  we  have 


tan  TT  =     -,  and  e  —  </p*  +  ?s  =  //Aa  +  B2  +  2  AB  cos  {  (g  —  h)  t  +  k  —  I  \ 
which  gives,  by  substituting  the  numbers  already  found, 


e  =  0-06021  .  A/1  —  0-95O09  x  cos  (83°  42'  —  18"-4054  x  t) 
which  is  the  eccentricity  of  Saturn's  orbit  ;  and  similarly, 

</=  0-04649  .  A/1  +  O-68592  .  cos   (83°  42'  —  18"-4054  x   t) 
for  that  of  Jupiter,  after  any  number  t  of  years  since  1700. 

The  longitudes  of  their  respective  aphelia,  will  also  be  known  by  the  formulae 

A  .  sin  (g-  1  +  k)  +  B  .  sin  (h  t  +  I) 
~  A  .  cos  (g  t  +  k)  +  B  .  cos  (A  <  -h  I)  ' 

A',  sin  (gt'+  k)  +  B'.  sin  (h  t  +  I) 
=  A>.coS(gt  +  k)+W.COa(ht  +  l)'> 
and  the  maxima  and  minima  of  these,  or  their  greatest  deviations  from  their  mean  places  will  take  place,  when 

g  A2  +  h  B2  +  AB  (g  +  h)  .  cos  {  (g  -  ft)  t  +  (k  -  1)  }  =-.  o 
that  is,  when 

cos  {(g  -  ft)  t  +  (k-  1)}  =  -  g(f++h)h™->  (223) 

If  this  fraction  be  less  than  unity,  the  aphelia  will  librate,  as  in  the  case  of  the  nodes  about  a  mean  position, 
if  not,  they  move  in  one  direction  continually.     In  the  case  of  Jupiter  and  Saturn  now  before  us,  g  A2  +  h  B2 
7  (g  +  h)  AB,  so  that  the  aphelia  go  on  for  ever  in  one  direction. 
The  period  in  which  the  eccentricities  go  through  all  their  evolutions,  and  return  to  the   same  state,   is 

^fiO^  ^?f-»O^ 

represented  by  --  —  =  —  -  -  —  =  70414  Julian  years. 

The  greatest  and  least  values  of  the  eccentricities  are  respectively  A  +  B  and  A'  +  B',  for  the  two  planets  ; 
and  in  the  case  before  us 

for  Saturn,  ----  0-08409  and  0-01345  "1  the  maximum  of  the  one  corresponding  to  the  minimum  of 
and  for  Jupiter,  ____  0'06036  and  0-02606  )      the  other  planet. 

Finally,  we  may  derive  relations  between  the  eccentricities,  masses,  and  semiaxes,  similar  to  those  obtained 
between  the  inclinations,  &c.  in  equations  (182,  183,  &c.)  for  since  p*  +  g2  =  e2  it  is  easy  to  see  that  we 
must  have 

m  .  e4  V1T+  mf  .  e'2  A/7  =  m  V~a(A*  +  B2)  +  m'  A/a7  (A'2  +  B'2)  (224) 

+  2  (m  A/IT.  AB  +  m'  A/17.  A'B')  .cos  [  (g  -  h)  t  +  (k  —  I)  } 
Now  we  have,  from  the  equations  of  condition, 


714  PHYSICAL    ASTRONOMY. 

Astronomy,  but  g  and  h  being  the  roots  of  (220) ,  we  have 


V  a  a'  */  acf 

Consequently,  our  equation,  on  executing  the  multiplications  indicated,  and  substituting,  reduces  itself  to 

m  <S~a~.  AB  +  m'  -/"a7  .  A'  B'  =  o 
so  that  the  equation  (224)  becomes  simply 

m  e" .  </~a~  +  m'  e'2  V~7  =  m  VT(AS  +  Ba)  +  m'  */~a?  (A'2  +  B'2)  =  constant,  (226) 

because  the  major  axes  are  constant ;  and  had   we  considered  a  greater  number  of  planets  than  two,  we 
should,  in  like  manner,  have  arrived  at  the  equation 

me*  V~~a  +  m'  if*  -/"a7  +  m" e"*  -/"a7'  -,u  &c.=  constant ;  (227) 


PART  III. 
OF  THE  THEORY  OF  THE  MOON. 

SECTION  I. 
Rigorous  investigation  of  the  differential  equations  of  the  moon's  motion,  and  general  expression  of  the  disturbing  forces. 

THE  extreme  minuteness  of  the  masses  of  the  planets,  as  well  as  their  great  distance,  renders  it  allowable  in 
the  theory  of  their  perturbations  to  neglect  altogether  the  squares  of  the  disturbing  forces,  and  affords  such 
facilities  to  the  whole  investigation,  as  to  permit  us  to  express  at  once  the  true  longitude  and  radius  vector  in 
explicit  functions  of  the  mean  longitude  or  the  time.  This  is  not  the  case  in  the  lunar  theory,  in  which  one 
of  the  most  remarkable  of  the  inequalities  depends  for  nearly  half  its  value  on  the  square  of  the  disturbing 
force,  and  in  which  the  whole  investigation  is  so  delicate,  as  to  render  it  necessary  to  abandon  the  direct 
process  followed  in  the  planetary  theory  and  ado_pt  a  route,  apparently  more  circuitous,  but  possessing  advan- 
tages of  a  peculiar  kind.  It  consists  in  expressing  the  radius  vector  and  the  time  or  mean  longitude  in 
functions  of  the  true,  and  thence  by  the  reversion  of  series  deducing  the  true  longitude  in  terms  of  the  mean. 
The  advantage  of  this  is,  that  we  are  thus  enabled  to  set  out  from  differential  equations  in  which  nothing  has 
been  neglected,  and  consequently  have  it  in  our  power  fully  to  appreciate  the  influence  of  all  the  terms  we 
may  afterwards  neglect  in  their  integration  on  the  final  result. 

Our  directing  principle  in  this  investigation  will  be  to  follow  out,  step  by  step,  with  the  proper  modifications, 
the  same  system  of  transformations,  by  which  the  differential  equations  1.  2,  3,  of  undisturbed  motion  were 
converted  into  (9)  and  (11)  expressing  the  radius  vector  and  time  in  terms  of  0. 

Our  first  step  will  consist  in  the  investigation  of  an  equation  corresponding  to  (9)  :  to  this  end  (as  we 
have  supposed  d  t  constant)  multiply  the  first  of  the  equations  (94)  by  y,  and  the  second  by  —  x,  and  add, 
and  we  get 

yiPx  —  xd*y  .     (      dO  d  Q 

-  --  •  -- 


and  integrating, 

y  d  x  —  x  d  y  <*  —  „  ..  , 

£ L  —   /,    4-  MI'   /    J    r H  V  rl  t  :  (228) 


but  y  dx  —  x  dy  =  p*  d  0,  so  that  this  equation  becomes 

ff  -ii-  =  h  +  m'j*T  P  d  t  (829) 

if  we  assume  d  Q  d  n 

T  =    X~^~~9^.  (230, 

P 

The  function  T  is  the  measure,  and  m'  .  T  the  quantity,  of  what,  in  the  lunar  theory,  is  called  the  tangential 
force,  or  that  part  of  the  disturbing  force  acting  on  m,  which  is  perpendicular  to  the  direction  of  the  radius  vector. 

This  is  evident,  if  we  consider  that  m' and  m'  representing  the  disturbing  forces  in  the  directions  QP 

d  y  d  x 

Fig.  A.    and  PM  (fig.  A)  if  we  resolve  these  each  into  the  direction  K  Q  perpendicular  to  QM  they  will  become 

respectively  ,  x      d  Q  ,  y       d  Q 

—  TO'  —  .  . and  +  m  —  .  — — • 

p       d  y  pax 

so  that  their  aggregate,  or  the  whole  tangential  force,  will  have  for  its  expression  [the  quantity  above  repre- 
sented by  Tin  equation  (23O). 

'['he  equation  (229),  if  m'  were  zero,  would  coincide  with  (9),  and  would  express  the  proportionality  of  tLe 


PHYSICAL    ASTRONOMY.  715 

13  areas  to  the  times.     The  termm'  /  T  p  d  t  then  expresses  the  momentary  effect  of  the  disturbing  forces  in  Astronomy. 
deranging  this  proportionality  ;  and  since 

f'p°-  d  0  =  h  (t  +  const.)  +  m'  i'd  tf*T  pdt  (231) 

the  term  mf  I  dt  I  T  p  dt  expresses  their  total  effect  after  the  lapse  of  the  time  t  ;  and  we  see  that  this  effect 

takes  place  solely  in  virtue  of  the  tangential  force,  agreeably  to  what  Newton  has  advanced  in  the  llth  section 
of  the  Principia. 

Our  next  step  is  to  investigate  an  equation  for  the  disturbed  motion,  corresponding  to  (11)  in  the  undis- 
turbed. The  process  we  followed  in  the  deduction  of  (11),  it  will  be  remembered,  consisted 

1st.  In  making  0  the  independent  variable. 

2dly.  In  eliminating  t  and  its  differentials. 

Now,  if  in  the  equations  (94)  instead  of  supposing  d  t  constant,  we  regard  it  as  variable,  we  must  change 

d1-  x  ldxd*xdxd*t  .        d2  y  , 

-  into  —  d  -  or  -  —  -  --  —  --  —  —  ,  and  so  for        *  ,  &c.     This  done,  let  the  first  of  the  equations 

(94)  be  multiplied  by  x,  and  the  second  by  y  ;  their  sum  will  be 


i-  yu>  ai  - 

tQ   -1—    7i2 

dt  )    d  t" 
dQ  ) 

r3 

m  .  <  j; 
j       a  i 

^^    dy] 

but  since  x  =  p  .  cos  0,  and  y  =  p  .  sin  0,  r  =  V  x'2  +  if  +  z% 

-  V  p*  +  2* 
this  equation  will  become  (supposing  dQ  constant), 


rf2  1 

We  have  only  now  to  find  the  value  of  —  —  on  the  supposition  of  d  6  being  invariable,  and  substitute  it  in 

U   t 

this  equation  ;  and  for  this  purpose  we  must  employ  our  equation  (229)  which  gives  (multiplying  it  by  Tpdt) 

Tpsd0  =  h.Tpdt+m'.Tpdt.  jTpd  t 
and  integrating, 

A>3  do  =  h  frpdt+  —  (  CT  Pdt\ 

from  which  we  get,  by  the  solution  of  a  quadratic  equation, 

m'  .  j\p  dt  =  -h+    V  fc=  +  2  m'  /T  p3dO 
•and  differentiating,  and  dividing  by  m'  T  p 

dt=  ^dd   t  (233) 

V  As  +  2m'  JTp^dO 

If  we  take  the  logarithmic  differential  on  both  sides  of  this,  supposing  d  0  constant,  we  get 

(P  t  dp  Tp3dO 

=  2—^-        TO'.  -  !      ;  -  ;  (234) 


d  t     '  a 

P  A*  +  ^  m' 


/' 

i"  /  r  p3  d  0 


de 


i  /i2  /  ' 

Substituting  this  then  in  equation  (232),  and  for       g  ,  writing  its  value  +  2  TO'.  —  given  by 

(233),  it  becomes 


/>d#H 

yj  X  ^3  X    J       '  I    \  ^  f         '      -       rl  «      /      '  rf  «     I    '  V!"'V 


716  PHYSICAL    ASTRONOMY. 

^  In  this  equation  let  —  =  s,  or  z  =  p  s,  so  that  s  ~  tan  heliocentric  elevation  of  m  above  the  plane  ot  the  Astronomy. 

rand  y,  =  tan  m  MQ  fig.  (A).     Moreover,  let 

dQ  dQ 


V=  -  -  -  L.J  (236) 

then  will  V  be  the  measure,  and  TO'  .  V  the  quantity,  of  that  part  of  the  disturbing  force  which  operates  to 
increase  the  gravity  of  m  towards  the  central  body.      For  if  we  resolve  the  forces  m'  .  -  —  —  and  m' 


dx  dy 

which  act  in  the  directions  of  the  co-ordinates  x  and  y  into  the  direction  of  the  distance  p,  they  will  become 

respectively  .    x       d  Q  y       d  Q 

+  TO'.  —  .  —  —  and  +  TO'  .  —  . 


.  .       . 

pax  p       d  y 

whose  sum  is  TO'  .  V. 

These  substitutions  made,  and  p  being  put  (as  in  the  elliptic  theory)  equal  to  —  ,  the  equation  will  become 

p 
when  multiplied  by  —  —  , 


_  2 


u  \  ri*i]  . 

)J    u3  }' 


h*{u*      u3  '  de          \dt 

which  equation  is  rigorous,  nothing  having  been  neglected  in  the  previous  process.     But  if  the  squares  of  the 

disturbing  forces  be  neglected,  we  have  — —  =  1  because  *  or  —  is  of  the  order  of  the  disturbing  forces  ; 

(1  +  s  )T  p 

and  since,  in  virtue  of  this  very  equation, 


*  •  (1  +  tfl  H 
.+ 

the  equation  becomes 


-} 


-  -77  '~   x   &c- 

n* 


T      du          2  ft,    S*Td0) 

~v?  '  ~d~0 W  J  ~tf~  I  '  (238) 

Now  ft  =  1  +  m,  and  if  we  neglect  the  powers  and  products  of  the  disturbing  forces  and  the  mass  m,  the 
equation  becomes 

ift  «  n.          m'   ( V  T       d  u  9.      f»T,l  01 

(239) 


' 


p.  _  m'  (  V       _T      du        _2_    f*TA  0\ 
'         ~  4«  ~  ~h?  {  a«  ~  ~u?  '  ~d~0         AV      «3    )  ' 


It  is  on  this  equation,  (or  the  rigorous  equation  (237))  that  the  perturbation  of  the  radius  vector  is  made  to 
depend  in  the  lunar  theory.  It  remains  to  derive  an  equation  from  which  the  perturbation  in  latitude  can  be 
obtained.  For  this  purpose,  all  that  is  necessary  will  be  to  regard  x  and  y  as  known,  and  to  employ  the  third 
of  our  general  equations  (94)  to  obtain  a  value  of  z,  or,  which  comes  to  the  same,  of  s,  since  z  =  p  s,  and  p  is 
given  by  the  previous  theory. 

Now,  if  in  our  equation  (94)  we  change  the  independent  variable  from  t  to  6,  it  becomes 


z 


but  since 


dt*  p*dO*  V;    ..............  (240) 


as  appears  from  equations  (233)  and  (234)  ;  if  we  substitute  these,  we  shall  get 


T      dz\ 
'.  —  .  -T-J-  ^ 
p       d  0  } 

Now  we  have  z  =  p  i,    dz  =  pds  +  sdp,     ds  z  —  p  d*s  +  2  dp  ds  +  *<P  p,  whence  we  find 


.      P-*  ,     dQ 

+  p3  {  ~  --  h  TO'.  —  --  h  m' 
r3  dz 


PHYSICAL    ASTRONOMY.  717 

Astronomy.  1      d*  z         1  d  p       dz          d2  s  p*  s     (  d? />       2d/>*"|  Physical 


P       d  0*        p*dO      dO          d  02 
But,  by  1,235)  we  have 


p»i      (  d?  p        2d/>2)  Physical 

H — —  <  — ; —  \  Astronomy 

p  d  0*  {   p*  p3      )  , \ 


so  that,  by  substitution  we  obtain, 


d  z         s  d  p       p  d  s 

but  dz  =  pds  +  sdp;  so  that  —  —  -  ---  —  f-  =  '•  --  ,  and  we 

d  o          dO          dO 


this  equation  (  writing  u  for  —  \  becomes 


h*  +  2 
and  when  the  square  of  the  disturbing  force  is  neglected, 

•-%+•+  sy^-ir+Tf! 

Such  are  the  fundamental  equations  on  which  the  problem  of  the  moon's  motion  depends.  They  were  first 
investigated  by  Clairaut,  in  a  piece  which  gained  the  prize  of  the  Petersburgh  Academy  for  1750,  and  have 
been  deduced  by  almost  every  writer  since  his  time,  as  the  groundwork  of  the  lunar  theory.  The  method 
we  have  followed  shews  how  the  expressions  of  the  tangential  and  centripetal  disturbing  forces  peculiar  to  this 
theory,  originate  in  the  general  equations  of  the  problem  of  these  bodies  ;  and  thus  connects  the  modern  with 
the  more  ancient  methods  followed  by  Newton,  Clairaut,  &c. 

Hitherto  we  have  only  considered  the  rigorous  equations  of  the  moon's  motion.  It  remains  to  apply  to  them 
the  methods  of  approximation  appropriate  to  the  case,  and  deduce  in  succession  the  equations  of  the  disturbed 
motion.  The  great  length  to  which  the  details  of  this  complicated  process  would  lead,  will  preclude  our 
entering  minutely  into  it.  We  must  content  ourselves  with  leading  results  and  general  principles.  One  great 
peculiarity  of  the  lunar  theory  consists  in  this  —  that  the  mass  of  the  disturbing  body,  instead  of  being,  as  in 
that  of  the  planets,  small  in  comparison  with  the  disturbed  and  central  bodies,  exceeds  them  both  several 
hundred  thousand  times.  Its  vast  distance,  however,  makes  up  for  this,  and  renders  its  disturbing  effect 
small  in  comparison  with  the  attraction  of  the  central  body.  In  the  foregoing  equations  then  the  quantities 
TO,  and  m',  are  not  to  be  our  guides  in  regulating  the  orders  of  the  corrections.  The  very  small  quantities, 
whose  powers  determine  the  convergency  of  our  approximations  are  !Tand  V,  or  rather  m'  T,  and  m'  V,  for  m' 
never  occurs  unmultiplied  by  one  or  other  of  these  quantities.  Let  us  therefore,  first  of  all,  consider  the 
nature  and  magnitude  of  these  forces. 


,    / 

v  =  "*  ( 


Expression  of  the  centripetal  atUHtbmg  force,  m'V. 

f   JL  AJL\   ^   —  (xS  +  yy'^S1  1  ^     '      *"' 

p     d  y  /          p 


Now,  if  we  neglect  the  inclination  of  the  moon's  orbit,  and  take  the  pline  of  the  ecliptic  for  our  fixed 
plane,  we  have 

z  =  o,     1=0,     r  =  p,     r  -   p',     s  —  o, 
and 

-=    \r*      2  rr1  .  cos  (0  —  0')  +rsi~i' 


718  PHYSICAL    ASTRONOMY. 

Agronomy 


/rx«M  ?*™- 

=  ->3        1    -    (  2  —  cos  (0  —  6')  —  (  —  J      V,  Astronomy. 


putting  w  =  0  —  0' ' ,  and  developing  in  powers  of  — .     Hence,  we  obtain  (writing  r  for  p  in   the  expression 

for  V)  after  all  reductions, 

,       _         m'   ( I  +  3  cos  2  w     r         9  cos  w  +  15  cos  3  «>   /  r  \  2  I 

Now  -n-  is  the  accelerating  force  exerted  by  the  sun  on  the  earth,  whose  intensity,  as  we  have  shewn  in 
r  2 

Section  1.  Part  I.,  is  represented  by  "2'17399  (in  its  mean  state)  on  the  supposition  of  the  earth's  attraction  on 

r         60-23799           1                         ,    /  r  \2             1 
the  moon  being  unity.     Also,  -j-  .=   -— — - —  =  — - —  nearly,  and  (—7  I    = — —  ;   so  that  we  may  safely 

(T  \  ^  111    T 

—r]    &c.     The  constant,  or  non-periodical  parts  of  m'V  is  therefore  equal  to  —  5  .  — —   =    —    £ 

/  r  \3         1  1  1 

'»    (-7)     x-=  nearly-  —  x  -. 

Let  a  and  o'  be  the  actual  mean  distances  of  the  moon  and  sun  from  the  earth,  and  supposing 


"'(7 


a  will  be  nearly         ^,  and  the  mean  effect  of  the  centripetal  disturbing  force  will  be  —    2        indicating  a 

178'7  r 

diminution  of  the  moon's  mean  gravity,  amounting  to  -j-J-j-th  of  its  whole  quantity. 

The  moon's  mean  gravity  being  diminished  by  the  action  of  the  sun,  and  the  velocity  remaining  nearly  the 
same,  the  centrifugal  force  which  would  be  balanced  by  the  centripetal  in  a  circular  orbit  without  the  sun's 
»ction,  will  obtain  the  preponderance  when  the  sun  acts  and  carry  the  moon  farther  out  ;  thus  the  disturbing 
force  has  the  effect  of  increasing  the  distance  and  periodic  time  above  what  they  would  be  in  the  undis- 
turbed orbit. 

Let  us  next  consider  the  tangential  disturbing  force  m'  T.     Now  we  have 

' 


jp      ay 


,   x  y  —  y 
=  m  ,  — 


-x' 


m'    (      r  fr\*i,\, 

=  T.  re  l  3  —r  cos  w  +  I  -y  I  .  &c.  i  .  sin  w 


,»T=|-£..l.sinS,; 

The  tangential  force  therefore  vanishes  both  in  syzigies  and  quadratures,  and  is  a  maximum  in  octants,  or 
at  45°  of  elongation .  Now  we  have  seen,  that  the  description  of  areas  would  be  equable,  were  it  not  for  this 
tangential  force.  Hence,  at  the  former  points,  the  equable  description  of  areas  still  holds  good  ;  while,  at 
the  latter,  an  acceleration  or  retardation  takes  place.  This,  of  course,  produces  an  equation  in  the  moon's 
longitude,  whose  period  is  semimenstrual,  as  during  the  first  quarter  of  its  synodic  revolution,  the  moon  is 
continually  retarded  in  its  orbit  (supposed  circular)  :  at  the  first  quadrature  the  momentary  retardation  ceases 
and  changes  to  an  acceleration  — and  here  the  effect  accumulated  through  the  whole  of  the  quadrant  has 
reached  its  maximum.  In  the  next  quadrant,  the  moon  is  gradually  accelerated  ;  and  at  the  full  moon,  its 
motion  has  regained  all  it  had  lost,  and  is  restored  to  its  mean  state,  so  that  here  the  equation  is  nothing,  and 
to  on.  This  is  the  origin  of  that  equation  of  the  moon's  motion  to  which  astronomers  have  attached  the 
naiue  of  the  variation. 


PHYSICAL    ASTRONOMY.  *19 

Astronomy  SECTION   II.  Physical 

Astronomy 


Approximate  integration  of  the  equation  of  the  moon's  orbit. 

IN  this  inquiry  we  will  commence  with  the  equation  (238),  in  which  the  square  of  the  disturbing  force  is 
neglected.  Now,  the  obvious  mode  of  beginning  the  approximation,  would  be  (in  analogy  with  what  we  did 

in  the  theory  of  the  planets)  to  substitute  for  u  its  undisturbed,  or  elliptic  value, .      But,  as  it  is 

a  (]  —  e2) 

obvious  that  the  more  nearly  our  first  approximation  approaches  the  truth,  the  more  rapid  will  be  the  conver- 
gency  of  all  the  succeeding  ones,  we  may  take  advantage  of  what  observation  has  taught  us  respecting  the 
form  of  the  lunar  orbit,  and  assume,  for  our  first  value  oft,  an  expression  representing  that  form,  more  nearly 
than  the  ellipse.  Now,  we  learn  from  observation,  that  during  a  single  revolution,  it  is  true,  the  orbit  does 
not  deviate  materially  from  an  ellipse,  but  that  in  a  very  few  revolutions  the  elliptic  radius  vector  differs 
very  sensibly  from  the  real  one,  by  reason  of  the  rapid  motion  of  the  lunar  apsides,  which  perform  a  whole 
circuit  in  little  more  than  nine  years.  Instead  then  of  representing  by  u  the  inverse  radius  vector  of  a  fixed 
ellipse,  we  will  give  it  such  a  form,  in  terms  of  d,  as  shall  express  that  of  an  ellipse  in  motion,  and  revolving 
in  its  own  plane  in  the  direction  of  the  moon's  motion,  as  observation  informs  us  it  does. 

Let  us  then  take  e,  such  that  1  :  1  —  c  '.'.  the  moon's  motion  in  longitude  to  the  motion  of  the  apsides, 
then  will  0(1  —  c)  be  the  angle  described  by  the  apsis,  while  the  moon  describes  0 ;  and,  consequently,  sup- 
posing (as  we  may)  that  the  origin  of  the  time  t  is  fixed  at  the  epoch  when  the  apsis  coincided  with  the  axis 
of  the  x,  we  shall  have  0  —  (1  —  e)  0  =  c  0  for  the  moon's  true  anomaly,  and  in  the  place  of  taking 

1  +  e  .  cos  0  I  +  e  .  cos  c  0  , 

u  = — — ,  we  may  take  u  = — —  for  our  initial  value  of  u. 

a  (I  —  f)    '  a  ( 1  —  t  ) 

Although  the  idea  of  commencing  the  approximation  with  this  value  of  u  is,  it  is  true,  taken  from  obser- 
vation, yet,  when  once  suggested,  no  matter  from  what  source,  we  may  look  on  it  as  a  mere  analytical  arti- 
fice; and  the  truth  or  falsehood,  convenience  oi-  inconvenience  of  the  assumption  will  be  tried  by  actual  substi- 
tution ;  when,  if  we  find  that  it  reproduces  the  original  expression  with  a  train  of  small  corrections  multiplied 
by  the  disturbing  force,  or  can  be  made  to  do  so  by  a  proper  assumption  of  the  constant  c,  we  may  rest  assured 
that  the  assumption  is  (mathematically  speaking)  a  legitimate  one,  and  concern  ourselves  no  farther  with  the 
way  in  which  we  arrived  at  it. 

Now,  neglecting  e2,  we  have 

u  =  —  (1  +  e  .  cos  c0) 
a 

whence  we  get  I  1 

—  =  a3  (1  —  3  e  .  cos  c  0)  ;  —  =  a4  (1  —  4  e  .cos  c9) 

du  ce 

= .  sin  c  0 


dO  a 

and,  by  substituting  this  in  (238)  which  is  equivalent  to 

d?  u  1  ,    u'3       1  +  3  cos  2  w        3  m'  u'3    ce    . 

o  =  — —  +  u +  m  . — - —  .  —  sm  c  0  .  sin  2  w 

d  e/1  IP  /i2  u3  2  2  ft2  u4      a 

3     /•      u'3 
+  —  I  m'  —  sin  Zw  .  d 0 

Now,  tfwe  neglect  the  eccentricity  of  the  earth's  orbit,  a'  =— ,   so  that  (  recollecting  that  m' .  I  — -  J    = 

"  =  -179-) 


—  -\ (1+3  cos  2  w)  (1  —  3  e  .  cos  c  i 

See 


2  A2 

3  a 


a  .  sin  c  9  .  sin  2  w  (244) 


.    /  ( 1  —  4  e  cos  c  0)  sin  2  w  .  d  0 

h*      J 


The  only  difficulty  now  in  the  way  of  integrating  this,  is  to  express  cos  2  w  and  sin  2  w  in  terms  of  6.  Now 
we  have  w  •=  0  —  ff  ;  and  since  we  neglect  the  eccentricity  of  the  earth's  orbit,  0'  is  proportional  to  the  time 
or  ff  =  n'  t;  also  we  have  0  by  the  equation,  hdt  =  r*dO,  or 

t  -  -i-  f-^4-  =  4-  / '<<  <>  (1  -  2  e  .  cos  c  0)  a2  =  ~  (0  -  -^-  .  sin  c  0 
hj      a2  hj  h    \  c 

Hrrce,  since  h=  •/  a  very  -\n.zr\y,  and  the  motion  of  the   apsides  being  slow  in  comparison  with  that  of 
tht  n.oon  itself,  c  is  nearly  unity,  at  least  sufficiently  so  for  a  first  approximation  ;   consequently, 
VOL,,  in.  5  A 


PHYSICAL    ASTRONOMY. 

t  =  cfi  (0  —  2  e  .  sin  c  0)  Phwical 

AstroDooiv 
or  «  t  =  0  —  2  e  .  sin  e  0 


and  eliminating  t  between  this  and  the  equation  0'  =  n'  t,  we  have 
and  therefore, 


-      0'  =  6  —  2  e  .  sin  c  0 
n 


0  —  ff  =  0(l  —  --\   +  2  e  .—  .  sinctf 

V.  r  ** 

/ 

putting  n  and  n'  for  the  mean  motions  of  the  moon  and  earth.  Hence  we  find,  putting  —  =  k 
cos  2u?  =  cos  2  (1  —  k)  0  —  sin  2  (1  —  k)  0  .  sin!  4  e    —  ,  sin  c  0  I 

L        '"  J 

=  cos  2  (1  —  A:)  0  +  2  k  c  { cos  (2  -  2  k  +  c)  0  —  cos  (2  —  2  k  —  e)  0  }  ;  V215) 

and  similarly, 

sin2«;  =  sin  2  (1  —  fc)  0  +  2  A:  e  {sin  (2  —  2k  +  c)  0  —  sin  (2  —  2  A:  —  c)  0}  (246) 

It  only  remains  to  substitute  these  values  in  (244)  where  it  becomes,  after  reducing  all  the  products  of  sines 
and  cosines  to  simple  multiples,  and  rejecting  e2, 

(Pu  IS  a  \          3  e 


o  = 


+  a.  {A  .  cos  (t  —  2fr)  0  +  Ze  .  cos  (2  —  2  A:  —  c)  0  +  C  e  cos  (2  —  a  A  +  c)  0} 
where  A,  B,  C,  are  co-efficients  easily  expressed  in  functions  of  k  and  c.     If  we  integrate  this,  we  get 


+  o  j  A',  cos  (2  —  2  *)  0  +  B'  e  .  cos  (2  —  2  *  -  c)  0  +  C'  .  e  .  cos  (2  -  2  k  +  c)  0  } 

1          e 

Let  us  now  consider  the  nature  of  this  result.     We  set  out  with  u  =  --  h  —  •  cos  c  0  for  a  first  assumed 

a          a 

value  of  u  ;  and  having  substituted  this,  have  deduced  another  value,  which  ought  to  be  a  nearer  approxima- 
tion, and  which  ought  to  coincide  with  the  former,  if  a  =  o. 

Now,  the  terms  within  the  brackets  being  all  multiplied  by  a,  vanish  as  they  should  do  when  a  =  o,  and 
the  two  first  terms  of  this  expression  will  coincide  with  the  uncorrected  value  OX  V,  provided  we  take 


/;°- 
and 

01 


neglecting  a4 ;  that  is,  since  a  = ,  c  =  0-99580,  and  I  —  c  —  0-O042.      Hence,  according  to  this  view 

178'7 
of  the  subject,  the  progression  of  the  apsides  in  one  revolution  of  the  moon  is  0  O042  x  360  ,  or  about  1°  31'. 

Now,  it  is  remarkable  that  this  progression  of  the  lunar  apsides,  as  determined  by  a  first  approximation, 
is  only  half  the  quantity  actually  observed  ;  and  this  is  the  conclusion  Newton  had  arrived  at,  where,  after 
going  through  a  process  capable  of  being  identified  with  that  now  under  consideration,  he  remarks  (in  the 
9th  section  of  the  Prindpia,)  "  Apsis  lunae  est  duplo  velocior  circiter."  Every  attempt  to  obtain  a  nearer 
coincidence,  by  taking  into  account  the  higher  powers  of  the  eccentricities,  the  inclination  of  the  moon's 
orbit  (which  is  pretty  considerable,  and  materially  modifies  some  of  its  inequalities'!  &c,  failed ;  and  this 
difficulty,  which  Newton  evidently  felt,  though  he  had  passed  it  in  this  apparently  negligent  manner,  a» 
being  at  that  time  beyond  his  reach,  or  deferred  for  farther  consideration,  became  so  great  a  stumbling-block 
in  the  way  of  succeeding  geometers,  as  to  shake  their  faith  in  the  theory  of  gravity,  till  Clairaut  shewed  that 
it  would  vanish  on  pushing  the  process  to  a  second  approximation,  and  that,  in  fact,  the  value  of  c  depends, 
for  nearly  half  its  quantity,  on  a  term  containing  the  square  of  the  disturbing  force. 

To  shew  this,  we  have  only  to  substitute  the  approximate  value  of  u  (248)  back  into  the  original  equation 

<Pv  I          (m'T    du         ro'V  /  d*  «  \    /'m'T        ) 

0  =  -       +  u  ~      +        ?-  TT-  1^  +  2  (T^  +  U)J  F*  rf'l 


=  — ,     or  1  -<?  =  —  a.— 


PHYSICAL    ASTRONOMY.  721 

Astronomy.  1  Physical 

•  _,_     ^_>  Supposing  now  we  represent  the  first  assumed  value,  or  —   (1  +  e  .  cos  c  0)  by  u  and  by  6  u  all  the  terms  Astronomy. 

added  by  the  first  approximation,  then  will  the  value  resulting  from  a  second  approximation  be  had  by  sub-  *"" 
stituting  u  +  S  u  for  u  in  all  the  part  of  the  above  equation  within  the  brackets,  and  integrating.     As  we  only 
require  at  present  to  know  the  terms  introduced  by  this  second  approximation,  let  us  call  them  S*  u,  and  the 
value  of  S2  u  will  be  had  at  once  by  integrating  the  equation 

T    du         „   V    .    „  /d2  a  «    .    .    \    />  T 

(249) 


a3 
Now.  since  ft2  =  a  very  nearly,  and  a  =  m'  .— ,  we  have 


m'   *  (  T    du  \         3        *  fsin  2  to    d  u  | 
A>      \tt»    d  6  )   ~~  T  "      \  (an)*     d  0  } 
3        (2  cos  2  w    d  u  sin  2  to     d  S  u  sin  2  w    d  u          ) 

=  ^a\-(^^Sw+-(a^-ir0-  -4a-^ur-dTSu} 

m'       V  a        1  +  3  cos  3  to          3  a    (  2  sin  2  u>  1  +  3  cos  2  u>        ) 

i  —  =  —  —  X  =  I 6  w  +  a    S  u  > 

A2       u8  2 a  (a  a)3  2  a     (     (a  u)3  (aw)4  j 


TOT         3  a       sin  2 


(a  w)4 


/„  m'  T  ,  /»fcos  2  MJ  sin  2  to   „     )  , 

5  —  —  d0=3a  /       -  —  Sw  —  2  a  -;  -  —  «M>d£» 
ASMS  ^7    1  (ai/)4  (au)5          J 

Let  these  values  be  substituted  in  the  general  equation  (249),  and  it  becomes 

) 

u  \ 


(P  &*  u  3(2  cos  Zw    du  sin  2  w  d  S  u  sin  2  w     cl  ti 

-      +  *  u  +     °  -~       Sw  +       "       '  -  4  a       - 


sin  2  w  ]  +  3  cos  2 


/  (P  u  \    /*(cos2to,  sin  2  w        ), 

+   6  °   (  ^TST  +  "    )   /  i  7  -  TT  ^  «>  —  2  a  -  -  —  S  u  [  d  0 
\  dO*  JJ    (.  (au)4  (au)s         J 


To  reduce  this  equation  to  the  degree  of  approximation  required,  let  us  agree  to  neglect  all  terms  which 
contain  a5  multiplied  by  any  power  of  the  eccentricity ;  or  in  that  part  of  the  approximation  which  depends  on 

the  square  of  the  disturbing  force,  to  regard  the  orbits  as  circular.     We  shall  have  then  •«  =  —  a  u  =  1,  and 

a 
a 

supposing  S  u  =  —  .0,  where  0  is  the  sum  of  any  number  of  terms  of  the  form  A  .  cos  i  6,  we  have 


1  A  d  e     —  l  r* s « _,          2  a*    /^ 

=-r-  I  s  -—.  =  — —  /  — _  d  0  = __  «  /  0  d  e 

hj         u*  h  J      u3  h       J  ^ 


or  since  n  =  — -  =  — - 
OT        a- 

/», 

c . n  t  ^  —  2«  /  0dp 

n' 
consequently,  since  w  =  0  —  0  =  0 t  =  0  —  k.nt,  we  get 

to  =  0  (1  -  Ar),      and£a>  =  —  A:.£n<  =  2fc«/0d0 

du                        1 
We  have  also  -—  =  o  and —  =  1 ;  so  that  the  equation  (260)  reduces  itself  to 

d*  «*  u  3  a*     .  d  0         6  k  as  f>  So2 

0  =    'sat      +  S    "  +  ~c   -  sm  2  to  .  --£-  +-     -sin2oi./0d0+  - —  (1  +  3  cos  2 

a »  a  »/  i  a 

3       a 
a" 


12  a2 


2    /•(  /'  ) 

/  Ucos2  w  /  0d0  —  sin  2  to  .0>d 


722  PHYSICAL    ASTRONOMY. 

Astronomy      Let  us  now,    more  particularly,  consider  any  one  of  the  terms  to  which  0  consists  ;  as,  for  instance,    Physical 
-•• ~V~-  A  .  COS  B  0.     This  gives  Astronomy. 

0  =  A.cosB0,      -~=  —  AB  .sin  B  0,    — £  =  -  AB'.cos  B  0  ;  I  <j>  dO  =  —  ,.sin  B  0,  &c. 
o  &  d  0  tj 

and  the  equation  will  give,  on  substitution  and  reduction, 

d*  S°~  u  f  6  ft  a2      A         3  a*  ) 

o  =  — — — -   -t-  a*  u  +  < .  -=r- .  A  B  >  sin  2  10  .  sin  B  0 

d  6*  [     a          B         2  a  J 

3  n*  3a*        cos  2  w  1 

-  (1  +  3.  cos  2  KJ)  +  (B5—  1)  . .  -  -U.cosB0 


or  resolving  the  products  of  sines  and  cosines  by  the  formula  (A),  page  690, 
o  =  — - — -  +  Pu  H — —  (  —  —  —  J  f  cos  (B  —  2  +  2 ft)  0  —  cos  (B  +  2  —  2 ft) 

u  (ft  2  (I  D  2  x 


3  A  a* 
H .  cos  B  0 

2  a 


(B  -2  +  2ft)0  +  cos  (B  +  2-2ft)0} 


That  is,  assembling  together  the  co-efficients  of  like  terms, 

rf2  &  u  3  A  o*  (       k        B         3  Ba  —  1  2  (ft  +  B) 


A  o*  (       k        B         3  Ba  —  1  2  (ft 

T1     B  -  T+  T+  4TT^)  -  B(B-2  +  2ft) 

3  A  «2  f       A         B         3          B*  -  1  2  (ft  -  B) 

—    -B  +  T 


cos      - 


3  A  a5 

H .  cos  B  0 

2  a 

Which  equation,  integrated,  gives 

3  A  «2     fcosB0       /     2fe       3  -  B        B*  -  1  4  (k  +  B)       \  cos  (B  —  2  +2  A)  0". 

<2a     '|B'-  1  +V     B"  "         2  2(1  -k)  ~  B  (B  -  2  +  2 /c)/ (B- 2  +  2  A)2- 1  I 

2*      3  +B        B2  —  1  4  (A  —  B)       NCOS  (B  +  2  — 2fe)0  I 

¥H        2        f  2(1  -if)  ~  B  (B  +  2-2  A)/  (B  +  2-24)*-!  J 

Such  is  the  value  of  that  part  of  8s  a,  which  originates  in  the  term  A  .  cos  B  0  in  the  value  of  o  or  in  the 
term  —  .  A  cos  B  0  in  u  itself.  A  similar  set  of  terms,  having  for  their  arguments  respectively, 

B'  0,     (B'  -  2  +  2  k)  0.     (B'   +  2  -  2  ft)  0 
and  B"  0,     (B"  -  2  +  2  k)  0,     (B"  +  2  -  2ft)  0  ;  &c. 

will   arise    from   the    other   terms  —  A'  .  cos  B'  0,     —  A" .  cos  B"  0,  &c.      Consequently,   by  substituting 

a  a 

for  B,  B',  B",  &c.  the  several  values  which  they  actually  have  in  the  first  approximate  value  ot  u  (equation 
248),  or  2  —  2  k,  2  —  2  ft  —  c,  2  —  2  ft  +  c,  we  shall  obtain  the  arguments  of  the  several  terms  in  the 
second  approximation,  and  so  on.  These  are  then  as  follows  : — 

From  c  0  arise  the  arguments    c  0,  2  c  0 

From  (2  —  2  ft)  0  arise  the  arguments  (2  —  2  k)  0 ;  c  0  ;     (4  —  4  ft)  0 
From  (2  -  2  ft  —  c)  0  arise  (2-2ft-c)0;    c  0  ;     (4-4ft-c)0 

From  (2  —  2ft  +  c)  0  arise  (2  -  2ft  +  c)  0 ;    c0  ;  (4  -  4  ft  +  c)  0 

Thus  we  see,  1st.  that  each  term  which  in  the  expression  of  u  is  multiplied  by  the  first  power  of  the  dis- 
turbing force  a,  reproduces  itself  on  a  second  approx'mation,  multiplied  by  the  square  of  a,  and  thus  the 
co-efficient  of  every  term  in  the  general  expression  of  i,  consists  in  fact  of  an  infinite  series  of  powers  of  a  ;  for 
what  takes  place  at  the  second  approximation  will,  of  course,  do  so  at  every  succeeding  step  ;  and,  in  fact, 
the  very  same  analysis,  and  the  same  resulting  formula,  may  be  applied  to  deduce  the  terms  depending  on 
a*,  o4,  &c.,  in  all  of  which  the  same  arguments  will  occur. 


PHYSICAL    ASTRONOMY.  723 


Astronomy.      Hence^  i(.  foyOWSj  that  if  we  set  out,  as  a  first  hypothesis,  with  u  =  --  (  I  +  e  .  cos  c  6)     the  process    of  Astronomy. 


approximation  leads  us  to  a  value  of  u,  of  the  form 

U  =  —  {  A   +  B  a  .   +  C  a*  4-  D  a-  +  &C.  }    -  — 
a  ft 

+  cos  c  0  {  A'  a  -4-  B'  a2  -•-  C'  a3  +  &c.  } 
+  cos  (2  —  2  fe)  6>  .  {  A  '  a  -r  Is"  a2  +  &c.  } 
+  cos  (4  —  4  A)  6)  .  {B"'  a5  +  &c.  } 
+  cos  (2  —  2  A;  +  c)  {  A"  a    t   Biv  a2  +  &c.  j 
+  cos  (2  —  2  k  —  c)  {  A."  a  }-  Bv  a2  +  &c.  } 
+  tic. 

We  have  therefore  only  to  take  care  that  this  shall  not  contradict  the  original  hypothesis,  and  therefore  we 
must  have 

1  =  A  +  B  a  -|-  C  a2  +  &c.  •    ^-  (253) 

A 
-  =  A'  a.   +  B'  a*  +  C'  a'  -|     ftc.  (254) 

These  equations  established,  the  two  first  terms  of  u  agree  with  the  original  value,  and  the  remaining  ones 
are  the  equations  due  to  the  effect  of  perturbation,  not  capable  of  being  expressed  by  the  hypothesis  of  a 
revolving  ellipse. 

The  equation  (253)  expresses  a  relation  between  a  and  h,  which  we  shall  consider  presently  ;  the  other 
(254)  gives  the  value  of  1  —  c,  or  the  velocity  of  the  motion  of  the  lunar   apogee.      We  have  already  seen 

3  e 
(248)  that  the  value  of  A'  is  —  —  -  —  ,  and  if  we  break  off  the  equation  (254)  at  the  first  power  of  a  we 

AI  f  I    (  I  ~-  C  ) 

have  also  seen  that  the  resulting  value  of  1  —  c  is  J  a,  amounting  to  only  half  the  value  given  by  observation. 
If  we  make  B  successively  equal  to  2  —  2  k  —  c  and  2       2  k  +  c  in  (252),  we  find  for  the  co-efficients  of 
the  argument  c  0,  respectively, 

1           3a*e  f2  +  6k  +  c  -  4k*  —  4kc  —  .-'  (2-2fc-c)*-l  4  (2  —  k  —  c)   ) 

c4  —  I  '    2  a  l~~            2  (2  -  2  h  —  c)  2  (1  —  k)  c  (2  —  2  k  —  c)  ) 
and 

1           3ase  f-2  +  6k-  c  -4k*  +  4kc  -c*  (2-2fc  +  c)2  —  1  4  (2  —  k  +  c)    j 

c4—  1  '    2a  V                2  (2  -  2  /c  +  c)  2  (1  —  k)  ~  c  (2  —  2  A:  +  c)  } 

when  B'  and  C'  are  the  co-efficients  of  e  .  cos  (2  —  2  k  —  c)  0  and  e  .  cos  (2  —  2  A:  +  c)  0  within  the 
parentheses  of  equation  (248). 

Thus  our  equation  (254)  carried  as  far  as  a?,  will  become 


/2-6Jf  +  c-4Ar"-4fec  —  c8       (2—  2A:  —  c)°-  1        4(2  —  fe-c)\ 
\  «(9-2*-c)  2(1  -k)  c(S-«*-c)/ 


1  -c*=—  a+  -a<W  W25*» 

^  ^2  +  6fr_c_4fr«  +  4ftc_c«      (2-2fe  +  c)a-l        4(2-/r  +  c)\    ; 
"!?  C    I  ^~7^       ^~;    ;     C  r  - 


2T2-2A  +  C)  2  (1  -  k)  c(2 

This  equation  is  to  be  resolved  by  approximation,  so  as   to  find  c  ;  and  as  a  is  a  very  small  quantity,  the 

2 
approximation  may  be  performed  easily  ;  for  we  have  only  to  rescind  oa,  and  take  c  =  1  --  —  a  for    a  first 

approximation  =  O9957,  and  then  substitute  this  for  c  in  the  co-efficient  of  a?,  which  will  then  become  a 
given  number  (B',  C',  being  previously  computed,  and  k  being  known,)  after  which,  if  this  co-efficient  be 
called  ft,  we  have 


and,  again  substituting  this  for  c,  we  find  a  nearer  approximation,  and  so  on.' 

Now,    it  is   a  remarkable   circumstance,  that   when  this   process   is  executed  in   numbers,   the   term 

—  '      .     a?  thus  added  to  the  value  of  c  is  found  to  be  very  nearly  equal  to  the  term  --  —a  introduced    by 

8  4 

the  first  step  of  the  approximation  ;   and  on  pushing  it  still  farther,  the  subsequent  corrections  are  found  to 
be   inconsiderable.     Now   the  velocity  of  the  apogee   of  the  lunar   orbit   is  expressed  by  1  —  c,  or  by 

—  a  +  —  -  a4  +  &c.    We  have  already  seen  that  the  value  of  this,  as  deduced  from  a  first  approximation 


724  PHYSICAL    ASTRONOMY. 

\stronomy.  3  Physical 

_,       _'.  (or  the  part  —  a)  came  out  only  half  what  observation  has  assigned  as  its  real  amount,  and  that  this  was  Astronomy. 
^v^  4 

regarded  as  a  great  difficulty  in  the  way  of  the  full  admission  of  the  Newtonian  law  of  gravity.  We  now  see 
the  reason  of  this  ;  the  remaining  half  is  accounted  for  by  the  term  —  -  o2  arising  from  a  second  approxi- 

o 

mation  ;  and  the  fact,  so  far  from  being  an  objection  against  gravity,  is  thus  converted  into  a  most  cogent 
argument  in  its  favour — its  effects  being  thus  shewn  to  correspond  not  merely  to  general  results  and  first 
approximations,  but  to  the  refinements  and  niceties  of  theory. 


SECTION  III. 

Expression  of  the  moon's  mean  longitude  in  terms  of  the  true,  and  vice  versd;  and  of  the  variation  and  evectwn 

of  the  moon. 

THE  purposes  of  astronomy  require  us  to  know  the  moon's  true  longitude  and   latitude  at  any  assigned 
instant,  or  to  know  the  angle  6  in  terms  of  the  time  t.     For  this  we  must  have  recourse  to  the  equation. 

/*  />*  d  e  /•  d  o 

t   —    /    -  .  -    ....  —  -T-    —    i     --  z  --------- 

Vft*  +  2m'  i'TP3de  «s    V  ft*  +  <>, 

If  we  develope  this,  and  retain  only  the  first  power  of  the  disturbing  force,  we  get 


because  V  =  a,  if  the  dict'>r^;n<r  force  and  square  of  the  eccentricity  be  neglected.     Now,  in  this  for  u  let  us 
substitute 

«  =  —  .  (1  +  e.cosctf)  +  «  {A',  cos  (2  —  2  A:)  0  +  B'e  .  cos  (2  —  2  A'  -  c)  0  +  C'e.cos  (2-2i  +  c)  0} 
a 

m     /»  T    ,  3  a     /'sin  2  w 

and  for  —  /  —  d  0  its  value  —  —  -  /  —.  —  —  d  0 
h*J    u3  2  J     (aw)4 

=  —?-  jsm  2  ic  (1  —  4  e  .  cos  c  6>)  d  0 

=  Ai.  C{  sin  (2  -  2  k)  0  -  (2  -  2  k)  e  sin  (2  -  2  k  +  c)  0  -  (2  +  2  k)  esin  (2  -  2*  -  c)  0  \  dO 


<2  (k  — 
whence  we  find 


s(,_2  ,  +  c)e  +  cos  (2_2fc_ 


or  putting  —  =  n 


n  t  -  0  —  — —  .  sin  c 0  +  P a .  sin  (2  —  2  k)  O  +  Qae .  sin  (2  —  Ik  +  c)  0  +  Roe. sin  (2  —  2 k  —  c)  0 
c 

A',  B',  C',  being  as  in  (248) 
where 


«-  -  ^ 

2_2^  +  c      r  «   ' 

_  a  (A/  -  2  B') 

2_2Ar-c       "   4  (A-  1)  (2-2^-c)    '     (2  -  2  A  -  c)4  . 
In  order  to  have  the  value  of  0  in  terms  of  n  t,  we  must  revert  this  series,  and  the  simplest  mode  of  accom- 
plishing this  will  be  to  follow  the  steps  by  which  we  resolved  the  equation  n  t  =  v  —  e  .  sin  v  in  the  elliptic 
theory  (vide  page  656,  col.  1,  equation  26).     Putting  it  therefore  into  the  form 


PHYSICAL    ASTRONOMY.  ,"25 

2  e  Physical 

0  =  n  t  +  .  sin  c  0  —  P  a  sin  (2  —  2  A)  0  (259)  Astronomy 

—  Q  a  e  .  sin  (2  —  2  K  +  c)  0 

—  R  a.  e  .  sin  (2  —  2  fc  —  c)  0 

«  /  appears  to  be  the  first  approximate  value  of  9,  neglecting  e  and  <i.  Let  this  be  substituted  in  the  whole 
of  the  second  member,  and  we  get 

2  e 

0  =  n  t  -\ .  sin  en  t—  P  a  .  sin  (2  —  2  A:)  nt  (260) 

c 

—  Q  a  e  .  sin  (2  —  2  k  +  c)  n  t 

—  R  a  e  .  sin  (2  —<2k—c)nt 

for  a  second  approximation  carried  as  far  as  the  first  powers  of  e  and  a.  If  we  now  substitute  this  value  in 
(259)  the  terms  multiplied  by  e  and  a  only  will  of  course  remain  ;  but  in  addition  to  those  multiplied  by  a  e, 
others  will  be  introduced ;  and  if  we  reject  e3  and  aa,  and  retain  only  a  e,  these  terms  will  be,  (putting  c  =  1) 

—  2  P  a  e  .  sin  (2  —  2  k)  n  t  .  cos  c  n  t 

—  2  P  a  e  (I  —  k)  .  cos  (2  —  2  k)  n  t  .  sin  c  n  t 
whose  sum,  after  the  usual  reductions,  is 

—  P(2—  k)  ae  .  sin  (2  —  2  k  +  c)  n  t  —  Pkae  .  sin  (2  —  2  fc  —  e)  nt 
Consequently,  if  we  push  the  approximation  only  to  such  terms,  we  get 

2  e 
6  =  n  t  +  -    -  .  sin  c  n  t  —  P  a  .  sin  (2  —  2  k)  n  t  (26 1 ) 

—  {Q  +  (2  —  k)  P}  ae  .  sin  (2  —  2  A:  +  c)  n  < 

—  { R  4-  k  ? }  a  e  .  sin  (2  -  2  t  -  e)  n  t 

The  variation  of  the  moon  is  that  inequality  which  is  represented  by  the  term  —  P  a  .  sin  (2  —  2  A)  n  t.  The 
numerical  magnitude  of  the  co-efficient  being  considerable,  (2146' )  it,  (as  well  as  the  inequality  represented 
by  the  term,  whose  argument  is  (2  —  2  k  —  c)  n  t,  which  is  called  the  evection,  and  whose  co-efficient  is  still 
greater  (4830'')  was  long  observed  before  its  cause  was  known.  If  we  observe  that  n  t  being  the  mean  longi- 
tude of  the  moon  k  n  t  will  be  that  of  the  sun,  and  if  we  put  ])  for  the  former,  and  O  for  the  latter,  the 
argument  of  the  variation  will  be  2  ])  —  2  O,  or  simply  ])  —  Q,  and  its  period,  half  a  synodic  revolution 
of  the  moon  =  14d  765.  The  argument  of  the  evection  is  in  like  manner 

2   D   -  2  O  —  c 

where  c  is  the  longitude  of  the  moon's  perigee,  and  its  period  is  nearly  that  of  the  variation,  a  little  longer, 
however,  on  account  of  the  progression  of  the  perigee  during  a  revolution  of  the  moon.  The  term  depending 
on  the  argument  (2  —  2  k  +  c)  n  t,  or  2  ])  —  2  O  +  c,  is  found  on  calculation  to  have  its  co-efficient  very 
small,  and  has  no  particular  name. 

In  the  theory  we  have  now  investigated  we  have  been  only  anxious  to  simplify  and  curtail  as  much  as  pos- 
sible the  developements,  it  being  far  beyond  our  design  to  enter  into  minutiae  on  so  complicated  a  problem. 
It  must  suffice  to  have  shewn  in  rather  more  than  a  general  way  the  origin  of  the  principal  inequalities,  and 
traced  them  from  the  differential  equations  to  their  final  expression  in  the  value  of  the  longitude.  In  so 
doing,  we  have  purposely  neglected  the  inclination  of  the  moon's  orbit,  and  its  effect  both  in  modifying  the 
disturbing  forces  which  act  in  the  plane  of  the  orbit,  and  in  altering  the  longitudes  by  reducing  them  from 
their  actual  values  to  their  projections  on  the  plane  of  the  ecliptic,  or  by  .what  is  called  in  astronomy,  the 
"  Reduction  to  the  ecliptic."  The  effect  of  the  inclination,  however,  will  be  briefly  touched  on  in  the  next 
section. 


SECTION  IV 

Of  the  effect  of  the  inclination  of  the  plane  of  the  moon's  orbit.  —  Of  the  motion  of  the  nodes,  and  the  precession  of  the 

equinoxes. 

IN  determining  the  effect  of  the  inclination  of  the  lunar  orbit  to  the  plane  of  the  ecliptic  and  the  motion  of 
its  nodes,  we  shall  suppose  the  orbit  circular,  and  neglect  the  square  of  the  disturbing  force,  the  cube  and 
higher  power  of  the  inclination,  and  the  product  of  the  disturbing  force  and  inclination.  Thus  our  equation 
(237)  will  become  simply, 

rf««  1  m    fV         2    /'Td0) 


The  value  of  the  part  depending  on  the  disturbing  forces  m'  Vandm'  Tin  this  equation  remains  unaltered,  if  we 
neglect  as4;  and  therefore  the  part  of  u  arising  from  perturbation  is  not  altered  when  we  only  push  the 
approximation  to  the  extent  now  proposed.  The  only  new  terms  arising  in  the  value  of  u  from  the  introduction 


/26  PHYSICAL    ASTRONOMY. 

'y<  "f  *>•»  inclination,  are  those  introduced  by  the  term  ^          ;    and  to  obtain  them,  we  have   only   to  Astronomy 

integrate  the  equation 

3 


neglecting  s4.     Now,   in  this  we  must  put  for  s  its  undisturbed  value,  which  is  given  by  the  equation 

<P  s 
— —  +  s  =  o  (making  m'  =  o,  in  equation  240)  und  which  is 

8  =  7.  sin  (0       N)  (263) 

where   7  is  the  tangent  of  the  inclination,   and  N  the  longitude  of  the  ascending   node.       This   gives 
s  _.    „    1  —  cos  2  (0  —  N) 

iPu  1 


which  integrated,  gives 

1  3   7"  ry2 

»  =  Tg- 75- -rr  cos  2  (0  —  N)  (264) 

Thus  we  see  that  the  effect  of  the  inclination  is  to  add  to 

u  =  —  or  u  =  —  the  terms ^—  (3  +  cos  2  (G  —  N)) 

n*  a  4  n' 

Now,  in  approximating  to  u,  we  assumed  in  the  foregoing  pages  an  expression  for  a  first  value  derived  not 
a  priori  from  theory,  but  from  the  observed  motion  of  the  lunar  apsis.  It  is  equally  a  matter  of  observation, 
that  the  node  of  the  lunar  orbit  does  not  remain  fixed,  but  is  continually  retreating  on  the  ecliptic,  in  a  direc- 
tion contrary  to  the  motion  of  the  moon  in  its  orbit.  Let  us  then  introduce  this  modification  into  our  expres- 
sion for  u  above  found,  and  taking  for  the  origin  of  the  time,  the  moment  when  the  node  coincided  with  the 
axis  of  the  x,  we  have  only  to  write  g  0  in  place  of  0  —  N,  when  we  get 

1    (          3  7*  I 

u  =  —  J\ —  7s — .  cos  2  g  0  >  Vi,65) 

«*    (  4  4  J 

Let  us  next  investigate  the  effect  of  the  inclination  on  that  part  of  u  which  arises  from  perturbation.  Now 
this  may  be  done  at  once,  by  merely  taking  in  250,  251, 


u=  —  0  = (—  7*  +  —  7s  cos  2  g  0  ) 

'•  a     \  4  4  / 


or                                                                     3    7s  I     7s 

</>  =  --  -—.cos  o.O        --  -cos2g0 

4     a  4      a 

3  <V9  -Y9 

tnat  is,  supposing  in  252,     1st.  A  =  --  —  ,  B  =  o,    and  2dly,  A  =  --  ;  —  and  B  =  2g 

4  a  4  a 

The  first  supposition  gives 


and  the  second  in  like  manner  introduces  into  the  value  of  «  the  new  arguments  cos  (2  g  —  2  +  2  k)  0  and 
cos  (2  g  +  2  —  2  k)  0  with  co-efficients  affected  with  a  73. 
These  arguments  existing  in  the  expression  of  u  will  in  like  manner  be  introduced  into  the  value  of  t  or 

y*dO 
-  —  -  and  thence,  by  the  process  of  reversion  before  explained,   will  arise  terms   of  the   form  a  7*  .  sin 


g  —  2+2/c)nt  and  a  7"  .  sin  (2  g  +  2  —  2  k)  n  t  in  the  longitude,  of  which  this  mention  must  suffice,  as 
their  co-efficients  may  easily  be  calculated  from  the  principles  above  laid  down,  and  nothing  very  remarkable 
in  the  lunar  theory  depends  on  them.  Other  terms  also  will  arise  from  the  part  of  (250)  multiplied  by 

d  U  .  .  fir  ry5 

—  —  -  which  in  this  case  is  not  =  o  as  was  supposed  in  (251)  but  is  equal  to  |—  sin  2  g  0. 

Ai    ft 

Let  us  now  examine  the  manner  in  which  the  disturbing  forces  affect  the  moon's  latitude.  For  this  purpose 
we  must  take  the  equation  (241)  in  which  writing  for  s  the  expression  7  .  sin  (gv  —  N),  for  —  —  its  value 

e  7  •  cos  (g  v  —  N)  for  m'  T  and  m'  V  their  values  —  -^  sin  2  (1  —  A)  0  and  -  —  -^  (1  +  3  cos  2  (1  —k)  0) 


I 

PHYSICAL    ASTRONOMY.  727 


Astronomy.  .    dQ    .  m' z          m'  p  s          m' a  s         m'as   /  a  \     .  Physical 

^^~J  and  for  m'  —  its  value  —  =  —~-  =  — 5^—  =  -^-  (\  +  3  -  coiw)    (see  page  718,    line  4,)  Astr^nomy 


a, 
or,  neglecting  the  product  s  x  —,  simply 

d  Q         m'as         a 

™   —: =  — TT-  =  -r  * 

d  z  a3          a3 

we  shall  find 

d*  s  3 

o  = +  s  +  —  a  7  .  g  sin  2  (1  —  k)  cos  0  (g  0  —  N) 

-I-  —  s  (1  +  3  cos  2  (1  -  k)  0)  +  as 
2 

and  resolving  the  products  of  sines  and  cosines  into  simple  sines, 
d8*  3  a 


o  = 


+  s  +  --o-y  •  {sin  (2-2fc+£.0-N)  +  sin  (2  -2  k  -  g  .  0  +  N)  } 


-1-  —  a  .  7  sin  (gO  -  N)  +  --  07  {sin  (2  -  2  A:  +  £  .  0  -  N)  -  sin  (2  -  2  A  -  g  .  0  +  N)  } 

4- 


T a  7 ' sin  (g  e  ~  N) 


+   —  a  -y  (1  +  g)  .  sin  (2  -  2  A:  +  g .  0  -  N) 


-   —  a-y  (1  -g)  .  sin(2-2/c-g.0 


The  integration  of  this  gives 
3  a  7 


.  _  •  sin  (g  6  —  N)    f  a  7  sin  (2  —  2  k  +  g  .  0  —  N)   X  &c. 

2  (g*  —  1) 

+  07  sin  (2  —  2  i  —  #  .  0  +  N)   X  &c. 

Now  the  process  of  approximation  ought  necessarily  to  reproduce  the  original  value  5=7.  sin  (g&  —  N). 
Hence,  exactly  as  in  the  case  of  the  apogee,  we  ought  to  have 

0  ,    °7-;x  •  sin  (g  0  -  N)  =  7  .  sin  (g0  -  N) 


and  3  a  3 

g*  =  1  +  T,     ^  =  1  +  T  « 

^ 
Thus  we  obtain  the  motion  of  the  node,  for  g  —  1  or  —  a  expresses  the  ratio  of  the  retrograde  velocity  of  the 

node  to  that  of  the  moon  in  its  orhit.  If  we  execute  the  calculation,  we  find  this  ratio  to  be  that  of  0-0042  :  I 
(see  page  720,  line  15  from  bottom,)  and  it  is  remarkable,  that  it  is  exactly  the  same  as  a  first  approximation 
gives  for  the  direct  motion  of  the  apogee  ;  but  there  is  one  remarkable  difference,  that  in  the  present  case,  this 
first  approximated  value  is  very  near  the  truth  as  given  by  observation,  and  is  little  altered  by  a  second 
approximation  ;  whereas,  in  the  other,  the  repetition  of  the  process  doubles  the  value. 

The  regression  of  the  moon's  nodes  is  a  circumstance  in  their  theory  which  admits  of  a  very  easy  and  fami- 
liar illustration. 

The  part  of  the  disturbing  force  which  acts  in  the  direction  of  the  z,  or  which  tends  to  draw  the  moon  out 

of  the  plane  of  its  orbit  is  m?  -  =  m'  .  (  —,  ---  —  |  .     If  we  wish  to  know  the  whole  force  which 

d  z  \r'3  X3     / 

tends  to  draw  the  moon  in  a  direction  at  right  angles  to  the  plane  of  its  own  orbit,  we  must  assume  that 
plane  for  the  plane  of  the  x,  y  ;  then  will  z  =  o,  and 


O  ,    ,  (\  1\ 

—  =  m'  z    [  -J-   —  —  I 

z  \rs        \s  / 


Now  w'    I  -jj.   —  —  -  )  =    --  r-rr-   cos  ic  —  nearly  --  -^r  cos  w,  and  zf  being  a  perpendicular  let  fall 
\r3          X  *  /  r4  a*  a 

from  the  sun  on  the  plane  of  the  moon's  orbit  is  equal  to  r'  or  a  multiplied  by  the  sine  of  the  sun's  angular 
distance  from  the  node  (sin  (0'  —  N))  and  by  that  of  the  inclination  or  by  7.     Hence,  the  expression  of  this 

torce  is  3  a  , 

--  —  x  cos  w  .  sin  (ff  —  N)  .  sin  7 
a* 

a-  is  the  moon  s  angular  distance  from  the  sun  as  seen  from  the  earth,  and  therefore  cos  w  is  equal  to  sine  of 
VOL.  in.  5.  B 


728  P  H  Y  S  I  C  A  L  A  S  T  R  O  N  O  M  Y. 

Astronomy.   ([  's  dist.  from  quadratures.      So  that  we  have  PbvsicmV 
3  a                                                                                                              -                                                                                                       Astronomy 
—  x  sin  <[  's  dist.  from  quadrat.   X  sin  O  s  dist.  from  8   X  sin.  inclin. ;                                    (268) 


for  the  value  of  the  force  in  question.  Let  us  now  examine  how  this  force  tends  to  produce  a  motion  in 
the  node.  To  this  end,  suppose  the  moon  to  set  off  from  its  ascending  node,  and  at  any  point  in  its  orbit 
(supposed  circular)  to  describe  an  infinitely  small  arc  which  will  be  a  portion  of  a  great  circle  seen  from  the 
earth's  centre.  If  the  disturbing  force  at  this  moment  ceased  to  act,  this  arc  would  be  a  portion  of  the 
undisturbed  orbit,  and  being  prolonged  backwards  would  cut  the  ecliptic  in  the  ascending  node,  Q,  whose 
position  therefore  would  be  the  same  at  the  beginning  and  end  of  this  infinitesimal  instant.  To  limit  our  ideas, 
let  us  conceive  the  moon  to  be  within  90°  from  the  Q,  and  therefore  its  motion  is  from  the  plane  of  the 
ecliptic.  Now,  let  the  disturbing  force  act,  and  suppose  its  direction  to  be  such  as  to  draw  the  moon 
out  of  its  orbit  towards  the  plane  of  the  ecliptic,  then  will  the  elementary  arc  actually  described  by  the 
moon  in  virtue  of  its  own  inertia  combined  with  the  new  impulse  given  by  this  force,  be  less  inclined  to  the 
ecliptic  than  the  last  described  portion  ;  and  therefore  being  produced  backwards,  will  cut  the  ecliptic  in  a 
point  behind  the  former  place  of  the  node.  This  point  is  the  new  or  consecutive  place  of  the  node,  which 
therefore  has  retreated  on  the  plane  of  the  ecliptic  by  the  action  of  the  disturbing  planet. 

On  the  other  hand,  had  the  disturbing  force  been  directed  from  the  plane  of  the  ecliptic,  the  new  path  of 
the  moon  would  be  more  inclined  than  in  the  preceding  instant  ;  and  therefore,  produced  backwards,  would 
cut  the  ecliptic  in  a  point  more/oricarrf  than  the  previous  place  of  the  node,  so  that  under  these  circumstances 
the  node  advances. 

Thus  we  see  that  when  the  moon  moves  from  the  ecliptic  and  the  force  acts  to  it,  the  node  retreats  ;  but 
when  the  force  acts  from  it,  the  node  advances. 

Again,  let  the  moon  be  approaching  the  ecliptic  ;  then,  if  the  force  act  to  that  plane,  it  will  approach  it 
more  rapidly,  and  will  cut  it  in  a  point  nearer  than  the  node  to  which  it  is  approaching.  This  node  (and  of 
course  both)  will  then  move  to  meet  the  moon,  or  in  a  direction  contrary  to  its  motion,  i.e.  will  retre.it  ;  and 
vice  versd,  if  the  force  act  from  the  plane,  the  node  advances. 

From  this  analysis  of  all  the  cases,  we  see  that  whenever  the  disturbing  force  tends  to  elevate  the  moon 
from  the  plane  of  the  ecliptic,  the  node  advances,  and  in  every  other  case  retreats.  Now,  it  is  easily  seen,  that 
the  former  condition  never  holds  good  unless  the  moon  is  between  the  node  and  the  quadrat  jres  ;  and  as  the 
extent  of  the  angle  in  which  this  can  happen  during  a  whole  revolution  of  the  moon  in  its  orbit,  is  necessarily 
less  than  two  right  angles,  the  preponderant  tendency  of  the  node  on  the  average  of  a  whole  revolution  is  always 
in  favour  of  its  retreat.  In  fact,  when  the  node  is  in  quadratures,  it  retreats  at  every  instant  of  the  lunation  ; 
and  in  the  most  unfavourable  case,  when  the  node  is  in  syzigies,  its  retreat  is  barely  counterbalanced  by  its 
advance,  and  the  node  only  rests  for  an  instant  ;  the  sun  being  then  for  a  moment  in  the  plane  of  the 
lunar  orbit. 

The  general  tendency  of  the  node  to  recede  on  the  ecliptic  is  thus  clearly  made  out  ;  but  we  may  go  further 
on  these  principles,  and  make  the  quantity  of  its  recess  a  matter  of  calculation.  For,  let  us  denote  by 

K  /  1  \ 

—  the  force  expressed  in  (268).     Then,  since  the  lunar  gravity  (—  I    draws   the   moon  in  the  instant  of 

time  d  t  through  the  versed  sine  of  an  arc  =  a  d  >>    or  through   a  space  equal  to  --  —  ,  the  force  —   will 

2a  a* 

(a  d  0)* 
draw  it  in  the  same  time  through  the  space  K  .  —  -  -  .     The  inclination  therefore  of  its  new  path  to  its  old 

will  be  represented  by  the  infinitely  small  angle 

{a  d  e]* 


K 


add 

Let  the  new  elementary  portion  be  prolonged  till  it  meets  the  ecliptic  in  Q',    Q  being  the  former  place  of  the 
node,  then  we  shall  have  Q  S,'  for  the  momentary  change  of  the  node's  place.     Now  this  is  the  side  of  a  sphe- 

rical triangle    opposite   to    the  infinitely  small  angle  K  d  0.     The   included  side  is  the  arc  of  the  moon's 

orbit  between  the  moon  and  node,  or  the  moon's  distance   from  its  node,  while  the  included  angle  is  /the 
inclination.     Hence,  if  we  call  L  the  longitude  of  the  node,  we  shall  have,  by  spherical  trigonometry, 


sin  I 
but  K  =  —  3  a  .  sin  (  <[    —  Q)  .  sin  (O  —  Q)  .  sin  I.     Hence,  we  have 

d  L  =  3  a  .  sin  (0  —  k  0)  .  sin  (k  0  —  L)  .  sin  (0  —  L)  d  0 

from  which  differential  eqar.tion  the  relation  between  Land  0  may  be  deduced. 

If  we  assume  L  as  constant  during  one  lunation  in  the  second  member,  we  get  by  integration  the  whole 
change  of  L  in  that  interval  approximately,  or  the  mean  motion  of  the  node  in  a  lunation,  which  we  will 
call  A  L 


PHYSICALASTRONOMY.  729 


Astronomv 


This  is  in  principle  the  method  followed  by  Newton  in  that  part  of  the  third  book  of  the  Principia,  where 
he  treats  of  the  motion  of  the  moon's  node  ;  the  most  elegant  and  satisfactory  instance  of  the  application  of 
his  geometry  to  the  lunar  theory. 

The  precession  of  the  equinoxes  is  explicable  on  the  same  principles  as  the  motion  of  the  moon's 
nodes.  The  centrifugal  force  at  the  earth's  equation  throws  out  a  portion  of  the  matter  of  which  it  consists 
into  the  form  of  an  oblate  or  flattened  spheriod  ;  and  we  may  conceive  this  redundant  matter,  as  a  spherio- 
dical  shell  investing  an  inscribed  sphere.  Suppose  now  every  particle  of  this  shell  at  liberty  to  obey  any 
impulse,  unfettered  by  the  others,  or  conceive  it  to  consist  of  an  infinite  number  of  infinitely  small  moons  : 
each  of  these  will  describe  an  orbit,  whose  nodes  have  a  tendency  to  recede  on  the  plane  of  the  ecliptic,  and 
though  some  (those  which  happen  to  lie  between  their  quadratures  with  the  sine  and  their  nodes,)  will  have 
their  nodes  in  a  state  of  advance  ;  all  the  rest,  which  constitute  the  greater  number,  will  have  theirs  in  a 
state  of  recess.  Conceive  now  the  particles  to  cohere  and  form  a  solid  ring,  unconnected  with  the  central 
globe,  the  motion  of  this  ring  will  be  a  mean  among  all  the  motions  of  its  parts,  and  the  nodes  of  the  ring 
will  in  consequence  continually  recede,  with  a  certain  velocity.  Now,  let  the  ring  adhere  to  the  sphere,  then 
must  all  its  motion  be  divided  between  itself  and  the  whole  mass  of  the  earth,  which  alone  has  no  such  ten- 
dency. The  redundant  matter  at  the  equator,  however,  bears  a  very  small  ratio  to  the  whole  mass  of  the 
earth  ;  so  that  owing  to  this  cause,  the  retrogradation  is  exceedingly  diminished  in  rapidity  ;  and  owing  to  the 

enormous  distance  of  the  sun  and  the  smallness  of  the  earth  (whose  radius  being  only  —  -  —  th   part   of   the 


distance  ;  so  that  here,  a  =  m'  .  (  —  J    =  a  quantity  quite  insensible,)   is   rendered  too  small  to 

x  Ct    s  {&tjQ\Jo) 


be  distinctly  perceived. 

But  the  moon  also  exerts  a  disturbing  force.  That  luminary  is  to  our  imaginary  moons,  or  terrestrial  mole- 
cules, what  the  sun  is  to  the  moon  itself  in  the  theory  of  the  lunar  perturbation.  If  we  reduce  these  principles 
to  calculation,  assuming  such  a  mass  and  distance  of  the  moon  as  we  know  to  be  near  the  truth,  we  shall 
find  that  a  retrograde  motion  of  the  earth's  equator  on  the  ecliptic  of  about  50"  per  annum  will  actually 
result.  Now  this  is  the  very  phenomenon  known  by  the  name  of  the  precession  of  the  equinoxes  ;  and 
though  its  strict  theory  is  much  more  complicated  and  difficult  than  the  general  view  here  taken,  its  accord- 
ance with  observation  is  perfect,  and  affords  one  of  the  most  refined  verifications  of  that  admirable  law  which 
holds  the  frame  of  nature  in  the  harmony  we  are  now  so  well  able  to  appreciate. 


TABLE 


OF    THE 


PRINCIPAL   MATTERS   IN   THE 

TREATISE    ON    PLANE   ASTRONOMY, 

Alphabetically  Arranged. 


Page. 

Aberration,  correction  for  -------  549 

proof  of  the  earth's  rotation  -     -     -  550 

in  latitude  and  longitude         -     -     -  551 

tables  of 552 

cause  of---------  553 

of  the  stars  discovered      -     -     -     -  499 

Albatenius            494 

Alexandrian  school    ---------  492 

Alfraganus - 494 

Alhazen      ----- 495 

Almagest   - 491 

Almamon  measures  a  degree    ------  494 

Almansor    ------------  — 

Anaximander  and  Anaximanes       -----  487 

Ancient  armillary  sphere     -------  488 

Apparent  time      ----------  533 

Arabs,  astronomy  of  the      -------  494 

Archimedes     -----------  489 

Arenarius   ------------  — 

Aristarchus      -------          ..-  488 

Arrangement  of  the  treatise  explained     -     -     -  503 

Arsachel 495 

Astronomical  instruments   -------  533 

Autolycus  ------------  487 


Brache,  Tycho      -     - 
Bradley,  discoveries  of 


495 
499 


Callippus     ------------  487 

Cassini,  discoveries  of--------  498 

Ceres  discovered        ---------  — 

Chaldeans,  claims  of  the      -------  485 

Characters  of  the  planets           .»..--  510 

Chinese,  astronomy  of  the    -------  493 

Circles  of  perpetual  apparition       -----  523 

Climates 491 

Clock,  astronomical  ---------  539 

Cometarium     -----------  519 

Comets,  general  view  of      -------510 

table  of 594 

Constellations,  tables  of       -------  506 

Copernicus       -----------  495 

Crepusculum   -----------531 


Earliest  work  on  astronomy      ----- 
Earth,  magnitude  of       ------       438, 

c  f 

figure  of     ---------- 

density  of        --------- 

diurnal  motion  of     ------- 

Eccentricity  of  the  planetary  orbits     -     -     - 
Eclipsarean      ----------- 

Eclipses,  general  view  of    ------- 

total,  annular,  &c.      ------ 

computation  of      ------- 

Ecliptic,  obliquity  of       ------       489, 

Egyptians,  their  astronomy       ------ 

Ele'ments  of  the  planetary  orbits 

Epoch  at  which  a  planet  is  in  aphelion    - 
Equation,  greatest,  of  the  centre    -     -     -     -     - 

of  time       --------- 

table  of      - 
Equinoxes,  precession  of     ------- 

Eratosthenes    ----------- 

Evection  discovered        -------- 

Euclid 

Eudoxus     ------------ 


Day,  mean  solar   -     -     -     - 

sidereal    -     -     -     -     - 

Density  of  the  earth        -     - 

Diurnal  motion  of  the  earth 


Page. 

487        Plane 
502  Astronomy. 

503 
511 
575 
520 
515 
516 
576 
541 
485 
575 

561 
564 
566 
541 

488 
492 

488 
487 


Figure  of  the  earth     ---------  502 

Fixed  stars,  division  into  constellations,  &c.      -  504 

Galileo,  discoveries  of--------  497 

Georgium  Sidus  discovered       ------  498 

Graphical  construction  of  an  eclipse  -     -     -     -  581 

Greatest  equation  of  the  centre      -----  563 

Greeks,  their  astronomy      ------  486 


Halley,  calculations  of  -     - 
Harding  discovers  Juno 
Heavens,  phenomena  of 
Hevelius     ------ 

Hipparchus      -     -     -     -     - 

History  of  astronomy  -  - 
Hook,  his  idea  of  gravitation 
Huygens,  discoveries  of 


Indian  astronomy 
Juno  discovered   - 


540 

503 
511 


Kepler 

Kepler's  laws  discovered 
illustrate'1 


499 
498 
511 
498 
490 
485 
500 
497 

493 
498 

495 
496 
559 


PLANE  ASTRONOMY.— NAUTICAL  ASTRONOMY. 


731 


Table.     Kepler's  problem 


Page. 
559 


Latitude,  aberration  in  - 
Laws  of  Kepler  -  -  - 
Light,  velocity  of  -  - 
Longitude,  aberration  in 


551 
559 
499 
551 


Magnitude  of  the  earth         -------  502 

Mass  of  the  planets  computed  ------  — 

Mean  time  - -----  533 

solar  day         ---------  540 

Mercury,  transit  of---------  498 

Milky  way 508 

Moon,  phases  of  ---------  514 

eclipse  of,  computed      ------  578 

Nebula;                                              5O8 

Newton's  discoveries      --------  500 

Nodes  of  the  planetary  orbits  ------  573 

Nutation  of  the  earth's  axis  discovered    -     -     -  499 

illustrated         554 

tables  of 556 


Obliquity  of  the  ecliptic 
Olbers  discovers  Pallas 
Orbits,  eccentricity  of    - 

elements  of 
Orrery        -     -     -     -     - 


541 

498 
575 

518 


Pallas  discovered       ---------  493 

Parallaxes       -----------546 

Penumbra  in  eclipses      --------  515 

Phases  of  the  moon         ------           -  514 

of  the  planets      --------515 

computed     -----  570 

Phenomena  of  the  heavens        ------  511 

Philolaus 487 

Physical  astronomy,  origin  of        -----  495 

Piazzi,  discoveries  of      --------  498 

Plane  astronomy  definitions      ------  521 

phenomena  illustrated  -     -     -  523 

problems  relating  to     -     -     -  528 

Planetarium    •                                         -     -     -     -  517 

Planetary  motions,  laws  of       ------  501 

PJanets,  new  ones  discovered         -----  493 

distance,  magnitude,  orbits,  &c.      -     -  5O9 

characters  of    --------  510 

proper  motion  of         ------  513 

delineation  of  phases        -----  573 

heliocentric  longitude  of       -  — 


Planets  to  find  the  periodic  time  of    -     -     -     - 

to  find  the  mean  distance  of        -     -     - 
Precession  of  the  equinoxes      ------ 

Ptolemy     ------------ 

system  of,  refuted       ------ 

Pythagoras      ----------- 


Quadrant,  astronomical 
Quadrantal  triangles 


535 
526 


Refraction       -----------  543 

table  of---------  543 

Roemer  discovers  the  velocity  of  light    -     -     -  499 
Royal  Observatory  founded      ------498 

Society  founded    --------  — 

Sidereal  year    -----------  540 

Solar  System  described        -------  508 

Sphere,  right,  oblique,  &c.  defined     -     -     -     -  523 

Stars,  fixed,  their  division,  arrangement,  &c.  -  504 

magnitude,  distance,  &c.  unknown  -  -  508 

Sun,  its  distance,  diameter,  &c.  -----  509 

eclipse  of,  computed  -  581 

Synopsis  of  spherical  trigonometry  -  -  -  -  525 

Table  of  constellations         -------  506 

of  aberration        --------  552 

of  nutation     ---------  555 

of  equation  of  time  -------  566 

of  the  planetary  orbits        -----  575 

of  comets        ---------  594 

Telescope,  invention  of        -------  496 

astronomical  use  of      -----  533 

Thales         486 

Thebit        ---- 494 

Transit  instrument         --------  535 

Trigonometry,  spherical,  synopsis  of       -     -     -  525 

Tropical  year        ----------  540 

Time,  mean  and  apparent    -------  533 

Twilight,  the  shortest     --------531 

Tycho  Brache 495 

Uranus  discovered     ---------  498 


Venus,  transit  of 
Vesta  discovered 

Year,  length  of 
sidereal 
tropical 


498 


491 
540 


NAUTICAL    ASTRONOMY. 


Adjustment  of  the  sextant         -  -     -     -     -  637 

Altitude  defined         - 607 

double,  rule  for      -------  618 

of  the  sun,  to  find,  having  the  apparent 

time         ---.  ....  628 

of  a  star,  to  find 629 

parallax  in         --------  613 

Amplitude  defined     ---------  607 

Angular  distance  how  observed     -----  637     Cardinal  points  denned        -------     goj- 


Apparent  time  converted  into  mean  time      -     - 
at  Greenwich,  to  find        - 
at  the  ship,  to  find  from  the  sun's 
altitude,  &c.  ------ 

from  the  altitude  of  a  star     -     - 
Azimuth  defined        --------- 

circles  defined       _-„-__. 


612 
609 

626 
627 
607 


732 


TABLES. 


Page. 

Table.     Chronometer,  to  find  mean  time  by         -     -     -  626 

to  find  the  rate  of        -     -     -     -  628 

to  find  the  longitude  by         -  626 

Circle,  Dollond's  reflecting             640 

Troughton's  reflecting              -     -     -     -  638 

Circles,  great,  defined     --------  607 

secondary,  defined       ------  — 

vertical  or  azimuth,  defined        -  — 

Clock,  sidereal     -     - -     -     -  608 

solar         ----------  — 

Compass,  variation  of--------  636 

Complement  of  latitude  defined     -----  COS 

Day,  mean  solar,  defined     -------  608 

sidereal,  defined     --------  — 

Declination  defined        --------  — 

sun's,  to  find         609 

moon's,  to  find      ------611 

Definitions,  preliminary       -------  607 

Degrees,  &c.  converted  into  time      -     -     -     -  608 

Dip  of  the  horizon     ---------  612 

Dollond's  reflecting  circle         ------  640 

Double  altitude,  method 618 


Ecliptic  defined 

obliquity  of 

Equator  defined  -  - 
Equation  of  time,  to  find 
Equinoctial  points  -  - 


608 

607 
609 
608 


Horizon,  dip  of  ----------6 12 

rational 6O7 

sensible  ---------  — 

Hour  angle --  626 

circles -  608 

Index  error     -----------  637 

Instruments  employed 

Instrument,  Robinson's, for  taking  altitudes  at  sea  641 


Latitude  defined        ..-- 60S 

by  meridian  altitude  of  a  star    -     -     -  615 

of  a  planet     -     -  616 

by  altitudes  of  two  fixed  stars        -     -  619 
ofthetnoon      -     -  616 
by  the  sun's  diameter  passing  a  hori- 
zontal line         -------  622 

by  the  Pole  star    -------  — 

Logarithms,  proportional     -------  643 

Longitude  defined      ---------  608 

to  find,  by  chronometers         ...  626 

by  lunars 630 


Mean  time  converted  to  apparent  time    - 
Meridian  defined        -------- 

altitude  defined     ------ 

Moon's  declination,  to  find        -     -     -     - 

real  ascension,  to  find        -     -     - 
Moon  and  stars,  their  distance  observed 


612 
607 

611 

638 


Nadir  defined  ----- 607 


Page 

Parallax  in  altitude    ---------  613    Nautical 

horizontal      --------  —   Astronomy 

Points,  equinoctial     ---------  608  v— '"V""* 

Pok  star,  latitude  by      -  622 

Polar  distance ..-.  60S 

Poles,  north  and  south,  defined     -----  607 

Preliminary  definitions        -------  . — 

Prime  vertical,  defined         -------  — 

Proportional  logarithms,  table  of       -     -     -     -  643 

Captain  Hall's  use  of  609 

Rate  of  chronometers     --------  628 

Reflecting  circle,  Troughton's       -----  638 

Dollond's 640 

Refraction 613 

Right  ascension  defined       -------  608 

sun's,  to  find        -----  o'O9 

moon's,  to  find  -----  611 

Robinson's  instrument  for  taking  altitudes  -  641 


Semidiameter        .--_-.----.  6 13 

oblique  -     -     -  630 

Sextant  described       ---------  637 

its  adjustment  and  use       -----  — 

Sidereal  clock,  to  find  mean  time  by        -     -     -  625 
time  reduced  to  mean  time         -     -  — 

day  defined -  608 

Sun's  declination,  to  find     -------  fiO9 

right  ascension,  to  find    ------      — 

Table  of  proportional  logarithms         -     -     -     -  g43 

for  the  latitude  by  the  pole  star     -     -     -  6'22 

depression  of  the  horizon    -     -     -  645 
augmentation  of  moon's  semidia- 

meter     --------  — 

diminution  of  equatorial  parallax  -  — 

reducing  sidereal  to  mean  time     -     -  — 

mean  to  sidereal  time     -     -  646 

sun's  parallax  in  altitude     -     -     -     -  645 

vertical  semidiameter     -----  — 

oblique  semidiameter     -     -     -     -     - 

Tables,  use  of,  for  oblique  semidiameter       -     -  631 

Time,  converted  into  degrees,  &c.      -     -     -     -  6O8 

apparent,  converted  into  mean  time    -     -  612 

at  Greenwich,  to  find     -     -     -  6O9 
at   the   ship,   to   find  from   the 

sun's  altitude,  &c.       -     -     -  626 

from  the  altitude  of  a  star  -     -  627 

equation  of,  to  find    -------  609 

mean,  to  determine  from  the  time  shewn 

by  a  sidereal  clock     ------  625 

ratio  of  mean  solar  to  sidereal       -     -     -  COS 

mean  solar      ---------  — 

sidereal      ----------  — 

to  reduce  to  mean  and  the  contrary  625 

Tropics  608 

Troughton's  reflecting  circle    ------  638 

Variation  of  the  compass  -     -  636 

Vertical  circles  defined  -----  607 


Oblique  semidiameter     - 
Obliquity  of  the  ecliptic 


630 
608 


Zenith  defined 607 

distance    ----------      — 

Zodiac  defined 608 


PHYSICAL    ASTRONOMY. 


N.  B.     The  numbers  in  parentheses  thus,  (22)  refer  to  the  Equations,  the  others  to  the  pages. 


Table.      Anomaly,  true,  how  dependent  on  the  mean  by  finite 
—*/-——'  equations,  (22)   (23) 

expressed    in    a   series    of    sines    and 


cosines  of  the  mean,  (2*) 
difference  between,  in 


a  parabola  and 
very  eccentric  ellipse,  (48) 
Aphelia    of   the   planetary   oroits,    secular    variations 

of  -     -     -     - 707,  713 

in  what  cases  they  librate,  and  in  what  cir- 
culate, for  ever  in  one  direction       -     713 
Apsides,    variation  of,    in   the  planetary   theory,    see 
Aphelia,  Lunar,  first  approximation  to  their 
motion    ---------     720 

second  approximation     -----     723 

Axes,   of  the   planetary   orbits,    expression    of   their 
variation,  (165,  2)       -----     700 

this  value   proved   to    vanish,    or   the   axes 
proved  invariable    ------     7O1 

Circular  motion,  laws  of     -------     649 

equations  of      --------     660 

Comet,  investigation  of  the  orbit  of  one,   from  three 
observations       -------     666 

of  109  days,  table  of  its  motion      -     -     661 

Disturbed  motion,  general  equations  of,  (94)  (96) 

form  assumed  by  them  in  the  lunar  theory, 

(229)  (233)  (241) 

Disturbing  forces,  investigation  of     -     -     -     -     674 
expression   of,    in    the  lunar   theory,   (242) 
(243) 

Eccentricities,     how    introduced     in    the    planetary 
theory     ---------     688 

squares  and  higher  powers,  how  taken  ac- 
count of       --------     694 

variation  of,  periodical,  (219)    -     -     -     712 

,  method  of  obtaining,   707,  713 

Elements  of  an  orbit,  enumeration  of      -     -     -     663 
determination  of,  a  priori  from  given  velo- 
city, and  direction  of  projection       -     658 
a  posteriori,  or  by  observation  i.  §  VI.      663 
of  a  comet's  orbit  determined      -     666,  669 
— ,  by  Olbers's  method       67 1 
variation  of,  by  perturbation,  general  view 

of 699 

Elliptic  motion,  §  III. 653 

equation  of  the  orbit,  (14)   -     -     -     -     654 
relation    between    the    anomaly    and  time, 

(21)  (22) 
in  an  ellipse  of  small  eccentricity,  (39)  (40) 

(41)  (42) 660 

great  eccentricity  -     -     662 

Equations,  linear  differential,  how  solved     -     -     677 
I'M 


Equations,  linear  differential,  failing  cases    -     -  679     Physical 

secular,  origin  of 691  Asttcoomy. 

•,  of  Jupiter  and  Saturn         -     -  698   """ ~V~~" 

Equinoxes,  precession  of     -------  728 

Evection  of  the  moon     --------  755 

Geocentric,     and     heliocentric    places,     relation  be- 
tween        664 

Gravity,  decreases  as  the  square  of  the  distance  650 

is  proportional  to  the  mass  attracting  561 

to  a  sphere       --------  651 

of  two  spheres  to  each  other     -     -     -  652 
of  spherical   shells  and  spheres  not  homo- 
geneous       --------  653 

Inclination  of  a  planet's  orbit,  subject  only  to  perio- 
dical variations       ------705 

of  Jupiter's  orbit,  its  limit  and  period       707 
of  Saturn's        --------      — 

of    the   planetary   orbits,    confined    within 
narrow  limits,  (182)  (182)  (183)  (195)  706 

Kepler's  laws  -     -  649 

Latitude  of  a  planet,  perturbation  in  (122) 
Longitude,  mean,  of  the  moon,  expressed  in  terms  of 

the  true,  (257)  724 

the  true,  expressed  in  terms  of  the  mean  of 

a  planet,  perturbation  in,  (142)  -     -     693 

Masses  of  the  planets,  determined  by  the  motion  of 
their  satellites         ------     656 

-  by  their  effect  in  producing 

perturbations  -     687 

Moon,  theory  of----------     714 

differential  equations  of  its  motion  (233)  (237) 

(250) 

first  approximation  to  its  orbit        -     -     719 
second  approximation  to  its  orbit  -     -     721 

Node  of  a  comet's  orbit,  how  determined,  (88) 

of  that  of  a  planet,  varies  by  the  effect  of 

perturbation       ------     .     702 

its  variation  expressed     -----     706 

is  libratory  or  circular,  and  how 

to  determine  which     -----     706 

Period   of  a  planet,   how  dependent  on   its  distance 
from  the  sun,  (24)       -----     655 

of  an  inequality,  depends  on  its  argument  678 

of  the  great  equation  of  Jupiter   and 
Saturn     ---------     699 

of  the  inclinations  and  nodes  of  the  orbits  of 
Jupiter  and  Saturn       -----     707 

of  their  eccentricities  and  aphelia  -     -     713 


734 


TAB  I/E  S. 


Table      Perturbations  of  the  planets,  ii. 675 

•^v™ ^  of  the  radius  vector,  finite  expression  foi 

the,  (113)  (I20) 

in  series,  in  circular  orbits,  (140)  (143) 
in  longitude,  general  formula  for,  (121) 

,  in  circular  orbits,  (142) 

in  latitude,  (122). 
Perturbative  function,  its  value,  ( 1 10) 

reduction  to  series  of  sines  and  cosines,   683 

•      in  circular  orbits,  (138,  1) 

Problem   of  three   bodies,  general  principles   of  its 
approximate  solution        -     -     -     -     676 

Radius  vector,  expressed  in  finite  terms  of  the  true 

anomaly,  (14) 
in   a   series  of  cosines,    &c.   of  the  mean 

anomaly,  (28) 
in  a  parabola,  (43) 
perturbation  of,  (113)  (120)  (141)  (143) 


Seculai  equations,  see  equations. 

Time,  expression  of,  in  an  ellipse,  (18)  (21) 

remarkable  expression  of,  in  any  parabolic 

arc,  (62)  665 

expression  of,  i.i  a  parabola,  (44) 

in  an  ellipse  of  small  eccentricity,  (42) 

great  eccentricity,  (46) 

Variation,  of  parameters,  method  of,  explained,  and 

applied  to  the  planetary  theory  -     -     ~OO 

of  the  inoon     -     -     -     -  -     -     725 

Velocities  of  the  celestial  motions,  i.  §  IV.  -     -     658 
Velocity,  its  law  of  variation    ------     658 

its  expression  in  any  conic  section,  (33) 
ratio    of,   to  that   in    a   circle   at    the   same 
distance  ---------     659 

in   a   circle,    parabola,    ellipse,    or   hyper- 
bola          -     660 


Physical 

Astronomy 


//,•/:.;  •l,r/..M. 


/)//-//../,  ',/  ,/  .     II,.-  .  I, -I  ,/„;-,  /.,     .)/,,,  .  ,    /.•,•;.,,,  /„  .,  /  .)/„„  ,„,„,     /.„./.,„,,..:,,.' 


II'.  l."n  i-i-  ..-,• 


SOUND. 

PART  I. 
OF  THE    PROPAGATION    OF   SOUND    IN  GENERAL. 

§  I.  Of  the  Propagation  and  Velocity  of  Sound  in  Air. 

Sound.       To  explain  the  nature  and  production  of  Sound,  the  laws  of  its  propagation  through  the  various  media  which      Part  I. 
— v"^^   convey  it  to  our  ears,  and  the  manner  of  its  action  on  those  organs ;  the  modifications  of  which  it  ig  suscep-  v— •v^-' 
tible  in  speech,  in  music,  or  in  inarticulate  and  unmeaning  noises;  and  the  means,  natural  or  artificial,  of  pro-          1. 
during,  regulating,  or  estimating  them,  are  the  proper  objects  of  Acoustics. 

Every  body  knows  that  Sounds  are  conveyed  to  our  ears  from  a  distance  through  the  air,  but  it  is  not  equally         3 
apparent  that  they  would  not  reach  us  as  well  through  a  space  perfectly  void;  or,  in  other  words,  that  the  air  Sound  is 
itself  is  the  vehicle,  or  active  agent,  by  whose  operation  they  are  conveyed  to  us.     Such,  however,  is  the  case,  conveyed  to 
Shortly  after  the  invention  of  the  air-pump,  it  was  found  that  the  collision  of  hard  bodies  in  an  exhausted  receiver  us  by  n>«an« 
produced  no  appreciable  Sound.     Hanksbee  (Philosophical  Transactions,  1705)  having  suspended  a  bell  in  the  of  the  air- 
receiver  of  an  air-pump,  found  the  Sound  die  away  by  degrees,  as  the  air  was  exhausted,  and  again  increase  on  Diminution 
its  readmission ;   and  when  made  to  sound  in  a  vessel  full  of  air,  the  Sound  was  not  transmitted  through   the  of  Sound  in 
interval  between  that  and  an  exterior  vessel  from  which  the  air  had  been  extracted,  though  it  passed  freely  when  rarefied  air. 
readmitted.     On  the  otber  hand,  when  the  air  was  condensed  in  a  receiver,  the  Sound  of  a  suspended  bell  was 
stronger  than  i«  natural  air,  and  its  intensity  increased  with  the  degree  of  condensation.     Roebuck,  (Transactions  -^  'c""e*se 
of  the  Royal  Society,  Edinburgh,  vol.  v.  p.  34,)  when  shut  up  in  a  cavity  excavated  in  a  rock,  which  served  as  densed  air. 
a  reservoir  of  air  for  an  iron  foundry  in  Devonshire  to  equalize  the  blast  of  the  bellows,  observed  the  intensity 
of  Sound  to  be  considerably  augmented  in  the  air  thus  compressed  by  their  action.     The  same  effect  has  been 
experienced  in  diving-bells.     More  recently  M.  Biot  has  repeated  the  experiment  of  the  exhausted  receiver,  with 
a  much  more  perfect  vacuum  than  could  be  procured  in   Hanksbee's  time  ;    and   found  the  Sound  to  be  quite 
imperceptible,   even   when  the    ear  was  held   close  to  the  receiver,  and  all  attention  paid.    (Mbm.  d'Arcueil, 
vol.  ii.  p.  97.) 

The  diminution  of  the  intensity  of  Sound  in  a  rarefied  atmosphere  is  a  familiar  phenomenon  to  those  who  are         3. 
accustomed  to  ascend  very  high  mountains.     The  deep  silence  of  those  elevated  regions  has  a  physical  cause,  On  high 
independent  of  their  habitual  solitude.     Saussure  relates,  that  a  pistol  fired  on  the  summit  of  Mont  Blanc,  pro-  mountains, 
duced  no  greater  report  than  a  little  Indian  cracker  (petit  petard  de  Chine)  would  have  done  in  a  room.   (Voyage 
dans  les  Alpes,  vol.  vii.  p.  337,  §  2020.)     We  have  ourselves  had  occasion  to  notice  the  comparatively  small 
extent  to  which  the  voice  can  be  heard,  at  an  altitude  of  upwards  of  13,000  feet  on  Monte  Rosa.     Observations 
on  this  point  in  the  elevated  passes  of  the  Himalaya  Mountains  would  be  interesting.     They  should  be  made  by 
the  explosions  of  a  small  detonating  pistol,  loaded  with  a  constant  charge,  and  the  distances  should  be  measured ; 
for  the  voice  loses  much  of  its  force  from  the  diminution  of  muscular  energy  in  rarefied  air,  and  distances  are  extrava- 
gantly underrated  by  estimation  in  such  situations.     The  height,  however,  to  which  an  atmosphere,  or  medium 
capable  of  conveying  Sound  extends,  far  exceeds  any  attainable  on  mountains,  by  balloons,  or  even  by  the  lightest 
clouds.     The  great  meteor  of  17S3  produced  a  distinct  rumbling  Sound,  although  its  height  above  the  earth's  Extent  of 
surface  was  full  50  miles  at  the  time  of  its  explosion.    (See  Sir  Charles  Blagden's  interesting  Paper,  Philoso-  .lhe  Solln(1- 
phical  Transactions,  1784.)     The  Sound  produced  by  the  explosion  of  the  meteor  of  1719,  at  an  elevation  of  at  ^ 
least  69  miles,  was  heard  as  "  the  report  of  a  very  great  cannon,  or  broadside;"  shook  the  doors  and  windows 
of  houses,  and  threw   a  looking-glass  out  of  its  frame  and  broke  it.     (Halley,   Philosophical  Transactions, 
vol.  xxx.  p.  978.)    "These  heights  are  deduced  by  calculation  from  observations  too  unequivocal,  and  agreeing 
too  well  with  each  other,  to  allow  of  doubt.     Scarcely  less  violent  was  the  Sound  caused  by  the  bursting  of  the 
meteor  of  July  17,  1771,  near  Paris;  the  height  of  which,  at  the  moment  of  the  explosion,  is  assigned  by  Le 
Roy  at  20,598  toises,  or  about  25  miles.    (Mem.  Ac.ad.  Par.  1771,  p.  668.)     The  report  of  a  meteor,  in   1756, 
threw  down  several  chimneys  at  Aix  in  Provence,  and  was  taken  for  an  earthquake.     These  instances,  and  others 
which  might  be  adduced,  are  sufficient  to  show  that  Sound  can  be  excited  in,  and  conveyed  by,  air  of  an  almost 
inconceivable  tenuity  (for  such  it  must  be  at  the  heights  here  spoken  of)  provided  tl^e  exciting  cause  be  suffi- 
ciently powerful  and  extensive,  neither  of  which  qualities  can  be  regarded  as  deficient  in  the  case  of  fire-balls, 
such  as  those  of  1719  and  1783,  the  latter  of  which  was  half  a  mile  in  diameter,  and  moved  at  the  rate  of  20 
miles  in  a  second.     It  may,  however,  be  contended,  and  not  without  some  probability,  that  at  these  enormous 
heights  Sound  may  owe  its  propagation  to  some  other  medium  more  rare  and  elastic  than  air,  and  extending 
beyond  the  limits  of  the  atmosphere  of  air  and  vapour. 

Sound  is  not  instantaneously  conveyed  from  the  sounding  body  to  the  ear.     It  requires  time  for  its  propagation.          , 
This  is  a  mattei  of  the  most  ordinary  observation.     We  hear  the  olows  of  a  hammer  at  a  distance,  a  very  sensible 
interval  of  time  after  WP.  see  them  struck.     The  report  of  a  gun  is  always  heard  later  than  the  flash  is  seen,  and 

VOL.  iv.  &  E  717 


748 


SOUND. 


Sound, 


the  wind 
on  it. 


fi. 


the  interval  is  longer  the  more  distant  the  gun.     We  estimate  the  distance  of  a  thunder-storm  by  the  Vngth  of 
the  interval  between  the  lightning  and  the  thunder-clap,  which  often  arrives  when  we  have  ceased  to  expect  it.  ' 
Sounds  not  j^\e  repOrt  of  the  meteor  of  1783  was  heard  at  Windsor  castle,  ten  minutes  after  its  disappearance.     This  is, 
iiwanta-       probably,  the  longest  interval  yet  observed. 

A  great  multitude  of  experiments  have  been  made  <o  determine  the  precise  velocity  of  Sound.     The  earlier 
results  differ  more  than  might  have  been  expected,  from  the  influence  of  several  causes  not  immediately  obvious, 
Velocity  of  but  chieflv  from  want  of  due  attention  to  the  influence  of  the  wind.     It  is  evident  from  the  mechanical  concus- 
Sound.         sion  attending  loud  noises,  that  Sound  consists  in  a  motion  of  the  air  itself  communicated  along  it  by  virtue  of 
t  of  its  elasticity,  as  a  tremor  runs  along  a  stretched  rope.     If,  then,  the  whole  body  of  the  air  were   moving  in  a 
contrary  direction,  with  the  velocity  of  Sound,  it  would  never  make  its  way  against  the  stream  at  all ;  and,  on 
the  other  hand,  when  the  wind  blows  from  the  sounding  body  direct  towards  the  ear,  it  is  equally  clear  that  the 
velocity  of  the  wind  itself  will  be  added  to  that  of  Sound  in  still  air.     If  a  stone  be  thrown  into  a  still  lake,  the 
waves  spread  with  equal  rapidity  in  all  directions,  in  circles  whose  centre  is  the  stone.     If  into  a  running  river, 
•  hey  still  form  circles,  but  their  centre  is  carried  down  the  stream  ;  and,  in  point  of  fact,  the  wave  arrives  oppo- 
site to  a  point  of  the  bank  above  the  place  where  the  stone  fell,  later  than  a  point  at  the  same  distance  below 
it  in  proportion  to  the  rapidity  of  the  stream.     Hence  all  experiments  on  the  velocity  of  Sound  ought  to  be  made, 
if  possible,  either  in  calm  weather,  or  in  a  direction  at  right  angles  to  that  of  the  wind. 

The  assumption  of  1300  feet  per  second  for  the  velocity  of  Sound  by  Roberts,  (Phil.  Trans.  1694,)  and  the 
Various  de-  inaccurate  determinations  of  Mersenne,  Bayle,  and  Walker,  (Phil.  Trans.  1698,)  which  give  respectively  1474, 
terminations  1200,  and  1305  feet,  (the  latter  by  a  mean  of  12  experiments  disagreeing  no  less  than  370  feet  inter  se,~)  scarcely 
of  the  velo-  jeserve  more  mention  than  the  rude  guesses  of  Roberval  and  Gassendi,  which  differ  by  an  amount  nearly  equal 
Sound.         to  the  whole  quantity  to  be  measured  ;  the  former  fixing  it  at  560  feet,  the  latter  at  1473.     The  first  experiments 
which  appear  to  have  been  made  with  any  degree  of  care,  were  those  instituted  by  the  Florentine  Academy  Dd 
Cimento.     It  was  observed  in  these  that  at  a  distance  equal  to  5739  English  feet,  the  Sound  of  a  harquebuss 
arrived  five  seconds  after  the  flash  ;  and  repeating  the  experiment  at  half  the  distance,  they  found  exactly  half  the 
time  to  be  required.     This  gives,  for  the  velocity  of  Sound,  1148  feet  per  second. 

Cassini  the  Elder,  Picard,  and  Roemer,  from  experiments  made  at  a  distance  of  1280  toises,  as  related  by 
Duhamel  in  the  Hist,  de  fAcad.  Par.  assign  1172;  while  Flamsteed  and  Halley,  from  a  series  of  observations 
at  the  Royal  Observatory,  the  origin  of  the  Sound  being  three  miles  distant,  concluded  the  velocity  to  be  1 142  feet 
per  second. 

In  a  Paper  communicated  to  the  Royal  Society  in  1708,  by  Dr.  Derham,  the  subject  of  the  velocity  of  Sound 
is  investigated  more  fully  and  distinctly  than  had  before  been  done,  and  with  some  degree  of  attention  to  a 
variety  of  circumstances  which  appear  likely  to  influence  its  propagation.  These  are  chiefly 

1.  The  direction  and  velocity  of  the  wind. 

2.  The  amount  of  barometric  pressure. 

3.  The  temperature  of  the  air  through  which  the  sound  is  conveyed. 

4.  Its  hygrometrical  state  of  moisture  and  dryness. 

5.  The  actual  weather,  whether  fog,  rain,  snow,  sunshine,  &c. 

6.  The  nature  of  the  Sound  itself,  whether  produced  by  a  blow,  a  gunshot,  the  voice,  a  musical   instru- 

ment ;  its  pitch,  quality,  and  intensity. 

7.  The  original  direction  impressed  on  the  Sound — by  turning,  for  instance,  the  muzzle  of  a  gun  one  way 

or  the  other. 

8.  The  nature  and  position  of  the  surface  over  which  the  Sound  is  conveyed ;  whether  smooth  or  rough, 

horizontal  or  sloping  ;  moist  or  dry,  &c. 

To  all  these  circumstances,  except  the  wind,  Derham  attributes  no  effect ;  and,  in  fact,  none  of  them,  except 
the  temperature  of  the  air,  have  been  ascertained  to  exercise  any  material  influence  on  the  velocity ;  though 
many,  indeed  all,  have  a  very  powerful  one  on  its  intensity,  or  the  loudness  of  the  Sound  as  it  reaches  the  ear 
from  a  given  distance.  The  quantity  of  aqueous  vapour  indeed  ought  (as  we  shall  see)  to  affect  the  velocity, 
but  in  a  degree  only  appreciable  in  the  most  delicate  experiments.  Derham  concludes,  from  the  whole  of  his 
observations,  that  Sound  is  propagated  at  the  rate  of  1142  feet  per  second,  agreeing  with  the  result  of  Flamsteed 
and  Halley,  and  with  that  of  the  Florentine  Academicians;  and  as  the  distances  of  the  stations  employed  were 
considerable,  in  one  case  amounting  to  upwards  of  12  miles,  this  determination  appears  deserving  of  some 
reliance.  The  temperature,  unfortunately,  was  not  registered  ;  so  that  the  experiment  loses  much  of  its  value 
from  the  impossibility  of  applying  with  certainty  the  requisite  correction. 

In  1737-1738,  the  Academy  of  Paris  directed  a  reinvestigation  of  the  subject,  and  Messrs.  Cassini  de  Thury, 
Maraldi,  and  La  Caille,  who  were  at  that  time  engaged  in  the  triangulation  of  France,  were  charged  with  the 
mentsof  the  coriduct  of  the  experiments;  an  account  of  which,  by  Cassini,  is  to  be  found  in  the  volumes  of  the  Histoire  de 
"'  VAcad.  for  the  latter  year  and  for  1739.  Their  observations  were  carefully  made,  and  the  distance  of  the  stations 
was  considerable,  (from  2931  to  16,079  toises.)  In  these  experiments  we  find  the  first  example  of  observations 
so  disposed  as  to  eliminate  in  some  measure  the  disturbing  effect  of  the  wind.  To  apprehend  how  this  may  be 
done,  let  us  suppose  a  current  of  wind  to  blow  uniformly  with  any  velocity  from  one  station  A  to  another  B  at 
any  distance,  and  at  these  two  stations  let  shots  be  fired.  The  Sound  of  the  shot  fired  at  A  will  then  be  accele- 
rated, and  that  of  the  signal  at  B  will  be  retarded,  in  traversing  the  interval,  by  equal  quantities ;  and  conse- 
quently (since  the  velocity  of  Sound  is  very  much  greater  than  that  of  the  most  violent  wind)  the  time  in  which 
the  Sound  runs  over  the  lino  A  B  will  be  diminished,  and  that  in  which  it  traverses  B  A  increased,  by  equal 
quantities ;  so  that  the  mean  will  be  unaffected  by  the  wind's  velocity.  In  fact,  supposing  V  to  be  the  velocity  of 
Sound,  v  that  of  the  wind,  and  S  the  space  described,  the  velocities  of  the  Sound  in  the  two  opposite  directions 


Part 


7. 

DerhanTs 
experi- 
ments. 
Circum- 
stances in- 
fluencing 
the  propa- 
gation cf 
Sound. 


9. 

Expert- 


1738. 
Mode  of 
observing 
by  recipro- 
cal signals. 


SOUND.  749 

'      will  be  V  -4-  v  and  V  —  » ;    and  the  times  of  description  of  the  space  A  B  will  be   — and whose  ', 

"•v-""  V  +  v  V  —  ti  ^v— • 

mean  is  equal  to  — ,  or  to  —  •!  1  +  (  --j-  )  +  ( -^r-  J  -f  &c.  > ,  which  when  v  is  small  with  respect  to 

c 
V,  reduces  itself  simply  to  T-T-.     The  most  violent  hurricane  moves  at  a  rate  less  than  one-tenth  that  of  Sound  ; 

so  that  in  the  worst  case  the  neglect  of  the  terms  depending1  on  the  velocity  of  the  wind  will  entail  an  error  less 
than  -rfo  of  the  whole  result,  or  about  11  feet  ;  and  under  ordinary  circumstances  such  as  are  likely  to  be  selected 
for  experiment,  their  influence  is  quite  inappreciable. 

It  is  evident,  however,  that  any  want  of  uniformity  in  the  rate  of  the  wind  will  destroy,  so  far  as  it  goes,  the          JQ, 
precision   of  the   result  so  obtained;    and   that,  in  consequence,   if  the  corresponding  signals  are   not  strictly  Influence  of 
simultaneous  so  as  to  make  the  Sound  traverse  the  same  identical  portion  of  the  aerial  current,  a  great  part  of  the  suddengusls 
advantage  of  this  mode  of  experimenting  is  lost.      M.  Arago  has  indeed  remarked,  that  even  in  that  case,  if  the  °f  wind- 
wind  be  very  irregular,  and  in  sudden  gusts,  it  vrill  still  uli'ect  the  result ;   to  conceive  which,  we  will  suppose  a 
gust  of  wind  to  arise  suddenly  at  the   station  A  at  the  moment  of  firing  the  signals  both  at  A  and   B.     The 
Sound  which  proceeds  in  the  direction  A  B,  as  it  runs  quicker  than  the  wind,  will  leave  it  behind,  and  be  propa- 
gated at  every  point  of  A  B  in  still  air,  before  the  agitation  of  the  wind  has  had  time  to  reach  it.     On  the  other 
hand,  the  Sound  from  B  will  meet  the  wind  ;    and,  during  the  latter  part  of  its  course,  at  least,  will  be  propagated 
in  a  moving  atmosphere.     Still,  it  will  be  observed,  that  it  can  be  only  the  latter  part  of  its  course  which  can  be 
thus  affected,  less,  at  all  events,  than  one-tenth  of  the  whole  space ;  and  the  effect  during  that  tenth  being  to 
retard  the  Sound  by  one-tenth  at  most  of  that  interval,  will  produce  a  total  effect,  not  exceeding  a   hundredth 
of  the  whole  time  of  traversing  A  B  ;  and,  consequently,  will  affect  the  mean  of  the  two  deduced  velocities  by  a 
quantity  not  exceeding  a  two-hundredth  part  of  its  value,  or  about  five  feet  per  second.     We  have  already  seen 
that  the  neglect  of  the  square  and  higher  powers  of  the  velocity  of  the  wind  may  in  the  same  extreme   case 
produce  double  this  amount  of  error.     This,  however,  is  the  error  produced  by  a  sudden  gust  equal  to  the  most 
violent   tornado.     In   ordinary  winds,  then,  it  must  be  quite  inappreciable  ;    and  the  method  of  simultaneous 
observations  at  opposite  stations,  provided  they  be  strictly  such,  may  be  regarded  as  completely  eliminating  the 
wind's  influence. 

In  the  experiments  of  Cassini  and  his  colleagues,  however,  none  of  these  niceties  were  attended  to;  a  long        11. 
interval  elapsed  between  the  corresponding  observations  when  obtained  ;    and,  indeed,  the  greater  part  of  their  Cassini's 
series  was  made  without  any  regard  to  correspondence  at  all.     They  conclude  the  velocity  of  Sound  to  be  173  resuU- 
toises,  or  1106  British  feet  per  second,  at  the  temperature  between  4°  and  6°  Reaum.  at  which  the  experiments 
were  made.     The  extreme  difference  of  velocities  due  to  a  favourable  and  a  contrary  wind  they  found  to  be  about 
one-eleventh  of  the  whole,  giving  -Jj  for  the  ratio  of  the  velocity  of  the  wind  to  that  of  Sound  as  their  maximum, 
or  50  feet  per  second.     When  the  correction  for  the  temperature  of  the  air  is  applied,  it  will  be  seen  presently 
that  their  result  justifies   the  reliance  placed  on  it  by  its   authors  ;    being,   in    fact,  within   about  a   yard  of 
the  truth. 

Nearly  about  the  same  time  Bianconi  in  Italy,  and  La  Condamine  at  Quito  and  at  Cayenne,  instituted  a  series        12. 
of  experiments  for  the  same  purpose,  of  which  accounts  will  be  found  in  the  Comment.  Bonon.  ii.  p.  365  ;  in  La  Other  deter- 
Condamine's  Introduction  Historique,  Sfc.   1751,  p.  98 ;    and  in  the  Mem.  Acad.  Par.   1745,  p.  448.      But  the  minations. 
theory  of  Sound  being  at  that  time  but  imperfectly  understood,  and  the  necessary  corrections  in  consequence 
being  not  sufficiently,  or  not  at  all  attended  to,  the  subject  has  been  regarded  as  still  open  to  further  discussion ; 
and  accordingly  a  great  number  of  researches  by  later  experimenters  have  been  instituted,  of  which  the  principal 
are  those  by  Muller  in  1791,  (Getting.  Gelehrte.  Anzeigen,  1791,  No.  159  ;)  by  Espinosa  and  Bauza,  in  Chili,  in 
1794,  (Ann.  de  Chim.  vii.  93;)  by  Benzenberg  in  1809,  (Gilbert's,  Annalen,  new  series,  v.  383;)  by  Arago, 
Bouvard,  Matthieu,  Prony  and  Humboldt,  and  Gay  Lussac,  in  lS22,(Connaiss.  des  Temps,  1825,  p.  361;)  by  Moll, 
Vanbeek,   and  Kujtenbronwer  in  Holland,  in  1822,  (Phil.  Trans.  1824,  p.  424  :)  by  Mr.  Goldingham,  in  1820, 
at  Madras,  (Phil.  Trans.  1823,  p.  96;)  by  Dr.  Gregory,  at  Woolwich,  in  1823,  (Trans,  of  Cambridge  Phil.  Soc. 
1824  ;)    and   by  General  Myrbaeh  and  Professor  Stampfer,  at  Saltzburg,  (Jahrburh  des  Polytekn.  Institute  su 
fPien,  vol.  vii.) 

Of  these  by  far  the  most  considerable  and  circumstantial,  as  well  as  in  all  probability,  from  the  instrumental         13. 
means  employed  and  precautions  used,  the  most  exact,  are  those  of  the  Dutch  and  the  Parisian  Philosophers.  Experi- 
Every  attention  was  paid  in  them  (at  least  in  the  case  of  the  Dutch   experimenters)  to  obtain  signals  strictly  ments  «f 
reciprocal,  by  guns  fired  at  the  same  instant  of  time  at  the  two  ends  of  the  line  of  observation  ;    all  those  cor-  jj^'  ^"" 
rections  depending  on  Meteorological  circumstances  which  theory  points  out,  and  which  it  will  be  the  object  of  an(j  Of 
subsequent  parts  of  this  Essay  to  explain,  being  carefully  applied  ;    and  the  distances  of  the  stations  being  at  Arago,  Mat- 
once  considerable,  and  determined  with  sufficient  exactness  by  Trigonometrical  operations.  tnieu>  to- 
One  very  material  difficulty  in  the  way  of  former  observers  (Benzenberg  excepted)  lay  in  the  want  of  adequate 
means  of  measuring  with  precision  intervals  of  time  to  a  minute  fraction  of  a  second.     This  difficulty  was  obviated  M  . J 
in  the  experiments  of  the  French  Commissioners,  by  the  use  of  the  stop-watch  of  Breguet,  and  the  chronograph  measuring 
of  Rieussec,  a  species  of  watch,  one  of  whose  hands  performs  a  revolution  per  second,  and  can  be  made  to  very  small 
touch  with  its  extremity  the  dial-plate,  at  any  instant,  and  leave  there  a  dot,  without  interrupting  its  motion  of  portions  of 
rotation,  by  the  sudden  pressure  of  a  small  lever  ;  to  effect  which  it  carries  with  it  a  drop  of  printer's  ink  in  a  time< 
peculiar  and  ingenious  species  of  dotting  pen.     In  the  Dutch  experiments,  a  clock  with  a  conical  pendulum  was 
used,  capable  of  determining  intervals  to  the  hundredth  of  a  second,  by  suddenly  suspending  the  motion  of  the 
index,  without  stopping  the  clock.     By  the  use  of  these  instruments  it  was  found  practicable  to  ascertain  the 

5  E  2 


750 


SOUND 


Sound, 


15. 

Their 

re  nulls. 


16. 

Synoptic 
view  of 
results 


interval  between  the  sight  of  the  flash,  and  the  arrival  of  the  report,  of  a  gun,  with  such  precision  as  to  destroy  all 
material  error  in  the  result  which  might  arise  from  this  cause  ;  an  improvement  of  great  importance,  when  we 
consider  that  an  error  of  a  single  tenth  of  a  second  in  the  measure  of  time  is  equivalent  to  110  feet  in  that  of 
distance. 

The  close  agreement  of  the  results  of  these  experiments  is  a  convincing  proof  of  their  accuracy.  The 
French  Philosophers  state  33T05  met.  =  1086'!  feet,  as  the  velocity  of  propagation  of  Sound  in  air  of  the  tern 
perature  of  freezing  water,  while  the  Dutch  experimenters  make  it  332'05  met.  =  1039-42  feet  in  perfectly  dry 
air  of  the  same  temperature.  The  latter  seems  to  deserve  the  preference,  if  only  from  the  circumstance  of  the 
signals  from  which  it  is  deduced  having  been  strictly  simultaneous,  the  guns  at  the  two  extremities  of  the  line 
(nine  miles  in  length)  having  been  fired  at  the  same  second  of  time,  while  in  the  former  series  this  exact 
coincidence  was  not  obtained. 

We  subjoin  a  list  of  the  results  arrived  at   in  the  various  determinations  above  enumerated,  with  their  dates, 
the  distances  of  the  stations  employed,  &c.  to  bring  the  whole  subject  under  one  view. 

TABLE  I. — Velocity  of  Sound  as  determined  by  various  Experiments. 


Part  I. 


Observers*  Names. 

Date  of 
Deter- 
mina- 
rion. 
A.  D. 

Distance  of 

Stations  in 
Feet. 

Velocity  in 
English  Fret 
per  second. 

Remarks. 

1474 
1148 

560 
1473 
1200 
1300 

1305 

1172 
1151 
1142 

114, 
1106 

1093 
1110 

1043 
1112 
1175 
1130 
1105 
1109 
1130 

1222-23 

1093 
1086-1 

1089-42 
1088-05 
1092-1 

1089-9 
1079-9 

Moll  and  Vaubeek  state  this  result  at  361  metres  = 
1  184  feet     Our  authorities  are  Derham  and  Walker. 

Essay  of  Languid  Motion. 
No  experiments  stated. 

By  return  of  echos  in  given  times  and  measuring  dis- 
tance. 

Duhamel. 
Moll  on  authority  of  Duhamel. 
As  stated  by  Derham. 

Near  Paris  at  Montlhery,  Dammartin,  &c.  Therm.  +5° 
Reaum.  consider  their  result  as  within  a  fathom  of 
the  truth. 
Do.  reduced  to  freezing  temperature. 
Between  Sette  and   Aiguesmorles,   Mtm.  Acad.  Par. 
1739,  p.    127,  temperature    not    stated,    probably 
about  +  6°  R. 

At  Quito. 
At  Cayenne. 
Cited  by  Goldingham,  (Phil.  Tram.  1823.) 

Cited  by  Dr.  Gregory,  (Trans.  Phil.  Sac.  Cambridge, 
ii.  120.) 
At   Chili,   at   a  temperature  =  74"  7'  Fahr.   mean   of 
four  determinations,   and    mean    temperature,    the 
mean  taken  giving  a  weight  to  each  proportional  to 
the  distance. 
At  freezing  temperature. 
At  freezing  temperature,  (between  Villejuif  and  Mont- 
lhery.) 
In  dry  air,  at  freezing  temperature. 

Mean  of  eight  results  given  by  Dr.  Gregory,  each  sepa- 
rately reduced  to  the  freezing  temperature. 
Mean  of  88  observations  reduced  to  the   freezing  tem- 
perature difference  of  level  of  stations  =  4474 
Hygrom.  20-31T       Reduced  to  the  freezing  tempera- 
Hygrora   11  '85  1  tar*.    The  mean  taken  by  attributing 

1660 

590K 

1694 
1698 

\Valker                    

{variable  -j 
600  to    \ 
2370     } 
8186 
9239 
15840 
f  5280  to  ) 
1    63360   ) 
(18744  to) 
1  102824  / 

1704 
1733 

Cassini  deThury,  Maraldi,  Lacaille    

1739 

1740 
1740 
1744 

144124 

78740 
67400 
129360 

T  F  Mayer      ..             

1778 
1791 

3412 
8530 

G   E  Miiller                  

1794 

•••Op 
li«2 

1823 
1823 
1822 

1821 

f  53626  to) 
\   14071     ) 

29764 
64064 

57839 
{Various  \ 
2700  to  } 
13460   > 
32615 

f  29547 
)    13932 

^  mean 

Arago,  Matthieu,  Prony,  Bouvard,  Uumuoidt 

Goldingham,  (Madras)    

1086-7 

Mean  .     17'4  J  portional  to  the  distance  of  the  sta- 
tions.    The  nature  of  the  hygrometer  not  stated. 

17.  The  agreement  between  such  of  the  above  results  as  are  reduced  to  the  standard  or  freezing  temperature,  i.  e. 

of  the  last  six,  and  the  first  determination  of  Cassini  at  Paris,  is  very  close;  their  extreme  discrepancy  being  less 
than  seven  feet,  or  a  160th  of  the  whole  amount,  and  their  mean  (1089'7)  agreeing  almost  precisely  with  the  result 
of  Moll,  Vanbeek,  &c. ;  we  may,  therefore,  adopt  1090  feet  without  hesitation  (as  a  whole  number)  as  no  doubt 


SOUND.  751 

Sound      within  a  yard  of  the  truth,  and  probably  within  a  foot.     The  reduction  to  the  zero  of  temperature  has  been  made       Part  I. 
-"~v~~-/  (when  not  performed  by  the  authors  themselves)  on  the  supposition  that  every  additional  degree  of  atmospheric  S-~  ^/-•— 
temperature,  on   Fahrenheit's  scale,  adds  1'14  foot  to  the  velocity,  a  correction  of  which   the  grounds  will  be  Vel°c>'y 
hereafter  explained.  (See  Art.  68.)  ado't^ 

It  may,  therefore,  be  stated  in  round  numbers,  that  Sound,  in  dry  air  and  at  the  freezing  temperature,  travels  ^rTos'u 
at  the  rate  of  1090  feet,  or  363  yards  per  second,  and  that  at  62°  Fahrenheit  (which  is  the  standard  temperature  feet  per 
of  the  British  metrical  system)   it  runs  over  9000  feet  in  eight  seconds,  12J  British  standard  miles  in  a  minute,  second. 
or  765  miles  in  an  hour,  which  is  about  three-fourths  of  the  diurnal  velocity  of  the  Earth's  equator. 

Hence,  in  latitude  42J°,  (42°  29'  40",)  if  a  gun  be  fired  at  the  moment  a  star  passes  the  meridian  of  any  APProxi; 
station,  the  Sound  will  reach  any  other  station  exactly  west  of  it  at  the  precise  instant  of  the  same  star's  arriving  rourid"n 
on  its  meridian.  bers. 

In  the  experiments  of  Dr.  Gregory,  the  velocity  of  the  wind   was   measured  by  an  anemometer,  and  allowed         ]  9, 
for.     The  close  agreement  of  their  results  with  those  of  the  Dutch  and  French  observers,  when  the  smallness  of  Comparison 
the  distances  is  taken  into  consideration,  is  a  strong  proof  of  the  care  and  accuracy  with  which  they  were  made.  "itl1  tlle 
The  observations  of  Mr.  Goldingham,  or  at  least  his  mode  of  stating  and  reducing  them,  has  been  strongly,  but      ,rttl 
we  think  undeservedly,  censured  in  PoggendorfPs  Annalen  der  Physik,  81.  Band.  s.  490.     He  takes  a  mean  of        2o'° 
all   the  velocities  observed  daily,  in  calm   weather,  during  a  very  long  time,   by  the  firing  of  a  morning  and  Remarks  on 
evening  gun  at  two  stations  visible  from  Madras,  and  a  mean  of  all  the  temperatures,  pressures,  and  hygrometer-  some  of  the 
readings.     All  that  we  have  done  is  to  apply  the  correction  for  this  mean  temperature  to  his  mean  velocities,  as  above 
if  they  had  been  given   by  a  single  observation,  a  course,  no  doubt,  perfectly  legitimate,  and  saving  a  world  of  results- 
calculation.     It  is  to  be  lamented  that  the  nature  of  his  hygrometer  is  not  stated,  as  its  indications  at  present 
are  perfectly  useless.     The  experiments  of  Espinosa  and  Bauza  differ  so  enormously  in  their  result  from   the 
rest,  even  when  reduced  to  the  freezing  temperature,  that  most  probably  some  fundamental  mistake,  either  in 
their  measurement  of  the  distances,  or  in  the  calculations  founded  on  them,  must  have  been  committed.     Our 
authority  is  the  Annales  de.  Chimie,  vol.  vii.  (N.  S.)  p.  93. 

Derham  found  that  fogs  and  falling  rain,  but  especially  snow,  tend  powerfully  to  obstruct  the  free  propagation        21. 
of  Sound,  and  that  the  same  effect  was  likewise  produced  by  a  coating  of  fresh  fallen  snow  on  the  ground,  though  Effect  of 
when  glazed  and  hardened  at  the  surface  by  freezing  it  had  no  such  influence.     Over  water,  or  a  surface  of  ice,  fogs,&c. 
Sound  is  propagated  with  remarkable  clearness  and  strength.     Dr.  Hutton  relates,  that  on  a  quiet  part  of  the  ^  ob^truct 
Thames,  near  Chelsea,  he  could  hear  a  person  read  distinctly  at   140  feet  distance,  while  on  the  land   the  same  Sound'  we) 
could  only  be  heard  at  76.     Lieutenant  Fosler,  in  the  third  Polar  Expedition  of  Captain  Parry,  found  that  he  conveyed 
could  hold  a  conversation  with  a  man  across  the  harbour  of  Port  Bowen,  a  distance  of  6696  feet,  or  about  a  mile  over  waier 
and  a  quarter.     This,  however  remarkable,  falls  far  short  of  what  is  related  by  Dr.  Young  on   the   authority  of  ?nd  sraooth 
Derham,  vii.  that  at  Gibraltar  the  human  voice  has  been  heard  ten  miles,  (perhaps  across  the  Strait.)     We  have  'ce- 
not  been  able  to  find  the  original  passage  either  in  his  Physico-Theology,  or  in  his  dissertation  De   Soni  Motu, 
in  both  which  very  remarkable  instances  are  adduced,  of  which  the  following  will  suffice  as  specimens. 

Guns  fired  at  Carlscroon  were  heard  across  the  southern  extremity  of  Sweden  as  far  as  Denmark  ;  SO  miles,        32 
as  Derham  states  from  memory,  but  according  to  the  map  at  least  120.  Distances 

Dr.  Hearn,  a  Swedish  physician,  relates  that  he  heard  guns  fired  at  Stockholm,  on  the  occasion  of  the  death  at  which 
of  one  of  the  Royal  family  in  1685,  at  the  distance  of  30  Swedish,  or  180  British  miles.  Sounds  have 

The  cannonade  of  a  sea-fight  between  the  English  and  Dutch,  in  1672,  was  heard   across  England  as  far  as  been  heard- 
Shrewsbury,  and  even  in  Wales,  a  distance  of  upwards  of  200  miles  from  the  scene  of  action. 

That  Sounds  of  all  pitches,  and  of  every  quality,  travel  with  equal  speed,  we  have  a  convincing  proof  in  the        23. 
performance  of  a  rapid  piece  of  music  by  a  band  at  a  distance.     Were  there  the  slightest  difference  of  velocity  in  All  sounds 
the  Sounds  of  different  notes,  they  could  not  reach  our  ears  in  the  same  precise  order,  and  at  the  exact  intervals  travel  with 
of  time  in  which  they  are  played,  nor  would  the  component  notes  of  a  harmony,  in  which  several  Sounds  of  different  etl,ual 
pitch  concur,  arrive  at  once.     M.  Biot  caused  several  airs  to  be  played  on  a  flute  at  the  end  of  a  pipe  951  metres,  V< 
or  3120  feet;  long,  which  were  distinctly  heard  by  him  at  the  other  end,  without  the  slightest  derangement  in 
the  order  or  intervals  of  sequence  of  the  high  and  low  notes.  (Mem.  d  Arcueil,  ii.  422.)     A  better  form   of  the 
experiment  would  have  been  to  strike  two  bells  of  very  different  pitch  one   against  the  other,  having  removed 
their  clappers.     Both  their  sounds  would  (no  doubt)  arrive  together. 

A  very  material  difference,  however,  is  observed  in  the  intensity  with  which  Sounds  are  propagated,  or  the         $4 
distances  to  which  they  may  be  heard  with   equal  distinctness   according  to  a  great  variety  of  circumstances.  Effect  0, 
Thus,  if  a  Sound  be  prevented  from  spreading  and  losing  itself  in  the   air,  whether  by  a  pipe,  by  the  vicinity  of  pipes  in 
an  extensive  flat  surface,  as  a  wall,  or  otherwise,  it  may  be  conveyed  to  very  great  distances  with  little  diminu-  conveying 
tion  of  force.     This  we  observe  familiarly  in  speaking  pipes   conducted  from  one  apartment  to  another  of  a  Sou'"i- 
building.     In  the  experiments  already  cited  of  M.  Biot,  a  person  being  stationed  at  one  end  of  the  enormous 
tube  above  mentioned,  (which  was  a  combination  of  cast  iron  conduit  pipes  laid  down  for  the   supply  of  Paris 
with  water,  forming  a  continuous  canal  of  equal  internal  diameter  throughout,  and  having  two  flexures  about  the 
middle  of  its  length)  the  lowest  whisper  at  one  end  was  distinctly  heard  at  the  other,  so  that,  in  fact,  the  only 
way  not  to  be  heard  was  not  to  speak  at  all.     Nay,  so  faithful   was  the  transmission  of  every  agitation  of  the 
air,  whether  sonorous,  or  otherwise,  along  the  pipe,  that  a  pistol  fired  at  one  end  actually  blew  out  a  candle  at 
the  other,  and  drove  out  light  substances  placed  there  with  considerable  violence. 

At  Carisbrook  Castle,  near  Newport,  in  the  Isle  of  Wight,  is  a  well,  210  feet  in  depth,  and  12  in  diameter,        25. 
into  which  if  a  pin  be  dropped,  it  will  be  distinctly  heard  to  strike  the  water.     The  interior  is  lined  with  very  Sou.nd  iu 
smooth  masonry.  Carisbrooii 

It  is  evident,  without  entering  into  any  nice  theoretical  considerations,  that  a  mechanical  impulse  of  whatever  ^^Q 
nature  impressed  on  any  portion  of  the  air  or  other  medium,  whether  fluid  or  solid,  and  thence  communicated 


752  SOU  N  D. 

Sound,     to  the  surrounding  parts,  if  allowed  to  spread  in   all  directions  as  from  a  centre,  must  reach  every  more  distant       Parti. 

^— v^"1—'  point  with  an  energy  continually  less  and  less,  because  the  same  quantity  of  motion  is  communicated  in  succes-  ^^^v*^ 

Conveyance  sjOn  to  a  larger  and  larger  sphere  of  inert  matter,  but  if  only  allowed  to  spread  in  certain  directions,  its  diminu- 

of  Sound      tjon  wj|]  ke  iess  rapid  in  proportion  as  the  quantity  of  matter  successively  put  in  motion  increases  less  rapidly. 

Hence  a  Sound  might  be  expected  to  be  conveyed  with  less  diminution  along  a  wall   than  in   the  open  air,  the 

trough  or  angle  between  the  wall  and  the  ground,  in  fact,  forming  two  sides  of  a  square  pipe,  and  the  divergence 

of  the  Sound  in  two  directions  being  thereby  in  great  measure  prevented.     Dr.   Hutton  relates  that  p;irt  of  the 

wall  of  a  garden,  formerly  in  the  possession  of  W.  Pitt,  Esq.  of  Kingston,  in  Dorsetshire,  conveys  a  whisper  in 

this  way  nearly  200  feet.   (Mathematical  Dictionary,  Article  Sound.)     It  is  probably  to  some  such  principle  that 

we  must  refer  a  fact  mentioned  by  the  last-named  author,  which  at  first  sight  appears  surprising  enough.      He 

relates   that  when   a  canal  of  water  was  laid  under  the   pit  floor  of  the  Theatre  Del  Argentina,  at  Rome,  a 

sunrisinsr  difference  was  observed.     The  voice  has  since  been  heard  very  distinctly  when  it  was  before  scarcely 

distinguishable.     It  is  a  general  remark  that  Sounds  are   well   heard  in  buildings  which  stand  on  arches  over 

water.     The  cause  of  this,  however,  seems  to  be  the  echo  produced  between  the  water  and  the  arch  which  unites 

with,  and  reinforces,  the  original  Sound.     The  Work  just  referred  to  contains  many  curious  instances  of  the  kind. 

27  When  Sound  in  the  course  of  its  propagation   meets  with  an  obstacle  of  sufficient  extent  and  regularity  it  is 

Echos.         reflected,  producing  the  phenomenon  we  call  an  Echo.     A  wall,  the  side  of  a  house,  or  the  surface  of  a   rock, 

Their  the  ceiling,  floor,  and  walls  of  an  apartment,  the  vaulted  roof  of  a  church,  all,  under  proper  circumstances,  give  rise 

nature.         to  Ecnos  more  or  less  audible.     The  reflected  Sound  meeting  another  such  obstacle  is  again  reflected,  and  thus 

the  Echo  may  be  repeated  many  times  in  succession,  becoming,  however,  fainter  at  each  repetition   till  it  dies 

away  altogether.     We  shall  here  set  down  a  tew  localities  in  which  Echos,  remarkable  either  for  distinctness,  or 

frequency  of  repetition,  may  be  heard. 

28.  An  Echo  in  Woodsiock  Park,  (Oxfordshire,)  repeats    17  syllables  by  day,  and  20  by  night,  (Plot,  Nat.  Hist. 

Instances  of  Oxford,  ch.  i.  p.  7.)     One  on  the  banks  of  the  Lago  del  Lupo,  above  the  fall  of  Terni,  repeats  15. 
remarkable       jn  jne  Abbey  Church  of  St.  Alban's  is   a  curious  Echo.     The  tick  of  a  watch   may  be  heard  from  one  end 
^  2Q         °f  l^e  church  to  the  other.     In  Gloucester  Cathedral,  a  gallery  of  an  octagonal  form  conveys  a  whisper  75  feet 
across  the  nave. 

30  An  Echo  on  the  north  side  of  Shipley  Church,  in   Sussex,  repeats   21  syllables.  (Cavallo,   citing  Plot  and 
Harris.) 

31  In  the  Cathedral  of  Girgenti,  in  Sicily,  the  slightest  whisper  is  borne  with  perfect  distinctness  from  the  great 
Echo  in  the  western  door  to  the  cornice  behind  the  high  altar,  a  distance  of  250  feet.     By  a  most  unlucky  coincidence  the 
Cathedral     precise  focus  of  divergence  at  the  former  station  was  chosen   for  the  place  of  the   confessional.     Secrets  never 
at  Girgenti.  ;ntentje(i   for  the   public  ear  thus  became  known,   to  the  dismay  of  the  confessors,  and  the  scandal   of  the 

people,  by  the  resort  of  the  curious  to  the  opposite  point,  (which  seems  to  have  been  discovered  accidentally,) 
till  at  length,  one  listener  having  had  his  curiosity  somewhat  over-gratified  by  hearing  his  wife's  avowal  of 
her  own  infidelity,  this  tell-tale  peculiarity  became  generally  known,  and  the  confessional  was  removed.  (Travels 
through  Sicily  and  the  Lipari  Islands,  in  the  Month  of  December,  1824.  By  a  Naval  Officer.  1  vol.  8vo. 
London,  1827.) 

32.  In  the  Whispering  Gallery  of  St.  Paul's,  London,  the  faintest  Sound  is  faithfully  conveyed  from  one  side  to 
the  other  <>f  the  dome,  but  is  not  heard  at  any  intermediate  point. 

33.  In  the  Manfroni  Palace  at  Venice  is  a  square  room  about  25  feet  high,  with  a  concave  roof,  in  which  a  person 
standing  in  the  centre,  and  stamping  gently  with  his  foot  on  the  floor,  hears  the  Sound  repeated  a  great  many 
times,  but  as   his  position  deviates   from  the  centre  the  reflected  Sounds  grow  fainter,  and   at   a   short  distance 
wholly  cease.     The  same  phenomenon  occurs  in  the  large  room  of  the  Library  of  the  Museum  at  Naples. 

34.  Southwell  (Phil.  Trans.  1746,  223.)  describes  an  Echo   in   an   old  Palace    near  Milan,  which   repeated  the 
Echo  in  the  report  of  a  pistol  56,  or  even  60  times.     His  description  is  singularly  confused,  but  the  palace  is  no  doubt  that 
Simonetta     Qf  Simonetta,  mentioned  by  Addison  in  his  Travels.     This  was  a  building  with  two  wings,  forming  three  sides  of 

a  square.  The  pistol  was  discharged  from  a  window  in  one  wing,  the  Sound  was  returned  from  a  dead  wall  in 
the  other  wing,  and  heard  from  a  window  in  the  back  front.  (Hutton,  Art.  Echo.  Misson,  Voy.  dllal.  ii.  196.) 
The  Palace  still  exists,  but  ear-witnesses  have  described  the  phenomenon  to  us  somewhat  differently.  The  Echos 
are  heard  at  the  window  whence  the  pistol  is  fired. 

35.  Beneath  the  Suspension  Bridge  across  the  Menai  Strait,  in  Wales,  close  to  one  of  the  main  piers,  is  a  remark. 
Echo  under  ably  fine  Echo.     The  Sound  of  a  blow  on  the  pier  with  a  hammer,  is  returned  in  succession  from   each  of  the 

e  Menai    cross-beams  which   support  the  roadway,  and   from  the  opposite  pier  at  a  distance  of  5~6  feet,  and,  in  addi- 
ge-        tion  to  this,  the  Sound  is  many  times  repeated  between   the  water  and  the  roadway.     The  effect  is  a  series  of 


Sounds  which  may  be  thus  written     ^     •  '" *""     |      '  I    j  •         '       I|HH>B^^^^J        &c.  ;  the  first  return 


is  sharp  and  strong,  from  the  roadway  over  head,  the  rattling  which  succeeds  dies  away  rapidly,  but  the  single  reper- 
cussion from  the  opposite  pier  is  very  strong,  and  is  succeeded  by  a  faint  palpitation,  repeating  the  Sound  at  the 
rate  of  28  times  in  five  seconds,  and  which  therefore  corresponds  to  a  distance  of  184  feet,  or  very  nearly  the 
double  interval  from  the  roadway  to  the  water.  Thus  it  appears,  that  in  the  repercussion  between  the  water  and 
roadway,  that  from  the  latter  only  affects  the  ear,  the  line  drawn  from  the  auditor  to  the  water  being  too  oblique 
for  the  Sound  to  diverge  sufficiently  in  that  direction.  Another  peculiarity  deserves  especial  notice;  viz.  that 
the  Echo  from  the  opposite  pier  is  best  heard  when  the  auditor  stands  precisely  opposite  to  the  middle  of  the 
breadth  of  the  pier  and  strikes  just  on  that  point.  As  he  deviates  to  one  or  the  other  side  the  return  is 
proportionably  fainter,  and  is  scarcely  heard  by  him  when  his  station  is  a  little  beyond  the  extreme  edge  of 


SOUND.  753 

Sound,     the  pier,  though  another  person  stationed  (on  the  same  side  of  the  water)  at  an  equal  distance  from  the  central       Part  '• 
•""v" ••'  point,  so  as  to  have  the  pier  between  them,  hears  it  well.     Thus,  in  the  reflexion  of  Sound,  there  is  an  evident  ^^^^"^ 
approach  to  the  law  of  equality  between  the  angles  of  incidence  and  reflexion  which  obtains  in  that  of  Light ;  and  t|^u. 
a  tendency  in  the  reflected  Sound  to  confine  itself  to  the  direction  which  a  ray  of  Light  regularly  reflected  at  the  Of  incidence 
echoing  surface  would  follow,  and  not  to  spread  into  the  surrounding  air  equally  in  all  directions.     This  expe-  and  reflex- 
rimeut  (which  we  had  an  opportunity  of  making,  with  the  assistance  of  Mr.  Babbage,  in  1827)  might  be  carried  'on- 
much  farther  under  more  favourable  circumstances ;   and,  we  doubt  not,  would  lead  to  remarkable  confirmations 
of  the  law  of  interference,  and  the  general  analogy  between  Sound  and  Light,  to  which  all  Optical  and  Acoustical 
phenomena  point,  and  of  which  we  shall  have  occasion  to  say  more  hereafter.     (See  also  our  Essay  on  LIGHT.) 
The  span  of  the  bridge  between  the  piers  is  576  feet,  and  the  breadth  of  each  pier  about  30  feet. 

The  most  favourable  position  for  the  production  of  a  distinct  Echo  from  plane  surfaces  is,  when  the  auditor  is        36. 
placed  between  two  such,  exactly  halfway.     In  this  situation  the  Sounds  reverberated  from  both  will  reach  him  Situations 
at  the  same  instant,  and  reinforce  each  other.     If  nearer  to  one  surface  than  the  other,  the  one  will  reach  him  favo"ra')le 
sooner  than  the  other,  and  the  Echo  will  be  double  and  confused.     If  the  Echoing  surface  be  concave  towards  tu 
him,    the   sounds   reflected    from    its  several    points    will,    after   reflexion,   converge   towards  him,  exactly  as 
reflected  rays  of  Light  do;   and  he  will  receive  a  Sound  more  intense  than  if  the  surface  were  plane,  and  the 
more  so  the  nearer  it  approaches   to  a  sphere  concentric  with  himself:  the  contrary  if  convex.     If  the  Echo  of 
a  Sound  excited   at  one   station    be  required    to  be  heard  most  intensely  at  another,  the  two  stations  ought 
to  be  conjugate  foci  of  the  reflecting  surface,  i.  e.  such  that  if  the  reflecting  surface  were  polished,  rays  of  Light 
diverging  from  one  would  be  made  after  reflection  to  converge  to  the  other.     Hence  if  a  vault  be  in  the  form  of 
a  hollow  ellipsoid  of  revolution,  and  a  speaker  be  placed  in  one  focus,  his  words  will  be  heard  by  an  auditor  in 
the  other  as  if  his  ear  were  close  to  the  other's  lips.     The  same  will  hold  good  if  the  vault  be  composed  of  two 
segments  of  paraboloids,  having  a  common  axis,  and  their  concavities  turned  towards  each  other ;  only  in  this 
case.  Sounds  excited  in  the  focus  of  one  segment  will  be  collected  in  the  focus  of  the  other,  after  two  reflexions. 

An  attention  to  the  doctrine  of  Echos  is  of  some,  though  we  think  a  rather  overrated,  importance  to  the  37. 
architect  in  the  construction  of  buildings  intended  for  public  speaking,  or  music,  especially  if  they  be  large.  In  Effect  °f 
small  buildings,  the  velocity  of  Sound  is  such  that  the  dimensions  of  the  building  are  traversed  by  the  reflected  churches 
Sound  in  a  time  too  small  to  admit  of  the  Echo  being  distinguished  from  the  principal  Sound.  In  great  ones,  an(j  pubiic 
on  the  other  hand,  as  in  Churches,  Theatres,  and  Concert  rooms,  the  Echo  is  heard  after  the  principal  Sound  has  buildings. 
ceased ;  and  if  the  building  be  so  constructed  as  to  return  several  Echos  in  very  different  times,  the  effect  will  be 
unpleasant.  It  is  owing  to  this  that  in  Cathedrals  the  service  is  usually  read  in  a  sustained  uniform  tone,  rather 
that  of  singing  than  speaking,  the  voice  being  thus  blended  in  unison  with  its  Echo.  A  good  reader  will  time  his 
syllables,  if  possible,  so  as  to  make  one  fall  in  with  the  Echo  of  the  last,  which  will  thus  be  merged  in  the  louder 
Sound,  and  produce  less  confusion  in  his  delivery.  For  music,  in  apartments  of  moderate  size,  all  objects  which  can 
obstruct  the  free  reflexion  of  Sound  from  the  walls,  floor,  and  ceiling  are  injurious.  The  Echo  is  not  sensibly 
prolonged  after  the  original  Sound,  and  therefore  only  tends  to  reinforce  it,  and  is  of  course  highly  advantageous. 
In  large  ones,  an  Echo  can  only  be  advantageous  in  the  performance  of  slow  pieces,  (as  Church  music.)  The 
prolongation  of  a  chord,  after  the  harmony  is  changed,  can  be  productive  of  nothing  but  dissonance.  When  ten 
notes  succeed  one  another  in  a  second,  as  is  often  the  case  in  modern  music,  the  longitudinal  Echo  of  a  room  55 
feet  long,  will  precisely  throw  the  second  reverberation  of  each  note  on  the  principal  Sound  of  the  following  one 
wherever  the  auditor  be  placed ;  which,  in  most  cases,  will  produce  (in  so  far  as  it  is  heard)  only  discord.  Much 
mistake  seems  to  be  prevalent  on  this  subject.  Thus  it  is  said  that  the  form  of  an  orchestra  should  be  para- 
bolic, &c.  that  the  rays  of  Sound  should  be  reflected  out  in  parallel  lines  to  the  audience.  But  even  if  they  were 
so,  the  reflected  Sound  cannot  possibly  reach  them  in  the  same  time  with  the  direct ;  and  in  Acoustics  it  is  of  little 
moment  in  what  direction  sounds  reach  the  ear,  which  is  not,  like  the  eye,  capable  of  appreciating  direction  with 
any  precision,  or  collecting  the  rays  or  waves  of  Sound  to  a  focus  within  the  ear.  It  is  not  possible  to  place  a 
whole  band  in  the  focus  of  a  parabolic  or  elliptic  orchestra,  or  a  whole  audience  in  that  of  a  corresponding  opposite 
segment.  We  may  add,  too,  that  an  apartment  would  be  worse  lighted,  were  its  internal  surface  a  polished 
semi-ellipsoid,  with  a  candle  in  the  focus,  than  if  it  were  of  the  usual  shape,  and  its  walls  and  ceiling  a  dead 
white.  The  object  to  be  aimed  at  in  a  Concert-room  is,  not  to  deafen  a  favoured  few,  but  to  fill  the  whole 
chamber  equally  with  Sound,  and  yet  allow  the  Echo  as  little  power  to  disturb  the  principal  Sound,  by  a  lingering 
after-twang,  as  possible.  But,  whether  for  music  or  for  oratory,  open  windows,  deep  recesses,  hangings,  or 
carpeting,  and  a  numerous  audience  in  woollen  clothing,  are  all  unfavourable  to  good  hearing.  They  are  to  Sound, 
what  black  spaces  in  an  apartment  would  be  to  light ;  they  return  back  none,  or  next  to  none,  of  what  falls  on 
them.  Their  fault  is  not  so  much  that  they  reflect  it  irregularly,  as  that  they  do  not  reflect  it  at  all. 

The  rolling  of  thunder  has  been  attributed  to  Echos  among  the  clouds ;  and  if  it  is  considered  that  a  cloud  is  a       38. 
collection  of  particles  of  water,  however  minute,  yet  in  a  liquid  state,  and  therefore  each  individually  capable  of  Reverbera- 
reflecting  Sound,  there  is  no  reason  why  very  loud  Sounds  should  not  be  reverberated  confusedly  (like  bright  lights)  g°°nj from 
from  a  cloud.     And  that  such  is  the  case,  has  been  ascertained  by  direct  observation  on  the  Sound  of  cannon,  the  clouds'" 
Messrs.  Arago,  Matthien,  and  Prony,  in  their  experiments   on  the  velocity  of  Sound,  observed,  that  under  a 
perfectly  clear  sky,  the  explosions  of  their  guns  were  always  heard  single  and  sharp,  whereas  when  the  sky  was 
overcast,  or  even  when  a  cloud  came  in  sight  over  any  considerable  part  of  the  horizon,  they  were  frequently 
accompanied  with  a  long  continued  roll  like  thunder,  and  occasionally  a  double  Sound  would  arrive  from  a 
single  shot. 

But  there  is,  doubtless,  also  another  cause  for  the  rolling  of  thunder,  as  well  as  for  all  its  sudden  and  capri-        39. 
cious  bursts  and  variations  of  intensity,  of  which   our  knowledge  of  the  velocity  of  Sound  furnishes  a  perfect  Explanation 
explanation.     To  understand  this,  we  must  premise  that,  cteteris  paribus,  the  estimated  intensity  of  a  Sound  will  °rthunder 


754  SOUND 

Sound.  ],e  proportional  to  the  quantity  of  it  (if  we  may  so  express  ourselves)  which  reaches  the  ear  in  a  given  tirne. 
•^-s.'-— '  Two  blows  equally  loud,  at  precisely  the  same  distance  from  the  ear,  will  Sound  as  one  of  double  the  intensity ; 
a  hundred,  struck  in  an  instant  of  time,  will  sound  as  one  blow  a  hundred  times  more  intense  than  if  they  followed 
in  such  slow  succession  that  the  ear  could  appreciate  them  singly.  Now  let  us  conceive  two  equal  flashes  of 
lightning,  each  four  miles  long,  both  beginning  at  points  equidistant  from  the  auditor,  but  the  one  running1 
out  in  a  straight  line  directly  away  from  him  ;  the  other  describing  an  arc  of  a  circle  having  him  in  its  centre. 
Since  the  velocity  of  Electricity  is  incomparably  greater  than  that  of  Sound,  the  thunder  may  be  regarded  as 
originating  at  one  and  the  same  instant  in  every  point  of  the  course  of  either  flash.  But  it  will  reach  the  ear 
under  very  different  circumstances  in  the  two  cases.  In  that  of  the  circular  flash,  the  Sound  from  every  point 
will  arrive  at  the  same  instant,  and  affect  the  ear  as  a  single  explosion  of  stunning  loudness.  In  that  of  the 
rectilinear  flash,  on  the  other  hand,  the  Sound  from  the  nearest  point  will  arrive  sooner  than  from  those  at  a 
greater  distance;  and  those  from  different  points  will  arrive  in  succession,  occupying  altogether  a  time  equal  to 
that  required  by  Sound  to  run  over  four  miles,  or  about  20  seconds.  Thus  the  same  amount  of  Sound  is  in  the  latter 
case  distributed  uniformly  over  20  seconds  of  time,  which  in  the  former  arrives  at  a  single  burst;  of  course,  it  will 
have  the  effect  of  a  long  roar,  diminishing  in  intensity  as  it  comes  from  a  greater  and  greater  distance.  If 
the  flash  be  inclined  in  direction,  the  Sound  will  reach  the.  ear  more  compactly,  («'.  e.  in  shorter  time  from  its 
commencement,)  and  be  proportionally  more  intense.  If  (as  is  almost  always  the  case)  the  flash  be  zigzag,  and 
composed  of  broken  rectilinear  and  curvilinear  portions,  some  concave,  some  convex  to  the  ear ;  and  if,  especially, 
the  principal  trunk  separates  into  many  branches,  each  breaking  its  own  way  through  the  air,  and  each  becoming 
a  separate  source  of  thunder,  all  the  varieties  of  that  awful  Sound  are  easily  accounted  for. 

40.  ^e  w'"  on'y  mention  one  other  phenomenon  which  is  accountable  for  on  the  same  principle.     In  the  eruption 

Pnenome-     of  a  volcano  it  is  often  remarked,  that  every  ejection  of  stones,  &c.   is   accompanied   with   an   explosion    like 

non  ob-         artillery  when  heard  at  a  distance ;  but  when  near,  the  Sound  resembles  rather  that  of  a  loud  and  deep  sigh, 

unaccompanied  with  any  sudden  burst.     In  both  cases  the  cause  of  Sound  is  the  same,  the  upward  rush  and 

tions  of'vol-  displacement  °f tne  a'r  %  tne  stone  ;  but  where  the  auditor  is  near  the  bottom  of  the  column  of  Sound,  it  reaches 

cann?.  his  ear  more  in  detail  than  when  at  a  distance,  and  therefore  nearly  equidistant  from  all  its  parts.     In  fact,  let 

t  =  the  time  taken  by  the  stone  to  rise  to  a  height  x,  and  let  a  be  the  distance  of  the  observer  from  the  bottom 

*J  (i*  -4-  x2 

of  the  column,  and  v  the  velocity  of  Sound,  then  will  t  -4-  • =   time  elapsed   from   the   moment   of 

v 

ejection  to  that  of  the  Sound  of  the  column  at  the  height  x  reaching  the  ear.     Hence  the  whole  Sound  of  the 

portion  x  of  the  column  will   arrive  in  an  interval  of  time  represented  by  t  -) — .     Now,  as  a 

v 

increases,  x,  and  therefore  t  remaining  constant,  this  function  diminishes  rapidly,  and  ultimately  approaches  t  as 
its  limit.  Thus  the  Sound  arrives  continually  more  and  more  condensed.  Should  any  discharge  be  made 
obliquely  towards  the  observer's  station,  a  still  greater  concentration  of  the  noise  will  happen,  as  may  be  easily 
seen  by  considering  that  if  shot  directly  towards  him,  with  the  velocity  of  Sound,  the  report  would  reach  him 
from  every  part  of  the  line  strictly  at  the  same  moment.  Now,  as  these  ejections  have  been  known  to  rise  to  a 
height  of  10,000  feet,  in  spite  of  the  resistance  of  the  air,  their  initial  velocity  must  be,  at  least,  equal  to  that  of 
Sound.  At  great  distances  it  is  probable  that  only  the  Sounds  produced  by  such  oblique  ejections  have  intensity 
or  (as  we  may  express  it)  body  enough  to  affect  the  sense. 

§  II.  Mathematical  Theory  of  the  Propagation  of  Sound  in  Air,  and  other  Elastic  Fluid  Media. 

^,  A  general  notion  of  the  mode  in  which  an  impulse  communicated  to  one  portion  of  the  air,  or  other  elastic 

General  no-  ^u't''  "s  diffused  through  the  surrounding  portions,  and  successively  propagated  to  portions  at  a  greater  and 
tionofihe  greater  distance  from  the  original  source  of  the  motion,  may  be  obtained  by  considering  the  way  in  which  a 
communica-  tremor  runs  along  a  stretched  cord,  or  in  which  waves  excited  in  the  surface  of  still  water  dilate  themselves 
tion  of  mo-  circularly,  and  propagate  a  motion  impressed  on  one  point  of  the  surface,  in  all  directions  to  a  distance.  In  the 
former  case,  conceive  a  blow  given  to  a  point  in  the  middle  of  the  cord,  transversely  to  its  length.  The  point  to 
media.  which  the  blow  is  given  will  be  thrown  out  of  the  straight  line,  and  a  flexure,  or  angle,  will  be  formed  in  that 
Propagation  part.  Owing,  however,  to  the  inertia  of  the  cord,  the  displacement  of  the  particles  in  the  first  instant  will  be 
of  tremors  confined  to  the  immediate  neighbourhood  of  the  point  of  impulse;  so  that  the  cord  will  not  at  once  assume  the 
^etctTed  State  rePresented  in  fiS-  *'  consisting  of  two  straight  portions  A  B,  B  C,  forming  a  very  obtuse  angle  ABC; 
vord.  but  rather  that  '"  %•  2-  in  which  the  greater  part  on  either  side  AD,  DC,  retain  their  original  position  ;  and  a 

Fig.  1.  small  part  DB  E,  proportioned  to  the  violence  and  suddenness  of  the  blow,  is,  as  it  were,  bulged  out  into  an 
angular  form  D  B  E.  The  particle  at  B  then  is  solicited  on  both  sides  by  the  tension  of  the  cord  in  directions 
B  D,  BE;  but  these  tensions,  which  in  the  quiescent  state  of  the  string  exactly  counteracted  each  other,  now 
only  do  so  in  respect  of  those  parts  of  each  which,  when  resolved,  act  in  directions  parallel  to  D  A,  EC  respec- 
tively. The  other  resolved  portions,  perpendicular  to  these,  conspire  and  urge  the  point  B  towards  its  point  of 
departure  6.  As  there  is  no  force  to  counteract  this  (the  impulse  being  supposed  momentary)  B  will  obey  their 
solicitation,  and  approach  6  with  an  accelerated  velocity.  But,  action  and  reaction  being  equal  and  contrary,  the 
same  force  by  which  the  molecule  E  drags  B  down,  will  be  exerted  on  E  to  drag  it  up,  or  out  of  the  line ;  so  that 


SOUND.  755 

Sound,  by  the  time  B  has  performed  half  its  course  towards  6,  E  will  have  been  raised  above  the  line,  and  will  have  part  I. 
— •v""''  acquired  a  velocity  capable  of  carrying  it  still  further  in  that  direction.  At  this  instant  the  cord  will  have  assumed  ^— ^— ^ 
the  fi<rure  A  D'  D  B  E  E'  C.  At  the  next  moment  the  forces  are  reversed,  B  then  tends  to  drag  both  D  and  E 
down  to  the  line  ;  but  its  own  acquired  momentum  is  expended  in  the  effort,  and  by  the  time  it  has  reached  its 
original  place  in  the  line,  its  inertia  is  destroyed,  and  it  rests  there  without  a  tendency  to  go  beyond  it  on  the 
other  side.  Meanwhile,  however,  D  and  E  have  attained  their  greatest  elevation  ;  and  thus  (he  protuberance 
D  B  E  is  resolved  into  two  D' D  B  and  BEE'  (of  less  height,  however)  on  either  side.  In  like  manner  the 
particles  D  and  E,  in  returning  to  their  places,  drag  up  the  next  adjoining  D'  and  E',  and  then  the  next,  and  so 
on  ;  and  thus  the  summits  of  the  protuberances  advance  along  the  line,  and  correspond  in  succession  to  all  its 
points  ;  and  the  visible  effect  is  an  undulation,  or  wave,  which  runs  along  the  cord  with  a  velocity  greater  the 
greater  is  the  force  with  which  the  cord  is  strained,  as  it  manifestly  ought  to  be,  since  the  rapidity  with  which  each 
molecule  returns  from  its  displaced  situation  is  greater  as  the  force  urging  it  is  so  ;  and  this  force  is  nothing 
more  than  the  resolved  part  of  the  tension. 

In  like  manner,  when  a  wave  is  excited  in  the  surface  of  water,  as  when  by  throwing  in  a  stone  one  portion        ^ 
is  violently  driven  down,  and  the  surrounding  part  heaped  up  above  its  natural  level  ;  this  subsides  and  fills  up  propagation 
the  vacuity  ;  but  as  its  pressure  takes  place  alike  on  both  sides  of  the  ridge,  the  fluid  on  the  outside  of  the  ridge  of  waves  in 
is  also  pressed  on,  from  below  upwards,  by  the  reaction  of  the  fluid  stratum  which  sustains  the  ridge,  and  whose  sl'U  water, 
pressure  is  propagated  equally  in  all  directions.     Thus  the  ridge,  in   subsiding,  not  only  fills   up   the  central 
vacancy,  but  forces  up  another  ridge  exterior  to  it;    and  this,  in   subsiding,  another,  and   so   on;   and  thus  an 
advancing  wave  is  formed  ;    and  the  same  action  taking  place  on  all  sides  of  the  centre,   the  wave  can  advance 
no  otherwise  than  in  the  direction  of  radii  on  all  sides  diverging  therefrom. 

It  is  by  no  means  intended,  in  what  is  here  said,  to  give  an  accurate  account  of  what  passes  in  either  of  these        43. 
cases,  (in  fact,  it  is  very  far  from  being  so,  as  the  reader  by  a  little  attention  will  soon  perceive,)  but  only  to  give 
a  first  conception  of  the  proposition  of  motion  by  undulations  or  waves. 

In  this  general  account  of  'he  above  cases,  one  thing,  however,  cannot  fail  to  strike  the  reader,  that  the  wave        44. 
which    advances  on   the  surface  of  water — the  sinuosity  which  runs  along  the  stretched  cord — are  neither  of  A  wave  not 
them  things,  but  forms.     They  are  not  moving  masses  advancing  in  the  direction  in  which  they  appear  to  run,  a  progress- 
but  outlines,  or  figures,  which  at  each  instant  of  time  include  all  the  particles  of  the  water  or  the  cord  which,  it  [vVt>1°v'n" 
is  true,  are.  moving,  but  whose  motion  is  in  fact  transverse  to  the  direction  in  which  the  waves  advance.     But  this  advanci'nj"1 
is  by  no  means  an  essential  condition.     We  may  generalize  this  idea  of  a  wave,  and  conceive  it  as  the  form,  form. 
space,  or  outline,  whether  linear  or  superficial,  comprehending  all  the  particles  of  an  undulating  body  which  are 
at  once  in   motion,   (supposing,  for  the  present,  that  the  motion  of  each  consists  of  a  simple  displacement  and 
return  to  quiescence,  and  not  in  a  repetition  of  several  such  displacements  and  returns  in  succession.) 

The  waves  in  a  field  of  standing  corn,  as  a  gust  of  wind  passes  over  it,  afford  a  familiar  example  of  the  rela-        45 
tion  between  the  motion  of  the  wave,  and  that  of  the  particles  of  the  waving  body  comprised  within  its  limits,  Example, 
and  of  the  mutual  independence  which  may  in  certain  cases  subsist  between  these  two  motions.     The  gust  in  Waves  in  a 
its  progress  depresses  each  ear,  in  its  own  direction,  which,  so  soon  as  the  pressure  is  removed,  not  only  re-turns,  '**'''  ?' 
by  its  elasticity,  to  its  original  upright  situation,  but  by  the  impetus  it  has  thus  acquired,  surpasses  it,  and  bends  st 
over  as  much,  or  nearly  as  much,  on  the  other  side  ;  and  so  on  alternately,  oscillating  backwards  and  forwards 
in  equal  times,  but  continually  through  less  and  less  spaces,  till  it  is  reduced   to  rest   by  the  resistance   of  the 
air.      Such  is  the  motion  of  each  individual  ear  ;   and  as  the  wind  passes  over  all   of  them   in  succession,  and 
bends  each  equally,  all  their  motions  are  so  far  similar.     But  they  differ  in  this,  that  they  commence  not  at  once 
but  successively.     Suppose  (to  fix  our  ideas)  the  wind  runs  over  100  feet  in  a  second,  and  that  the  ears  stand 
one  foot  asunder,  and  each   makes  one  complete  vibration  to  and  fro  in  a  second.      Suppose  A  (fig.  3)  to  be  the  pig.  3. 
furthest  point  which  the  wind  at  any  given  instant  of  time  has  reached,  or  the  last  ear  which  it  lias  just  bent,  and 
let  the  action  of  the  wind  be  regarded  as  lasting  only  for  a  single  instant.     Then  will  the  next  preceding  ear  B 
have  already  begun  to  rise  from  its  bent  position,  the  next  C  will  have  risen  rather  more,  and  the  25th  ear  G 
(since  the  distance  A  F  is  25  fctt,  and  consequently  since  -^fjj  =  j  of  a  second  have  elapsed  since  the  wind  was 
at  G)  will  have  gone  through  one-fourth  of  its  complete  vibration  to  and  fro,  and  will  have  therefore  just  attained 
its  upright  position  ;  so  that  the  ears  F,  E  immediately  adjacent  towards  A  will  not  yet  have  quite  recovered  their 
perpendicularity,  but  still  lean  somewhat  forwards;  while  those  on  the  other  si.le  H,  I  will  have  surpassed  the 
perpendicular,  and  have  begun  to  sway  backwards  ;  consequently  at  G  the  stalks  will  on  both  sides  be  convex 
towards  G,  and  the  ears   in  that  place  will  be  further  asunder  than  in  their  state  of  rest,  and   will  appear  as  it 
were  rarefied  when  viewed  by  a  spectator  so  distant  as  to  take  in  a  great  extent  at  once.      Still  further  in  rear  of 
the  wind,  as  50  feet,  at  L,  the  50th  ear  will  have  swung  backwards  as  far  as  possible,  and  will  just  have  its  motion 
destroyed.     The  preceding  stalk,  K,  will  still  want  somewhat  of  its  extreme  backward  flexure ;   the  subsequent 
one,  M,  will  already  have  risen  a  little,  and  therefore  the  interval  of  the  ears  K,  N  will  be  just  what  it  was  in  the 
state  of  rest.     At  L,  then,  the  spectator  will  see  the  ears  at  their  natural  distances  from  each  other.     Again  the 
75th  stalk,  Q,  in   rear  of  the  wind  will  have  had  time  to  rise  again  erect  from  its  backward  inclination,  three- 
fourths  of  a  second  having  elapsed  since  its  first  bending  forward.     The  74th,  P,  will  not  be  quite  erected  ; 
the  76th  will   have  surpassed  the  erect  state,  and  have  again   begun  to  lean  forward.     The  stalks  then  on   both 
sides  of  Q  will  curve  towards  Q,  and  their  ears  will  therefore  be  closer  together  than  in  their  natural  state,  and 
will  appear  condensed  to  the  spectator  above  mentioned.     Finally,  the  99th,  100th,  and  101st  ears  will  be  again 
in  the  same  relative  state  as  the  49th,  50th,  and  51st ;   only  leaning  forwards  instead  of  backwards,  and  therefore 
neither  condensed  nor  rarefied.     The  field,  then,  will  present  to  the  spectator  a  series  of  alternate  condensation!* 
and  rarefactions  of  the  corn  ears,  separated  by  intervals  in  their  natural   state  of  density  ;  and  this  series  will 
extend  so  far  in  rear  of  the  wind,  till  the  resistance  of  the  air  and  want  of  perfect  elasticity  in  the  stalks  shali 
VOL.  iv.  5  p 


756 


SOUND. 


Sound,      have  reduced  them  to  rest,  and  these  alternations,  by  the  difference  of  shading  they  offer,  will  become  apparent 
v— v>—- x  to  his  sight  as  dark  and  bright  zones. 

46.  It  matters  not,  for  our  present  purpose,  that,  the  impulse  is,  in  the  case  here  taken,  not  propagated  mechani- 

Velocity  of  cally  from  ear  to  ear  by  mutual  impulse,  but  that  each  moves  independently  of  all  the  rest.  All  we  want  to 
illustrate  is  the  distinction  between  the  wave  and  the  moving  matter,  and  the  independence  of  their  motions. 
The  waves  here  run  along  with  the  speed  of  the  wind,  whatever  that  may  be  ;  for  it  is  always  the  point  25  feet 
in  rear  of  the  wind  that  is  most  rarefied,  and  that  at  75  that  is  most  condensed  ;  and  the  interval  between  the 
first  and  100th  ear,  comprehending  ear?  in  every  state  or  phase  of  their  vibrations,  is  what  we  term  a  wave.  The 
velocity  of  the  wave,  then,  is,  in  this  case,  that  of  the  wind  ;  and  is  totally  distinct  from,  and  independent  of,  that 
of  each  or  any  particular  ear.  The  one  is  a  constant,  the  other  a  variable  quantity  ;  the  one  a  general  result- 
ing phenomenon,  the  other  a  particular,  individual,  mechanical  process,  going  on  according  to  its  own  laws. 

Neither  is  it  of  the  least  consequence  whether  the  excursions  of  the  several  stalks  from  their  position  of  rest 
be  great  or  little  ;  whether  the  degree  of  bending,  or  force  of  the  wind,  be  great  or  small,  provided  its  velocity 
be  constant.  In  the  case  of  wind,  indeed,  the  force  depends  on  the  velocity  ;  but  if  we  conceive  the  impulse 


Part  I. 


the  wave 
distinguish- 
ed from  that 
of  its  com- 
ponent 
parts. 


the  waves. 


Various 
species  of 
\v;uos. 


4t>. 

Sonorous 
waves  pro- 
uagaled  in 


47. 

And  inde- 
pendent of 

their  excur-  gjvcn  jjy  a  rj<r|d  rod  made  to  sweep  across  the  field,  any  greater  or  less  degree  of  flexure  might  he  given,  with 
*ne"state"of  l^e  same  velocity,  by  a  mere  change  of  its  level ;  but  the  velocity  of  the  wave  would  still  be  that  of  the  rod  in 
rest.  every  case. 

48.  But  with  respect  to  the  breadth  of  the  wave,  or  the  magnitude  of  that  interval  which  comprises  particles  in 

Breadth  of  every  phase  or  state  of  their  motion,  going  and  returning,  it  is  otherwise.  This  is  a  result  depending  essentially 
on  the  motions  of  the  particles  themselves  ;  for  we  see  evidently  in  the  above  instance,  that  this  breadth,  which 
is  100  feet,  is  equal  to  the  space  run  over  by  the  wind  in  a  time  equal  to  that  of  one  complete  vibration,  going 
and  returning,  of  each  individual  ear.  Now  this  time  depends  only  on  the  elasticity  of  the  stalk,  and  the  weight 
of  the  ear  it  carries.  In  general  then  we  may  state,  that  "  The  breadth  of  a  wave  is  equal  to  the  space  run  over 
by  it  in  a  time  equal  to  that  in  which  any  molecule  of  the  waving  body  performs  one  complete  vibration,  going 
and  returning,  through  all  the  phases  of  its  motion."  In  the  case  here  taken,  the  motion  of  the  individual  mole- 
cules is  not,  as  in  the  former  instances,  transverse  to  that  of  the  wave,  but  parallel  to  it.  It  is  then  hardly  to 
be  termed  a  form,  or  an  outline.  To  such  a  wave,  the  term  pulse  is  often  applied.  Whatever  be  the  nature  of 
the  internal  motions,  however,  the  general  name  wave  or  undulation  will  equally  apply,  and  will  be  used  in 
future  indiscriminately  for  all  sorts  of  propagated  impulses.  It  is  not  even  necessary  that  the  motions  of  the 
constituent  particles  should  be  rectilinear,  or  even  lie  in  one  plane.  We  may  suppose  the  impelling  cause  to  be  a 
whirlwind.  In  this  case  each  ear  will  have  a  rotatory  or  twirling  motion,  or  the  stalk  a  conical  one,  simply,  or 
in  addition  to  its  flexure  in  a  vertical  plane  ;  hut  the  wave  is  independent  of  these  particularities. 

In  the  case  just  described,  each  particle  is  supposed  to  be  set  in  motion  by  an  external  cause,  and  to  be  unin- 
fluenced in  its  motions  by  the  rest.  It  is,  therefore,  not  a  case  of  the  propagation  of  motion  at  all.  It  is  quite 
otherwise  with  Sound,  or  other  similar  cases,  where  every  particle  of  a  medium  receives  its  whole  motion  from 
those  which  were  moving  before,  and  transmits  it  to  others  previously  at  rest.  The  problem  to  investigate  the 
general  laws  of  the  communication  of  motion  under  such  circumstances  is  one  of  the  utmost  complexity,  and 
at  present  has  been  only  resolved  under  very  restricted  conditions  ;  enough,  however,  to  verify  principal  facts, 
and  establish  leading  points,  in  the  doctrine  of  Acoustics.  We  shall  be  far  from  attempting  to  present  here  any 
thing  approaching  to  a  sketch  of  the  profound  geometrical  researches  which  have  been  bestowed  on  this 
department  of  Physics,  contenting  ourselves  with  referring  the  reader  for  a  knowledge  of  them  to  the  various 
Memoirs  of  Euler,  D.  Bernouilli,  Lagrange,  Poisson,  &c.  See  (I.)  Recherches  sur  la  Nature  et  la  Propaga- 
tion du  Son,  par  L.  de  Lagrange,  Mem.  Acad.  Turin,  i.  247.  (2.)  Euler,  Recherches  sitr  la  Propagation  des 
Ebranlemens  dans  un  Milieu  Elastique,  Misci'l.  Turin,  ii.  (3.)  Nouvel/es  Recherches  sur  la  Propagation  du  Son, 
par  M.  Lagrange,  Miscel.  Turin,  ii.  (4.)  Euler,  De  la  Propagation  du  Son,  Mem.  Acad.  Berlin,  1759,  p.  185, 
and  Supplement,  p.  210,  and  Continuation,  p.  241.  (5.)  Enler,  Eclaircissemens  plus  detaill-s  sur  la  Propagation 
du  Son  et  sur  tEcho,  Mem.  Acad.  Berlin,  1765,  p.  335.  (6.)  Poisson,  Sur  (Integration  de  quelques  Equations 
Lineaires  des  Differens  Partielles,  Mem.  de  I'Acad.  Paris,  1818,  p.  121.  (7.)  Poisson,  Sur  la  Theorie  du  Son, 
Journal  de  I'Ecole  Polytechn.  xiii.  319.  :  while  we  confine  ourselves  to  just  so  much  developement  of  the 
mathematical  analysis  of  the  subject  as  will  suffice  for  the  demonstration  of  the  chief  theoretical  propositions  we 
shall  have  occasion  for  in  the  sequel. 

Let  us  then  consider,  as  the  simplest  case,  the  propagation  of  Sound  in  a  straight  canal  of  equal  bore  through- 
Propagation  out,  filled  with  air  or  any  other  elastic  fluid  of  equable  density  and  elasticity,  unacted  on  by  gravity,  and  of  which 
ln  the  transverse  section  is  so  small,  and  the  sides  so  perfectly  polished,  that  we  may  regard  the  motions  of  all 
particles  in  the  same  section  as  exactly  similar;    so  that  each  section  shall  merely  advance  and  recede  in  the 
pipe,  without  any  lateral  change  of  place  of  its  constituent  molecules  inter  se.     Let  A  B  (fig.  4)  be  such  a  pipe, 
and  let  any  section  of  it,  as  A,  be  agitated  by  an  external  cause,  with  any  arbitrary  motion,  i.  e.  one  whose  dura- 
tion and  extent,  and  whose  velocity  at  every  instant,  shall  be  entirely  dependent  on  the  will,  or,  if  we  please,  the 
caprice  of  an  external  operator  sufficiently  powerful  to  command  it  ;  and  let  us  inquire  how  any  other  section 
whatever,  situated  at  any  assigned  distance,  x,  from  A,  will  move  in  consequence  of  this  arbitrary  motion  of  A. 

Let  us  then  conceive,  that,  in  general,  the  section  or  stratum  of  molecules  a  abb,  whose  distance  from  the 
initial  place  A  of  the  section  A  is  represented  by  x,  shall,  after  the  lapse  of  any  time  t,  have  been  transported 
into  the  situation  a  o.  ft  ft,  at  a  distance  A  «  =  y  from  the  same  fixed  point  A.  Let  x1  x'',  &c.  be  the  distances  of 
the  next  consecutive  sections  from  the  fixed  point  A,  in  their  state  of  rest,  and  y',  y",  &K.  their  distances  after 
the  lapse  of  the  same  time  t.  Then  wil.  x  -  x  =  d  x,  x"-  x'=  d  x',  x'"  -  x"  —  d  x",  &c.  be  the  thicknesses  (sup- 
posed infinitely  small)  of  these  strata,  or  the  spaces  occupied  by  them  (taking  the  area  of  the  section  for  unity) 
in  their  quiescent  state,  and  /  —  y  =  dy,  y"  -  y'  —  dy',  y"' —  y"=dy",  &c.  the  same  in  their  state  o<~ 


50. 


air  in  one 
dimension. 
Fig.  4. 


51. 

Analysis  of 
this  caae. 


SOUND.  757 

Sound.      motion.     Now  as  these  strata  were  in  contact  at  the  origin  of  the  motion,  and  are  held  together  by  the  pressure      Hart  1. 
_-^^— _>  of  the  surrounding  fluid,  they  will  remain  in  contact,  and  advance  and  recede  along  the  pipe  as  one  mass,  only   ^-•"V"-' 
the  space  they  will  occupy  at  different  points  of  their  motion  will  be  variable,  according  to  the  degree  of  conden- 
sation or  dilatation  they  may  have  undergone  in  virtue  of  their  motion  itself.     If,  for  instance,  at  any  moment 
the  hinder  of  them  dy  be  in  the  act  of  urging  forward  the  next  dy',  it  will  be  condensed;  if  retreating,  rarefied 
in  comparison  with  the  state  of  the  preceding  one  dy'1. 

Now  any  stratum  of  molecules  dy'  interjacent  between  two  others  dy  and  dy"  can  only  undergo  a  change        52. 
in  its  velocity  when  urged  by  some  force,  und  the  only  force  which  can  urge  it  is  the  difference  of  pressures  it  Expression 
may  experience  on  its  two  faces  by  the  difference  (if  any)  of  the  elasticities  of  the  adjacent  strata  dy''  and  dy.  "! 'heading 
If  we  can  estimate  this,  the  laws  of  Dynamics  will  enable  us  to  express  the  consequent  change  of  motion.     To   ° 
this  end,  then,  let  the  elasticity  of  the  air  in  its  quiescent  state  be  represented  by  E,  which  is  a  given  quantity, 
and  is   measured  by  the  weight  of  a  column  of  mercury  sustained  by  it,  or  by  the  length  of  a  homogeneous 
column  of  air  of  the  same  density,  whose  weight  shall  suffice  to  keep  it  so  compressed,  or  be  equal  to  that  of 
the  column  of  mercury  in  the  barometer.     Then,  since  the  elasticity  of  air  is  inversely  as  the  space  it  occupies, 
(casleris  paribus,)  the   elasticity  of  the   air  when  occupying  the   stratum  dx  :  its    elasticity  when  occupying 

dx 
dy  ::  dy:  dx,  and  therefore  the  elasticity  when  occupying  the  space  dy  =  E  .    — — .     Similarly  the  elasticities 

dx'  dx" 

of  the  air  occupying  d  y'  and  d  y"  will  be  represented  by  E  .  — — j-  and  E  .        „  .     Hence  the  plane  separating 

j  y 

d  x  d  x1 

the  strata  dy  and  dy'  will  be  pressed  forward  by  the  elasticity  E  .    — — ,  and  backward  by  E  .  -7-7-     So  that 

y  y 

it  will,  on  the  whole,  be  urged  forward  by  —  E  (  — - — -.  —  — —  Ithal  is,  by —  E  d  —. — ,  the  differentials  being  all  on 

\dy'        dy  J  dy 

the  supposition  of  t,  the  time  being  constant,  and  x  andy  only  variable.  Now,  if  we  denote  by  H  the  length  of  a 
homogeneous  column  of  air  necessary  to  counterbalance  the  elasticity  of  the  quiescent  air,  and  by  D  its  density, 
we  have  H  D  =  its  weight  =  the  elasticity  E,  and  dx' .  D  =  the  weight  of  the  stratum  d  x',  which,  substituting 

for  D  its  value—,  becomes  d  x'  .   —.     Thus,  then,  the  moving  force  —  Ed  — —  is  exerted  in  urging  for- 


ward  a  weight  =  d  x' .  — ,  and  is  therefore  equivalent  to  an  accelerating  force 

'(r) 

V.»  /  =.  +  H  Y4^f.  4&. 


IT  \"»    /  .        I 

-d*~  '\-jj 

regarding  d  x  as  constant,  or  all  the  strata  dx,  d  x',  dx",  &c.  as  originally  equal. 

Now  the  distance  of  the  mass  thus  urged  from  the  fixed  point  A,  at  the  expiration  of  the  time  t,  is  y1.     Hence,        53. 
if  we  regard  only  the  motion  of  the  particle  d  y'  (or  which  comes  to  the  same)  of  d  y,  which  is  in  contact  with  Equation 
it,  we  have  by  Dynamics  deduced. 

=  —  -^     w 

where  2g  =  9met- 8088  =  32-180  British  standard  feet  and  gravity,  for  the  unit  of  accelerating  force,  and  in 
which  equation  t  is  expressed  in  mean  solar  seconds  ;  and  all  linear  quantities,  such  as  H,  ae,  y,  in  metres  or 
feet,  according  as  we  take  the  metre  or  foot  as  the  unit  of  linear  measure. 

This  is,  in  fact,  an  equation  of  partial  differentials,  y  being  at  once  a  function  both  of  *  the  original  distance        54. 
of  the  stratum  d  x  from  the  origin  of  the  motion,  and  of  t  the  time  elapsed.     In  its  present  form,  simple  as  it  .Limita'ion 
appears,  it  is  altogether  intractable  and  incapable  of  integration.     Jn  fact,  it  embraces  a  class  of  dynamical  '„" 'sound* 
problems  of  very  great  complexity ;  for  it  is  evident  that  since  no  hypothesis  has  been  made  in  any  way  limiting  the 
extent  of  the  excursions  of  the  original  or  subsequent  strata  from  their  points  of  quiescence,  this  equation  must 
contain  the  general  expression  of  all  possible  motions  of  elastic  fluids  in  narrow  pipes,  whether  great,  as  when 
urged  by  pistons  or  driven  by  bellows,  or  small,  as  are  the  tremors  which  cause  Sound.     In  the  theory  of  Sound 
we  suppose  the  agitations  of  each  molecule  so  minute  as  not  to  move  it  sensibly  from  its  point  of  rest.     Expe- 
rience confirms  this.     Sounds  transmitted  through  a  smoky  or  dusty  atmosphere  cause  no  visible  motion  in  the 
smoke  or  floating  dust,  unless  the  source  of  Sound  be  so  near  as  to  produce  a  wind,  which,  however,  is  always 
insensible  at  a  very  moderate  distance. 

If  we  introduce  this  condition,  the  equation  (a)  admits  of  integration  ;  for  the  whole  amount  of  motion  of  each        55. 
molecule  being  extremely  minute,  their  differences  for  consecutive  molecules,  or  the  amount  of  the  rarefactions  Simplifica- 

,  tion  of  the 

and  condensations  undergone,  must  be  much  more  so.     Hence  the  value  of   —-.  which  expresses  the  ratio  of  1nal  e<lu'' 

dx  'ion. 

the  condensation  of  the  stratum  d  y  in  motion  and  in  rest,  may  be  regarded  as  equal  to  unity,  and  the  equation 
becomes  simply, 


(6) 
6  r'2 


758  SOU  N  D. 

Sound,     which  is  the  equation  of  Sound  regarded  as  propagated  in  one  dimension,  that  of  length,  only  ;  or,  as  prevented 
— •v'"--'  from  spreading  laterally  by  a  pipe. 

56.  The  complete  integral  of  this  equation  is  well  known  to  be 

Its  Integra-  r\  s       t         j\      i     ^  /  j\  /  \ 

tion  y  =  F  (x  -$-  at)  -\-  f  (x  —  at),  (e) 

where  F  and  f  denote  arbitrary  functions  of  the  quantities  within  the  parenthesis,  and  which  must  be  determined 
by  a  consideration  of  the  initial  state  of  the  fluid,  or  by  the  nature  of  the  motion  originally  communicated  to  its 
molecules. 

57  Let  us  then  suppose,  that,  at  the  commencement  of  the  motion,  we  have  impressed  on  each  section  of  the 

Deiermina-  fluid,  along  its  whole  extent,  any  arbitrary  velocities  and  condensations,  by  any  means  whatever,  so  as  to  com- 
tion  of  the  prehend  in  our  investigation  all  possible  varieties  of  initial  motion,  whether  expressible  by  regular  analytical 
arbitrary  functions,  or  depending  on  no  regular  law  whatever.  It  is  manifest  that  these  conditions  will  be  expressed  by 
functions.  assunljnrr  arbitrary  functions  of  x,  such  as  0  (x)  and  fy  (x)  for  the  initial  values  of  the  two  partial  differentials 

and  — — ,  whereof  the  former  represents  in  all  cases  the  velocity  (v)  of  a  particle  which  would  be  at  the 


—  — 
dt  d  x 

distance  x  from  the  origin  of  the  coordinates  in  the  state  of  equilibrium,  and  the  latter  the  linear  extent  (e)  of 
that  particle  compared  with  its  original  extent,  to  which  its  density  and  elasticity  are  reciprocally  proportional. 
Now,  differentiating  (c)  we  get  for  the  general  values  of  e  and  e 

v=  4r  =  «{F'O  +  aO-/'  O-«0};  (<0 

a  t 

F'(*  +  «0  +  f'(x-at);  00 


d  x 
consequently  their  initial  values,  making  t  =  0,  will  be 

0  (*)=  a.  {F<  (a)  -/'(*)} 
y,  GO  =  F'  (r)  +  /'  (*). 
whence  we  get  immediately 


and  multiplying  by  dx  and  integrating 

F  (x)  =  J-  /{  a^  (x)  +  0  (*)  }  dx;        f  (x)  =    :  -/{  «  V  (*)  -  0  (*)}<*  */ 

&    (t  -i   (I 

and  thus  the  forms  of  the  functions  F  and  /become  known  when  those  of  0  and  ^  are  given. 

58  The  question  of  the  propagation  of  Sound,  however,  does   not  require  us  to  concern  ourselves  with   these 

Expression  functions,  as  a  knowledge  of  the  actual  velocity  aud  density  of  any  molecule  at  any  instant  is  sufficient  for  our 
of  the  state   purpose.     Substituting  then  in  (d)  and  (e)  for  F'  and  /',  the  forms  corresponding  in  0  and  ^,  we  get 
of  any  mole- 

iaU'.eantany  t,  =  =          {  *  (*  +  °  <>  ~  +  C*  ~  «  0  }  +    4    {  0  C*  +  «  0  +  0  (*  ~  «  0  }  5  (ff) 


(A) 


or,  as  it  may  also  be  written, 


v  =  —  {  0  (x  +  a  t)  +  a  y-  (*  +  a  t)  }  +  —  {  0  (j;  -  a  t)  -  a  y-  (x  -  a  t)  }  ;  (i) 

e  =  {  0  (.r  +  a  t)  +  a  ty  (x  +  a  t)  }  -  —  {  0  (x  -  a  t)  —  a  -f  (x  -  a  t)  }  .  (j) 

%  a  &  & 

59  These  are  essentially  the  same  expressions  with  those  given  by  Euler  in  his   Paper  on   the  Propagation  of 

Sound,  in  the  Berlin  Memoirs  for  1759,  and  by  Poissnn  in  his  elaborate  Memoir  on  the  Motion  of  Elastic  Fluids 
in  Pipes,  and  on  the  theory  of  Wind  Instruments,  and  they  comprise  the  whole  theory  of  the  linear  propagation 
of  Sound.  But  before  we  proceed  to  the  interpretation  of  their  meaning  in  particular  cases,  we  have  a  few  remarks 
to  make  on  their  general  form. 

60.  And,  first,  it  is  evident,  that  since  the  variable  quantity  x  enters  into  all  the  terms  both  of  v  and  c  under  the 

functional  characteristics,  these  quantities,  regarded  as  functions  of  t,  are  modified  essentially  by  the  value  of  .r. 

Remarks  on  which  may  be  regarded  as  a  parameter,  or  constant  element  in  the   composition  of  the  functions  expressing  the 

these  ex-      nature  of  the  motion  of  any  assigned  molecule.     If  only  x  +  a  t,  or  only  x-at,  separately  entered  under  the 

characteristics,  since  x  +  at  —  a  (t  4-  —  )  and  x  -  at  =  -  a( t  -  --)  the  variation  of  x  would  only  vary 

\  (I  /  \  Q>  / 

the  origin  of  t ;  and  the  motions  of  all  the  successive  molecules  would  be  performed  according  to  the  same 
laws,  only  commencing  at  a  different  epoch  for  each  molecule  ;  but,  as  both  these  quantities  are  involved,  that 
will  not  be  universally  the  case.  Consequently,  in  general,  it  appears  that  the  undulation,  or  pulse,  as  it  is 
oropagated  onward,  becomes  modified  essentially  in  its  quality  by  the  distance  it  has  passed  over,  it  is  no  longer 


SOUND.  759 

Sound,      the  same  sound,  i.  e.  not  identical   with  what  would  be  produced  by  shifting   the  initial   stratum   forward.     Its      pari  I. 

-»v— *•'  velocity,  intensity,  and  pitch,  it  is  true,  will  remain  (as  we  shall  see)  unaltered  ;    but  its  quality,  its  mode  of  ^- v-- — ' 

action  on  the  ear,  (which  must  be  differently  affected  by  changes  in  the  nature  of  the  impulse  made  on  it,)  will 

undergo  a  change.     This  establishes  an  essential  difference  between  a  Sound  wave  and  such  a  wave  as  we  took 

for  an  illustration  in  Art.  45,  where  every  point  was  in  succession  agitated  by  the  same  identical  motion. 

Consequently  every  theory  of  Sound  in  which  it  is  assumed  that  the  several  particles  in  a  sounding  column  are        fil 
all  in  succession   agitated   alike,  is   defective.      This   is  the  case  with  Newton's  doctrine  of  the  propagation  of  Inaccura- 
Sound  as  delivered  in  the  47th  proposition  of  the  2nd  hook  of  the  Principia,  and,  were  there  no  other  objection  ciesiuNew. 
against  it,  would  suffice  to  vitiate  the  whole.     This,  and  other  unsatisfactory  points  in  the  celebrated  theory  'f"cS 
alluded  to,  were  first  distinctly  perceived  and  pointed  out  by  Lagrange,  in  the  first  volume  of  the  Turin  Miscel- 
lanies, and  an  exact  and  rigorous   investigation  substituted  in  its  place,  in  which  the  sounding  column  is  regarded 
as  consisting  of  a  series  of  finite,  insulated   particles,  mutually  repelling  each   other  ;    a  mode   of   conception 
which  leads,  by  a  very  complicated  analysis,  to  the  same  results  with  that  above  stated,  but  which  has  the  advan- 
tage of  setting  in  a  distinct  light  the  internal  mechanism,  if  we  may  so  term  it,  by  which  Sound  is  propagated, 
and  to  which  we  therefore  willingly  refer  the  reader. 

Moreover,  since  by  differentiating  the  equation  (d)  we  get  62. 

^  =  a»{  F"(*  +  at)  +f"(x  -at)}, 

this  will  be  proportional  to  the  accelerating  force  acting  on  the  molecule.  It  is  therefore  by  no  means  universally 
proportional  to  y  —  x,  the  distance  of  the  molecule  from  its  point  of  rest ;  and  therefore  another  assumption  on 
which  the  Newtonian  doctrine  of  Sound  rests,  viz.  that  the  motion  of  each  molecule  necessarily  follows  the  law  of  a 
vibrating  pendulum,  is  equally  destitute  of  foundation.  In  fact,  Cramer  had  shown,  befoie  the  examination  of 
Lagrange,  that  any  other  law  of  molecular  motion  might  be  substituted  in  Newton's  enunciation  of  his 
general  proposition,  and  the  demonstration  would  be  equally  conclusive,  and  the  resulting  velocity  of  Sound  the 
same. 

Let  us  now  descend  more  into  particulars  ;  and,  first,  let  us  suppose  the  initial  state  of  the  fluid  to  consist  in         63, 
a  general  repose  of  the  whole  of  an  infinitely  extended  column,  except  a  very  small  portion  at  A  the  origin  of  Case  °"  a 
the  coordinates,  which  we  will  suppose  agitated  with  any  arbitrary  motion.     This  is,  in  fact,  the  simplest  case  of  j^jiJa*™^ 
the  production  of  Sound  ;  the  initial  disturbance  of  the  air  being  always  confined  within  extremely  small  limits  turbance. 
compared  to  the  distances  to  which  the  Sound  is  propagated.     Let  us  then  conceive  the  initial  disturbance  to 
take  place  over  a  minute  length  2  a  of  the  column,  whose  middle  we  will  suppose  to  be  in  the  origin  of  the  x. 
This  amounts  to  supposing  <p  (x)  =  0,  and  ty  (x)  =  1,  for  every  value  of  x  not  comprised  within  the  limits 
x  =  —  a  and  x  =  -J-  a,  admitting  them  to  have  any  arbitrary  values  between  these  limits. 

If  we  suppose  now  t  to  be  less  than    ,  and  regard   at  first  what  happens  only  on  the  positive  side  of        KA 

& 

Propagation 

the  origin  of  the  .T,  since  t  < we  have  a  t  <  x  —  a,  and  therefore  x  —  a  t  >  -\-  a,  and,  d,  fortiori,  °n^\l\n^ 

turbance, 

x  +  a  t  >  -f-  °.  consequently  for  all  values  of  t  less  than  —  -  we  have  0  (x  —  a  t)  =  0,  0  (*  -f-  a  t)  =  0  ;  a^'"",l^. 

Rating  sud- 

Y-  (jc  —  a  f)  =  Y'  (*  +  a  t)  —  1  !  an£l  therefore  for  all  values  of  t  less  than  —  we  have  v  =  0,  and  e=l. 

a 

Consequently  the  molecule  at  the  distance  x  from  the  origin  of  the  coordinates,  will  remain  at  rest  and  uncon- 
densed,  or  expanded,  so  long  as  t  remains  less  than  --  -  ;  that  is,  for  a  time  proportional  to  the  distance 

CL 

from  the  nearest  point  of  the  initial  disturbance.  But  the  moment  t  has  attained  this  limit,  0  (x  ±  a  f)  will  have 
finite  values,  and  ty  (x  i  a  t)  values  differing  from  unity,  and  v  and  e  will  consequently  have  such.  The  particle 

X  -\-  a 
then  will  begin  to  move,  and  to  undergo  a  change  of  density,  and  will  continue  to  do  so  till  t  — .      At 

this  limit  we  have  x  -  at  =  -  a,  x-\-at=2at-a  =  2(x+a)-a=:2x  +  a,  and  consequently  x  -f  a  t  > 
+  «.  Hence  at  this  limit  we  have  again  0  (x  —  at  )  =  0  {x  -(-  a  t)  =  0,  and  ty  (.x  ~~  a  0  =  Y'  (*  +  a  0  =  1> 
and  the  motion  and  condensation  of  the  particle  will  cease  ;  and  will  not  be  resumed  afterwards,  because  the 

supposition  t  >  —     —  gives  x  —  at  <  —  a,  and  x-\-at>%x-\-a.  and,   a  fortiori,   >  +  a,  so   that   the 
a 

functions  retain  their  values  0  and  1  from  this  moment  for  ever. 

Thus  we  see  that  the  molecule  distant  by  x  from  the  origin  of  the  coordinates  will  remain  at  rest  for  a  certain        55 

X  —  a  »  4-  •  *  —  a      Velocity  of 

time  I  = ,  will  then  begin  to  move,  and  continue  moving,  during  a  time  equal  to  ' —   — Sound  uni- 

a  a  a         form 

=:  or  till  t  —  ,  and   will  then   return  to  a  state  of  permanent   rest.     A  similar  reasoning  will 

a  a 

apply  for  negative  values  of  x.     Hence  if  we  consider  any  two  molecules  at  distances  x,  x  from  A,  we  see  that 


760  S  O  U  N  D. 

Souna.      the   more   distant  will   commence   and  terminate   its    motion   later  than   the   nearer,   by    an   interval  of   time     1'an  I 

— ^^  -^•fc--'  J  __  *^^*V^* 

=   .     This  then  is  the  time  required  for  the  propagation  of  the  impulse,  or  Sound,  o"er  <he  intermediate 

a 

space  x1  —  x,  and  being  proportional  to  that  space,  the  velocity  of  propagation  must  be  uniform,  and  must  be 

represented  by  the  quantity  a  (  =  — ; —  =  -       :  ).     Hence  it  follows  that  the  velocity  of  Sound  is  uni- 

V         L    •""  L  time   / 

form, — is  independent  of  the  nature,  extent,  and  intensity  of  the  primitive  disturbance,  (for  the  arbitrary  func- 


expressej  ty  (/le  quantity  we  have  called  a,  that  is  >J  Zg  H- 

65.  Let  us  reduce  this  to  numbers,  in  order  to  compare  theory  with  observation.     To  this  end,  if  we  call  A  the 

First  ap-       density  of  mercury,  h  the  height  of  the  mercury  in  a  barometer  exposed  to  the  same  pressure  as  Uie  sounding 
proximation  column,  and  D  the  density  of  the  air  in  it,  we  have  for  the  height  of  a  homogeneous  column  of  such  air  capable 


p 
t 
rical  value. 


to  its  nume-  o,-  counterbalancing  the  elasticity  of  the  sounding  fluid,  the  following  value 


and,  calling  V  the  velocity  of  Sound,  we  should  have 


Now,  at  the  freezing  temperature,  and  in   a  mean  state   of  barometric  pressure,  we  have,  according  to  Biot, 

A 
A  =  Omet.  76  ;  2  g  =  9m<*-.  8088  ;  and  —  =  10463  ;  so  that  we  obtain,  by  executing  the  numerical  operations, 

V  =  279met.  29  =  916*" .  322. 

67.  The  actual  value  of  V  obtained  by  experiment  is,  as  we  have  seen,  1089.42.     The  difference,  173  feet,  is  nearly 

Great  differ-  one-sixth  of  the  whole  amount ;  a  discrepancy  far  too  great  to  be  attributed  to  any  inaccuracy  in  the  determi- 

ence  be-       nation  of  the  data,  which  are  all  of  the  utmost  precision.     It  is  evident,  then,  that  there  is  something  radically 

tween  the-    insuffjcjent  in  the  theory,  as  above  delivered  ;    and,  accordingly,  Geometers  for  a    long  while  endeavoured  to 

pe^imenu*"  account  for  it  on   various   suppositions.     Newton,   who,   by    a  singularly  happy  coincidence,   which   certainly 

on  the  as-     deserves  to  be  called  a  divination,  had,  from  a  theory  totally  inapplicable  in  all   its  points,  elicited  the  correct 

sumed  hy-    expressjon    V  2  g-  H   above  demonstrated,   for   the   velocity  of  Sound,   and  who  immediately  encountered  this 

se8'      difficulty  on  deducing  its  numerical  value,  endeavours  to  account  for  the  deficient   173  feet  by  supposing  the 

molecules  of  the  air  to  be  actual  spherical  solids  of  a  certain  diameter,  (-j^f  of  the  interval  between  them,)  and 

"tiTto  "   ^*a'  *ne  Sound  is  propagated  through  them  instanter.     It  is  needless  to  comment  on  this  explanation.    Lagrange 

accmint        treats  the  whole  matter  lightly,  and  seems  inclined  to  attribute  the  deviation  of  fact  from  theory  to  erroneous 

for  it.  data  ;    in   other  words,  dissembling  the  difficulty,  which   Euler,   on   the  contrary,  broadly  acknowledged  ;    and 

considered  that  it  might  possibly  arise  from  an  incorrectness  of  analysis,  in  assuming  the  factor  (  — —  )  =  1  in 

\dx  J 

the  equation  (a)  Art  53,  previous  to  integration.  The  true  explanation  was  reserved  for  the  sagacity  of  Laplace. 
But  before  we  state  it,  it  will  be  necessary  to  consider  what  will  be  the  effect  of  variations  of  temperature  and 
pressure  on  the  velocity,  according  to  the  principles  already  laid  down,  and  the  formula  arrived  at. 

68.            With  regard  to  an  increase  of  pressure,  its  effect  is  to  increase  the  density  of  the  air ;   but  since  at  the  same 
Effect  of  va-  time  it  increases  its  elasticity,  and  in  exactly  the  same  ratio;    the  mass  to  be  moved,  and  the  moving  force,  are 
nations  of     increased  alike,  and  therefore  the  accelerating  force  remains  unaltered.     The  velocity,  therefore,  ought  to  undem-o 
emperi    re  nQ  cnange  ^y  this  alteration.     On  the  other  hand,  an  increase  of  temperature,  under  a  constant  pressure,  tends 
sure'oTthe  to  dilate  the  air,  and  either  renders  it  more  elastic  in  the  same  space,  or  more  rare  with  the  same  elasticity, 
velocity  of     Hence,  on  a  variation  of  temperature,  the  moving  force  remains  unaltered,  while  the  mass  moved  decreases,  and 
Sound.         therefore  an  acceleration  in  all  the  resulting  motions  must  arise.     The  velocity  of  Sound  then  ought  to  be  greater 
in  warm  than  in  cold  air,  cteteris  paribus.     These  two  conclusions  are  both  amply  confirmed  by  experiment. 
They  agree  too  with  the  formula  above  stated  ;    for,  if  we  denote  by  (A)  the  mean  height  of  the  mercury  in  the 
barometer  (0°"-.76),  and  by  (D)  the  density  of  air  under  this  pressure  at  the  freezing  temperature,  since,  by  the 
experiments  of  Gay  Lussac,  air  expands  0'00375  of  its  volume  by  every  degree  centigrade  of  increase  of  temper- 
ature, its  density  under  the  pressure  (A)  at  any  other  temperature  -f  T°  (centig.)  will  be    -  - —          ,  and 

I  +  T  .  0*00375 

under  the  pressure  A  it  will  be  —    X    .    .      '  Q.no375    =    D  '     conse(luently  tne    expression    (Art.  66  )    for 
the  velocity  becomes 


V  =  \/  2  g  (A)  .  ^y  X  (1  +  -r .  0-00375). 


Now,  if  we  call  (V)  the  velocity  under  the  mean  pressure  (A),  and  at  the  freezing  point,  this  gives 

'"<*>•  c5T 


=  y/s 


SOUND.  761 

Sou"d-      and  therefore  V  =  (  V)  .  V~]  +  T  .  0-00375  =  (V)  {  I  +T  .  0-001875  }  ,  vJ^-L' 

or  if  T  be  expressed  in  degrees  of  Fahrenheit's  scale, 

V=  (V)  {  I  +  T.  0-001042}; 

which  shows,  first,  that  the  velocity  is  independent  of  the  pressure,  since  h  is  not  contained  in  its  expression  ; 
and  that,  secondly,  ^  increases  by  very  nearly  the  0'001875  part  of  its  whole  quantity  for  every  degree  centi- 
grade, or  4-  X  U'001875  =  0'001042  for  every  degree  Fahrenheit  above  the  freezing  point,  that  is  in  feet  1'136, 
(see  Art.  17.)  and  decreases  by  the  same  quantity  for  each  degree  below  freezing. 

The  law  of  Mariotte,  which  makes  the  elastic  force  of  the  air  proportional  to  its  density,  and  which  has        69. 
been  employed  in  estimating  the  elasticity  with  which  each  molecule  of  the  aerial  column  resists  condensation,  Laplace's 
and  transmits  it  to  its  neighbour,  assumes  that  the  temperature  of  the  whole  mass  of  air  is  alike,  and  undergoes  e^planat'OB 
no  change  in  the  act  of  condensation,  and  is  therefore  only  true  of  masses  of  air  which,  after  compression,  are  malv'abm" 
of  the  same   temperature  as  before.     But  it  is  an  ascertained  fact,  that  air  and  all  elastic  gaseous  fluids  give  mentioned. 
out  heat  in  the  act  of  compression,  i.  e.  actually  become  hotter,  a  part  of  their  latent  heat  being  developed,  and 
acting  to  raise  their  temperature.     This  is  rendered  evident  in  the  violent  and  sudden  condensation  of  air  by  a  Heat  deve- 
tight-fitting  piston  in  a  cylinder  closed  at  the  end.     The  cylinder,  if  of  metal,  becomes  strongly  heated;  and  if  a  !°p*d  ln.a'r 
piece  of  tinder  be  enclosed,  on  withdrawing  the  piston  it  is  found  to  have  taken  fire  ;  thus  proving  that  a  heat,  compression 
not  merely  trifling,  but  actually  that  of  ignition,  has  been  excited,  of  at  least  1000°  of  Fahrenheit's  scale.     Now 
when  we  consider  how  small  the  mass  of  air  in  such  an  experiment  is,  compared  with  that  of  the  including 
vessel,  which  rapidly  carries  off  the  heat  generated,  it  is  evident  that  if  air  by  any  cause  could  be  compressed  to 
the  same  degree  without  contact  of  any  other  body,  a  very  enormous  heat  would  be  generated  in  it.     It  would, 
therefore,  resist  the  pressure  much  more  than  if  cold  ;    and,  consequently,  would  require  a  much  more  powerful 
force  to  bring  it  into  that  state  of  condensation  than,  according  to  Marriotte's  law,  would  be  necessary. 

Air,  then,  when  suddenly  condensed,  and  out  of  contact  with  conducting  bodies,  resists  pressure  more  (i.  e.        70. 
requires  a  greater  force  to  condense  it  equally)  than  when  slowly  condensed,  and  the  heat  developed  carried  off  Influence  of 
by  the  contact  of  massive  bodies  of  its  original  temperature.     In  other  words,  it  is  under  such  circumstances  'J"5  cause  '" 
more  clastic,  and  our  analytical  expression  for  its  elasticity  must  be  modified  accordingly.     In  fact,  the  conrien-  y^^f' 
sation  of  the  aerial  molecules  in  the  production  of  Sound  is  precisely  performed  under  the  circumstances  most  Sound. 
favourable  to  give  this  cause  its  full  influence  ;  the  condensations  being  so  momentary  that  there  is  no  time  for 
any  heat  to  escape  by  radiation  ;  and  the  condensed  air  being  in  contact  with  nothing  but  air,  differing  infinitesi- 
mully  from  its  own  temperature  ;  so  that  conduction  is  out  of  the  question.     Let  us  see  now  how  this  will  affect 
the  matter  in  hand. 

It  was   assumed  in  Art.  35,  that  the  elasticity  of  the  air  occupying  the  space  dx,  or  (E)  :  its  elasticity  when        71. 
occupying  d  y  :  :  d  y  :  d  x.     But,  in  fact,  the  varied  temperature  being  taken  into  account,  the  latter  ratio  should  Modiflca- 
have  stood  :  :  dy  (1  +  o  T)  :  dx  (1  -f  ar'),  where  a  denotes  the  coefficient  0-00375,  and  T  and  T'  the  original  ^"ysisand 

d  X  d  X     \  -4-  a  7f      formulae  re- 

and  altered  temperatures  in  centrigrade  degrees.     Hence  in  place  of  E  .  -  —  we  must  have  E  .  —  —  .  —  —    -  ,  quired  by  it. 

that  is,  E  .  —  —   {1  +  a  (T'  —  T)  }  ,  for  the  elasticity  of  the  molecule  of  air  when  occupying  the  space  d  y, 

Ui  y 

because,  the  condensations  being  all  along  supposed  exceedingly  small,  T'  differs  from  T  only  by  a  quantity  of  the 
same  order  as  the  condensations  ;  so  that  (T'  —  T)*  and  its  higher  powers  may  be  neglected. 

Now,  whatever  may  be  the  law  according  to  which  the  temperature  of  a  mass  of  air  is  increased  by  a  sudden        72. 
diminution  of  its  volume,  it  is  obvious  that  for  very  small  condensations,  such  as  those  considered  in  the  theory  Analysis. 
of  Sound,  the  rise  of  temperature  will  be  proportional  to  the  increase  of  density  ;  because,  the  quantity  of  latent 
heat  having  sustained  only  a  very  minute  diminution,  by  a  given  extremely  small  condensation,  a  repetition  of 
the  same   condensation   will  develope  a  quantity  of  heat  falling  short  of  the  first  only  by  a  quantity  of  the 
second  order  ;   so  that,  neglecting  such  quantities,  double  the  condensation  will  develope  double  the  heat,  and  so 

in  proportion.     Hence  we  must  have  T'  —   T  —  k  \l  --  —  >  where  k  is  a  constant  coefficient,  whose  magni- 

l         d  x  ) 

tude  may  become  known  either  by  direct  experiment,  or  by  the  very  phenomena  under  consideration.     Substi- 
tuting this  for  i'  —  T,  we  get,  for  the  elasticity  of  the  condensed  molecule, 


-.  . 

dy 

And  the  difference  of  elasticities  on  either  side  of  the  plane  separating  the  molecules  dy  and  dy',  instead  of 

being,  as  in  (Art.  35.)  —  E  .  d  -y—  ,  will  be  now  represented  by  —  d  {  E  (I  -{-  k  a)  ---  —  A  a  E  }  ,  that  is, 

ay  d  y 


by  -  E(l  +ka).d  ---. 

d  y 

This  differs  from  the  expression  originally  obtained  only  by  the  constant  factor  (1  +  A  a).  Without,  therefore, 
going  again  through  all  the  foregoing  analysis,  we  see  at  once  that  the  general  equations  of  Sound  will  be 
precisely  as  before,  writing  only  (1  +  k  a)  ,  H  for  H  throughout  ;  and,  therefore,  if  instead  of  putting,  as  before, 


62  SOUND. 

II,   we   put  a  =  ^2gU(l+ku)  =  ^  2  g-  H  .  K. ;    when   K  =  1  -I-  k  a    the   equation    (a)    will 


/  dy  V   d'y  ,       dfy 

I  — r—  ) .  -~  =  a'  .  -    —  ; 
\dxjdt'  (1  xl 


all  the  other  equations  will  remain  unaltered,   and  the   velocity   of  Sound  on  this  new  hypothesis  will  be 
veiodtUy"of     expressed  by  the  new  value  ascribed  to  a,  that  is,  by 
Sound.  

v  = 


„/, 


~K(l   -far) 


where  a  =  U'00375. 

74.  The  actual  numerical  value  of  the  constant  coefficient  K  may  be  determined,  as  we  have  before  said,  in  two 
Value  of       ways;  either  by  direct  experiment  on  the  increase  of  temperature  developed  in  a  given  volume  of  air  by  a  give:: 
K  how  de-    condensation,  or  by  a  comparison  of  the  formula  to  which  we   have  arrived  with  the  known  velocity  of  Sound. 

mmable.  ^g  we  nave  a]rea(]y  observed,  ho.vever,  the  circumstances  under  which  Sound  is  propagated  are  far  more  favour- 
able to  the  free  and  full  production  of  the  whole  effect  of  the  cause  in  question  than  those  of  any  experiments  in 
close  vessels.  We  must  not,  therefore,  be  surprised,  if  the  value  of  K  as  derived  from  such  experiments  should 
differ  materially  from  its  value  deduced  from  the  velocity  of  Sound  ;  nor  vice  versa,  if  the  observed  velocity  of 
Sound  should  differ  materially  from  that  obtained  by  calculation,  from  an  experimental  value  of  K.  It  is  suffi- 
cient, in  a  philosophic  point  of  view,  to  have  pointed  out  a  really  existing  cause,  a  vera  causa,  which  must  act  to 
increase  the  velocity,  and  is  fully  adequate  to  do  so  to  the  extent  observed. 

75.  We  have  seen  that  the  numerical  value  of  V  neglecting  K  is  equal  to  916-322  feet.     The  observed  value  on 

Determined  tjje  ct]ler  hand,  is  10S9-42.     Hence  we  have  the  following  equation  for  determining  K  and  k, 
from  the 

se0^dy  °f  1089-42  =  916-322  x  V  1  +  k  .  000375  =  916-322  X  V~K, 

'ts"lf-  /"1089'4'2\! 

whence  we  obtain  K  =  (  -  -  I  =  1-4132, 

\  91o'32_/ 

1          [  /1089-42V          1 
and  ^O^^tblfrS*)-'}-110-26- 

Difficulty  of  The  actual  amount  of  heat  given  out  by  a  given  amount  of  condensation  is  not  an  element  very  easily  or 
its  direct  de-  exactly  determinate  by  direct  experiment  with  thermometers.  If  a  common  mercurial  thermometer  be  enclosed 
termination.  jn  a  recejver>  an(j  t],e  a}r  suddenly  compressed,  the  thermometer,  it  is  true,  rises  ;  but  the  amount  of  its  rise  is 
evidently  far  inferior  to  the  actual  increase  of  temperature ;  for,  first,  its  mass  is  enormously  greater  than  that  of 
the  air  immediately  in  contact  with  it ;  secondly,  it  is  brovight  into  contact  successively  with  an  unknown,  and,  no 
doubt,  a  variable  quantity  in  different  experiments,  by  the  effect  of  circulation ;  thirdly,  the  vessel  used  carries 
off  by  far  the  greater  part  of  the  heat,  and  one  which  we  have  no  means  of  estimating.  It  is  accordingly  found 
that  by  increasing  the  sensibility  of  the  thermometer,  by  extending  its  surface  compared  to  its  mass,  higher  and 
higher  degrees  of  temperature  are  indicated  for  the  same  condensation  ;  and  highest  of  all  when  the  delicate 
pyrometer  of  Breguet  is  used,  which  consists  of  two  extremely  thin  strips  of  platina  and  palladium  soldered 
together  over  their  whole  surface,  and  coiled  up  in  a  spiral,  which  twists  and  untwists  by  the  different  expansions 
of  the  metals  constituting  its  inner  and  outer  face.  Still,  however,  though  almost  all  surface,  the  materials  of 
which  this  instrument  consists  are  so  infinitely  denser  than  air,  that  its  indications  must  fall  far  short  of  the 
truth. 

'6.  Another  very  ingenious  method  has  been  practised  by  Messrs.  Clement  and  Desormes.    (Journal  de.  Physique, 

Experiment  November,  1919,   p.  334.)     Suppose  we  have  any  quantity  of  air  enclosed  in  a  receiver  communicating,  first, 
'    with  an  air-pump,   by  a  valvular  orifice,  (A)  ;  second,  with  the  upper  part  of  a  barometer  tube  containing  tner 
srmes.          cury,  whose  height  therefore  measures  the  elasticity  of  the  air  in  the  receiver  by  its  depression  below  the  baro- 
metric level  of  the  external  atmosphere;  thirdly,  with  the  external  air,  by  a  stopcock,  or  valve,  (B,)  so  large  that 
the  pressure  within  may  be  instantaneously  restored  to  an  equilibrium  with  that  without,  on   opening  it.     Let 
the  whole  apparatus  be  at  the  temperature  of  the  atmosphere,  (T,)  and  suppose  the  valve  (B)  open,  then  will  the 
internal  elasticity,  or  pressure,  (P,)  be  equal  to  that  without,  and  also  the  density  (D.)     Close  the  valve  B,  and 
open  A,   and,  by  means  of  the  air-pump,  exhaust   a  small  portion  of  the  air;    and,  again  closing  the  valve  A,  let 
the  apparatus  remain  at  rest  till  the  whole  has  attained  the  temperature  r  of  the  atmosphere.     In  this   state  let 
the  internal  pressure  be  observed  by  the  barometer,  which  call  P' ;  and  D',  the  density,  will,  of  course,  be  equal  to 

P1 

D  .  ,  and  is  therefore  known.     Now  suddenly  open  the  valve  B.     The  external  air  will  rush  in  and  restore 

the  equilibrium.  The  moment  this  is  done  (which  will  be  known  by  the  cessation  of  the  inward  current)  let  the 
valve  B  be  closed.  It  will  then  be  found  that  the  internal  temperature  is  raised  by  the  condensation  thus  effected, 
and  has  become  T' ;  and  the  increase  of  temperature  T'  —  ^  may  be  measured  by  a  delicate  thermometer,  and 
that  with  the  more  precision  the  greater  the  capacity  of  the  receiver.  But  it  will  be  much  more  exactly  measured 
bv  the  following  process,  which,  in  fact,  amounts  to  making  the  receiver  itself  an  air  thermometer.  At  the 
moment  of  closing  the  valve  the  internal  pressure  is,  of  course,  P.  But  as  the  air  cools,  its  elasticity  diminishes, 
and,  being  cut  off  from  a  fresh  supply  from  without,  the  mercury  wrill  rise  in  the  barometer  tube  till  the  whole 


SOUND.  763 

Sound,     of  the  heat  evolved  is  dissipated.     Let  the  internal  pressure,  then,  be  again  observed  when  this  state  is  attained,      Parti. 

— v— <~J  -pit  **~*^~~ 

and  call  it  P",  then  will  the  corresponding  density,  or  D",  be  equal  to  D  .   — .     It  is  required  from  these  data 


(P,  P',  P".  being  given  by  observation)  to  deduce  the  value  of  T'  —  T  and  the  coefficient  k. 

Now,  this  is  easy;  for,  first,  since  in  the  final  state  of  the  receiver  the  density  is  D"  sustaining  a  pressure  P" 


77. 


therefore, 

P  —  P" 

P  =  P"  {  1  +  a.  (T>  -  T)  }  ,  whence  T'  -  T  =  —  og)/    -. 

Now,  secondly,  this  is  the  elevation  of  temperature  due  to  the  sudden  transition  of  the  air  from  the  density  D' 
to  the  density  D",  by  the  introduction  of  that  portion  of  external  air  which  rushed  in  on  opening  the  valve. 
Calling  1  the  capacity  of  the  receiver,  1  X  D'  =  D'  expresses  the  quantity  of  air  in  it  before  the  valve  was  opened, 
and  1  X  D"  or  D"  the  quantity  after,  so  that  D"  —  D'  expresses  the  quantity  of  air  admitted.  Its  density  before 

D" 

admission  being  D,  and  afterwards  D",  it  had  undergone  a  dilatation  equal  to  1  —   -=-,  and  therefore  its  tem- 

(D"\ 
1  --  =7-  1  .     On  the  other  hand,  the  quantity  of  air 

in  the  receiver  before  opening  the  valve  was  1  X  D'  =  D',  and  this  quantity  having  changed  its  density  suddenly 

(D'  \ 
1  --  =rj-  J  .     These  two 

masses  of  air,  the  one  cooled  by  dilatation,  the  other  heated  by  condensation,  became  suddenly  mixed,   and 

sum  of  products  of  masses  and  changes  of  temp. 
therefore  must  have  undergone  a  mean  rise  ot  temperature  =    -  -  — 

sum  of  masses 

and,  consequently, 

mean  elevation  of  temperature  =  —  —  D'  +  D"  —  D'  =  D" 


D/'-jyf    D1          D-D"  )  D"  -  D'  )  D'         JD»_         ) 

D"      1  ~W  '          D        j  D"     ~  t  D"       '    D  j    ' 

D"       /         D'\l,       ,D»/        D'\        /        D'V 


D"        P"         ,  D'        P1  D'        F 

But  we  have  -=r-   =  -p-,  and  —  =:  — ,  so  that  jy7  =  pF  > 

and  therefore  substituting,  we  find  for  the  value  of  the  above  expression,  or  T'  —  T, 

(  P"  /        F  \       /         F  \*1  P"  —  P'  f          P  (P"  —  P')  ) 

If  we  suppose  the  changes   of  pressure  sufficiently  small  to  allow  of  their  squares  being  neglected,  the  value  v»hieof* 

pii    /          P'  \             P"  —  P'                                      P  _  p" 
of  /  -  T  is   reduced  to  k  .  —  f  1  —   ^-  1  =  k. j, .     Equating  this  to 5^—  ,  the    previously 


expressed. 
|i-     pT  I  —  K.  — -p .     equaling  mis  to  — —  ,  tne    previously 

determined  value  of  T'  —  T,  we  get 


_   1    P-P"    _P_  P(P-F') 

=  7  '  P"  -  P' '  P"  '  P"  (P"  -  p')    ' 

In  an  experiment  of  Messrs.  Clement  and  Desormes,  on  which  M.  Poisson  has  grounded  his  computation  of       78. 
the  theoretical  velocity  of  Sound,  the  values  of  P,  P',  P"  were  Numerical 

P  =  0""7665  ;         P  -  P'  =  0""01381 ;         P  -  P"=  0--00361 ; 
and,  consequently,  F'  -  P'  =  0-01020  , 

which  gives,  by  the  approximate  formula, 

k  a.  =  0-3492,         and  1  +  k  a  =  1  "3492  ; 
whence  the  velocity,  at  a  mean  pressure  and  freezing  temperature,  comes  out 

916ft"-322--/l-3492=  1064-35, 

which  falls  short  of  the  actually  observed  velocity  only  by  about  25  feet.     If  the  rigorous  value  of  ka  be 
employed,  the  deficiency  is  rather  less,  the  velocity  coming  out  1066'2.    In  this  experiment,  the  time  occupied 
vol.  IT.  5  a 


764 


SOUND. 


Sound,     by  the  intromission  of  the  air  was  about  half  a  second  ;  the  whole  elevation  of  temperature,  computed  from  the 
formula  T'  —  T  = =— ,  must  have  been  lfr"321  centig.  (=  2°'378  Fahr.)     M.  Poisson  has  shown  (A/males 

aP" 

de  Chim.  xxiii.  1823,  p.  11)  that  an  absorption  of  ^  of  a  degree  (cent.)  by  the  receiver,  which  might  very  well 
happen,  would  completely  reconcile  the  observed  and  theoretical  velocities.  Laplace,  calculating  on  the  experi- 
ments of  Messrs.  Welter  and  Gay  Lussac,  has,  since,  obtained  a  still  nearer  approximation  to  the  theoretical 
velocity,  the  difference  amounting  only  to  about  3  metres.  In  inquiries  of  such  delicacy,  and  where  the  effects 
of  minute  errors  of  experiment  become  so  much  magnified,  it  seems  hardly  candid  to  desire  a  more  perfect 
coincidence. 
79.  Laplace,  guided  by  peculiar  theoretical  considerations  respecting  the  constitution  of  gaseous  fluids,  has  been 

Another       induced  to  put  the  foregoing  expression  for  the  velocity  of  Sound  under  a  somewhat  different  form.   Let  K  denote 

mode  of       the  ratio  of  the  specific  heat  of  air  under  a  constant  pressure  to  its  specific  heat  if  retained  at  a  constant  density ; 

expressing    tjjat  jS)  a  fraction  whose  numerator  is  the  quantity  of  heat  requisite  to  raise  a  given  mass  of  air  1°  in  temperature 
e  deve'      under  a  constant  pressure,  (its  volume   being  permitted   to  increase,)  and  whose   denominator  is  the  quantity 

loped  heat    necessary  to  raise  it  1°  in  a  constant  volume,  or  when  so  confined  as  not  to  dilate.     Then  will  the  velocity  of 

on  the  velo-  Sound  be 

city  of 


Part  I. 


To  show  this,  let  Q  and  q  be  the  quantities  of  heat  above  mentioned.  It  is  evident,  first,  that  when  forcibly 
prevented  from  expanding,  and  thereby  absorbing  heat  and  rendering  it  latent,  a  less  quantity  of  heat  will  suffice 
to  raise  the  temperature  of  a  given  mass  of  air  any  given  quantity,  as  1  ,  than  if  unconfined.  In  fact,  suppose 
it  heated  1°,  and  allowed  meanwhile  to  dilate,  so  that  the  temperature  of  the  dilated  air  shall  be  1°  above  its 
primitive  state,  then,  if  compressed  back  into  its  original  volume,  the  whole  quantity  of  heat  developed  by  the 
condensation  will  be  employed  in  raising  the  temperature  still  higher.  If  then  the  quantity  Q  of  caloric  raise 
the  temperature  1°  under  a  given  pressure,  it  will  raise  it  more  than  1°  when  confined  to  a  given  volume,  by  the 
whole  amount  of  temperature  due  to  a  compression  equal  to  its  dilatation  in  the  former  case.  Suppose  the 
initial  temperature  freezing,  then  if  a  =  0'00375,  an  increase  of  temperature  of  1°  cent,  will  produce,  under  a 
constant  pressure,  a  dilatation  =  a,  and  the  volume  from  1  will  become  1  +  a.  Let  the  air  so  dilated  and  raised 
in  temperature  be  compressed  back  to  its  former  volume,  then  will  its  temperature  be  further  increased  by  A  a,  k 
denoting  as  before  ;  so  that  the  quantity  of  caloric  Q  will  have  ultimately  produced  a  rise  of  temperature 

=  1  +  k  a,  under  a  constant  volume  ;  and  therefore  a  quantity  =:  — l    <    ^ —  only  would  be  required  to  raise 

r\  r\ 

This  demonstration  assumes,  as  an  axiom,  that  the 


it  1°.     Hence  q  = 


,  and  1  +*  a  =  —  =  K. 
9 


temperature  produced  by  the  introduction  of  the  same  quantity  of  caloric  is  the  same,  whether  it  be  introduced 
into  air  confined  in  a  given  space,  or  into  air  allowed  to  expand  freely,  and  then  forcibly  compressed  back ;  which 
it  evidently  is,  since  the  heat  given  out  by  the  compression  must  of  necessity  exactly  equal  that  absorbed  and 
rendered  latent  in  the  act  of  expansion. 


§  II.     Of  the  Linear  Propagation  of  Sound  in  Gases  and  Vapours. 


80. 

Interpreta- 
tion of  the 
formula  in 
the  case  of 
gases  and 
vapours. 


The  analysis  by  which  we  have  in  the  foregoing  articles  determined  the  laws  and  velocity  of  the  propagation 
of  Sound  in  air,  applies  equally,  mutatis  mutandis,  to  its  propagation  in  all  permanently  elastic  fluids,  and  in 
vapours,  in  so  far  as  their  properties  are  the  same  as  those  of  gases.  The  formula  so  often  referred  to  then 


V  = 


—    .K.(1+.T) 


expresses  the  velocity  of  Sound  in  all  such  media,  provided  for  (D)  we  write  instead  of  the  density  of  atmo- 
spheric air  that  of  the  gas  at  the  freezing  temperature,  and  under  the  mean  pressure  (A).  In  the  case  of  vapours, 
we  must  suppose  in  calculating  the  value  of  (D)  that  they  follow  the  law  of  gases  in  their  condensation,  and 
that  no  portion  of  them  undergoes  a  change  of  state  to  a  liquid,  by  reduction  to  the  standard  temperature  and 
pressure.  Suppose,  then,  the  specific  gravity  of  atmospheric  air  to  be  denoted  by  s,  and  that  of  any  gas  or  vapour 
under  the  same  temperature  and  pressure  by  s1 ;  then  if  V  and  V  be  the  velocities  of  Sound  in  air,  and  in  the 
gas  or  vapour,  we  have 


=  v/ 


«*<*>.  T 


ar), 


V'  = 


because  (see  PNEUMATICS,  HEAT)  the  law  of  dilatation,  or  the  value  of  o,  is  alike  in  all.     Consequently,  we  have 

JL-  \/_L     51 

V    '  '  V    j        K  ' 


SOUND. 


765 


Sound. 


V :  V  : 

whence  the  ratios  of  s' :  »  and  of  K' :  K  being  known,  the  ratio  of  the  velocities  is  also  known,  being,  cteteris 
paribus,  in  the  inverse  subduplicate  ratio  of  the  specific  gravities. 

To  compare  this  with  experiment  directly  is  impracticable,  as  no  column  of  any  gas  but  atmospheric  air  can 
be  obtained  of  sufficient  length  and  purity  to  determine  the  velocity  of  Sound  in  it  by  direct  measure.  Indirectly, 
however,  the  comparison  may  be  performed  by  comparing  the  Sounds  of  one  and  the  same  organ-pipe,  filled 
with  the  gases  to  be  compared,  successively,  or  by  other  means  of  a  similar  kind,  of  which  more  hereafter.  (See 
INDEX,  under  the  heads  Gases,  Vapours,  Sounds  of  Pipes.) 

The  following  Table  exhibits  the  Velocities  of  Sound,  as  deduced  from  theory,  and  compared  with  experiments 
instituted  by  M.  Van  Rees,  in  conjunction  with  Messrs.  Frameyer  and  Moll. 


GM,  or  Vapour. 

Velocity  of  Sound, 
reduced  to  0°  R. 
(freezing.) 

Velocity  of  Sound, 
reduced  to  0°  R. 

Velocity  assigned  by 
Chladni,  Acoiatict, 
p.  274. 

Theory. 

Experiment. 

Oxygen  (from  Manganese,  therefore  impure)..  .  ' 

Confined  Confined  over  water, 
over 
mercury. 

Metres. 
317-7 
339-0 

1233-3 

270-7 
341-1 
270-6 
317-4 
337-4 
305-7 
229-2 
298-8 
432-0 
422-6 
262-7 

Metres. 
316-6 
338-1 

914-2 

275-3 
316-9 
281-4 
309-8 
317-8 
318-7 
229-2 
309-3 
399-4 
369-6 
289-1 

Metres. 
310 
310 
(680)  according  to 
1  820  J  its  purity. 
269 

320 

Hydrogen.  .  . 

Protoxide  of  azote  (from  nitrate  of  ammon.).  .  .  . 

Parti. 


81. 

How  com- 
parable with 
experiment. 

82. 

Velocity  of 
Sound  in 
various 
media. 


We  give  this  Table,  to  the  best  of  our  comprehension,  from  a  very  imperfect  and  obscure  abstract  of  an  inaugural 
dissertation  of  M.  Van  Rees,  (printed  in  1819,)  given  in  the  Journal  de  Physique,  1821,  p.  40.  We  have  not 
been  able  to  procure  the  original.  The  differences  of  the  columns  probably  arise  from  impurities  in  the  gases, 
or  difficulty  in  estimating  the  exact  pitch  of  Sounds  propagated  by  them. 

These  determinations  are,  of  course,  liable  to  considerable  errors ;  but  the  difference  between  the  results  of  theory        83. 
and  experiment  in  the  case  of  hydrogen  is  so  great  as  to  warrant  a  conclusion,  otherwise  not  improbable,  that  the  Peculiarity 
value  of  the  coefficient  K  in  that  gas  (at  least)  is  materially  different  from  what  it  is  in  others.     Experiments  ln  hydrogen. 
are  hardly  yet  sufficiently  multiplied  to  enable  us  to  speak  with  certainty  on  this  point ;  but  if  by  any  means  we 
are  enabled  to  determine   precisely  the  velocity  of  Sound,  in  a  gas,  or  indeed  in  any  medium,  the  ratio  of  the 
values  of  this  coefficient  in  it,  and  in  air,  may  be  obtained  by  the  analogy 


which  expresses  that  the  value  of  K  is  as  the  square  of  the  velocity  of  Sound,  and  the  specific  gravity  of  the 
medium  jointly.  Thus  the  specific  gravity  of  pure  hydrogen  being  to  that  of  air  as  0-0694  :  1,  (Thomson, 
Attempt  to  establish  the  first  Principles  of  Chemistry,  i.  72.)  and  the  velocity  of  Sound  in  it  being  to  that  in  air 
as  2999-4  to  1089-4,  we  have 

K  in  hydrogen  :  K  in  air  :  :  (2999'4)s  X  0'0694  :  (1089-4)'  X  1, 
:  :  0-526  :!::!:  1-901. 

But  not  only  the  velocity  of  Sound  differs  in  media  of  different  chemical  and  mechanical  natures.  Its  inten- 
sity, i.  e.  the  impression  it  is  capable  of  producing  on  our  organs  of  hearing,  cteteris  paribus,  also  varies 
extremely  with  a  variation  in  the  density  of  the  transmitting  medium.  This  we  have  already  remarked  in  the 
case  of  air,  whether  rarefied  or  condensed.  Priestley  {Observations  and  Experiments,  Ki.  355.)  enclosed  a  piece 
of  clockwork,  by  which  a  hammer  could  be  made  to  strike  at  intervals,  in  a  receiver  filled  successively  with 
different  species  of  gas.  The  distances  at  which  the  Sound  ceased  to  be  heard  were  measured.  He  thus  found 
that  in  hydrogen  the  Sound  was  scarcely  louder  than  in  a  vacuum,  (such  a  one  as  he  could  produce.)  In 
carbonic  acid  it  was  louder  than  in  air,  and  somewhat  louder  also  in  oxygen.  Perolle  (Mem.  Acad.  Toulouse, 
1781  ;  Mem.  Acad.  Turin,  1786-1787)  has  described  some  experiments  not  altogether  in  agreement  with  these. 
The  distance  at  which  a  given  Sound  ceased  to  be  heard  in  atmospheric  air  being1  56  feet,  he  found  that  in 
carbonic  acid  it  was  48  only;  while  in  oxygen  and  nitrous  gas  the  distance  was  63,  and  in  hydrogen  only  11. 
Chladni  found  the  Sound  of  hydrogen  gas  in  an  organ-pipe  remarkably  feeble  and  difficult  to  distinguish,  and 
that  of  oxygen  stronger  than  that  of  atmospheric  air,  but  remarked  nothing  particular  in  the  case  of  carbonic 
acid.  (Acoust.  281.) 

Leslie  (Camb.  Phil.  Trans,  i.  267.)  relates  some  very  curious  experiments,  by  which  it  should  appear  that 

5a  2 


Value  of  K 
in  hydrogen 
nearly  dou- 
ble of  its 
value  in  air. 

84. 

Intensity  of 
Sound  dif- 
fers in  differ- 
ent media. 


85. 


766  SOUND. 

Sound,     hydrogen  gas  is  peculiarly  indisposed  for  the  conveyance  of  Sound.     He  rarefied  the  air  of  a  receiver  in  which      Part  I 
N— "N/— -^  a  piece  of  clockwork  was  enclosed,  striking  a  bell  every  half  minute,  100  times;  and  then  introduced  hydrogen  ^—v- 
Singular       gaSi  when  no  augmentation  whatever  of  the  Sound  took  place.     Yet  more  ;  when  the  air  in  the  receiver  was  only 
h^dro°enin  ^      exhausted,  and  the  deficiency  filled  up  with  hydrogen  gas,  not  only  the  Sound  was  not  increased,  but  was 
enfee°Wingin  actually  diminished  so  as  to  become  scarcely  audible.     If  this  last  fact  be  correctly  stated,  (which  from  the  high 
Sound.         character  of  Mr.  Leslie,  as  an  experimenter,  we  must  not  doubt,)  some  peculiar  modification  of  the  usual  process 
by  which  Sound  is  propagated  must  have  taken  place.     It  is  much  to  be  regretted  that  the  circumstances  are 
not  more  fully  stated ;  the  pitch  of  the  bell,  in  air,  in  the  mixed  gases,  and  in  hydrogen  alone  ;    the  dimensions 
of  the  receiver ;  the  distances  at  which  the  Sounds  ceased  to  be  heard ;  and  whether  the  same  effect  took  place 
when  bells  of  different  pitch  were  struck,  and  when  the  bell  was  muffled  so  as  to  produce  no  musical  Sound, 
are  all  particulars  of  essential  consequence  to  enable  us  to  form  a  judgment  of  what  really  took  place  in  this 
interesting  experiment,  which  we  venture  to  express  a  hope  will  be  repeated  and  varied  by  its  author  on  a  scale 
proportioned  to  its  importance.     We  shall  have  occasion  again  to  refer  to  this  subject.     (See  Index,  Interference 
of  Sonorous  Vibrations  and  Propagation  of  Sound  in  Mixed  Media) 

86.  When  hydrogen  is  breathed  (which  may  be  done  for  a  short  time,  but  not  altogether  without  inconvenience 
Effect  of      and  even  danger)  the  voice  is  singularly  affected,  being  rendered  extremely  feeble,  and  at  the  same  time  raised 
hydrogen  on  jn  pjtcn_    (Odier,  Journal  de  Physique,  vol.  xlviii.)     This  is  just  what  ought  to  arise  from  the  lungs,  larynx,  and 
wheii°'         fauces  being  filled  with   an  exceedingly  rare  medium  ;   but  if,  as  some  experimenters  relate,  the  effect  subsists 
breathed,      long  after  the  hydrogen  is  expired,  and  the  lungs  completely  cleared  of  it,  this  can  only  be  ascribed  to  some 
Musical        physiological  cause  depending  on  its  peculiar  action  on  the  organs  of  the  voice.     The  singular  Sounds  produced 

Sounds  ex-  by  burning  this  gas  in  pipes  of  proper  construction  have  nothing  to  do  with  the  propagation  of  Sound  in  the  gas 
1  by  the  jtse]f_ 

ofhydrogen.      The  propagation  of  Sound  in  vapours  offers  two  distinct  cases  in  which  it  would  at  first  appear  that  very 

87.  different  effects  should  take  place.     In  the  first,  in  which  the  vapour  is  subjected  to  a  less  compression  than  what 
Propagation  is  sufficient  to  reduce  a  portion  of  it  to  the  liquid  state,  experiments  have  sufficiently  proved  the  identity  of  the 
of  Sound  io  laws  which  regulate  the  compression  and  dilatation  of  this  species  of  elastic  fluids  with  those  which  prevail  in 
vapours.       ^e  case  of  ordinary  gases  ;    and,  indeed,  recent  researches  have  proved  that  a  great  number,  and  rendered  it 

probable  that  all  the  latter,  are  in  fact  only  vapours  of  certain  liquids  capable  of  sustaining  a  very  much  greater 
than  the  ordinary  atmospheric  pressure ;  or,  which  is  the  same  thing,  habitually  maintained  at  a  temperature  far 
above  their  boiling  point.  In  this  state,  then,  the  propagation  of  Sound  in  vapours  differs  in  no  respect  from  that 
in  gases.  But  when  the  pressure  sustained  by  the  vapour  is  sufficient  to  condense  a  portion  of  it,  as,  for  instance, 
in  the  upper  part  of  a  vessel  in  which  water  is  kept  boiling,  and  which  is  therefore  full  of  steam  at  212°  Fahr.,  it 
would  seem,  at  first  sight,  that  no  Sound  could  be  propagated  through  such  a  medium  ;  for,  since  the  slightest 
additional  pressure  is  sufficient  to  reduce  a  portion  of  the  vapour  to  the  liquid  state,  it  would  appear  that  the 
whole  effect  of  an  impulse  suddenly  communicated  to  any  portion  of  the  vapour,  urging  it  towards  the  adjacent 
stratum,  would  be,  not  the  compression  of  the  whole  of  such  portion  into  less  dimensions,  accompanied  with 
increased  elastic  force,  but  the  absolute  condensation  of  a  small  portion  into  inelastic  water,  the  remainder 
retaining  precisely  the  same  elasticity  as  before.  Thus  the  necessary  conditions  for  the  propagation  of  the 
impulse  are  nullified,  and  it  should  seem,  therefore,  that  no  Sound  could  be  excited  in  such  a  case. 

But  if  in  vapours,  as  in  gases,  the  act  of  compression  developes  a  certain  portion  of  heat,  it  is  evident  that  this 
Experiment  may  be  such  as  to  prevent  altogether  the  mechanical  condensation  of  the  compressed  vapour,  and  maintain  it  in 
by°Biot  'ts  e'astic  state  even  under  the  increased  pressure  ;  and  therefore  Sound  ought  on  this  supposition  to  be  propa- 
gated freely.  Thus  it  appears  that  we  are  furnished  with  an  experimentum  crucis  for  deciding  on  the  validity  of 
the  explanation  above  stated  of  the  excess  of  the  observed  above  the  theoretical  velocity  of  Sound.  If  the 
momentary  condensations  and  dilatations  of  an  elastic  fluid  do,  as  supposed  in  that  explanation,  give  out  and 
absorb  heat,  Sound  should  be  freely  propagated  in  a  saturated  vapour,  (i.  e.  a  vapour  in  contact  with  liquid,  or 
under  a  pressure  which  it  can  just  sustain.)  If  not,  no  Sound  can  be  transmitted  by  it.  The  experiment  has 
been  made  with  care  by  M.  Biot,  assisted  by  Messrs.  Berthollet  and  Laplace,  (Mem.  dArcueil,  ii.  99.)  by  means 
of  a  bell  suspended  in  a  large  glass  balloon.  When  completely  exhausted,  no  Sound  was  heard  on  striking  the  bell ; 
but  on  the  admission  of  a  little  water  it  was  feebly  heard,  and  as  the  water  and  balloon  were  warmed,  became 
stronger  and  stronger.  When  allowed  to  cool,  the  vapour  condensed,  and  the  Sound  became  enfeebled  by  the 
same  degrees.  When  alcohol  was  used  instead  of  water  the  Sound  was  more  powerful,  and  still  more  so  when 
ether  was  introduced,  the  vapours  of  these  liquids  at  a  given  temperature  being  more  dense  than  that  of  water. 
As  in  these  experiments  care  was  taken  to  keep  the  inside  of  the  balloon  constantly  wet  with  the  liquid,  it  is 
evident  that  the  only  condition  requisite  to  be  observed,  that  of  maintaining  the  vapour  in  the  interior,  at 
its  maximum  of  pressure,  was  completely  fulfilled.  The  reader  is  referred  to  the  original  Memoir  for  an 
account  of  the  details  of  this  elegant  experiment.  The  reasoning  above  stated  is  M.  Blot's.  We  would 
Remarks  remark,  however,  on  it,  that  the  developement  of  the  latent  heat  of  a  vapour  on  its  condensation  into  a  liquid, 
thereon.  though,  no  doubt,  analogous  to,  is  still  in  a  material  point  different  from,  the  developement  of  heat  in  a  gas  by 
mere  compression,  unaccompanied  with  a  change  of  state.  If  the  latent  heat  of  steam  at  212°  (amounting  to 
about  94  5°)  be  not  conducted  away,  the  steam  cannot  be  condensed  into  water  of  2L2°.  A  portion  will  be 
condensed,  but  its  latent  heat  will  be  employed  in  raising  the  temperature  of  the  water  produced  and  of  the 
remaining  steam,  and  thus  increasing  its  elasticity  and  resistance  to  the  pressure.  Thus,  the  propagation  of 
Sound  in  saturated  vapour  is  not  incompatible  with  the  reduction  of  a  portion  of  the  vapour  to  a  liquid  state  at 
every  condensation  caused  by  the  sonorous  pulse,  and  its  reconversion  into  vapour  when  the  condensation  goes 
off  :  nor  is  it  to  be  assumed  as  proving  any  thing  with  respect  to  gases  or  vapours  under  less  than  their  maxi- 
mum pressure.  The  heat  developed  may  (for  any  thing  this  experiment  proves)  come  entirely  from  the  liquefied 


SOUND.  767 

portion,  and  have  no  existence  when  no  portion  is  liquefied.     We  do  not  make  this  remark  as  detracting     Part  I. 
'  from  the  merit  of  M.  Biot's  ingenious  views,  in  which,  on  the  contrary,  we  fully  coincide  as  to  their  result,  V-~~V"~*' 
but  as  an   instance  of  the  circumspection  requisite  in  drawing  conclusions  in  a  theory  so  delicate  as  that 
of  the  propagation  of  Sound. 

§  IV.     Of  the  Propagation  of  Sound  through  Liquids. 

The  experiments  of  Canton,  and  the  more  recent  ones  of  Perkins,  Oersted,  Colladon,  and  Sturm,  have  shown        89. 
that  water,  alcohol,  ether,  and,  no  doubt,  all  other  liquids,  are  compressible  and  elastic,_though  requiring  a  very  Liquids 
much  greater  force  to  produce  a  given  diminution  of  bulk  than  air.     Water,  according  to  the  experiments  of  c°mP";ssi- 
Perkins,  (Phil.  Trans.  1820,  p.  234.)  as  computed  by  Dr.  Roget,  suffers  a  condensation  of  -^.-^g  by  a  pressure  el^" 
of  100  atmospheres.     This  result  agrees  sufficiently  well  with  that  of  Canton,  which  gave  a  condensation  of 
0-000046  for  every  atmosphere  of  pressure,  (Phil.  Trans.  1764,)  and  has  been  since  confirmed  by  Oersted's 
researches. 

Since  water,  then,  and  other  liquids  have  the  essential  property  of  elastic  media,  on  which  the  propagation  of       90. 
Sound  depends,  it  may  be  presumed,  a  priori,  that  Sounds  are  capable  of  being  conveyed  by  them  as  well  as  by  And  there- 
the  air ;  and,  indeed,  better,  by  reason  of  their  greater  density,  pursuant  to  the  same  law  which  obtains  in  gases.  f°re  capable 
This  conclusion  is  abundantly  confirmed  by  experiment.     Hauksbee  (Phil.  Trans.  1726,  371.)  ascertained  that  °fcon*eying 
water  would  transmit  a  Sound  excited  in  air.     Anderon  (Phil.  Trans.  1748,  p.  151.)  describes  a  number  of  expe- 
riments on  the  hearing  of  fishes,  from  which,  indeed,  he  concludes,  that  they  are  altogether  devoid  of  this  sense. 
But  a  very  different  conclusion  really  follows  from  them.     Fishes  enclosed  in  a  glass  jar  appeared  (says  Anderon)  Hearing  of 
utterly  insensible  to  any  Sound  excited  in  the  air  without  them,  (if  unaccompanied  with  motion,)  but  the  slightest  fishes. 
tap  with  the  nail  on  the  edge  of  the  jar,  although  made  in  such  a  situation  that  the  motion  could  not  be  seen  by 
them,  immediately  disturbed  them.     This  is  easily  explicable ;   and  is,  in  fact,  just  what  ought  to  happen.     The 
intensity  of  Sound  excited  in  aay  medium  must  evidently  be  proportioned  to  the  energy  of  the  original  impulse, 
and  must  therefore  be  much  greater  when  arising  from  the  direct  impact  of  a  solid  body  on  the  water,  or  its  con- 
taining vessel,  than  from  that  of  the  particles  of  the  air  in  a  sonorous  wave,  whose  momentum  is  necessarily  very  small. 
As  fishes  have  no  external  organs  of  hearing,  Sounds  must  be  conveyed  to  their  sensorium  by  direct  propagation, 
through  the  bones  of  their  heads ;    and  they  are  probably  insensible  to,  or  habitually  careless  of,  those  feeble 
impulses  which  are  communicated  from  the  air.     But  that  the  latter  impulses  do  exist,  and  are  audible  by  our 
ears,Anderon'sPaper  furnishes  proof  enough.    He  made  three  people,  stripped  quite  naked,  dive  at  once, and  remain  g       , 
about  two  feet  below  the  surface  of  the  water.     In  this  situation  he  spoke  to  them  as  loud  as  he  was  able.     At  cjted  in  atr 
their  coming  up  they  repeated  his  words,  but  said  he  spoke  very  low.     He  caused  the  same  persons  to  dive  heard  under 
about  12  feet  below  the  surface,  and  discharged  a  gun  over  them,  which  they  said  they  heard,  but  that  the  noise  water, 
was  scarce  perceivable.     He  further  caused  a  diver  to  halloo  under  water,  which  he  did  ;    and  the  Sound  was 
heard,  though  faintly.     A  grenade,  exploded  about  nine  feet  below  the  surface,  gave  a  prodigious  hollow  Sound, 
with  a  most  violent  concussion  of  the  earth  around.     Lastly,  he  caused  a  diver  to  descend  with  a  bell  in  his 
hand,  whose  ringing  he  (the  diver)  assured  him  he  could  hear  distinctly  at  all  depths  ;  adding,  also,  that  he  could 
hear  the  rushing  of  the  water  through  a  flood-gate  at  20  feet  distance  from  the  place  he  was  in. 

The  Abbe"  Nollet  having  descended  to  various  depths,  from  4  to  24  inches,  could  hear  all  Sounds  made  in  the        91. 
air  (as  a  clock  striking,  a  hunter's  horn,  the  human  voice,  &c.)  distinctly,  but  faint  and  attenuated.     (Brocklesby,  Nollet's  ex 
Phil.  Trans,  1748,  p.  237.)  periments. 

Franklin,  having  plunged  his  head  below  water,  caused   a  person  to  strike  two  stones  together  beneath  the        93. 
surface  ;   and  at  more  than  half  a  mile  distance  heard  the  blows  distinctly.     These  instances  are  sufficient  to  Franklin's, 
show  that  Sound  is  audibly  conveyed  through  water  as  well  as  through  air ;    and,  indeed,  if  properly  excited, 
much  better. 

A  series  of  experiments  on  the  velocity  of  Sound  in  sea-water  was  instituted  by  M.  Beudant,  at  Marseilles.        93. 
Two  observers,  with  regulated  watches,  were  stationed  in  boats  at  a  known  distance.     Each  was  accompanied  Velocity  of 
by  a  diver.     A  bell  was  struck  at  stated  intervals  at  one  station ;  and  at  the  instant  of  its  being  heard  by  the  Souud  in 
diver  at  the  other  he  made  a  signal,  and  the  time  was  noted  by  the  observer  in  the  boat.     Of  course,  time  was  pat^T'    , 
lost.     The  mean  result  of  these  observations  gives  1500  metres  =  4921  feet  per  second  for  the  velocity.  experiments 

A  more  careful  and  no  doubt  more  exact  determination  was  undertaken  and  executed  in  1826,  by  M.  Colladon,  in        94 
the  Lake  of  Geneva.     After  trying  various  means  for  the  production  of  the  Sound,  as  the  explosion  of  gunpowder,  Colladon 
blows  on  anvils,  and  bells  ;  the  latter  were  preferred,  as  giving  the  most  instantaneous,  and,  at  the  same  time,  and  Sturm's 
most  intense  Sound,  the  blow  being  struck  about  a  yard  below  the  surface  by  means  of  a  metallic  lever.     The  exPcriment» 
experiments  were  all  made  at  night,  to  avoid  the  interference  of  extraneous  sounds,  and  for  the  better  observing 
of  the  signals  made  at  each  blow  by  the  flash  of  gunpowder. 

To   render  audible  to  an  observer  out  of  water  (in  which   situation  only  can  any  observations  worthy  of       95. 
confidence  be  made)  sgunds  excited  at  a  great  distance,  a  very  ingenious  method  was  practised  by  M.  Colladon.  Method 
He  found,  that  although  the  Sound  of  the  blow  was  well  heard  directly  above  the  bell,  yet  the  intensity  of  the  practised  by 
Sound  so  propagated  into  the  air  diminished  with  great  rapidity  as  the  observer  removed  from  its  immediate  ™cm'ohea* 
neighbourhood,  and  at  two  or  three  hundred  yards  it  could  no  longer  be  heard  at  all.     This  fact  renders  it  water  at" 
probable,  that  the  waves  of  Sound,  like  those  of  light,  in  passing  from  a  denser  into  a  rarer  medium,  undergo,  great  dis- 
at  a  certain  acuteness  of  incidence,  a  total  reflexion;  (see  LIGHT,  Art.  (184)  ;  see  also  Index  to  this  Article —  unces. 


708  SOUND. 

Sound.  Reflexion  of  Sound  —  Echo  ,•)  and  cease  to  penetrate  the  surface,  so  that  the  Sound  heard  beyond  that  limit  is  Part  I. 
v—-v—  •'  merely  that  which  diverges,  in  the  air,  from  the  point  immediately  above  the  bell.  Acting  on  this  idea,  M.  Col-  ^-•v"* 
ladon  plunged  vertically  into  the  water  a  thin  tin  cylinder,  about  three  yards  long  and  eight  inches  in  diameter, 
closed  at  the  lower  end,  and  open  to  the  air  above  ;  thus  forming  an  artificial  surface  on  which  the  sonorous 
waves,  impinging  perpendicularly,  might  enter  the  air,  and  be  thence  propagated  freely  as  from  a  new  origin  ; 
just  as  we  may  look  into  water  at  any  obliquity  by  using  a  hollow  tube  with  a  glass  plate  at  the  end  perpendi- 
cular to  the  axis.  This  contrivance  succeeded  completely,  and  he  was  enabled  by  its  aid  to  hear  the  strokes  of  a 
bell  under  water  at  a  distance  of  2000,  6000,  and  even  14,000  metres,  (about  9  miles,)  viz.  across  the  whole 
breadth  of  the  lake  of  Geneva,  from  Rolle  toThonon.  A  better  spot  could  not  have  been  found,  the  water  being 
exceedingly  deep,  without  a  trace  of  any  current,  and  of  the  most  transparent  purity.  The  signals  were  made 
by  the  inflammation  of  gunpowder,  which  being  performed  by  the  same  blow  of  the  hammer  by  which  the  bell 
Was  struck,  all  loss  of  time  was  effectually  avoided.  The  time  was  reckoned  by  a  quarter  second  stop-watch,  from 
the  appearance  of  the  flash  to  the  arrival  of  the  Sound. 

96.  By  the  mean  of  44  observations  on  three  different  days,  it  appeared  that  a  distance  of  13-487  metres  was 
Result  of      traversed  in  9'295  seconds,  the  greatest  deviation  being  less  than  three-tenths  of  a  second.     M.  Colladon 
their  expe-    assumes  9'4  as  the  true  interval,  regarding  it  as  probable  that  a  minute  portion  of  time  is  necessarily  lost  in  the 

estimation  of  the  interval.  The  mean  temperature  of  the  water,  from  trials  made  at  both  stations,  and  half  way 
between,  was  found  to  be  8°'l  cent.  (=  46°'6  Fahr.)  At  this  temperature,  then,  the  velocity  of  Sound  in  the 
water  of  the  Lake  of  Geneva  was  1435  metres  =  4708  feet  per  second. 

97.  To  compare  this  result  with  theory,  we  will  take  the  data  afforded  by  the  experiments  of  Messrs.  Colladon 
Comparison  and  Sturm  on  this  very  water;  whose  foreign  contents,  as  appears  by  the  analysis  of  M.  Tingry,  amount  only 
with  theory.  to  ^^  of  its  wejght,  and  which  may,  therefore,  be  regarded  as  pure  water,  (though,  of  course,  saturated  with 

air.)     They  state  the  compressibility,  both  at  this  and  at  the  freezing  temperature,  at  0-0000495  for  every  atmo- 

sphere; i.  e.  that  an  increase  of  pressure  of  one  atmosphere  produces  a  diminution  of  bulk  equal  to  —  : 

1  '000*000 

of  the  whole,  or  very  nearly  one  two-hundred-thousandth.  But  as  the  atmospheric  pressures  used  in  their  expe- 
riments were  not  standard  ones,  but  each  equal  to  a  column  of  mercury  0-76  metres  long,  at  a  temperature  of 
10°  cent.,  instead  of  0°  the  compressibility  by  one  standard  atmosphere  must  be  equal  to 

specific  gravity  of  mercury  at    0°  cent.  1-0018 

0-0000495  X  —r—  lno         -  =  0-0000495  X  —  -  -  =  0-000049589. 

specific  gravity  of  mercury  at  10°  cent.  1-0000 

98.  To  apply  the  general  analysis  by  which  the  velocity  of  Sound  in  an  elastic  medium  was  deduced  (Art.  52.)  to 
Different       this  case,  we  must  express  the  elasticity  in  a  form  somewhat  different  from  that  before  employed  in  the  case  of 
mode  of  ex-  aerial  fluids.     Let  us  then  put  e  for  the  compressibility  of  any  elastic  medium,  or,  the  diminution  of  bulk  it  will 

1  e  sustain  by  an  additional  pressure  of  a  single  atmosphere;  or  by  immersion  to  the  depth  of  0-76  metres  (•=  29-927 

of  elasticity.  Incnes)  m  mercury  of  the  freezing  temperature,  (so  that  in  water  e  =  0-000049589.)     Then,  if  we  neglect  the 

heat  disengaged  by  compression,  an  infinitesimal  column  d  x  of  the  medium,  when  compressed  into  a  space 


•=  d  y,  will  exert  a  resistance  on  the  compressing  column  equal  to  one  atmosphere  x    --  .     Let  A  be  the 

e 

area  of  the  section  of  the  sounding  column,  then  will  the  weight  of  the  particle  dx  be  represented  by  A.dx  X  D, 
where  D  is  the  density  of  the  medium  ;  and  its  elastic  pressure  on  the  section  A,  which  separates  it  from  the 

l_*» 
preceding  particle,  will  be  A  X    -  X  (A)  A,  where  (A)  =  the  standard  height  of  mercury  in  the  barometer 

and  A  the  density  of  mercury  at  the  freezing  temperature.     This,  then,  is  the  force  mutually  exerted  between 
dx  and  the  particle  immediately  preceding  it.     Similarly  the  force  exerted  between  dx1  and  the  particle  (dx) 

1  _   dy< 
immediately  preceding  it  is  represented  by  A  x  -  -  X  (A)  A  ;  and  the  difference  of  these,  or  the  whole 

force  by  which  d  x  is  urged  forwards,  is  therefore 

WAJAl^  Ad,  ,W±  .**. 

e    \        dxj)  e  da'' 

so  that  the  accelerating  force  acting  on  d  x  is 

(A)       A_        fy 
D 

' 

or   c'en*  K.     Finally,  therefore,  if,  as  before,  we  represent  by  a  the  velocity  of  Sound,  we  shall  have 


If  we  take  into  consideration  the  heat  developed  by  compression,  we  have  only  to  multiply  this  by  the  coefn- 


/  2  g  (A)  A         _        /  9-8083  X  0"76  X  13-568 
:  V         eD  V      ~  0-000049589  x  D 


SOUND.  769 

Sound.      The  specific  gravity  of  the  water  of  the  lake  at  the  temperature  of  the  experiment  was  found  to  be  exactly  that      Part  I. 
— •V~'  of  distilled  water  at  its  maximum  density,  the  trifling  expansion  due  to  the  excess  of  temperature  being  exactly  ^-^v*"-' 
counterbalanced  by  the  superior  density  due  to  the  saline  contents,  so  that  D  =  1.     Reducing,  then,  the  value  j^jj"^  to 

of  a  to  numbers,  we  find  a  =  1428-2  met.  (=  4685'6  feet)  x  ^"iL 

As  we  have  seen,  the  velocity  actually  observed  was  1435  metres.     The  agreement  of  this  with  the  coefficient       \QQ. 
of  V  K  within  7  metres  (a  space  run  over  by  the  aqueous  pulse  in  one  200th  of  a  second)  is  so  near,  as  to  autho-  Heat  deve- 
rize  the  conclusion  that  in  water,  at  least,  the  heat  developed  by  compression,  and  consequent  increased  resistance 
to  sudden  condensation,  is  insensible.  of  Wjter" 

In  the  course  of  these  experiments,  M.  Colladon  was  led  to  remark  some  very  curious  particulars  respecting       jQl. 
the  nature,  intensity,  and  duration  of  Sounds  propagated  by  water.     He  observed,  first,  that  the  Sound  of  a  Curious 
bell  struck  under  water,  when  heard  at  a  distance,  has  no  resemblance  to  its  Sound  in  air.     Instead  of  a  con-  phenomena 
tinued  tone,  a  short  sharp  sound  is  heard,  like  two  knife-blades  (messerklingen)  struck  together.     The  effect  ODserved  m 
produced  by  hearing  such  a  short  dry  sound,  at  a  distance  of  many  miles  from  its  origin,  he  compares  to  that  of  JimentT1'8 
seeing,  for  the  first  time,  very  distant  objects  sharply  defined  in  a  telescope.     When  tried  at  different  distances, 
it  preserved  this  character,  varying  only  in  intensity,  so  as  to  render  it  impossible  to  distinguish  whether  the 
sound  heard  arose  from  a  violent  blow  at  a  great  distance,  or  a  gentle  one  near  at  hand.     It  was  only  when 
within  200  metres  (about  a  furlong)  that  the  musical  tone  of  the  bell  was  distinguishable  after  the  blow.     In  air 
the  contrary  takes  place,  as  every  one  knows  ;  the  shock  of  the  first  impulse  of  the  hammer  being  heard  only 
in  the  immediate  neighbourhood  of  the  bell,  while  the  continued  musical  Sound  is  the  only  one  that  affects  the 
hearing  at  a  distance.     The  reason  of  this  curious  difference  will  be  apparent  when  we  come  to  speak  of  Musical 
Sounds.     (See  Index.     Musical  Sounds.     Vibrations  of  Bodies  in  different  media.) 

Another  very  curious  and  important  observation  of  M.  Colladon,  is  that  of  the  effect  of  interposed  obstacles.       102. 
Sounds  in  air  spread  round  obstacles  with  great  facility,  so  that  by  a  hearer  situated  behind  a  projecting  wall,  or  Non-diver- 
the  corner  of  a  building,  sounds  excited  beyond  it  are  heard  with  little  diminution  of  intensity.     But  in  water  genceof  the 
this  was  far  from  being  the  case.     When  the  tin  cylinder,  or  hearing-pipe,  already  mentioned,  was  plunged  into  gSund 
the  water,  at  a  place  screened  from  rectilinear  communication  with  the  bell  by  a  projecting  wall  running  out  from  round  obsta- 
the  shore,  whose  top  rose  above  the  water,  M.  Colladon  assures  us,  that  a  very  remarkable  diminution  of  intensity  clesinwatei 
in  the  Sound  was  perceived,  when  compared  with  that  heard  at  a  point  very  near  the  former,  but  within  reach  of 
direct  communication  with  the  bell ;  or,  so  to  speak,  out  of  the  acoustic  shadow  of  the  wall.     Thus  the  phenomena 
of  Sound  in  water  approximate  in  this  respect  to  the  rectilinear  propagation  of  light,  and  may  lead  us  to  presume, 
that  in  a  medium  incomparably  more  elastic  than  water,  the  shadow  would  be  still  more  perfect  and  more  sharply 
defined.     A  material  support  is  thus  afforded  to  the  undulatary  doctrine  of  Light,  against  one  of  its  earliest  and 
strongest  objections — the  existence  of  shadows. 

It  appears,  from  these  experiments,  that  the  velocity  of  Sound  in  water  may  be  correctly  computed  when  its       103. 
compressibility  is  known,  without  the  necessity  of  having-  regard  to  the  heat  developed  during  compression.  Velocity  of 
From  all  direct  experiments  hitherto  made,  it  appears  that  in  water,  and  all  other  liquids,  the  quantity  of  heat  Sound  in 
thus  developed  is  either  altogether  insensible,  or  at  least  very  minute ;  so  that,  most  probably,  the  same  thing  °ttle.r 
will  hold  in  other  liquids.     The  Memoir  of  Messrs.  Colladon  and  Sturm,  then,  which  contains  a  very  elaborate 
determination  of  the  compressibility  of  a  variety  of  liquids,  will  afford  the  means  of  computing  the  velocity  of 
Sound  in  them.     We,  therefore,  subjoin  a  Table  of  their  results,  and  of  such  others  as  we  have  been  able  to 
collect. 


770 


SOUND. 


Sound. 


Table  of  ab. 
solute  com- 
pressibili- 
ties of  va- 
rious bodies. 


Substance  compressed. 

Authority. 

Absolute  com- 
pression in  mil- 
lionth pane  of 
the  original  vo- 
lume. 

Pressure  by  which  the  compression  in  the 
last  column  was  produced. 

Mercury  at  0°  cent  

Colladon  and  Sturm. 
Ditto. 
Ditto. 
Ditto. 
Ditto. 
Ditto. 
Ditto. 

Ditto. 

Ditto. 
Ditto. 

Ditto. 

Ditto. 
Ditto. 

Ditto. 

Galy-Calazat. 
Ditto. 
Colladon  and  Sturm. 
Galy-Calazat 
Ditto. 
Ditto. 
Canton. 
Perkins. 
Oersted. 

5-03 
51-30 
49-50 
73-0 
35-5 
32-0 
42-2 

J96-2 

\  93-5 
I  89-0 
(  133-3 
I  118-5 
(  149-3 
I  141  3 

38-0      J 

71-5 
f  79-3 
I  71-3 
/  85-9 
t  82-25 
46-8 
47-0 
3-30 
2-84 
7-09 
0-18 
46-00 
47-09 

Acolumn  of  mercury  Om-76  high  at  10°C 
Ditto. 
Ditto.    (If  correctly  computed.) 
Ditto. 
Ditto. 
Ditto.    (Query  strength.) 
Ditto.            Ditto. 
Ditto.    (Under  an  initial  pressure  of  one 
atmosphere.) 
Ditto.    (Under  8  atmosph.  init.  press.) 
Ditto.     (Under  20               ditto.) 
Ditto.     (Under  3                 ditto.) 
Ditto.    (  Under  24  atmospheres.) 
Ditto.    (  Under  3        ditto.) 
Ditto.    (Under  24       ditto.) 
Ditto.     (Under  a  mean  pressure  of  about 
10  atmospheres.    Diminishes 
rapidly   as   the   pressure   in- 
creases.) 
Ditto. 
Ditto.     (Under  4  atmospheres.) 
Ditto.    (Under  16       ditto.) 
Ditto.     (Under  2         ditto.) 
Ditto.     (Under  9         ditto.) 
One  atmosphere,  (doubtful.) 
Ditto               ditto. 
0-76  met.  of  mercury  of  10°  cent. 
One  atmosphere  (1) 
Ditto.              ditto. 
Ditto.              ditto. 
One  atmosphere  at  50°  Fahr. 
As  computed  by  Dr.  Roget. 

Water  saturated  with  air  at  0°  
Oil  of  turpentine  at  0°  

Nitric  acid  S.  G.  1  •  403  at  0°   
Concentrated  sulphuric  acid  at  0°.  . 

Alcohol  at  1  1°-6  cent  

at  11"-1 

Water  saturated  at  20°  cent,  with  1 

Nitric  ether  0°  

Acetic  ether  at  0°  

Muriatic  ether  at  1  1"  2  

Olive  oil  

Glass  

LeadP  ...... 

Water  

Pait  I. 


Of  the  Propagation  of  Sound  in  Solids  and  in  Mixed  Media. 

104.  Solids,  if  elastic,  are  equally  well,  or  better,  adapted  for  the  conveyance  of  Sound  with  fluids.     By  elasticity 
Elasticity  in  m  a  so''^  's  not  meant  a  power  of  undergoing  great  extensions  and  compressions,  after  the  manner  of  air,  or 
solids,  what.  Indian  rubber,  and  returning  readily  to  its  former  dimensions ;  but  rather  what  is  commonly  called  hardness,  in 

contradistinction  to  toughness,  a  violent  resistance  to  the  displacement  of  its  molecules  inter  se  in  all  directions. 
Thus  the  hardest  solids  are,  generally  speaking,  the  most  elastic,  as  glass,  steel,  and  the  hard  brittle  alloys  of 
copper  and  tin,  of  which  mirrors  are  made  ;  and  in  proportion  as  they  are  so,  they  are  adapted  to  the  free  propa- 
gation of  Sound  through  their  substance. 

105.  But  an  important  condition  in  their  constitution  is  homogeneity  of  substance ;  and  in  a  substance  perfectly 
Effect  of  in-  homogeneous,  we  may  add,  too,  uniformity  of  structure.     The  effect  of  want  of  homogeneity  in  a  medium,  on  its 

.    power  of  propagating  Sound,  is  precisely  analogous  to  that  of  the  same  cause  in  obstructing  the  free  passage  of 
'oHds  oiftne    &^>  an<^  (as  ^e  undulatory  doctrine  of  light  teaches)  for  the  very  same  reason.     The  sonorous  pulses,  in  their 
propagation  passage  through  it,  are  at  every  instant  changing  their  medium.     Now,  at  every  change  of  medium,  two  things 
of  Sound,     happen ;  first,  a  portion  of  the  wave  is  reflected,  (see  Reflexion  of  Sound, — Echo,  in  the  Index,)  and  the  intensity 
of  the  transmitted  part  is  thereby  diminished ;  secondly,  the  direction  of  propagation  of  the  transmitted  part  is 
changed,  and  the  sonorous  rays,  like  those  of  light,  are  turned  aside  from  their  direct  course.     (See  Refraction 
of  Sound,  in  the  Index.)     Thus  the  general  wave  is  broken  up  into  a  multitude  of  non-coincident  waves,  ema- 
nating from  different  origins,  and  crossing  and  interfering  with  each  other  in  all  directions.     Now,  whenever  this 
takes  place,  a  mutual  destruction  of  the  waves,  to  a  greater  or  less  extent,  arises,  and  the  Sound  is  stifled  and 
obstructed.     Further  yet : — as  the  parts  of  a  non-homogeneous  medium  differ  in  elasticity,  the  velocities  with 
which  they  are  traversed  by  the  sonorous  pulses  also  differ ;  and  thus,  among  the  waves  which  do  ultimately  arrive 
at  the  same  destination  in  the  same  direction,  some  will  arrive  sooner,  some  later.     These,  by  the  law  of  inter- 
ference, tend  mutually  to  destroy  or  neutralize  each  other. 

106.  But  of  all  causes  which  obstruct  the  propagation  of  Sound,  one  of  the  most  effective  is  a  want  of  perfect  adhe- 
Effect  of  im-sion  at  the  junctures  of  the  parts  of  which  such  a  medium  consists.     The  effect  of  this  may  be  conceived,  by 
^unction  of   reffar("nS'.  tne  superficial  strata  of  molecules  of  each  medium  when  in  contact,  as  forming  together  a  thin  film  of 
parts.           'ess  elast'ci'y  than  either  ;    at  which,  therefore,  a  proportionally  greater  reflexion  of  the  wave  will  take  place 

than  if  the  cohesion  were  perfect, — just  as  light  is  much  more  obstructed  by  a  tissue  of  cracks  pervading  a 
piece  of  glass,  than  it  would  be  by  any  inequality  in  the  composition  of  the  glass  itself. 


S  0  U  N  D.  771 

Sound.          A  pleasing  example  of  tlie  stifling  and  obstruction  of  the  pulses  propagated  through  a  medium,  from  the  effect       1>art 
••-v  ••  '  of  ils  non-homogeneity,  may  be  seen   by  filling  a  tall  glass  (a  Champagne  glass,  for  instance)  half  full  of  that  s 
.sparkling  liquid.     As  long  as  its  effervescence  lasts,  and  the  wine   is  full  of  air-bubbles,  the  glass  cannot  be    , 
made  to  ring  by  a  stroke  on  its  edge,  but  gives  a  dead,  puffy,  disagreeable  Sound.     As  the  effervescence  subsides  j 
the  tone  becomes  clearer,  and  when  the  liquid  is  perfectly  tranquil  the  glass  rings  as  usual ;   but  on  reexciting  the  tjon- 
bubbles   by  agitation,  the  musical  tone  again  disappears.     To  understand  the  reason  of  this,  we  must  consider 
what  passes  in  the  communication  of  vibrations  through  the  liquid  from  one  side  of  the  glass  to  the  other.     The 
glass  anil  contained  liquid,  to  give  a  musical  tone,  must  vibrate  regularly  in  unison  as  a  system  ;  (see  Vibrations 
of  a  System  of  bodies  ;)    and  it  is  clear,  that  if  any  considerable  part  of  a  system  be  unsusceptible  of  regular 
vibration,  the   whole  must   be  so.      This  neat   experiment  seems  to   have  been  originally   made   by  Chladni, 
(Acoiutique,  §  214,)  and  has  been  employed   by  Humboldt,  to  illustrate  by  it  a   natural  phenomenon  equally 
familiar  and   striking  ;    we   mean,   the  greater  audibility  of  distant  Sounds   by  night  than  by   day.      This  he 
attributes   to    the    uniformity    of   temperature    in    the    atmosphere    by    night,   when    upward    currents   o,'   air,  Greater 
healed    by    their   contact  with   the    earth   under  the    influence   of  the   sun's   rays,   are    no  longer   continually  a' 
mixing  the  lower  with  the  upper  strata,  and  disturbing  the  equilibrium  of  temperature.       It   is  obvious  &*t  night  than 
Sound,  as  well  as  light,  must  be  obstructed,  stifled,  and  dissipated  from  its  one  original  direction,  by  the  mixture  by  day. 
of  air  of  different  temperatures,  (and  consequently  elasticities;)  and   thus  the  same  cause  which  produces  that  Humboldi's 
extreme  transparency  of  the  air  at  night,  which  astronomers  only  fully  appreciate,  renders  it  also  more  permeable  e!CPlanaU(in 
to  Sound.     There  is  no  doubt,  however,  that  the  universal  and  dead  silence  generally  prevalent  at  night  renders 
our  auditory  nerves  sensible  to  impressions,  which  would  otherwise  escape  them.     The  analogy  between  Sound 
and  light  is  perfect  in  this  as  in  so  many  other  respects.     In   the  general   light  of  day  the  stars  disappear.      In  Anolller 
the  continual  hum  of  noises  which  is  always  going  on  by  day,  and  which  reach  us  from  all  quarters,  and  never 
leave  the  ear  time  to  attain  complete  tranquillitv,  those  feeble  Sounds  which  catch  our  attention  at  night  make  no 
impression.     The  ear,  like  the  eye,  requires  long  and  perfect  repose  to  attain  its  utmost  sensibility. 

To  a  caus<>  of  the  same  kind,  particularly  modified,  may  possibly  be  attributable  the  singular  effect  of  hydrogen       108. 
gas  when  mixed  with  air,  already  described,  Art.  85,  in  unfitting  it  for  the  free  propagation  of  Sound.     Chemists  Sounds  in 
maintain  that  when   gases  are  mixed,  the  molecules  of  each   form    separate    and  independent   systems,   being  ""xe(l 
mutually  inelastic,  and  each  sustaining  a  part  of  the   pressure   proportional  to   its  own  density.     They  admit,  "3 
l.owever,  that  the  molecules  of  one  gas  (A)  act  as  obstacles,  to  obstruct  the  free  motion  of  those  of  another  (H  ;) 
and  on  this  principle  they  explain  the  slow  mixture  of  two  gases  in  separate  vessels  communicating  by  a  narrow 
aperture.     Granting  these  postulates,  let  us  conceive  a  pulse  excited  in  a  mixture  of  equal  volumes  of  two  gases. 
If  the  velocity  of  Sound  in  both  be  alike,  the  pulse  will  run  on  in  each,  although  independently,   yet  with  the 
same  speed,  and  at  any  instant,  and  at  any  point  of  the  medium,  the  contiguous  molecules  of  both  gases  will  be 
moving  in  the  same  direction  and  with  the  same  velocity.    They  will,  therefore,  offer  no  mechanical  obstruction  to 
each  other's  motion,  and  Sound  will  be  freely  propagated.     But  if  they  differ  in  their  specific  elasticity,  the  case 
will   be  altered.     Each   being  non-elastic  to  the  other,  two  distinct  pulses  will  be  propagated,  and  will  run  on 
uith  different  velocities  ;    the  molecules  of  either  gas,  at  different  points   beginning,  and  ceasing  to  be   agitated 
with   the  pulsation   at  different  instants.     Thus  an  internal  motion,  a   change  of  relative   position   among  the  oiisln,,,,1(,n 
molecules  of  the  gas  (II)  and  those  of  the  gas  (A)  will  take  place,  the  one  set  being  obliged  to  force  themselves  Of  Suund  liv 
a  passage  between  the  other;  in  which,  of  course,  a  portion  of  their  motion  will  be  diverted  in  all  sorts  of  lateral  hydrogen 
directions,  and  will  be  mutually  destroyed.      It  is  evident  that  the  greater  the   difference  of  specific  elasticities,  g;|s  mixed 
the  greater  will  be  the  effect  of  this  cause.      In  hydrogen  the  velocity  of  the  pulse  is  nearly  three  times  its  velocity  w."!'  *'?  ex" 
in, atmospheric  air;   and,  of  course,  it  may  be  expected  in  this  case  to  act  with  great  efficacy.     In   azote  and  " 
oxygen  the  velocities  are  so  nearly  alike,  that  very  litlle  obstruction  can  arise  from  its  influence ;  so  that,  in  so  far 
as  the  phenomena  of  Sound   are  concerned,  atmospheric  air  may  be  looked  upon  as  a   homogeneous  medium. 

If  saturated  with  aqueous  vapour,  at   high   temperatures,  however,  it   is   possible  that  the  effect  may  become        109. 
sensible,  and,  perhaps,  to  this  cause  may  be  attributed  a  phenomenon,  mentioned  by  more  than  one  experimenter  Duplication 
on  this  branch  of  Physics,  of  the  occasional  duplication  of  the  Sound  of  a  gunshot  heard  from  a  great  distance,  of  Sound- 
a  part  of  the  Sound  being  transmitted  quicker  than  the  rest  by  aqueous  vapour,  or  even  by  water  in  the  liquid  "j^ved''"'' 
state  suspended  in  the  air.      If  this  be  the  case.  Sounds  might  be  expected   to  be  heard  double  in  thick  fogs,  or 
in  a  snow-storm.    But  the  remarkable  obstruction  to  Sound  caused  by  fog,  and  especially  by  snow,  (see  Art.  21,) 
would,  probably,  prevent  any  Sonnd  from  being  heard  far  enough  to  permit  the  interval  of  the  tivo  pulses  to  be 
distinguishable.     This  latter  phenomenon,  we  may  here  observe,  affords  another  and  very  satisfactory  illustration  £fl-ect  Or 
of  the  general  principle  explained  in  Art.  107.     To  it  we  may  add  the  well-known  effect  of  carpeting,  or  woollen  carpeting, 
cloth  of  any  kind,  in  deadening  the  Sound  of  music  in  an   apartment.      The  intermixture  of  air  and  solid  fibres  &c.  in  dea- 
in  the  carpets  through  which  the  Sound  has  to  pass,  deadens  the  Echo  between  the  ceiling  and  floor  by  which  dening 
the  original  Sound  is  swelled. 

A  phenomenon  noticed  by  every  traveller  who  visits  the  Solfaterra  near  Naples,  but  whose  true  nature  has  been       no. 
much  misconceived,  is  e.isily  explicable  on  this  principle.     The  Solfaterra  is  an  amphitheatre,  or  extinct  crater,  Pheno- 
surrounded  by  hills  of  lava,  in  a  rapid  state  of  decomposition  by  the  action  of  acid   vapours  issuing  from  one  menon 
principal  and  many  subordinate- vents  and  cracks.     The  whole  soil  of  the  level  at  its   bottom  consists  of  ibis  obser'f(l  »i 
decomposed  lava,  whose  disintegration,  however,  is  not  so  complete  as  to  reduce  it  to  powder;   but  leaves  it  in  Solraterra- 
coherent  white  masses  of  a  very  loose  friable   structure.     At  a  particular  spot,   a  large  stone  violently  thrown 
against  the  soil,  is  observed  to  produce  a  peculiar  hollow  Sound,  as  if  some  great  vault  were  below.     Accordingly 
it  is  usually  cited  as  a  proof  of  the  existence  of  some  vast  cavity  below,  communicating  with  the  ancient  vent  of 
the  volcano,  and  perhaps  with  subterraneous  fires;  while  others  ascribe  it  to  a  reverberation  from  the  surrounding 
hills,  with  which  it  is  nearly  concentric  ;    and  others  to  a  variety  of  causes  more  or  less  fanciful.     It  seems  most  Kxi'!ain'c' 

VOL.  iv.  5  H 


772 


SOUND. 


Sound. 


111. 

Essential 
diffe  fence 
in  the  con- 
stitution  nf 
fluids  and 
solids. 


Propagation 
of  oblique 
or  transverse 
undulations. 


Propagation 
of  pulses 
in  crystal- 
lized media. 

112. 

Wood  an 
excellent 
conductor 
of  Sound. 


113. 

Conduction 
of  Sound 
along  a  \vire. 


Through 
rocks. 


Through 
cast  iron. 
Biot's  ex- 
periments. 


probable,  however,  that  the  hollow  reverberation  is  nothing  more  than  an  assemblage  of  partial  echoes  arising 
from  the  reflexion  of  successive  portions  of  the  original  impulse  in  its  progress  through  the  soil  at  the  innume- 
rable half-coherent  surfaces  composing  it ;  were  the  whole  soil  a  mass  of  sand,  these  reflexions  would  be  so 
strong  and  frequent  as  to  destroy  the  whole  impulse  in  too  short  an  interval  to  allow  of  a  distinguishable  after- 
sound.  It  is  a  case  analogous  to  that  of  a  strong  light  thrown  into  a  milky  medium,  or  smoky  atmosphere  ; 
the  whole  medium  appears  to  shine  with  a  nebulous  undefined  light.  This  is  to  the  eye,  what  such  a  hollow 
Sound  is  to  the  ear. 

The  general  principle  on  which  the  conveyance  of  Sound  through  solids  depends,  is  precisely  the  same  as  in 
fluids  ;  and  the  same  formuhi  may  be  used  to  express  its  velocity  when  the  specific  elasticity  is  known.  There 
are,  however,  two  very  important  particulars  in  which  they  diH'er ;  first,  the  molecules  of  fluids  are  capable  of 
displacement  inter  se.  Those  of  solids,  on  the  other  hand,  are  subjected  to  the  condition  of  never  changing  their 
order  of  arrangement.  Secondly,  each  molecule  of  a  fluid  is  similarly  related  to  those  around  it  in  all  directions ; 
in  solids  each  molecule  has  distinct  sides,  and  different  relations  to  space  and  to  the  surrounding  particles. 
Hence  arise  a  multitude  of  modifying  causes,  which  must  necessarily  affect  the  propagation  of  sonorous  pulses 
through  solids,  which  have  no  place  in  fluids,  and  modes  of  vibration  become  possible  in  the  former,  which  it  is 
difficult  to  conceive  in  the  latter,  whose  parts  have  no  lateral  adhesion.  Thus  we  may  conceive  pulses  propa- 
gated in  solids,  like  those  of  a  cord  vibrating  transversely,  in  which  the  motion  of  each  molecule  is  transverse, 
or  oblique,  to  the  direction  in  which  the  general  pulse  is  advancing.  Again,  the  cohesion  of  the  molecules  of 
crystallized  bodies  is  different  on  their  different  sides,  as  their  greater  facility  of  cleavage  in  some  directions 
than  in  others  indisputably  proves.  They  must,  in  consequence,  have  unequal  elasticities  in  different  directions  ; 
and  thus  the  velocity  of  the  pulse  propagated  through  a  crystallized  solid  will  depend  on  its  direction  with 
respect  to  the  axes  of  crystallization.  Among  uncrystallized  solids,  too,  there  are  many,  such  as  wood,  whale- 
bone, &c.  which  have  a  fibrous  structure,  in  virtue  of  which,  it  is  evident,  they  are  very  differently  adapted  to 
convey  an  impulse  longitudinally  and  transversely. 

Interruptions  of  crystalline  structure,  then,  ought  to  produce  an  effect  on  the  conveyance  of  Sound  analogous 
to  that  of  the  mixture  of  extraneous  matter  in  a  medium.  The  conducting  power  of  wood  along  the  grain  is 
certainly  very  surprising.  A  simple  experiment  will  show  it.  Let  any  one  apply  his  ear  close  to  one  end  of 
the  longest  stick  of  sound  timber,  and  let  an  assistant  at  the  other  end  scratch  with  the  point  of  a  pin,  or  tap  so 
lightly  with  its  head  as  to  be  inaudible  to  himself.  Every  scratch  or  tap  will  be  distinctly,  nay  loudly,  heard  at 
the  other  end,  as  if  close  to  the  head.  In  general,  however,  all  solids  tolerably  compact  conduct  Sound  well, 
and  transmit  it  rapidly. 

Chladni  relates  an  experiment  made  by  Messrs.  Herhold  and  Rafn,  in  Denmark,  where  a  metallic  wire 
600  feet  long  was  stretched  horizontally.  At  one  end  a  plate  of  sonorous  metal  was  suspended,  and  slightly 
struck  ;  an  auditor  placed  at  the  other,  and  holding  the  wire  in  his  teeth,  heard  at  every  blow  two  distinct 
sounds;  the  first  transmitted  almost  instantaneously  by  the  metal,  the  other  arriving  later  through  the  air. 
Messrs.  Hassenfratz  and  Gay  Lussac  made  a  similar  experiment  in  the  quarries  :<t  Paris;  a  blow  of  a  hammer 
against  the  rock  produced  two  Sounds,  which  separated  in  their  progress ;  that  propagated  through  the  stone 
arriving  almost  instantly,  while  the  Sound  conveyed  by  the  air  lagged  behind.  The  same  thing  has  been  observed 
in  the  blasting  of  rocks  in  the  deep  mines  of  Cornwall.  These  experiments  were,  however,  made  at  intervals 
too  short  to  give  any  numerical  estimate  of  the  velocity  of  transmission  of  Sound  in  the  iron  or  stone.  The  only 
direct  experiments  we  have  on  this  subject  are  those  of  M.  Biot  himself,  who,  assisted  by  Messrs.  Bouvard, 
Malus,  and  Martin,  ascertained  the  interval  required  for  the  Sound  of  a  blow  on  the  cast-iron  conduit  pipe  already 
spoken  of,  Art.  24,  to  traverse  measured  lengths  of  it.  The  pipe  consisted  of  joints  of  cast  iron,  each  2'ra"'515  = 
S'2514  feet  long,  and  connected  by  flanches  with  collars  of  lead  covered  with  tarred  cloth  interposed,  and  strongly 
screwed  home;  each  collar  measured  O'met  14256  =  0'f'46773.  A  blow  being  struck  at  one  end, and  heard  at  the  other, 
the  interval  between  the  arrival  of  the  Sound  through  the  air  and  through  the  iron  was  noted.  The  length  being 
known,  the  time  required  for  the  transmission  of  the  aerial  Sound  became  known  with  great  precision,  and  thence 
the  time  of  transmission  through  the  iron  became  known  also.  The  following  is  a  statement  of  the  results  : 


Observers'  names. 

Number  of 
iron  joints. 

Number  of 
leaden 
collars. 

Total   length 
when  con- 
nected in 

metres. 

Observed  in- 
terval of  the 
sounds  in 
seconds. 

Number  of 
observa- 
tions. 

Computed 
time  of  trans- 
mission in  air. 
Seconds. 

Deduced  time 
of  transmis- 
sion through 
the  compound 
solid. 

Biot,  Bouvard  ..  . 

78 

77 

197-27 

0-542 

53 

0-579 

0-037 

Bouvard,  Malus.  . 

156 

155 

394-55 

0-810 

64 

1-158 

0-348 

Biot,  Martin  .... 

376 

375 

951-25 

2-500 

200 

2-790 

0-290 

Ditto     do  

ditto. 

ditto. 

ditto. 

Time  directly  observed  by  a  different  method. 

0-260 

The  last  result  was  obtained  as  follows.     Each  observer  holding  in  one  hand  a  chronometer  and  in  the  other  a 

ob'    hammer,  (the  chronometers  being  carefully  compared,)  the  one  (M)  at  the  precise  beats  of  0'-  and  30'-  struck  on 

eity  of      °"  tne  P'Pe-  anc^  *'le  otner  noted  the  moment  of  arrival  of  that  Sound  only  which  was  propagated  through  the  solid, 

Sound  m       (*•  e-  the  first.)     At  every  15"  and  45s",  and  also  precisely  on  the  beat  of  his  chronometer,  the  observer  (B)  struck 

c»it  iron.      the  pipe,  and  (M)  noted  in  the  same  manner  the  moment  of  arrival  of  the  metallic  Sound  by  hit  watch.     From 


SOUND.  773 

such  reciprocal  observations,  a  very  little  consideration  will  show  that  the  exact  time  required  for  the  Sound's      ''art  '• 
•*  propagation  through  the  solid  may  be  obtained,  independent  of  any  observation  of  the  aerial  Sound,  as  well  as  v-"-v"~-/ 
of  the  rates  of  the  watches.     The  agreement  of  the  results  obtained  by  the  two  methods  sufficiently  proves  that 
the  result  of  Messrs.  Bouvard  and  Malus,  in  the  above  Table,  is  too  large ;  rejecting  this  for  that  reason,  and 
the  first  on  account  of  the  shortness  of  the  pipe,  we  have,  as  a  mean  result,  0's'275  for   the  time  required  to 
traverse  951'25  metres,  which  gives  a  velocity  of  3459'"""  1  =  11090  feet  per  second  for  the  velocity  of  Sound 
in  cast  iron  at  the  temperature  of  the  experiment,  (11°  cent.  =  51°'8  Fahr.)  and  neglecting  the  very  small  retard- 
ation due  to  the  collars,  whose  united   thickness  was  5-me'61=  18'f-41  only.    This  is  about 10J  times  its  velocity  Its  velocl'y 
in  air.     Chladni  assigns  3597  metres  for  the  velocity  of  Sound  in  brass.     Laplace,  calculating  on  an  experiment  "' 
of  Borda,  on  the  compressibility  of  brass,  makes  it  3560'4.     According  to  Chladni,  the  following  are  the  velo- 
cities of  Sound  in  different  solids,  that  in  air  being  taken  for  unity  :   tin  =  7^,  silver  =  9,  copper  =  12,  iron  = 
17,  glass  =  17,   baked  clay  (porcelain  >)  10.  ...  12,   woods  of  various  species  =11....  17.     The  error  in  the 
case  of  iron  throws  a  doubt  on  all  the  rest;  unless,  perhaps,  steel  be  meant.     (Acoust.  §  219.) 

From  this  determination  we  may  estimate  the  time  it  requires  to  transmit  force,  whether  by  pulling,  pushing,       115. 
or  by  a  blow,  to  any  distance,  by  means  of  iron  bars  or  chains.     For  every  11090  feet  of  distance  the  pull,  push,  Time  re- 
or  blow,  will  reach  its  point  of  action  one  second  after  the  moment  of  its  first  emanation  from  the  first  mover.  <]u"'ei1  '" 
In  all  moderate  distances,  then,  the  interval  is  utterly  insensible.     But  were  the  sun  and  the  earth  connected  by  ^™'tjron 
an  iron  bar,  no  less  than  1074  days,  or  nearly  three  years,  must  elapse  before  a  force  applied  at  the  sun  could  reach  m(|s  |jvtre 
the  earth.     The  force  actually  exerted  by  their  mutual  gravity  may  be  proved  to  require  no  appreciable  time  for  &c.' 
its  transmission.     How  wonderful  is  this  connection  ! 

§  VI.   Of  the  Divergence  and  Decay  of  Sound. 

Hitherto  we  have  taken  no  account  of  the  lateral  divergence  of  Sound,  which  we  have  supposed  confined  by       \  \c} 
a  pipe ;   but  it  is  evident  that  condensation  taking  place  in  any  section  of  such  a  channel  will  urge  the  contained  Divergence 
air  laterally  against  the  side  of  the  pipe,  as  well  as  forward  along  its  axis ;  and,  consequently,  if  the  pipe  were  of  Somul 
cut  off  at  any  point,  the  Sound  would  diverge  from  that  point  into  the  surrounding  air.     Accordingly,  when  any  from  l[leir"i 
one  speaks  through  along  straight  tube  the  voice  is  heard  laterally,  as  if  proceeding  from  the  mouth  of  a  speaker  °  a  l"'10' 
at  the  orifice. 

In  general,  a  Sound  excited  in,  or  impulse  communicated  to,  any  portion  of  the  air  or  other  elastic  medium,       117 
spreads,  more  or  less  perfectly,  in  all  directions  in  space.     We  say  more  or  less  perfectly  ;  for  though  there  are  Unequal 
Sounds,  as  the  blow  of  a  hammer,  the  explosion  of  gunpowder,  &c.  which  spread  equally  in  all  directions,  yet  divergence 
there  are  others  which  are  far  from  being  in  that  predicament.     For  instance,  a  common  tuning-fork  (a  piece  of0/  certa'" 
steel  in   the  shape  represented  in  fig.  6)  being  struck  sharply,  when  held  by  the  handle  (A)  against  a  substance,  p™"'^' 
is  set  in  vibration,  the  two  branches  of  the  fork  alternately  approaching  to  and  receding  from  each  other.     Each 
of  them,  consequently,  sets  the  air  in  vibration,  and  a  musical  tone  is  produced.     But  this  Sound  is  very  unequally 
audible  in  different  directions.     If  the  axis  of  the  fork,  or  the  line  to  which  it  is  symmetrical,  be  held  upright 
about  a  foot  from  the  ear,  and  it  be  turned  round  this  axis  while  vibrating,  at  every  quarter  revolution  the  Sound  Exemplified 
will  become  so  faint  as  scarcely  to  be  heard,  while  in  the  intermediate  axes  of  rotation  it  is  heard  clear  and  strong.  in  Sound  °f 
The  audible  situations  lie  in  lines  perpendicular  and  parallel  to  the  flat  faces  of  the  fork,  the  inaudible  at  45°  ?  *? 
inclined  to  them.     This  elegant  experiment,  due  originally  to  Dr.  Young,  has  recently  been  called    into  notice 
by  Weber.     (Wellenlehre,  §  271.) 

The  non-uniformity  of  the  divergent  pulses  which  constitute  certain  Sounds  is  easily  demonstrated  by  con-       118. 
sidering  what  happens  when  a  small  disc  is  moved  to  and  fro  in  a  line  perpendicular  to  its  surface.     The  aerial  A  Priori 
molecules  in  front  of  the  disc  are  necessarily  in  an  opposite  state  of  motion  from  those  similarly  situated  behind  c.ons'dera- 
it.     Hence,  if  we  conceive  a  wave  propagated  spherically  all  around  it,  the  vertices  of  the  two  hemispheres  in 
front  and  behind  are  in  opposite  motions  with  respect  to  the  centre.     But  with  regard  to  that  wave  of  the  sphere 
where  the  vibrating  plate  prolonged  cuts  it,  there  is  evidently  no  reason  why  its  molecules  should  approach  to  or 
recede  from  the  centre,  or,  rather,  there  is  as  much  reason  for  one  as  for  the  other.     They  will   therefore  either 
remain  at  rest,   or  move  tangentially ;  so  that  the  motion  of  the  whole  sounding  surface,  or  wave,  will,  in  this 
case,   be  rather  as   in  fig.  7  than  in  fig.  8  ;  and  a  corresponding  difference,  both  in  the  intensity  and  character  Fjg- 1 '• 
of  the  Sound  heard  in  different  directions,  may  be  fairly  expected.  Fl?-  ®' 

The  mathematical  theory  of  such  pulses  as  these  is  of  the  utmost  complication  and  difficulty,  depending  on       ; '^' 
the  integration  of  partial  differential  equations  with  four  independent  variables,  viz.  the  time  and  the  three  coor-  y  i  ^L" 
dinates  of  the  moving  molecules.     It  is  therefore  of  much  too  high  a  nature  to  have  any  place  in  an  Essay  like  Sound'  m° 
the  present.     We  shall  merely  content  ourselves  with  stating  the  following  as  general  results  in  which  mathema-  free  air. 
ticians  are  agreed.  121. 

1st.  The  velocity  of  propagation  of  a  sonorous  pulse  is  the  same,  whether  we  regard  it  as  propagated  in  one,  Lawof  (lie 
two,  or  three  dimensions,  i.  e.  in  a  pipe,  a  lamina,  or  a  mass  of  air.  decay  nf 

2nd.  Sounds  propagated  in  a  free  mass  of  air  diminish  in  intensity  as  they  advance  further  from  the  sonorous    "Tn^ 
centre,  and  their  energy  is  in  the  inverse  duplicate  ratio  of  this  distance,  cateris  paribus.  Case  ~,' 

We  shall  not  attempt  a  proof  of  these  propositions  in  the  general  cases,  but  content  ourselves  with  Illustrating  spherical 
them  in  one  particular  but  important  case,  viz.  when  the  initial  impulse  is  confined  to  a  very  small  space,  undulation 
and  consists  in  any  small  radiant  motion  of  all  the  particles  of  a  spherical  surface  in  all  directions  equally  from  alike  on  a|i 
the  centre.  sides- 

Since  the  initial  wave  is  spherical,  and  similar  in  all  its  parts,  it  will  evidently  retain  this  property  as  it  dilates       123 

b  H  2 


774  S  O  U  N   I). 

Sound,      by  the  progress  of  the  impulse.     It",   then,  it  be  conceived  to  be  divided  into  its  infinitesimal  elements  In  a      Part  I. 
••—  v—  x  system  of  pyramidally  disposed  plane  surfaces,  having  the  common  vertex  in  the  centre  of  the  sphere,  each  of  these  ~—  —  ~— 
elements  will  form  the  base  of  one  of  the  pyramids,  and  its  molecules  will  advance  and  recede  along  its  axis,  aa 
the  pulse  traverses  them,  without  any  change  of  their  relative  positions,  inter  se  ;  so  that  the  whole  wave  may  be 
regarded  as  broken  up  into  partial  waves,  each  advancing  as  if  confined  within  a  pyramidal  pipe,  independently 
of  all  the  rest. 

124  Now  in  any  one  of  these  imaginary  pipes  the  pulse  will  be  propagated  from  layer  to  layer  of  the  included 

Velocity  of    particles  with  the  same  velocity  as  if  the  pipe  were  cylindrical,  for  the  divergence  of  the  sides  of  the  pipe  can  only 

a  pulse  in      cause  a  lateral  extension,  and  thence  a  diminished  thickness,  of  the  stratum,  and  will,  therefore,  alter  the  velocity 

any  pyra-     of  each  of  its  molecules  and  the  extent  and  law  of  its  motion  from  what  it  would  be  in  a  cylindrical  pipe.     But 

nuilal  pipe.    ^  we  consj<jer  a  row  of  particles  situated  in  the  axis  of  the  pyramid,  the  propagation  of  a  pulse  along  them 

depends,  as  we  have  seen,  neither  on  the  velocity  nor  extent  or  law  of  excursion  of  the  individual  molecules,  but 

only  on  their  intrinsic  elasticity.    The  latter,  however,  is  not  altered  by  the  shifting  of  the  whole  vibrating1  fibre  into 

a  w'ider  or  narrower  part  of  the  pipe,  since,  from  this  cause,(its  excursions  from  its  original  place  beiiu-  supposed 

infinitely  small,)  the  whole  dilates  or  contracts  together,  as   if  by  an   external   compressing  or  rarefying  force. 

Now  we  have  seen  that  a  variation  in  the  general  density  of  the  medium  in  which  a  pulse  is  propagated  from 

external  pressure  makes  no  change  in  its  velocity.    It  follows,  then,  that  the  pulse  will  be  propagated  with  equal 

velocity  along  the  line  of  molecules  in  question,  whether  the  pipe  be  cylindrical  or  pyramidal,  or,  indeed,  of  any 

shape  ;  and  as  it  runs  equally  fast  in  each  of  the  imaginary  pyramids  into  which  the  sphere  is  divided,  the  wave,  of 

which  it  is  an  element,  will  dilate  itself  spherically  with  the  uniform  velocity  of  Sound  in  a  straight  tube.     See  also 

Euler,  Comm.  Petrop.  1771,  cap.  iv.  &c.,  where  the  general  equations  for  the  motions  of  air  in  tubes  of  any  figure 

are  deduced,  and  the  above  proposition  proved  therefrom,  in  the  case  of  hyperbolic  tubes  (p.  391)  and  conical  or 

pyramidal  ones  (p.  418.) 

125.  Let  us  now  conceive  a  spherical  wave  by  any  means  excited,  such  that  the  whole  interval,  reckoned  along  its 

Application   radius,  within  which  the  motion  to  and  fro  of  the  molecules  is  comprised,  shall  be  equal   to  2  a.     This,  then,  will 

01  the  law  of  ^e  tjle  Drea(jth  of  the  wave,  and  as  all  its  parts  dilate  equally  fast,  this  will  continue  to  be  its  breadth  throughout 

its  whole  progress.     Its  surface  increases  in  the  ratio  of  the  square  of  the  radius,  and,  therefore,  calling  r  this 

radius,  2  o  r8  will  represent  the  quantity  of  matter  in  motion  at  the  moment  the  Sound  has  reached  the  distance  r 

from  its  origin.     Now,  as  all  the  air  within  and  beyond  the  wave  is  quiescent,  the  whole   impulse,  or  via  viva, 

originally  communicated  to  the  sphere  first  set  in  motion,  is  successively  transferred  to  all  the  rest  without  loss 

or  increase,  (by  the  general  law  of  the  conservation  of  the  vis  viva.  See  MECHANICS.)     And  since  it  is  distributed 

equally  over  the  whole  spherical  surface,  any  portion  of  it,  of  given  magnitude,  (that  of  the  aperture  of  the  ear, 

for  instance,)  will  receive  a  part  of  the  whole,  proportional  to  -  —  —  ,  or  to  —  .    Thus  the  whole  shock  or  impulse 

given  to  the  ear,  while  the  wave  passes  over  it,  is  as  the  inverse  square  of  the  distance  from  its  origin,  and  the 
absolute  velocity  of  each  molecule  in  any  determinate  phase  of  its  motion  inversely  as  the  distance  itself. 
1P6.  In  the  theory  of  Sound,  as  in  that  of  Light,  the  intensity  of  the  impression  made  on  our  organs  is  estimated 

KiT«;  mi       by  the  shock,  impetus,  or  vis  viva,  of  the  impinging  molecules,  which  is  as  the  square  of  their  velocity  ;   and  not 
tiie  ear  esti-  by  t|,cir  inertia,  which  is  as  the  velocity  simply.     Were  the  latter  the  case,  there  could  be  no  such  thing  as  Sound 
J  by  the  or  Light,  since  the  negative  inertia  of  the  receding  molecules  would  exactly  equal  and  destroy  their  positive  effect 
in  their  advance.   (Fide  LIGHT,  Art.  578.)     We  conclude,  then,  that  the  intensity  of  Sound  decays,  in  receding 
Law  of  de-    from  j)s  orjgin>  as  the  square  of  the  distance  increases.     It  is  exceedingly  diflicult  to  subject  this  law  to  satisfac- 
g'-v  °j          tory  experimental  tests,  and  we  know  of  no  attempt  that  has  yet  been  made  tor  the  purpose. 

§  VII.     Of  the  Reflexion  and  Refraction  of  Sound,  ami  ofEchos. 

127.  As  there  is  no  body  in  nature  absolutely  hard   and  inelastic,   whenever  the   particles  of  a  vibrating  medium 
Reflexion  of  impinge  on  the  solid  or  fluid  matter  which  contains  or  limits  it,  they  will  agitate  those  of  the  latter  with  motions 
Sound  atthe  similar  to  their  rwn,  but  modified  by  their  greater  or  less  density  and  mobility.     A  pulse,  then,  will  be  propa- 
ronfmesof    ^a^e(j  jn(O  t|,e  so]|d  or  fluid  according  to  its  own  laws,  but  this  will  not  take  place  without  the  propagation  back 

'  again  of  a  pulse  in  the  original  medium,  which  may  be  regarded  as  the  reflexion  or  echo  of  the  first.  To  under- 
stand how  this  happens,  let  us  consider  what  takes  place  when  a  motion  is  first  impressed  on  any  small  stratum, 
whose  thickness  is  2  a  (as  in  Art.  63)  of  a  sounding  column,  and  let  its  law  be  as  there  expressed,  i.  c.  that  the 
velocity  of  any  one  of  its  particles  at  the  distance  x  from  its  middle  shall  be,  at  the  first  instant,  represented  by 
0(,r),  and  the  linear  extent  of  the  same  molecule,  compared  with  its  original  length,  ore,  shall  equal  ifr(x)  where 
<p  (x)  —  o,  and  ijr  (x)  =  1  from  x  =  —  oo  to  ;r  =  —  n,  and  from  x=  +  a\.ox=+cc;  while  from  x  =  —  a  to 
x  =  -j-  o  they  may  have  any  arbitrary  values. 

128.  Since    t  is  always  positive,   if   we   take  x  >  +  o    we  have,    of   necessity,  x-}-at>  +  a,   and,  therefore, 
Condiiion     0  (x  +  a  t)  =  o,  and  4/  (x  -f  a  t)  =  1,  so  that  the  values  of  c  and  e  in  equations  (t)  (j)  become 


of  the  single 


-  «  Q  - 


•, 

-«  0} 


ae  =  -\a  — 


SOUND.  775 

Souml  v  Part  '• 

.  which  gives  ae=  a  —  v,  or  1  —<•=  —  .  v_ir..-i_ 

^*v^  cc 

Again,  on  the  negative  side  of  the  x  we  take  x  <  —  a,  we  have,  of  necessity,  x  —  a  t  <  —  a,  and,  therefore, 
<f>(x  —  at)  =0  ;  Y'  (*  —  «  0  =  1  >  ano",  consequently, 


if  l   [' 

-|  a  +  0  (>  +  a  0  +  y,  (*  +  a  t)  j  J 


«>"- 


«e  =  — i  a 

t) 
and,  therefore,  in  this  case,  1  —  e  == . 

Gi 

}  —  e  —  1 —  =  — expresses  the  condensation  the  molecule  dx  has  undergone  in  its  disturbed 

dx  d  x 

state.  Hence  we  see,  that  in  each  of  the  two  waves  into  which  the  primary  impulse  separates  itself,  one  running 
towards  the  positive,  the  other  towards  the  negative  side  of  the  x,  there  obtains  this  condition,  viz.  that  the 
condensations  of  the  aerial  molecules  are  proportional  to  their  actual  velocities,  the  fluid  being  condensed  wherever 
the  molecules  are  moving  from  the  origin  of  the  first  impulse,  and  dilated  when  returning  to  it. 

This  remarkable  relation,  which  does  not  of  necessity  hold  good  within  the  limits  of  the  first  disturbance,       129. 
establishes  a  distinction  equally  marked  between  the  initial  impulse  and  the  waves  freely  propagated  from  it.    The  "'  ihis  « 
former  is  subject  to  no  law,  the  latter  must  obey  this  condition.     Any  impulse,  then,  in  which   this  condition  is  f^0"  ob*. 
not  satisfied,  will  immediately  divide  itself  into  two  pulses  running  opposite  ways,  in  each  of  which  the  condition  ^jji'n'ot'du" 
in  question  holds,  but  so  long  as  this  condition  obtains,  no  subdivision  of  the  pulse  will  take  place.     This  is  vide  iueli: 
easily  shown,  for  if  we  suppose  an  initial  impulse  communicated  to  any  portion  (2  a)  of  the  fluid  in  which  this 
relation  is  purposely  maintained,  such  supposition  is  equivalent  to  making 

which,  substituted  in  (i)  and  (j),  give,  for  all  values  of*  and  t, 

p  =  a  —  a^r  (x  —  at) 
ae=:  Y"  (x  —  a  t), 

in  which,  whenever  x  is  negative  and  <  —  a,  we  have  v  =  0  and  1  —  e  =  0 ;  thus  indicating  that  the  molecules 
on  the  negative  side  of  such  a  primitive  disturbance  as  supposed  will  remain  constantly  at  rest,  in  other  words, 
that  the  pulse  will  only  be  propagated  on  the  positive  side. 

Whenever,  then,  in  the  progress  of  a  pulse  through  a  medium,  it  receives,  by  extraneous  causes,  any  modifica-       '30. 

Ifditturbed 

tion  which  disturbs  the  condition  1  —  e  =  — ,  it  will  undergo  subdivision,  and  a  portion  will  run  backward,  or  be  ^jli^y, 

and  a  part 

reflected.     Similarly  this  portion  may  be  again  subdivided  and  undergo  partial  reflexion,  and  so  on  ad  iiifinitum,  rlln  '''"'k 
giving  rise  to  a  continual  series  of  repetitions  or  Echos  of  the  original  Sound. 

Let  us  now  examine  more  closely  what  passes  at  the  junction  of  two  media  when  the  pulse  arrives  there  ;  and,       131. 

first,  in  the  equations  (t)  and  (j)  let  us  write,  instead  of  0  (.r)  and  ^  (x),  which  are  arbitrary,  the  combinations,  General 

equally  arbitrary,  equations  of 


F(x)  -  f(x)  along  pipes 

y,  (T)  =  I  _J_      *•*'  -  J  (x't  filled  with 

a  different  m^ 

when  it  is  to  be  observed  that  F  and/  are  not  the  same  with  the  F  and  /of  Art.  57,  which  we  shall  have  no  more 
occasion  to  refer  to.  If,  then,  we  put  s  =  1  —  e,  so  that  s  shall  represent  the  infinitely  small  condensation  under- 
gone by  the  molecule  d  x  in  its  troubled  state,  those  equations  will  bpcome 


as=f(x  -  at)-Y(x  +  at)  )' 

These  represent  the  state  of  the  molecules  of  the  first  medium.     Similarly,  the  state  of  those  in  the  second  will, 
of  necessity,  be  represented  by  another  system  similar  in  form, 

»'=/'(*-  a'O+F  (a  +o'0  I 
i's'=f(x  -  a't)  -  '         ' 


where  a'  represents  the  velocity  of  Sound  in  the  second  medium,  but  the  functions  /'  and  F'  (which  are  not  here 
intended  to  represent  the  derived  functions  or  differential  coefficients  of/  and  F,  but  others  quite  distinct)  are 
here  no  longer  arbitrary,  because  the  motion  of  the  particles  of  the  second  medium  must  evidently  depend  on  that 
of  the  first,  and  on  their  relative  elasticities,  densities,  &c.  Let  us  see,  then,  what  conditions  the  nature  of  the 
case,  and  their  mutual  action  at  their  point  of  junction,  will  enable  us  to  assign  for  deducing  the  form?  of  these 
functions  from  those  of  /  and  F,  supposed  to  remain  arbitrary. 


776  SOUND. 

Sound.          Now,  first,  the  condition  of  continuity  of  the  two  media  requires  that  the  strata  in  contact  should  always  have      Pan  I. 
'•—  —v—  "^  a  common  motion,  or  that  for  the  value  x  =r  I,  corresponding  to  the  place  of  junction,  we  should  have  v  =  v',  v—  -v"" 

132.       which  gives 

Condition  of  f(l-at)  +  F(l  +  at)=:f'(l    -  a'  t)  +  F'  (I  +  a'  t);  (C). 

continuity. 

133  Again,  they  must  not  only  have  a  common  motion,  but  a  common  elasticity,  at  this  point.    Now,  if  we  call  E 

Must  have  a  the  natural  elasticity  of  the  first  medium  and  E'  that  of  the  second,  the  elasticities  in  the  disturbed  state  will  be 
common  expressed  by  E  (1  +  /3s)  and  E'  (1  -j-  /3's'),  where  ft  and  ft'  are  constant  coefficients  depending  on  the  nature  of 
elasticity  at  the  medi  and  the  heat  developed  in  them  by  compression,  and  which  would  each  be  unity  were  no  heat  so 
|;h0enl  "  developed.  Hence  we  must  have  E  (1  +  /3s)  =  E'  (1  +  /3s'),  and  since  in  the  state  of  equilibrium  E  =  E',  we 
must  also  have  /3  *  =  ft'  s,  that  is 


p       a 
or,  putting  c  =  —  x  - 

p        d 


-  at)-F(l-l-af)=c  {/(Z-a'O-F  (/  +  «'<)};  (D). 

134.  Suppose  the  whole  extent  of  both  media  to  be  initially  at  rest  (and,  therefore,  v  =  v'  —  s  =  «*  =  0),  for  every 

value  of  x,  but  those  comprised  within  the  region  of  the  primitive  disturbance  (x  =  +  a),  supposed  very  minute 
and  situated  at  the  origin  of  the  x,  we  shall  have  then 

f  (x)  =  0  and  F"  (x)  =  0  from  x  =  I  to  x  =r  CD, 
and  since  t  is  necessarily  positive,  and  also  a',  therefore 

/'  (l-\-a'i)  =  Q  and  P  (I  +  a'  <)  =  0. 
The  equations  C  and  D  then  become 

t)  =f'(l  -  a'  t) 


and,  consequently, 

f(l-af)  -  V(l+af)  =  c{f(l-  al)  +  F(l  +  at)}.  (E). 

135.  Now  this  equation  is  equivalent  to  as  =  CD  ;  x  being  supposed  =  I,  (equation  A.)     Consequently,  whatever 
Division  of    be  the  motion  of  the  first   medium,  the  existence   of  a  second,  in  contact  with  it,   establishes  at  their  point  of 
.he  pulse  on  junctjon  a  relation  between  the  velocity  v  and  the  condensation  s  of  its  terminal  stratum,  which  is  incompatible 
in'gan'oh-"    w'tn  ^ie  condition  as=  v,  (unless  in  the  very  peculiar  case  where  c  =  1,)  which  we  have  shown  to  be  essential 
stacle.          to  the  total  propagation  of  the  pulse  forward.     It  will,  therefore,  divide  itself  conformably  to  what  was  said  in 

Art.  129,  and  a  portion  will  run  back  in  the  first  medium  and  cause  an  Echo. 

136.  In  the  second  medium,  on  the  other  hand,  we  have  constantly  x  >  I,  and,   therefore,  x-\-a't>l,  so  that 
But  is  pro-   F  (x  +  a'  t)  =  0,  and,  therefore,  the  equations  (B)  give 

sTn^te.  </  =  «V  =  /'(*-«'<);  (F.) 

'*'  The  condition  of  the  single  propagation  of  the  pulse  onward  in  this  medium  c'  =  aV  being  therefore  satisfied,  no 
further  subdivision  of  the  pulse  will  take  place,  and  each  particle  of  the  second  medium  will  be  agitated  once  and 
no  more.  The  reader  who  would  pursue  this  discussion  (a  very  delicate  one)  further,  is  referred  to  M.  Poisson's 
Memoir,  Sur  le  Mouvement  des  Fluides  Elastiques  dans  des  Tvyaux  Cylindriques,  Mem.  Acad.  Par.  1818,  1819. 
See  also  a  very  curious  Paper  by  Euler,  Sur  la  Propagation  du  Son  et  sur  la  Formation  de.  tEcho,  Mem.  Acad. 
Berlin,  1765,  p.  355  ;  where  he  shows  how  an  echo  may  be  formed  at  the  open  mouth  of  a  tube,  by  the  mere 
conditions  to  be  satisfied  by  the  arbitrary  functions,  and  without  any  reflexion  properly  so  called.  It  is  enough 
that  the  condition  as  =  v  should  be  disturbed  (as  it  will  by  the  sudden  breaking  off  of  the  pipe)  to  cause  an 
echo.  See  also  Weber,  Wellenlehre,  §  276,  who  shows  how  this  disturbance  takes  place,  owing  to  the  greater 
freedom  of  motion  suddenly  attained  by  the  particles  when  the  pulse  reaches  the  free  air. 

,07  If  we  suppose  a  plane  wave  of  indefinite  extent  to  fall  obliquely  on  the  surface  of  a  second  elastic  medium,  each 

Oblique  re    Particle  of  this  surface  may  be  regarded  as  being  put  in  agitation  by  it  and  becoming  a  separate  and  independent 

fraction  of    centre,  from  which   spherical  waves  originate  and   are  thence  propagated  in  either  medium  with  the  velocity 

Sound.         peculiar  to  it.     Now,  if  we  investigate  the  surfaces  which  in  either  medium  are  common  tangents  to  all  these 

spheres,  and  which,  therefore,  will  be  the  form  of  the  general  or  resulting  waves  in  each,  we  shall  find  them  to  be 

planes  ;  that  in  the  medium  of  incidence  being  inclined  to  the  surface  at  an  angle  equal  to  that  made  with  it  by  the 

incident  wave,  and  that  in  the  other  medium  at  an  angle  whose  cosine  is  to  the  cosine  of  that  made  with  it  by  the 

incident  wave  as  the  velocity  of  propagation  of  the  wave  in  the  first  medium  to  that  in  the  second.     For  the 

Internal        demonstration  of  these  propositions  we  shall  refer  to  our  article  on  LIGHT,  Art.  586.     Thus  the  reflexion  and 

total  refrac    refraction  of  Sound  at  oblique  surfaces  obeys  the  same  geometrical  laws  with  those  of  Light.     The  observation 

of  Messrs.  Colladnn  and  Sturm,  above  cited,  Art.  95,  shows   th^t  this  analogy  extends  to  the  case  of  oblique 

internal  reflexion  at  the  surface  of  a  less  elastic  medium,  which,  at  a  certain  incidence,  becomes  total. 


PART  II. 

OF  MUSICAL   SOUNDS. 

§  I.     Of  the  Nature  and  Production  of  Musical  Sounds. 

EVERV  impulse  mechanically  communicated  to  the  air,  or  other  sonorous  medium,  is  propagated  onward  by  its      Part  "• 
'  elasticity  as  a  wave,  or  pulse ;   but,  in  order  that  it  shall  affect  the  ear  as  an  audible  sound,  a  certain  force  and  N~— "v""' 
suddenness   is  necessary.     Thus  the   slow  waving  of  the  hand  through  the  air  is  noiseless,   but  the  sudden       138. 
displacement  and  collapse  of  a  portion  of  that  medium  by  the  lash  of  a  whip  produces  the  effect  of  an  explosion.  Perceplion 
It  is  evident  that  the  impression  conveyed  to  the  ear  will  depend  entirely  on  the  nature  and  law  of  the  original  ° 
impulse,  which  being  completely  arbitrary,  both  in  duration,  violence,  and  character,  will  account  for  all  the 
variety  we  observe  in  the  continuance,  loudntss,  and  quality  of  Sounds.     The  auditory  nerves,  by  a  delicacy  of 
mechanism,  of  which  we  can  form  no  conception,  appear  capable  of  analyzing  every  pulsation  of  the  air,  and 
appreciating  immediately  the  law  of  motion  of  the  particles  in  contact  with  the  ear.     Hence  all  the  qualities  we 
distinguish  in  Sounds — grave  or  acute,  smooth,  harsh,  mellow,  and  all  the  nameless  and  fleeting  peculiarities 
which  constitute  the  differences  between  the  tones  of  different  musical  instruments — bells,  flutes,  cords,  &c.,  and 
between  the  voices  of  different  individuals  or  different  animals. 

Every  irregular  impulse  communicated  to  the  air  produces  what  we  call  a  noise,  in  contradistinction  to  a      139. 
musical  Sound.     If  the  impulse  be  short  and  single  we  hear  a  crack,  bounce,  or  explosion ;  yet  it  is  worthy  of  M»*e,  as 
remark,  as  a  proof  the  extreme  sensibility  of  the  ear,  that  the  most  short  and  sudden  noise  has  its  peculiar  d's'lnguisl1 
character.     The  crack  of  a  whip,  the  blow  of  a  hammer  on  a  stone,  and  the  report  of  a  pistol,  are  perfectly  musrj°a( 
distinguishable  from  each  other.     If  the  impulse  be  of  sensible  duration  and  very  irregular  we  hear  a  crash,  if  Sound, 
long  and  interrupted,  a  rattle  or  a  rumble,  according  as  its  parts  are  less  or  more  continuous,  and  so  for  other 
varieties  of  noise. 

The  ear,  like  the  eye,  retains  for  a  moment  of  time,  after  the  impulse  on  it  has  ceased,  a  perception  of  excite-       140. 
ment.     In  consequence,  if  a  sudden  and  short  impulse  be  repeated  beyond  a  certain  degree  of  quickness,  the  ear  Continuous 
loses  the  intervals  of  silence  and  the  Sound  appears  continuous.     The  frequency  of  repetition  necessary  for  the  Sound- 
production  of  a  continued  Sound  from  single  impulses   is,  probably,  not  less  than  sixteen  times  in  a  second, 
though  the  limit  would  appear  to  differ  in  different  ears. 

If  a  succession  of  impulses  occur,  at  exactly  equal  intervals  of  time,  and  if  all  the  impulses  be  exactly  similar      ^1- 
in  duration,  intensity,  and  law,  the  Sound  produced  is  perfectly  uniform  and  sustained,  and  has  that  peculiar  and  .Perlotl'cal 
pleasing  character  to  which  we  apply  the  term  musical.     In  musical  Sounds  there  are  three  principal  points  of  p™^^8 
distinction,  the  pitch,  the  intensity,  and  the  quality.     Of  these,  the  intensity  depends  on  the  violence  of  the  musical 
impulses,  the  quality  on  their  greater  or  less  abruptness,  or,  generally,  on  the  law  which  regulates  the  excursions  Sounds, 
of  the  molecules  of  air  originally  set  in  motion.     The  pitch  is  determined  solely  by  the  frequency  of  repetition  of  Pitch, 
the  impulse,  so  that  all  Sounds,  whatever  be  their  loudness  or  quality,  in  which  the  elementary  impulses  occur 
with  the  same  frequency,  are  at  once  pronounced  by  the  ear  to  have  the  same  pitch,  or  to  be  in  unison.    It  is  the 
pitch  only  of  musical  Sounds  whose  theory  is  susceptible  of  exact  reasoning,  and  on  this  the  whole  doctrine  of 
harmonics  is  founded.     Of  their  qualities  and  the  molecular  agitations  on  which  they  depend,  we  know  too  little 
to  subject  them  to  any  distinct  theoretical  discussion. 

The  means  by  which  a  series  of  equidistant  impulses,  or,  to  speak  more  generally,  by  which  an  initial  impulse       142. 
of  a  periodical  nature  (i.  e.  capable  of  being  represented  by  a  periodical  function)  can  be  produced  mechanically,  Means  of 
are  extremely  various.    Thus,  if  a  toothed  wheel  be  turned  round  with  uniform  velocity,  and  a  steel  spring  be  made  producing 
to  bear  against  its  circumference  with  a  constant  pressure,  each  tooth,  as  it  passes,  will  receive  an  equal  blow  j^™],^" 
from  the  spring,  and  the  number  of  such  blows  per  secqjid  will  be  known,  if  the  velocity  of  rotation  and  number 
of  teeth  in  the  wheel  be  known. 

The  late  Professor  Robison  devised  an  instrument  in  which  a  current  of  air  passing  through  a  pipe  143. 
was  alternately  intercepted  and  permitted  to  pass  by  the  opening  and  shutting  of  a  valve  or  stopcock.  When  The  Sirene. 
this  was  performed  with  sufficient  frequency  (which  could  only  be  done,  we  presume,  by  giving  a  rapid  rotatory 
motion  to  the  stopcock  by  wheelwork)  a  musical  tone  was  produced,  whose  pitch  became  more  acute  as  the 
alternations  became  more  frequent.  This  is  precisely  the  principle  of  the  Sirene  of  Baron  Cagniard  de  la  Tour. 
In  this  elegant  instrument  the  wind  of  a  bellows  is  emitted  through  a  small  aperture,  before  which  revolves 
a  circular  disc,  pierced  with  a  certain  number  of  holes  arranged  in  a  circle  concentric  with  the  axis  of  rotation, 
exactly  equidistant  from  each  other,  and  of  the  same  size,  &c.  The  orifice,  through  which  the  air  passes,  is 
so  situated,  thai  each  of  these  holes,  during  the  rotation  of  the  disc,  shall  pass  over  it  and  let  through  the  an,  uui 
the  disc  is  made  to  revolve  so  near  the  orifice,  that  in  the  intervals  between  the  holes  it  shall  act  as  a  cover  and 
intercept  the  air.  If  the  holes  be  pierced  obliquely,  the  action  of  the  current  of  air  alone  will  set  the  disc  in 
motion  :  if  perpendicular  to  the  surface,  the  disc  must  be  moved  by  wheelwork,  by  means  of  which  its  velocity  of 
rotation  is  easily  regulated  and  the  number  of  impulses  may  be  exactly  counted.  The  Sound  produced  is  clear 
and  sweet,  like  the  human  voice.  II,  instead  of  a  single  aperture  for  transmitting  the  air,  there  be  several,  so 
disposed  in  a  circle  of  equal  dimension  with  that  in  which  the  holes  of  the  disc  are  situated,  that  each  shall  be 

777 


f78  S  O  U  N  D. 

Sound.      opposite  one  corresponding  hole  when  at  rest,  these  will  all  form  Sounds  of  one  pitch,  and  being  heard  together 
>-— ^/- ~^  will  reinforce  each  other.     The  Sirene  sounds  equally  when  plunged  in  water,  and  ted  by  a  current  of  that  fluid,  ^ 
as  in  air  ;  thus  proving  that  it  is  the  number  of  impulses  alone,  and  nothing  depending  on  the  nature  of  the 
medium  in  whicl)  the  Sound  is  excited,  that  influences  our  appreciation  of  its  pitch. 

144.  In  general,  whatever  cause  produces  a  succession  of  equidistant  impulses  on  the  ear,  causes  the  sensation  of  a 
Echos  from   musical  Sound,  whether  such  periodicity  be  a  consequence  of  periodical  motions  in  the  origin  of  the  Sound,  or  of 
a  series  of     the  mode  in  which  a  single  impulse  is  multiplied  in  its  conveyance  to  the  ear.  For  example,  a  series  of  broad  palisades 
palisades.      set  ecj^eways  jn  a  ]ine  directed  from  the  ear,  and   equidistant  from  each  other,  will  reflect  the  Sound  of  a  blow 

struck  at  the  end  of  the  line  nearest  the  auditor,  producing  a  succession  of  echos,  which  (by  reason  of  the  equidistance 

/  distance  of  palisades  \ 

of  the  palisades)   will  reach  his  ear  at  equal   intervals  of  time,  I  =  2  X  —  I,  and  will 

\  velocity  of  Sound      / 

therefore  produce  the  effect  of  a  number  of  single  impulses  originating  in  one  point.  Thus  a  musical  note  will 
be  heard  whose  pitch  corresponds  to  a  number  of  vibrations  per  second,  equal  to  the  quotient  of  the  velocity  of 
Sound  by  twice  the  distance  of  the  palisades. 

145.  A  similar  account  may  be  given  of  the  singing  Sound  of  a  bullet,  or  other  missile,  traversing  the  air  with  great 
Singing  of  a  rapidity.     The  bullet  being  in  a  state  of  rapid  rotation,  and  not  exactly  alike  in  all  its  parts,  presents,  periodically, 
bullet.          at  equal  intervals  of  time  and  space,  some  protuberance  or  roughness  first  to  one  side,  then  to  the  other.     Thus 

an  interruption  to  the  uniformity  of  its  mode  of  cutting  through  the  air  is  periodically  produced,  and 
reaches  the  ear  in  longer  or  shorter  equal  intervals  of  time,  according  as  the  rectilinear  velocity  of  the  bullet 
bears  a  greater  or  less  ratio  to  the  velocity  of  its  rotation  about  its  axis. 

145.  The  echos  in  a  narrow  passage,  or  apartment  of  regular  figure,  being  regularly  repeated  at  equal  very  small 

Echos  in  a    intervals,  always  impress  the  ear  with  a  musical   note  ;  and  this  is,  no  doubt,  one  of  the  means  which  blind 

chamber.      persons  have  of  judging  of  the  size  and  shape  of  any  room  they  happen  to  be  in.     But  the  most  ordinary  ways 

Vibratioi.s    jn  w|,jc|j   musical  Sounds  are   excited  and  maintained  consist  in  setting  in  vibration  elastic   bodies,   whether 

bodies          flexible,  as  stretched  strings,  or  membranes ;   or  rigid,  as  steel  springs,  bells,  glasses,  &c.  or  columns  of  air  of 

determinate  length  enclosed  in  pipes.     All  such  vibrations  consist  in  a  regular  alternate  motion  to  and  fro  of  the 

particles  of  the  vibrating  body,  and  are  performed  in  strictly  equal  portions  of  time.     They  are,  therefore,  adapted 

to  produce  musical  sounds  by  communicating  that  regularly  periodic  initial   impulse  to  the  aerial   molecules  in 

contact  with  them  which  such  sounds  require.     We  shall,  therefore,  proceed  to  consider  more  particularly  the 

principal  of  these   modes  of  production  ;    but  especially,  at  present,  the  first  and  last,  being  the  most  simple 

cases. 

§  II.     Of  the.  Vibrations  of  Musical  Strings  or  Cords. 

147.  If  a  string,  or  wire,  be  stretched  between  two  fixed  pins,  or  supports,  and  then  struck,  or  drawn  a  little  out 

Vibration  of  of  its  straight  line,  and  suddenly  let  go,  it  will  vibrate  to  and  fro,  till  its  own  rigidity,  and  the  resistance  of  the 

•'  ^retched    ajri  reduce  it  to  rest ;  but  if  a  bow  (which  is  an  instrument  composed  of  a  bundle  of  fibres  of  horse  hair,  loosely 

stretched,  and  rendered  adhesive  by  rubbing  with  rosin)  be  drawn  across  it,  the  vibrations  are  continually  renewed, 

and  may  be  maintained  for  any  length  of  time,  and  a  musical  Sound  is  heard  corresponding  to  the  rapidity  of  the 

vibration. 

148  The  mathematical  theory  of  the  vibrations  of  a  stretched  cord  is  remarkable,  in  an  historical  point  of  view,  as 

having  given   rise  to  the  first  general  solution  of  an  equation  of  partial  differences ;  and  led  geometers  to  the 

consideration  of  the  nature  and  management  of  the  arbitrary  functions  which  enter  into  the  integrals  of  these 

equations.     Such  functions,  as  we  have  seen,  enter  into  the  general  expressions  for  the  motion  of  the  air  in  Sound  ; 

and  such,   as  we  shall   presently  show,  into  that  of  the  molecules  of  a  vibrating  cord  ;    i.nd  a  long  and  lively 

discussion,  on  the  degree  of  generality  which  ought  to  be  attributed  to  them,  soon  arose  between  Euler,  D'AIem- 

bert,  D.  Bernoulli,  and  Lagrange.     It  is  not,  however,  our  intention  in  this  Article  to  enter  into  any  points  of 

historical  detail,  and  we  shall  content  ourselves  with  a  reference  to  the  principal  Memoirs,  &c.  on  the  subject, 

which  the  reader  may  consult  for  himself;  while  we  proceed  to  give  such  a  view   of  the  subject  as  is  consistent 

with  the  present  state  of  knowledge  on  this  delicate  point,  and  sufficient  for  the  purpose  we  have  in  hand.     See 

References.  Taylor,  De  Motu  Nervi  Tcnui,  Phil.  Trans.  1713-26  ;   D'Alembert,  Mem.  Acad.  Berl.  1747  ;   Ditto,  1753;   Ditto 

Opuscules,  torn.  i. ;   Euler,  Mkm.  Arad.  Berl.  1753;  Daniel   Bernouilli,  Ditto;    Liigrange,  Miscellanea  Taurin. 

vol.  i.     See  also  Sauveur,  Mem.  Acad.  for  1713,  p.  324  :   J.  Bernouilli,  on  Vibrating  Cords,  Pi-trap.  Comm.  !ii. 

13  ;    Daniel  Bernouilli,  Ditto,  p.  62;    Ditto,  on  Vibrations  of  Unequal  Cords,  Acad.  Berl.  1765,   p.  81  ;    Ditto, 

on  Vibrations  of  Compound  Cords,  N.  Comm.  Pelropp.  xvi.  257  ;   Euler,  Acad.  Berl.  1748,  p.  69  ;   Ditto,  Ditto, 

1765,  p.  307,  335;    Ditto,  on  Unequal  Vibrating  Cords,  N.  Comm.  Pttropp.  xvii.  381  ;    Ditto,  1780,  iv.  ii.  99. 

149.  LetM  N  (fig.  9)  be  a  cord  maintained  by  any  means  in  a  const  ant  state  of  equal  tension  throughout,  and  disturbed 

Solution  of  by  any  external  cause  from  its  rectilinear  position,  and  then  left  to  take  Us  own  form  and  motion  in  consequence 

the  problem  Ofits  tensjon  .   its  gravity,  however,  being  neglected.     Let  MABCDN    be    the  figure  of  the  cord  after  the 

"ionsVfT    l°Pse  °f  anv  l'me  t  from  l'le  initial  disturbance;   respecting  which  we  will  only  suppose  that  the  distance  of  all 

wretched      its  points  from  the  axis  VT  (the  undisturbed  rectilinear  position  of  the  cord)  is  extremely  small  ;   so  that  in  this 

.-.'nl.  theory,  as  in  that  of  the  sonorous  vibrations  of  the  air,  we  concern   ourselves  only  with  such  excursions  of  the 

F'S-  9-         vibrating  molecule  as  may  be  considered  infinitely  minute.     Let  A  B  C  D  be  points  of  the  cord  infinitely  near 

each  other  ;    and  erecting  the  ordinates  A  P,  B  Q,  C  K,  DS,  and  drawing  A  a,  B  6,  C  c,  D  d,  parallel  to  V  T 


SOUND. 


7/9 


Sound, 


put  V  P  =  x,  VQ  =  x',  V  R  =  x",  &c.  and  A  P  =  y,  B  Q  =  y',  C  R  =  y",  &c.     Let  the  tension  of  the  cord  at    P"t  II. 
rest  be  represented  by  c,  which  (since  the  cord  is  infinitely  little  disturbed  from  its  position  of  repose)  will  also  v"~"v"™ 
be  its  tension  in  its  disturbed  state  ;  and  will  in  this,  as  in  the  former  state,  be  uniform  over  its  whole  length, 
the  curvature  being  evanescent.     The  point  B  of  the  cord  then  will  be  solicited  towards  the  axis  by  the  tension 
c  applied  at  B,  and  acting  in  the  direction  B  A,  and  whose  resolved  value  is,  therefore, 


BA 


d  x'  -(-  d  y* 
dy 


'\dx 


2 


., 

d  x 


neglecting  the  higher  powers  of  the  quantity  ,  which  (being  the  tangent  of  the  inclination  of  the  element 

u  JC 

A  B  to  the  axis)  is  infinitely  small.     Similarly  the  point  B  will  be  solicited  from  the  axis  by  the  tension  c  applied 

d  y1 
at  B  in  the  direction  B  C,  whose  resolved  part  in  the  direction  of  the  ordinate  is  equal  to  c  .-T*-J.     The  resolved 

parts  in  directions  parallel  to  the  axis,  being  equal  and  parallel,  destroy  each  other ;   consequently,  the  whole 

(d  y1         d  y\  d?y 

—?- — —  I;    or,  supposing  dx  constant,  c.  dx,  tending  to  increase 

(L  JC  Ct  if  f  Qi  OC- 

the  value  of  y. 

Now  the  motion  of  the  cord  will  be  the  same,  whether  we  regard  it  as  a  continuous  mass,  or  compound  of      150 
detached  particles  situated  at  A,  B,  C,  D,  &c.  and  connected  by  filaments  A  B,  B  C,  &c.  without  weight.     Thus  Its  equation 

AB  +  BC  BC-fCD  derived  a"d 

at  B  we  may  conceive  to  be  placed  a  weight  equal  to  —  —  ,  at  C  the  weight ~ ,  and  so  on,  that  integrated. 

&  'it 

is,  neglecting  {  — —  I ,  simply  a  constant  weight  d  x  in  each  point.     This,  then,  is  the  mass  to  be  moved  by  the 

d1  y  d'  y 

moving  force  c  — —  dx,  and  the  accelerating  force  is,  therefore,  simply  c  — — .     Hence,  calling  t  the  time 
d  x1  d  x* 

and  regarding  d  t  as  constant  as  well  as  d  x,  x  and  t  being  independent  variables,  and  putting  2  g  —  9-m"'8088 
=  32-fat18169,  or  g  =  16'fe"090845,  we  have 

d*y   _ 


or,  putting 


2  gc  =  af;         a  =  V  2  gc, 

d*y  _   ,   d* y 

dt*  ~  a  '  dx*  ' 


This  equation  is  precisely  similar  to  that  above  obtained  for  the  propagation  of  Sound  along  a  cylindrical  pipe, 
and  its  integral  will,  of  course,  be  of  the  same  form,  viz. 

y  —  F  (x  -f  a  0  +  f(x  —  a  <.) 

The  determination  of  the  arbitrary  functions  in  this  equation  will  depend  on  the  conditions  we  may  set  out  from.       151. 
Now,  first,  when   the  cord  is  supposed  to  be  of  indefinite  length,  and  the  part  initially  disturbed  to  be  compa-  Determina- 
ratively  very  small  ;  and  having  an  indefinite  undisturbed  portion  on  either  side.     In  this  case,  it  is  evident  by  tion  of  the 

the  very  same  reasoning  as   that  of   Articles  63   and  64,   that  a  pulse  or  undulation  will  run  out  both  ways  arbitrary 

_  J     functions. 

along  the  cord  from  the  point  of  initial  disturbance,  with  a  velocity  represented  by  a  =  "J  2  g  c,  every  molecule 
of  the  cord  being  once  agitated  during  the  time  the  pulse  runs  over  it,  and  no  more.  Moreover,  a  condition 
similar  to  that  which  ensures  the  single  propagation  of  the  pulse  when  once  it  has  proceeded  beyond  the  limits 
of  the  initial  disturbance  (x  =  rt  a)  in  the  theory  of  Sound,  holds  good  in  the  present  case  ;  for  we  have 


-  =  F  (»  +  a  t) 
ct  x 


'(*•  -  at) 


-  af'(x-at). 


So  that  on  the  positive  side  of  the  ar,  when  x  >  a,  and  therefore  x  +  a  t  =  a,  and  F  (x  -f-  a  f)  and  F'  (i  -)-  a  t) 

d  y  d  y 

=  0,  we  shall  have  —  —        —    —  - 


a  —  —  = 


which  expresses  that  the  tangent  of  the  obliquity  of  the  cord  to  the  axis  in  its  disturbed  state,  at  any  point,  is 
proportional  to  the  absolute  velocity  of  that  point  in  its  motion,  or  putting  0  =  angle  B  A  a, 

a  .  tan  0  •=  —  v  ; 

and   when  this  conditon  ceases  to   hold  good,  as  it  does  when  the  pulse  encounters  an  obstacle  either  fixed 
or  less  movable  than  the  rest  of  the  cord,  it  will  be   either   wholly  reflected,  or  divide  itself  into  two,  one 
running  back,  and  producing  a  species  of  imperfectly  echoed  or  reflected  wave,  just  as  in  the  theory  of  Sound. 
VOL.  iv.  5  I 


r 

for  the  sin- 
gle  pl.opa. 

gation  of  a 
Wave' 


the  wavebv 
anobstacK 
M'n  Soun<l- 


780  SOU  N  D. 

Sound.          Since,  in  the  above  investigation,  c  represents  &  force  equal  to  the  tension  on  the  same  scale  that  dx  represents 
^^V^s  a  weight  equal  to  that  of  the  element  d  x,  we  have 

152.  weight  of  d  x  :  tension  ::  d  x  :  c. 

Velocity  

with  which  Hence  c  represents  the  length  of  a  portion  of  the  cord  whose  weight  is  equal  to  the  tension,  and  v  2  g  c  the 

a  wave  runs  ve]ocijy  which  would  be  acquired  by  a  body  falling  freely  by  gravity  through  that  length.     Hence  this  theorem, 

stretched  The  velocity  °f  a  Pu^  or  undulation  propagated  along  a  tended  cord,  z's  equal  to  that  which  a  heavy  body  would 

cord.  acquire  by  falling  freely  through  the  length  of  a  portion  of  the  cord  whose  weight  is  equal  to  its  tension. 

153.  Let  us  next  suppose  the  cord  attached  at  one  of  its  extremities  to  an  immovable  point,  and  let  the  undulation 
Case  when  be  supposed  to  reach  this  point,  at  which  suppose  x  =  I,  then,  whatever  be  the  value  of  t,  y  =  0,  when  x  =  I. 
one  extre-  go  tna(  we  must  have 

mity  of  the  pi  ft    j    ^  *•>     t     f  ft  n  t\  —  0 

cord  is  J     t   J 

fixed-  Since  at  may  have  any  positive  value,  and  since  on  the  positive  side  of  the  x  (at  which  we  have  supposed  the 

fixed  end  situated)  x  <  I,  therefore  I  —  x  is  in  all  cases  positive,  and  therefore  may  be  one  of  the  values  of  a  t. 
We  may  substitute,  then,  I  —  x  for  a  t  in  this  equation,  when  we  get  for  positive  values  of  x  less  than  I,  and  for 
all  negative  ones 

F(2l-x)  +f(x)  =  0;         or  F  (2  I-  x)  =  -  f(x).  (p) 


Now,  in  general,  y  =  F  (x  +  at)  +  f(x  —  at). 

If,  then,  we  make  a  t  =  x-\-  ui,  where  <i>  is  any  quantity  between  +  a  and  —  a,  at  which  values  of  x  both  f(x) 
and  F  (x)  may  be  supposed  to  vanish, 

y  =  F(Zx  +  w)+f(-w), 

and  if  we  make  a  <  =  2  /  —  a?+a>,  we  have 

y  =  F  (2  I  +  a,)  +  f  (2  x  -  2  I  -  ui), 

but  by  (p)  F(2l+  w)=  -/(--  a,), 

so  that  for  the  latter  value  of  t  we  have 

y  =  —/(—«")  +  /  (2  x  -  2  I  —  u). 


Now  since  when  t  =  0,  we  have  y  =  F  (x)  +  f(x)  and  ^  =  F'  (x)  +  /'  (x)  ;  ^f  =  a  {  F  (,r)  -  /'  (x)  }  , 

d  x  at 

all  these  values  must  vanish  unless  x  lies  between  the  limits  +  a  and  —  a.  Consequently,  for  all  values  but 
those  comprised  within  such  limits,  we  now  have  F  (i)  =  0  andy(j)  =  0.  From  the  above  equations,  then, 
supposing  x  >  a,  or  <  —  a ;  and,  therefore,  F  (2  x  -f-  o>)  =  0,  and  f  (2  x  —  2  I  —  w)  =  0,  we  see  that  for 
values  of  a  t  between  x  -f-  «  and  x  —  a,  y  will  have  real  values  ;  and  that  when  a  t  attains  any  value  between 
2  I  —  x  +  a,  and  2  I  —  x  —  a,  y  will  again  have  real  values,  the  same  as  the  former,  only  with  contrary  signs. 
Thus  the  reflected  pulse  runs  back  with  the  same  velocity  as  the  direct,  and  is  in  all  respects  similar  and  equal 
to  it,  only  that  it  lies  on  the  opposite  side  of  the  axis.  A  reasoning  precisely  similar  applies  to  the  case  of  an 
aerial  pulse  reflected  from  the  bottom  of  a  stopped  pipe,  supposed  perfectly  rigid. 

154.  If  tne  COI"d  De  fixed  at  both  ends,  the  two  pulses  into  whieh  the  initial  pulse  has  separated  itself,  will  each  be 
Case  when    totally   reflected,   and   will  run  along  the  whole  length,  being  reflected  again  at  the  other  end,  and  thus  run 
both  ends     backwards   and  forwards  for  ever,  at  least  if  we  neglect  the  effect  of  the  stiffness  of  the  cord   and   resistance 
are  fixed.      0(-  tne  ajr .  crossjng  each  other  at  each  traverse. 

155.  Suppose  the  whole  length  of  the  cord  to  be  /  -f-  f  =  L,  of  which  I  lies  on  the  positive,  and  I'  on  the  negative 
In  this  case  side  of  the  origin  of  the  x.     That  portion  of  the  subdivided  primitive  pulse  which  runs  towards  the  positive  side 

of  the  x  will  describe  the  length  I  in  a  time  =    —  ;  being  then  reflected  it  will  describe  the  whole  length  I  -f-  f 
vibratory  /  _1_  7'  II 

motion.        jn  a  time   —21 — ;  and  being  again  reflected,  it  will  describe  I'  in  a  time  — ,  so  that  after  a  time 
a  a 

-  L  4-   l  +  1'  4-  JL         2L 

_     I      -  I     ~  » 

a  a  mm 

it  will  reach  its  first  starting  point ;  and  having  been  twice  inverted  by  reflexion,  will  lie  now  on  the  same  side 
of  the  axis  it  originally  was.  Similarly,  the  negative  portion  of  the  original  pulse  will  describe  V,  I'  -J-  I,  and  /, 
and  reach  its  starting  point  after  two  reflexions  in  the  time 

a  a  a          a 

the  same  as  the  other,  and  will  also  have  recovered  its  original  situation  with  respect  to  the  axis.  Thus  at  the 
end  of  this  time  the  two  pulses  will  precisely  reunite,  and  constitute  a  compound  pulse  in  all  respects  similar  to 

the  initial  impulse.     The  state  of  the  cord,  then,  after  the  lapse  of  the  time    —  ,  will  (abstracting  the  effects  of 

0 

resistance,  &c.)  be  precisely  what  it  was  at  first ;  and  so  again,  after  the  lap^e  of  time  —  , ,  &c.  the  same 

a         a 

state  will  recur,  so  that  if  left  to  itself  it  will  continue  to  vibrate  for  ever. 


SOUND.  781 

Sound.          Thus  we  see  that  what  in  an  indefinite  cord  was  merely  a  pulse  running  along  it  and  never  returning,  becomes,     Part  II. 
y-"~v^~-'  by  the  reaction  of  the  fixed  extremities  of  a  finite  one,  a  regular  vibration,  in  which  each  molecule  repeats  its  •>—• *^~-s 
motion  to  and  fro  on  either  side  of  the  axis,  at  equal  intervals,  for  ever.     In  the  foregoing  reasoning  no  particular      156. 
assumption  has  been  made  respecting  the  value  of  a.      It  has   not  been  supposed  small  with  respect  to  Z,  I',  Passage 
and,  consequently,  the  above  conclusion  applies  equally  to  the  case  where  the  initial  disturbance  is  confined   .m  atrj>n- 
to  a  minute  portion  of  the  cord,  and  where  a  large  portion,  or  even  its  whole  length,   is   disturbed  at  once.  *^  jT^j6 
Only  in  the  former  case  the  motions  of  the  individual  molecules  of  the  cord  will  be  performed  by  starts  interrupted  nent  ,ibra- 
by  intervals  of  absolute  rest  in  the  axis.     In  the  latter  there  will  be  no  moments  of  rest  but  those  when  the  tion. 
direction  of  the  motion  changes  at  the  extreme  points  of  their  excursions. 

Hence  we  conclude  that  when  a  stretched  cord,  whose  length  =.L,  is  struck,  or  forcibly  drawn  out  of  its  straight      157. 

situation  into  any  form  and  let  go,  it  will  continue  to  vibrate  to  and  fro,  and  that  the  time  of  one  complete  Time  of  "• 

bration  of  a 

vibration,  after  which  it  resumes  its  initial  state,  is  represented  by •  =  ,  being  equal  to  the  time  of  a  cor(j 

a          V  2gc 

pulse  running  over  double  the  length  of  the  cord,  or  to  the  time  in  which  a  body  would  describe  such  double 
length  with  the  velocity  acquired  by  falling  down  a  height  equal  to  the  length  of  a  portion  of  the  cord  whose  weight 
is  the  tension. 

Hence  the  times  of  vibration  of  different  cords  are,  as  their  lengths  directly,  and  the  square  roots  of  the  tending      158. 
forces  inversely,  and  the  number  of  vibrations,  data  tempore,  as  the  lengths  inversely,  and  the  squaje  root  of  the  '"  different 
tensions  directly. 

The  equations  which  express  the  conditions  arising  from  the  immobility  of  the  ends  of  the  cord  so  far  limit      159. 
the  arbitrary  functions  F  and  f,  that  when  the  figure  of  the  cord  between  its  two  extremities  is  given  it  may  be  Prolonga- 
prolonged  beyond  them  to  any  extent.     To  show  this,  let  y,  represent  the  ordinate  P,  M,  of  the  curve  supposed  to  tion  of  the 
be  continued  beyond  B,  one  of  the  fixed  extremities,  at  a  distance,  B  P,,  beyond  that  end  equal  to  B  P,  the  "S"reof  tlle 
distance  from  it  of  the  ordinate  y,  and,  for  simplicity,  suppose  I1  =  0,  or  let  the  origin  of  the  x  be  at  the  other  eit|ler  s;,je 
fixed  extremity,  A,  (fig.  10.)     Then  we  have  of  its  fixed 

y=F(*  +  at)+f(X-at).  ^T(T' 

yl  —  F  (2  L  -  x  +  a  t)  +  /(2  L  —  x  -  a  t). 
Now  the  condition  of  Art.  153,  derived  from  the  fixity  of  the  point  B,  viz. 

gives,  if  we  write  for  x  successively  x  —  at  arid  2  L  —  x  —  a  t,  the  following  equations, 

F(2L-,r  +  at)  +  f(x  -  af)  =  Q, 

F  (,r  +  a  0  +  /(2  L  -  a  -  a  t)  —  0, 
whose  sum  is  no  other  than 

y  "f*  y\ =  o,  or  Vi  =  —  y- 

Thus  we  see  that  the  curve  A  M  B  will  be  continued  beyond  B  by  merely  reversing  it  from  right  to  left  and 
transferring  it  to  the  other  side  of  the  axis.  Again,  if  we  put  yt  for  the  ordinate  P8  M,  at  a  distance  =  x  beyond 
C,  we  have 

and  F(-  x  +  at)+f(2L.  +  x  —  at)  =  0  ] 

On  the  other  hand,  the  condition  of  the  immobility  of  the  point  A  gives,  as  we  have  seen, 


in  which,  writing  successively  for  x,  -\-x-\-at,  and  —  x  -f-  a  t,  we  get 

F(x+at)+f(-x-at)  =  0  1 

F  (  -  x  -f-  a  t)  +f  (x  -  a  t)  =  0  I 

and  subtracting  the  sum  of  these  from  that  of  the  two  former,  we  find  ultimately 

y«  -  V  =  0,  or  y,  =  y, 

so  that  the  portion  of  the  curve  C  M8  D  is  the  very  same  with  the  first  portion  A  M  B.  And  thus  we  may  go  on 
as  far  as  we  please,  repeating  the  same  curve  alternately  in  a  direct  and  reverse  position,  and  the  same  manifestly 
holds  good  on  the  other  side  of  the  point  A. 

A  very  simple  consideration  will  show  »hat  such  ought  to  be  the  case  ;  for  if  we  conceive  two  equal  and  similar      160. 
cords,  A  M  B,  B  M,  C,  (fig.  10,)  both  attached  to  the  same  point,  B,  and  vibrating  simultaneously,  the  strain  on  Origin  oi 
B,  from  both  their  tensions,  will  be  always  equal  and   opposite,  provided  the  curves  be  so  related  as   above  noda' 
described,  and  B,  therefore,  will  be  retained  in  equilibrium,  independently  of  its  attachment  to  any  extraneous  g"");8'  in 
body,  so  that  were  it  detached,  or  if  the  two  cords,  instead  of  being  fixed  to  one  immovable  point,  were  merely       jg  j 
linked  together  at  B,  so  as  to  form  one  cord  of  double  the  length,  their  vibrations  would  be  the  same.  A  cord  may 

Some  curious  and  important  consequences  follow  from  this.     And,  first,  a  cord,  although  vibrating  freely,  may  have  any 
yet  have  any  number  of  points,  equally  distributed  at  aliquot  parts  of  its  whole  length,  which  never  leave  the  number  of 
axis,  and  between  which  the  vibrating  portions  are  equal  and  similar,  and  lie  alternately  above  and   below  the  tliem- 

5  i  2 


782  SOUND. 

Sound,      axis,  and  in  reversed  positions  as  to  right  and  left.     Such  points  of  rest  are  called  nodes  or  nodal  points,  the      Part  II. 
v»- v— '   intermediate  portions  which  vibrate  are  termed  bellies  or  ventral  segments.  v--^/"~- 

162.  Secondly,  if  a  string  in  the  act  of  vibration  be  touched  in  any  point  so  as  to  reduce  that  point  to  rest  and  retain 
Division  ofa  it  in  the  axis,  then  if,  after  the  contact,  it  vibrate  at  all,  it  will  divide  itself  into  a  certain  number  of  ventral  parts 
vibrating       similar  and  equal  to  each  other  and  separated  by  nodes,  and  each  of  these  will  vibrate  as  if  the  others  had  no 

>t  ™\se<r  ex'stence>  but  instead  the  nodes  were  fixed  points  of  attachment.  Hence,  if  L  be  the  whole  length  of  a  cord,  n  the 
mentsvibra-  number  of  ventral  segments  into  which  it  divides  itself,  and,  therefore,  n  —  1,  the  number  of  its  nodes,  the  time  of 
ling  simul-  2  L 

taneously.     one  complete  vibration  (going  and  returning)  will  be and  the  number  of  vibrations  per  second  will  be 

n  *  2gc, 

represented  by  the  reciprocal  of  this  fraction. 

163.  Experience  confirms  this.     If  the  string  of  a  violin,  or  violoncello,  while  maintained  in  vibration  by  the  action 
Production    of  the  bow,  be  lightly  touched  with  the  finger  or  a  feather  exactly  in  the  middle,  or  at  one-third  of  the  length,  it 
of  harmonic  wjn  not  cease  to  vibrate,  but  its  vibrations  will  be  diminished  in  extent  and  increased  in  frequency, 'and  a  note 

1  s-  will  become  audible,  fainter  but  much  more  acute  than  the  original,  or,  as  it  is  termed,  the  fundamental  note  of 
the  string,  and  corresponding  in  the  former  case  to  a  double,  in  the  latter  to  a  triple  rapidity  of  vibration.  The 
note  heard  in  the  former  case  being  the  octave,  in  the  latter  the  twelfth,  above  the  fundamental  tone  (See  Index, 
Musical  Internals.)  If  a  small  piece  of  light  paper,  cut  into  the  form  of  an  inverted  V,  be  set  astride  on  the  string, 
it  will  be  violently  agitated,  and,  probably,  thrown  off'  when  placed  in  the  middle  of  a  ventral  segment,  while  at 
a  node  it  will  ride  quietly  as  if  the  string  were  (as  it  really  is  at  those  points)  at  perfect  rest.  The  Sounds  thus 
produced  are  termed  harmonics. 

164.  But,  further,  any  number  of  the  different  modes  of  vibration,  of  which  a  cord  is  thus  susceptible,  may  be  going 
Coexistence  On  simultaneously,  or  be,  as  it   were,   superposed  on  each  other.     This  is  a  consequence  of  the  principle  in 
nodes'of       mechanics  of  " tlle  superposition  of  small  motions,"  which,  when  the  excursions  of  the  parts  of  a  system  from  their 
ribration  in   p'aces  of  rest  are  infinitely  small,  admits  of  any  or  all  the  motions  of  which,  from  any  causes,  they  are  susceptible, 
sa«cord.       to  go  on  at  once  without  interfering  with  or  disturbing  each  other.     In  the  particular  case  before  us  it  is  easily 

shown,  for  since  the  general  integral  of  the  equation 

,  d'y 


y 

where  F  and  f  denote  arbitrary  functions,  we  may  suppose 

F  (x)  =  F,  (*)  +  F,  (,)  -f  F.  (*)  +  &c. 

/(*)=/  (*)  +  /.  (*)  +  /.  (*)  +  &c. 
where  F,,  F*  &c.,  and_/i,/2,  &c.,  denote  functions  equally  arbitrary,  and  we  get 

-«0}  +  &c. 


Now  each  of  the  expressions  within  brackets  is  the  integral  of  an  equation  exactly  similar  to  the  original  one. 
Therefore,  if  we  put 


we  shall  have  y  =  ^  +  yf  -f  y,  +  &c. 

Thus,  if  the  several  particular  modes  of  vibration,  y=yly  =  yi,  &c.,  be  possible,  y  =  y,  +  y,  -j-  &c.  will   also 

be  possible  :  the  ordinate  of  the  curve  into  which  the  cord  at  any  moment  forms  itself  in  virtue  of  the  compound 

vibration  will  be  the  sum  (algebraically  understood)  of  the  ordinates  it  would  have  in  virtue  of  each  simple  one, 

separately:  the  compound  curve  will  be  formed  by  first  constructing  on  the  abscissa,  as  an  axis,  any  one  of  the 

simple  ones,  then  on  that  curve,  as  an  abscissa,  any  other,  on  the  new  curve  thence  arising  any  other,  and  so  on. 

165.  Hence  it  is  evident  that  if  we  suppose  the  curve,  whose  ordinate  is  y,,  to  be  of  the  form,  fig'  11,  (a)  having  no 

Fig.  11.        node,  and  that,  whose  ordinate  is  y*  to  have,  for  instance,  one  node,  as  fig.  11,  (6)  the  corresponding  modes  of 

Curves  aris-  vibration,  when  coexisting,  will   produce  a  curve,  such   as  (c).     On  these  we  may  superpose  a  third   mode  of 

superposi-     vibration'  where  the  strin£  divides  itself  into  three  ventral  segments,  as  (d),  and  the  result  will  be  a  curve,  such  as 

tion  of  seve-  (e)'  and  so  on  to  anv  extent.     The  reader  may  exercise  himself  in  tracing  the  variations  of  form  in  these  curves 

ral  coexist-    as  they  go  through  the  several  phases  of  their  periodic  excursions  during  one  complete  period  of  a  vibration  of  the 

ent  vibra-     whole  string  as  one  cord. 

Experience  again  confirms  this  result  of  theory.     It  was  long  known  to  musicians  that,  besides  the  principal  or 

Harmonic     f"Iulamental  note  of  a  string,  an  experienced  ear  could  detect  in  its'  Sound  when  set  in  Vibration,  especially  when 

Sounds         very  "gMy  touched  in  certain  points,  other  notes,  related  to  the  fundamental  one  by  fixed  laws  of  harmony,  and 

heard  with     which  are  called,  therefore,  harmonic  sounds.    They  are  the  very  same  which,  by  the  production  of  distinct  nodes, 

the  funda-     may  be  insulated,  as  it  were,  and  cleared  from  the  confusing  effect  of   the  coexistent  Sounds,  as  in  Art.  163. 

They  are,  however,  much  more  distinct  in  bells,  and  other  sounding  bodies,  than  in  strings,  in  which  only  delicate 

ears  can  detect  them. 

167.  The  inonochord  is  an  instrument  well  adapted  to  exhibit  these  and  all  other  phenomena  of  vibrating   strings. 

The  mono-    It  is  nothing  more  than  a  single   string  of  catgut  fixed  at  one  end  immovnbly,  and  at  the  other  strained  over  a 

well-defined  edge,  which  effectually  terminates  its  vibrations,  either  by  a  known  weight  or  by  screws.     A  similar 

well-defined  edge  is  also  interposed  between  its  fixed  end  and  the  vibrating  portion,  and  the  interval  between  the 

tw:>  edges  is  graduated  into  aliquot  parts,  or  in  any  other  convenient  way,  and  it  is  provided  with   a  movable 


SOUND.  783 

Sound,      bridge,  or  piece  of  wood  capable  of  being1  placed  at  any  division  of  the  scale,  and  abutting  firmly  against  the     Part  II. 

j  Y-«_'  string  so  as  to  stop  its  vibrations,  and  divide  it  into  two  of  equal  or  unequal  lengths,  as  the  case  may  be.  v— •*/-"-' 

By  the  aid  of  this  instrument  we  may  ascertain  the  number  of  vibrations  which  belongs  to  any  assigned  musical       108. 
note,  or  which  correspond  to  the  notes  of  any  musical  instrument,  as  a  piano-forte,  &c.     For  when  we  have  ascer-  Applied  to 
tained  the  weight  of  a  known  length  of  the  catgut,  of  which  the  string  is  formed,  and  the  weight  which  must  be  "^"j^'j,* 
applied  to  stretch  the  cord,  so  as  to  make  its  fundamental  tone  coincide  with  any  given  note,  (as  the  middle  C  of  agivennote 

V  '2  rr  C 

a  piano-forte,)  then  by  the  formula  — — —  we  know  the  nutnner  of  complete  vibrations  going  and  returning,  and 


by  the  formula *JL!L  the  number  of  oscillations  from  rest  on  one  side  of  the  axis  to  rest  on  the  other,  that 

lj 

is,  the  number  of  impulses  made  on  the  ear  per  second  corresponding  to  that  fundamental  tone.  To  determine 
the  same  for  any  note  sharper,  higher,  or  more  acute  than  the  fundamental  note,  we  have  only  to  apply  the 
bridge,  and  move  it  backwards  and  forwards  till  the  sound  of  the  vibrating  part  of  the  string  is  in  unison  with  lhat 
of  the  note  to  be  compared,  of  which  the  ear  judges  with  the  greatest  precision  ;  then  if  the  length  of  this  part, 
read  off  on  the  divided  scale,  be  called  I,  the  number  of  its  vibrations  per  second  will  be  to  that  of  the  whole 
string  L  :  !  L  :  /,  and  is  therefore  known. 


of  the  doctrine  of  Acoustics  in  a  clear  point  of  view.  The  contact  of  a  solid  obstacle  is  not  the  only  means  °f  haveVmdde 
producing  them.  If  two  cords  equally  tended,  and  in  all  other  respects  similar,  but  one  only  half,  one  third,  or  of  vibration 
other  aliquot  part  of  the  length  of  the  other,  be  placed  side  by  side,  and  the  shorter  be  struck  or  sounded,  the  in  common, 
vibration  will  be  communicated  to  the  longer  by  the  intervention  of  the  air,  which  will  thus  at  once  be  thrown  into 
a  mode  of  vibration,  in  which  the  whole  length  is  divided  into  ventral  segments,  each  equal  to  the  shorter  string. 
To  understand  how  this  may  happen,  let  us  conceive  first  two  strings  of  equal  length,  one  at  rest  the  other  vibrat- 
ing, and  let  them  be  placed  parallel,  and  side  by  side,  then  the  sonorous  pulses  diverging  at  any  instant  from  each 
point  of  the  moving  string,  will  arrive  at  once  at  each  corresponding  point  of  the  other.  The  aerial  molecules  in  their 
progress,  while  condensed,  will  press  on  the  string  and  give  it  a  very  slight  motion  in  their  own  direction  ;  in  their 
retreat  they  will  be  followed  by  the  string,  whose  vibrations  by  hypothesis  are  synchronous  with  their  own,  but  it 
will  not  follow  them  so  fast  as  they  retreat,  and  it  will  be,  therefore,  urged  and  accelerated  by  those  behind.  It 
will,  however,  come  to  rest,  in  its  furthest  point  of  excursion,  at  the  same  time  with  the  aerial  molecules,  when  its 
elasticity  will  begin  to  urge  and  accelerate  it  in  the  contrary  direction.  But  now  also  the  direction  of  the  motion 
of  the  air  has  changed,  and  again  conspiring  with  that  of  the  cord  still  continues  to  accelerate  it,  and  so  on,  till, 
after  a  very  great  number  of  repetitions  of  this  process,  the  cord  will  be  set  in  full  vibration  and  will  become 
itself  a  source  of  Sound.  But  its  Sound  will  always  be  much  fainter  than  that  of  the  original  vibrating  cord,  for 
this  reason,  viz.  that  its  acquired  motion  is  perpetually  dissipated,  laterally,  into  the  surrounding  air,  for  no  cord 
is  so  exactly  uniform,  or  so  equally  tended  in  every  part  of  its  transverse  section,  that  it  can  vibrate  rigorously  in 
one  plane.  Hence  it  will  inevitably  begin  to  rotate,  or  to  describe  vibrations  whose  plane  is  continually  shifting, 
(see  Art.  177,)  and  thus  it  will  throw  off  laterally  a  great  part  of  the  motion  it  receives  from  the  air;  just  as  a 
body  exposed  to  the  radiation  of  a  hot  fire  never  acquires  a  temperature  equal  to  that  of  the  fire,  part  of  the  heat 
communicated  being  dissipated  by  lateral  radiation. 

Just  as  a  small  pull,  repeated  exactly  in  the  time  of  its  natural  swing,  will  raise    a  great  bell,  or  a  trifling       170. 
impulse  a  heavy  pendulum,  so  the  molecules  of  the  air,  in  a  state  of  sonorous  vibration,  will  impress  on  any  body  Establish. 
capable  of  vibrating  in  their  own  time  an  actual  vibratory  motion,  and  if  a  body  be  susceptible  of  a  number  of  nl     a°fa- 
modes  of  vibration  performed  in  different  times,  that  made  only  will  be  excited  which  is  synchronous  withthe  aerial  brationby 
pulsations.   All  other  motions,  though  they  may  be  excited  for  a  moment  by  one  pulsation,  will  be  extinguished  by  sympathy 
a  subsequent  one.     Hence,  if  two  cords  have  any  mode  of  vibration  in  common,  that  mode   may  be  excited  by  illustrated, 
sympathy  in  either  of  them  when  the  other  is  sounded,  and  that  only.      For  example,  if  the  length  of  one  cord  be 
to  that  of  the  other  as  2  :  3,  and  if  either  be  set  vibrating,  the  mode  of  vibration,  corresponding  to  a  division  of 
the  former  into  two,  and  of  the  latter  into  three  ventral  segments,  will,  if  it  exist  in  the  one,  be  communicated  by 
sympathy  to  the  other.     Nay,  if  it  do  not  originally  exist,  it  will,  after  a  while,  establish  itself;  for  all  accidental 
circumstances  which  may  favour  such  a  division  have  their  efftcts,  however  minute,  continually  preserved  and 
accumulated,  till  at  length  they  become  sensible. 

In  the  vibrations  of  cords,  which  from  their  small  surface  can  receive  but  a  trifling  impulse  from  the  air,  the        171, 
Sounds  and  motions  excited   by  this  sort  of  sympathetic  communication  are  feeble,  but  in  vibrating  bodies,  which  Remarkable 
present  a  large  surface,  they  become  very  great.      It  is  a  pretty  well  authenticated  feat  performed  by  persons  of  effects  of 
clear  and  powerful  voice,  to  break  a  drinking-glass  by  singing  its  proper  fundamental  note  close  to  it.     (See  s^m[ 
Chladni,  Acoust.  §  224.)     Looking-glasses  also  are  said  to  have  been  occasionally  broken  by  music,  the  excur-  tionofvibra 
sions  of  their  molecules  in  the  vibra'.ions  into  which  they  are  thrown  being  so  great  as  to  strain  them  beyond  tion 
the  limits  of  their  cohesion.  172. 

The  coincidence  of  the  theory  above  stated,  of  the  propagation  of  a  wave  along  a  stretched  cord,  with  experi-  Numerical 
ment,  has  been  put  to  careful  trial  by  Weber.  (See  his  Wdlenlehre  auf  Experimente   Gegrundet,  8vo.   Leipsig.  c«mPanso" 
1825,  p.  460,  a  most  instructive  work.)     He  stretched  a  very  equal  and  flexible  cotton  thread,  51  feet  2  inches  theorWith8 
in  length,  weighing  864  grains,  horizontally,  by  a  known  weight.     The  thread  was  struck  at  6  inches  from  one  experiment 
end  at  the  instant  of  letting  go  a  stop-watch  of  peculiar  and  delicate  construction,  marking  thirds,  (sixtieths  of  by  Weber. 


784 


SOUND. 


S.and.      seconds,)  whose  motion  was  instantaneously  arrested  when  the  wave  had  run  a  certain  number  of  times  over  the 
— •V™"'  length  of  the  string1  backwards  and  forwards.     The  mean  of  a  great  many  observations,  agreeing  well  with  each 
other,  gave  as  follows  : — 


Part  II. 


Tension  in 
Grains. 

Length  run  over 
by  the  Wave. 

Time  of  its  de- 
scription in 
Thirds. 

Time  of  running  over    The  same  ,;          , 
the  :  length  102  /.       |ated  from  the  for. 
4  in.    in     Third*,                           /  
by  observation.           nmIaV  =  V2yc. 

10023 
10023 

102/  ten. 

204     8 

46 
92 

46                          46-012 
46                           46-012 

10023 

409     4 

184 

46                          46-012 

33292 

409     4 

99 

24-72                      25-246 

69408 

409     4 

65 

16-25                     17-485 

173. 

Ocular  evi- 
dence of 
the  transi- 
tion from 
progressive 
pulses  to  a 
permanent 
state  of  vi- 
bration. 

174. 

Difference 
in  the  quali- 
ty of  the 
tone  of 
stringed  in- 
struments 
whence 
arising. 
Fig.  12. 


Phases  of  a 

vibration' 

traced. 


175. 


176. 

Vibrations 
out  of  one 
plane. 
Forms  of 
orbits  ile 
icribed  ex- 
hibited. 
Fig.  13. 

177. 


A  completer  coincidence  could  not  have  been  wished  for.  The  slight  discrepancies  may,  perhaps,  arise  from  the 
want  of  uniformity  in  the  tension  of  so  long  a  thread,  which  would,  of  course,  form  a  catenary  of  sensible  curva- 
ture. We  should  observe,  however,  that  M.  Weber  has  reckoned  here  the  weight  (864  grains)  of  the  thread  as 
part  of  the  tension,  a  proceeding  whose  legitimacy  may  be  questioned. 

The  way  in  which  the  permanent  or  vibratory  oscillations  of  a  cord  arise  by  reflexion  at  its  fixed  extremities 
from  a  wave  propagated  along  it  progressively,  may  be  rendered  a  matter  of  ocular  inspection  if  we  take  a  long 
and  pretty  thick  cord,  fasten  it  at  one  end,  and  holding  the  other  in  our  hands,  give  it  a  regular  motion  to  and 
fro,  transverse  to  the  length  of  the  cord.  Progressive  waves  will  thus  arise,  which,  as  soon  as  they  reach  the 
fixed  end,  and  are  reflected,  will  be  observed  to  interfere  with  those  still  on  their  way,  and,  as  it  were,  to  arrest 
them,  producing  a  series  of  nodes  and  ventral  segments,  whose  number  will  depend  on  the  tension  and  frequency 
of  the  alternate  motion  communicated  to  the  movable  end.  In  this  arrangement  the  continual  periodic  renewal 
of  the  primary  impulse  by  the  hand  supplies  the  place  of  a  reflecting  obstacle  at  that  end. 

The  pitch  of  the  Sound  of  a  vibrating  string  depends  only  on  the  number  of  vibrations  made  dato  tcmpore,  its 
quality  will  depend  partly  on  the  nature  of  the  string,  and  especially  on  its  equality  of  thickness,  besides  which, 
much  may  depend  on  the  form  and  extent  of  the  wave  excited,  or  of  the  curve  into  which  it  is  thrown.  In 
instruments,  like  the  violin  or  violoncello,  played  with  a  bow,  or  the  guitar  or  harp,  where  the  string  is  drawn 
softly  out  of  its  position  and  suddenly  let  go,  this  curve  is,  probably,  single,  and  occupies  the  whole  length  of  the 
string ;  but  in  the  piano-forte,  where  the  strings  are  struck,  near  one  extremity,  with  a  sharp  sudden  blow,  there 
can  be  little  doubt  that  the  vibration  consists  in  an  elevation  or  bulge,  more  or  less  extensive,  running  backwards 
and  forwards.  Fig.  12  represents  the  different  phases  of  a  single  complete  vibration  of  a  string  so  struck.  The 
first  wave  (1)  is  a  single  elevation,  it  divides  in  (2)  into  two  running  contrary  ways  ;  in  (3)  that  nearest  the  end 
A  is  reflected  and  takes  a  reversed  position  ;  in  (4)  they  advance  the  same  way  towards  B  ;  in  (5)  the  unreflected 
portion  reaches  B,  is  there  reflected  and  reversed,  as  in  (6).  In  (7)  it  meets  and  coincides  with  the  former  reflected 
portion,  there  forming  a  depression  equal  and  similar  to  the  original  elevation  in  (1),  and  as  far  distant  from  the 
end  B  as  the  former  from  A.  After  this  the  same  steps  are  repeated  in  the  reverse  direction,  till  the  original 
elevation  is  reproduced  again,  as  in  (1).  The  waves,  however,  must  be  supposed  to  bear  a  much  more  consi- 
derable ratio  to  the  whole  string  than  in  the  figure.  It  is  evident  that  the  magnitude  of  this  ratio  must  influence 
the  quality  of  the  tone,  and  thus  a  difference  of  character  in  the  tone,  according  as  the  keys  are  struck  with  quick 
short  brilliant  blows,  or  gently  pressed,  and  the  duration  of  the  contact  of  the  hammers  with  the  strings  prolonged 
for  an  instant  of  time,  giving  rise  to  a  more  moderate  but  sustained  tenuto  effect,  by  bringing  a  larger  portion  of 
the  string,  or  even  the  whole  into  motion  at  once. 

But  whether  the  portion  disturbed  at  once  be  large  or  small,  whether  it  occupy  the  whole  string,  or  run  along  it 
like  a  bulge  in  its  line,  whether  it  be  a  single  curve,  or  composed  of  several  ventral  segments  with  intervening 
nodes,  we  must  never  lose  sight  of  the  fact  that  the  motion  of  a  string  with  fixed  ends  is  no  other  than  an  undula- 
tion <;r  pulse  continually  doubled  back  on  itself  and.  retained  constantly  within  the  limits  of  the  cord,  instead  of 
running  out  both  ways  to  infinity. 

It  very  seldom  (for  the  reasons  mentioned  in  Art.  170)  can  happen  that  the  vibrations  of  a  string  actually  lie  in 
one  plane.  Most  commonly  they  consist  of  rotations  more  or  less  complicated,  except  when  produced  by  the 
sawing  of  a  bow  across  the  string,  when  they  are  forcibly  limited  to  the  plane  of  motion  of  the  bow.  The  real 
form  of  the  orbit  described  by  any  molecule  may  be  made  matter  of  ocular  inspection,  by  letting  the  sun  shine 
through  a  narrow  slit  so  as  to  form  a  thin  sheet  of  light.  Let  a  polished  wire  be  placed  so  as  to  penetrate  this 
sheet  perpendicularly  to  its  plane,  and  the  point  where  it  cuts  the  plane  will,  at  rest,  be  seen  as  a  bright  speck,  but 
when  set  vibrating  it  will  form  a  continued  luminous  orbit,  just  as  a  live  coal  whirled  round  appears  as  a  circle 
of  fire.  Fig.  13  exhibits  specimens  of  such  orbits,  observed  by  Dr.  Young. 

A  very  curious  case  of  a  mode  of  vibration,  by  which  a  string  may  be  made  to  produce  a  Sound  graver  than 
its  fundamental  tone,  is  mentioned  by  M.  Biot.  If  an  obstacle  be  placed  below  the  middle  point  of  a 
vibrating  string  so  as  just  to  touch,  but  not  to  press  against  it,  and  the  string  be  then  drawn  up  vertically  and 
let  go,  it  will  strike  at  every  oscillation  upon  this  obstacle,  and  bend  over  it,  as  in  fig.  14,  at  every  blow  ;  thus 
resolving  itself  into  two,  of  half  the  length.  Thus  the  first  semi-oscillation  will  be  performed  as  a  whole,  the 


SOUND.  785 

Sound,      next  as  a  subdivided  string.     Let  unity  represent  the  time  of  one  complete  oscillation  from  rest  to  rest  of  the      Part  II. 
— v^^  whole  string ;    then  will  the  times  in  which  the  different  phases  of  the  motion  now  in  question  are   performed  v""v^'' 
be  as  follows  : 

From  the  position  A  B  C  to  the  straight  line  AC....  =  £. 

From  the  position  A  C  to  the  position  A  E  D,  D  F  C  =  £. 

Back  to  the  straight  line    —  i- 

Back  to  the  original  position  ABC =  J. 

Sum  =  J  +  £+i  +  i  =  3. 

Thus  the  interval  between  two  consecutive  blows  made  by  the  string  on  the  bridge  is  ^  of  the  time  of  oscillation 
of  the  string  as  a  whole,  from  rest  on  one  side  of  the  axis  to  rest  on  the  other,  or  of  the  impulses  made  by  it  on 
the  ear  when  so  vibrating.  Hence,  the  blows  on  the  bridge  will  be  heard  as  a  continued  note,  (though  extremely 
harsh  and  disagreeable,)  graver  than  that  of  the  string  vibrating  as  a  whole,  by  the  musical  interval  called  a 
fifth.  (See  Index,  Musical  Intervals.) 

§  III.     Of  the  Vibrations  of  a  Column  of  Air  of  Definite  Length. 

The  general  equation  representing  the  motions  of  the   molecules  of  a  tended  cord  of  indefinite  length  is,  as       |^g 
we  have  seen,  precisely  similar  in  its  form,  and  in  that  of  its  complete  integral,  to  that  of  the  particles  of  air  in  a  Analogy  be- 
sounding  column.    There  subsists>  of  course,  a  perfect  analogy  between  the  two  cases,  and,  mutatis  mutandis,  all  tweenvib™- 
propositions  which  arc  true  of  a  vibrating  cord  are  also  true  of  a  vibrating  cylindrical  volume  of  air.  ti.msof  air 

Thus,  if  such  a  cylindrical  column  be  enclosed  in  a  pipe,  whose  length  =  I  +  I'  =  L,  stopped  at  both  ends  by  '°»P'PJ»n»l 
perfectly  immovable  stoppers,  and  if  we  suppose  any  single  impulse  communicated  to  one  of  its  sections  at  the  stretched* 
distance  /  from  one  of  its  extremities  (A),  this  will  immediately  divide  itself  into  two  pulses  running   opposite  cord, 
ways;  they  will  be  totally  reflected  at  the  two  extremities,  the  one,  after  describing  the  space  I  before  and  I'  after       179. 
reflexion,  will  meet  the  other  which  has  described  V  before  and  I  after  reflexion,  at  a  distance  =  I  from  the  other  Transition 
extremity  B,  and  produce  a  compound  agitation  in  the  section  at  that  place  similar  to  the  primitive  disturbance ;  from  a  Prn" 
thence  the  partial  pulses  will  again  diverge,  and  after  each  undergoing  another  reflexion  will  again  unite  in  their  Pufsae'et'u  , 
original  point  of  departure,  constituting  a  repetition  of  the  first  impulse,  and  so  on,  till  the  motion  is  destroyed  permanent 
by  friction  and  by  the  imperfect  fixity  and  rigidity  of  the  stoppers,  allowing  some  of  it  to  pass  into  them  and  be  vibration. 
lost  at  each  reflexion. 

But  if  the  section  first  set  in  motion  be  maintained  in  a  state  of  vibration  synchronous  with  the  return  of  the       ISO. 
reflected  pulse,  it  will  unite  with  and  reinforce  it  at  every  return,  and  the  result  will  be  a  clear  and  strong  musical  Effect  of  H 
Sound  resulting  from  the  exact  combination  of  the   original  periodic  impulse  with  all  its  echos.     This  will  be  permanent 
transmitted  through  the  pipe  to  the  outer  air,  and  thus  dissipated  and  lost. 

For  simplicity,  let  us  suppose  the  section  primitively  set  in  vibration  and  so  maintained,  to  be  situated  just  in  „"„[!*  "' 
the  middle  of  the  pipe.     Then,  when  once  the  regular  periodic  pulsation  of  the  contained  air  is  established,  it  is       \$\ 
evident  that  the  motion  of  the  column  will  consist  of  a  constant  and  regular  fluctuation  to  and  fro  within  the  pipe  Simplest 
of  the  whole  mass,  the  air  being  always  condensed  in  one  half  of  the  pipe  while  it  is  rarefied  in  the  other.     The  mocle  of  vi- 
greatest  excursions  from  their  place  will  be  made  by  the  molecules  in  the  middle,  while  those  at  the  extremities,  b'atlon  of 
being  constantly  abutted  against  the  stoppers,  remain  unmoved,  and  the  excursions  made  by  each  intermediate  piL^iosed 
molecule  will  be  greater  the  nearer  it  is  to  the  middle.     On  the  other  hand,  the  rarefactions  and  condensations  at  both 
are  greatest  at  the  extremities,  and  diminish  as  we  approach  the  middle  of  the  pipe,  where  there  is  neither  conden-  ends. 
sation  nor  rarefaction.     The  analogy  of  this  ease  with  the  case  of  the  vibrating  cord  will  be  evident  if  we  consider  Anal°gy 
that  the  condensation  in  the  former  is  represented  by  the  angle  of  inclination  of  the  vibrating  curve  to  its  abscissa 
in  the  latter,  and  that  the  mode  of  vibration  now  contemplated  in  the  aerial  molecules  is  analogous  to  that  of  a  [ng^aseofa 
cord  vibrating  as  a  whole,  and  having  its  two  halves  symmetrical.  stretched 

In  the  same  way  as  a  vibrating  cord  is  susceptible  of  division  into  its  several  aliquot  parts  all  vibrating  simul-  cortl- 
taneously,  so  may  the  aerial  column  in  our  stopped  pipe  vibrate  in  distinct  ventral  segments.     The  manner  in       !%-• 
which  this  may  take  place  will  be  evident  on  inspection  of  figs.  15  and  16,  where  the  arrows  denote  the  directions  of  Sul)division 
the  motions  of  the  vibrating  molecules,  and  where  we  see  the  immobility  of  the  nodal  sections  is  secured  by  the  "„  Vo'lum 
equal  and  opposite  pressures  of  the  molecules  on  either  side  of  them.     At  these  nodal  sections,  too,  the  same  by  nodei." 
thing  holds  good  as  at  the  stopped  extremities,  their  molecules  remain  constantly  at  rest  while  yet  they  undergo  Figs.  15, 16. 
greater  vicissitudes  of  compression  and  dilatation  than  those  in  any  other  parts  of  the  column. 

Precisely,  too,  as  in  the  vibrations  of  strings,  any  number  of  these  modes  of  vibration  may  go  on  simultaneously.       183. 
Such  combined  modes  may  be  produced  by  an  expert  flute  player,  by  a  nice  adjustment  of  the  force  of  his  breath  ;  Coexistem 
at  least  the  octave  of  any  note  may  be  obtained  without  difficulty,  and  distinctly  heard  with  the  fundamental  tone.  °f  several 

Half  way  between  two  nodes  (regarding  the  stopped  ends  as  nodes)  the  condensations  and  rarefactions  are  m°dei!  ot 
evanescent,  and  the  amplitudes  of  the  molecular  excursions  are  at  a  maximum.     Now  at  such  a  point  let  us  V'  ?gj 
conceive  a  narrow  ring  of  the  cylindrical  pipe  in  which  the  vibrating  column  is  contained  to  be  cut  away,  so  as  to  vibration  of 
open  a  free  communication  with  the  outer  air.     There  will  be  no  tendency  for  air  to  pass  in  or  out,  because  the  air  in  a  pip« 
air  within  is  constantly,  at  these  points,  in  its  natural  state  as  to  density;  neither  will  its  motion  be  impeded,  "pen  at  o» 
being  parallel  to  the  axis  of  the  column  and  without  any  lateral  bias.     The  detachment  then  of  such  a  ring  will  end- 
no  way  alter  the  vibrations  of  the  column,  nor,  a  fortiori,  will  the  opening  of  a  hole  in  the  pipe  at  this  place  affect 


786  SOUND. 

Sound.      them.     Suppose,  now,  at  this  hole,  a  vibrating  body  placed,  whose  vibrations  are  executed  in  equal  times  with     Part  tl. 
^•"v-^  those  in  which  the  excursions  to  and  fro  of  the  included  aerial  sections  are  performed  in  the  stopped  pipe.     They   ^-••v* 
will  be  communicated  to  them,  and  thus  the  Sound  of  the  pipe  will  be  excited  and  maintained.    Such  an  aperture 
is  called  an  embouchure. 

185.  But  let  us  now  conceive  the  one  half  (A)  of  the  pipe  entirely  removed,  and  in  its  place  a  disc  substituted  exactly 
Case  where  closing  the  aperture,  and  maintained,  by  some  external  cause,   constantly  in  a  state  of  vibration,  such,  that  the 
the  vibra-     performance  of  one  complete  vibration,  going  and   returning-,  shall  exactly  occupy   as  much  time  as  a  sonorous 
-ited"6""  Pu'se  would  take  to  traverse  the  whole  length  of  the  stopped  pipe  (A  +  B),  or  double  that  of  the  open  one  (B). 
vibrating       Its  first  impulse  on  the  air  will  be  propagated  along-  the  pipe  (B)  and  reflected  at  the  stopped  end,  and  will  again 
disc.            reach  the  disc  just  at  the  moment  when  the  latter  is  commencing  its  second  impulse.     But  the"  absolute  velocity 

of  the  disc  in  its  vibrations  being1  excessively  minute  compared  with  that  of  Sound,  the  reflected  pulse  will 
undergo  a  second  reflexion  at  the  disc  as  if  it  were  a  fixed  stopper.  It  will,  therefore,  in  its  return  exactly 
coincide  and  conspire  with  the  second  original  impulse  of  the  disc,  and  the  same  process  being  repeated  on  every 
impulse,  each  will  be  combined  with  all  its  echos,  and  a  musical  tone  will  be  drawn  forth  from  the  pipe  vastly 
superior  to  that  which  the  disc  vibrating-  alone  in  free  air  would  produce.  This  is,  in  fact,  the  simplest  instance 
of  the  resonance  of  a  cavity,  of  which  more  hereafter.  (See  Index,  Resonance.}  Now,  it  is  manifestly  of  no 
importance  whether  the  pulses  reflected  from  the  closed  end  of  the  pipe  (B)  undergo  a  second  reflexion  at  the  disc, 
and  are  so  returned  back  by  the  pipe,  or  whether  we  regard  the  disc  as  penetrable  by  the  pulse,  (i.  e.  a  mere 
imaginary  vibrating  section,)  and  suppose  the  pulse  to  run  on  and  be  reflected  at  the  extremity  of  the  other  half 
(A)  of  the  bisected  pipe  (A-j-  B),  and  on  its  return  again  to  pass  freely  through  the  disc  and  be  again  reflected 
at  the  stopped  extremity  of  (B).  The  Sounds  produced  will  be  the  same,  on  the  principle  of  the  superposition  of 
vibrations.  Thus  we  see  that  the  fundamental  Sound  of  a  pipe  open  at  one  end  is  the  same  with  that  of  a  pipe 
closed  at  both  ends,  and  of  double  the  length. 

186.  The  mode  here  supposed  of  exciting  and  maintaining  the  vibrations  of  a  column  of  air  in  a  pipe  is  easily  put  in 
Mow  per-     practice.     Let  any  one  take  a  common  tuning-fork  and  on  one  of  its  branches  fasten  with  sealing-wax  a  circular 

m  disc  of  card  of  the  size  of  a  small  wafer,  or  sufficient  nearly  to  cover  the  aperture  of  a  pipe.  The  sliding  joint  of 
the  upper  end  of  a  flute,  with  the  mouth-hole  stopped,  is  very  fit  for  the  purpose;  it  maybe  tuned  in  unison  with 
the  loaded  tuning-fork  (a  C  fork)  by  means  of  the  movable  stopper,  or  the  fork  may  be  loaded  till  the  unison  is 
perfect.  If  the  fork  be  then  set  in  vibration  by  a  blow  on  the  unloaded  branch,  and  the  disc  be  held  close  over 
Fig.  16.  tne  mouth  of  the  pipe,  as  in  fig.  16,  a  note  of  surprising  clearness  and  strength  will  be  heard.  Indeed,  a  flute 
may  be  made  to  "speak"  perfectly  well  by  holding  close  to  the  embouchure  a  vibrating  tuning-fork  while  the 
fingering  proper  to  the  note  of  the  fork  is  at  the  same  time  performed.  We  shall  have  further  occasion  to  refer 
to  this  point.  (Resonance,  Index.) 

187.  But  the  most  usual  means  of  exciting  the  vibrations  of  a  column  of  air  in  a  pipe  is  by  blowing  into,  or  rather 
Action  of  a   over  it,  either  at  its   open  end  or  at  an  orifice  made  for  the  purpose  at  the  side,  or  by  introducing  a  small 
reed.            current  of  air  into  it  through  an  aperture  of  a  peculiar  construction  called  a  reed,  provided  with  a  "  tongue" 

or  flexible  elastic  plate  which  nearly  stops  the  aperture,  and  which  is  alternately  forced  away  by  the  current  of 
air  and  returns  by  its  elasticity,  thus  producing  a  continual  and  regularly  periodic  series  of  interruptions  to  the 
uniformity  of  the  stream,  and,  of  course,  a  Sound  in  the  pipe  corresponding  to  their  frequency.  Except,  however, 
the  reed  be  so  constructed  as  to  be  capable  of  vibrating  in  unison,  or  nearly  so,  with,  at  least,  one  of  the  moilc^ 
of  vibration  of  the  column  of  air  in  the  pipe,  the  Sound  of  the  reed  only  will  be  heard,  the  resonance  of  the 
pipe  will  not  be  called  into  play,  and  the  pipe  will  not  speak;  or  will  speak  but  feebly  and  imperfectly,  and  yield 
a  false  tone. 

188.  But  of  reeds  more  hereafter;  (see  Index,  Reeds ;)  at  present  let  us  consider  what  takes  place  when  the  vibrations 
Excitement  of  a  column  of  air  are  excited  by  blowing  over  the  open  end  of  a  pipe  or  an  aperture  in  its  side.     To  do  it  effec- 
of vibrations  tually  the  air  must  be  directed  in  a  small  current,  not  into,  but  across  the  aperture,  as  in  fig.  18,  so  as   to  graze 

I  the  opposite  edge.  By  this  means  a  small  portion  will  be  caught  and  turned  aside  down  the  pipe,  thus 
"  giving  a  first  impulse  to  the  contained  air  and  propagating  down  it  a  pulse  in  which  the  air  is  slightly  condensed. 
Fig.  18.  This  will  be  reflected  at  the  end  as  an  echo,  and  return  to  the  aperture  where  the  condensation  goes  off,  the 
section  condensed  expanding  into  the  free  atmosphere.  But  in  so  doing,  it  lifts  up,  as  it  were,  and  for  a 
moment  diverts  from  its  course  the  impinging  current,  and  thus,  while  it  passes,  suspends  its  impulse  on  the  edge 
of  the  aperture.  The  moment  it  has  escaped  the  current  resumes  its  former  course,  again  touches  the  edge  of 
the  aperture,  creates  there  a  condensation,  and  propagates  downwards  another  condensed  pulse,  and  so  on.  Thus 
the  current  passing  over  the  aperture  is  kept  in  a  constant  state  of  fluttering  agitation,  alternately  grazing  and 
passing  free  of  its  edge,  at  regular  intervals,  equal  to  those  in  which  a  sonorous  pulse  can  run  over  twice  the 
length  of  the  pipe ;  or,  more  generally,  in  which  the  condensations  and  rarefactions  recur  at  its  aperture  in  virtue 
of  any  of  the  modes  of  vibration  of  which  the  column  of  air  in  the  pipe  is  susceptible. 

189.  In  general,  wherever  there  is  a  communication  opened  between  the  column  of  air  in  a  pipe  and  the  free  atmo- 
Practical       sphere,  that  point  will  become  a  point  of  maximum   excursion  of  the  vibrating  molecules,  or  the  middle  of  a 
betvvee'n'vi-  ventra'  segment.     In  such  a  point  the  rarefactions  and  condensations  vanish,  the  air  reducing  itself  constantly 
brations  of    to  an  equilibrium  of  pressure  with  the  free  atmosihere  with  which  it  is  in  contact.     Hence,  if  the  pipe  speak  at 
air  in  an       all,  it  will  take  such  a  mode  of  vibration  as  to  satisfy  this  condition,  but,  consistently  with  this,  it  may  divide 
open  pipe     itself  into  any  number  of  ventral  segments.    But  here  there  is  a  point  of  practical  difference  between  the  affections 

"stretched0*'  °^a  v'*)rat'"g'  aerial  column  and  those  of  a  tended  cord.     The  tension  of  the  cord  can  only  be  maintained  steadily 

cord.        *    ln  practice,  by  fixing  its  two  ends  ;  so  that  the  case  of  one  extremity  fixed,  the  other  free,  can  have  no  existence 

but  in  imagination,  where  the  cord  may  be  conceived  as  of  indefinite  length  in  one  direction,  so  that  the  out- 

-unning  pulses  may  lose  themselves,  or,  "at  least,  never  return.     It  is  true  they  might  be  stifled  by  wrapping  one 


SOUND.  787 

Sounu.      end  of  a  very  long  cord  in  cotton,  but  whether,  under  such  circumstances,  any  mode  of  producing  and  maintaining     Part  H. 
— v-^-'  an   initial  periodical   impulse  sufficiently  regular   to  produce  musical   Sounds   could  be  found  remains   to  be  v"— "v™"-" 
tried.     The  nearest  approach  to  the  case  in  question  is  when  one  end  of  a  long  cord  is  held  in  the  hand  and 
agitated  while  the  tension  is  maintained,  as  in  Art.  173. 

In  cords  with  fixed  extremities,  however,  all  the  ventral  segments  must,  of  necessity,  be  complete,  no  half      190. 
segments  can  exist.     In  pipes  it  is  otherwise.     The  air  in  a  pipe  closed  at  one  end  vibrates  as  a  half,  not  the  T'mes  and 
whole  of  such  a  segment.     It  is  owing  to  this  that  a  pipe  open  at  both  ends  can  yield,  if  properly  excited,  a'"?     .  °  . 
musical  Sound.     The  column  of  air  in  it  vibrates  in  the  mode  represented  in  fig.  19,  where  there  is  a  node  in  the  pipesopen" 
middle  and  each  ventral  segment  is  only  half  a  complete  one.     In  general  it  is  easy  to  represent,  in  an  algebraic  a!  both  ends 
formula,  the  time  of  vibration,  or  the  number  of  vibrations  per  second  corresponding  to  any  mode  of  vibration.  V'S- 19- 
For,  first  in  a  pipe  open  at  both  ends  let  the  number  of  nodes  be  n,  then  there  will  be  n  —  1   complete  ventral 
segments  between  them,  as  in  fig.  20,  and  a  moiety  of  one  at  each  end.     If,  then,  we  call  L  the  whole  length  of  F'S-  20- 
such  a  pipe  in  feet,  V  the  velocity  of  Sound  in  feet  per  second,  the  length  of  one  complete  ventral  segment  will 

be  —  .     This  length  is  traversed  by  a  sonorous  pulse  in  a  time  —  .  rr ,  and  this  is  the  time  of  vibration  of  the 
n  n     V 

middle  section  of  it  to  which  the  Sound   corresponds.     The  pipe,  then,  vibrating  according  to  this  mode,  will 

V 

yield  a  Sound  whose  pitch  is  that  of  a  cord  making  n  .  -=—  vibrations  per  second  ;  and  the  series  of  tones  it  can 

L 

produce  is  expressed  by  the  following  series  of  numbers  of  vibrations, 

V  V  V 

!-T;      2'T;      3T'&C' 

In  the  case  of  a  pipe  closed  at  one  end,  the  stopped  end  must  be  regarded  as  a  node.   (Fig.  21.)     Calling  the       191, 
whole  number  of  nodes,  thus  included,  n,  the  number  of  complete  ventral  segments  will  be  n  —  1,  and  one  half  In  pipes 

j^  j^  closed  at 

segment  will  terminate  at   the  open   end.      Therefore  ,    or  — —  ,  will  be  the  number  of one  end- 

2  (n  —  1)  +  1  271+1  Fig.  21. 

O  T 

such  halves  contained  in  the  length  L,  and  — will,  therefore,  be  the  length  of  each  complete  one ;   so 

&  71  ~(~    1 

that  each  will  make .  —   vibrations  in  one  second,  and  thus  the  series  of  tones  such  a  pipe  can  yield 

will  be  expressed  by  the  series  of  vibrations, 

IV  3V  5V 

— *  --TT.   •  —  --,   -  Rrr* 

2  '    L  '          2   '    L  '          2    '    L  '  & 

Lastly,  the  number   of  nodes,  including  the  two  stopped  ends  of  a  pipe  closed  at  both  ends,  being  n,  the       192. 
number  of  segments  (all  complete)  into  which  it  will  be  divided  will  be  n  —  1,  and  the  length  of  each  will  be  In  pipes 
j  U  closed  at 

—    .    —  ;    so  that  the  series  of  Sounds,  of  which  such  a  pipe  is  susceptible,  is  represented  by  the  series  both  ends- 

of  vibrations, 

V  V  V 

I-T'    2-r;    3-T'&c- 

Taking,  therefore,  unity  for  the  number  of  vibrations  per  second  in  the  fundamental  tone,  the  series  of  harmonics 
will  run  as  follows  : 

In  a  pipe  stopped  at  both  ends 1,  2,  3,  4,  5,  &c 

—  open  at  both  ends 1,  2,  3,  4,  5,  &c. 

—  stopped  at  one  end,  open  at  the  other.  ...    1,  3,  5,  7,  9,  &c. 

It  being  recollected,  however,  that  in  the  last  series  the  fundamental  note  1  is  an  octave  lower  than  in  the  others, 
i.e.  performs  its  vibrations  only  half  as  rapidly. 

To  produce  these  Sounds  by  blowing  into  a  pipe,  it  is  only  requisite  to  begin  with  as  gentle  a  blast  as  will       193 
make  the  pipe  speak,  and  to  augment  its  force  gradually.     The  fundamental  tone  will  be  heard  first;   and  as  Method 'o. 
the  strength  of  the  blast  increases,  will  grow  louder,  till  at  length  the  tone  all  at  once  starts  up  an  octave,  i.  e.  exciting 
the  interval  between  notes  whose  vibrations  are  as  1  :  2.     By  blowing  still  harder,  the  next  harmonic,  1  :  3,  or  tllese  dif~ 
as  it  is  called  in  Music,  the  octave  of  the  fifth,  or  the  twelfth  of  the  fundamental  tone,  is  heard  ;  but  no  adap-  ^e'    f 
tation  of  the  embouchure,  or  force  of  the  wind,  will  produce  any  note  intermediate  between  these.     The  next  vibration 
harmonic  is  1  :  4,  and  corresponds  to  the  double  octave,  or  fifteenth  of  the  fundamental  tone;   and  the  next,  or 
1  ;  5,  to  the  seventeenth,  or  major  third  above   the  double  octave.     (See  the  explanation  of  these  terms  in 
Art.  210,  et  seq.)     The  next,  1  :  6,  corresponds  to  the  nineteenth,  or  double  octave  of  the  fifth,  and  so  on.     All  the  Senes  of 
notes  here  enumerated  are  very  readily  produced  on  the  flute,  without  changing  the  fingering,  from  the  lower  C  harmonic 
or  D  upwards,  by  merely  varying  the  force  of  the  blast,  and  a  little  humouring  the  form  of  the  lips  and  their  (°nes  of  • 
position  with  respect  to  the  embouchure.     The  reader  may  consult  on  this  subject  D.  Bernoulli!,  Sur  le  Son  et  E'pe  or 
sur  les  TOJIS  des  Tvyaux  d'Orgiies,  Mem.  Acad.  Paris,  1762  ;   in  which   the  true  theory  of  wind-instruments  is    "'*' 

VOL.   IV.  5  K 


788 


SOUND. 


194. 

Ration 
the  action 


inonic 
sounds. 


195. 


Sound,      first  clearly  stated,  though  pointed  out  by  Sauveur,  in  a  Paper  published  in  the  Mem.  de  VAcad.  1701.      Also  a     Part  I', 
*V*P'  Very  instructive  Paper  by  Lambert,   Observations  sur  les  Fifties,  Mem.  Acad.  Berl.  1775.     (See  also  Euler,  De  — —  v— 
Motu  Aeris  in  Titbis  Petrop.  Comm.  xvi.   and  Lagrange's  Memoirs  already  cited,  Mel.  de  Turin,  i.   and  ii.) 
M.  Biot,  Hv  adapting  an  organ  bellows  to  regulate   the  blast,  and  give  it  the   requisite  force  and  uniformity, 
succeeded  in  drawing  from  a  pipe  furnished  with  a  proper  embouchure,  not  only  these,   but  also   the   notes 
represented  in  the  harmonic  series  by  7,  8,  11,  and  12,  but  not  9  or  10,  (the  reason  of  which  vacancy  does  not 
appear.)    Traits  de  Physique,  ii.  126. 

The  rationale  of  the  continual  subdivision  of  the  vibrating  column,  as  the  force  of  the  blast  increases,  is  very 
obvious.     A  quick  sharp  current  of  air  is  not  so  easily  driven  aside  by  an  external  disturbing  force  ;   and  when 
ie  embou-  so  driven,  returns  more  rapidly  to  its  original  course,  than  a  slow  and  feeble  one.     A  quick  stream,  when  thrown 
chure  in  ex-  into  a  ripple  by  an  obstacle,  undulates  more  rapidly  than  a  slow  one.     Consequently,  on  increasinj;  the  force  of 
citing  bar-    the  blast,  a  period  will  arrive  when  the  current  cannot  be  diverted  from  its  course  and  return  to  it  so  slowly  as  is 
required  for  the  production  of  the  fundamental  note.     The  next  higher  harmonic  will  then  he  excited,  until,  the 
force  of  the  blast  increasing,  it  becomes  once  more  incapable  of  sympathizing  with  the  excursions  of  the  aerial 
molecules  at  the  embouchure  in  this  mode  of  vibration,  and  so  on. 

If  we  know  the  velocity  of  Sound  in  the  column  of  air  included  in  a  pipe,  the  length  of  the  pipe,  and  the 
Velocity  of  mode  of  vibration,  the  number  of  vibrations  may  be  computed,  and  vice,  versa,  if  we  know  the  number  of  vibrations 
i",1™.  con-  made  in  a  given  pipe,  vibrating  in  a  known  manner,  we  may  thence  compute  the  velocity  of  Sound.     This 
the  note™"1  furnishes  a  ready  and  simple  method  of  determining  the  velocity  of  Sound  in  any  gas  or  vapour.     We  have  only 
sounded  by  to  fill  a  pipe  first  with  air,  and  then  with  the  gas  or  vapour  in  question ;   and  having  set  them  vibrating  by  any 
agivcnpipe.  proper  means,  so   as  to  draw  forth  their  fundamental  tone,  to  compare  this  with  a  monochord,  or  with  any 
Velocity  of   mllsical  instrument  possessing  a  regular  scale  or  progression  of  notes  where  vibrations  are  known;  and  having 
s^thence  t'lus  ascerta>ne<l  the  number  of  vibrations  per  second  performed  by  a  column  of  each  medium,  the  velocities  of 
deduced.       Sound  in  the  respective  cases  will  be  in  the  direct  ratio  of  their  numbers.     It  is  thus  that  Chladni,  and,  more 
lately,  Vanrees,  Frameyer,  and  Moll  have  ascertained  the  velocities  of  Sound  in  the  various  media  enumerated 
in  Art.  82. 

196.  That  it  is  really  the  air  which  is  the  sounding  body  in  a  flute,  organ-pipe,  or  other  wind-instrument,  appears 

The  column  from  the  fact,  that  the  materials,  thickness,  or  other  peculiarities  of  the  pipe,  are  of  no  consequence.  A  pipe  of 
of  air,  not  paper  an(j  one  of  lead,  glass,  or  wood,  provided  the  dimensions  be  the  same,  produce,  under  similar  circum- 
the  sound'-  stances,  exactly  the  same  tone  as  to  pitch.  If  further  proof  were  necessary,  the  difference  of  pitch  produced  by 
filling  the  pipe  with  different  gases  would  place  the  point  beyond  a  doubt.  If  the  qualities  of  the  tones  produced 
by  different  pipes  differ,  this  is  to  be  attributed  to  the  friction  of  the  air  within  them,  setting  in  feeble  vibration 
their  own  proper  materials. 

The  influence  of  the  size  and  situation  of  the  embouchure  of  a  pipe,  and  still  more  of  the  manner  of  exciting 
the  vibrations  of  the  sections  of  the  aerial  column  near  that  place,  are  very  material  in  determining  the  pitch  of 
hure  of tne  *one  "ttered.  Were  it  possible  to  excite  the  aerial  column  to  vibration  by  setting  in  motion  a  single  section 
a  pipe  on  its  of  it  by  a  wish,  we  should  obtain,  doubtless,  Sounds  always  strictly  conformable  to  the  length  of  the  pipe  and 
pitch.  its  harmonic  subdivisions  as  above  ;  but,  in  fact,  the  vibrating  column  of  air  and  the  extraneous  body  (be  it 

reed,  tuning-fork,  or  stream  of  air)  which  sets  it  in  motion  exercise  on  each  other  a  mutual  influence  ;  they  vibrate 
as  a  system,  (see  Index,  Vibrations  of  a  System  of  Bodies,)  and  the  resulting  tone  may  be  made  to  deviate  more 
or  less  from  the  pure  fundamental  tone  of  the  pipe,  according  to  the  greater  or  less  mass  of  matter,  and  the  fixity 
of  the  vibrations  of  the  apparatus  by  which  the  pipe  is  made  to  speak.  When,  for  example,  the  cause  of  vibration 
is  the  mere  passage  of  a  stream  of  air  over  the  orifice,  whose  motions  are  almost  entirely  commanded  by  the 
condensations  and  rarefactions  within  the  pipe,  (Art.  194,)  but  little  deviation  can  take  place.  Yet,  by  varying 
the  inclination  of  the  stream,  (as  in  the  case  of  the  flute  by  turning  the  mouth-hole  more  inwards  or  outwards 
with  respect  to  the  lips,)  and  thus  giving  it  a  greater  or  less  obliquity  to  the  edge  against  which  it  strikes,  we 
may  alter  the  note  very  sensibly,  as  is  known  to  all  flute-players,  who  use  this  means  of  humouring  the  instru- 
ment, and  playing  in  tune  in  keys  which  would  otherwise  be  insupportable. 

198.  In  the  diapason  organ-pipe,  whether  open  or  stopped,  a  stream  of  air  is  admitted  at  the  vertex  of  the  conical 
Diapason       lower  end,  but  is  prevented  from  passing  through  the  whole  length  of  the  pipe  by  a  plate  of  metal  separating  the 
organ-pipe.    eone  from  the  pipe,  and  is  forced  to  escape  through  a  narrow  slit  transverse  to  the  axis  of  the  pipe,  in  doing 

which  it  strikes   against  the  edge  of  a  thin  piece  of  lead,  or  other  flexible  metal.     This  disposition   will   be 

Fig.  22.        understood  by  inspection  of  the  figure  22,  in  which  B  B  is  the  organ-pipe,  and  bob  the  conical  appendage  at  its 

foot  by  which  the  air  is  admitted.     One  side  of  the  pipe  B  L  M  is  flattened  and  a  little  bent  inwards,  and  at  L 

a  narrow  slit  is  made,  just  opposite  to  the  lower  edge  of  which  is  the  plate  of  metal   b  b,  which   has   its  edge 

nearest  the  orifice  a  little  cut  away,  so  as  not  quite  to  fill  the  whole  section  of  the  pipe,  but  to  leave  a  narrow 

slit  parallel  to  the  slit  FF  in  the  side  of  the  pipe.     Through   this  the  air  admitted  at  c  escapes,  and  is  directed 

in  a  thin  sheet  against  the  upper  lip  L  of  the  lateral  slit ;   against  which  it  breaks,  as  described  in  Art.  188,  and 

Its  harmo-    sets  in  vibration  the  column  of  air  contained  in  the  pipe.     If  the  stream  of  air  be  too  strong  the  pipe  will  yield 

rics-  the  octave  and  harmonic  of  its  fundamental  note,  forming  the  series  1,  2,  3,  4,  &c.     If,  on  the   other  hand,  the 

current  of  air  remaining  constant,  the  breadth  of  the  slit  through  which  the  air  escapes  be  diminished,  according 

to  the  experiments  of  MM.  Biot  and  Hamel,  harmonics  will  also  be  produced,  but  in  the  progression  1,  3,  5,  7, 

&c.  the  octaves  of  the  fundamental  tone  and  of  all  the  others  being  entirely  wanting. 

199.  In  reed-pipes,  or  those  in  which   the  vibrations  are  excited   and  maintained  by  passing  a  current  of  air  into 
Reed-pipes.  the  pjpe  through  a  reed,  (Art.  187,)  the  influence  of  the  reed  on  the  pipe  is  very  great.     The  most  perfect  and 

pure  tone  is  produced,  of  course,  when  the  reed  and  the  pipe  separately  are  pitched  in  unison,  but  a  considerable 
latitude  in  this  respect  exists  ;    and  within  certain  limits,  depending  on  the  mass  and  stiffness  of  the  reed,  as 


ing  body. 


197. 

Influence 
of  the  em- 


SOUND.  789 

Sound,      compared  with  the  dimensions  of  the  pipe,  a  power  of  mutual  accommodation  subsists,  and  a  mean  tone  is  pro-      Part  II. 
•— V—--'  duced,  less  powerful  and  less  pure  and  pleasing,  however,  as  the  pipe  is  more  forced  from  its  natural  pitch,  until    ^— ~-s—J 
it  ceases  to  sound  altogether,  and  the  note  produced,  if  any,  is  that  of  the  reed  alone.     In  this  respect  there  is,  Mutual 
however,  a  great  difference  in  pipes  of  different  sizes.     In  large  organ-pipes  the  reed  vibrates  with  nearly  the  Jhe™"jeajjj 
same  freedom  as  in  the  open  air,  and  will,  therefore,  apeak  when  the  pipe  has  ceased  to  resound;    but  iif  small  piptl 
and  narrow  pipes,  as  in  oboes,  and  other  similar  wind-instruments,  a  much  closer  correspondence  between  the 
pitches  of  the  reed  and  pipe  is  required,  or  the  reed  will  not  vibrate.     Messrs.  Biot  and   Hamel  adapted  to  a  fig.  23. 
glass  pipe  a  reed  of  the  ordinary  construction  represented  in  fig.  23,  in  which  the  vibrating  tongue  L  (by  whose  9oim"'c~ 
oscillations  the  opening  of  the  reed  at  R  is  alternately  opened  and  closed)  could  be  lengthened  or  shortened  at  reej 
pleasure  by  thrusting  in  or  withdrawing  a  wire  F/",  which  bears  with  a  slight  spring  against  the  tongue  at  f. 
The  blast  of  wind  being  maintained  constant,  the  reed  was  made  to  yield  its  gravest  note,  by  withdrawing  the 
wire  as  far  as  possible,  after  which,  by  pushing  it  in,  the  pitch  of  the  reed  was  gradually  raised.     It  was  observed  ^nd  its 
thou  that  the  tone  of  the  pipe  grew  constantly  more  acute,  but  that  after  a  certain  point,  it  began  to  diminish  in  manner  of 
intensity,  till  at  length  no  Sound  could  be  heard.     At  this  point,  the  tongue  of  the  reed,  being  narrowly  examined  vibrating. 
through  the  glass,  was  observed  to  be  still  in  rapid  vibration  ;   but  its  vibrations  were  performed  entirely  in  the 
air,  so  as  not  to  strike  upon  and  close  the  orifice.     A  constant  passage  then  being  left  for  the  air,  the  vibrations 
of  the  pipe  could  not  be  excited.     But  this  state  of  things  continued  only  so  long  as  the  tongue  was  of  that 
precise  length.     The  moment  the  wire  was  pushed  in  by  the  smallest  quantity,  the  Sound  sprung  forth  anew  of  a 
pitch  still  corresponding  with  the  shortened  state  of  the  tongue. 

The  influence  of  the  air  in  a  pipe  on  the  reed  by  which  it  is  set  in  vibration,  causes  the  quality  of  the  tone  of      200. 
a  reed-pipe  to  depend  materially  on  its  figure.     Thus  it  is  found  that  a  reed-pipe  of  the  funnel-shaped  form,  Influence  of 
fig.  24,  composed  of  two  cones,  one  more  divergent  than  the  other,  set  on  the  orifice,  gives  the  clearest  and  most  tlle  form  "' 
brilliant  tone ;  but,  on  the  other  hand,  if  the  upper  cone  be  reversed,  so  as  to  contract  the  aperture,  fig.  25,  the  j,    P'P0  "" 
Sound  is  .stifled.     But  when  two  similar  cones,  placed  base  to  base,  are  adapted  to  the  aperture  of  a  long  conical  pj,,   2j 
pipe,  as  in  fig.  26,  the  Sound  acquires  remarkable  fullness  and  force.     This  belongs,  however,  to  a  most  intricate  Fig.  25. 
part  of  the  theory  of  Sound,  the  vibrations  of  masses  of  air  in  cavities  of  any  form.  Fig-  26. 

The  quality  of  the  tone  produced  by  reed-pipes  will  also  of  course  materially  depend  on  the  construction  of      201. 
the  reed  itself,  and  the  substance  of  which  it  is  composed.     If  the  vibrating  lamina  be  of  metal,  and  at  every  Influence  ol 
vibration  it  strikes  on  a  metallic  orifice,  these  blows  will  be  heard,  and  will  give  a  harsh,  rude,  and  screaming  'jj 
character  to  the  Sound.     If  the  edges  of  the  aperture  be  covered  with  soft  leather,  this  is  much  alleviated.     Hut  or  th'ea ' 
if,  instead  of  covering  the  aperture  by  striking  on  it,  the  tongue  is  so  constructed  as  merely  to  obstruct  it  by  passing  Sound. 
backwards  and  forwards  through  it  at  each  oscillation,  care  being  taken  to  make  it  fit  without  touching  the 
edges  of  the  aperture,  these  blows  are  avoided  altogether  ;   the  tongue  cominar  in  contact  with  nothing  but  air 
during  its  whole  motion.     In  consequence,  its  tone  is  remarkably  soft  and  pure,  and  fret;  from  any  harshness. 

The  invention  of  this  reed  is  ascribed  by  Biot  to  M.  Grenie,  who  has  taken  out  a  patent  for  it ;  but,  without      202. 
erecting  a  prior  claim  on  the  part  of  Kratzenstein,  we  may  bring  forward  a  very  familiar  instrument,  the  Jew's-  Grenie's 
harp,  as  offering,  at  least,  an  apparent  analogy  with  M.  Greni6's  reed.     The  construction  of  this  instrument  is  so  .r4?    ,     , 
well  known  that  there  is  no  need  to  describe  it;   and  though  the  theory  of  it  be  somewhat  obscure,  there  can  be  uarp 
little  doubt  that  its  action  is  that  of  a  reed  which  calls  into  play  the  resonance  of  the  cavity  of  the  month,  and 
sympathizes  with  it  in  its  vibrations,  at  least  in  some  of  their  modes.     The  Jew's-harp  is  an  instrument  much 
mistaken  and  unjustly  contemned.     Nothing  can  exceed  the  softness,  sweetness,  and  delicacy  of  this  instrument, 
when  carefully  constructed  and  well  played  ;*  as  might  be  expected  from  a  reed  in  which  the  tongue  is  perfectly 
at  liberty.     That  the  instrument  itself  vibrates  in  unison  with  the  note  it  calls  forth,  is  evident  from  the  fact, 
that  when  merely  held  before  the  open  mouth,  or  lightly  retained  between  the  lips,  its  Sound  is  feeble  and  scarce 
audible;  but  acquires  a  great  accession  of  force  when  brought  in  contact  with  and  firmly  held  between  the  teeth  , 
the  note  is  still  further  sustained  and  reinforced  by  directing  a  current  of  air  forcibly  through  it.     It  is  not  here 
meant  to  say,  that  the  great  oscillations  to  and  fro  of  the  tongue  are  commanded  by  the  resonance  of  the  cavity, 
or  are  performed  in  the  same  time  with  its  vibrations.     On  the  contrary  the  spring  is  far  too  strong  and  large  to 
admit  of  this.     It  is  more  probably  by  a  series  of  subordinate  vibrations  going  on  in  the  tongue  while  oscillating, 
that  the  sympathy  is  established. 

The  instrument  called  the  German  harmonica  is  a  reed,  on  M.  Grenie's  principle,  consisting  of  nothing  but  a      203. 
very  thin  lamina  of  brass,  of  the  form  of  an  oblong  parallelogram  fixed  by  one  of  its  narrow  ends  in  a  frame  of  TheGemian 
its  own  shape,  but  just  so  much  larger  as  to  allow  of  its  free  motion.     This  instrument  vibrates  by  a  blast  urged  u' 
through  it  yielding  a  clear  musical  tone  of  a  very  pleasing  character  and  fixed  pitch.     If  placed  at  the  end  of  a 
pipe  it  performs  the  office  of  a  reed,  and  its  tone  commands,  or  is  commanded,  by  the  pipe  according  to  circum- 
stances, as  above  explained. 

When  the  action  of  the  embouchure  of  a  pipe  is  so  decided  as  to  be  incapable  of  being,  to  any  sensible  extent,       204. 
commanded  or  influenced  by  the  resonance  of  the  pipe  ;  as,  for  instance,  when  the  column  of  air  in  a  stopped  Case  where 
pipe  is  set  in  vibration  by  a  tuning-fork  furnished  with  a  disc,  as  described  in  Art.  119,  the  pipe  will  sound,  and  the  P'Pe  ' 
reinforce  the  Sound  of  the  tuning-fork,  but  more  and  more  feebly,  as  the  pitch  of  the  latter  departs  more  from  b"'"^™6 
that  of  the  pipe.    The  experiment  is  easily  made  by  tuning  the  upper  joint  of  a  flute  with  the  mouth-hole  stopped  bouchure. 
exactly  in  unison  with  a  fork,  and  then  moving  the  piston  of  cork  at  the  end  of  the  pipe  to  and  fro,  or  loading 
the  fork  with  wax,  so  as  to  put  it  more  or  less  out  of  tune.     The  fork  and  aerial  column  \ibrate  as  a  system,  in 
which  the  former  has  so  much  the  preponderance  as  to  command  the  latter  completely. 

We  may  here  notice  a  very  remarkable  experiment,  which  we  do   not  remember  to  have  seen  elsewhere       205. 

*  As  we  have  heard  it  done  by  M.  Eulenstein. 
b&2 


790  S  O  t;  N  D. 

Sound,      described,  and  which  shows  to  what  an  extent  the  principle  of  the  superposition  of  vibrating  motions  and  the     Par'  "• 
•;—  -s^—  —  -   simultaneous  coincidence  of  different  modes   of  vibration   in  the  same   vibrating   body,  must  be  admitted  in  *~~  "V1"1 
Semnent'-k'  Acoustics-     If'  '"stead  of  one,  two  such  disked  tuning-forks  be  held  over  the  mouth  of  a  pipe  side  by  side,  both 
a'do'ubl'e       nearly  in  unison  with  the  pipe,  but  purposely  tuned  out  of  unison  with  each  other,  by  an  interval  so  small  (see 
Sound  Index,  Musical  Intervals)  as  to  produce   strong  beats,  (see  Index,  Beats,)  both  Sounds  at  once  will   be  rein- 

yielded  by  forced  by  the  pipe,  and  the  beats  will  be  heard  with  the  same  degree  of  distinctness  as  if  two  pipes,  each  in 
one  pipe.  unison  with  one  of  the  forks,  were  sounding  side  by  side.  The  same  column  of  air,  then,  at  the  same  time,  is 
vibrating  as  a  part  of  two  distinct  systems,  and  each  series  of  vibrations,  however  near  coincidence  they  may  be 
brought,  continues  perfectly  distinct  and  absolutely  free  from  any  mutual  influence.  To  those  who  have  not  tried 
the  experiment,  the  fact  of  a  pipe  actually  out  of  tune  with  itself,  and  yielding  two  notes  in  irreconcilable 
discord  with  one  another,  yet  both  equally  clear  and  loud,  will,  at  first  sight,  appear  not  a  little  extraordinary. 

206.  One  of  the   most  singular  species  of  pipe  is  that  employed  in  the  organ  to  imitate  the  human  voice.     It  is 
Vox  huma-  composed  of  a  very  short  conical  pipe,  the  base  upwards  surmounted  by  a  short  cylinder,  and  the  pitch  is  regu- 
na  organ-      lated  entirely  by  the  reed.     (See  fig.  27.)     There  is  a  circular  ojierculum  which  half  closes  the  open  end  of  the 
FV%7         cylinder,  to  imitate  the  lips,  the  reed  performing  the  part  of  the  larynx,  and  the  pipe  itself,  of  the  cavity  of  the 

throat  and  mouth.     (See  Index,  Voice.)     This  pipe,  when   well  executed,  imitates  the  human  voice  extremely 
well  ;  but  with  a  peculiar  nasal  twang,  and  somewhat  of  a  screaming  tone. 

207.  Chimney-pipes  are  those  which  are  closed  at  the  upper  end  by  a  cover,  through  the  centre  of  which  a  pipe  of 
Chimney-      smaller  diameter  is  passed,  as  a  continuation  of  the  lower  one.    (Fig.  28.)     Their  Sound  is  intermediate  between 
pipes.           those  of  open  and  slopped  pipes  of  the  same  length.     Bernonilli  (Mem.  Acad.  Par.  1762)  has  investigated  the 
F'S-  28<        laws  of  their  vibrations.     (See  also  Biot,  Traits  de  Phya.  ii.  p.  153.)     If  we  call   L   and   /  the  lengths  of  the 

greater  and  smaller  pipes  respectively,  and  X  that  of  a  pipe  closed  at  one  end  capable  of  yielding  the  same  funda- 
mental note,  n  :  I,  the  ratio  of  their  diameters,  Bernoulli  finds  the  following  equation  for  determining  X, 


tan 


/   T  L  \  /  7T  l\  1 

I  —  .  —  I  X  tan  I  —  .   —  I  =  —  . 
\2         X  /  \  2         X  /        n* 


This  equation  holds  good  when  the  lower  end  of  the  great  pipe  is  closed  ;  if  it  be  open,  the  equation  is 

L-X\          /*        f\         1 


§  IV.  Of  Musical  Intervals,  of  Harmony,  and  Temperament. 

208.  Our  appreciation  of  the  pitch  of  a  Musical  Sound  depends,  as  we  have  seen,  entirely  on  the  number  of  its  vibrations 

I  nisons.  performed  in  a  given  time.  Two  Sounds  whose  vibrations  are  performed  with  equal  rapidity,  whatever  be  their 
difference  in  quality  or  intensity,  affect  the  ear  with  a  sentiment  of  accordance  which  we  term  a  Wlixon,  and  which 
irresistibly  impresses  on  us  the  conviction  of  a  perfect  analogy,  or  similarity  between  them,  which  we  express  by 
saying  that  their  pitch  is  the  same,  or  that  they  sound  the  same  note.  In  fact,  their  impulses  on  the  air,  and 
on  the  ear,  through  its  medium,  occurring  with  equal  frequency,  blend,  and  form  a  compound  impulse,  differing 
in  quality  and  intensity  from  either  of  its  constituents,  but  not  in  the  frequency  of  its  recurrence  ;  and,  therefore, 
the  ear  will  judge  of  it  as  of  a  single  note  of  intermediate  quality. 

•?09  But  when  two  notes  not  in  unison  are  sounded  at  once,  the  ear  distinctly  perceives  both,  and  (at  least  with 

Musical         practice,  and  some  ears  more  readily  than  others)  can  separate  them,  in  idea,  and  attend  to  one  without  the  other, 
concords       But  besides  this,  it  receives  an  impression  from  them  jointly,  which  it  does  not  acquire  when  sounded  singly,  even 
and  <lis-        jn  c]ose  succession,  an  impression  of  concord,  or  dissonance,  as  the  case  may  be  ;  and  is  irresistibly  led  to  regard 
some  combinations  as  peculiarly  agreeable  and  satisfactory,  and  others  as  harsh  and  grating.     Now  it  is  inva- 
riably found  that  the  former  are  those,  and  those  only,  in  which  the  vibrations  of  the  individual  notes  are  in  some 
very  simple  numerical  proportion  to  each  other,  as  1  to  2,  1  to  3,  1  to  4,  2  to  3,  &c.,  and  that  the  concord  is  more 
satisfactory  and  more  pleasing,  the  lower  the  terms  of  the  proportion  are,  and  the  less  they  differ  from  each  other. 
While,  on  the  other  hand,  such  notes  as  vibrate  in  times  bearing  no  simple  numerical  ratio  to  each  other,  or  in 
which  the  times  of  the  ratio  are  considerable,  as  8  :  15,  for  example,  when  heard  together  produce  a  sense  of 
discord,  and  are  extremely  unpleasant.     This  simple  remark  is  the  natural  foundation  of  all  harmony. 
210.  Next  to  a  unison,  in  which  the  vibrations  of  the  two  notes  are  in  the  ratio  of  1  :  1,  the  most  satisfactory  concord 

The  octave.  js  the  octave,  where  the  vibrations  are  as  1  :  2,  or  one  note  performs  two  vibrations  to  each  single  one  of  the 
other.  The  octave  approaches  in  its  character  to  a  unison,  and  indeed  two  notes  so  related  when  played  together 
can  hardly  be  separated  in  idea ;  and  when  singly,  appear  rather  as  the  same  note  differently  modified,  than  as 
Kig.  2S.  independent  Sounds.  The  reason  of  this  will  be  evident  on  inspecting  fig.  29,  where  the  dots  in  the  upper  line 
represent  the  periodically  recurring  impulses  on  the  ear  produced  by  the  vibrations  of  the  acuter  note,  while  thos» 
in  the  lower  represent  the  same  impulses  as  produced  by  those  of  the  graver  ;  as  the  ear  receives  these  all  in  the 
order  they  are  placed,  it  wfll  be  the  same  thing  as  if  they  were  produced  by  two  Sounds  both  of  the  graver  pitch, 
but  one  of  a  different  intensity  and  quality  from  the  other ;  the  one  having  its  impulses  (represented  by  :  )  the 
sum  of  two  separate  impulses  of  the  octave  Sounds  ;  the  other  consisting  of  the  alternate  impulses  (.)  of  the 
acuter  only. 
21 1  In  like  manner  the  octave  of  the  octave,  or  the  fifteenth,  as  it  is  called  in  Music,  which  consists  of  notes  whose 


SOUND.  791 

Sound,      vibrations  are  as  1  :  4,  is  a  very  agreeable  and  perfect  concord  ;   as  are,  indeed,  all  the  scaie  of  octaves  1  :  8,  J^      ^ 
— /-— '  1  :  16,  &c.  they  all  partake  of  the  peculiar  character  of  the  octave,  a  sense  of  perfect  adjustment  or  identity.          ThTdChil 

The  next  in  order  is  the  combinations  1  :  3,  where  the  vibrations  of  the  graver  note  are  trisected  by  those  of  octave)  ur 
the  acuter,  as  in  fig.  30,  which  gives  a  concord  called  the  twelfth,  a  very  agreeable  one.     In  this,  if  we  substitute  fifteenth,&c. 
for  the  note  1  its  octave  2,  we  shall  have  the  concord  whose  vibrations  are  in  the  ratio  of  2  :  3  ;  or,  as  we  shall       212. 
call  it  for  brevity,  the  concord  2  :  3,  whose  pulsations  are  represented  in  fier.  31.     This  concord  is  termed  the  The 
fifth,  and  is  a  remarkably  perfect  and  agreeable  one,  even  more  so  than  the  'twelfth,  which  although  simpler  in  pp11^ 
a  numerical  estimate,  yet  from  the  greater  interval  between  its  component  notes  allows  them  to  be  more  readily  ^  fift(j 
distinguished,  while  the  notes  of  the  fifth  blend  much  more  perfectly.  Fig.  31. 

If,  instead  of  substituting  for  1  its  octave  2,  we  substitute  its  double  octave  4,  we  get  the  concord  4  :  3,  or       313 
the  fourth,  which  may  be  regarded  as  a  sort  of  complement  of  the  fifth,  and  is  also  very  agreeable.  The  fourth. 

The  concords  1:5,   2  :  5,  and  4  :  5,  especially  the  latter,  in  which  the  tones  approach  pretty  near  to  each  Fig.  32. 
other,  are  all  remarkably  agreeable.     The  last  is  called   a  major  third,  and  the  two  former  are  regarded  rather       214. 
as  varieties  of  it  than  as  independent  concords.     The  concord  8  :  5  (which  is  its  complement  in  the  same  sense  ^dmajor 
as  4  :  3  the  fourth  is  to  2  :  3  the  fifth)  is  called  the  minor  sixth,  and  is  almost  equally  agreeable  with  the  major  ^  '33 
third,  to  which  it  is  related.  The  minor 

The  concord  3  :  5  is  called  the  major  sixth,  and,  as  well  as  its  complement  6  :  5,  or  the  minor  third,  though  sixth. 
pleasing,  is  decidedly  less  satisfactory  than  the  foregoing  ;  and,  as  we  see  by  casting  our  eyes   on   the  figure,       21 5. 
the  periods  of  recurrence  of  their  combined  pulses  in  the  same  order  is  longer  and  more  complex. 

Higher  primes  than  5  enter  into  no  harmonic  ratios.     Such  combinations,  for  instance,  as  1  :  7,  5  :  7,  or  6  :  7,  ^JL,"^ 
are  altogether  discordant.     The  same  may  be  said  of  the  more  complicated  combinations  of  the  lower  primes       gig. 
1,  2,  3,  5.     The  ear  will  not  endure  them,  and  cannot  rest  upon  them.     When  sounded,  a  sense  of  craving  for  Discords 
a  change  is  produced,  and  this  is  not  satisfied  but  by  changing  one  or  both  of  the  notes  so  as  to  fall  as  easily  as  and  their 
the  case  will  permit  into  some  one  of  the  concords  above  enumerated.     This  is  called  the  resolution  of  a  discord  ;  resolution. 
and  such  is  the  constitution  of  our  minds  in  this  respect,  that  a  concord  agreeable  in  itself  is  rendered  doubly       217. 
so  by  being  thus   approached  through   a  discord.     For  example,  let  us  take  the  ratio  5  :  9,  which  is  called  a  {j^"^^" 
flat  seventh,  a  combination  decidedly  discordant.     If  we  multiply  the  terms  of  this  ratio  by  5,  we  get  25  :  45.  tion  of  the 
A  small  change  in  one  of  the  notes  will  reduce  this  to  27  :  45,  or  3  :  5,  a  major  sixth — an  agreeable  concord,  discord  of 
Now  this  will  be  done,  if,  retaining  the  lower  note  5  or  25,  we  change  the  upper  from  45  to  45  X  $4 !   that  is  to  the  flat 
say,  to  a  note  whose  vibrations  are  to  its  own  as  25  :  27.     This  ratio  corresponds  to  a  musical  interval  called  a  s«ve"th. 
semitone.    Hence  the  discord  in  question  will  be  satisfactorily  resolved  by  holding  on  its  lower  note,  and  making 
its  upper  one  descend  a  semitone.. 

On  the  proper  alternation  of  concords  and  discords  the  whole  of  musical  composition  depends,  but  though  the 
principle  above  stated  must  be  satisfied  in  the  resolution  of  every  discord,  there  are  other  rules  to  be  attended  to 
by  which  our  choice  is  limited  to  some  modes  rather  than  others  ;  for  example,  in  the  foregoing  instance  it  is 
the  upper  note  which  must  descend  a  semitone.  The  ascent  of  the  lower  by  the  same  interval,  which  would 
equally  change  the  ratio  as  above  indicated,  would  offend  against  other  precepts  with  which  we  have  here 
nothing  to  do. 

The  interval,  as  it  is  called  in  Music,  between  the  two  notes  of  which  any  simple  concord  or  discord  consists,        ' 
depends  not  on  the  absolute  number  of  vibrations  which  either  makes  in  a  given  time,  but  on  their  relative  pro-  JJ"^*1^"' 
portion.     For  it  is  no  matter  how  slowly,  or  how  rapidly,  the  vibrations  take  place,  provided  the  order  in  which  ppml  on  e" 
their  impulses  reach  the  ear  be  the  same.     Hence,  if  the  vibrations  4  and  5  produce  on  the  ear  the  agreeable  effect  ratios  Of  vj_ 
of  a  major  third;  two  notes,  each  an  octave  higher,  or  having  their  vibrations  respectively  8  and   10;    or   in  brations, not 
general  any  two  having  their  vibrations  in  this  ratio,  will  produce  the  same  effect.     This  is  a  matter  of  expe-  "{J^j1*^ 
rience,  but  the  inspection  of  the  figures  representing  the  older  of  succession  of  the  individual  vibrations  enables  *u°n°b"re 
us  to  understand  its  reason. 

If  we  take  any  note  for  a  fundamental  Sound,  and  tune  a  string  or  a  pipe  so  as  to  vibrate  with  the  degree  ot       2 
rapidity  corresponding  to  that  Sound,  and  represent  by  unity  the  number  of  vibrations  it  makes  per  second ;  and  ^""P1^ 
if  we  also  tune  other  strings  to  make  in  the  same  time  respectively  the  numbers  of  vibrations  represented  by  °   ()  ^  sc 
5          A          Q          fj  second  and 

T'     T'     ~Z'     IT'  2;    and  then  sound  a11  these  strings  in  succession,  beginning  with  the  fundamental  note,  seventh. 
4        o        2        3 

we  shall  perceive  that  two  of  the  sequences,  the  first  and  last,  are  much  wider  than  the  rest,  and  would  admit 
the  interpolation  of  a  note  between  each.  But  it  is  no  longer  possible  to  choose  for  these  interpolated  notes 
such  as  will  make  concordant  intervals  with  any  of  the  rest,  or  their  octaves.  But  in  order  to  obtain  as  many 
concords  as  possible  in  the  scale,  so  as  to  produce  the  most  harmonious  music,  they  are  made  to  harmonize 
with  that  note  which  bears  the  nearest  relation  to  the  fundamental  one,  (for  its  octave  is  regarded  as  a  mere 
repetition  of  itself,)  i.  e.  the  fifth.  The  vibrations  of  a  note  a  fifth  higher  than  the  fifth  are  represented  by 

—  ),  or  — ;  and  as  this  is  greater  than  2,  it  lies  beyond  the  octave.     We  must,  therefore,  tune  our  interpo- 
2  /  4 

g 
lated  string  an  octave  lower,  or  to  the  vibration   — ,  and  thus  we  get  the  second.     Again,  if  we  tune  another  to  The  second. 

the  vibration  - ,  or  —  X  — ,  it  will  form,  with  the  fifth  of  the  fundamental  note  (  —  Y  a  major  third — the 
next  most  harmonious  interval  on  the  scale.  The  note  thus  interpolate',!  is  the  seventh. 


792  SOUND. 

Sound.          The  interpolated  scale,  with  the  vibrations  of  its  respective  notes,  will  stand  thus :  Part  II. 

^£~  Sisns  W-   <•*>'   <8>-    <4>-   <5>-   (6>-   (7>-   W- 

Names  of  intervals 1st.     2d.      3d.      4th.     5th.     6th.     7th.     8th. 

9         5         4          3          5          15 
Ratios  of  v.bration 1.       -.     -,      -,      -,      -,      _,      2. 

or  multiplying  all  by  24,  to  avoid  fractions, 

24,     27,     30,      32,      36,      40,       45,       48. 

Diatonic  This  is  called  the  natural,  or  diatonic  scale ;  when  all  its  notes  are  sounded  in  succession,  whether  upwards  or 
downwards,  the  effect  is  universally  acknowledged  to  be  pleasing.  The  ear  rests  with  perfect  satisfaction  on  the 
fundamental  note,  and  the  intervals  succeed  each  other  gracefully,  with  sufficient  variety  to  avoid  monotony. 
Accordingly  all  ages  and  nations  have  agreed  in  adopting  this  scale  as  the  foundation  of  their  music. 

222.  This  scale  consists  of  seven  distinct  notes,  for  the  eighth  being  the  octave  of  the  first  is  regarded  as  a  mere 
Continua-     repetition  of  it.     And  if  we  add  to  it  on  both  sides  the  octaves  of  all  its  tones  above  and  below,  and  again  the 
tion  of  the   octaves  of  these,  and  so  on,  we  may  continue   it   indefinitely  upwards  and  downwards.     Not  that  the  ear  will 

!  UP~.    follow  these  additional  tones,  to  an  unlimited  extent.     When  the  vibrations  are  less  numerous  than  about  16  per 

downwards,  second,  the  ear  loses  the  impression  of  a  continued  Sound  ;   and  perceives,  first,  a  fluttering  noise,  then  a  quick 

Limits  of      rattle,  then  a  succession  of  distinct  Sounds  capable  of  being  counted.     On  the  other  hand,  when  the  frequency 

audibility.     of  the  vibrations  exceeds  a  certain  limit  all  sense  of  pitch  is  lost ;  a  shrill  squeak,  or  chirp,  only  is  heard  ;    and, 

what  is  very  remarkable,  many  individuals,  otherwise  no  way  inclined  to  deafness,  are  altogether  insensible  to 

certain  ears  Vei7  acu*e  Sounds,  even  such   as  painfully  affect  others.     This  singular  observation  is  due  to  Dr.  Wollaston. 

(See  his  Paper  on   Sounds  inaudible  to  certain  ears,  Phil.  Trans.  1820.)     Nothing  can  be  more  surprising 

than  to  see  two  persons,  neither  of  them  deaf,  the  one  complaining  of  the  penetrating  shrillness  of  a  Sound, 

while  the  other  maintains  there  is  no  Sound  at  all.     Thus  while  one  person  mentioned  by  Dr.  Wollaston  could 

but  just  hear  a  note  four  octaves  above  the  middle  E  of  the  piano-forte,  others  have  a  distinct  perception  of 

Sounds  full  two  octaves  higher.     The  chirp  of  the  Sparrow  is  about  the  former  limit ;   the  cry  of  the  Bat  about 

one  octave  above  it ;   and  that  of  some  insects,  probably,  more  than  another  octave.     Dr.  Wollaston' s  sense  of 

hearing  terminated  at  six  octaves.     The  whole  range  of  human  hearing  comprised  between  the  lowest  notes  of 

the  organ  and  the  highest  known  cry  of  insects,  seems  to  include  about  nine  octaves. 

223.  It  is  probable,  however,  that  it  is  not  alone  the  frequency  of  the  vibrations  which  renders  shrill  Sounds  inau- 
Remark.       dible.     There  is  no  reason  why  an  impulse,  if  strong  enough  singly  to  affect  the  ear,  should  lose  its  effect  if 

s  repeated  many  thousand  times  in  a  second.  On  the  contrary  such  repetition  would  render  the  noise  intolerable. 
govue^*cu  e  But  this  is  not  the  case  with  musical  Sounds;  thc'ir  individual  impulses  would,  probably,  be  quite  inaudible 
singly,  and  only  impress  by  repetition.  Now,  as  vibrating  bodies  have  only  a  certain  degree  of  elasticity, 
extreme  swiftness  of  vibration  can  only  take  place  when  their  dimensions  are  very  minute,  and  consequently  the 
excursions  of  their  molecules  from  rest,  and  their  absolute  velocities,  excessively  minute  also.  Thus  in  proportion 
as  Sounds  are  more  acute  their  intensity  (which  depends  wholly  on  the  extent  and  force  of  their  vibrations) 
diminishes.  No  doubt,  if  by  any  mechanism  a  hundred  thousand  hard  blows  per  second  could  be  regularly 
struck  by  a  hammer  on  an  anvil,  at  precisely  equal  intervals,  they  would  be  heard  as  a  most  deafening  shriek  ; 
but  in  natural  Sounds  the  impulses  lose  in  intensity  more  than  they  gain  in  number,  and  thus  the  Sound  grows 
feebler  and  feebler  till  it  ceases  to  be  heard. 

224.  "  As  there  is  nothing  in   the   nature  of  the  atmosphere  (remarks  Dr.  Wollaston)  to  prevent  the  existence 
Possible       of  vibrations  incomparably  more  frequent  than  any  of  which  we  are  conscious,  we  may  imagine  that  animals 


limits  of       ijke  tne  Gj-yii^  whose  powers  appear  to  commence  nearly  where  ours  terminate,  may  have  the  faculty  of  hearing 

0  still  sharper  Sounds,  which  we  do  not  know  to  exist ;  and  that  there  may  be  other  insects  hearing  nothing  in 

different' "     common  with  us,  but  endued  with  a  power  of  exciting,  and  a  sense  which  perceives  vibrations  of  the  same  nature 

animals.       indeed  as  those  which  constitute  our  ordinary  Sounds,   but  so  remote  that  the  animals  who  perceive  them  may 

be  said  to   possess  another  sense  agreeing  with  our  own  solely  in  the  medium  by  which  it  is  excited."     The 

same  may,  no  douljt,  be  true  of  aquatic  animals.     The  shrimp  and  the  whale  may  have  no  Sound  in  common. 

225.  By  the  aid  of  the  ascending  and  descending  series  of  Sounds  in  the  natural  scale  thus  obtained,  pieces  of  music 
Key-note  of  perfect]y  pleasing.,  both  in  point  of  harmony  and  melody,  may  be  played  ;   and  they  are  said  to  be  in  the  key  of 

that  which  is  assumed  as  the  fundamental  note  of  the  scale,  or  whose  vibrations  are  represented  by_l.  If  such 
Its  analysis  a  piece  be  analyzed,  it  will  be  found  to  consist  entirely,  or  chiefly,  of  triple  and  quadruple  combinations,  or 
into  chords,  chords,  such  as  the  following  : 

226.  1.  The  common,  or  fundamental  chord,  or  chord  of  the  tonic,  or  the  1st,  3d,  and  5th,  (1,  3,  5,)  or  the  3d,  5th, 
The  funda-   an(j  octave  (3,  5,  8)  sounded  together.     This  is  the  most  harmonious  and  satisfactory  chord  in  music,  and  when 

mental  or  SOunded  the  ear  is  satisfied,  and  requires  nothing  further.     It  is,  therefore,  more  frequently  heard  than  any  other ; 
common  ..  /,  •      i  -.,,..  ,. 

chord.  an°  lts  continual  recurrence  in  a  piece  of  music  determines  the  key  it  is  played  in. 

227.  2.  The  chord  of  the  dominant.     The  fifth  of  the  key-note  is  called,  by  reason  of  its  near  relation  to  the  funda- 
Chord  of  mental  note,  the  dominant.     The  chord  in  question  is  the  common  chord  of  the  dominant,  or  the  notes  (2,  5, 
the  domi-  antj  7)  sounded  together. 

"""pno  3.  The  chord  of  the  subdominant,  or  the  note  4,  consisting  of  the  notes  1,  4,  6,  being  the  common  chord  of  4 

p.    ',    ,  as  a  fundamental  note. 

ihe°3iib-  4.  The  false  close,  or  the  combination  (1,  3,  6)  or  (3,  6,  8)  which  is,  in  fact,  the  common  chord  of  the  note  6, 

dominanf.  only  with  a  minor  third  (6,  8)  instead  of  a  major.     The  term  false  close  arises  from  this,  that  a  piece  of  music, 

229  frequently  before  its  final  termination,  (which  is  always  on  the  fundamental  chord,)  comes  to  a  momentary  close 


SOUND.  793 

Sound.      on  this  chord,  which  pleases  oniy  for  a  short  time,  but  requires  the  strain  to  be  taken  up  again  and  closed  as  usual      Part  II. 
•— v— •''  to  give  full  satisfaction.  •nTTT"' 

5.  The  discord  of  the  7th,  or  (2,  4,  5,  7.)     It  consists  of  four  notes  ;  and  is,  in  fact,  the  common  chord  of  the    . 
dominant,  with  the  note  immediately  below  it,  or  the  seventh  in  order  above  it.     The  interval,  however,  between     °^g0 
the  notes  (4)  and  (5),  or  between  (5)  and  the  octave  of  (4)  next  above  it,  is  represented  by  the  ratio  chord  (or 

discord)  of 

2  X  4      ,    3          16  the  seve'ntK 


3  2          9 

or  (taking  24  as  the  number  of  vibrations  in  a  unit  of  time  corresponding  to  the  note  (1))  =  42  — .     This 

3 

interval,  then,  is  less  than  the  seventh  of  the  diatonic  scale,  and  is  about  half-way  intermediate  between  the  sixth 
and  seventh  of  that  scale.  It  is,  therefore,  called  the  flat  seventh.  (See  Arts.  231  and  232.)  This  discord 
resolves  itself  into  the  chord  (3,  5,' 8  ;)  and  unless  that  combination,  or  one  equivalent  to  it,  follows,  the  ear  is 
not  satisfied.  The  notes  (4)  and  (5)  are  the  essential  ones  of  this  discord,  and  the  others  are  regarded  as 
accompaniments.  If  played  together,  the  ear  requires  that  in  the  next  chord  4  should  descend  or  be  succeeded 
by  (3,)  while  the  note  7  is  required  to  rise  or  be  succeeded  by  (8.) 

With  these  chords  and  a  few  others,  such  as  the  chord  of  the  9th,  whose  essential  notes  are  1  and  2,  or  1  and  9,       231. 
may  a  great  variety  of  music  be  played,  but  it  would  be  found  monotonous.     The  ear  requires,  in  a  long  piece,  Modulation 
a  variety  of  key.     The  fundamental  note  occurs  so  often  that  it  seems  to  pervade  the  whole  of  the  composition, 
and  must  therefore  be  changed.     But  this  change  of  key,  which  is  called  modulation,  is  not  possible  without 
introducing  other  notes  than  those  already  enumerated.     It  is  true  the  chord  (2,  5,  7)  is  the  perfect  fundamental 
chord  in  the  key  of  (5  ;)   but  the  other  chords  in  that  key  corresponding  to  those  already  enumerated  cannot  be 
formed,  with  the  exception  of  its  sub-dominant,  which  is,  in  fact,  the  common  chord  of  1.     Take,  for  instance, 
its  dominant.     This  would  be  formed,  if  it  could  be  formed  at  all,  of  the  notes  (2,  4,  6)  or  (4,  6,  9.)     But  if  we  Necessity 

come  to  analyze  the  intervals  of  these  notes,  we  find  that  °f  intro- 

ducing in- 
(4)      _    32  (6)    _    40  termediate 

(2)"~27;          (2T~"27"  E°teS- 

5  3 

Now  these  differ  from  the  ratios  —    and    —   which  exist  between  the  notes  (3,  1)  and   (5,  1)  of  the  perfect 

common  chord.  Consequently,  if  we  would  play  equally  well  in  tune  in  the  key  (5)  we  must  introduce  these  new 
ratios  ;  and,  in  fact,  we  ought  to  have  for  that  purpose  notes  corresponding  to  all  the  ratios 

39  35  34 

2~xl~'        Y  x  7'        T  :<  3  '  & 

and  similarly  for  every  other  key  we  might  choose  to  play  in.  But  this  would  require  an  enormous  number  of 
notes,  and  would  render  the  generality  of  musical  instruments  too  complicated.  It  becomes  necessary,  then,  to 
consider  how  the  number  may  be  reduced,  and  what  are  the  fewest  notes  that  will  answer. 

3 
Let  us  take  for  example,  as  above,  the  dominant  of  the  note  (5.)     The  number  of  its  vibrations  is  36  X  — ,  or      333 

54,  the  half  of  which  (because  it  surpasses  the  octave  of  1)  is  27.     This  is  correctly  the  number  corresponding  fied*!™  a 

5  1  numerical 

to  (2.)     Now,  taking  this  for  a  key-note,  the  major  third  of  (2)  has  27  x    —  =  34  —   for    the  number  of  its  example. 

vibrations  in  a  unit  of  time.     Now  in  the  scale  as  it  stands  we  have  32  and  36,  so  that  the  note  in  question  is 
almost  just  half-way  between  them,  and  must  therefore  be  interpolated.     It  will  stand  between  (4)  and  (5)  on  Flats  and 
the  scale,  and  is  denoted  in  Music  by  the  sign  tt  sharp,  or  ^  fiat;  thus  it  is  written  either  as  (4)  sharp,  (4)8,  or 
as  (5)  flat,  (5)  K     With  regard  to  the  fifth  of  the  new  fundamental  note  (2)  its  representative  number  is 


This  comes  so  near  40,  that  the  ear  hardly  perceives  the  difference  ;  and  though  a  small  error  of  one  vibration  Inl 
in  80  is  introduced  by  using  the  note  (6)  as  the  dominant  of  (2,)  yet  it  is  not  fatal  to  harmony  ;  and  there  is  no  tio"f  of '"'" 
necessity  for  encumbering  ourselves  with  new  names  of  notes,  and  additional  pipes  or  strings  to  our  instruments  ha-monv 
for  its  sake.     In  practice  these  errors  are  modified  and  subdued  by  what  is  called  temperament,  of  which  this 
is  the  origin,  and  of  which  more  presently. 

81 
The  interval  -  -  being  the  difference  of  two  notes,  one  of  which  is  the  octave  of  the  perfect  sixth  of  the       233. 

TH 

fundamental  note,  and  the  other  arises  by  reckoning  upwards  three  perfect  fifths  from  the  same  origin,  is  called 
a  comma.     The  former  note  is  represented  by 

5          10  /  3  V       27  ,  27       10        81 

2  x  —  =  ,         the  latter  by  I  —  I  =  —  ,  and 1 =  — . 

3  3  J  \  2  /         8  83        80 

In  like  manner,  if  we  would  choose  any  other  note  for  the  fundamental  one,  similar  changes  will  be  required,       2^' 


794  SOUND 

Sound,     and  no  two  keys  will  agree  in  giving  identically  the  same  scale.     All,  however,   will  be  nearly  satisfied  by  the      Pan  II. 
>— N-— ••'  interpolation  of  a  new  note  half-way  between  each  of  the  larger  intervals  of  the  scale,  thus  '^••V*' 

Thechr°,  1*1  2*1  4*1  5*1  6*1 

malic  scale. 

1,        «r    },       2,       or   },     3,     4,      or  V,       5,        or  \,       6,       orV,       7,       8; 
2  t»  J  8  •  J  5  *  J  6  b  J  7  M 

and  the  scale  so  interpolated  is  called  the  chromatic  scale. 

235.  Musicians  have  long  been  at  issue  on  the  most  advantageous  method  of  executing  this  interpolation.     If, 
tempe-     indeed,  it  were  intended  to  give  such  a  preference  to  the  natural  scale  1,  2,  3,  4,  &c.  as  to  make  it  perfect,  to  the 

sacrifice   of  all  the  other  keys,  there  would   be  little  difficulty,   as  a  mere   bisection  of  the  intervals  would, 

probably,  answer  every  practical  purpose  :  thus  1  *  or  2  ^  might  be  represented  by   \/  1   x  -—  ;   2  *  =  3  b  by 

9          5~ 

•jr    X  —  f  and  so  on  ;    but  as  in  practice  no  preference  is  given  to  this  particular  key,  (which  is  denoted  in 

Music  by  the  letter  C,)  but,  on  the  contrary,  variety  is  purposely  studied,  it  is  found  necessary  to  depart  from  the 
pure  and  perfect  diatonic  scale,  even  in  tuning  the  natural  notes;  and  to  do  so  with  the  least  offence  to  the  ear 
is  the  object  of  a  perfect  system  of  temperament.  If  the  ear  absolutely  required  perfect  concords  there  couln 
be  no  music,  or  but  a  very  limited  and  monotonous  one.  But  this  is  not  the  case.  Perfect  harmony  is 
never  heard,  and  if  heard  would  probably  be  little  valued,  except  by  the  most  refined  ears;  and  it  is  thif 
fortunate  circumstance  which  renders  musical  composition,  in  the  exquisite  and  complicated  state  in  which  it 
at  present  exists,  possible. 

236.  In  order  to  judge  of  the  limits,  however,  to  which  the  ear    vill  bear  a  deviation  from  exact  consonance  of 
musical  vibrations,  we  must  first  see  what  takes  place  when  tw     notes  nearly,  but  not  quite,  in  unison  or  concord 
are  sounded  together.     Conceive  two  strings  exactly  equal  and  similar,  and  equally  drawn  out  from  the  straight 
line,  to  be  let  go  at  the  same  instant ;  and  suppose  one  to  make  100  vibrations  per  second,  the  other  101  ;   let 
them    be  placed    side  by  side,  and  at   the  same   distance   from  the    ear.     Their  first  vibrations  will  conspire 
in  producing  a  Sound-wave  of  double  force,  and  the  impression  on  the  ear  will  be  double.     But  at  the  50th 
vibration  one  has  gained  half  a  vibration  on  the  other,  so  that  the  motions  of  the  aerial  particles,  in  virtue  of 
the  two  coexistent  waves  emanating  from  either  string,  are  now  not  in  the  same  but  in  opposite  directions;   and 
the  two  waves  being  by  supposition  of  equal  intensity,  they  will  instead  of  conspiring  exactly  destroy  each  other, 
and   this   will   be  very  nearly  the   case  for  several   vibrations   on  either   side  of  the  50th.     Consequently,   in 
approaching  the  50th  vibration,  the  joint  Sound  will  be  enfeebled  ;  there  will  be  a  moment  of  perfect  silence,  and 
then  the  Sound  will  again  increase  till  the  100th  ;   when  the  one  string  having  gained  a  whole  vibration  on  the 
other,  the  motions  of  the  particles  of  air  in  the  two  waves  will  again  completely  conspire,  and   the  Sound  will 
attain  its  maximum.     The  effect  on  the  ear  will  therefore  be  that  of  an  intermitting  Sound  alternately  loud  and 
faint.     These  alternate  reinforcements  and  subsidences  of  the  Sound  are  called  beats.     The  nearer  the  Sounds 
of  the  strings  approach  to  exact  unison,  the  longer  is  the  interval  between  the  beats.     If  we  call  n  the  number 
of  vibrations  in  which  one  string  gains  or  loses  exactly  one  vibration  on  the  other,  and  m  the  number  of  vibrations 

per  second  made  by  the  quicker,  will  be  the  interval  between  two  consecutive  beats.     When  the  unison  is 

m 

complete,  no  beats  are  heard.  On  the  other  hand,  when  it  is  very  defective  they  have  the  effect  of  a  rattle  of  a 
very  unpleasant  kind.  The  complete  destruction  of  the  beats  affords  the  best  means  of  attaining  by  trial  a 
perfect  harmony. 

237.  Beats  will  likewise  be  heard  when  other  concords,  as  fifths,  are  imperfectly  adjusted.     Suppose  one  string  to 
Beats  heard  make  201  vibrations,  while  the  other  makes  300;  then,  at  and  about  the  100th  of  one,  and  the  150th  of  the 
with  all        other,  the  former  will  have  gained  half  a  vibration,  and  those  vibrations  of  the  one  which  fall  exactly  on  those 

jCt  of  the  other,  (see  fig.  31,)  being  performed  with  contrary  motions  will  destroy  each  other;  those  which  fall 
intermediate  only  partially.  The  beats  then  will  be  heard,  but  with  less  distinctness  than  in  the  case  of 
unisons. 

238.  This  seems  the  proper  place  to  notice  an  effect  which  takes  place  in  perfect  concords,  and  only  in  those  which 
Resultant     are  very  perfect,  viz.  the  production  of  a  grave  Sound  by  the  mere  concurrence  of  two  acute  ones.     If  we  examine 
Sounds.        ^e  f,gure  212,  which  represents   the  succession  of  vibrations  in  a  perfect  fifth,  we  shall  see  that  every  third  of 

the  one  coincides  exactly  with  every  second  vibration  of  the  other.  These  coincidences  (so  delicate  is  the  ear) 
are  remarked  by  it,  and  a  Sound  is  heard,  besides  the  two  actually  sounded,  of  a  pitch  determined  only 
by  the  frequency  of  the  precise  coincidences  ;  that  is,  in  this  case,  a  precise  octave  below  the  lowest  tone  of  the 
concord. 

239.  In  general,  if  one  note  makes  m  vibrations  and  the  other  n,  while  another,  of  which  they  may  both  be  regarded 
General  de-  as  harmonics,  makes  one,  that  one  will  be  the  resultant  tone,  provided  m  and  n  be  prime  to  each  other;   so  that 

in  t\,~  — i_  difficulty  in  determining  the  resultant  of  two  notes,  is  to  determine  of  what  they  are  both  harmonics. 


This  will  be  done  by  reducing  m  and  n,  if  fractions,   to  a  common  denominator   •=-   and    — ;  then,  if  m' 

and  n'  have  no  common  factor,  —  will  represent  the  fundamental  tone.     If,  then,  m  and  n  be  integers,  and 
without  any  common  factor,  the  resultant  will  be  represented  by  1. 


SOUND.  795 

Hence  follows  a  very  curious  fact,  viz.  that  if  several  strings,  or  pipes,  be  tuned  exactly  to  be  harmonics  of  one    fart  If. 
of  them,  or  to  have  their  vibrations   in  the  ratios  1,  2,  3,  4,  5,  &c.,  then  if  they  be  all,  or  any  number  of  them,  •— v— ' 
from  'he  first  onward,  sounded  together,  there  will  be  heard  but  one  note,  viz.  the  fundamental  note.     For  they      240. 
are  all  harmonics  of  the  first  note  1  ;  and,  moreover,  if  we  combine  them  all  two  and  two,  we  shall  find  compa-  f^J'"^ 
ralively  but  few  which  will  give  other  resultants,  so  that  these  will  be  lost,  us  well  as  the  individual  Sounds  of  th"con™m 
the  strings,  all  but  the  first,  in  the  united  effect  of  all  the  resultant  unit  Sounds.     But  to  produce  this  effect,  the  course  Of  , 
strings,  or  pipes,  must  be  very  perfectly  tuned  to  the  strict  harmonics.    The  effect  can  never  take  place  by  touching  great  many, 
the  keys  of  a  piano-forte  corresponding  to  the  harmonic  notes,  because  they  are  always  of  necessity  tempered. 

To  return  to  our  temperament.     If  we  count  the  semitones  in  the  chromatic  scale  between  (1)  and  (8)  we  Th;  ^ca',e 
shall  find  the  number  of  such  intervals  12.     If,  then,  we  would  have  a  scale  exactly  similar  to  itself  in  all  parts,  rf  „„,!  j, 
and  which  should  admit  of  our  playing  equally  perfectly  in  every  key,  we  have  only  to  compute  the  values  of  tervals,  or 
the  fractions  iso-harmo- 

nic  scale. 

1  =  2°,  2^,  2  A,  2* * 2T^,  2, 

which  may  readily  be  done  by  logarithms,  and  we  shall  find  the  ratios  of  the  vibrations  which  will  give  what 
may  be  termed  the  scale  of  equal  intervals. 

If  we  examine  the  chromatic  scale,  and  consider  it  as  appnximatively  composed  of  equal,  or  nearly  equal,      242. 
intervals  called  semitones,  the  following  will  be  the  uumber  of  semitones  in  each  interval :  Sy«te:n  of 


In  the  semitone 1 

second  or  tone 2 

minor  third    3 

major  third 4 

minor  fourth  or  fourth 5 


In  the   major  fifth  or  fifth 7 

minor  sixth 8 

• major  sixth 9 

• flat  seventh 10 

seventh  .  .  .11 


major  fourth  or  minor  fifth    ....      6          octave 12 

If  then  we  reckon  upwards  from  the  note  (l)by  fifths,  viz.  from  (1)  to  (5),  from  (5)  to  (9),  (or,  which  comes  to 
the  same,  descending  an  octave,  to  (2),)  from  (2)  to  (6),  from  (6)  to  (10),  that  is  to  (10  —  7),  or  (3),  and  so  on, 
we  shall  find  that  after  taking  twelve  such  steps  as  these  we  shall  have  fallen  upon  every  note  in  the  scale,  and 
come  back  to  the  fundamental  note  or  its  octave.  But,  since  no  power  of  2  is  exactly  the  same  with  any  power  of 
g 
—  ,  it  is  evident  that  no  series  of  steps  by  perfect  fifths  can  ever  bring  us  to  any  one  of  the  octaves  of  the 

fundamental  note.     Were  the  chromatic  scale  perfect,  twelve  fifths  should  exactly  equal  seven  octaves,  and  three 
major  thirds  should  precisely  make  one  octave.     Neither  of  these,  however,  can  be  true  of  perfect  fifths  or  thirds, 

(3  \18  /   5  \3 

—  I   =  129'74,  and  2'  =  128,  giving  a  difference  of  nearly  one  vibration  in  64   and  I   —  1  =  T953 

instead  of  2.     Thus,  if  we  reckon  upwards  by  perfect  fifths,  we  surpass  the  octaves  ;  if  by  thirds,  we  fall  continually 
below  them.     In  this  dilemma  it  has  been  proposed  to  diminish  all  the  fifths  equally,  making  a  fifth  instead  of 

3  — 

— ,  to  be  equal  to  2  ' J  ;   and  tuning  regularly  from  the  note  (1)  upwards  by  such  fifths   and  from  the  notes 

so    tuiivd    doTcii wards   by    perfect   octaves.     This   constitutes   what  has  been  called   the  system   of  equal 
temperament 

L 

It  is  evident  that  in  this  system  the  notes  will  all  of  necessity  be  represented  by  powers  of  2      ;  and  that      243. 
therefore  the  scale  resulting  from  this  system  is  identical  with  that  of  equal  intervals,  or  the  iso-harmonic  scale  Defect  of 
described  in  the  last  article.     Theoretically  speaking,  it  is  the  simplest  that  could  be  devised  ;   and,  practically,  tnis  system 
(though  fastidious  ears  may  profess  to  be  offended  by  it,)  it  must  produce  no  contemptible  harmony.  It  has,  how- 
ever, one  radical  f'ault.,;it  gives  all  the  keys  one  character.  In  any  other  system  of  temperament  some  intervals,  though 
of  the  same  denomination,  must  differ  by  a  minute  quantity  from  each  other ;  and  this  difference  falling  in  one 
part  of  the  scale  in  one  key,  in  another  in  another,  gives  a  peculiarity  of  quality  to  each  key,  which  the  ear  seizes 
and  enjoys  extremely.     This  fact,  in  which,  we   believe,  all  practical  musicians  will  agree,  is  alone  sufficient  to 
prove,  that  perfect  harmony  is  not  necessary  for  the  full  enjoyment  of  music.     Most  practical  musicians  seem  to 
have  no  fixed  or  certain  system  of  temperament ;  at  least  very  few  of  them  when  questioned  appear  to   have 
any  distinct  ideas  on  the  subject. 

It   is   a  mistake  to  suppose,  as  some   have  done,  that  temperament  applies  only  to  instruments  with  keys       244. 
and  fixed  tones.     Singers,  violin  players,  and  all  others  who  can  pass  through  every  gradation  of  tone,  must  all  Occasional 
temper,  or  they  could  never  keep  in  tune  with  each  other  or  with  themselves.     Any  one  who  should  keep  on  'empera- 
ascending  by  perfect  fifths,  and  descending  by  octaves  or  thirds,  would  soon  find  his  fundamental  pitch  grow  ment' 
sharper  and  sharper  till  he  could  at  last  neither  sing  nor  play ;  and  two  violin  players  accompanying  each  other, 
and  arriving  at  the  same  note  by  different  intervals,  would  find  a  continual  want  of  agreement. 

Musical  intervals  may  be  numerically  represented  by  the  logarithms  of  the  fractions   expressing  the  ratios  of      245. 
the  vibrations  of  the  notes  between  which  the  intervals  are  comprised  ;  for  the  interval  depending  only   on  this  Musical 
ratio,  and  the  sum  of  any  two  intervals  corresponding  to  the   product  of  their  respective  ratios,  the   logarithms  intervals  re- 
of  the  latter  are  the  proper  measures  of  the  magnitudes  of  the  former.     Thus  an  octave   corresponds  to  a  ratio  Pre!jenteJ 
of  2  :  1  of  the  vibrations  of  its  extreme  Sounds  ;  two  octaves  to  the  ratio  4  :  1  or  2J  :  1,  three  to  8  :  1  or  21  :  1,  rj{nl^f 
and  so  on ;  so  that  log.  2,  2.  log.  2,  3.  log.  2,  &c.  or  any  numbers  in   that   proportion,  are  proper   numerical 
representatives  of  these  intervals.  The  intervals  of  the  diatonic  scale  will,  therefore,  be  represented  logarithmically 
68  follows  • 

v<ji,.  iv.  5  L 


796 


S  O  U  N  D. 


Sound. 


Interval. 

Ratio. 

Logarithm. 

Approx. 

Differences. 

(l)to(l) 

0 

1 

000000 

0 

(1)  to  (2) 

Major  tone=  T. 

1 

005115 

51 

(1)  to  (3) 

Major  3d    =  III 

f 

009691 

97 

40  —  t 

(1)  to  (4) 

Minor4th=  IV. 

i 

012494 

125 

(1)  to  (5) 

Major  5th  =  V. 

t 

017609 

176 

51    =    T 

(1)  to  (6) 

Major  Cth  =  VI. 

022185 

222 

4o  -  —  t 

(1)  to  (7) 

Major  7th  =  VII. 

v 

027300 

273 

(1)  to  (8) 

Octave       =  VIII. 

2 

030103 

301 

Rift  H, 


The  approximate  values  of  the  intervals  being  all  true  to  a  500th  of  a  tone,  an  interval  far  too  minute  for  the 
nicest  ear  to  appreciate,  may  be  used  in  all  musical  calculations  where  high  multiples  of  them  are  not  taken. 

246.  It  will  be  observed  that  the  diatonic  scale  so  constructed,  consists  of  three  different  intervals  between  conse- 
Sequence  of  cutive  notes.     Thus,  the  interval  from  (1 )  to  (2)  is  5 1  parts  of  a  scale  on  which  the  octave  measures  301.     This 
intervals  in    js  called  a  major  tone,  T,  and  the  same  interval  occurs  again  between  (4)  and  (5),  and  between  (6)  and  (7),  as 
the  diatonic  w;jj  appear  by  referring  to  the  column  of  differences.     Again,  the  interval  from  (2)  to  (3)  is  46  such  parts  only, 

and  this,  which  occurs  again  between  (5)  and  (6),  is  called  a  minor  tone.,  (t).  Lastly,  the  interval  between  (3) 
and  (4)  and  between  (7)  and  (8)  is  28  such  parts,  and  is  called  (but  more  improperly)  a  semitone,  (9),  being  in 
fact  much  more  than  the  ha'If  of  either  a  major  or  a  minor  tone.  The  term  limma,  which  has  been  used  by 
some  authors,  is  much  preferable. 

247.  This  is  the  origin  of  what  is  called  the  enharmonic  diesis,  and  of  the  distinction  existing  between  the  sharp  of 
Enharmonic  one  note  and  the  flat  of  that  next  above  it;  a  distinction  essential  to  perfect  harmony,  but  which  cannot  be  main- 
diens.          tained  in  practice,  except  in  organs  and  complicated  instruments,  which  admit  a  great  variety  of  keys  and  pedals, 

or  in  stringed  instruments  or  the  voice,  where  all  gradations  of  tone  can  be  produced,  and  then  only  when  used 
without  a  fixed  accompaniment.  To  explain  this  distinction,  suppose,  in  the  course  of  a  piece  of  music,  com- 
menced in  the  key  of  (1),  we  should  modulate,  as  it  is  called,  into  the  key  of  (4),  its  sub-dominant;  that  is, 
change  our  key,  and  adopt  a  new  scale,  having  (4)  for  its  fundamental  tone.  To  make  the  new  scale  perfect, 
the  intervals  should  be  the  same,  and  succeed  each  other  in  the  same  order  as  in  the  original  key  (1).  That  is, 
setting  out  from  (4)  we  ought  to  have  for  our  sequence  of  intervals  T  t  0  T  t  T  0.  Now,  this  sequence  does 
not  takj  place  in  the  unaltered  scale  of  (1),  when  we  set  out  from  any  note  but  (1),  and  if  we  prolong  it  back- 
ward to  (4),  they  will  stand  thus, 

(4)        (5)        (6)        (7)        (1)        (2)        (3)        (4) 


' 

t 

T 

0 

T 

t 

e 

whereas  they  ought  to  stand  tl 
( 

1US, 

*)       ( 

T 

i)        (< 
t 

5)       (1 

0 

)        (1 

T 

)        (2 

t 

)      (a 

T 

)      0 

e 

)  &c. 
&c. 

The  first  two  intervals  are  the  same  in  both.  The  two  next  will  also  agree  if  vie  flatten  the  note  (7),  so  as  to 
make(7)b  —  (6)  =  0  and(l)  —  (7)b  =  T,  which  leaves  the  interval  (1)  —  (6)  the  same  as  before,  viz.  T  +  0, 
or  a  perfect  minor  third.  The  quantity  by  which  (7)  must  be  flattened  for  this  purpose,  or  (,")  —  (7)b,  is  equal 
tor  —  0=51  —  28  =  23,  and  this  is  the  amount  by  which  in  this  case  a  note  dilFei-s  from  its  flat.  As  to 
the  remaining  three  intervals,  the  difference  between  T  and  t  being  small,  amounting  only  to  5,  (which  is  the 

81 
logarithmic  representative  of  the  ratio  — ,  or  a  comma,)  the  sequence  t  r  6  is   hardly  distinguishable  from  ^  t  0, 

T  —  t 
and  if  the  note  (2)  be  tempered  flat  by  an  interval  =  — — — ,  or  half  a  comma,  this  sequence  will  in  both  cases 

P 

be  the  same,  and  our  two  scales  of  (1)  and  (4)  will  be  rendered  as  perfect  as  the  nature  of  the  case  will  permit, 
by  the  interpolation  of  only  one  new  note.  But,  on  the  other  hand,  suppose  we  would  modulate  into  the  key  (7). 
In  this  case  the  scales  will  stand  thus : 


(7)       (1)       (2)        (3)        (4)        (5)        (6)       (7) 


0 


t 

0 


0 


t 


Perfect  scale  of  (1). 
Perfect  scale  of  (7). 


T     |     0 

(7)  (!)»  (2)»  (3)  (4)jf  (5)S  (6)$  (7) 
This  change  will  require  the  interpolation  of  no  less  than  five  new  notes,  the  notes  (7)  and  (3)  being  the  only 
ones  that  remain  unchanged.  But  to  confine  ourselves  to  the  change  from  (6)  to  (6)8,  we  have  (7)  —  (6)  =  T 
and  (7)  -  (6)jf  =  0.  Consequently  (6)fl  -  (6)  =  T  -  e  =  23  =  (7)  -  (7)\>,  as  befure  determined.  But 
since  the  whole  interval  between  (6)  and  (7),  or  (7)  —  (6),  which  is  =  T  =  51,  is  more  than  double  of  this 
quantity,  the  flattened  note  (7)b  will  lie  nearer  to  the  higher  note  (7),  and  the  sharpened  one  (6)J  nearer  lo  the 
lower  one  (6)  than  a  note  arbitrarily  interpolated  half  way  between  them,  to  answer  both  purposes  approxi- 
mately, would  do,  and  thus  a  gap,  or,  as  it  is  termed,  a  diesis,  would  be  left  between  (6)S  and  (7)b. 
The  diesw  in  this  case  amounts  only  to  a  comma  (=  5),  or  the  tenth  part  of  a  major  tone.  (T)  (=  51),  in 


SOUND.  797 

Souna.     other  cases  it  would  be  greater.     But  in  all  cases  the  interval  between  any  note  and  its  sharp  is  considered  to  be 
j-v-^  equal  to  that  between  the  same  note  and  its  fiat.     Assuming  this  as  a  principle,  a  variety  of  systems  of  tempera- 
ment have  been  devised  for  producing  the  best  harmony  by  a  system  of  21  fixed  Sounds,  viz.  each  note  of  the  __ 
seven  in  the  scale,  with  its  sharp  and  flat,  (regarded  as  different).  MB*™ 

The  first  and  most  celebrated  is  that  of  Huygens.     He  supposes  the  octave  divided  into  31  equal  parts.     Of  Sou'nds'in 
these  a  whole  tone,  whether  ^  or  t,  (for  he  makes  all  his  tones  equal,)  consists  of  5,  a  limma  (or  an  approximate,  the  octave. 
or  tempered  value  of  G)  =  3,  the  interval  between  each  note  and  its  sharp  or  flat  =  2,  and  the  diesis  =  1.     This       249. 
gives  the  following  scale  of  intervals.  Huygens's 

(1)  (1)8  (2)b  (2)  (2)8   <S)b   (3)  (4)b   (3)8   (4)  (4)3  (5)b   (5)  (5)8   (6)b   (6)  (6)8  (7)b  (7)  (l)b   (7)8  (1)  'ys'em- 

212      2        12111212       2       ,12212111 
and  by  picking  our  notes  among  these,  we  may  obtain  a  scale  approaching  extremely  near  to  a  perfect  diatonic 
scale,  whichever  we  may  choose  for  our  key-note. 

Instead  of  dividing  the  octave  into  31  equal  parts,  Dr.  Smith  proposes  to  divide  it  into  50,  of  which  8  shall  con-       250. 
stitute  a  tempered  tone,  and  5  a  limma,  or  tempered  value  of  9,  and  the  interval  between  each  note,  and  its  sharp  Dr-  Smith's 
or  flat,  shall  =  3.     This  will  give  the  sequence  of  intervals  as  below.  system. 

(1)  (1)8  (2)b  (2)  (2)8  (3)  b  (3)  (4)  b  (3)8  (4)  (4)8   (5)  b  (5)  (5)8  (6)  b  (6)  (6)8  (7)  b  (7)  (1)  b  (7)8  (1). 

3233232         1232332332321         2 
This  scale,  he  observes,  approximates  insensibly  near  to  what  he  terms  the  system  of  equal  harmony,  a  system,  in 
our  opinion,  uselessly  refined,  and  founded  on  principles  for  which  the  reader  is  therefore  referred  to  his  Work  on 
Harmonics,  (Cambridge,  1749.) 

Either  system,  no  doubt,  will  give  very  good  harmony  ;  but  as  on  the  piano-forte  only  12  keys  can  be  admitted,       251. 
and  as  this  instrument  is  now  become  an  essential  element  in  all  concerts,  and  indeed  the  chief  of  all,  a  tempera-  Tempera- 
ment must  be  devised  which  will  accommodate   itself  to   that  condition.     Of  the  division  of  the  octave  into  12  ™ents 
equal  parts  we  have  already  spoken.     Its  fifths  are  all  too  flat,  and  its  major  thirds  all  too  sharp ;  and  the  harmony  the  piano- 
is  equally  imperfect  in  all  keys.     But  it  has  generally  been  considered  preferable  to  preserve  some  keys  more  free  ibrte. 
from  error,  partly  for  variety,  and  partly  because  keys  with  five  or  six  sharps  or  flats  are  comparatively  little  used, 
so  that  these  may  safely  be  left  more  imperfect,   (which   is   called  by  some  throwing  the  wolf  into  these  keys.) 
Dr.  Young  recommends  as  a  good  practical  temperament  to  tune  downwards  six  perfect  fifths  from  the  funda- 
mental note,  and  upwards   six  fifths  equally  imperfect   among  themselves.     Or,  as  he  observes  is  more  easily 
executed,  to  make  the  third  and  fifth  of  the  natural  scale  perfectly  correct,  to  interpose  between  their  octaves  the 
second  and  sixth,  so  as  to  make  three  fifths  equally  tempered,  and  to  descend  from  the  key-note  by  seven  perfect 
fifths,  which  will  complete  the  scale.  (Lectures  on  Natural  Philosophy,  vol.  i.  lect.  33.) 

The  system  called  by  Dr.  Smith  that  of  mean  tones,  or  the  vulgar  temperament,  supposes  the  octave  divided 
into  five  equal  tones  and  two  equal  limmas,  succeeding  each  other  in  the  order  a  a  fi  a  a  a  /3  instead  of  Ttd  T  trQ  m^an' 
as  in  the  diatonic  scale,  and  such  that  the  third  shall  be  perfect  and  the  fifth  tempered  a  little  flat.     These  con-  Or  vulgar"8' 
dilions  suffice  to  determine  a  and  J3,  for  we  have  tempera- 

5a-f  2/3  =  1  octave  =  3  T-f  2t  +  20,  meilt 

a  a  =1  third   =  T  -f  t, 

and  consequently  a  = ;  ft  :=  0  H ; 

or,   (since  r  +  t  =  09691,  and  T  -  t  —  00540,)  a  =  04845  and  /3  =  02938.     And  since  the  interval  from  the 
first  to  the  fifth  of  the  scale  in  this  system  is  =  3a  +  /3  =  2T-j-2-|-0 — ,  it  appears  that   this  is  flatter 

than  a  perfect  fifth  by  the  quantity  \  (r  —  t),  or  a  quarter  of  a  comma.     In  this  system  the  sharps  and  flats  may 
be  inserted  by  bisecting  the  larger  intervals. 

Mr.  Logier  has  lately,  in  a  Work  of  great  practical  utility  and   very  extensive  circulation  among  musical       253, 
students,  endeavoured  to  place  the  interpolation  of  the  intermediate  notes  between  those  of  the  natural  scale  on  Logier's 
d  priori  grounds,  by  assuming  the  flat  seventh  (7)b  as  the  seventh  harmonic  of  the  fundamental  note  (1),  that  is  system  m 
to  say,  the  note  produced  by  subdividing  into  seven  equal  parts  the  length  of  a  string  whose  fundamental  tone  harraony 
is  (1),  or  at  least  one  of  the  octaves  of  that  note.     There  is  something  ingenious  in  this  idea.     In  the  first  place 
it  completes  the  series  of  the  10  first  harmonics  or  notes,  whose  vibrations   are  multiples  by  1,  2,  3,  4,  5,  6,  7, 
8,  9,  10,  of  those  of  the  fundamental  tone,  which  would  thus  be,  in  their  order,  (1),  (1),  (5),  (1),  (3),  (5),  (7)b, 
(1),  (2),  (3),  or  octaves  of  these,  and  thus  derives  five  out  of  the  twelve  notes  of  the  octave  from  one  uniform 
principle.     Again,  it  gives  something  like  a  plausible  reason  for  the  prominent  importance  of  the  chord  of  the  flat 
seventh  (see  Art.  230.)  in  music.     This  chord,  in  fact,  which,  if  we  take  (1)  for  a  fundamental  note,  consists  ot 
the  notes  (1),  (3),  (5),  (7)b,  becomes  in  this  point  of  view  a  perfect  concord,  consisting  entirely  of  harmonics  of 
(1),  and  its  pulses  will  succeed  each  other  on  the  ear  in  a  cycle  comprising  four  vibrations  of  the   fundamental 
tone  (1),  five  of  the  next  (3),  six  of  the  next.  (5),  and  seven  of  the  essential  note  (7)b,  as  represented  in  fig.  35. 
The  succession  of  pulses  in  the  common  chord  is  also  represented  in   the  same  figure,  and  its  regularity  and 
pleasing  variety,  even  to  the  eye,  explains  its  agreeable  effect  en   the  ear.     It  is   for  musicians  to   say,  whether 
they  can  make  up  their  minds  to  regard  the  discord  of  the  sev.enth  in  the  light  of  a  perfect  concord  or  no.   There  is 
certainly  nothing  at  all  discordant  in  the  vulgar  sense  of  the  word,  i.  e.  unpleasant  in  its  Sound,  and  so  far  it  may 
be  regarded  as  at  least  "  discordia  concors,"  but  so  far  from  possessing  the  essential  character  of  a  concord,  that 
the  ear  is  satisfied  in  hearing  it,  and  expects  and  desires  no  more ;  there  is  no  discord  which  calls  so  urgently  for 
resolution.     But,  although  it  be  true,  that  the  seventh  harmonic  of  the  fundamental  note  lies  between  its  natural 

&  L  2 


798 


SOUND. 


Sound,  seventh  and  its  octave,  (it  must  lie  somewhere,)  yet,  in  fact,  it  is  materially  too  flat  a  Sound  to  be  used  as  a  good  flat 
^t^mm->  seventh  (7)b.  Its  actual  Sound  coincides  much  more  nearly  with  the  (6)8  of  Huygens  and  Smith  ;  and  this 
defeat,  though  it  might  be  tolerated  in  quick  compositions,  and  especially  in  piano-forte  music  where  the  notes 
are  not  held  on,  but  degrade  rapidly  in  intensity,  would  be  at  once  felt  in  a  slow  piece  on  the  organ.  It  is 
still  worse  if  we  derive  from  it,  by  a  similar  process,  the  intermediate  note  between  (5)  and  (6),  or  (6)b,  and 
thence  again  (5)b,  and  complete  the  chromatic  scale  of  twelve  notes  by  deriving  (3)b  according  to  the  same 
principle  from  (4),  and  (2)b  from  (3)b,  according  to  Mr.  Logier's  system  as  laid  down  by  him.*  The  (2)b 
thus  derived  would  hardly  be  distinguished  from  (1)  natural,  or  the  (5)b  from  (4)  natural,  as  the  following 
scale  will  show,  where  the  fractions  represent  the  ratios  of  the  vibrations  of  the  notes  above  them  to  those  of  the 
fundamental  tone  (1). 

(1).    (2)b,    (2).    (3)b,    (3),     (4),    (5)b,    (5),    (6)b,    (6),     (7)b,    (7),     (8). 

i.     tt.      *.     *.       *.      f    Mi!-     -I      tf,      f      t,       V.     2- 

Sevenths,  then,  tuned  on  Mr.  Logier's  principle,  will  require  a  much  more  violent  temperament  than  either  fifths 
or  thirds,  either  of  which  might  be  used  as  a  means  of  introducing  the  intermediate  notes  ;  and  the  system  must 
in  consequence  be  abandoned,  as  must  every  system  which  professes  to  render  musical  arithmetic  any  thing  more 
than  a  matter  of  convention  and  approximation. 

We  annex  here,  for  comparison,  a  Table  of  the  logarithmic  values  of  the  intervals  from  (1)  the  fundamental 
tone  to  all  the  other  notes  in  the  several  scales  of  21,  or  of  12  notes,  according  to  the  different  systems  and 
principles  above  mentioned. 

The  numbers  marked  thus  (*)  are  what  would  be  given  by  pushing  the  application  of  Mr.  Logier's  principle 
through  the  whole  scale,  and  are  inserted  only  to  show  the  rapid  progressive  effect  of  flattening  by  a  series  of 
untempered  harmonic  sevenths. 


Part  il. 


Com  para- 
live  Table  ol 
different 
scales  of 
tempera- 
ment, *c 


Designation  of  Note. 

Intervals  in  the  perlect 
Diatonic  Sea  e. 

Diatonic 
Scale  with 
its  tones 
biatcttd. 

ft/stem  of 
mean 
tones  or 
•mlgar 
tempera- 
ment, 
Art  «» 

System  of 
equal 
Tempera- 
ment, 
Art.  242. 

Dr. 
Younj'l 
first 

SjU.-rn, 
Art.  231. 

Dr. 
Yoting-I 
second 
System, 

An.  251. 

Mr. 

Scale.' 

i 

Hujgent's 

System  of 
21  naes, 
Art.  240. 

Smith's 
approxi- 
mate 
Svstem 
of  *l  notes, 
Art.  250. 

Mean 

Ratio. 

Log. 

AST 

(1) 

1 

00000 

0 

0 

0 

0 

0 

5 

0 

0 

0 

0 

(1)3 

19 

18 

(1)8  =  (2)b 

— 

_ 

— 

25 

24 

25 

23 

23 

9* 

24 

(2)b 

29 

30 

(2) 

3. 

05115 

51 

51 

48 

50 

49 

49 

51 

49 

•18 

49 

(2)3 

68 

66 

(2)5  =  (3)b 

— 

— 

_ 

74 

73 

75 

74 

74 

67 

73 

(3)b 

78 

78 

(3) 

5 

09691 

97 

97 

97 

100 

98 

97 

97 

97 

96 

98 

(4)b 

107 

108 

(3)3 

117 

114 

(4) 

J 

12494 

125 

125 

126 

125 

125 

125 

125 

126 

126 

125 

146 

144 

(4)3  =  (5)b 

_ 

.^ 

„!.._. 

150 

150 

151 

148 

148  i    127* 

150 

(5)b 

155 

157 

(5) 

! 

17609 

176 

176 

175 

176 

175 

176 

176 

175 

175 

176 

194 

193 

(5)»  =  (6)b 





^^ 

199 

199 

201 

199 

199 

185* 

199 

(6)b 

204 

205 

«. 

22185 

222 

222 

223 

226 

224 

224 

222 

223 

223 

224 

(6)3 

243 

241 

(6)3  =  (7)b 

— 

— 

— 

247 

247 

251 

250 

250 

243 

248 

(7)b 

252 

253 

(7) 

V 

27300 

273 

273 

272 

276 

273 

273 

273 

272 

271 

273 

(8)b  =  (l)b 

282 

283 

(7)3 

291 

289 

(8)  =  (1) 

2 

30103 

301 

301 

301 

301 

301 

301 

301 

301 

301 

301 

Comma  .  .  . 

81 

00540 

5i 

Limma  =  0 

T^ 

02803 

28 

Minor  tone  t 

10 

04576 

46 

Major  tone  T 

0 

05115 

51 

Minor  third 

* 

07918 

79 

Major  third 

A 

09691 

97 

Fourth  

I 

12494 

125 

Fifth  

• 

I 

17609 

176 

» I 

»  Syjtemof  the  .trim*  of  Mutic  and  Practical  Competition,  p.  50.     We  should,  however,  remark  that  the  powerful  descending  tendency 
of  the  chord  of  'he  flat  seventh  is  necessarily  much  augmented  by  tuning  the  (7)b  too  flat. 


S  O  U  N  D. 


799 


Sound.          The  last  column  contains  a  scale  derived  by  taking  the  mean  of  all  those  in  the  other  columns  which  differ  in     part  II. 
— >v— '  the  principle  of  their  origin,  (excepting  those  in  the  tenth  column,  for  obvious  reasons.)     It  approaches  through   ^— ~v— J 
its  whole  extent  so  near  to  the  system  of  mean  tones  in   col.  6,  as  to  be  quite   undistinguishable  from  it ;    the       255. 
deviation  in  no  case  exceeding  a  single  unit,  or  a  fiftieth  part  of  a  tone.    This  system,  then,  though  the  most  iuarti-  Remarks, 
ficial,  is  probably  as  good  as  any  which  the  nature  of  music  admits,  holding  a  sort  of  mean  between  the  advan- 
tages and  defects  of  all  the  rest.     Consult  on  Temperament  and  on  Musical  Scales,  Salinas,  de  Musica,  (1577  ;) 
Zarlino,  Dimostrazione  Armoniche,  and  Istituzione  Armoniche;  Deschales,  Cursus  Mathematicus  de  Progressu 
Musico  ;  Sauveur,  Mem.  Acad.  Par.    1700;  Smith,  Harmonics;  Pepusch,  Phil.  Trans.  Lond.   1746;  Farey, 
Phil.  Mag.  xxviii.  ;  Young's  Lectures  and  his  Catalogue  of  Authors  in  vol.  ii. 

For  one  purpose,  that  of  explaining  to  beginners  the  notes,  intervals,  and  rules  of  music,  the  system  of  equal       256. 
temperament,  which  supposes  the  octave  divided  into  12  equal  parts,  which  in  this  system  only  are  really  semi-  Expression 
tones,  has  the  advantage  of  avoiding  all  discussions  and  puzzling  explanations  on  the  nature  of  harmony,  as  it  oftheprin- 
makes  all  intervals  which  are  called  by  the  same  name  strictly  alike.     Regarding  the  octave  as  consisting  of  J-'P3' cnor<ls 
12  semitones,  and  designating  its  notes  in  succession,  beginning  with  the  fundamental  note,  by  0,  1,  2,  3,  4,  5,  6,  numbers!1 ' 
7,  8,  9,  10,  11,  12,  &c.  it  will  not  be  amiss  if  we  write  down  in  this  notation   the  principal  scales,  chords,  &c. 
which  occur  in  music. 

2,    3 

2, 

2,         3; 

10, 


1, 


4, 

5, 

6,    7, 

8,    9, 

10,  11,  12. 

4, 

5, 

7, 

9, 

11.  12. 

5, 

7, 

9, 

11,  12.( 

8, 

7, 

5, 

3, 

2,    0. 

0, 

4, 

7, 

0, 

3, 

7, 

0, 

4, 

7. 

10. 

o, 

4. 

7. 

9. 

0, 

2, 

4, 

7. 

0, 

3, 

7, 

9. 

°. 

3, 

c, 

9. 

o, 

2, 

6, 

8. 

Notation  of 
chromatic 
numbers 
explained. 


Chromatic  scale 0, 

Diatonic  scale 0, 

Minor  scale  ascending 0, 

Minor  scale  descending 12, 

Common  major  chord 

Minor  chord 

Fundamental  discord  of  the  flat  seventh  . 
Chord  of  the  added  sixth  (Logier,  Ex.  117.) 

Chord  of  the  ninth   

Minor  chord  with  added  sixth 

Diminished  seventh    

Chord  of  the  sharp  sixth  (Logier,  Ex.  197.) 

These  are  all  the  chords,  consisting  of  four  different  notes  (or  tetrachords)  in  common  use  in  music.     As  to        257. 
pentachords,  such  as  what  have  been  called  the  major  and  minor  nil  th,  and  compound  sharp  sixth,  whose  notes  Triads, 
are  respectively  0,  2,  4,  7,  10  ;  0,  1,  4,  7,  10,  and  0,  4,  6, 7,  10,  (Logier,  Ex.  212,  158,  and  202.)  they  are   in  fact,  tetrachords, 
only  chords  of  the  seventh  (0,  4,  7,  10)  with  a  fifth  note  violently  forced  in  ;  the  effect  being  to  distract  the  ear  by 
aharsh  discoid,  out  of  which  it  is  but  too  glad  to  escape,  to  be  very  nice  about  its  resolution.    In  like  manner  the 
pentachord  0,  2,  5, 7, 11,  or  the  chord  of  the  eleventh,  is  the  chord  of  the  seventh  accompanied  by  the  sub-dominant  of 
its   radical  note,    and  thus  anticipating  its  resolution  ;  as  is  easily  seen  by  adding  5  to  each  of  its  component 
numbers,  when  it  becomes  5,  7, 10,  12,  16,  or,  which  is  the  same  thing,  5,  7,  10,  0,  4,  or  0,  4,  5.  7,  10,  (since  the 
addition  or  subtraction  of  12  semitones,  or  an  octave,  does  not  alter  the  character  of  the  Sounds  ;)  and  in  the 
same  way  may  other  pentachords  be  formed,  as  0, 3,  4, 7,  10 ;  0,  4,  7,  8, 10  ;  (Ex.  dementi's  Sonatas,  Op.  22. 
son.  1.  bars  68,  69.)  0,4,7,9,10.     As  to  such  combinations  as  the  hexaehord  0,  2,  5,  7,  9, 11,  or  "  the  chord  of 
the  13th,"   (Logier,  Ex.  273.)    in    which  only  one  note  of  the  whole  natural  scale  (4)  is  wanting,  they  are 
abominable  jangles,  as  offensive  to  a  simple  and  unvitiated  ear,   as  the  mixed  flavours  and  haitt-goiits  of  the 
palled  epicure  are  to  an  appetite  not  spoiled  by  artificial  excitement. 

The  reader  who  would  try  these  chords  on  the  piano-forte,  has  only  to  place  his  finger  on  any  black  or  white 
key  as  a  radical  note,  and  also  on  the  keys  distant  from  that  one  by  the  numbers  of  semitones  (reckoning 
upwards)  marked  in  the  designation  of  the  chord  as  above.  Thus  to  produce  the  chord  of  the  sharp  sixth 
having  Db  for  its  radical  note.  The  note  0  corresponds  to  Db,  2  to  Eb,  6  to  Gb,  and  8  to  Ab,  which  are,  there- 
fore, the  notes  to  be  struck  together,  (to  whatever  octaves  of  the  instrument  they  may  be  afterwards  transferred, 
as  the  rules  of  composition  may  dictate,)  and  so  of  others. 

Any  of  these  chords  is  said  to  be  inverted,  when,  instead  of  taking  0  lor  the  initial  note,  we  regard  any  other 
of  its  component  Sounds  as  such.  On  the  system  of  notation  here  employed,  (which  we  will  term  the 
system  of  Chromatic  numbers  to  distinguish  it  from  those  in  Art.  234,  to  which  the  term  Diatonic  numbers  may  be 
applied,)  nothing  is  easier  than  to  represent  the  inversions  of  any  chord.  Take,  for  instance,  the  major  concord, 
0,  4,  7.  The  addition  of  12  (the  octave  of  0)  does  not  change  the  chord ;  so  that  it  may  be  written  thus, 
0,4,7,  12,  or,  leaving  out  the  first  note,  and  adopting  4  for  the  initial  note,  4,  7,  12.  If,  now,  we  choose  to 
regard  the  note  4  as  an  initial  one,  and  count  upwards  from  it,  we  have  only  to  subtract  4  from  each  of  these 
numbers,  and  we  get  0,  3,  8  for  a  first  inversion.  Appending  12  to  this  again,  and  rejecting  the  initial  0,  it 
becomes  3,  8,  12,  from  each  of  which  numbers  subtracting  3  we  get  the  second  inversion,  0,  5,  9.  If  we  repeat 
the  same  process  on  this  we  fall  back  on  the  original  combination.  Thus  we  see  that  this  chord  admits  of  only 
two  inversions.  Again,  suppose  we  would  find  the  inversions  of  the  chord  of  the  added  sixth,  or  0,  4,  7,  9.  The 
process  will  stand  thus: 


258. 


259. 

'nvenioni 


8,  11,  12,  according  to  Logier  and  ethers.     Fide,  on  this  point,  Weber's  excellent  and  scientific  work, 


300 


SOUND. 


Sound. 


0,  4,  7,  9,  12 
4,  7,  9,  12 
4,  4,  4.  4 

1),  3,  5,    8         1st  inversion. 
3,  5,    8,  12 
3,  3,    3,    3 

2d  inversion. 


"art  II. 


0,2, 
2, 
2, 


9 
9,  12 

2,    2 


260. 


0,    3.    7,  10         3d  inversion. 
3,    7,  10,  12 
3.    3,    3,    3 

Original  chord  again 0,    4,    7,    9 

Thus  we  see  that  this  chord  admits  of  three  distinct  inversions.  In  general,  a  triad  admits  of  three  forms,  or 
one  original,  and  two  inversions,  a  tetrachord  of  4,  a.  pentachord  of  5,  and  so  on  ;  though  it  may  happen,  as  in 
the  case  of  the  triad  0,  4,  8,  or  the  tetrachords  0, 2,  6,  8,  and  0,  3,  6,  9,  that  some  or  all  of  the  inversions 
reproduce  the  original  chord. 

If  we  go  through  the  same  process  for  other  triads  and  tetrachords,  we  get  their  inversions  as  follows: 

Triads. 


1st  form,  or 
radical. 

2d  form,  or 
1st  inversion 

3d  form,  or 
2d  inversion 

0,   4,    7 

0,  3,  8 

0,  5,  9 

0,  3,  7 

0,  4,  9 

0,  5,  8 

Equivocal  triad,  or  double  third  

0,  4,  8 

0,  4,  8 

0,  4,  8 

Tdrachords. 


1st  form. 

2d  form. 

3d  form. 

4th  form. 

0    4    7    10 

036      8 

0    3    5,     9 

0,  2    6      9 

0.  4    7      9 

0    3    5,     8 

0,  2,  5,    9 

0,  3,  7    10 

024      7 

0,  2    5    10 

0    3    8    10 

057      9 

0,  3    7      9 

0    4,  6,     9 

0.  2,  5,     8 

0,  3    6    10 

0    2    7.     9 

0    5,  7,  10 

0,  2,  5,     7 

0,  3,  5    10 

Equivocal. 

0,  3    fi,     9 

0    3,  6      9 

0,  3,  6,     9 

0,  3    6,     9 

0    4    6    10 

0-26      8 

0.  4    6    10 

0,  2    6      8 

Pentachords, 


1  st  form. 

2d  form. 

3d  form. 

4th  form. 

5lh  form. 

Minor  ninth  (Logier, 
Ex.  158.)  
Compound        sharp 
sixth  (Loff.Ejr.202) 
Major   ninth    (Log. 
Ex.  212)  

0,  1,  4,  7,  10 
0,  4,  6,  7,  10 
0,  2,  4,  7,  10 

0,  3,  6,  9,  11 
0,  2,  3,  6,     8 
0,  2,  5,  8,  10 

0,  3,  6,  8,     9 
0,  1,  4,  6,  10 
0,  3,  6,  8,  10 

0,  3,  5,  6,     9 
0.  3,  5,  9,  11 
0,  3,  5,  7,     9 

0,  2,  3,  6,     9 
0,  2,  6,  8,     9 
0,  2,  4,  6      9 

Eleventh  (Log.  Ex. 
267). 

0,  4,  5,  7,  10 

0,  1,  3,  6,     8 

0.  2,  5,  7,  11 

0,  3,  5,  9,  10 

0,  2,  6.  7,     9 

261. 

Remirks. 


Aliquot 


These  chords,  thus  figured  and  arranged,  afford  room  for  some  remarks  of  importance.     In  the  first  place  we 
observe  that  they  all,  with   the   exception  of  the  triad  0,  4,  8,   and   the  tetrachords  0,  2,  6,  8,  and  0,  3,  6,  9, 
contain  a  major  or  minor  concord,  0,  4,  7,  or  0, 3,  7.    This  seems  necessary  to  give  any  chord  a  decided  character ; 
for  the  excepted  cases  above  specified  have  all  an  equivocal  effect  and   leave  the  ear  in  suspense  whither  the 
modulation  will  lead.     For  with  respect  to  the  chords  0,  4,  8,  and  0,  3,  6,  9,  they  divide  the  octave  equally,  the 
division  of    one  into  three  major  thirds,  the  other  into  four  minor,  as  is  immediately  seen  if  we  write  them  thus,  0,  4,  8,  12, 
the  octave,    and  0,  3,  6,  9,  12.     In  consequence,  all  their  inversions  are  similar  to  the  original  chords,  and  they  are  equally 
related,  the  former  to  three,  and  the  latter  to  four  different  keys,  and  may  lead  into  either  of  them,  according  as  a 


SOUND.  801 

Sound.     note  added  so  as  to  form  a  dominant  seventh,  or  anticipative  sub-dominant,  or  some  other  powerful  leading    p.irt  II. 
interval,  or  with  either  of  their  component  notes,  shall  decide.     This  is  one  mode  of  conceiving  the  chord  of  the  --— ~  -— ' ' 
minor  ninth,  which  may  be  either  regarded  as  a  chord  of  the  seventh,  with  the  first  semitone  1  added,  (as  in  it« 
first  form  above,)  or  as  a  diminished  seventh,  0,  3,  6,  9,  with  the  note  11  added,  as  in  the  first  inversion,  with  8  as 
in  the  second,  or  with  2  as  in  the  third,  either  of  which  makes  aflat  seventh  with  one  or  other  of  its  notes. 

The  transitions  thus  produced  by  means  of  the  tetrachord  0,  3,  6,  9,  are  peculiarly  graceful.     It  is  otherwise       262. 
with  the  equivocal  triad  0,  4, 8,  which   is  essentially  harsh  and  unpleasing,   (in  spite  of  the  perfect  harmony  Equivocal 
which,  if  we  were  to  leave  out  the  octave  and  tune  its  thirds  perfect,  its  members  must  produce  with  each  other,  cnords- 
since  it  would  be  in   that  case  an  absolute  concord.)     Whether  this  chord,  or  that  which  we  have  called  the 
triple  fifth,  has  ever  been,  or  can  be,  used  in  music,  we  know  not,  though  perhaps,  properly  handled,  it  might 
become  a  source  of  modulation  ;  which,  however,  is  for  practical  musicians  to  consider. 

The  chord  of  the  sharp  sixth  0,2,  6,  8  is  also  equivocal,  arising  from   a  double  aliquot  division  of  the  octave,       263 
and  the  two  last  of  its   inverted  forms  being  therefore  merely  repetitions  of  the  two  first.     Like  the  diminished  Relations 
seventh,  then,  it  holds  the  ear  in  suspense,  till  the  addition  of  another  note  decides  the  course  the  modulation  shall  t^e™™r'^ 
take,  and  the  chord  so  arising  is  the  compound  sharp  sixth.  (Seethe  inversions  of  this  latter  chord  compared  with  chords  and 
those  of  the  former.)  pentachord! 

In  like  manner  the  major  ninth  contains  both  a  ninth  and  a  seventh,  though  not  the  other  accompaniments  of 
the  seventh.  The  tetrachord  which  (for  want  of  another  name,  we  have  called  the  minor  added  sixth,  from  its 
being  a  minor  concord  with  a  sixth  added)  is  related  to  this  compound  ninth  in  the  same  way  as  has  just  been 
pointed  out  with  respect  to  the  chords  of  the  diminished  seventh  and  minor  ninth,  and  to  those  of  the  sharp  sixth 
and  compound  sharp  sixth  ;  the  character  of  the  tetrachord,  which  is  undecided  of  itself,  and  admits  of  more  than 
one  resolution,  being  determined  by  the  note  added  in  the  pentachord  so  as  to  form  a  dominant  seventh  with  some 
one  or  other  of  its  other  members. 

The  chord  of  the  eleventh  otters  room  for  a  remark   analogous  to  what  we  have  before  observed  (Art.  262.)        264. 
respecting  the  equivocal  triad  0,4,  8.     It  contains  within  itself  three  fifths  and  a  major  third;  as  is  obvious  if  we  <?llolj<J  of 
take  its  fifth  form  0,  2,  6,  7,  9,  and  transfer  the  notes  2,  6,  and  9  to  the  next  octave  above,  when  it  will  become  j^j^d1"1 
0,  14,  18,  7,  21,  or  0,  7, 14, 18,  21.     The  notes  0,  7,  14,  21,  in  this  arrangement,  make  fifths  with  each  other,  and 
the  note  18  forms  with  14  a  major  third  ;  if,  then,  the  intervals  were  tuned  perfect,  their  vibrations  would  succeed 
each  other  in  a  regular  cycle,  but  if  the  cycle  formed  by  two  perfect  thirds,  which  requires  only  25  vibrations 
of  its  highest  note,  or  16  of  its  lowest  to  complete  it,  is  too  complex  for  the  ear  to  relish,  the  cycle  of  three  perfect 
fifths,  which  requires  27,  will  already  be  too  complex  ;  and  if  we  add  to  this  a  major  third,  the  ear  will  lose  all 
ense  of  recurrence,  and  only  discord  will  result. 

But  to  place  this  in  clearer  evidence,  we  need  go  no  further  than  the  chord  of  the  ninth,  which,  when  written        255 
thus,  0,  4, 7,  14,  manifests  a  major  third,  (0, 4,)  a  fifth,  (0,  7,)  and  a  double  fifth,  (0,  7  +  7,)  of  the  fundamental  chords  o{ 
note,  and  therefore,  if  tuned  perfect,  would  excite  a  sense  of  perfect  concord,  wtre  not  the  period  of  recurrence  of  the  ninth 
the  vibrations  too  long  for  the  ear  to  seize  ;  and  a  similar  remark  applies  to  the  discord  of  the  seventh,  which  consists  an'I  seventh 
of  a  major  third,  a  fifth,  and  a  double  fourth,  from  the  fundamental   tone  (0,  4, 7,  5  +  5.)     It  may  be  that  the  analvsed- 
harshness  of  the  triad  0,  4,  8,  and  of  the  tetrachord  0,  2,  7,  9,  the  former  consisting,  if  tuned  perfect,  of  a  third 
and  double  third,  the  latter  of  a  fifth,  a  double  fifth,  and  a  triple  fifth,  may  arise  from  an  imperfect,  or  obscure, 
and   therefore  unsatisfactory,  perception  of  the  cycles  of  their  vibrations  by  the  ear,  the   former,   as  before 
remarked,  occupying  25,  and  the  latter  27,  single  vibrations  of  the  highest  note.     But  it  is  time  to  leave  these 
speculations. 

§  V.  Of  the  Sonorous  Vibrations  of  Bars,  Rods,  and  Plates 

The  vibrations  of  all  bodies,  if  of  a  proper  degree  of  frequency,  and  of  sufficient  force  to  be  communicated  £66 
through  the  air,  or  any  other  intermedium,  to  our  organs  of  hearing,  produce  Sounds  whose  pitch  depends  on 
their  frequency ;  and  their  force  and  quality  on  the  extent  and  other  mechanical  circumstances  of  the  vibrations, 
and  the  nature  of  the  vibrating  body.  The  mathematical  investigation  of  these  vibratory  motions  is  altogether 
foreign  to  our  purpose.  It  is  a  branch,  and  one  of  the  most  intricate  and  least  manageable  branches,  of 
Dynamics,  and  we  shall,  therefore,  refer  our  readers  for  its  theory  and  details  to  the  writings  of  the  various 
eminent  authors  who  have  discussed  it.  See  Bernoulli!,  Com.  Petrop.  vol.  xiii.  On  the  Vibrations  of  Laminae ; 
and  Nov.  Com.  Petrop.  vol.  xv.  :  Euler,  Com.  Petrop.  vol.  vii.,  Nov.  Com.  vol.  xvii.,  and  Act.  Petrop.  vol.  iii.  Referenct,Sl 
On  the  Vibrations  of  Plates ;  Riccati,  Soc.  Ital.  vol.  i.  p.  444  ;  Lexell,  On  the  Vibrations  of  Rings,  Act.  Petrop. 
1781  ;  Lambert,  On  the  Sounds  of  Elastic  Bodies,  N.  Act.  Helv.  vol.  i.  ;  J.  Bernoulli!,  On  the  Vibrations  of 
Rectangular  Plates,  N.  Act.  Petrop.  1787  ;  Biot,  On  the  Vibrations  of  Surfaces,  Mem.  Inst.  vol.  iv. 

A  solid  body  may  vibrate,  either  in  consequence  of  its  inherent  elasticity,  by  which  it  tends  to  return  to  its  own        267 
proper  figure  and  state  when  forcibly  deranged,  or  in  consequence  of  an  external  tension.     To  the  former  sort  of  Various 
vibrations  helong  those  of  rods,  tuning-forks,  plates,  rings,  bells,  gongs,  and  vessels   of  all  shapes,  or  generally,  ways  in 
of  all  solid  masses  which  ring  when  struck.     To  the  latter,  those  of  vibrating  strings  and  membranes,  such  as  which 
the  parchment  of  a  drum  or  tambourin,  &c.     But,  further,  a  solid   may  vibrate  by  its  own   proper  elasticity  in  so'l(is  m*T 
two  very  different  ways.     First,  an  undulation  may  be  propagated  through  it,  as  through  an  elastic  compressible 
medium,  and,  in  this  case,  the  waves  will  consist  of  alternate  strata  of  condensed  and  rarefied   solid  matter, 
precisely  similar  to  those  of  an  elastic  fluid.     If  the  solid  be  homogeneous,  such  as  the  metals,  glass,  £c.,  the 
<>lasticity  being  the  same  in  all  directions,  the  waves  will  be  propagated  from  the  centre  of  disturbance,  according 


802  SOUND. 

Sound       to  exactly  the  same  laws  as  in  a  mass  of  air  of  the  same  shape.     But  if  crystallized,  this  may  not  be  the  case,  01 
v.—  ^-~-*   the  vibrations  instead  of  being  in  the  direction  of  the  propagated  wave,   may  be   transverse,  or  oblique  to  it,  or  V 

may  even  not  be  confined  to  one  plane,  but  may  be  performed  in  circles  or  ellipses.     See  Article  LIGHT. 
.jgg  If  a  straight  rod  of  glass,  or  a  metal,    to   be  struck   at  the  end   in   the  direction  of  its  length,   or  rubbed 

LongitudV  lengthways  with  a  moistened  finger,  it  will  yield  a  musical  Sound,  which,  unless  its  length  be  very  great,  will  be 
ml  vibra-  of  an  extremely  acute  pitch  ;  much  more  so  than  in  the  case  of  a  column  of  air  of  the  same  length.  The  reason 
tions  of  a  of  this  is  the  greater  velocity  with  which  Sound  is  propagated  in  solids  than  in  i\ir.  Thus  the  velocity  of 
straight  rod.  propagation  in  cast-iron  being  10J  times  that  in  air,  a  rod  of  cast-iron  so  excited  will  yield  for  its  fundamental 

note  a  Sound  identical  with  that  of  an  organ  pipe  of  —  -  of  its  length  stopped  at  both  ends,  or  —  of  its    length 

if  open  at  one  end.  See  §  III.,  all  the  details  of  which  are  applicable  to  the  present  case.  To  such  vibrations, 
how  pro-  C'hladni,  who  first  noticed  them  in  long  wires,  has  applied  the  term  longitudinal.  (Art.  Acad.  Erfurt,  1796.)  To 
duced.  produce  the  harmonics  of  such  a  rod  or  wire  he  held  it  lightly  at  the  place  of  one  of  its  intended  nodes  between 
the  finger  and  thumb,  and  applied  the  friction  in  the  middle  of  one  of  the  vibrating  segments.  If  the  rod  be  of 
metal,  the  friction  which  he  found  to  succeed,  was  that  of  a  bit  of  cloth  sprinkled  with  powdered  rosin,  if  of 
glass,  the  cloth,  or  the  finger,  may  be  moistened  and  touched  with  some  very  fine  sand  or  pumice  powder.  It 
may  be  observed  here,  that,  generally  speaking,  a  fiddle-bow  well  rosined  is  the  readiest  and  most  convenient 
means  of  setting  solid  bodies  in  vibration.  To  educe  their  gravest  or  fundamental  tones,  the  bow  must  be 
pressed  hard  and  drawn  slowly,  but  for  the  higher  harmonics,  a  short  swift  stroke  with  light  pressure  is  most 
proper.  In  all  cases  the  point  intended  to  be  a  node  must  be  lightly  touched  with  the  finger,  and  the  vibration 
must  be  excited  (as  above  said)  in  the  middle  of  a  ventral  segment.  Such  is  the  case  analysed  by  Chladni.  In 
general,  however,  the  vibrations  of  a  cylindrical  rod  or  tube  so  excited  are  more  complex.  See  Art.  286,  Index, 
Art.  Longitudinal  Vibrations. 

2fig  Hut  by  far  the  most  usual  species  of  vibration  executed  by  solid  bodies  is  that  in  which  their  external  form  is 

Transverse    f°rc'b'y  changed,  and  recovered  again  by  their  spring.     The  simplest  case  is   ihut  of  a  rod  executing  vibrations 
vibrationsoi  to  and  fro  in  a  direction  transverse  to  its  length.     This  case  has   been  investigated  mathematically  by  D.  Ber- 
a  rigid  rod.  nouilli  and  Euler,  as  also  by  Riccati  ;  (see  the  list  of  authors  above  cited,  Art.  266  ;)  and  their  results  have  been 
compared   with  those  of  experiment  by  Chladni,  Acovst.  sec.  5,  and  found  correct.     The   cases  enumerated  by 
Chladni  are  six  in  number. 

Fig.  36.  1.  When  one  end  of  the  rod  is  firmly  fixed  in  a  vice  or  let  into  a  wall,  the  other  quite  free.     In  this  case  the 

Enumera-     curvature  assumed  by  the  rod  in  its  vibrations  must  of  necessity  have  its  axis  or  position  of  rest  for  a  tangent,  as 

"'          fi£-  36' 


cases 


cases 

Fig.  37.  2.  One  end  applied  or  pressed  perpendicularly  against  an  obstacle,  the  other  free.     In  this  case,  the  excur- 

sions of  the  applied  end  to  and  fro  are  prevented  b\  the  friction  and  adhesion  to  the  obstacle,  but  the  axis  is  not 

of  necessity  a  tangent.  See  fig.  37. 
Fig.  38.  3.  Both  ends  free.  Fig.  38. 

Fig.  39.  4.   Both  ends  applied.  Fig.  39 

Fig.  40.  5.  Both  ends  fixed.  Fig.  40. 

Fig  41.  6.  One  end  fixed,  the  other  applied.  Fig.  41. 

270.  All  these  cases  have  been  examined  by  Chladni  at  length.     We  shall,   however,   select  only  the  fourth  case 
Examina-     where  both  ends  are  applied,  because  it  will  afford  room  for  an  important  remark.    In  this,  then,  the  several  modes 
turn  of  one   of  vibration  corresponding  to  1,2,  3,  4,  5   vibrating  or  ventral  segments  of  the  rod  will  be  as  in   figs.  39,  43,  44. 
rigV'sa  43  ^*ow  tnese  are  s'm'lar  to  tne  curves  which  would  be  assumed  by  a  vibrating  string  under  the  same  circum- 
41           "'  stances  of  subdivision.     But  the  notes  produced  are  very  different.     For  whereas   in  the  case  of  a  string  the 

vibrations  of  the  successive  harmonics  are  represented  by  1,  2,  3,  4,  5,  &c.  ;  in  that  of  a  rod  they  arc  represented 
by  the  squares  of  these  numbers  1,  4,  9,  25,  &c.,  which  correspond  to  double  the  former  intervals.  In  all  the  other 
cases  the  series  is  still  less  simple. 

271.  This  alone  suffices  to  shew  the  insufficiency  of  any  attempt  to  establish,  as  some  have  wished  to  do,  the  whole 
Remark  on    theory  of  harmony  and  music  on  the  aliquot  subdivision  of  a   vibrating  string.     Had  vibrating  rods  or  steel 
ilie  origin     springs  (which  yield  an  exquisite  tone)  been  always  used  instead  of  stretched  chords,  such  an  idea  would  never 

harmony.  nave  suggested  itself,  yet  no  doubt  our  notions  of  harmony  would  have  been  what  they  now  are.  The  same 
remark  applies  still  more  forcibly  to  the  modes  of  subdivision  of  vibrating  surfaces,  which  in  many  cases  have 
their  harmonics  altogether  irreducible  to  any  musical  scale. 

272.  A  rectangular  plate  may  be  regarded  as  au  assemblage  of  straight  rods  of  equal  length,  ranged  parallel  to  each 
Vibrations    other.     Supposing  such  an  assemblage  all  set  in   vibration  similarly  and  at  once,  they  will  retain  their  parallel 
<>f  a  rectan-  juxtaposition  during  their  vibration,  and  may,  therefore,  be  supposed  to  adhere,  and  form  a  plate.     Consequently, 

:    among  the  possible  series  of  vibrations  of  a  rectangular  plate  will  be  found  all  those  of  a  rigid  rod.     Accordingly. 

simplest        when  fixed,  (for  instance,)  by  one  of  its  edges  in  a  vice,  with  its  plane  parallel  to  the  horizon  and  strewed  over 

case.  with  sand,  if  it  be  set  in  vibration  by  a  fiddle-bow  and  touched  in  one  of  its  possible  nodes,  its  subdivisions  will 

be  rendered  visible  to  the  eye,  by  the  sand  being  thrown    away  from  the  vibrating  'parts  and  accumulating  on 

those  at  rest.     Thus  the  plate  will  be  crossed  transversely  by  a  series  of  nodal  lines  marked  in  sand,  and  whose 

distances  from  each  other  and  from  the  ends  of  the  plate  may  be  measured  at  leisure. 

273.  But  besides  these,  rectangular  plates  are  susceptible  of  other  modes  of  subdivision,  having  two  sets  of  nodal 
Other  cases,  lines,  straight  or  curved,  crossing  at  right  angles,  or  otherwise,  and  dividing  the  plate   into   smaller  plates,  each 

vibrating  in  its  middle,  and  at  rest  at  its  edges,   and   every  two  contiguous  plates   separated    by  a   nodal  line 
making  their  simultaneous  excursions  on  contrary  sides  of  their  slate  of  rest. 
l!74.          To  produce  these  subdivisions,  and  to  render  them  visible,    Uke  a   rectangular  plate  (for  simplicity  we  will 


SOUND. 


803 


Sound,  suppose  it  a  square)  of  glass,  or  metal,  of  an  even  thickness,  not  too  thick,  and  holding  it  firm  between  the 
— •v—'  points  of  the  finger  and  thumb  of  the  left  hand,  or  between  two  points  of  a  clamp-screw  covered  with  cork  or 
leather  so  as  not  to  jar,  taking  care  to  keep  the  pressure  confined  to  as  small  a  space  as  possible,  draw  a  rosined 
bow  over  the  edge,  which  should  be  smoothed  and  a  little  rounded.  If  then  the  point  where  it  is  held  be  the 
centre  of  the  plate,  and  the  bow  be  applied  close  to  one  of  the  angles,  sand  strewed  over  it  will  arrange  itself  on 
the  two  diameters  which  divide  it  into  four  equal  squares  as  in  fig.  44.  Each  of  these,  in  the  act  of  vibration, 
becomes  a  surface  of  double  curvature,  and  their  motions  are  conlrary  to  each  other;  those  marked  -f-  making 
their  excursions  on  one  side  of  the  plane  of  repose,  while  those  marked  —  are  on  the  other.  This  mode  of 
vibration  corresponds  to  the  gravest  tone  produced  by  the  plate. 

If,  the  plate  being  still  held  in  the  centre,  the  bow  be  applied  at  the  middle  of  one  side,  the  sand  will  occupy 
the  diagonals  of  the  plate,  which  are  the  nodal  lines  corresponding  to  this  mode.  In  this,  as  in  the  former  case, 
the  plate  subdivides  itself  into  four  equal  vibrating  segments  as  in  the  fig  45,  but  the  tone  is  different,  being  a 
fifth  higher  than  in  the  former  case,  the  distribution  of  the  inertia  with  respect  to  the  elastic  power  of  the  plate 
being  such  as  to  admit  a  quicker  motion. 

If  the  plate  be  held  at  a,  the  intersection  of  two  nodal  lines  fig.  46,  and  the  bow  be  still  applied  at  the  middle 
of  one  side,  or  at  t!ie  angle  adjacent  to  a,  the  plate  will  vibrate  as  there  represented.  In  this  subdivision,  the  four 
small  squares  at  the  angle  and  the  large  one  at  the  centre  vibrate  on  one  side,  or  negatively,  while  the  four  in- 
termediate oblong  rectangles  adjacent  to  the  sides  vibrate  positively. 

These  instances  may  serve  to  show  the  mode  of  proceeding  in  more  complicated  cases,  and  with  plates  cf 
other  figures.  Among  these,  circular  ones  hold  the  chief  place  both  for  symmetry  and  variety.  The  examples, 
fisjs.  47 — 93,  are  selected  from  those  described  by  Chladni,  who  has  determined  by  experiment  the  tones 
corresponding  to  each  mode  of  division  in  plates  of  a  great  variety  of  figures.  Of  these  we  shall  only  give  some 
examples  in  the  case  of  a  square  plate,  of  which  we  shall  suppose  the  gravest  or  fundamental  tone  to  be  repre- 
sented by  1.  This  premised,  if  we  regard  the  plate  as  subdivided  into  n  X  n'  rectangles  by  n  nodal  lines 
parallel  to  one  side,  and  n'  parallel  to  the  other,  the  notes  corresponding  will  be  as  in  the  following  Table  : 


Part  II. 


Values  of  n. 


0 

1 

2 

3 

4 

5 

0 

or*  + 

Cfe 

0 

• 

1 

(0 

(3)" 

(3)«-b  - 

B 

n 

2 

(3)'" 

(3)'"b  - 

CO* 

(1)VS  + 

3 

(l)'"b  + 

(7)1, 

(4)' 

(7)' 

4 

(3)hfc  - 

(7)' 

(3)t 

5 

' 

0)T*  + 

Mode  of 
producing 
their  several 
subdivi- 
sions. 
First  mode 
of  a  vibra- 
tion of  a 
square 
plate. 
Fig.  44. 

275. 
Second 
mode. 
Fig.  45. 

276. 
Fig.  46. 
Other 
modes. 

277. 

Fig.  47— 93 
Modes  cf 
vibra'ion  of 
square  and 
circular 
plates  ob- 
served by 
Chladui. 

Series  of 
Sounds  pro- 
duced by  a 
square 

plate. 


The  vibrations  of  triangular,  hexagonal,  elliptic,  and  semicircular  plates  have  also  been  investigated  by  Chladni,      278 
and  fig.  94 — 123  exhibit  some  out  of  a  great  variety  of  nodal  figures,  to  which  they  give  rise  in  their  various  Fig.! 
modes  of  vibration.  J23, 


toi.  rv 


a  * 


PART  III. 
§  I.     Of  the  Communication  of  Vibrations  and  of  the  Vibrations  of  Systems 

Sound      THE  subject  of  the  sonorous  vibrations   of  solids   has   recently  been  taken  up  in  a  more  general  and  extended     Part  III. 
y_r-^-^y  point  of  view  by  M.  Felix  Savart,  in  a  series  of  Memoirs  communicated  by  him  to  the  Royal  Academy  of  Sciences  ^-•••v"-- 

279.  of  Paris,  and  of  which  copies,  or  copious  extracts,  are  printed  in  the  Annales  de  Chimie.     We  regret   that  the 
Savart's        narrow  limits  which  remain  to  us  in  this  volume,  will  allow  little  more  than  a  slight  sketch  of  the  contents  of  the 
researches     principal  of  these  most  interesting  papers,  the  whole  of  which  are  models  of  experimental  research,  and  indeed, 

"",  so  full  of  new,  curious,  and  instructive  matter,  that  it  is  next  to  impossible  either  to  condense  or  abstract  them;  for 
which  reason  we  earnestly  recommend  our  readers,  who  may  be  led  to  take  an  interest  in  the  subject  of  this  Essay, 
not  to  content  themselves  with  the  meagre  statements  here  offered,  but  to  procure  and  study  diligently  the  origi- 
nal Memoirs. 

280.  In  order  to  a  regular  analysis  of  this  intricate  subject,  it  was  first  requisite  to  obtain  some  certain  mode  of  corn- 
Method  of    municating  to  any  given  point  of  a  solid  vibrations  confined  to  one  plane,  and  whose  period  of  recurrence,  as  well 
communi-     as  the  plane  in  which  they  were  performed,  and  the  amplitude  of  their  excursions,  could  be  varied  at  pleasure.  The 
caiing  a        vibrations  of  a  stretched  string  set  in  motion  by  a  fiddle-bow,  afford  the  means  of  doing  this.     Such  are  necessa- 
rily confined  to  the  plane  in  which  the  motion  of  the  bow  is  performed,  because  any  vibratory  motion  out  of  this 

given  point  plane  is  prevented,  or  immediately  stifled  by  the  pressure  of  the  bow  ;  and  as  the  plane  of  its  motion  may  be 
of  a  solid,  varied  at  pleasure,  and  the  amplitude  of  excursion  may  be  increased  or  diminished  by  a  change  of  pressure,  and 
velocity  of  stroke,  all  the  requisite  conditions  are  here  obtained.  Accordingly,  if  the  vibrating  part  of  such  a  string 
be  brought  to  press  on  a  solid  not  too  massive,  or  if  the  end  of  the  string  be  attached  to  a  point  in  the  solid, 
M.  Savart  has  found  that  the  regularly  repeated  impulses  of  the  string  are  transferred  to  the  solid  with  perfect 
fidelity. 

281.  A  familiar  example  of  this  communication  of  impulses  is  found  in  the  violin.     In  that  instrument,  fig.  124,  the 
Vibrations     strings  which  are  stretched  from  end  to  end  of  it,  are  divided  into  two  uneqal  parts  by  the  bridge,  A,  on  which 
of  a  violin     tnev  ajj  press  strongly,  and  at  the  same  time  rest  in  small  notches,  so  as  not  to  slip  laterally  on  it.     The  portion, 

B,  of  the  string  which  lies  towards  the  handle,  C,  of  the  instrument,  is  free,  and  is  set  in  vibration  by  the  bow  in  its 
cated  to  the  own  plane  ;  but  that  on  the  other  side  of  the  bridge,  D,  is  loaded  with  a  mass  of  horn  or  whalebone,  E,  to  which  all 
wood.  the  other  strings  are  also  attached,  and  which,  being  only  tied  to  the  wood-work,  cannot  propagate  the  vibrations 

Fig.  124.      of  any  one  string  sounding  separately,  by  reason   of  the  contradictory  and   unequal  tensions  of  the  other  three. 
Thus  the  bridge  is  in  fact  acted  on  only  by  the  vibrations  of  that  part,  A  B  C,  of  the  string  which  is  crossed  by  the 
bow,  as  if  it  terminated  abruptly  at  its  point  of  pressure,  A.     These  vibrations  constantly  lend,  therefore,  to  tilt  the 
bridge  laterally  backwards  and  forwards,  and  to  press  up  and  down  alternately  the  two  little  prominences  or  feet, 
F  G,  by  which  it  rests  on  the  belly  of  the  violin.     It,  therefore,  sets  the  wood  of  the  upper  face  in  a  state  of  regular 
vibration,  and  this   again  is  communicated  to  the  back   through  a   peg  set  up  in   the  inside  of  the  fiddle,  and 
^'hral™'sd  through  its  sides,  called  the  soul  of  the  fiddle,  or  its  sounding  post.     In  consequence,  if  the  upper  surface  be 
how  ob-     '  strewed  with  sand,  it  will  assume  a  regular  arrangement  in  nodal  lines  when  the  bow  is  drawn  ;  and  the  same  sub- 
served,        division  is  also  observed  in  the  wood  of  the  under  surface,  if  the  sounding-post  be  exactly  placed  in  the  centre  of 
p.  ,    symmetry  of  the  nodal  figures.     The  experiment  can  hardly  be  made,  however,  with  a  common  fiddle,  by  reason 

to  vibrate  *  of  the  convexity  of  its  surface,  on  which  sand  will  not  rest;  but  if  one  be  constructed  with  plane  boards,  or  if, 
by  commu-  abandoning  the  fiddle,  a  string  be  stretched  on  a  strong  frame  over  a  bridge,  which  is  made  to  rest  on  the  centre 
mcationwith  of  a  regularly  formed  plate  or  disc  of  metal  or  wood,  strewed  with  sand,  the  surface  thus  set  in  vibration  by  the 
strings.  string  will  be  seen  to  divide  itself  by  regular  nodal  figures. 

282.  Now  M.  Savart  has  observed  this  remarkable  fact,  viz.  that  if  the  tension  or  length  of  the  string  thus  placed  in 
Joint  vibra-  vibratory  communication  with  a  plate,  be  changed,  so  as  to  vary  the  note  it  speaks,  the  nodal  figures  on  the  plate 
lions  of  a      undergo  a  corresponding  variation,  and  the  plate  still  vibrates  in  unison  with   the  string  ;  or,  which  is  the  same 

thing,  the  two,  together  with  the  interposed  bridge,  form  a  vibrating  system,  in  which,  though  the  vibrations  of 
•y"tem?S  '  the  several  parts  are  necessarily  very  different  in  their  nature  and  extent,  yet  they  have  all  the  same  periods.  This 
experiment  is  very  important.  It  shows  that  the  Sounds  of  such  thin  plates  are  not  like  those  of  strings  confined  to 
certain  fixed  harmonics,  but,  according  to  the  forms  of  their  nodal  lines,  and  the  proportions  of  the  vibrating  areas  in 
opposite  states  of  excursion,  may  assume  any  assigned  period ;  in  other  words,  given  the  vibrating  plate  and  the 
pitch,  a  nodal  figure  may  be  described  on  it,  which  shall  correspond  to  that  pitch,  and  the  plate  (with  more  or 
less  readiness,  however)  is  always  susceptible  of  such  a  vibration  as  shall  yield  that  note,  and  produce  that  nodal 
figure.  How  far  this  proposition  is  general,  and  with  what  limitations  it  is  to  be  understood,  we  shall  soon  see. 
Meanwhile  this  remark,  it  will  be  observed,  furnishes  a  complete  explanation  of  the  effect  of  sounding-boards  in 
musical  instruments.  It  is  not,  as  some  have  supposed,  that  there  ex-ist  in  them  fibres  in  every  state  of  tension, 
some  of  which  are  therefore  ready  to  vibrate  in  unision  with  any  proposed  Sound,  and,  therefore,  reinforce  it.  Such 
a  cause  could  at  best  produce  but  a  very  feeble  effect.  It  is  the  whole,  board  which  vibrates  as  part  of  a  svstem 

804 


SOUND  805 

So  md.     with  every  note,  and  (as  vibrations  may  be  superposed  to  any  extent)  the  same  sounding-board  may  at  once  form     Parj  "'• 
-^v" "^  a  part  of  any  number  of  systems,  and  vibrate  in  unison  with  every  note  of  a  chord.     Still  some  modes  will  always  v— •~v~-' 
be  more  difficult  than  others,  and  no  sounding-board  will  be  perfectly  indifferent  to  all  Sounds. 

The  longitudinal  vibrations  of  a  rod  of  glass,  excited  by  rubbing  it  with  a  wet  cloth,  may  also  be  used  to  excite       283. 
vibrations  in  a  given  point  of  a  solid  perpendicular  to  its  surface,  by  applying  its  end  to  it,  or  cementing  it  to  the  Longitudinal 
nolid  by  mastic.     In  this  way  Chladni  applied  it  to  draw  forth   the  Sounds  of  glass  vessels,  (which  when  hemi-  ^lhr»tlons  u( 
spherical,  and  of  sufficient  size  and  even  thickness,  are  remarkably  rich  and  melodious,)  in  an  instrument  which  he  n]0ye()  to 
called  the  Euphone,  exhibited  by  him  in  Paris  and  Brussels.     The  principle  of'this  instrument  was  at  the  time  con-  communi- 
cealed  ;  hut  the  enigma  was  subsequently  solved  hy  M.Blanc,  who  on  his  part  independently  made  the  same  remark,  ca'e  vibra- 
and  applied  it  to  a  similar  purpose.  tio"s  to 

If  the  solid  (a  circular  glass  disc  for  instance)  to  which  such  a  vibrating  rod  or  tube  is  fastened,  be  of  small  chiadni's 
comparative  dimensions,  its  vibrations  are  commanded  by  those  of  the  rod,  and  the  Sound  yielded  will  be  that  of  Euphont. 
the  rod  alone;  and  vice  versa,  if  the  disc  be  large,  and  the  rod  small,  the  note  sounded  will  be  that  of  the  disc,       284. 
which  will  entirely  command  the  rod  ;  but  in  the  intermediate  cases,  both  M.  Savart  and  M.  Blanc  have  observed,  Mutual  in- 
the  note  will  be  neither  that  of  the  disc  or  the  rod  separately,  but  the  two  will  vibrate  together  as  a  system,  each  fluer)ce  of » 
yielding  somewhat  to  the  other.     It  is  a  case  exactly  analogous  to  that  of  a  reed-pipe,  in  which  the  reed  and  j!1" 
column  of  air  mutually  influence  each  other's  note.     See  Art.  199.     This  mutual  influence  of  propagated  motion,  rod  con. 
by  which  two  periodically  recurring  impulses  affect  each  other's  period,  and  force  themselves  into  synchronism,  nected  wit 
extends  to  cases  where  at  first  sight  it  would  hardly  be  suspected.    Thus  Ellicot  observed  that  two  clocks  fastened  ''• 
to  the  same  board,  or  even   standing  on  the  same  stone  pavement,  beat  constantly  together,  though  when  sepa-  ^'7° 
rated   their  rates   were  found   to  differ  very  considerably ;  and   Breguet  has  since  made  the  same  remark  on  piaceii  near 
watches.     Thus  also  two  organ-pipes  vibrating  side  by  side,  if  very  nearly  in  unisor,  will  under  certain  circum-  together. 
stances  force  themselves  into  exact  concord,  as  has  been  observed  by  Hudlestone,  (Nicholson's  Journal,  i.  329.)  Of  organ- 
and  lately  recalled  to  notice  by  some  experiments  made  in  Copenhagen.     The  experiment  with  the.  disked  tuning-  P'Pes  nearl. 
fork  and  pipe,  related  in  Art.  204.,  may  here  again  be  referred  to. 

The  longitudinal  vibrations  of  a  rod  have  also  been  used  by  M.  Savart,  to  communicate  vibrations  from  one       285. 
solid  to  another ;   as  for  instance,  from  the  upper  to  the  under  of  two  circular  discs  cemented  at  their  centres  to  Fig.  125. 
the  two  ends  of  the  rod,  at  right  angles  to  their  planes,  as  at  fig.  125.     If  the  two  discs  be  of  the  same  dimen-  Vibrations 
sions  and  materials  so  as  to  yield,  when  separately  vibrating,  the  same  note,  the  vibrations  of  one  of  them,  (the  ''om"1""1 
upper  for  instance,)  excited  by  a  bow,  will  be  exactly  imitated   by  the  other,  and  sand  strewed  over  both  will  t^gen  t^(l 
arrange  itself  in  precisely  the  same  forms  in  both  discs,  and  that,  into  whatever  number  of  vibrating  segments  that  platas  by  a 
immediately  excited  be   made  to   subdivide  itself.     But  if  the  discs  separately  do  not  agree  in  their  tones,  the  rod. 
system  may  yield  a  tone  intermediate,  and  each  being  differently  forced  from  its  natural  pitch,  the  nodal  figures  on 
them  will  then  no  longer  correspond. 

The  st-ite  of  vibration  in  which  the  molecules  of  the  connecting  rod  are  thrown  in  such  cases,  deserves  a  nearer       286 
examination.     For  simplicity  let  us  suppose  the  discs   equal,  the   rod  cylindrical,  and  the  vibration  of  the  system  State  of 
such  that  each  disc  shall  subdivide  itself  into  four  quadrantal  segments.     In  this  case  it  is  clear  that  as  the  form  vibration  of 
assumed  at  any  instant  hy  the  upper  disc  is  undulated  or  wrinkled,  as  represented  in  fig.  12*5,  the  section  of  the  fhe connect, 
rod  in  immediate  contact  with  it,  and  which  obeys  all  its  motions,  must  assume  a  similar  form,  and  so  of  all  the  J^™n  j 
rest.    Thus  if  we  conceive  the  rod  split  into  infinitesimal  columns,  parallel  to  its  axis,  all  the  columns  in  two  oppo-  f\,  j2g. 
site  quadrants  will  be  ascending,  while  those  in  the   other  two  are  descending ;   and  thus  the  two  corresponding 
opposite  quadrants  of  the  lower  plate  will  be  drawn  upwards,  while   the   alternate  ones  are  forced  downwards, 
giving  a  similar  distortion  to  its  figure,  and  disposing  it  to  a  similar  vibration  only.     It  will  depend  on  the  length 
of  the  rod,  and  the  time  taken  by  an  undulation  to  run  over  its  length,  compared  with  that  of  a  vibration  of  either 
disc,  whether  the  phases  of  vibration   in  the  two  discs  shall   be  the  same  at  the  same  instant  or  not.     It  may 
happen  that,  for  instance,  the  quadrant,  D  B,  of  the  upper  disc  shall  have  completed  its  downward  motion,  and 
begun  to  return  before  .he  pulsation  propagated  through  the  rod  has  arrived  at  the  lower  disc  ;   and  in  that  case 
the  corresponding  quadrants  of  the  two  discs  will  be  always  in  opposite  phases  of  their  periodic  motion.     But  the 
nodal  lines  will  of  necessity  correspond  in  both. 

When  the  two  discs  are  unequal,  the  propagation  of  the  pulses  through  the  rod  must  of  course  cease  to  be       267. 
uniform,  and  each  section  of  it  down  its  whole  length  will  have  its  own  peculiar  law  of  form  and  motion,  which  Case  where 
it  is  beyond  our  power  to  investigate.     In  that  case  its  molecules  must  have  lateral  as  well  as  vertical  motions,  th 
and  its  vibrations   must  be  partly  longitudinal  and  partly  twisting,  in  a  way  easier  imagined  than  described.     If ar 
the  discs  be  dissimilar  in  form  as  well  as  unequal  in  dimension,  the  vibrations  of  the  connecting  rod  will  of  course 
be  very  complicated. 

These  principles  have  been  applied  by  M.  Savart,  and  apparently  with  success,  (as  appears  by  the  very  able       288. 
report  of  M.  Biot  on  his  Paper,)  to  the  improvement  of  violins,  and  the  construction  of  these  delicate  instruments  M.  Savarfs 
on  scientific  and  experimental  grounds.     Every  one  is  aware  of  the  difficulty  of  procuring  perfect  violins,  and  vlolln*- 
the  enormous  prices  they  bear,  so  that  fixed  rules,  by  which  any  ordinary  artist  can  with  certainty  produce  an 
excellent  one,  are  evidently  highly  valuable.     We  long  to  see  M.  Savart's  construction  tried  in  this  country,  but 
must  refer  to  his  Paper  (Annales  de  Chimie,  vol.  xii.  p.  225,  &c.)  for  the  details. 

It  appears  from  what  we  have  said,  that  the  motions  of  the  molecules  of  a  rod  which  communicates  the  vibrations       289. 
of  one  disc  to  another,  or,  more  generally,  which  vibrates  longitudinally  by  any  exciting  cause,  are  not  of  necessity  Longjtadi 
analogous  to  those  of  the  air  in  a  cylindrical  pipe,  at  least  not  to  that  simple  case  of  the  latter  vibrations,  which  we  have  "jonj'0fa~ 
heretofore  considered  in  our  3d  Section.     The  several  transverse  sections  of  such  a  rod,  in  the  act  of  vibration,  do  rod  further 
not  necessarily  merely  advance  and  recede  longitudinally,  but  may  become  curves  of  double  curvature  ;  in  short,  examined. 
such  a  rod  may  be  considered  as  an  assemblage  of  vibrating  discs,  ranged  along  a  common  axis,  along  which 

5  M  2 


806 


S  O  U  N  D. 


Origin  of 
nodal  sur- 
faces. 


290. 
Of  nodal 
lines  in 
general. 

•291. 

Mow  such 
nodal  lines 
are  distin- 


from  each 
other. 

Motions  of 
-.iinl  agi- 
tated by 
normal  and 
by  tangen- 
tial vibra- 
tions. 


Sound,  they  may,  it  is  true,  be  also  carried  backwards  and  forwards'with  a  vibratory  motion,  while  at  the  same  time 
— s^— •>  their  flexure  is  changing  from  convex  to  concave,  and  vice  versa.  Now  it  may  happen  that  a  point,  or  a  line,  ' 
(straight  or  curved,)  in  any  one  of  such  discs,  may  be  advancing  in  the  direction  of  the  axis  in  consequence  of  the 
bodily  motion  of  the  whole  disc,  while,  in  virtue  of  its  flexure  in  the  act  of  changing  its  figure,  it  may  be 
receding  ;  and  this  advance  and  recess  may  so  balance  each  other,  that  the  point  or  line  shall  be  at  rest.  If 
this  be  true  at  one  instant,  it  will  be  so  at  all  instants,  because  the  vibrations  have  all  one  period,  and  follow  the 
same  law  of  increase  and  decrease  in  their  phases.  Thus  we  have  a  nodal  point,  or  a  nodal  line  ;  and  as  each 
disc,  by  reason  of  the  law  of  continuity,  must  have  a  similar  one,  the  assemblage  of  such  lines  will  mark  out 
within  the  rod  a  nodal  surface,  dividing  it  into  separate  solids  whose  molecules  on  either  side  of  such  surface  are 
in  opposite  phases  of  their  motion. 

What  is  here  said  of  rods,  applies  of  course  to  solids  of  any  figure  and  dimension,  neither  is  there  the  slightest 
reason  why  it  should  not  apply  to  vibrating  masses  of  air,  or  any  other  elastic  fluid.  Any  such  mass  may  be 
conceived  as  cut  up  into  two  or  more  oppositely  vibrating  portions  pervading  it  according  to  certain  laws. 
Where  these  surfaces  out-crop  or  intersect  the  external  surface  of  the  mass,  there  will  be  a  nodal  Ihie. 

Such  nodal  lines,  formed  on  the  surfaces  of  bodies  by  the  longitudinal  vibrations  of  their  molecules,  (i.  e.  by 
vibrations  parallel  to  their  surfaces,)  may  be  detected  and  rendered  visible  to  the  eye  by  fine  dry  sand,  or  the 
powder  of  Lycoperdon,  strewed  over  them  ;  and  the  motions  of  the  particles  in  the  act  of  forming  them  will  easily 
distinguish  such  vibrations  as  are  executed  parallel  to  the  surface  (in  which,  of  course,  the  surface  is  not  thrown 
into  waves)  from  such  as  take  place  at  right  angles  to  it,  where  the  surface  itself  leaps  up  and  down.  In  the 
latter  case,  the  particles  of  sand  dance,  and  are  violently  thrown  up  and  down  over  the  whole  extent  of  the 
vibrating  portions,  till,  at  length,  they  are  entirely  dispersed  from  them.  In  the  former,  they  only  glide  along 
close  to  the  surface,  and  meet  and  settle  on  the  nodal  lines,  and  that,  sometimes,  with  incredible  swiftness. 
The  reason  why  they  retreat  to  the  nodal  lines  is  easily  understood.  The  amplitude  of  the  excursions  of  the 
vibrating  molecules  of  the  surface  diminishes  as  we  approach  a  nodal  line.  Hence  a  particle  of  sand  anywhere 
situated,  if  thrown  by  an  advancing  vibration  towards  this  line,  will  not  be  thrown  quite  so  far  back  by  the  sub- 
sequent retreating  vibration,  because  its  then  situation  is  one  less  agitated.  Thus  the  motion  of  each  particle  of 
sand  is  one  of  alternate  advance  towards  the  node  and  recess  from  it,  but  the  advances  are  always  greater 
than  the  recesses.  In  consequence,  it  creeps  along  the  surface,  and  will  not  rest  till  it  has  attained  the  node. 
When  a  large  disc  of  glass  is  set  vibrating  vigorously  by  a  bow,  perpendicular  to  its  plane,  the  grains  of  sand 
will  fly  up  some  inches  from  it  and  be  scattered  in  all  directions.  M.  Savart  has  distinguished  by  the  name 
tangential  vibrations  all  such  motions  of  the  superficial  particles  of  a  body  as  are  performed  parallel  to  the  sur- 
face ;  while  those  executed  at  right  angles  to  it,  in  virtue  of  which  the  surface  itself  heaves  and  sinks,  he  calls 
transverse. ;  and  to  motions  compounded  of  both  these,  where  the  surface  both  swells  and  falls  and  shifts  laterally 
backwards  and  forwards,  he  gives  the  term  "  oblique  vibrations.''  In  this  we  shall  follow  him. 

This  acute  experimenter  has  investigated  with  great  minuteness  the  tangential  vibrations  of  long  flat  rods  or 
rulers  of  glass,  as  well  as  of  cylinders  and  tubes.  They  are  extremely  complicated,  and  offer  most  singular 
na!-tangen-  phenomena,  some  of  which  we  shall  now  describe.  If  we  take  a  rectangular  lamina  of  glass  0'"'70  (=  27m-56) 
tial  vibra-  long,  Om'015  (=  Oin-59)  broad,  and  0'n'0015  (=  Oin-06)  thick,  and  holding  it  by  the  edges  in  the  middle,  between 
the  finger  and  thumb  with  its  flat  face  horizontal,  strewed  with  sand,  and,  at  the  same  time,  set  it  in  longitudinal 
vibration,  either  by  rubbing  its  under  side  near  either  end  with  a  bit  of  wet  cloth,  by  tapping  it  on  the  end  with 
light  blows,  or  by  rubbing  lengthwise  a  very  small  cylinder  of  glass,  cemented  on  to  its  end  in  the  middle  of  its 
breadth,  and  parallel  to  its  length  ;  in  whatever  way  the  vibration  be  communicated,  we  shall  see  the  sand  on 
its  upper  surface  arrange  itself  in  parallel  lines,  at  risrht  angles  to  its  longer  dimension,  and  always,  in  one  or  the 
other  of  the  two  systems,  represented  in  figs.  127  and  128.  Now  it  is  very  remurkable  that  although  the  same 
one  of  these  two  systems  will  always  be  produced  by  the  same  plate  of  glass,  yet  among  different  pJatcs  of  the 
aboTe  dimensions,  even  though  cut  from  the  same,  sheet,  side  by  side,  one  will  invariably  exhibit  one  system,  and 
the  other  the  other,  without  any  visible  reason  for  the  difference.  Moreover,  in  the  system,  fig.  127,  the  disposi- 
tion of  ihe  nodal  lines  is  unsymmetrical,  one  of  them,  a,  being  nearer  to  one  end,  and  the  closer  pair,  ff,  not  being 
situated  in  the  middle  ;  and  this  too  is  peculiar  to  the  plate,  for  wherever  it  be  rubbed,  whichever  end  be  struck, 
still  the  line  a  will  always  be  formed  nearest  to  the  same  extremity. 

Now  let  the  positions  of  the  nodal  lines  be  marked  on  the  upper  surface,  and  then  let  the  plate  be  turned  till 
the  lower  surface  becomes  the  upper,  and  this  being  sanded,  let  the  vibrations!  again  be  excited  just  as  before.  The 
nodal  lines  will   now  be  formed  quite   differently,  and  will  fall   on   the   points  just  intermediate  between  those  of 
the  other  surface ;  i.  t.  on  the  points  of  greatest  excursion  of  its  vibrating  molecules.     In  a  word,  if  n,  n,  n,  n,  &c. 
the  opposite  in  fig.  128,  or  130,  represent   the  places   of  the    nodes  on  the  one  surface,  then  will  n',  n',  &c.  be  those  of  the 
tides.  other.     Thus  all  the  motions  of  one  half  the.  thickness  of  the  lamina  are  exactly  contrary  to  those  of   the  corre- 

Kigs.  1129,     spending  points  of  the  other  half.    This  property,  indeed,  is  general,  whatever  be  the  material,  length,  breadth,  or 
thickness,  of  the  lamina. 

If,  the  other  dimensions  remaining,  the  thickness  be  increased,  the  Sound  will  remain  the  same,  but  the  numb'r 
of  nodal  lines  will  be  less.  This  fact  alone  is  sufficient  to  prove  an  essential  difference  between  the  vibrating  por- 
tions of  such  a  plate,  and  the  ventral  segments  of  an  organ-pipe  harmonically  subdivided. 

If  the  breadth  of  a  plate  of  the  above  length  be  greater  than  Oin'6  the  nodal  lines  cease  to  be  straight,  and  ranged 
across  the  breadth  at  right  angles  to  the  sides.  They  pass  into  curves,  and,  when  the  breadth  is  increased  to 
Om'04,  (=  1'"'57,)  they  assume  the  forms  in  figs.  131,  132,  the  former  representing  the  lines  on  the  upper,  the 
latter  those  on  the  under  surface.  If  the  breadth  be  enlarged  to  Om'06,  (=  2ln'36,)  the  figures  on  the  two  facws 
will  be  as  in  figs.  133,  134.  If  the  dimensions  be  so  varied  as  to  convert  the  plate  into  a  square,  the  nodal 
figures  will  assume  the  forms  in  figs  135,  136.  If  the  form  of  the  plate  pass  into  the  circular  or  •riangular,  the 


Part  III 


292. 

LongitucTi- 


lion  of 

rectangular 

rods. 


Figs.  127, 
128. 


293. 

Different 
arrange- 
ment of 
nudes  on 


130. 

294. 

Effect  of  a 
varied 
thickness. 
29a. 

Tangential 
vibrations 
'  '  broad 
rcctangulai 
plates. 


SOUND.  807 

Sound.     same  mode  of  vibration  (longitudinal-tangential)  being  preserved,  still   the  opposite  sides  of  the  plates  will  pre-     Part  HI. 
-^v~-/  gent  different  nodal  figures,  as  in  figs.  137,  138,  and  139,  140.  -— • ~y~~^ 

To  examine  the  longitudinal-tangential  vibrations  of  cylindrical  tubes  or  rods,  as  sand  will  not  lie  on  their  con-  O 
vex  surfaces,  M.  Savar*.  employed  the  ingenious  artifice  used  by  Sauveur  to  exhibit  the  harmonic  nodes  of  a  vibrating  p™"^^ 
string.     For  this  purpose  the  latter  set  astride  on   the   string  a  small  bit  of   paper   cut  into  the  form  of  an  140 
inverted  A.  But  in  this  case  it  is  found  *o  answer  better  to  encircle  the  vibrating  cylinder  with  a  narrow  ring  of  paper,       296. 
whose  internal  diameter  is  three  or  four  times  that  of  the  cylinder,  and  which,  therefore,  hangs  quite  loosely  on  it.  Longitudi- 
Jf  a  cylinder  of  glass  about  two  metres  (6  J  feet)  long  be  encircled  by  several  such  rings,  or  riders,  and,  being  held  nal-tangen- 
horizontally  by  the  middle,  as  lightly  as  possible,  be  rubbed   in  the  direction  of  its  length  with  a  wet  cloth,  (it  '!a 
should  be  very  wet,)  it  will  yield  a  musical  Sound,  and  all  the  riders  will  glide  rapidly  along  it  to  their  nearest  nodal  cyiin()cr^ 
points  on  the  upper  surface,  where  they  will  rest.     Now  let  all  these  points  be  marked,  and  then  let  the  cylinder  be 
turned  so  as  to  bring  the  opposite  portion  of  its  circumference  uppermost  and  horizontal,  and  let  the  vibration  be 
again  excited  in  the  same  manner.     Then  we  shall   remark  the  very  same  phenomenon  as  in  rectangular  plates, 
viz.  that  the  nodal  points  on  this  edge  correspond  nearly  to  the  middles  of  the   intervals  between  those  of  the 
opposite  one. 

If  the  cylinder,  instead  of  being  turned  at  once  half  round,  be  turned  only  a  little  at  a  time,  and  a. ways  in   the       297. 
same  direction,  the  riders  will  come  to  points  of  rest  constantly  more  and  more  towards  one  or  the  other  end  of  Nodal  Imei 
the  cylinder,  according  as  it  is  turned  to  the  right  or  to  the  left;  and  if  the  locus  of  all  the  nodal  points  be  traced  spirally 
by  this  means,  it  will  be  found  to  be  a  species  of  spiral  line  or  screw,  making  one  or  more  turns  round  the  cylinder  "ranged, 
according  to  its  length. 

But  there  exist?  here  a  peculiarly  bearing  an  obvious  relation  to  what  we  have  observed  already  in  the  case  of       ggg 
rectangular  plates.     The  continuity  of  this  spiral  is  interrupted  near  the  middle  of  the  cylinder,  or  rather  it  stops  TWO  spiral 
short  at  a  point  n,  on  one  side  of  the  central  point,  and  recommences  at  N,  a  point  equidistant  on  the  other  side  j  nodal  lines 
but  in  a  contrary  direction,  so  as  to  form  on  the  two  moieties   of  the  length  of  the  cylinder  a  right  and  a  left-  running 
handed  screw.     Again,  these  spirnls  are  not  equally  inclined  to  the  axis  in  all  parts  of  their  course.     They  consist  °PP°slte 
of  portions  alternately  much   and  little  inclined,  having  points  of  maximum  and  minimum  inclination  alternately 
atevery  90°  of  their  course  round   the  cylinder,  as  in  fig.  141  ;  thus  dividing  the  cylinder   into  four  quadrantal  Fig.  141. 
portions,  which  are  related  to  each  other  in  the  same  manner  as  the  upper  and  under  faces,  and  the  right  and  left 
sides  of  the  vibrating  parallelepipeds,  examined  in  Arts.  292.  et  seq. 

It  appears  then  that  when  a  cylinder  is  set  in  a  state  of  longitudinal-tangential  vibration,  it  assumes  of  itself       299. 
(by  reason  no  doubt  of  some  casual  inequality  in  its  form  or  structure,  giving  it  a  bias  one  way  or  the  other)  four  Fourprin- 
pnncipal  edges,  dividing  it  into  quadrantal  portions.     Of  these,  two  opposite  ones  (which  we  will  designate  by  ™^* ^  ^*s 
the  numbers  1  and  3,  and  call  the  upper  and  under  edge)  are  divided  by  the  nodal  lines  in  points  rc,  nt  N,  N',,  ;„„  Cyrn^e, 
and  7i3  n't  N,  N',,  where  their  inclination  is  a  maximum,  and  the  others  2,  4,  which  we  may  call  the  sides, 
at  n,  n's  N,  N',,  and  nt  n't  N4  N't,  where  it  is  a  minimum. 

What  we  have  said  relates  to  the  disposition  of  the  nodal  lines  on  the  exterior  surface  of  a  tube,  or  of  a  solid       %QQ 
cylinder.     In  the  case  of  a  hollow  tube,  the  nodal  lines  of  the  internal  surface  may  be  examined  by  strewing  in  it  Nodal  lines 
a  little  fine  sand,  provided  its  diameter  be  so  large  as  not  to  drive  all   the  sand   into  a  crowded  line   along  the  in  the  in- 
bottom.     We  shall  thus  detect  a  spiral   in  all  respects  similar  to  that  on  the  external  surface ;  only  that  its  coils  '«rior  of  a 
run  exactly  along  the  intervals  of  those  of  the  external  one.     So  that  in  all  cases,  those  points  of  the  internal  sur-  cy'llulr'cal 
face  are  most  strongly  agitated  bj  the  vibration  which  correspond  to  points  at  rest  on  the  outer,  and  vice  versa. 
M.  Savart  has  noticed  a  very  curius  phenomenon  in   this  case.     At  the  points  of  maximum  inclination  the  sand 
gathers  itself  up  in  a  circular  heap,  and  remains  at  perfect  rest;  but  at  those  of  minimum  inclination  it  forms  a 
long  ellipse,  the  borders  of  which  keep  constantly  circulating  in  one  direction;  and  if  instead  of  sand,  a  small  globe 
of  ivory  or  wax  be  put  into  the  tube  ;  at  these  points  it  remains,  it  is  true,  without  shifting  \isplace,  but  spins  con- 
stantly in  one  direction  round  a  vertical  axis,  so  long  as  the  vibration  continues. 

We  have  all  along  supposed  that  the  state  of  vibration  into  which  the  cylinder  or  tube  is  thrown,  is  that  corre-       30i_ 
sponding  to  the  gravest  tone  it  can  yield  by  vibrations  of  the  kind  in  question.    M.  Savart  has  examined  its  higher  Higher 
modes,  and  has  pointed  out  other  peculiarities,  but  for  these  we  must  refer  the  reader  to  his  Memoir,  Ann.  de  modes  of 
Chim.  vol.  xxv.  p  236.     We  will  merely  remark  that  in  these  modes,  the  threads  of  the  screw  break  off,  and  re-  vlbra''°n 
verse  their  direction  at  the  points  of  union  of  the  several  ventral  segments.  vision  of  the 

cylinder 
into  ventral 
segments. 

§  II.      Of  the  Cummunication  of  Vibrations  from  one  Vibrating  Body  to  another. 

We  have  already  seen  that  a  rod  placed  between  two  discs,  one  of  which  is  set  in  vibration,  becomes  the  means       302. 
of  communicating  its  vibrations  to  the  other.     But  it  may  be  announced  as  a  general  fact,  that  whenever  a  vibrating  General  law 
body  is  brought  into  intimate   contact  with   another,  it  communicates  to   it  its  own  vibrations,    more   or  less  of  the  c<?m" 
effectually  as  their  union  is  more  perfect.     Tins  proposition  has  been  carried  still  further  by  M.  Savart,  whose  ex-  oMbT'0" 
periments  show  that  all  the  particles  of  the  body  thus  set  in  vibration  by  communication  are  agitated  by  motions  motion"'' 
not  merely  similar  in  their  periods,  but  actually  parallel  in  their  directions,  to  those  of  the  original  source  of  the       303. 
motion.     Examples  will  best  explain  the  meaning  of  this.  Fij.  142. 

Example  1.  Let  A,  fig    142,  be  along  flat  glass  ruler  or  rod,  cemented  with  mastic  to  the  edge  of  a  large  bell-  vibrali°n' 
glass,  such  as  is  used  for  the  harmonica,  or  musical  glasses,  or  a  large  hemispherical  drinking-glass,  perpendi-  ofaflat  T0"1 
cular  t(    Us  circumference.     Let  it  be  very  lightly  supported  in  a  horizontal  position  on  a  bit  of  cork  at  C,  and  cateTfrom 
then  lev  the  boll-glass  be  set  in  vibration  by  a  bow,  at  a  point  opposite  the  place  where  the  rod  meets  it.     It  will  »  bell -glut. 


808 


SOUND. 


Sound. 


304. 

Vibrate 
together  as 
a  system. 
s    305. 
fig.  143. 
Joint  vibra- 
tions of  two 
rods  trans- 
verse to 
each  other. 


306. 
Fig.  144. 
Joint  libra 
tions  of  a 
system  of 
<li*cs  and 
rods. 


307. 

Fit;.  145. 
Vibrations 
•}(  a  disc 
excited  by 
communica- 
tion from  a 
string. 


308. 
Fig.  146. 
Passage  of 
oblique  vi- 
brations 
Into  tan- 
gential, or 
into  trans- 
verse. 
Fifr.  147— 
150. 


309. 

Vibrations 
of  a  mem- 
brane ex- 
cited by 
communi- 
cation from 
the  air. 
Case  I. 
Transverse 
vibrations. 

310. 

Oblique 

1  ihrations. 
Tangential. 


vibrate  transversely,  i.  e.  the  motions  of  its  molecules  will  he  perpendicular  to  its  surface;  and  these  motions  will 
he  communicated  to  the  rod,  without  any  change  in  their  direction,  whose  vibrations  will  be  longitudinal-tan- 
gential, as  will  he  rendered  evident  by  strewing  its  surface  with  sand,  when  the  nodal  lines  will  be  formed  as 
in  Art  292,  and,  if  the  apparatus  be  inverted,  and  the  sand  strewed  ou  the  under  side  of  the  rod,  the  nodal  lines 
will  be  seen  to  correspond  to  the  points  of  greatest  excursion  on  the  other  side,  as  in  that  article. 

In  this  combination  the  original  tone  of  the  bell-glass  is  altered,  and  the  note  produced  differs  both  from  that 
yielded  by  it,  or  by  the  glass  rod  vibrating  alone.  The  two  vibrate  as  a  system  together  and,  what  is  singular, 
the  Sound  of  the  glass  is  considerably  reinforced  by  the  combination. 

Example  2.  Let  A'  be  a  rectangular  strip  of  glass  firmly  cemented  at  right  angles  to  another  strip,  A,  across 
its  breadth.  Let  the  latter  be  lightly  supported  on  two  bits  of  cork,  C,  fastened  to  a  wooden  piece,  B,  so  as  just 
to  touch  A  in  the  places  of  two  of  its  nodes  when  vibrating  transversely.  Then,  if  A  be  placed  horizontally,  and 
strewed  with  sand,  and  A'  be  set  in  longitudinal-tangential  vibration,  either  by  nibbing  with  a  wet  cloth,  or  by 
any  other  means,  A  will  vibrate  transversely,  as  will  be  known  by  the  dancing  of  the  sand  and  its  settling  on  the 
nodes  C  C'.  On  the  other  hand,  if  A  be  held  vertically,  and  agitated  transversely  by  a  bow,  while  A'  is  hori- 
zontal and  strewed  with  sand,  the  latter  will  indicate  longitudinal-tangential  vibrations,  both  by  the  creeping  of 
the  sand,  and  by  the  difference  of  the  nodal  figures  on  its  two  faces. 

Example  3.  Let  M  be  a  rectangular  plate  (fig.  144)  mounted  like  A  in  the  last  example,  but  instead  of 
carrying  a  simple  plate  A',  let  it  carry  a  system  of  circular  discs  traversed  by  a  lamina,  as  in  the  figure.  Then, 
if  the  faces  of  these  discs  and  of  the  lamina  M  be  horizontally  placed  and  strewed  with  sand,  and  the  lamina  M 
be  set  in  longitudinal-tangential  vibration,  all  the  discs  will  be  so  too,  and  the  sand  will  arrange  itself  in  figures 
which,  on  every  alternate  disc,  1, 3,  b,  &c.  will  be  of  one  species,  (such  as  at  a  for  instance,)  but  on  every  other, 
2,  4,  6,  &c.  will  be  of  a  different  species,  as  b.  Now  if  the  whole  apparatus  be  inverted,  so  as  to  place  the  lamina 
M  uppermost,  and  let  the  system  of  discs  hang  down,  the  then  upper  surfaces  of  the  discs  wifl  exhibit  the  same 
system  of  nodal  figures,  but  in  the  reverse  order :  ».  e.  the  discs  1,  3,  5,  &c.  will  give  the  figure  b,  and  2,  4,  6,  &c. 
the  figure  a.  In  this  apparatus,  if  the  connecting  piece  which  traverses  all  the  discs  be  examined,  it  will  be 
found  to  vibrate  transversely,  while  the  discs  and  lamina  M  vibrate  tangentially,  and  vice  versa. 

Example  4.  Let  A  be  a  strong  frame  of  wood  of  the  form  [,  across  the  extreme  edges  of  which  is  stretched  a 
strong  catgut  or  other  chord,  and  let  L  L'  be  a  circular  disc  of  glass,  or  metal,  retained  between  the  chord  and 
back  of  the  frame  by  the  pressure  of  the  former.  Then,  if  the  chord  be  set, in  vibration  by  a  bow  drawn  trans- 
versely across  it  in  one  steady  direction,  the  Vibrations  of  the  chord  will  all  lie  in  the  plane  of  the  bow,  and  will 
be  communicated  in  the  same  direction  to  the  disc,  which  will  execute  tangential  vibrations,  each  of  its  molecules 
moving  to  and  fro  in  lines  parallel  to  the  bow  through  the  whole  extent  of  the  disc.  This  is  easily  verified  by  the 
direction  in  which  sand  strewed  on  it  creeps.  Conceive  the  whole  apparatus  placed  with  the  chord  vertical,  and 
projected  on  the  plane  of  the  horizon.  If,  as  in  fig.  145,  a,  F  F'  be  the  projection  of  the  bow,  the  surface  of  the 
disc  will  be  marked  with  nodal  lines  parallel  to  it,  the  sand  there  being  left,  while  that  in  the  intermediate  spaces 
creeps  along1  to  the  edffes,  as  marked  by  the  arrows,  and  runs  off.  If  the  projection  of  the  bow  FF'  be  oblique  to 
the  line  joining  the  points  of  support  of  the  disc,  as  in  fig.  145,  c,  the  nodal  line  will  be  curved,  as  there  shown,  but 
the  motion  of  the  molecules  of  sand  going  to  form  it  will  still  be  parallel  to  FF*.  Finally,  if  the  bow  be  drawn 
parallel  to  the  line  joining  the  points  of  support,  as  in  fig.  145,  d,  the  nodal  line  will  be  formed  of  two  arcs  making 
a  cusp,  but  the  same  law  of  molecular  motion  will  still  hold  good,  as  the  arrows  indicate. 

Example  5.  Let  LL'  be  a  rectangular  lamina  fastened  at  one  end  into  a  block,  T,  and  at  the  other  attached  to 
a  chord,  c  e,  stretched  parallel  to  its  length,  over  a  bridge,  e,  and  put  in  vibration  by  a  bow  perpendicular  to  it, 
FF'.  Then,  if  the  plane  of  the  bow  and  string  coincide  with  the  plane  of  the  surface  of  the  lamina,  the  latter 
will  execute  tangential  vibrations  across  its  breadth,  and  will  exhibit  on  its  upper  surface  a  single  nodal  line, 
n  n'  nfl,  as  in  fig.  147,  but  on  its  under  none,  all  the  sand  being  driven  off'.  Now  incline  the  bow  to  the  surface 
of  the  lamina  as  represented  in  fig.  146,  c,  at  an  angle  of  about  20°,  still  keeping  it  perpendicular  to  the  string,  and 
the  nodal  line  will  assume  the  curvature  represented  in  fig.  148.  If  the  bow  be  still  more  inclined,  the  curve  breaks 
up,  and  at  45°  of  inclination  becomes  changed  into  transverse  and  oblique  lines,  as  in  fig.  149  ;  and  it.  is  now 
observed  that  the  sand  not  only  runs  in  the  direction  of  the  arrows,  but  also  begins  to  leap,  indicating  an  oblique 
vibration  of  the  surface.  Lastly,  when  the  bow  is  inclined  90°  to  the  plane  of  the  lamina,  as  in  fig.  IbO,  the 
vibration  becomes  altogether  transverse,  the  nodal  lines  are  similarly  disposed  on  both  sides  of  the  plate,  and  the 
sand  merely  leaps  up  and  down  till  it  is  danced  off  the  vibrating  parts,  without  any  tendency  to  creep. 

Example  6.  If  a  very '  hin  membrane  be  stretched  horizontally  over  the  orifice  of  a  circular  bowl,  as  a  drinking- 
cup.  or  harmonica-glass,  (extremely  thin  paper  wetted  and  glued  to  the  edges,  and  then  suffered  to  become 
tight  by  drying,  answers  very  well,)  and  if  fine  sand  be  strewed  on  it,  it  becomes  a  most  delicate  detector  of 
aerial  vibrations.  Suppose  now  a  circular  disc  of  glass  held  concentrically  over  it  with  its  plane  parallel  to  that 
of  the  membrane,  and  set  in  transverse  vibration  so  as  to  form  any  of  Chladni's  acoustic  figures,  as  for  instance 
fig.  (99).  Then  will  this  figure  he  imitated  exactly  by  the  sand  on  the  membrane.  Now  let  the  vibrating  disc 
be  shifted  laterally,  so  as  no  longer  ID  have  its  centre  vertically  over  that  of  the  membrane,  but  keeping  its 
plane,  as  well  as  that  of  the  membrane,  horizontal.  Still  the  figures  marked  out  on  the  latter  will  be  fac-similes  of 
those  on  the  disc,  and  that,  whatever  be  the  extent  of  lateral  removal,  till  the  vibrations  become  too  much 
enfeebled  by  distance  to  have  any  effect  at  all. 

But,  in  place  of  shifting  the  disc  laterally,  let  its  plane  be  inclined  to  the  horizon.  Immediately  the  figures 
on  the  membrane  will  change  though  the  vibrations  of  the  disc  remain  unaltered,  und  the  cliange  will  be  the 
greater,  the  greater  be  the  inclination  of  the  plane  of  the  disc  to  that  of  the  membrane.  And  when  th..- 
former  plane  is  perpendicular  to  the  horizon,  the  nodal  figure  on  the  membrane  is  found  to  be  transformed 
into  a  system  of  straight  lines  parallel  to  the  common  intersection  of  the  two  planes,  and  the  particles  of  sand 


Part  HI. 


SOUND.  809 

Sound,     instead  of  dancing,  creep  in  opposite  directions  to  meet  in  these  lines.     One  of  these  always  passes  through  the     Part  III. 
— ^v~*      centre,  and  the  whole  system  is  analogous  to  what  would  be  produced  by  attaching  a  cord  to  the  centre  of  a  v^~v~"' 
disc,  and,  having  stretched  it  very  obliquely,  setting  it  in  vibration  by  a  bow  drawn  parallel  to  the  surface.     In  a 
word,  the  vibrations  of  the  membrane  are  now  tangential,  and  they  preserve  this  character  unchanged,  however 
the  disc  be   now  shifted  laterally,  provided  its  plane  be  not  turned  from  the  vertical  position.     If  the  disc  be 
made  to  revolve  about  its  vertical  diameter,  the  nodal  lines  on  the  membrane  will  rotate,  following  exactly  the 
motion  of  the  disc. 

Nothing  can  be  more  decisive  or  instructive  than  this  experiment.     We  here  see  evidently,  that  the  motions  of      311 
the   aerial  molecules  in  every  part  of  a  spherical  wave,    propagated  from  a  vibrating  body  as  a  centre,   instead  Nature  of 
of    diverging   like    radii    in   all   directions    so    as    to    be    always   perpendicular    to   the    surface   of  the    wave,  the  aerial 
are  all  parallel  to  each  other;   in  a  word,  they  are  disposed,  not  as  in  fig.  8,  but  as  in  fig.  7 ;  and  thus  the.  motions  in 
hypothesis  of  Art.  113.  is  found  to  he  completely  verified.     Arid  the  same  thing  holds  good  not  only  in  air,  but  a  *our"i' 
in  liquids,  as  the  experiments  hereafter  to  be  related  (due,  like  all  those  just  cited,  to  M.  Savart)  satisfactorily  "' 
demonstrate. 

This  experiment  is  also  remarkable  in  several  other  points  of  view.     So  long  as  the  Sound  of  the  disc,  and  its       312 
mode  of  vibration,  as  well  as  its  inclination  to  the  plane  of  the  membrane,  and  the  tension  of  the  latter,  continue  Data  on 
unchanged,  the  nodal  figure  on  the  membrane  will  continue  the  same ;  but  if  either  of  these  be  varied,  the  mem-  which  die 
brane  will  not  cease  to  vibrate,  but  the  figure  will  be  modified  accordingly.     Let  us  consider  separately  the  effect  VIDratlons 
of  each  of  these  changes.  brat  e™^" 

And  first,  cecteris  immvtatis,  let  the  pitch  of  the  Sound  whose  vibrations,  communicated  through  the  air  to  the  pe-nd. 
membrane,  excite  its  motions,  be  altered,  as   by  loading  the  disc,  or  increasing  or  diminishing  its  size,  (or,  if  the       313. 
Sound  be  excited  by  any  other  cause,  as  a  pipe,  the  voice,  &c.,  then   by  varying  its  pitch  by  any  appropriate  First,  the 
means.)     The  membrane  will  still  vibiate,  differing  in  this  respect  from  a  rigid  lamina,  which  will  only  vibrate  Pitcn  of  the 
by  sympathy  wiih  Sounds  corresponding  to  its  own  subdivisions.     The  membrane,  he  it  observed,  will  vibrate  in  Sounc'' 
sympathy  with  any  Sound,  but  every  particular  Sound  will  mark  out  on  it  its  own  particular  nodal  figure,  and  as 
the  pitch  varies  the  figure  varies.     Thus  if  a  slow  air  be  played  on  a  flute  near  it,  each  note  will  call  up  a  parti- 
cular form,  which  the  next  will  efface,  to  establish  its  own. 

Secondly.     Suppose  the  exciting  cause  be  the  vibration  of  a  disc,  or  lamina  of  any  form.     I?  its  mode  of  vibru-       314 
tion  be  varied  so  as  to  change  its  nodal  figures,  those  on  the  membrane  will  vary  ;  and  if  the  same  note  be  pro-  Secondly, 
duced  by  different  sabdivisions  of  different  sized  discs,  the  nodal  figures  on  the  membrane  will  be  different.  the  nature 

Again,  if  the  tension   of  the  membrane  be  varied  ever  so  little,  most  material  changes  will  take  place  in  the  a"ll.moc'e 
figures  it  exhibits.     If  paper  be  the  substance  employed,  mere  hygrometric   changes  affect  it  to  such  a  degree,  °f  thVexci"- 
that  if  moistened  by  breathing  on  it,  and  allowed  to  dry  while  the  exciting  Sound  is  continued,  the  nodal  forms  ing  cause, 
•will  be  in  a  constant  state  of  fluctuation,  and  will  not  acquire  permanence  till  the  paper  is  so  far  dried  as  the  state       315. 
of  the  surrounding  atmosphere  will  permit.     Indeed,  this  fluctuation  is  so  troublesome  in  experiments  of  this  Thirdly, 
kind,  that  to  avoid  them  it  is  necessary  to  coat  the  upper  or  exposed  side  of  the  paper  with  a  thin  film  of  varnish.  tne  tens'on 
Of  all  substances  which  can  be  employed  for  the  exhibition  of  these  beautiful  experiments,  M.  Savart  observes,  f^*  mem" 
by  far  the  best  is  such  a  varnished  paper  stretched  on  a  frame  and  moistened  on  the  under  side.     The  moisture  Effect  of 
diminishes  the  cohesion  of  the  fibres,  and  renders  them   nearly  independent  of  each  other,  and  indifferent  to  all  hygrome- 
impulses.     As  a  proof  of  this,  he  observes,  that  he  has  frequently  obtained,  on  a  circular  membrane  of  paper  so  tric  changes 
prepared,  a  nodal  figure  composed  of  no  fewer  than  twenty  concentric  annul!,  which  is  far  beyond  what  can  be  on  PaPer 
obtained  in  any  other  way.  membranes. 

In  some  cases,  a  very  curious   and  instructive  phenomenon   is  obseived   in  these  experiments.     Between  the       3jg 
nodal  lines  formed  by  the  coarser  and  middle-sized  grains  of  sand,  others  will  be  occasionally  observed,  formed  Secondary 
only  of  the  very  finest  dust,  of  microscopic  dimensions.     This  phenomenon  will  be  seen  to  greater  advantage  if  a  nodal 
little  dust  of  Lycoperdon  be   mixed  with  the  sand.     These  intermediate  lines  M.  Savart  explains,  by  referring  figures. 
them  to  different  and  higher  modes  of  subdivision,  coexisting  with  that  by  which  the  principal  figure  is  formed. 
The  more  minute  particles  are  proportionally  more  resisted  by  the  air  than  the  coarser  ones,  and  are  thus  pre- 
vented from  making  those  great  leaps  which  throw  the  coarser  ones  into  their  nodal  arrangement.     They,  there- 
fore, rise  and   fall  with   the  surface,  to  which   the/  are   as  it  were  pinned   down.     But  they  are  affected  by  the 
minuter  waves  which  have  a  smaller  amplitude  of  excursion,   and  occur  more  frequently,  and  form  their  figures 
under  the  influence  of  these  us  if  ihe  greater  ones  did  not  exist.     These  secondary  figures  often  appear  as  concentric 
rings  between  the  primary  ones,  and  not  (infrequently  the  centre  of  the  whole  system  is  occupied  by  a  secondary  point. 

Figures  151—161  are  specimens  of  the  nodal  figures  thus  formed  on  circular  membranes.     Of  these,  fig.  161       3^17 
shows  the  modification  which  is  apt  to  take  place  when  the  tension  of  the  membrane  is  not  quite  equable.     Figs.  f,g  151_ 
162,  163,  are  figures  exhibited  by  square  membranes,  and  fig.  164 — 166  by  triangular  ones.  166. 

A  very  important  application   of  these  properties   of  stretched   membranes  has  been  made  by  M.  Savart,  by        3jg 
employing  surh  a  one  as  an  instrument  for  detecting  the  existence  and  exploring  the  extent  and  limits  of  conti-  Stretched 
guous  and  oppositely  vibrating  portions  of  masses  of  air.     For,  since  such  a  membrane  is  thrown  into  vibration  membranes 
by  all  aerial  vibrations  of  a  certain  force,  the  fact  of  the  existence  or  not  of  a  vibratory  motion  in  any  point  of  the  employed- 
air,  of  a  chamber  for  instance,  or  a  box,  or  large  organ-pipe,  maybe  ascertained  by  observing  whether  said  strewed  to  delcct 
on  it  is  set  in  motion,  and  arranged  i'i  regular  forms,  on  holding  the  membrane  at  that  point.   Thus  if  an  organ-pipe  t'ibratJons 
be  made  to  sound  with  a  constant  force,  and  the  exploring  membrane  be  so  far  removed  from  it  that  the  mem 
brane  shall  just  cease  to  be  agitated  visibly,  the  force  of  the  Sound  being  increased  by  a  quantity  not  sensible  to 
the  ear,  the  sand  will  recommence  its  motion.     Nay,  if  two   such  pipes,  placed  close  together,  be  made  to  beat, 
(sec  Index,  Beats,)  the  membrane  will  be  seen  to  be  agitated  at  the  coincidences,  and  at  rest  in  the  interferences 
*f  their  vibrations.     We  shall  presently  return  to  this  part  of  our  subject. 


810 


SOUND. 


Sound. 
^-»V^— 

319. 
Use  of  the 
membrana 
tympani. 


820. 

Fig.  167. 
Construc- 
tion of  the 
ear. 


Fig.  168. 


321. 

Vibrations 
of  an  un- 
elastic 
membrane 
how  pro- 
duced. 
Fig.  170. 


Anotner  highly  interesting  application  of  the  same   properties,  is   tne   view  which  M.  Savart  has  taken  of  the     Part  III. 
use  of  the  "  membrana  tympani"  in  the  ear.     Of  all  our  organs,  perhaps,  the  ear  is  one  of  the  least  understood.  v-~v"~- 
It  is  not  with  it  as  with  the  eye,  where  the  known  properties  of  light  afford  a  complete  elucidation  of  the  whole 
mechanism  of    vision,  and  the  use  of  every  part  of  the  visual  apparatus.     In  the  ear  every  thing  is  on  the 
contrary  obscure  ;  anatomists,  it  is  true,  have  scrupulously  examined  its  construction,  and  many  theories  have 
been  advanced  of  the  mode  in  which  Sounds  are  conveyed  by  it  to  the  auditory  nerve,  (where  of  course,  as 
with  the  optic  nerve  in  the  eye,  all  inquiry  terminates,  for  to  trace  the  progress  of  sensation  along  the  nerve  to  the 
brain,  and  thence  to  the  sentient  soul,  it  is  needless  to  remark,  is  altogether  beyond  our  reach.)     But  nothing 
certain  can    be    said    to    be    known,  though  it    is    to  M.  Savart  that    we  owe    the  most  rational  hypotheses 
hitherto  proposed. 

Fig.  167  represents  the  auditory  apparatus.  It  consists  externally  of  a  wide,  conch-shaped  opening,  K  L, 
which  contracts  into  a  narrow  pipe,  A  B,  defended  from  the  entry  of  dust  and  insects  by  hairs,  and  a 
viscous  exudation  which  is  slowly  secreted,  and  terminated  liy  a  thin  elastic  membrane,  called  the  Tympanum, 
F,  or  drum  of  the  ear.  Behind  this  there  is  a  cavity  which  communicates  with  the  mouth  by  a  small  duct  called 
the  Eustachian  tube,  H  G  I.  If  this  be  stopped,  deafness  is  said  to  ensue,  but,  as  Dr.  Wollaston  has  shown,  only 
to  Sounds  within  certain  limits  of  pitch.  In  the  cavity  behind  the  tympanum  is  placed  a  mysterious  and  com- 
plicated apparatus,  B  C  P  S,  represented  complete,  and  on  an  enlarged  scale,  in  fig.IGS,  consisting  of  four  little  bones, 
of  which  the  first,  S  C,  is  called  the  hammer,  and  rests  with  its  smaller  end  in  contact  with  the  tympanum,  and  its 
larger  on  the  second  bone,  B  P,  called  the  anvil,  between  which,  and  the  last,  V,  called  the  stirrup,  a  little  round 
bone,  P,  forms  a  communication.  These  bones  form  a  kind  of  chain,  and  no  doubt  vibrations  excited  in  the  tympa- 
num by  vibrating  air,  as  in  the  experiments  above  detailed,  are  somehow  or  other  propagated  forward  through  these; 
but  they  are  so  fur  from  being  essential  to  hearing,  that  when  the  tympanum  is  destroyed,  and  the  chain  in  con- 
sequence hangs  loose,  deafness  does  not  follow.  The  last  of  this  chain  of  bones,  however,  is  attached  to  another 
membrane,  p,  which  closes  the  orifice  of  a  very  extraordinary  system  of  canals,  excavated  in  the  bony  substance  of 
the  skull,  called  the  Labyrinth,  represented  separately  in  fig.  169,  which  consists  of  three  semicircular 
arcs,  (1,  2,  3,)  originating  and  terminating  in  a  common  canal,  which  is  prolonged  into  a  spiral  cavity  (4)  called 
the  cochlea.  The  whole  cavity  of  the  labyrinth  is  rilled  with  a  liquid,  in  which  are  immersed  the  branches  of  the 
auditory  nerve,  in  which,  no  doubt,  resides  the  immediate  seat  of  the  first  impression  of  Sound,  as  that  of  sight  does 
in  the  retina.  If  the  membrane  which  closes  the  labyrinth  be  pierced,  and  this  liquid  let  out,  complete  and  irreme- 
diable deafness  ensues.  It  appears  from  some  most  extraordinary  experiments  by  M.  Flourens  on  the  ears  of 
birds,  (of  which,  however,  the  details  are  too  revolting  to  find  a  place  in  any  but  works  on  anatomy  and 
physiology,)  that  the  nerves  enclosed  in  the  several  canals  of  the  labyrinth,  have  other  uses  besides  their 
services  as  organs  of  hearing,  and  serve,  in  some  unaccountable  and  mysterious  manner,  to  give  to  animals 
their  faculty  of  balancing  themselves  on  their  feet,  and  directing  their  motions.  On  this  point  we  refer  the 
reader  to  M.  Cuvier's  report  on  M.  Flourens's  Memoir,  Annales  de  C/iimie,  vol.  xxxix.  p.  104,  and  of  course  to 
the  Memoir  itself,  whenever  and  wherever  it  may  appear  ;  and  for  other  not  less  interesting  and  extraordinary 
facts  of  a  similar  nature,  to  M.  Majendie's  Paper  on  the  functions  of  the  two  great  divisions  of  the  spinal  column, 
and  the  influence  of  the  cerebrum  and  cerebellum  on  voluntary  motion,  abstracted  by  himself  in  a  late  volume  of 
that  collection.* 

To  understand  how  the  vibrations  of  a  disc  may  be  conceived  to  be  communicated  by  the  air  to  a  membrane 
in  M.  Savart's  experiments,  let  us  take  a  simple  case,  and  suppose  A  B  C  D  to  he  a  horizontal  circular  disc, 
vibrating  in  that  mode  which  gives  a  subdivision  into  four  quadrantal  segments,  A  C,  C  B,  B  D,  DA;  and  let 
abed  be  an  infinitely  thin  circular  membrane  placed  under  it,  which  we  will  suppose  to  be  barely  coherent  so  as 

•  From  the  painful  subject  of  knowledge  of  the  most  interesting  and  practically  useful  kind,  to  be  purchased  only  by  the  extremity  of 
animal  suffering,  we  turn  with  gladness  to  a  pleasing  duty.  We  have  drawn  largely,  both  in  the  present  Essay,  and  in  our  Article  on  LIGHT, 
from  the  Annales  de  Ckimie,  and  we  take  this  on/.y  opportunity  distinctly  to  acknowledge  our  obligations  to  that  most  admirably  conducted 
work.  Unlike  the  crude  and  undigested  scientific  matter  which  suffices  (we  are  ashamed  to  say  it)  for  the  monthly  and  quarterly  amuse- 
ment of  our  own  countrymen,  whatever  is  admitted  into  its  pages  has  at  least  been  taken  pains  with,  and,  with  few  exceptions,  has  sterling 
merit.  Indeed,  among  the  original  communications  which  abound  in  it,  there  are  few  which  would  misbecome  the  fiist  academical 
collections  ;  and  if  any  thing  could  diminish  our  regret  at  the  long  suppression  of  those  noble  Memoirs  which  aie  destined  to  adorn  future 
volumes  of  that  of  the  Institute,  it  would  be  the  masterly  abstracts  of  them  which  from  time  to  lime  appear  in  the  Annales,  either  from  the 
hands  of  the  authors,  or  from  the  reports  rendered  by  the  committees  appointed  to  examine  them,  which  latter,  indeed,  are  universally 
models  of  their  kind,  and  have  eontriouted,  perhaps  more  than  any  thing,  to  the  high  scientific  tone  of  the  French  xavani.  What  author 
indeed,  but  will  write  his  best  when  he  knows  that  his  work,  if  it  have  merit,  will  immediately  be  reported  on  by  a  committee  who  wil 
enter  into  all  its  meaning,  understand  it  however  profound,  and  not  content  with  mertty  understanding  it,  pursue  the  trains  of  thought  to 


been  often  astonished  to  see  with  what  celerity  every  thing,  even  moderately  valuable  in  the  scientific  publications  of  this  country,  finds 
its  way  into  their  pages.  This  ought  toencouiage  our  men  of  science.  They  have  a  larger  audience,  and  a  wider  sympathy  than  they  are, 
perhaps,  aware  of ;  and  however  disheartening  the  general  diffusion  of  smatterings  of  a  number  of  subjects,  and  the  almost  equally  geneial 
indifference  to  profound  knowledge  in  any,  among  their  own  countrymen,  may  be,  they  may  rest  assured  that  not  a  fact  they  may  discover, 
no'  a  good  experiment  they  may  make,  but  is  instantly  repeated,  verified,  and  commented  upon,  in  Germany,  and  we  may  add  too  in  Italy. 
We  wish  the  obligation  were  mutual.  Here,  whole  branches  of  continental  discovery  are  unstudied,  and  indeed  almost  unknown  even  by 
name.  It  is  iu  vain  to  conceal  the  melancholy  truth.  We  are  fast  dropping  behind.  In  Mathematics  we  have  long  since  drawn  the  rein  and 
given  over  a  hopeless  race.  In  Chemistry  the  case  is  not  much  better.  Who  can  tell  us  any  thing  of  the  Sulfo-salts?  Who  will  explain 
to  us  the  laws  of  Isomorphism?  Nay,  who  among  us  has  even  verified  Thenard's  experiments  on  the  oxygenated  Acids — Oersted's  and 
Berzelius's  on  the  radicals  of  the  Earths — Balard's  and  Serrulas's  on  the  combinations  of  Brome — and  a  hundred  other  splendid  trains  of 
research  in  that  fascinating  science?  Nor  need  we  slop  here.  There  are.  indeed,  few  sciences  which  would  not  furnish  matter  lor  similar 
remark.  The  causes  are  at  once  obvious  and  deep  seated.  But  this  is  not  the  place  to  discuss  them. 


S  O  U  N  D.  811 

Somd.  to  be  impervious  to  air,  but  to  have  no  tension  of  its  own.  Its  molecules  will,  therefore,  obey  implicitly  all  the  Par-  'M. 
-^^m^>  motions  of  the  aerial  ones  adjacent  to  them,  and  its  figure,  at  any  instant,  will  be  that  assumed  by  a  stratum  of  v^~v^»- ' 
the  air  originally  plane,  and  parallel  to  A  B  C  D,  in  consequence  of  the  displacement  of  its  particles  by  the  undu- 
lation propagated  from  all  parts  of  A  B  C  D,  as  they  reach  it  at  once,  allowing  for  the  time  taken  to  traverse  their 
respective  distances  from  it.  Let  us  now  consider  how  a  molecule  of  air,  M,  placed  any  where  in  the  plane  a  eft 
perpendicular  to  the  disc,  and  intersecting  it  in  A  B,  will  be  affected.  Since  the  disc  vibrates  transversely,  all 
its  particles  on  one  side  of  A  B  (as  towards  C)  will  be  at  any  instant  in  a  precisely  opposite  phase  of  their  excur- 
sion from  the  corresponding  particle  on  the  side  towards  D,  and  moving  with  equal  velocity.  Therefore,  the 
undulations  propagated  simultaneously  from  both  these  particles  will  reach  the  molecule  M  in  question  at  once, 
(being  equidistant  from  it.)  and  being  (at  least  in  so  far  as  their  direction  is  not  modified  in  their  passage,  and 
at  all  events  as  to  that  part  ot  them  which  is  at  right  angles  to  the  plane  a  /B  -/  S)  equal,  and  contrary,  destroy 
each  other,  so  that  in  virtue  of  these  the  molecule  M  acquires  no  transversal  vibration.  And  since  the  same  is 
true  of  every  other  corresponding  pair  of  molecules  into  which  the  two  halves  A  C  B  and  A  D  B  of  the  vibrating 
disc  can  be  divided,  the  molecule  M  will  not  vibrate  (or  at  least  not  transversely)  in  virtue  of  the  vibration  of  the 
whole  disc.  The  same  is  true  of  every  other  molecule  situated  in  the  plane  a  eft,  and  also  by  a  similar  reasoning 
in  the  plane  ^/eS  at  right  angles  to  it.  There  will  then  be  two  nodal  planes  pervading  the  whole  atmosphere,  in 
which  the  aerial  molecules  have  no  transverse  (i.  e.  vertical)  motion.  But  if  we  suppose  the  molecule  M  situated 
anywhere  out  of  these  planes  the  case  is  otherwise.  Suppose  it,  for  instance,  situated  at/  in  the  quadrant  ceb 
of  the  membrane.  This  being  nearer  to  each  molecule  of  the  quadrant  CEB  of  the  disc  than  to  the  corresponding 
molecules  of  the  others,  the  influence  of  the  former  will  predominate,  and  the  molecule  f  will  be  agitated  by  a 
transverse  motion  similar  to  that  of  the  quadrant  of  the  disc  vertically  over  it.  If  then  the  membrane  be  strewed 
with  sand,  it  will  be  thrown  off  from  the  vibrating  quadrants,  and  arranged  on  two  rectangular  nodal  lines  ab,cd 
parallel  to  those  of  the  disc,  just  as  ifh  vibrated  by  its  own  tension,  while  yet  it  is  obvious  that  all  the  while  it 
has  only  obeyed  implicitly  the  motions  of  the  adjacent  air. 

If,  however,  the  membrane  has  tension  and  thickness,  this  will  modify  the  effects  of  the  direct  aerial  action,  and      33^ 
that  in  a  way  far  too  complicated  for  us  to  enter  into  here  in  detail.     We  may  remark,  however,  that  in  that  case,  Effects  of 
each  individual  aerial  impulse  must  be  regarded  as  an  arbitrary  initial  disturbance  of  its  state  of  equilibrium,  in  tension  and 
virtue  of  which  it  will  be  thrown   into   periodic   vibrations;  and  these  again   will  propagate  similar  vibrations  thickness  of 
back  through  the  air  to  the  disc  A  B  C  D ;  and  this  being  constantly  repeated  the  result  may  be  the  establishment  (,rjn'"em~ 
of  a  joint  resultant  periodic  vibration,   by  the   destruction  of  every  motion  not  periodic,    from  the  innumerable 
repetitions  of  the  impulses  and   the  consequent  infinite  superposition  of  plus  and  minus  excursions.     But  this 
interchange,  of  course,  will  be  the  more  energetic  the  thinner  is  the  interposed  lamina  of  air;  for  if  its  thickness  be 
great,  the  vibrations  excited  in   the  membrane,  or  semi-rigid  disc,  abed,  (as  we  will  now  suppose  it,)  will  be 
feeble,  and  when  propagated  back  through  the  air  will  be  still  further  enfeebled  so  as    to  aflect  the  motion  of 
A  B  C  D  but  little.     In  this  case  then,  supposing  the  two  discs  to  be  out  of  unison  with  each  other,  and  to  have 
no  common  mode  of  vibration,  the  disc  abed  will  become  the  seat  of  two  distinct  systems  of  vibration.     The  first, 
regularly  periodical,  being  that  directly  communicated  by  sympathy.     The  other,    the  resultant  of  an  indefinite 
number  of  vibrations  kept  up  by  means  of  the  tension,  in  all  phases  and  stages  of  degradation. 

Now,   provided  the  time  elapsed  since  the  commencement  of  the  vibrations  be  long  enough  to  allow  of  our       323. 
regarding  the  number  of  previous  vibrations  as  infinite,  or  which  comes  to  the  same,  long  enough  to  have  allowed  General 
all  traces  of  the  initial  vibrations  to  have  been  destroyed  by  resistance,  friction,  &c.,  these  last  will  either  exactly  tlleorei>' 
destroy  each  other,  or,  if  they  leave  a  residue,  that  residue  will  consist  in  a  vibratory  motion,  having  the  same  forcerl'v'i 
period  with  the  primary  impulse.  brations. 

As  this  is  a  proposition  of  great  importance,  not  only  in  the  theory  of  Sound,  but   in  many  other  physical      324. 
theories,  such  as  that  of  the  Tides,  for  example,  we  must  not  let  it  rest  on  a  vague  assumption,  but  demonstrate  it  Demonstri- 
rigorously.     Let  then  t  represent  the  time  elapsed  since  the  commencement  of  the  vibrations,  t  being  so  large  t'oa- 
that  it  may  be  considered  as  infinite  in  comparison  of  the  duration  of  a  single  vibration.     Then  if  we  call  T  the 
time  of  one  complete  vibration,  or  one  period  of  the  primary  vibrations,  the  impulse  communicated  through  the 
air,  or  otherwise,  to  any  point  of  the   membrane,  or  other  vibrating  body,  will  at  any  instant  be  represented  by 

some  periodic  function  of  the  form  F(  cos  2  it  .  —   1,  or  F  (cos  n  f)  putting  ^=-  =  n,  which  function  may  always 

be  resolved  into  a  series  of  periodical  terms  of  the  form  A  .  cos  int,  i  being  an  integer,  of  which  each  may  be 
considered  as  the  representative  of  a  single  vibratory  motion  of  the  simplest  kind,  whose  superposition  forms  the 
actual  vibration  in  question.  Consequently,  we  may  content  ourselves  with  considering  any  one  of  them  as 
A  .  cos  n  t,  since  all  the  rest  are  subject  to  the  same  argument. 

Next,  let  0  be  the  time  of  one  complete  unforced  vibration   of  the   membrane  or  elastic  body  in  virtue  of  its      325. 

2  T 
natural  elasticity,  and  let  v  =  —  .  i,  so  that  a  .  cos  »  t  would  denote  the  general  term  of  a  series  expressing  the 

velocity  of  any  one  of  its  molecules  in  a  state  of  unforced  vibration,  and  let  F  (t)  be  a  function  expressive  of  the 
law  of  diminution  of  the  vibrating  motion  by  friction,  resistance,  and  imperfect  elasticity.  So  that  if  t  be  the  time 
since  a  certain  velocity  V  was  communicated  to  it,  V  .  F  (t)  .  cos  v  t  will  be  its  velocity  after  the  expiration  of  t 
as  it  will  then  subsist,  modified  by  the  elastic  forces  and  mechanical  state  of  the  membrane. 

Conceive  the  aerial  impulse  to  act  not  continuously,  but  at  equal  infinitely  small  intervals  of  time  T,  (infinitely       326 
small  relative  not  only  to  t  but  to  T  and  0.)     Then,  first,  the  impulse  A  .  cos  n  t,  acting  durinn-  the  time  T   will 
produce  the  velocity  A.  .  T  .  cos  n  t. 

VOL.   IV.  9  N 


812  S  O  TJ  N  E 

Sound.          Secondly.  The  impulse  A  .  cos  (n  t  —  n  T)  which  acted  at  the  moment  immediately  preceding,  produced  in  the     Pan  1  '• 
^^—     -_i  first  instance  the  velocity  A  .  T  .  cos  (re  t  —  n  T).     But  this,  once  produced,  was  immediately  modified  by  the  ^^^^—' 
inherent  elasticity  of  the  membrane,  and  in  the  subsequent  moment  became 

A  .  T  .  cos  n  (t  —  T)  .  F  (T)  .  cos  v  T. 

Similarly  the  impulse  A  .  cos  n  (t  —  2  T)  acting  at  the  instant  preceding  this  generated  the  velocity  A  T  .  cos 
(<  —  2T),  which,  in  like  manner,  (being  regarded  as  an  arbitrary  initial  disturbance,)  became  modified  in  the  time 
2  T  to  A  T  cos  n(t  —  2  T)  .  F  (2  T)  .  cos  2  v  t.  And  so  on.  Thus,  the  whole  accumulated  velocity  at  the  instant 
t,  arising  from  all  the  preceding  impulses,  will  be  expressed  by 

A  T  .  {  cos  nt  +  cos  n  (t  —  T)  .  cos  v  -r .  F  (T)  -f-  cos  n  (t  —  2  T)  .  cos  2  v  T  .  F  (2  T)  -J-  &c.  }  , 

which  series,  since  the  function  expressed  by  F  (t)  is  supposed  to  decrease  constantly  as  t  increases,  and  since 
the  whole  number  of  vibrations  is  supposed  so  great  that  the  terms  of  the  series  F  (t),  F  (2  T),  F  (3  T),  &c.  shall 
at  length  become  perfectly  insensible,  may  be  regarded  as  continued  ad  injinitum. 

In  fact,  whatever  supposition  we  may  make  as  to  the  law  of  degradation  of  the  motion  within  the  limit  of  a 

327.      single  period,  it  must  evidently  diminish  in  geometrical  progression  in   similar  phases   of  successive  periods,  so 
General         tnat  W(J  must  naye 

!£;  of "  F  (T  +  *)  =  J  .  F  (T)  ;   F  (T  +  2  0)  =  <f  •  F  (T)  Ac. 

vibratory      rp^  premisedt  the  series  in  question,  by  merely  changing  the  arrangement  of  its  terms,  and  grouping  together 
"eometric     those  equidistant  from  each  other  by  the  interval  0,  will  become  resolved  into  partial  series  thus, 
progression.  (cosw  t  -\-  cos  n  (t  —  0)  .  cos  v  6 .  F  (0)  +  cos  n  (t  -  2  0)  .  cos  2  v  0 .  F  (2  0)  -f-  &c.   ] 

A  T  .  J  +  cos  n  (t  -  T)  .  F  (T)  +  cos  n  (t  -  -r  -  0)  .  cos  (v  6  -|-  v  T)  .  F  (0  -f  T)  -f-  &c. 

(+&c.  J 

But  we  have,  first,  F(o)=:l,  F  (0)  =  q,  F(26»)=gf,  &c. ;  and,  moreover,  since  v  6  =  2  iir,  therefore 
cos  i>0—  1,  and  cos  2  vO  =r  1,  &c.  Consequently  the  above  expression  becomes 

f  (cos  n  t  +  q  .  cosn  (t  -  t»)  -f-  q* .  cos  n  (t  -  2  0)  +  &c.)  1 

A  T  J  -f  F  (T)  .  (cos  n  (t  -  T)  +  q  .  cos  n  (t  -  T  —  0)  +  q*.  &e.)  I. 

(-f-  F  (2  T)  .  (cos  n  (t-  2  T)  +  q  .  cos  n  (t  —  2  T  -  0}  +  <f  .  &c.  -f  &c.)  J 
Snmmatlon  Now,  e««ch  of  these  series  is  readily  summed,  for  we  havu  by  well-known  trigonometrical  fonnulcB 

°Jal  sencs"  cos  nt  +  q.  cos  (n t  —  n  G)  -j-  q* .  cos  (re  t  —  2  n  6)  -j-  &c. 

1  —  q  .  cos  nO  q  .  sin  n  0 

=  cos  n  t . — *-  — — .  +  sin  n  t  . ;•  — r~- — .. 

1  —  2  g  .  cos  re  0  -f-  g«  1  -  2q  .  cosn  0  +  q* 

Each  of  the  fractions  being  constant,  and  independent  of  t,  if  we  call  them  M  and  N,  our  series  will  become 

|(M  .  cos  n  t  +  N  .  sinn  0  +  F  (T)  {  M  .  cos  n  (t  —  T)  -f  N  .  sin  n  (t  -  T)  }          1 
+  F(2  T)  {  M  .  cos  n  (t  -  2  T)  +  N  .  sin  n  (t  -  2-r)  }  \. 
+  &c. 

328.  Let  us  now  consider  the  area  of  a  curve  whose  abscissa,  x,  is  divided  into  equal  elements  each  equal  to  r,  while 
Summation   its  successive  ordinates,  y,  are  represented  by  0  (o),  (j>  (T),  0  (2  T),  &c.     It  is  evident  that  its  ureufy  dx  will  be 
of  the  whole  equa]  to 

%£!?  T  .  0  (o)  +  T  .  0  (T)  +  T  .  0  (2  T)  +  &c. ; 

and,  therefore,  the  sum  of  this  series,  from  the  term  -r.0  (o)  to  T.  0C0).  will  be  equal  to  the  integral^  0(x)  dt, 
from  x  =  0  to  x  =  6.     Thus  our  series  will  assume  the  form  of  a  definite  integral,  w'z. 
A  ./."  d  x  .  F  (x)  {  M  .  cos  n  (t  —  x)  +  N  .  sin  n  (t  -  x)  }  , 

expressing  in  the  manner  now  pretty  general  the  limits  of  the  integral  by  indices  attached  to  the  integral  sign. 
Resolving  now  the  sines  and  cosines  of  n(t  —  x),  this  becomes  (T  and  t  being  independent  of  each  other) 

A .  cos  n  t .  fl  d  x  .  F  (x)  {  M  .  cos  n  x  —  N  .  sin  n  r  } 
—  A  .  sin  n  t  f?  d  x  .  F  (JT)  {  M  .  sin  n  x  -  N  .  cos  n  x  }  . 

Now,  whatever  be  the  law  of  degradation  denoted  by  the  function  F,  it  is  clear  that  these  definite  integrals  must 
at  last  reduce  themselves  to  certain  constants  independent  of  t,  which,  if  we  call  P  and  Q,  the  whole  takes  the 
simple  form 

P  .  cos  n  t  —  Q .  sin  n  t, 

which  is  a  periodic  function  having  the  same  period  as  the  primary  vibrations.* 

329.  In  the  limiting  case,  when  the  elasticity  of  the  body  on  which  the  forced  vibrations  are  impressed  is  perfect, 
Caieofper-   and  resistance,  friction,  and  every  other  cause  of  loss  of  motion   is  prevented,  F  represents  a  constant,  and  is 
feet  elasti-     equal  to  unity.     In  this  case  both  the  constants  P  and  Q  in  the  above  expressions  vanish,  and  the  whole  motion 

city  of  the 

iody.  *  This  demonstration   being  general,  we  may  here  observe,  that,  on  the  undulatory  theory  of  light,  rays  ot  one  refraugibility  can  never 

excite  by  any  combination  of  their  own  vibrations  with  those  of  the  bodies  they  may  traverse  or  impinge  on,  any  resultant  rays  of  a  different 
refrangibility,  at  least  to  long  as  the  exciting  light  continues  in  action      When  it  has  ceased,  the  case  may  be  otherwise. 


S  O  U  N  D.  813 

Sound,      of  (rtf  body,  after  a  great  numbci  of  vibrations  have  elapsed,  is  zero.     In  this  case  then,  the  elastic  body  is     Part  III. 
-~**/—s  completely  incapable  of  vibrating  in  sympathy  with  any  other  not  having  a  common  mode.     In  all  others  P  ^^•v*^-' 
and  Q  have  finite  values,  which  will  be  greater,  or  less,  according  to  the  circumstances  of  the  case. 

Thus  we  see   that  imperfect  elasticity,  or  other  equivalent  causes  of  the  gradual  loss  and  dissipation  of  the       330^ 
impressed  impulses,  is    the    essential   condition    on   which  forced  vibrations  in  general  depend,  and  that  in  General 
proportion  as  a  disc  or  membrane  is  devoid  of  tension  it  should   be  more  readily  susceptible  of  such  vibrations :  effect  of  im- 
precisely what  M.  Savart  has  shown  to  be  really  the  case  in  fact. 

It  may  be  objected  to  what  is  said  in  Art.  321,  that  it  would  follow  from  that  reasoning  that  the  Sound  of  a    'C'QQJ 
vibrating  disc  should  be  inaudible  whenever  the  ear  is  situated  in   a  plane  passing  through  one  of  its  nodal  object'ons 
diameters,  and  at  right  angles  to  the  disc.   But,  in  the  first  place,  what  is  there  said  applies  only  to  such  motions  of  considered. 
the  aerial  particles  as  are  performed  in  those  planes.     But,  in  fact,  a  lateral  motion,  or  one  parallel  to  the  disc's 
surface,  must  also  exist,  by  reason  of  the  alternate  tilting  up  and  down  of  adjacent  ventral    segments,  which 
must  give  the  whole  body  of  air  terminated  by  them  a  small   reciprocating   rotatory  motion  about  the  nodal  line 
separating  them  as  an  axis.     Thus,  though  the  transverse  vibrations  are  here  destroyed,  the  sensation  of  Sound 
may   still  be  excited  by  tangential  ones.     And,  secondly,  though  alternate  motion  were  altogether  destroyed, 
condensations  and  rarefactions  still  subsist. 

But,  in  fact,  there  »*  observed  a  difference  in  the  intensity  of  Sound  emanating  from  vibrating  bodies  in  certain       J.QO 
cases,  according  to  their  angular  position  with  respect  to  the  line  joining  them  and  the  ear.     We  have  already  phenomena 
(Art.  117.)   described  Dr.  Young's  remarkable  experiment  of  the  tuning-fork.     It  is  precisely  a  case  in  point,  ofatuning- 
and  a  circumstantial  explanation  of  it  will  be  at  once  interesting  for  its  own  sake,  and  illustrative  of  the  general  fork  ex- 
argument.     Let  then  A,  B,  fig.  171,  be  sections  of  the  two  branches    of  the  fork  in  its  state  of  rest,  and  since  Plained- 
when  set  in  vibration  they  alternately  approach  to  and  recede  from  each  other,  let  us  consider  them  first  in  their     '*>' 
state   of  approach,  as  at  a  b.     In  this  state  they  compress  the  air  between  them,  and  squeeze  it  out  laterally  in 
the  direction  of  the  arrows  P,  Q,  while,  at  the  same  instant,  the  aerial  particles  adjacent  to  the  flat  outward  faces 
of  the  two  branches,  and  which  of  necessity  follow  their  motions,  are  urged  inwards  as  indicated  by  the  arrows 
R  S.     Thus  the  four  quadrants  of  the  initial  circular  wave  propagated  round  the  fork,  are  alternately  in  opposite 
states  of  motion,  the  molecules  at  P  Q  receding  from  the  centre,  while  those  at  R,  S,  approach  to  it,  and  Dice 
versa,  when   the  branches  of  the  fork  having  dosed  to  the  utmost  begin  to  open  again.     In  this  case  the  latera1 
air  will  rush  in  to  fill  the  gap,  while  that  in  contact  with  the  broad  faces  will  be  forced  outwards.     If  then  we 
consider  any  intermediate  point  C,  about  45°  distant  from  Q  and  R,  this,  in  virtue  of  both  impulses,  will  acquire 
equal  tendencies  in  opposite  directions,  and  will  rest,  or  at  least  will  acquire  only  a  small  tangential  motion,  in 
consequence  of  the  reciprocating  eddies  of  the  air  round  the  angles  of  the  branches.     That  these  motions  really 
do  take  place  as  here  pointed  out,  any  one  may  have  ocular  demonstration  by  imitating  the  opening  and  shutting 
of  the  branches  with  his  hands  near  the  flame  of  a  candle  burning  steadily,  taking  care  not  to  mukepitffs  of  wind 
but  regular  removals  of  the  air  to  and  fro. 

One    of  the    most    curious  and    interesting  purposes    to   which   M.   Savart  has    applied   the  properties  of       333. 
membranes,  is  to  explore  the  actual  state  of  the  air  in  different  parts  of  a  vibrating  mass  of  determinate  figure,  Membranes 
as  to  motion  or  rest.     For  this  purpose,  the  Sound  should  be  excited  and  maintained  by  a  constant  cause  at  a  Uj     'Ju"" 
high  degree  of  intensity,  especially  if  the  mass  of  air  be  large,  as  in  a  chamber  or  gallery  ;  and   to  give  the  ("^rations 
membrane  the  greatest  possible  sensibility,  it  ought  to  be  stretched  so  as  to  be,  naturally,  in  unison  with  the  note  Of  masses 
sounded,  so  as  to  act  as  a  receiver  and  condenser  of  the  small  aerial  motions.     The  greatest  purity  and  intensity  of  air. 
of  the  Sounds  to  be  employed  for  this  purpose,  may  be  obtained  by  a  harmonica  glass,  or  the  bell  of  a  clock,  'nte"« 
maintained  in  vibration  by  a  bow;   and  this  may  be  still  further  augmented  by  adapting  to  it  a  resonant  cavity,  (jucedbPr° 
as,  for  instance,  a  large  cylindrical  vase  of  considerable  diameter,  closed  at  one  end,  and  of  such   dimensions  as  resonance, 
separately  to  vibrate  the  same  note.  (See  Art.  338.)     The  tones  thus  produced,  especially  when  large  harmonica 
glasses  are  used,  as  M.  Savart  remarks,  are  of  such  intensity,  that  no  ear  can  long  support  them,  and,  at  the 
same   time,   of  such    a    rich  and    mellow    quality,  that  all  other  musical   Sounds  appear  poor  and  harsh  in 
comparison.     In  order  yet  more  to  increase  the  sensibility  of  the   membrane,  the  frame  on  which  it  is  stretched  Mode  of 
should  be  fitted  over  the  orifice   of  a  similar  resonant  cavity.     For  convenience,  and  lest  the  tension  of  the  "sm§a 
membrane  should  vary  by  hygrometric  changes,  it  is  proper  to  have  means  of  varying  this  at  pleasure,  a  mode  of 
which  is  described  by  M.  Savart  in  the  Memoir  from  which  we  draw  our  information.  (Annales  de  Chimie,  vol. 
sxiv.  p.  76.) 

Suppose  now  that,  being  provided  with  such  an  apparatus  as  here  described,  we  shut  ourselves  up  in  an        334. 
apartment  of  regular  figure,  and  free  from  furniture  or  projections  from  the  walls,  recesses,  &c.,   and  place  one  Vibrations 
of  our  resonant  cylinders  with  its  axis  horizontal,  and  the  vibrating  bell   or  glass  opposite  its  orifice.     In  the  °f  thha°mab'rr"1 
direction  of  its  axis  place   the  membrane  horizontally,  with  its  proper  frame  and  resonant  cylinder  below  it,  and  examine" 
strew  the  horizontal  surface  with  saml.     If  now,  first,  we  place  the  membrane  thus  armed  very  near  the  source 
of  the  Sound,  it  will  vibrate  with  great  force      As  we  withdraw  it,  (keeping  it  still  in  the  line  of  the  axis  of  the 
first  resonant  cylinder,)  its  vibrations  will  diminish  gradually,  and  at  length  cease,  after  which  (still  continuing  to 
remove  it  along  this  line)  they  will  recommence  and  reach  a  maximum,  at  a  point  when   their  intensity  is  nearly 
equal  to  that  close  to  the  source  of  Sound.     Removing  the  membrane  yet  further,  a  new  point  of  indifference  is 
found,  and  so  on  till  we  reach  the  end  of  the  chamber.     If  we  walk  along  the  same  line,  keeping  the  ear  in  the 
plane  of  the  horizontal  axis  of  '/he  resonant  cylinder,  we  shall  perceive  the  Sounds  to  be  much  louder  in  the 
places  where  the  vibrations  of  the  membrane  attain  their  maxima,  than  at  the  intermediate  points  where  they  are 
at  a  minimum.     At  these  latter,  a  very  curious  phenomenon  has  been  observed  by  M.  Savart.     When  the  auditor 
moves  his  head  away  from  such  a  point,  towards  the  right,  (always  supposing  it  to  remain  in  the  line  of  the  axis 
above  mentioned,)  the  Sound  will  appear   to  come  from  the  right,  and  if  towards  the  left,  it  will  seem  to  come 

5  N  2 


814  SOUND 

Sound,     from  the  left,  whether  the  original  source  of  Sound  be  to  the  one  or  the  other  side.     This  singular  effect  shows  that     Part  PI. 
V^^v^*''  the  aerial  molecules  on  either  side  of  the  point  of  indifference,  are  in  opposite  states  of  motion   at  any  given   ^^v""— 
instant.     In    making  this  experiment,  the  head  should  be  so  turned,  that  the  axis   of  the  resonant   cylinder 
prolonged  shall  pass  through  both  ears.     Suppose,  for  instance,  the  Sounding  apparatus  to  be  to  the  observer's 
left,  and  that  his  head  be  very  near  it.     The  Sound  will  appear  to  enter  at  his  left  ear.     As  he  removes  further 
away,  so  as  to  pass  one  of  the  nodes,  it  will  seem  as  if  the   Sound  had  changed   sides,  and  now  came  from  the 
right.     When  another  node  is  passed,  it  will  appear  to  have  again  shifted  to  the  left,  and  so  on. 

335.  But,  if  we  quit  the  axis  of  the  cylinder,  and  carry  an  exploring  membrane,   such  as  already  described,  about 
Spiral  form  the  apartment,  noting  all   the  points  where  it  vibrates  most  forcibly,  allowing  ourselves,  as   it  were,  to   bo   led 
of  the  nodal  from  SpOt  to  SpOt  by  jts  indications,  we  shall  trace  out  in  the  air  of  the  room  a  curve  of  double  curvature  marking 
rectangular  tne  max'ma  °'  tne  excursions  of  the  aerial  molecules.     If  the  experiment   be   made   in  a  gallery,  or  passage, 
chamber,  or  whose  length  is  its  principal  dimension,  this  curve  will  be  found  to  be  a  kind  of  spiral,  creeping  round  the  walls, 
gallery.        floor,  and  ceiling,  obliquely  to  the  axis  of  the  gallery,  thus  presenting  a  marked  analogy  to  the  disposition  of  the 

nodal  lines  in  a  long  rod  vibrating  tangentially,  {vide.  Art.  313.)  ;  which  are  also,  it  should  be  remarked,  imitated, 
with  modifications  more  or  less  complicated,  in  square  or  rectangular  rods. 

336.  A  still  more  remarkable  effect  was  observed  by  M.  Savart,  in  thus  exploring  the  vibrations  of  the  air  in  an 
Their  con-    apartment  with  an  open  window.     The  spiral  disposition  of  the  vibrating  portions  was,  found  to   be  continued 
tinuation      out  of  the  window   into  the  open  air,  the  lines  of  greatest  intensity  running  out  in  great  convolutions  which 
wimimv  nito  seeme(^  '°  omw  wider,  on  receding  from  the  window,  and  could  be  traced  to  a  great  distance  from  it. 

the  air.  The   vibrations  of  the   air  in  an  organ-pipe  were   explored  by  M.  Savart,  by  lowering  into  the  pipe,  placed 

337.  vertically  with  its  upper  end  open,  a  thin  membrane   stretched  on  a  light   ring,  and  suspended   by  a  fine  silk 
Vibrations  thread,  and   strewed  with  sand.     Thus  ocular  demonstration  of  the  existence  of  its  subdivision   into  distinct 
of  air  in  ventral  segments  was  obtained,  the  sand   remaining  undisturbed   when   the  membrane   occupied  precisely  the 
pipes  ex-  place  of  a  node.     By  this  means,  too,  the  influence  of  the  embouchure  on  the  places  of  the  nodes,  a  curious  and 

delicate  point  in  the  theory  of  pipes,   which  we    have    not    before  alluded    to,  may  be    subjected   to    exact 
membranes.  .       *  „,       •    „       .    J  r  ,  .  .  J  •' 

examination.      JLhus,  tor  instance,  when  the  column  ot  air   in  the  pipe  vibrates  in  the  manner  described.  Art. 

190,  fig.  19.  having  two  half  ventral  segments,  and  one  node  in  the  middle,  it  is  found  that  the  node  is  only 
approximately  so  placed,  being  always,  in  fact,  nearer  to  the  embouchure  than  to  the  open  end. 

333  It  is  well  known  that  if  we  sing  near  the  aperture  of  a   wide-mouthed   vessel,  some  one  note  (which  is  in 

unison  with  the  air  in  the  vessel)  will  be  reinforced  and  augmented,  and  sometimes  to  a  great  degree.  This  is 
what  is  meant  by  the  resonance  of  the  mass  of  air  contained  in  the  cavity  of  the  vessel,  or  as  it  may  be  termed, 
the  resonance  of  the  cavitv.  This  has  been  known  from  the  earliest  times.  The  ancients  are  said  to  have 
placed  large  brass  jars  under  the  seats  of  their  immense  theatres  to  reinforce  (one  does  not  well  see  how)  the 
Resonance  voices  of  the  actors.  Any  vessel  or  cavity  may  be  made  to  resound  by  placing  opposite  its  orifice  a  vibrating 
of  cavities,  body,  having  a  surface  large  enough  to  cover  the  aperture,  or  at  least  to  set  a  considerable  portion  of  the  aerial 
stratum  adjacent,  to  it  in  regular  oscillation,  and,  at  the  same  time,  pitched  in  unison  with  the  note  which  the 
cavity  would  of  itself  yield.  The  experiment  of  the  disked  tuning-fork,  in  Art.  204,  is  a  case  exactly  in  point. 
The  pipe  which  resounds  in  that  experiment,  may  be  pitched  precisely  in  unison  with  it  by  its  stopper,  and  in 
proportion  as  it  departs  from  a  perfect  unison  the  resonance  is  feebler.  A  series  of  disked  tuning  forks,  or 
vibrating  steel  springs,  thus  placed  over  the  orifices  of  pipes  carefully  tuned,  constitutes  a  very  pretty  musical  instru- 
ment, capable  of  a  fine  swell  and  fall  according  as  the  discs  are  brought  nearer  to,  or  further  from,  the  orifices 
of  the  pipes,  or  inclined  to  their  axes,  and  of  remarkable  purity  and  sweetness  of  tone.  A  similar  adaptation  of 
resonant  cavities  to  a  series  of  harmonica  glasses  fixed  on  a  common  revolving  axis,  has  been  recommended  by 
M.  Savart  as  the  principle  of  a  musical  instrument,  whose  effect,  should  it  be  found  to  answer  the  expectations  his 
description  of  the  tones  thus  drawn  forth  is  calculated  to  excite,  would  probably  surpass  that  of  all  others  yet 
invented.  See  Art.  333.  The  cavities  best  adapted  to  this  purpose  are  short  cylinders  of  large  diameters  with 
movable  bottoms  fitting  by  tight  friction  by  which  they  may  be  tuned. 

339.  Such  cavities  may  be  regarded  as  short  organ-pipes.     When  the  diameter  of  a  pipe  is  greatly  increased  in 
Resonance   proportion  to  its  length,  so  that  it  becomes  a  box,  the  law  of  the  proportionality  of  the   time  of  vibration  to  the 
and  vibra-    ieng(h  ceases  to  hold  good,  and  the  note  yielded  is  flatter  than  that  of  a  narrow  pipe  of  equal  length,  and  the 
b'oiTsha  ed  more  so  tne  w'('er  the  pipe.     Thus  M.  Savart  found  that  a  cylinder  of  4^  inches  in  length,  and  5  in  diameter, 
cavities.       resounded  in  unison  with  a  narrow  pipe  6  inches  long,  making  1034  vibrations  per  second.     That  sagacious 
Resonance    experimenter  has  found,  that  cubical  boxes  speak  with  surprising  promptitude   and  facility,  and  yield  Sounds 
of  cubical    extremely  pure,  and  of  a  peculiar  quality,  on  which  account,  and  by  reason  of  the  little  height  in  which  they  may 
boxes.         ^  pufif^  ne  recommends  them  for  organ-pipes.     A  cube  of  53  or  54  lines  ( =  4^'"-)  in  the  side  yields  the  same 

note  as  a  pipe  10  or  11  inches  long,  and  2  or  2£  inches  diameter.  They  may  be  excited  by  an  embouchure  at 
one  of  their  lower  edges,  precisely  similar  to  that  of  an  organ-pipe.  But  they  will  also  speak  if  the  embouchure 
be  situated  in  the  middle  of  the  side.  M.  Savart  has  also  examined  the  vibrations  of  a  great  variety  of  different- 
shaped  pipes,  boxes,  or  cavities,  for  which  see  Annahs  de  Chimif,  vol.  xxix.  p.  404. 

340.  There  is  yet  another  remarkable  case  of  vibrations  communicated  between  the  different  members  of  a  system 
( .  iimmuni-    of  which  we  have  not   yet  spoken,  though  offering  a  good   example  of  the  verifioation  of  the   general    law  of 
vibrations     eqlla'itv  of  period    and  parallelism  of  direction  of  the  vibratory  motions  of  all    the  molecules  ot  a  system  laid 
through        down  in  Art.  302.     It  is  when  vibrations  are  communicated  through  a  liquid.     The  following  expeilments  of 
liquids.        M.  Savart  will  show  the  mode  in  which  this  is  accomplished. 

341.  He  took  a  cylindrical  tinned  iron  vessel  whose  bottom  was  placed  parallel  to  the  horizon,  and  having  cemented 
Exaeriment.  to  its  centre  a  glass  rod,  so  as  to   hang  perpendicularly  down  from  it,  he  covered  the   bottom  to  the  depth  of 

about  an  inch  and  a  half  with  water,  on  which  was  floated  a  thin  disc  ot  varnished  wood,  covered  en   its  upper 


SOUND.  815 

face  with  sand.     The  apparatus  thus  prepared,  he  impressed  on  the  glass  rod  a  longitudino-tangential  vibration,     IVt  III. 
'   (Art.  296.)  which  of  course  became  normal  when  communicated   to  the  bottom  of  the  vessel,  and  observed  the  •— • — v— - 
sand  on  t-he  upper  face  of  the   disc  to  be  also  agitated   with   normal  motions,   and  to  assume   nodal   figures  (: 
according  to  the  laws  of  that  species  of  motion.     To  show  more   clearly  the  nature  of  the  communication,  he  J^"'"^! 
threw  out  the  water,  and  supported  the  wooden  disc  by  a  small   solid   stem  perpendicular  to  its  surface,  and  the  brations. 
bottom  of  the  vessel,  and  attached  to  the  centres  of  both,  when  it  «as  found  that  the  disc  was  affected  precisely 
in  the  same  way  as  before. 

On  a  vessel  of  water,  whose  rim  is  maintained  in  a  state  of  normal  vibration  by  a  bow  drawn  perpendicularly       342 
across  it  at  any  point,  let  a  thin  rectangular  lamina  of  wood  be  floated,  having  its  length  parallel  to  the  bow,  and  its  Communi 
extremity  opposite  to  the  point  of  the  circumference  excited.     The  lamina  will  be  seen  (as  usual  by  sand  strewed  calion  of 
on  its  upper  face)  to  execute  longitudino-tangential  vibrations,  and  will  be  crossed  by  nodal  lines  at  right  angles  '*Kf't"o*s 
to  its  length.     But  if,  instead  of  directing  the  axis   or  longer  edges  of  the   lamina  perpendicularly  towards  the 
vibrating  point  of  the  side  of  the  vessel,  we  incline  it  obliquely  to  the  direction  of  the  vibrations,  still    the  sand 
on  its  upper  face  will  continue  to  glide  in  the   same  direction  as  before,  that  is,  parallel   to  the  vibrations  of  ihe 
side  of  the  vessel,  so  that,  if  the  floating  lamina  be  made  to  revolve  slowly  in  a  horizontal  plane,  the  direct  ion  of 
the  creeping  motion  of  the  sand  on  its  surface  will  continually  vary  with  respect  to  the  position  of  its  edges,  though 
constant  with  regard  to  the  sides  of  the  vessel. 

Not  only  arc  the  vibrations  thus  faithfully  tranrferred  through  the  water  to  bodies  floating  on  its  «urface,  hut       343. 
even  to  such  as  arc  totally  immersed  in  it.     The   experiment  is   easily  made  by  suspending  in   such  a  vessel  as  Commum- 
above  described  under  the  water,  and  not  in  contact  with  the  sides  or  bottom,  a  disc  of  glass,  by  means  of  fine  calion  ol  > 
silk  threads,  and  strewing  sand  on  the  surface  of  the  water  which   sinks  and  spreads  evenly  on  the  disc.     This  b;a''?ns '" 
will  be  observed  to  be  agitated  with  very  decided  normal,  or  tangential  motions,  according  as  the  former  or  latter  fire"^^'™-" 
of  the  modes  of  excitement  used  in  the  experiments,  Arts.  341,  342,  is  employed;  and  to  arrange  itself  in  nodal  mersed. 
figures  accordingly. 

From  these  and  similar  experiments  it  appears  that  vibratory  motions  are  communicated  through  liquids  344. 
precisely  as  through  gases  aud  solids,  without  change  of  character  or  direction.  This,  observes  M.  Savart,  How  we 
explains  how  the  nerves  of  hearing,  extended  throughout  the  convolutions  of  the  labyrinth  and  immersed  in  the  J"d8e .°' th^ 
liquid  which  fills  it,  transmit  to  the  sensorium,  not  only  the  general  impression  of  Sound,  but  of  the  direction  in  c' 
which  it  comes. 

These  remarkable  and  striking  results  all  tend  to  confirm  and  strengthen  the  analogy  between  Sound  and  Light.       345. 
The  luniinilerous  ether,  like  air  and  liquids,  transmits  vibrations  without  altering  their  direction,  as  the  phenomena  Further 
of  polarized  light  demonstrate.     The  additional  weight  of  evidence  thus  thrown  into  the  scale  of  the  nndulatory  an»l"si«s 
theory  of  light  did   not  escape  the  penetrating  mind  of  Dr.  Young,  to  whom  that  theory  was  so  deeply  indebted.  ^eluee" 
Doubtless  the  analogy  thus  ascertained  would  not  have  remained  idle  in  his  hands,  had  not  death  snatched  him  too  [join 
from  science  while  in  the  vigour  of  his  intellect,  and  when  so  much  might  have  yet  been  hoped  from  him.      It  has  OILightand 
been  our  unprecedentedly  unfortunate  lot,  while  composing  these  Essays  on  the  sister  sciences  of  Light  aud  Sound,  so""d 
to  have  to  deplore  the  loss  of  nearly  all  the  great  modern  contributors  to   their  advancement.     A   Fraiieuhofer,  a 
Fresnel,  a  Wollaston,  and  a  Young,  names  forming  an  epoch  in   the   history  of  human  knowledge,  have  been 
snatched  away  in  quick  and  alarming  succession,  not  enfeebled  by  age  or  with  faculties  weakened  by  disease,  but 
all  in  the  meridian  of  their  intellectual  powers,  or  in  that  rich  maturity  when  practice  had  only  familiarized  them 
with  their  resources,  and   perfected   them   in  their  use.     To  Dr.  Young  the  theory  of  Sound  is  in  many  respects 
deeply  indebted,  and  it  richly  repaid  the  attention  he  devoted  to  it  by  furnishing  him  with  the  pregnant  idea  of 
his  principle  of  interferences  ;  a  principle  which  has  proved  the  key  of  all  the  more  abstruse  and  puzzling  properties 
of  light,    and    whose  establishment  would   alone  have  sufficed  to  place  him   in  the  highest  rank  of  scientific 
immortality,  even  were  his  other  almost  innumerable  claims  to  such  distinction  disregarded. 

§111.     Ofthf.roice. 

Almost  every  animal  has  a  voice  or  cry  peculiar  to  itself,  originating  in  an  apparatus  destined  for  that  purpose  340 
of  more  or  less  complexity.  The  voice  is  most  perfect  and  varied  in  man  and  in  birds,  which,  however,  differ 
extremely  in  the  degree  in  which  they  possess  this  important  gift.  In  quadrupeds,  it  is  limited  to  a  few  uncouth 
screams,  bcllowings,  and  other  noises,  perfectly  unmusical  in  their  character,  while  in  many  birds  it  assumes  the 
form  of  musical  notes  of  great  richness  and  power,  or  even  of  articulate  speech.  In  the  human  species  alone, 
and  that  only  in  some  rare  instances,  we  find  the  power  of  imitating  with  the  voice  every  imaginable  kind  of 
noise,  with  a  perfect  resemblance,  and  of  uttering  musical  tones  of  a  sweetness  antl  delicacy  attainable  by  no 
instrument.  But  in  all,  without  exception,  (unless,  perhaps,  the  chirp  of  the  grasshopper,  or  cricket,  be  one,) 
the  Sounds  of  the  voice  are  produced  by  a  wind  instrument,  by  the  column  of  air  contained  in  the  mouth,  throat, 
and  anterior  part  of  the  windpipe,  set  in  vibration  by  the  issue  of  a  stream  of  air  from  the  lungs  through  a 
membranous  slit  in  a  kind  of  valve  placed  in  the  throat.  In  man  and  in  quadrupeds,  this  organ  is  single,  but 
in  birds,  as  M.  Savart  has  shown,  it  is  double  ;  a  valve  of  the  kind  abovementioned  being  placed  at  the  openinn-  of 
each  of  the  two  great  branches  into  which  the  trachea  first  divides  itself  as  it  enters  the  lungs,  iust  before  they 
unite  into  one  common  windpipe. 

The  organs  of  the  voice,  in  man,  consist  of 

1.  The  thorax,  which,  by  the  aid  of  the  diaphragm  and  the  24  intercostal  muscles  acting  on   the  lungs  within,       347. 
anil  alternately  compressing  and  dilating  them,  performs  the  office  of  a  bellows  The  thorax. 


816  S  O  U  N  D. 

Sound.          2.  The  trachea,  a  cartilaginous   and  elastic  pipe  which  terminates  in   the  lungs  by  an  infinity  of  roots,  or     P«'  "I- 
-•"v^-'  bronchiae,  and  whose  upper  extremity  is  formed  into  a  species  of  head  called  the  larynx  situated  in  the  throat,  *"• "V" 

348.  composed  of  five  elastic  cartilages,  of  which  the  uppermost  is  called  the  epiglottis,  whose  office  is  to  open  and 
The  trachea  shut,  like  a  valve,  the  aperture  of  the  exterior  glottis,  and  which  constitutes  the  orifice  of  the  larynx 

JVfl"5  3.  The  epiglottis,  where  it  adheres  to  the  larynx,  is  also  united  to  the  tongue,  and  forms  a  somewhat  concave 
The  i-  valve,  of  a  parabolic  form,  whose  base  is  towards  the  tongue,  and  which,  by  its  convexity,  resists  the  pressure  of 
glottis  tne  f°°d  ar|d  liquids  as  they  pass  over  it  in  the  act  of  swallowing. 

350.  4.  Within  the  larynx,  rather  above  its  middle,  between  the  thyro'id  and  arytenoi'd  cartilages,  are  two  elastic 
The  glottis,   ligaments  like  the  parchment  of  a  drum  slit  in  the  middle,  and  forming  an  aperture  making  a  right  angle  with 

the  exterior  glottis,  and  which  is  called  the  interior,  or  true  glottis.  This  slit,  in  adults,  is  about  four-fifths  of  an 
inch  long,  and  a  twelfth  of  an  inch  broad.  This  aperture  is  provided  with  muscles  which  enlarge  and  contract 
it  at  pleasure,  and  otherwise  modify  the  form  of  the  larynx. 

351.  5.  The  tongue,  the  cavity  of  the  fauces,  the  lips,  teeth,  and  palate,  with  its  velum  pendulum,  and  the  uvula,  a 
The  mouth,  pendulous,  conical,  muscular  body,  which  performs  the  office  of  a  valve  between  the  throat  and  nostrils,  as  well 
u     a,           as,  perhaps,  the.  cavity  of  the  nostrils  themselves,  are  all  concerned  in  modifying  the  impulse  given  to  the  breath 

as  it  issues  from  the  larynx,  and  producing  the  various  consonants  and  vowels,  according  to  the  different 
capacities  and  shapes  of  their  internal  cavity. 

352_  In  speaking  or  singing,  the  glottis,  it  has  been  generally  supposed,  performs  the  part  of  a  reed.    The  membranes 

Glottis  sup-  of  which  it  is  composed  being  kept  at  a  greater  or  less  state  of  tension  by  the  muscles  with  which  it  is  provided,  and 

posed  to  act  its  opening  expanded  or  contracted  according  to  the  degree  of  gravity  or  acuteness  of  the  Sound  to  be  uttered. 

as  a  reed       3ut  the  tone  thus  originally  produced  by  the  glottis  is  sustained  and  reinforced  by  the  c  liunn  of  air  in  the  larynx, 

throat,  and  mouth,  whose  dimensions  and  figure  are  susceptible  of  great  variation  by  the  action  of  the  innumerable 

muscles  which  give  motion  to  this  complicated  and  intricate  part  of  our  frame.     Thus  in  a  general  way  we  may 

conceive  how  the  voice  is  produced  and   modified  ;  but  when  we  would  penetrate  further  into  particulars,  the 

difficulties  presented  by  the  organs  of  the  voice  are  even  greater  than  those  which  beset  the  investigation  of  those 

of  hearing. 

353.  One  material  one  has  been  lately  much  elucidated  by  the   experimental  researches  of  M.  Savart.     How,  we 
Difficulty      may  naturally  ask,  can  tones  of  such  gravity  as  we  hear  produced  by  the  human  voice,  be  excited  in  so  short  a 
from  the       column  of  air  as  that  contained  in  the  throat  of  a  man  ?     The  vibrating  column  here  hard-ly  exceeds  a  few  inches 
fhTnotes      'n  'en{?th,  yet  the  notes  produced  by  a  bass  singer  are  those  which  would  require  a  pipe  of  several  feet  in  length 
produced,     sounded  in  the  usual  manner.     That  it  is  not  a  mere  relaxation  of  the  membrane  of  the  glottis   is  evident ;    the 

dropping  of  the  lower  jaw,  and  the  effort  made  in  every  possible  way  to  increase  the  dimensions  and  diminish  the 
tension  of  the  throat  and  fauces  generally,  in  singing  the  lower  notes  of  the  scale,  sufficiently  prove  that  the  note 
of  the  glottis  is  reinforced  in  this. case,  as  in  that  of  acuter  Sounds,  by  the  resonance  of  the  cavity  in  which  it 
sounds. 

354.  From  M.  Savart's  experiments  it  appears  that  in   short  pipes,   and  cavities  whose  other  dimensions  bear  a 
Explained     considerable  ratio  to  their  length,  the  tone  yielded  is  rendered  much  graver  when  the  pipe  or  cavity  is  constructed 
>yM.Sa-     of  a  flpxjble  material  capable   of  being  agitated  and   set  in  vibration   by  the  air,  than  when  made  of  more  rigid 

materials.  He  constructed  a  cubic  box-pipe  with  paper  stretched  on  slight  square  frames  (if  wood,  joined  together 
at  the  edges,  and  made  it  speak  by  an  embouchure  at  the  edge.  He  then  observed,  that  so  long  as  the  paper 
was  tightly  stretched  the  Sound  yielded  by  the  cube  was  nearly  as  acute  as  it  would  have  been  had  the  whole 
been  rigid,  but  that  when  its  tension  was  diminished  by  exposing  it  to  moist  vapour,  or  even  by  wetting  it,  the 
Sound  descended  in  the  scale  by  an  interval  proportioned  to  the  degree  of  moisture  the  paper  had  imbibed.  It 
was  thus  lowered  even  two  whole  octaves,  when  it  grew  so  feeble  as  to  be  no  longer  audible  ;  but,  repeating  the 
experiment  in  the  still  of  night,  it  could  yet  be  heard,  and  no  limit  indeed  then  seemed  set  to  the  descent  of  the 
Sound  ;  and  even  when  no  longer  audible  the  vibration  of  the  paper  sides  could  still  be  made  sensible  by  sand 
strewed  on  them,  which  arranged  itself  in  nodal  lines,  for  the  most  part  elliptic  or  circular. 

355.  The  relaxation  then,  or  increase  of  tension  of  the  soft  parts  which  form  the  cavity  of  the  mouth  and  larynx,  is 
Tension  and  no  doubt  a  principal  cause  of  the  graduation  of  its  tones.     Whoever  will   sing  open-mouthed   before  a  looking- 
relaxatioii     g]ass  wjn  not  fail  to  be  struck  with  the  extraordinary  contraction  of  the  uvula  (a  small  pendulous  substance  which 
and  mouth*  seems  to  na"ff  down  from  the  roof  of  the  mouth)  which  takes  place  in  the  higher  notes.     It  shrinks  up  almost  into 

a  point,  and  every  surrounding  part  seems  to  partake  its  tension. 

356.  We  have  observed  that   the  glottis  has  been   most  generally  regarded  as  performing  the  functions  of  a  reed, 
Savart's  ob.  especially  since  the  free  reed  (anche  libre)  invented  by  Kratzenstein,  and  revived  by  Grenie",  (probably  without 
jee'ions        knowledge  of  Kratzcnstcin's  prior  invention  ;  vide  Willis,  Phil.  Trans.  Camb.  vol.  iii.)  has  been   brought  into 
ned-Me"    Peneral  no*'ce-     This  idea  is  strongly  advocated  among  others  by  Biot.     But  M.  Savart  professes  himself  dissatis- 
act;onof  the  ^ed  wi'h  such  an  explanation  of  its  use.     He  remarks,  and   seemingly  with  justice,  that  the  essential  principle  of 
glottis.          a  reed,  the  periodical  opening  and  closing  of  the  orifice  through  which  the  stream  of  air  passes,  is  wanting  in  the 

glottis.  Were  the  glottis  a  reed,  the  edges  of  the  vocal  ligaments  which  form  the  slit  through  which  the  air  passes 
would  require  to  be  almost  in  contact,  and  should  be  alternately  forced  asunder  by  the  effort  of  the  air,  and 
brought  together  by  their  tension.  But  on  the  contrary  he  lound  that  the  larynx  of  the  dead  subject,  when  left 
in  its  natural  state,  and  gently  blown  into  through  the  trachea,  yielded  Sounds  approactiing  to  those  of  the  voice, 
although  the  opening  left  between  the  borders  of  the  glottis  was  as  much  as  one-sixth,  or  even  one-fourth  of  an 
357_  inch  across,  and  more  than  half  an  inch  long. 

M.  Savart  s  The  instrument  to  which  M.  Savart  attributes  the  greatest  analogy  to  the  larynx,  is  a  species  of  whistle,  common 
explanation  enough  as  a  children's  toy  or  even  as  a  sportsman's  call,  in  the  form  of  a  hollow  cylinder  about  three-fourths  of 
of  the  voice.  an  jncn  jn  diameter,  closed  at  both  ends  by  flat,  circular  plates,  having  holes  in  their  centres.  The  form  is  not  of 


SOUND.  817 

Sound      murh  importance,  it  may  be  made  hemispherical,  &c.     Being  held    between  the  teeth  and  lips,  the  air  is  blown      Part  III. 

— ^-*_j  through  it,  and  Sounds  are  produced  which  vary  in  pitch  with  the  force  of  the  blast.     If  the  air  be  conducted  to  v-— v— ' 
it  through  a  porte  vent,  and  cautiously  graduated,  all  the  Sounds  within  the  compass  of  a  double  octave  may 
readily  be  obtained  from  it ;  and  if  great  precautions  are  taken  in  the  management  of  the  wind,  tones  even  yet 
graver  may  be  educed,  so  as  to  admit,  in  fact,  no  limit  in  this  direction. 

When  we  come  to  investigate  the  nature  of  articulate  Sounds,  and  of  speech,  the  difficulties  are  much  greater.       357. 
Conrad  Amman,  in  his  work  on  the  Voice,  first  attempted  to  explain  the  manner  in  which  the  vowels  and  Amman's 
consonants  are  formed.     With  regard  to  the  vowels,  he  regards  them  as  mere  modifications  of  the   continued  wo.rk  °"  ''" 
tone  produced  by  the  larynx,  depending  on  the  configurations  of  the  mouth.     Thus  to  pronounce  A   (the  broad  " 
A  in  Ah !)  the  tongue  must  be  laid  flat  in  the  lower  jaw,  and  the  mouth  opened  wide,  and  lips  turned  outwards. 
Any  musical  or  continued  tone  produced  in  the  throat  will  then  have  the  character  of  the  vowel  A.     If  the  tongue 
be  gradually  elevated   so  as  to  bring  its   middle  nearer  the  palate,   and  at  the  same  time  thrust  forwards,  its 
extremity  approaching  the  upper  teeth,  the  Sound  will  deviate  from  the  broad  A  into  a  (hate,)  e  (peep.)     These 
Sounds  therefore  (the  a  in  hate,  and  the  e  in  peep)  he  calls  dental  vowels.     On  the   other  hand,  if,  the  tongue 
remaining  as  before,  the  lips  be  thrust  out  and  drawn  together,  preserving  as  great  an  interior  cavity  of  the 
fauces  as  possible,  we  shall  have  the  Sounds  of  the  vowels  in  all,  hope,  poor,  wood.     These  he  calls  labial  vowels, 
&c.     These  distinctions  are  to  a  certain  extent  correct  and  reasonable,  but  they  give  us  no  insight  into  the 
question,  ffhat  it  is  which  constitutes  the  essential  distinction  between  vowel  and  vowel,  and  on  what  part  of  the 
mechanism  of  the  voice  do  vowel  Sounds  depend  ? 

In  1779,  the  Imperial  Academy  of  Petersburgh  proposed  as  one   of  their  prize  questions,  an  inquiry  into  the       35S. 
nature  of  the  vowel  sounds  A  E  I  O  U,  and  the  construction  of  an   instrument  capable  of  artificially  imitating  Kratzen- 
them.     The  prize  was  awarded  to  M.  Kratzenstein,  whose  curious  Memoir  on  the  subject   the  reader  may  find  s.teln's '"'**• 
in  the  XXIst  volume  of  the  Journal  de  Physique,  p.  358     His  principle  consisted  in  the  adaptation  of  a  reed  in  all  the*  vowels 
essential  respects  similar  to  Greni^'s,  where  the  tongue  passes  to  and  fro  through  the  slit  without  contact,  to  a 
set  of  pipes  of  peculiar  forms,  some  of  them  very  odd  ones,  and  for  whose  shapes  no  other  reason  could  be  given 
than  their  success  on  trial.     This,  however,  was  a  great  step.     It  showed  the  vowel  quality  of  a  Sound  to  be 
something  distinct  from   mere  pitch,   and  susceptible  of  being  produced  at   pleasure  by  mechanical   artifice. 
Pursuing  this   idea.  Mr.  Willis  has  lately  entered  more  extensively  into  the  subject,  and,  in   a  Paper  recently 
printed  in  the  Hid  volume  of  the  Transactions  of  the  Cambridge  Philosophical  Society,  has  succeeded  in  educing 
all  the  vowel  Sounds  by  a  mere  combination  of  a  reed  on  Kratzenstein's  construction  with  a  cylindrical  pipe  of 
variable  length,  and  investigating  the  laws  of  their  production. 

This  may  be  the  place  to  remark  the  extreme  imperfection  of  our  written  language  in  its  representation  of       359. 
vowels  and  consonants.     We  have  six  letters  which  we  call  vowels,  each  of  which,  however,  represents  a  variety  of  Remarks  on 
Sounds  quite  distinct  from  each  other,  and  while  each  encroaches  on  the  functions  of  the  rest,  a  great  many  very  w 
good  simple  vowels  are  represented  by  binary  or  even  ternary  combinations.     On  the  other  hand,  some  single 
vowel  letters  represent  true  diphthongs,  (as  the  long  sound  of  i  in  alike,  and  that  of  u  in  rebuke,)  consisting  of  two 
distinct  simple  vowels  pronounced  in  rapid  succession,  while,  again,  most  of  what  we  call  diphthongs  are  simple 
vowels,  as  bleak,  thief,  laud,  &c.     This  will  render  an  enumeration  of  our   English  elementary  Sounds,  as  they 
really  exist  in  our  language,  no   matter  how   written,  not  irrelevant.      We   have  therefore  assembled   in  the 
following  synoptic  table  sufficient  examples  of  each  to  render  evident  their  nature,  accompanied  with  occasional 
instances  of  the  corresponding  Sounds  in  other  languages.     The  syllables  which  contain  the  Sounds  intended  to 
be  instanced  are  printed  in  italics  where  words  of  more  than  one  syllable  are  instanced. 

I     (""Rook;  Julius;  Rude;  Poor;  Womb;  Wound;   Oi/vrir,  (Fr  1  360. 

"  \~  Good;  Cushion;  Cuckoo;  Rund,  (Germ.);  Gusto,  (Ital.)  Synoptic 

2.  '    Spurt ;  Assert  ;  Dirt ;   Kirtue ;  Dove ;  Doable ;  Blood.  lable  of 

3.  Hole ;  Toad.  jjjjjj1* 
All ;  Caught ;   Organ  ;  Sought ;  Broth ;  Broad.  Sounds 
Hot ;  Comical ;  Kommen,  (Germ.) 

5.  Hard  ;  Braten,  (Germ.)  ;  Charlatan,  (Fr.) 

6.  Laugh  ;  Task. 

7.  Lamb ;  Fan  ;  That. 

8.  Hang  ;  Bang  ;  Twang. 

9.  Hare;   Hair;   Heir;  Were;  Pear;   Hier,  (Fr.)  ;   LeAren,  (Germ.) 

10.  Lame;  Tame;  Crane;   Faint;  Lawman;  Meme,  (Fr.);  Slddchen,  (Germ.) 

11.  Lemon;  Dead;  Said;  Any  ;  Every  ;  Friend;  Besser  (Germ.)  i  Eloigner,  (Fr.) 

12.  Liuer  ;  Diminish;  Persevere;  Believe. 

13.  Peep  ;  Leave  ;  Beliece ;  Sieben,  (Germ.)  ;  Coqui\\e.  (Fr.) 

14.  s;  sibilus  ;  cipher  ;  the  last  vowel  and  the  first  consonant 

True  Diphthongs. 

1.  Life;  The  Sounds  No.  5  and  No.  13,  slurred  as  rapidly  as  possible,  produce  our  English  i,  whv  h       361 

is  a  real  diphthong.  Diphthongs. 

2.  Brow ;  Plough  ;  Lrtttfen,  (Germ.)     The  vowel  Sound  No.  5  quickly  followed  by  No.  1 

3.  Oil ;  Kauen,  (Germ.)  ;  No.  4  succeeded  by  No.  13. 

4.  Rebuke;  Yew;  You;  No.  13  succeeded  by  No.  1. 

5.  Yoke  ;  No.  13  succeeded  by  No.  3. 

6    Young;   Yearn  ;  Hear;  Here  ;  No.  13  succeeded  by  No  2  more  or  less  rapidK 


818  S  O  U  N  D. 

Sound.          The  consonants  present  equal  confusion.     They  may  be  generally  arranged  in  three   classes :  sharp  Sounds,     part  III. 
^— ~v  •• '  flat  ones,  and   indifferent  or  neutral.     The  former  two   having  a  constant  relationship  or  parallelism  to  each   k— -v--— • 

362.      other,  thus  : 
Con*°"*nU-       SHARP  CONSONANTS      S.  sell,  cell ;  a.  (as  we  will  here  denote  it)  shame,  sure,  schirm,  (Germ.)  ;  0.  thing  ;   F 

fright,  enough,  phantom  ;  K.  king,  coin,  quiver ;  T.  talk;  P  papa. 

sonants.  FLAT  CONSONANTS.     Z.  zenith  ;  casement ;  '£.   pleasure,  jarAm,  (French)  ;  Q.  the  th  in  the  words  the,  that, 

364.  thoii.;  V.  vile;  G.  good;  D.  duke  ;  B.  babe. 

ogj  NEUTRAL  CONSONANTS.     L.  lily  ;  M.  mamma;  N.  Nanny;  v.  hang;   to  which  we  may  add  the  nasal   N  in 

Xeutra|  '  gnu,  JEtna,  Dnieper,  which,  however,  is  not  properly  an  English  Sound.     R.  rattle;  H.  hard. 

366.  COMPOUND  CONSONANTS.      C,  or  fa.  church,  cicerone,  (Ital.)  and  its  corresponding  flat  sound  J.  or  D  £".  \est, 
Compound.  gender ;  \.  extreme,  Xerres ;  f.  exasperate,  exalt,  Xerxes  ;  &c.  &c. 

367.  We  have  here  a  scale  of  13  simple  vowels   and  21  simple  consonants,  33  in  all,  which  are  the  fewest  letters 
Remark  on     with  which  it  is   possible  to  write  English.     But  on  the  other  hand,  with   the  addition  of  two  or  three  more 
*n  un'versa'  vowels,  and  as  many  consonants,  making  about  40  characters  in  all,   every  known  language  might  probably  be 

e  '  effectually  reduced  to  writing,  so  as  to  preserve  an  exact  correspondence  between  the  writing  and  pronunciation ; 
which  would  be  one  of  the  most  valuable  acquisitions  not  only  to  philologists  but  to  mankind,  facilitating  the 
intercourse  between  nations,  and  laying  the  foundation  of  the  first  step  towards  a  universal  language,  one  of  the 
great  desiderata  at  which  mankind  ought  to  aim  by  common  consent. 

ggg  This  enumeration  will  serve  to  show  what  are  the  difficulties  which  any  one  must  contend  with  in  constructing, 

Difficulties  wnat  has  been  often  attempted,  a  talking  engine.  Still  the  partial  success  obtained  by  Kratzenstein,  and  about 
of  imitating  the  same  time  by  Kempelen,  who  has  given  a  very  curious  account  of  his  experiments  in  Mecanismede  la  Parole, 
speech  by  ought  to  encourage  further  trials. 

mechanism.        rpQ  return)   however,   to   Mr.  Willis's  curious  and   novel   researches.     He  relates   that,  having  provided  an 
WIT '     aPParatus  consisting  of  a  wind-chest,  or  reservoir,  connected  with  a  pair  of  double  bellows,  and   opening  into  a 
experiments  Por^ven^  having  a  free  reed,  on  Kratzenstein's,  or  Greni6*s  construction,  at  its  termination,  his  first  object  was  to 
on  the  verify  Kempelen's  account  of  the  vowels.     He  therefore  adapted  his  reed  to  the  bottom  of  a  funnel-shaped  circular 

vowel  cavity,  open  at  top,  as  in  fig.  172,  which   represents  a  section  of  the  apparatus,  and  on  making  the  reed  speak, 

Sounds.  and  placing  his  hand  in  various  positions  pointed  out  by  Kempelen  within  the  funnel,  he  obtained  the  vowels 
Reetition  A  (No.  5.),  E  (No.  10.),  I  (No.  13.),  O(No.  3.),  U(No.  J.)  very  distinctly,  On  using,  however,  a  shallower 
of  Kempe-  cavity  these  positions  became  unnecessary,  and  the  hand  might,  he  found,  be  replaced  by  a  flat  board  slided  over 
len's  expe-  the  mouth  of  the  cavity  ;  and  by  using  a  very  shallow  funnel,  as  represented  in  fig.  173,  he  succeeded  in  obtaining 
rimcnts.  the  whole  series  in  the  order  U  (No.  1.),  O(No.  3.),  A  (No.  5.),  E  (No.  10.),  I  (No.  13.) 

Being  thus  led  away  from   Kempelen's  experiment,  he   proceeded  to  try  the  effect  of  adapting  to  the  reed 

370.  cylindrical  tubes,  whose  length  could  be  varied   at  pleasure  by  sliding  joints.     This  was  easily  accomplished  by 
Mr.  Willis's   fixjng  die  reed  with  its  port-vent  into  the  end  of  a  pretty  long  horizontal  pipe  coming  off  from  the  wind -chest, 
rm'reed  ptoes  over  which  on  its  outside  a  tube,  open  at  both  ends,  was  made  to  slide  on  leather  wrapped  round  it  in  the  manner 
of  variable      of  a  piston,  and  capable  of  being  lengthened,  by  the  attachment  of  pieces  of  similar  tube  of  its  own  length,  to 
length.          any  extent.     He   thus   describes   the   results   so  obtained.     Let  abed  represent  the  length  of  the  outer,  or 
Fig  '74-       sounding  pipe,  projecting  beyond  the  reed,  and  take  a  b,  be,  c  d,  &c.  equal  to  the  length  of  a  stopped   pipe  in 
Vowel  unison  with  the  reed  employed,  that  is  equal  to  half  the  length  of  the  sonorous  wave  of  the  reed.     If,  now,  the  pipe 
Sounds  ap-     be  drawn  out  gradually,  the  tone  of  the  reed,  retaining  its  pitch,  first  puts  on  in  succession  the  vowel  qualities 
pear  in  re-     I  E  A  O  U.     As  the  length   approaches  to  a  c  the   same  series  makes  its  appearance  in  an  inverted  order,  as 
gular  sue-       represented  in  the  diagram,  then  on  passing  the  length  a  c  in  direct  order  again,  and  so  on  in  cycles,  each  cycle. 

wion  and    heing  rnerely  a  repetition  of  the  foregoing,  but  the  vowels  becoming  less  and  less  distinct  in  each  successive  cycle, 
and  the  distance  of  any  given  vowel  from  its  respective  central  points  a,  c,  &c.  being  the  same  in  all  the  cycles. 

371.  If  another  reed  be  adapted   to  the  same  pipes  having  a  different  fundamental  Sound   or  sonorous  wave,  the 
Law  of  the     same  phenomena  will  be  produced,  only  that  the  central  points  of  the  new  cycles  will  now  be  at  a  distance  from 

"  °f  d  eac^  ot^er  emla'   to   tf'6   sonorous  wave  of  the  new  reed,  but  the  distamvs  of  the  several  vowel  points  from  the 
"  "  centres  of  the  respective  cycles  will  be  the  same  as  before;  so  that,  generally,  if  the  reed  wave  ac  =  '2  a,  and  the 
length  of  the  pipe  which  first  produces  any  given  vowel,  from  a,  bv  equal  to  u,  the  same  vowel  will  be  constantly 
reproduced  by  a  pipe  whose  length]=  2n  a  i  v,  n  being  any  whole  number. 

When  the  pitch  of  the  reed  is  high,  so  that  the  length  a  c  of  its  wave  is  less  than  twice  the   distance  a  U  corre- 

I;'m'tsjj|!"h   spending  to  any  vowel,  all  the  vowels  beyond  that  distance  become  impossible.     If,  for  instance  ac  be  less  than 

certain*  "      ^a  ^'  ')ut  £reater  than  2  a  O,  the  series  will  never  extend  so  far  as  U,  but  on  lengthening  the  pipe  indefinitely 

vowels  be-    the  succession  of  vowels  I  E  A  O  A  E  I  will  be  repeated.     If,  in  like  manner,  still  higher  notes  be  taken  for  the 

come  im-      reed,  more  vowels  will  be  cut  off.     This,  Mr.  Willis  remarks,  is  exactly  the  case  with  the  human  voice :  female 

possible.        singers  being  unable  to  pronounce  U  and  O  on  the  higher  notes  of  their  voices.    For  example,  the  proper  length 

for  a  pipe  to  produce  O  is  that  which  corresponds  to  the  note  C"  two  octaves  above  the  middle  C  of  a  piano-forte, 

and  beyond  this  note  in  singing  it  will  be  found  impossible  to  pronounce  a  distinct  O 

Cylinders  of  the  same  length,  or  more  generally  cavities  of  any  figure  resounding  tne  same  note,  give  the  same 
374.       vowel  when  applied  to  one  and  the  same  reed. 

lenetb>of          The  mllowin£  tanle  is  Siven  Dv  Mr-  Willis  as  expressing  the  distances  from  the  central  points  of  the  cycles  at 
rowel  pipe».  which  the  several  vowels  are  produced  in  inches. 


SOUND. 


819 


Sound. 


Vowel  Sound 
according  to  its 
place  in  the  Scale. 
Art.  360. 

Example. 

Length  of  Vowel 
Pipe  in  inches  and 
decimals. 

Piano-forte 
note  cor- 
responding. 

I 

No.  13. 

Set 

0.38? 

Gv 

E 

11 

Pet  (  ?  Pay) 

0.6 

C' 

A 

10 

Pay  (?  Pet) 

1.0 

Div 

6  or  7 

Paa 

1.8 

pi: 

5 

Part 

2.2 

D»b 

A° 

4 

Paw 

3.05 

D'»b 

0 

4 

Nought 

3.8 

G» 

3 

No 

4.7 

E»b 

U 

2 

But 

Indefinite. 

C" 

1       ' 

Boot 

Part  III. 


On  this  Table  Mr.  Willis  observes  that  he  does  not  despair  of  its  completion  and  extension  by  future  experiments, 
eventually  furnishing  Philologists  with  a  correct  measure  for  the  shades  of  difference  in  the  pronunciation  of  the 
vowels  by  different  nations.  One  source  of  fallacious  decision,  however,  it  must  be  remarked,  will  subsist  in  its 
application,  in  the  effect  of  contrast,  on  which  much  of  the  difference  between  vowels  depends.  Its  influence 
indeed  may  be  traced  in  the  above  Table  itself.  Thus  Mr.  Willis,  assisted,  no  doubt,  by  the  contrast  arising  from 
rapid  and  frequent  transition,  has  been  able  to  discriminate  between  the  vowel  Sounds  yielded  by  pipes  of  the 
lengths  3'05  and  3'S,  though  the  Sounds  in  the  exemplifying  words  Paw  and  Nought,  which  he  lias  chosen,  are 
so  closely  allied  that  we  confess  our  own  inability  to  detect  any  shade  of  difference,  for  which  reason  we  have 
designated  them  by  the  same  number.* 

Mr.  Willis  terminates  this  highly  interesting  Paper  with  some  curious  experiments  and  remarks  on  the  murtual 
influence  of  a  reed  and  a  pipe  with  which  it  is  connected,  as  also  of  the  port-vent,  which  conducts  the  air  to  it.  If 
a  reed  be  made  to  sound  in  a  pipe  of  variable  length  (/),  the  Sound  yielded  by  it  will  remain  constant  till  the 

length  (I)  (beginning  we  will  suppose  from  o)  becomes  nearly  eaual  to  — ,  or  one  quarter  the  length  of  the  Sound 

wave  of  the  reed  ;  here  it  begins  to  flatten,  ana  as  I  is  still  increased,  continues  to  do  so  till  the  length  somewhat 
exceeds  \  o,  when  it  suddenly  jumps  back  to  a  note  about  a  quarter  of  a  tone  sharper  than  the  original  Sound  of 
the  reed,  to  which  it,  however,  soon  again  descends,  and  continues  stationary  till  the  length  I  becomes  nearly 
equal  to  2  a  -j-  \  a,  when  the  flattening  again  commences,  and  continues  till  /  exceeds  2  a  +  J  a,  and  so  on 
periodically,  but  less  decidedly.  The  total  amount  of  flattening  is  usually  a  whole  tone.  A  jerk  of  the 
bellows,  or  a  too  hasty  lengthening  of  the  pipe,  will  make  the  pitch  spring  hack  much  sooner  than  it  would  do 
with  cautious  management,  nay,  with  proper  dexterity  it  may  be  made  to  yield,  just  about  the  point  of  junction,  a 
double  note,  composed  of  one  flatter  and  one  sharper  than  the  reed  would  yield  alone.  Mr.  Willis  seems  to  think 
that  in  this  case,  however,  the  two  Sounds  are  onlv  quickly  alternated  so  as  to  seem  to  go  on  at  once.  Examining 
the  reed  in  a  glass  pipe  with  a  magnifier,  he  found  its  excursions  diminished  when  the  note  was  flattened  or 
sharpened  ;  but  when  the  double  Sound  was  educed  they  were  no  longer  well  defined,  but  the  tongue  of  the  reed 
seemed  thrown  into  strange  convulsions.  This  recalls  the  experiment  of  Biot  and  Hamel  described  in 
Art  199,202. 

Being  thus  brought  back  to  the  subject  of  reeds  and  forced  vibrations,  we  must  not  omit  to  recommend  to  our 
reader's  attention  the  curious  and  elaborate  dissertation  of  MM.  Weber  and  Floss,  entitled  Leges  oscillationis 
oriundcc  si  duo  corpora  diversa  edentate  osciHanlia  ita  cnnjungantur  ut  oscillare  non  possint  nisi  sirmd  et  syn- 
cfironice  excmplo  illustratas  TUBORUM  LINGUATORUM.  A  detailed  comparison  of  their  results  with  those  of 
Mr.  Willis,  which  the  necessity  of  bringing  this  Essay  to  a  close  forbids  us  to  enter  into,  would  be  very  interesting. 
MM  Weber  and  Floss  agree  with  him  in  the  periodical  recurrence  of  the  note  of  the  reed  at  equal  intervals,  and  in 
its  flattening  up  to  a  certain  point,  &c.  ;  while  in  other  points  there  is  diversity  of  result  enough  to  make  a  careful 
revision  of  the  whole  subject  well  worth  while;  though,  perhaps,  it  is  not  more  than  maybe  accounted  for  by  the 
different  constructions  of  their  reeds;  in  the  one  set  of  experiments  the  oscillations  of  the  tongue  of  the  reed 
having  been  executed  parallel,  in  the  other  at  right  angles  to  the  axis  of  the  cylinder.  It  is  somewhat  curious,  that 
they  seem  to  have  entirely  oveiluoked  the  vowel  qua  ity  of  the  Sounds  educed,  perhaps  from  not  having  employed 
sliding  tubes,  and  thus  missing  the  effect  of  contrast. 

We  had  proposed  to  have  devoted  a  section  to  M.  Savart's  recent  elegant  application  of  his  delicate  methods 
of  detecting  and  exploring  sonorous  vibrations  to  the  determination  of  the  law  of  elasticity  in  different  directions 
with  respect  to  the  axes  of  crystallized  bodies;  but  it  would  lead  us  too  far,  and  we  must  be  content  to  refer  our 
readers  to  the  XLIId  volume  of  the  Annales  de.  Chimie  for  information.  The  field  is  a  wide  one,  and  it  will,  we 
doubt  not,  be  long  before  it  is  fully  explored. 


375 

Remarks 


376. 

Mr.  Wii:_- 
experi- 
ments tin 
mutual  in* 
fluence  of 
reed  and 
pipe. 


Production 
of  a  double 
note. 


377. 

Weber  and 
Floss'sWork 
on  reed- 
pipes. 


378. 
Savart  on 
elasticity  of 
crystallized 
bodies. 


VIM 


*  Let  the  reader  pronounce  slowly,  and  distinctly,  the  words  Paw,  Gnaw,  Naugnty,  Nought,  for  nis  own  satisfaction. 
IV.  5  O 


820  SOUND 

Sound.          Neither  shall  we  devote  a  separate  section  to   the  description   and  explanation   af  acoustic  phenomena  which 

•~-^s—s  occur  in  Nature.     Many  such,  indeed,  have  been  sufficiently  noticed  already.     In  Art.  23  we  have  explained 

37!).       satisfactorily  the  origin  of  thunder,  and  we  shall  here   only  remark  that  the  subterraneous  thunder  which  aecom- 

Sounduf       panics  earthquakes  may  (at  least  in  some  cases)  be  ascribed  to  a  general  cause  not  very  dissimilar,  the  successive 

earthquakes.  arrjvaj  at  tne  ear  of  undulations  propagated  at  the  same  instant  from  nearer  and  remoter  points,  or  if  from  the 

same  points,  arriving  by  different  routes,  through  strata  of  different  elasticities. 

380.  The  concise  and  unblunted  propagation  of  Sound  through  water,  remarked  by  Messrs.  Colladon  and  Sturm,  is 

curiously  exemplified  by  the  shock  of  an  earthquake  felt  and  heard  at  ?ea.  The  sensation  is  always  described  as 
that  of  striking  on  a  rock;  the  Sound  as  that  of  grating  on  a  gravelly  bottom  ;  none  of  the  hard,  rough  Sounds 
of  the  first  impulse  being  at  all  softened  or  rounded  by  the  distance. 

There  is,  however,  one  natural  phenomenon  so  very  surprising,  and  to  us,  we  confess,  so  utterly  inexplicable, 
though  resting  on  the  authority  of  ear-witnesses  of  such  credit  that  it  is  impossible  to  disbelieve  the  facts,  that  we 
cannot  forbear  inserting  a  short  description  of  it,  with  which  we  shall  conclude. 

Description       There  is  a  place  about  three  leagues   to  the  North   of  Tor,  in  the  neighbourhood  of  Mount  Sinai  in  Arabia 

of  the  place  Petraea,  called  El  Nakoiis,  (Nakous  is  the  name  of  a  sonorous  metal  plate  used  in  the  Greek  convents  in  the  East 

called  Na-    instead  of  a  bell)  from  musical  Sounds  of  a  very  singular  and  surprising  character  heard   there.     It  has  been 

visited  by  very  few  Europeans,  two  of  whom,  however,  Mr.  Seetzen  and  Mr.  Gray  of  Oxford,  have  published  accounts 

of   it,   the  former  in   the  Monatliche  Correspondenz,   (Oct.    1812;)    the   latter  in   Dr.    Brewster's  Edinburgh 

Philosophical  Journal,  where  also  Mr.  Seetzen's  account  of  it  will  be  found  translated,  which  is  as  follows : — 

"  After  a  quarter  of  an  hour's  walking,  (from  Wody  El  Nachel?)  we  reached  the  foot  of  a  majestic  rock  of  hard 
sandstone.  The  mountain  was  quite  bare,  and  composed  entirely  of  it.  I  found  inscribed  on  it  several  Greek 
and  Arabic  names,  and  also  some  Koptic  characters,  which  showed  that  the  place  had  been  visited  for  centuries. 
At  noon  we  reached  the  part  of  the  mountain  called  Nakous.  There,  at  the  foot  of  the  ridge,  we  beheld  an 
isolated  peaked  rock.  Upon  two  sides  this  mountain  presented  two  surfaces,  so  inclined,  that  the  white  and 
slightly  adhering  sand  which  covers  it  scarcely  supports  itself,  and  slides  down  with  the  smallest  motion,  or  when 
the  burning  rays  of  the  sun  complete  the  destruction  of  its  feeble  cohesion.  These  two  sandy  declivities  are  about 
150  feet  high.  They  unite  behind  the  insulated  rock,  and  forming  an  acute  angle,  they  are  covered  like  the 
adjacent  surfaces  with  steep  rocks,  which  are  mostly  composed  of  a  white  and  friable  free-stone. 

"  The  first  Sound  was  heard  an  hour  and  a  quarter  after  noon.  We  climbed  with  great  difficulty  as  far  as  the 
sandy  declivity,  a  height  of  70  or  80  feet,  and  stopped  under  the  rocks  where  the  pilgrims  are  in  the  habit  of 
placing  themselves  to  listen.  In  climbing,  I  heard  the  Sound  from  beneath  my  knees,  and  this  made  me  think 
that  the  sliding  of  the  sand  was  the  cause,  not  the  effect,  of  the  sonorous  motion.  At  three  o'clock  the  Sound 
was  heard  louder,  and  it  lasted  six  minutes,  when,  having  ceased  for  ten  minutes,  it  began  again.  It  appeared 
to  me  to  have  the  greatest  analogy  to  the  humming-top ;  it  rose  and  fell  like  the  Sound  of  the  jEolian  harp.  To 
ascertain  the  truth  of  my  discovery,  I  climbed  with  the  utmost  difficulty  to  the  highest  rocks,  and  I  slid  down  as 
fast  as  I  could,  and  endeavoured,  with  the  help  of  my  hands  and  feet,  to  set  the  sand  in  motion.  This  produced 
an  effect  so  great,  and  the  sund  in  rolling  under  me  made  so  loud  a  noise,  that  the  earth  seemed  to  tremble,  and 
I  certainly  should  have  been  afraid,  had  1  been  ignorant  of  the  cause. 

"  But  how  can  the  motion  of  the  sand  produce  so  striking  an  effect,  and  which  is,  I  believe,  produced  nowhere 
else?  Does  the  rolling  layer  of  sand  act  like  a  fiddle-bow,  which,  on  being  rubbed  upon  a  plate  of  glass,  raises 
and  distributes  into  determinate  figures  the  dust  with  which  the  plate  is  covered  ?  Does  the  adherent  and  fixed 
layer  of  sand  perform  the  part  of  the  plate  of  glass,  and  the  neighbouring  rocks  that  of  the  sounding  body  ? 
Philosophers  must  decide  this." 

We  give  here  M.  Seetzen's  account  in  preference  to  Mr.  Gray's  as  being  the  earliest,  and  in  his  own  words, 
preserving  even  his  own  conjectures  (not  the  most  plausible)  on  its  cause,  and  we  shall  be  glad  if  the  visits 
of  future  travellers  to  the  spot  shall  throw  further  light  on  this  very  strange  phenomenon. 

Slouch,  ftb.  3,  1830.  J.  F.  W.  HERSCHEL. 


SOUND 


821 


INDEX. 


The  Numbers  refer  to  the  Articles  according  to  the  Marginal  Number*  of  the  Text. 


Sound.  Am.  The  medium  of  conveying  sounds,  2.  RurtftdoTcompreued, 
SJ_  ,.^ ,  diminution  or  increase  of  sound  in,  2.  Gives  out  he;it  by  cou- 
den*aticn,  its  effect  on  velocity  of  sound,  69,  70.  In  pipes,  vibra- 
tions of.  See  Pipe.  l'it>rtiti"ii*.  communicates  vibrations  between 
solids,  and  how,  309,  310.  Masses  of  their  vibrations  explnn-d  by 
membranes,  318.  In  an  apartment  its  vibrations  and  nodal  lines, 
334. 

Alcohol,  velocity  of  sound  in  its  vapour,  82.  Propagation  of  sound 
in.  88. 

Alphabet,  English,  its  imperfections,  359.     Universal,  367. 

Analogy  between  sound  and  light,  137.     See  Light. 

ANDERON,  on  hearing  of  fishes,  90. 

ARAOO,  his  determination  of  the  velocity  of  sound,  13.  His  remark 
on  corresponding  observations  used  for  that  purpose,  10. 

Arbitrary  funclinna  in  the  integrals  of  the  equations  of  sound,  57. 
Of  vibrating  strings,  151. 

Atmosphere  sounding,  its  probable  extent,  3. 

Bat,  its  cry,  222. 

BAVJZA,  his  determination  of  the  velocity  of  sound,  12,  If). 

Beats,   236.      Of    imperfect    concords,    237.      Rendered   ocularly 

visible,  318. 

BENZENBEHG,  on  the  velocity  of  sound,  12.  16. 
BERNOUILLI,  his  theorems  respecting  chimney-pipes,  207. 
BETDANT,  his  experiments  on  the  propagation  of  sound  in  sea-water 

93. 

BIANCONI.  on  the  velocity  of  sound,  12.  16. 
BIOT,  liis  experiments  on  propagation   of  sound   through  lung  pipes, 

23,  24.     In  steam,  88.     In  cast  iron,  113. 
Boards,  sounding  rationale  of  their  effects,  288. 
BOUVARD,  determination  of  the  velocity  of  sound  by  him  and  others, 

13,  16. 

Zfor-shaped  cavities.     See  Resonance. 
BOYI.E,  his  determination  of  the  velocity  of  sound,  6. 
Bullet^  singing  of  explained,  145. 

CAGNARD  DE  LA  TOUR,  his  sirene,  143. 

Cannon  used  for  signals,  9, 13.    Cases  of  double  sound  from  a  single 

shot,  38. 

CANTON,  his  experiments  on  the  compressibility  of  water,  89. 
Carisbruok  well.     Remarkable  effect  of  the  propagation  of  sound  in 

it,  25. 

Carpets,  their  effect  in  deadening  sound  in  a  room,  lU9. 
CASSINI  and  others'  determination  of  the  velocity  of  sound,  9. 
Cavities.     See  Resonance,  Cube,  Box 
CIILADNI,    his   acoustic    researches,   passim.     His   experiments   on 

intensity  of  sound  in  gases,  82.  84.     On  propagation  of  sound  in  an 

effervescing  liquid,  107.     His  euphone,  283.     His  researches  on 

acoustic  figures  and  vibrating  surfaces,  272,  &c. 
Chimney-pipes,  207. 
Chords,   the   chief  musical.  225.  et  seq.  256,  257.      Expressed    in 

chromatic    numbers,  256.      Inversions    of  and   tables,  259,  260. 

Equivocal,  262. 

Chromatic  scale,  234.     Numbers,  256. 
CLEMENT  and  DESORMES,  their  experiments  on  heat   given  out  by 

condensed  air,  76. 
Close,  false,  228. 

Cfouds,  reverberation  of  sound  by,  38. 
Coexistence  of  different  modes  of  vibration  in  strings.  164.     In  pipes, 

183.     See  Superposition. 
COLLADON  and  STURM,  their  experiments  on  the  propagation  of  sound 

in  water,  94. 
Comma,  233. 
Communication  of  vibrations,  general  law  of.  302.     Through  liquids, 

340.     Read  also  §  II.  Part  III. 
Compressibility  of  water  and  other  bodies.  103. 
Concord*,  what.  209.     Principal,  enumerated,  210 — 215. 
Concourse  of  sounds,  sounds  resulting  from,  238 — 240. 
CoNHAMtNE,  L.\,  his  experiments  on  the  velocity  of  sound,  12,  If). 
Condensation  of  air,  heat  given  out  by  increases  velocity  of  sound,  48, 


and  rarefaction,  alternate  of  air  in  a  pipe,  181.    None  alan  orifice, 

Ciini/ition*  for  determining  arbitrary  functions  in  the  equations  of 
sound,  38.  Of  the  undivided  propagation  uf  a  pulse,  129  Of  conti- 
nuity of  two  media,  132.  Of  equal  elasticity  at  their  junction, 

1JJ.     "I  the  single  propagation  of  a  wave  along  a  cord    151 
Conductors  of  sound,  112,  113. 

Consonants,  enumeration  of,  362.     Their  classification,  363 365 

Continuity  of  a  sound  produced  by  successive  impulses  140 
Cord.     See  String. 

Corresponding  observations  for  determining  the  velocity  of  sound  9 
Crystallized  media,  propagation  of  a  pulse  through,  1 1 1.     Sava'rt  on 

their  elasticities,  378. 

Cubes  of  air,  and  cubical  cavities,  their  vibrations,  339. 
Curve  of  a  vibrating  string,  159.     Its  prolongation  by  repetition   159 

Arising  from  superposed  vibrations,  165. 
Cylind-r,  vibrations  of,  296.     Higher  modes  of,  301.     Nodal  lines 

in,  297.     In  interior  of  a  hollow  one,  300. 

Decay  of  tound,  §  VI.  p.  773.  Art.  1 16,  et  seq.     Its  law,  121.     Of 

vibratory  motion  in  geom.  progression,  327. 
DSRHAM,  Dr.,  his  determination  of  the   velocity   of  sound,  7.     His 

researches  on  sound,  id. 
DESORMES.  See  Clement. 
Diapason  organ-pipe,  198.  A  musical  instrument,  see  Tuning. 

fork. 

Diatonic  scale,  221. 
Diesis,  enharmonic,  247. 

Diminution  of  sound  in  rarefied  air  or  on  high  mountains,  3. 
Diphthongs,  English,  enumeration  of,  361. 
Direction  of  sound,  how  judged  of,  344.     Curious  case  of  inis-iudir- 

ment  of,  334. 
Disc,  circular,  its  vibrations  and  nodes.  277.    Excited  to  vibration  by 

a  coru,  o,*7. 
Discords,  musical,  what,  209.  216.      Resolution    of,   217.     Names 

and  notes  of,  260,  261. 
Distance  at  which  sounds  have  been  heard,  22.     Decay  of  tound  by 

See  Decay. 
Divergence  of  sound  not  alike  in  all  directions,  117.     From  the  end 

of  a  pipe,  116. 

Divers,  their  hearing  under  water,  90.  93. 
Dominant,  and  sub  dominant,  chords  of,  227,   228.     Seventh   217 

230. 
Double  sound  from  a  sinole  source,  38.     Explained,  109.     Another 

case,  113.     Produced  by  one  pipe,  205.     By  a  reed-pipe,  376. 

Ear.  its  structure,  320. 

Earthquakes,  sounds  heard  in,  379. 

Echo,  how  produced,  27.  liemarkable  ones  instanced,  29,  et  s*y. 
Situations  favourable  to,  36.  In  churches  and  public  buildings, 
37.  Laws  of  reflection  of  sound  in,  35.  95.  See  Reflexion. 
Partial,  at  fissures  of  a  solid,  110.  From  palisades,  144.  In  a 
chamber,  146. 

Effervescing  liquids,  propagation  of  sound  obstructed  by,  107. 

Eleventh,  chord  of,  264.     Its  inversions,  260. 

Embouchure  of  a  pipe,  184,  185.  193.  Rationale  of  its  action.  194. 
Influence  of  on  pitch,  197.  Case  of  a  pipe  commanded  by,  204. 
See  Pipe. 

Enharmonic  scale  and  diesis,  247. 

Equation  of  sound,  55,  56.  Of  vibrating  chords,  150.  Of  the  pro- 
pagation of  sound  along  pipes  filled  with  different  media,  131. 

Equivocal  chords,  260,  262. 

ESPINOSA,  bis  determination  of  the  velocity  of  sound,  12.  16.  20. 

EULENSTEIN,  his  performance  on  the  Jew's  harp,  202.     Note. 

Euphone,  a  musical  instrument  of  Chladni,  283. 

False  i-hse,  228 

Kftk.     See  Interval.     Chord  of  triple,  260, 26i 

Fishes,  their  hearing,  90. 


Index. 


822 


SOUND. 


Scund.      FLAMSTKBD,  his  determinations  of  the  velocity  of  sound,  6. 
—     .— _  -  Flails  and  sharps  in  roiisic,  232. 

Florentine  Academy,  their  determination  oftlie  velocity  of  sound,  6. 

FLOSS  and  WEBIR  on  reed-pipes,  377. 

FLOURENS,  his  experiments  on  nerves  of  the  ear,  320. 

Flute.     See  Pipe. 

Fog,  its  effect  in  obstructing  sound,  21. 

Forced  vibrations,  323.     Depend  on  imperfect  elasticity,  330. 

Fork,  tuning.     See  Tuning-fork. 

Formula  for  velocity  of  sound,  6!>.  For  effect  of  temperature  and 
pressure,  68.  For  effect  of  developed  heat,  73.  Laplace's,  for  the 
velocity,  79.  For  time  of  vibration  of  a  string  harmonically  divided, 
162  For  velocity  of  a  wave  in  a  string,  157. 

FRANKLIN,  his  experiment  on  hearing  under  water,  92. 

Functions,  arbitrary,  in  vibrations  of  a  string,  151.     In  sound,  57. 

Gam,  propagation  of  sound  in,  80.  82.     In  mixed,  108. 

GAY  LUSSAC'S  determination  of  their  expansion,  113. 

Geneva,  Lake  of,  experiments  on  the  propagation  of  sound  in,  95. 

Girijente,  remarkable  echo  in  the  cathedral  of,  31. 

data,  velocity  of  sound  in,  82.     Compressibility  of,  103. 

Glasses  broken  by  sounds,  3. 171. 

Glottis,  350.  Assimilated  to  a  reed,  352.  This  analogy  disputed  by 
Savart,  356. 

Gloucester  cathedral,  echo  in,  29. 

GOLDINGHAM,  his  velocity  of  sound,  12.  16.  His  computations  de- 
fended, 20. 

Grasshopper,  pitch  of  its  cry,  224. 

GREGORY,  Dr.,  his  experiments  on  velocity  of  sound,  12.  16.  20. 

GRENIE,  his  reed,  202. 

HALLEY,  his  determination  of  velocity  of  sound,  6.  His  account  of 
the  report  of  a  meteor  in  1719,  3. 

Harmonics  of  a  string,  163. 166. 

Harmonica  Glasses,  303.  333.     German,  203. 

Harmony,  §  IV.  Part  I.  Imperfect,  how  introduced  into  music,  232. 
Perfect,  its  origin,  209.  Remarks  on  the  origin  of  harmony,  271. 

Harp,  Jew's,  202. 

HASSKNFRATZ,  his  experiment'on  the  conveyance  of  sound  by  stone, 
113. 

HALKSBEE'S  experiment  on  sound  in  exhausted  receiver,  3. 

Hearing  of  fishes,  90.  Of  sounds  under  water,  90,  91,  92.  Bynight 
more  delicate  than  by  day,  and  why,  107.  Organs  of  See  F,ar. 

Heat  given  out  by  condensed  air,  69.  By  water,  100.  Its  influence 
on  the  velocity  of  sound,  70,  el  seq.  Its  amount,  how  best  deter- 
mined, 75,  76.  Developed  by  hydrogen  in  condensation,  83. 

HRRHOLoand  KAPN,  their  experiments  on  propagation  of  sound 
through  wires.  1 13. 

Hexachoras,  257. 

HUMCOLDT,  his  explanation  of  the  audibility  of  sounds  by  night,  107. 

HUYOENS,  his  system  of  21  notes  in  the  octave,  249. 

Hydrogen,  peculiarities  in  its  transmission  of  sound,  83.  Its  effect 
in  destroying  sound  when  mixed  with  air,  85.  Explanation,  108. 
Its  singular  effects  on  the  voice  when  breathed,  86. 

Jcf,  sounds  well  conveyed  over,  21. 

Impulses  periodic.     See  Periodic  Impulses. 

Influence,  mutual,  of  vibrating  bodies,  284.  Of  two  clocks,  organ- 
pipes  or  strings,  284.  Of  a  reed  and  pipe,  376,  377. 

Insects,  their  hearing,  224. 

Intensity  of  sound  on  high  mounlains,  2.  Law  of  its  decay  by  dis- 
tance, 126.  In  gases,  84.  In  hjdrogen,  83.  Is  as  the  vis  viva 
not  as  the  inertia,  126. 

Interference  of  sonorous  vibrations,  85. 

Interruptions  in  media,  their  efTrct  in  obstructing  sound,  105. 

Intervals,  musical,  enumerated,  210 — 2'JO.  On  what  depend,  219. 
Represented  by  logarithms,  '245.  Their  values,  ihiil.  Their 
sequence  in  the  diatonic  scale,  246.  Tahle  of  their  values  in 
various  systems  of  tempered  scales,  254. 

Inversions  of  chords,  2W,  260. 

Iso-tiarmonic  scale,  241. 

Inn,  cast,  velocity  of  sound  in,  1 14.     Biot's  experiments  on,  113. 

Jew's  harp,  202. 

KEMPELEN,  his  imitation  of  the  voice  and  the  vowel  sounds,  36 

Key-note,  22ft. 

KKATZENSTEIN,  his  free   reed.  202.     His   researches  on   articulats 

sounds,  357.     On  the  vowels,  3.i8. 
KUYTF.MIB  jcjwEit  and  others,  their  determination  of  the  velocity  of 

(ound,  13.  16. 


labyrinth  of  the  ear,  320. 

I. AC-AIM. E  and  others  on  the  velocity  of  sound,  6. 

language,  written,  its  imperfections  as  at  present  existing,  359. 

LAPLACE,  his  explanation  of  the  excess  of  velocity  of  sound  over 
New.on's  formula,  69.  His  own  formula,  79. 

larynx,  348. 

LESLIE  on  the  propagation  of  sound  in  hydrogen,  85.  108. 

light,  its  analogy  with  sound,  35.  38.  102.  105—110.  137.  145. 

Limits  of  audibility,  223,  224.  320. 

Liquids,  propagation  of  sound  through,  §  IV.  89,  el  s*q.  Velocity  of 
sound  in  various,  103.  Communication  of  vibrations  through,  340, 
et  seq. 

Logarithms  used  to  measure  musical  intervals,  245. 

LOOIEK,  MR.,  his  system  of  harmony,  253. 

longitudinal  vibrations  of  a  rod,  268.  Ufa  glass  stick  used  to  com- 
municate vibrations  to  solids.  283,  et  tea.  Its  state  examined. 
286.  289. 

Lycoperdon,  powder  of,  its  use  in  acoustic  experiments,  291. 

MALIIS,  an  experiment  of,  on  the  conduction  of  sound,  113. 

MAHALDI,  his  experiments  on  the  velocity  of  sound,  9. 

MATHIEU,  with  Bouvard,  Araxo,  &c.,  determines  velocity  of  sound,  13. 

Membrana  tympuni  of  the  ear,  its  use,  319,  320. 

Membranes,  theii  vibrations,  309,  et  seq.     Used  to  explore  vibrations 

of  the  air,  318.  333.     Imperfectly  elastic,  their  vibrations,  321 . 
Menai  bridge,  echo  under,  35. 

MERSENNE,  his  determinations  of  the  velocity  of  sound,  6. 
ftleteors,  sounds  produced  by  their  explosion,  3. 
Mixed  media,  propagation  of  sound  in,  85.  104,  et  seq. 
Modes  of  vibration  of  a  cord,  163.     Of  an  aerial  column,  183. 
Modulation,  231. 
MOLL,  Vaubeek  and  Kuytenbrouwer's  determination  of  the  velocity  of 

sound.  13.     On  the  velocity  of  sound  in  gases,  82. 
Mnivirhnril,  167. 
Mont  H/anc,         "1 

Monte  HUSH,          Meebleness  of  sound  observed  on,  3. 
Mountains,  high,J 

Musical  instruments,  new,  proposed,  338. 
Musical  intervals.      See  Internal. 
Musical  lountli  under  water,  101.      In  general,   Part  II.  §  I.  el  seq 

See  the  several  heads. 

Nnkovs,  extraordinary  acoustic  phenomena  of,  380. 

Nerves  of  the  ear,  tl.eir  offices,  320. 

NEWTON,  his  theory  of  sound  examined,  61. 

Ninth,  chord  of,  256.  265.     Its  inversions  260. 

Nodal  points  of  a  vibrating  string,  1GO.     uf  an  aerial  column,  182. 

How  established,  189. 
Nodal  lines,   their  origin,  290.     Of  tangentially   vibrating  surfaces, 

their  peculiar  character,    292,  293,     Spiral  form   of  in   vibrating 

cylinders,  298.  In  hollow  cylinders,  300.  Of  the  air  in  a  chamber, 

334,  335. 

Nodal  figures  on  vibrating  surfaces,  274,  st  srq.     Secondary,  316. 
Nodal  surfaces  in  a  vibrating  solid  mass,  289. 
Noise,  how  different  from  musical  sound,  139. 
NOLLET,  ABBE,  his  experiment  on  'hearing  under  water,  91. 
Numbers,  chromatic,  256.     Diatonic,  ibid. 

OOIER,  his  experiments  on  the  effect  of  hydrogen  gas  on  the  voice,  86. 
Organ-pipes.     See  Pipe,  Diapason,  Vox  humana/  Kmbouchure,  &c. 

Palisades,  echos  from,  produce  a  musical  soun<J,  144. 

Pentachords,  257.  2GO.     How  related  to  tetrachords,  263. 

PEROLLE,  his  experiments  on  intensity  of  sound  in  gases,  84. 

Periodic  impulses.  How  executed.  142.  Produce  musical  sounds, 
141. 

Phenomena,  natural,  explained. Thunder,  23.  Volcanic 

sounds,  24.  Sounds  in  Carisbrook  well,  16.  Stifling  of  sound 
by  hydrogen  gas,  108.  Resonant  sounds  at  Solfaterra,  110. 
Why  very  acute  sound.-'  are  at  length  inaudible,  223.  Increased 
audibility  of  sounds  by  nighl,  107 

PICARD,  his  determination  of  the  velocity  of  sound.  6. 

Pipes,  their  effect  in  convejing  sounds  to  great  distances,  24.  I'yra- 
midal,  propagation  of  sound  in,  124.  Open,  vibrations  of  air  in, 
184.  186.  187.  193.  Open  at  both  ends,  190.  Closed  at  both 
ends,  192.  Chimney,  207.  Harmonics  of,  193.  Proofs  that  it 
is  the  air  enclosed  which  sounds,  196.  Influence  of  their  forms 
on  the  quality  of  their  sounds,  201.  Case  of  pipes  commanded 
by  their  embouchure,  214.  Double  sound  yielded  by  one,  205. 
Pax  huaiana,  20fi.  Vibrations  of  air  in  explored,  337.  K"id,. 
ItiU  '470  ei  seq.  Lengths  of,  which  yield  the  vowels,  374. 


Indei. 


SOUND. 


823 


Sound. 


Pucn  of  a  musical  sound.  141. 

Plates,  rectangular,  tlieir  vibrations,  272,  273.     Circular,  triangular, 

and  elliptical,   277,  ft  seq.     Set  in  vibration  by  communication 

witn  a  string,  281.     Joint  vibration  as  a  system,  282.     Tangential 

vibrations  ol.  2(J5. 

POISSOK,  his  computation  of  the  velocity  of  sound,  78. 
Post,  sounding  of  a  fiddle,  281. 
PRONY,    and  others,  determination  of  th«    velocity  of   sound    by, 

1.1.  16. 

PRIESTLEY,  his  experiments  on  intensity  of  sound  in  gases,  84. 
Propagation  of  sound,  generally,  mathematical   theory  of,  41.     In 

air  of  one   dimension,  50.      Of  a  single   initial   disturbance,  64. 

Linear  in  gases  and  vapours,  80,  ct  seq.     In  mixed  media,  85.  104, 

et  seq.     In  solids,  104,  et  seq.     Not  alike  in  all  directions,    117, 

118.     Of  waves  along  a  stretched  string,  151. 
Pulses,   aerial,   49.       Propagation    of    in    crystallized    media,    111. 

Condition  of  their  undivided  propagation,  128.     Division   of  by 

an    obstacle,  or  change  in  the  propagating  medium  or  mode  of 

propagation,  129. 

Quality  of  lone  of  stringed  instruments  whence  arising,  174. 

RAFN.     See  Herhold. 

Rarefied  air,  feebleness  of  sound  in,  2. 

Receiver,  exhausted,  sounds  quite  inaudible  in,  2. 

Reeds,  187.  199.     Forced  vibration  of,  how   peiformed,  199.     Free, 

of  Kratzenstein  and  Greni6,  202. 
Reed-pipes,  199.     Mutual  influence  of  reed  and  pipe,  376.     Willis's 

researches  of  sounds  of,  370  376.     Weber  and  Floss  on,  377. 
Reflexion  of  sound,  127.     Analogous  to  thai  of  light,  35.     At  inter- 
nal surface  of  water,  95.     At  junction  of  two   media,  105.  127. 

Internal  total,  137.     Ol  a  wave  along  a  cord   by  an  obstacle,  159. 

See  Echo. 

Refraction  of  sound  oblique,  137. 

Resonance  of  cavities,  186.  338.     Of  box-shaped  cavities,  339. 
Resultant  sounds,  238,  et  seq. 
RORISON,  his  invention  of  an   instrument  analogous  to    the  sirene, 

143. 

Roots,  conduction  of  sound  through,  1 13. 
Roils,   longitudinal   vibrations  of,  292.  296.     Spiral   form  tf  nodal 

lines  in,  297.     Communication  ol  their  vibrations  to  glasses,  &c. 
303,     From  one  to  another,  306. 

RoEitvcK,  his  observations  on  sound  in  compressed  air,  3. 
ROEMF.R,  determination  of  the  velocity  of  sound  by,  6. 
Rotatory  vibrations  of  a  cord,  176. 

Saint  Allan's  church,  remarkable  echo  in.  29. 

Saint  Paul's  church,  whispering  gallery  in,  32. 

Sand  used  to  render  nodal  lines  viable,  272.  To  distinguish  nor- 
mal from  tangential  vibrations,  291. 

SAUSSURE,  his  observations  on  sounds  on  Mont  Blanc,  3. 

SAVART,  his  acoustic  discoveries,  passim.     See  the  several  heads. 

Scale,  diitonic.  220.  Chromatic,  234.  Isoharmonic.  241.  Of  mean 
temperament.  242.  Table  of  notes  in  various  ones,  25-1.  Major 
and  minor,  256.  Of  21  sounds  in  the  octave,  249,  250. 

Secondary  nodal  lines,  316. 

Seventh,  216,  217.  230.  Its  inversions,  260.  Examined,  265. 
Flat,  its  origin  according  to  Logier,  25'i. 

Sharps  and  flats,  232. 

Shipley  church,  remarkable  echo  at,  30 

Silence,  at  great  elevation.  2.      B/  night,  effect  of,  107. 

filmonetta  palace,  remarkable  echo  in,  34. 

Sirene,  143. 

Sink.     See  Intervals.  Concords. 

Smith,  Dr.,  his  system  of  equal  harmony,  260.  His  scale  of  21 
sounds  in  the  octave.  250. 

Snow,  its  effect  in  stifling  sound,  21. 

Solfuterra,  the  hollow  sounds  heard  in  it  explained,  110. 

Solids,  propagation  of  sound  in,  104.  Effect  of  interrupted  struc- 
ture in,  105.  Ways  in  which  they  may  vibrate,  287.  In  what 
different  from  fluids  in  their  relations  to  sound,  111. 

Sountl,  (see  the  general  heads,)  causes  which  obstruct  its  propa- 
gation, HI.  85.  105.  108,  109.  Analogy  of  with  light,  (see  LIGHT  ) 
345. 

Sounds,  inaudible,  to  certain  ears,  222.  Very  acute,  their  feeble- 
ness, 223.  Resultant,  238,  239.  At  a  place  called  Nakous.  380. 
Greater  audibility  by  night  than  day,  107.  At  Solfaterra,  110. 

Sparrow,  pitch  of  its  chirp,  222. 

Speech,  organs  of,  357,  et  seq.   Imitations  of  by  mechanism,  357.  3G8 

Spiral  form  of  nodal  lines  in  vibrating  cylinders,  297. 

Springs,  vibrations  of,  269. 

Stone  conducts  sound  well,  Exp.  113. 


Strings  tended,    their  vibrations  investigated,   149.     Propagation  of 

waves  along,  152.     How  it  passes  into  a  periodic  vibration.    155. 

Communicate  their  vibrations  to  solids,  281.  307.     Used  to  excite 

regular  vibrations  in  other  bodies,  307. 
STURM.     See  Colladon,  94. 
Subdivision  of  a  vibrating  string  into  harmonics,  160,  et  seq.     Of  a 

pipe.   182.     Of  a   vibrating  surface,  274,  etseq.     Oi  a  cylinoric 

solid,  301. 
Sun,  time   required   to  transmit   force  from  to  earth  by  iron   bars, 

115. 
Superposition  of  vibrations,   164.     Of  modes  of  subdivision,  316. 

Of  sounds  from  a  pipe,  205,  376. 
Sympathy  of  vibrations  of  two  cords,  169,  170. 
System   of  mean   tones,   252.     Of  harmony,  (Logier's,)  253.     Of 

bodies  vibrating  in  communication,  282,  and  Part  III. 

Tables  of  different  assigned  velocities  of  sound  in  air,  16.  In  ga<e» 
and  vapours,  82.  Of  compressibility,  103.  Of  logarithms  of  the 
musical  intervals,  245.  Of  vowel  sounds,  360.  Of  lengths  of 
vowel-pipes,  374. 

Tangential  vibrations,  291.  Of  rods  and  rulers,  292.  Peculiar 
character  of  nodal  lines  produced  by  them,  292.  Of  broad  rectan- 
gular plates,  295. 

Temperament,  §  IV.  Origin  of,  235.  System  of  equal,  242.  Its 
delects,  243.  Occasional,  244.  Adapted  for  piano-forte,  251 
Dr.  Young's  system,  251.  Vulgar,  252. 

Tension  of  a  membrane,  its  effect  on  the  nodal  lines,  315. 

Third,  major,  214.  Minor,  255.  See  Intervals.  Double,  chord  of, 
260,  261. 

Thunder,  its  phenomena  explained,  38,  39. 

Timber,  good  conductor  of  sound,  112. 

Time  of  vibration  of  a  stretched  cord,  157. 

Transmission  of  force  through  bars,  115.  Of  sound,  see  Propaga- 
tion. 

Transverse  vibrations  of  a  rod,  269. 

Triads,  257.     Table  of  their  inversions,  260. 

Tuning-fork,  unequally  audible  in  different  directions,  117.  Ex- 
plained, 332.  Used  to  excite  vibrations  in  a  pipe,  186.  Disked. 
186.  205. 

Tuning  a  pianoforte,  practical  easy  rules  for,  251. 

Uvuia,  351.  Its  tension  and  relaxation,  355. 
fr»(<  i.i  ions.  See  Pulses,  Vibrations,  Waves. 
Unisons,  208. 

VANBEEK  and  olhers,  determination  of  velocity  of  sound  by,  13.  16. 

Vapour,  propagation  of  sound  in,  80,  82.  87.  At  their  maximum 
pressure,  88.  Aqueous  in  air,  iis  ett'tcl  on  sound,  109. 

Velocity  of  sound,  4,  5,  et  seq.  Influence  of  wind  on,  5.  Various 
determinations  of,  6,  7.  9.  16.  Standard  velocity  at  freezing  tem- 
perature, and  mean  pressure  =  1089  42  leet.  Compared  with 
the  earth's  diurnal  motion,  19.  Same  for  all  sounds,  23.  Uni- 
form, 65.  Its  analytical  expressions,  65.  68.  Eflect  of  heat  deve- 
loped, 69.  Corrections  for  temperature,  &c  68.  Numerical 
value  computed  from  theory,  78.  In  sea-water,  93.  In  fresh 
95  In  other  liquids,  103.  In  cast  iron  and  solids,  114.  In 
airs,  or  gases,  195 

f-'r/ocity  of  waves  running  along  a  tended  cord,  152. 

Central  legments  of  a  string,  162.     Of  a  pipe,  18^ — 189. 


reflexion  of  a  pulse,  179.  Of  bars,  rods,  and  plates,  §  V.  Part  II. 
Various  modes  of  in  solids,  267.  Longitudinal,  of  a  rod,  268. 
Transverse,  269.  Of  plales,  see  Plates.  Communication  of, 
Part  III.  Methods  of  communicating  in  any  required  plane,  280. 
Of  a  fiddle-string  how  communicated  to  its  support,  281.  Of  the 
sounding-board  bow  observed.  282.  Tangential,  291.  How  dis- 
tinguished, 291.  Of  long  rectangular  rods,  292.  Normal,  291. 
Oblique,  and  their  passage  iu'o  normal  and  tangential,  308.  Of  a 
system  of  rods,  discs,  &c.  305,  306.  Of  membranes,  307,  et  seq. 
Of  masses  of  air,  318.  333.  Of  an  inelastic  membrane  how  pro- 
duced, 321.  Forced,  general  theorem  respecting,  323.  Of  air 
in  a  chamber,  or  gallery,  334.  In  pipes,  explored  by  a  membrane, 
337.  Their  communication  through  liquids,  340,  et  seq.  Of  »ir 
in  cavities  with  flexible  sides,  354. 

iolin,  Savart's  improvements  on  its  construction,  288.  Communi- 
cation of  vibration  of  strings  to  the  body,  281.  Sound-post  ot 
281. 


824 


SOUND. 


Sound 


jicf,  human,   distances   at   which  audible,  21.    Organs  of,  346. 

Mode  of  its  production,  352 — 357. 
Volcanoes,  noises  heard  in  their  eruptions  explained,  40. 
f-'owel',  enumeration  of,  360.     Willis,  Kratzenstein,  and  Kempelen'.s 

researches  on,  369,  ttseq. 
{'oxhuinami  organ-pipe,  206. 

Walts,  conveyance  of  sounds  along,  26. 

WALKER,  determination  of  velocity  of  sound  by,  6. 

H'altr,  sounds  well  conveyed  over,  21.  26.  Compressibility  of,  89. 
Propagation  of  sound  through,  §  IV.  passim.  Velocity  of  sound  in, 
96.  Hearing  unuer,  90.  Curious  phenomena  obsened  in  trans- 
mission of  sound  by,  101.  Heat  developed  by  its  compression, 
100.  Non-divergenie  of  sound  from  straight  line  in,  102.  Sirsne 
sounding  in,  143. 

Wiirc»,  in  still  water  how  propagated,  42.  Wha>,  44.  In  a  field  of 
corn,  45.  Their  velocity  distinguished  (mm  that  of  their  mole- 


cules,  46.     Their  breadth,  47.     Various  species  of,  48.     Sonorous 

how  propagated  in,  49.     Along  a  cord,  149,  &c. 
WEHEK,  his    researches  on   waves,  and  on  reed-pipes,  171.377,  el 

passim.     His  experiment  on  a  tuning-fork,  117. 
HW/,  Caiisbrook,  echo  in,  25. 
Whispering  Gallery.     See  St.  Paul's. 
Wn.i.ic,  Mr.,   his  researches  on  the  vowel  sounds,  ,369,  ft  ley.     On 

sounds  of  reed-pipes,  370. 

IVind,  its  influence  on  the  propagation  of  sound,  9 —  11. 
Wire,  a  good  conductor  of  soun<l,  113. 
WOI.I.ASTON.  on  the  limits  of  hearing,  222,  320. 
Wuud,  a  good  conductor  of  sound,  1 12. 
Woodstock  Park,  echo  in,  28. 
Written  characters  to  express  sounds,  359. 

Yoi'Nc,  Dr.,  his  experiments  and  discoveries   pattim.     His  experi- 
ment on  the  sound  of  a  tuning-fork,  117.     Its  explanation, 332. 


Inde». 


END  OF  VOLLME  IV. 


ERRATA  IN  THE  ESSAY  ON  SOUND 


Art. 
2, 
2, 

104, 

117, 
151, 
151, 
178, 
186, 
19.% 
209, 
212, 
219, 
223, 
224, 


235, 
236, 
238, 


Line. 

5, 
15, 

10* 

11, 
16, 

4, 

7,  and  margin, 

7, 
10, 

1, 
4, 

8, 
7, 


10, 
17, 
3, 


Correction. 
Hauksbee. 
Hauksbee. 
quick  and  copious, 
positions  of  the  axis. 
<c  _|_  a  t  >  «. 
condition, 
column. 
fig.  17. 
whose, 
terms. 

combination, 
impulses. 

the  number,  extent, 
sense  r 


Error. 
Hanksbee, 
Hanksbee, 
free, 
axes, 

x  -)-  a  t  =:  «, 
conditon, 
volume, 
fig.  16, 
where, 
times, 

combinations, 
impulse, 
the  extent, 

Sound  sense  ? 

Nevertheless  spiders  hear  the  sound  of  music.     Vide  Latreille's  anecdote  of  PelissoD,  who 

tamed  one  in  the  Bastille, 
more,  most, 

vibrations,  vibrations  of  the  quicker, 

figure  figure  31. 


ADDITIONAL  ERRATA  IN  THE  ESSAY  ON  LIGHT. 


Page. 

Line. 

Em 

Correction. 

416, 

19, 

«,ftr 

t,  0,  C. 

434, 

5, 

organ, 

origin. 

434, 

9, 

maxima, 

nature. 

439, 

electric  attractions, 

elective  attractions. 

452, 

18, 

1200, 

1100. 

456, 

18  frombott. 

on, 

or. 

457, 

6, 

two  canals, 

dele.      . 

473, 

33, 

regularly, 

irregularly. 

518, 

35, 

mackled, 

macled. 

518, 

11  frombott. 

(«•+>•+*)> 

(**  +  /  +  ««)«. 

518, 

3  frombott 

holes, 

poles, 

537, 

2  from  bott.  note, 

prelate, 

prolate. 

541, 

12  frombott. 

(B"t>-(-C"H>)  v 

(B"t)  +  C"«>)«. 

544, 

4. 

second  plane, 

secant  plane. 

551, 

34  frombott. 

dele  what  is  said  about  camphor,  as  that  substance  in  & 

solid  state  docs  possess  the  rotatory  property. 

556, 

7  frombott. 

rays, 

rings. 

557, 

6, 

rays, 

rings. 

559, 

17, 

o  .  cos  0  A8  + 

y  .  cos  O  A8  +. 

560, 
561, 

17, 

17, 

s  +  y 

undulatory, 

Y  +  y. 

undulated. 

571, 
571, 

4,  col.  1,1 
7,  col.  1,) 

extra, 

extreme. 

574, 

10, 

»% 

** 

577, 

&, 

hyposulphate  of  lime  and, 

dele. 

1'latp  3 


A- 


1)     h      F. 


-c      /;//  •  //•/.  ii 


//./    li    ./ft   117 


/«/;;   .iii.iii.t 


//././.•:     .In. 17,, 


/;;///  .in. i;7 

B 


r- 


//,/  /('•     In  !;:•_' 


H 


sor 


/•';,;.  17.  .In  .  inii. 


J       <  l-'liit,l,iiii,.llt,il          I,'T.  •///// 

r,.,i,ir,l,;l        ,i.<       ./         *,'•//,-,  T./ 


,      rf,Y/,T>,/, 


Harp 


trt.2S9. 


in  ;',;<>. 


hi  ;•<;:>. 


Kq.Sfl.  _4rt?269.  270. 


.In. 


'm  •//  Art. 'M 


i/.  •/;'.  .4rt:Z70. 


Eq.  43.  Alt.  270. 


I'u/.-ll  .Irl 


I'm  •:.,  ./if  ZU. 


J-iii.  -it;  .ir;  276. 


Jy.  47.  Ait.  277.          Fig. 


SOUND. 


Han-     1 


FLO.   7", 


/•',,i.  7,':  Fig.    79.  Fig.  80. 


Fy.  HL  /w.  82.  Fur.  S3. 


Fig.  ,U.  /•/,/.  35.  Fig.  #6.  Fig 


Fig.  8S. 


Fig.  39.  Fig.  90. 


/•',„   "1 


Fig 


Arti 


Fig.  Hi 


Fy.   94. 


Fig.  95. 


J 


Fig  96. 


7 


Fig.  97. 


Fin.  96 


Fiq.  99. 


Fig.  100        Fy.  1O1.         Fy.  1O2.        Fig.  103.        Fig.  104. 


Fig  110. 


Fig.  105.  Fig.  106.  Fy  107.  Fig.  1O8.  Fig.  1O9 


Fig.  HI 


l-'i.i  II' 


I'm    II; 


Fiq.  117. 


In 


Fuj.  123. 


, . ~ts,  JtotZ  1930.  by  Balckvi/v  &   Cran-  '••/•  Mow, London/. 


..  to  /.•.•!.  /..'/(./.•/< 


./iiirjtiin 


ri.it.'  (, 


LIGHT. 

PART  I. 

Of  unpolarized  Light. 
§  I.    INTRODUCTION. 

Light.      IN    this  article  we  propose  to  give  an  account  of  the  properties  of  light ;  of  the  physico-mathematical  laws     Part  I 
'  which  regulate  the  direction,  intensity,  state  of  polarization,  colours,  and  interferences  of  its  rays ;    to  state  the  -~~-^-~~ 
theories  which  have  been  advanced  for  explaining  the  complicated  and   splendid  phenomena  of  optics;    to 
explain  the  laws  of  vision,  and  their  application,  by  the  combined  ingenuity  of  the  philosopher  and  the  artist,  to 
the  improvement  of  our  sight ;   and  the  examination  and  measurement  of  those  objects  and  appearances  which, 
from  their  remoteness,  minuteness,  or  refinement,  would  otherwise  elude  our  senses. 

The  sight  is  the  most  perfect  of  our  senses  ;  the  most  various  and  accurate  in  the  information  it  affords  us  ; 
and  the  most  delightful  in  its  exercise.  Apart  from  all  considerations  of  utility,  the  mere  perception  of  light  is 
in  itself  a  source  of  enjoyment.  Instances  are  not  wanting  of  individuals  debarred  from  infancy  by  a  natural 
defect  from  the  use  of  their  eyes,  whose  highest  enjoyment  still  consisted  in  that  feeble  glimmering  a  strong 
sunshine  could  excite  in  their  obstructed  organs  ;  but  when  to  this  we  join  the  exact  perception  of  form  and 
motion,  the  wonderous  richness  and  variety  of  colour,  and  the  ubiquity  conferred  upon  us  by  just  impressions  of 
situation  and  distance,  we  are  lost  in  amazement  and  gratitude. 

What  are  the  means  and  mechanism  by  which  we  receive  this  inestimable  benefit  ?  Curiosity  may  well  prompt 
the  inquiry,  but  a  more  direct  interest  urges  us  to  pursue  it.  Knowledge  is  power ;  and  a  careful  examination  of 
the  means  by  which  we  see,  not  only  may,  but  actually  has  led  us  to  the  discovery  of  artificial  aids  by  which 
this  particular  sense  may  be  strengthened  and  improved  to  a  most  extraordinary  degree ;  giving  to  man  at  once 
the  glance  of  the  eagle,  and  the  scrutiny  of  the  insect — by  which  the  infirmities  of  age  may  be  deferred  or 
remedied — nay,  by  which  the  sight  itself  when  lost  may  be  restored,  and  its  blessings  conferred  after  long  years 
of  privation  and  darkness,  or  on  those  who  from  infancy  have  never  seen.  But  as  we  proceed  in  the  inquiry 
we  shall  find  inducements  enough  to  pursue  it  from  purely  intellectual  motives.  A  train  of  minute  adaptation 
and  wonderful  contrivance  is  disclosed  to  us,  in  which  are  blended  the  utmost  extremes  of  grandeur  and 
delicacy ;  the  one  overpowering,  the  other  eluding,  our  conceptions.  In  consequence  of  those  peculiar 
and  singular  properties  which  are  found  to  belong  to  light  in  its  various  states  of  polarization,  it  affords 
to  the  philosopher  information  respecting  the  intimate  constitution  of  bodies,  and  the  nature  of 
the  material  world,  totally  distinct  from  the  more  general  impressions  of  form,  colour,  distance,  &c. 
which  it  conveys  to  the  vulgar.  Its  notices,  it  is  true,  in  this  respect,  are  addressed  rather  to  the  intellect 
than  the  sense  ;  but  they  are  not  on  that  account  less  real,  or  less  to  be  depended  on.  Polarized  light 
is,  in  the  hands  of  the  natural  philosopher,  not  merely  a  medium  of  vision  ;  it  is  an  instrument  by  which  he 
may  be  almost  said  to  feel  the  ultimate  molecules  of  natural  bodies,  to  detect  the  existences  and  investigate  the 
nature  of  powers  and  properties  ascertainable  only  by  this  test,  and  connected  with  the  more  important  and 
intricate  inquiries  in  the  study  of  nature. 

The  ancients  imagined  vision  to  be  performed  by  a  kind  of  emanation  proceeding  from  the  eye  to  the  object 
seen.  Were  this  the  case,  no  good  reason  could  be  shown  why  objects  should  not  be  seen  equally  well  in  the 
dark.  Something  more,  however,  is  necessary  for  seeing  than  the  mere  presence  of  the  object.  It  must  be  in  a 
certain  state,  which  we  express  by  saying  that  it  is  luminous.  Among  natural  bodies  some  possess  in  themselves 
the  property  of  exciting  in  our  eyes  the  sensation  of  brightness,  or  light ;  as  the  sun,  the  stars,  a  lamp,  red-hot 
iron,  &c.  Such  bodies  are  called  self-luminous ;  but  by  far  the  greater  part  possess  no  such  property.  Such 
bodies  in  the  dark  remain  invisible,  though  our  eyes  are  turned  directly  towards  them ;  and  are  therefore  termed 
dark,  non-luminous,  or  opaque,  though  this  word  is  also  used  occasionally  to  express  want  of  transparency.  All 

Opaque  bo-  bodies,  however,  though  not  luminous  of  themselves,  nor  capable  of  exciting  any  sensation  in  our  eyes,  become. 

die.  become  so  on  being  placed  in  the  presence  of  a  self-luminous  body.     When  a  lamp  is  brought  into  a  dark  room,  we  see, 

luminous  m  not  oll|y  jne  ialnp)  1^,4  an  the  other  bodies  in  the  room.     They  are  all,  so  long  as  the  lamp  remains,  rendered 
1*  luminous,  and  are  in  their  turn  capable  of  illuminating  others.     Thus  a  sunbeam  passing  into  a  darkened  room 

nous  body,  renders  luminous,  and  therefore  visible,  a  sheet  of  paper  on  which  it  falls  ;  and  this,  in  its  turn,  will  in  like 
manner  illuminate  the  whole  apartment,  and  render  visible  every  object  it  contains,  so  long  as  it  continues  to 
receive  the  sunbeam.  The  moon  and  planets  are  opaque  bodies ;  but  those  parts  of  them  on  which  the  sun 
shines  become  for  the  time  luminous,  and  perform  all  the  offices  of  self-luminous  bodies.  Thus  we  see,  that  the 
communication  which  we  call  light,  subsists  not  only  between  luminous  bodies  and  our  eyes,  but  between 
luminous  and  non-luminous  bodies,  or  between  luminous  bodies  and  each  other. 

2Y* 


342  LIGHT. 

Light.          Many  bodies  possess  the  property  of  intercepting  this  peculiar  intercourse  between  luminous  bodies  and  our     Part  L 
t^~~v~m^  eyes,  or  other  bodies.     A  screen  of  metal  interposed  between  the  sun  and  our  eyes  prevents   our  seeing  it  ;  ^—  •-  v~~ 

•T.         interposed  between  the  sun  and  a  sheet  of  white  paper,  or  other  object,  it  casts  a  shadow   on  such  object:    i.e. 
Opaque  bo-  renders  it  non-luminous.     By  this  power  of  bodies  to  intercept  light,  we  learn  that  the  communication  which 

~ 


~ 


n     H»ht     const'tutes  ^  takes  place  in  straight  lines.     We  cannot  see  through  a  bent  metallic  tube,  nor  perceive  the  least 
glimpse  of  light  through   three  small   holes  in  as  many  plates  of  metal   placed  one  behind  the   other  at  a 
distance,  unless  the  holes   be  situated  exactly  in  one  straight  line.     Moreover,  the  shadows  of  bodies,  when 
Light  ema-  fairly  received  on  smooth  surfaces   perpendicular  to  the  line  in  which  the  luminous  body  lies,  are  similar  in 
•btdaht*       figure  to  the  section  of  the  body  which  produces  them,  which  could  not  be,  except  the  light  were   commu- 
lines  nicated  in  straight  lines  from  their  edges  to  the  borders  of  the  shadow.     We  express  this  property  by  saying 

that  light  emanates,  or  radiates,  or  is  propagated  from  luminous  bodies  in  straight  lines  ;  by  which  expressions 
nothing  more  is  to  be  understood  than  the  mere  fact,  without  in  any  way  prejudging  the  question  as  to  the 
in  all  di-     intimate  nature  of  this  emanation.     Moreover,  it  emanates  from  them  in  all  directions,  for  we  see  them  in  all 
rections,       situations    of  the  eye,  provided    nothing    intervene    to  intercept  the  light.      This  is  the  essential   distinction 
between    luminous  bodies  and   optical   images  ;    from  which,  as   we   shall  see,  light  emanates   only  in  certain 
directions.     Whether  it  emanates  equally  in  all  directions  will  be  considered  farther  on. 

6.  Light  also  radiates  from   every  point  (at  least  from  every  physical  point)  of  a  luminous  body.     This   may, 

»nd  from      perhaps,  be  regarded  as  a  truism  ;    for  those  points  of  a  luminous  body  from  which  (as  from  the  spots  in  the 

lerypnysi-  sun)  no  light  emanates,  are,  in  fact,  non-luminous,  and  the  body  is  only  partially  so;  the  figure  of  the  spots  is 

a  luminous'  on'v  seen>  because  it  is  also   necessarily  that  of  the  luminous  surface  which  surrounds  them.     Still  it  should 

surface.        be  borne  in  mind,  for  reasons  which  will  appear  when  we  come  to   speak  of  the  formation  of  images.    It  is 

possible  (nay,  probable)  that  a  luminous  surface,  such  as  that  of  the  flame  of  a  candle,  may  consist  only  of  an 

immense  but  finite  number  of   luminous   points,  surrounded   by  non-luminous  spaces  ;    but  it  is   not   ocular 

demonstration  this  idea  admits  of;  and  it  is  sufficient  for  our  purpose  that,  so  far  as  our  senses  inform  us,  every 

physical  point  of  a  luminous  surface  is  a  separate  and  independent   source   of   light.     We   may  magnify  in  a 

telescope  the  sun's  disc  to  any  extent,  and  intercept  all  but  the  very  smallest  portions  of  it,  (spots  excepted,) 

yet  the  visibility  of  one  part  is  no  way  impaired  by  the  exclusion  of  the  rest.     In  this  sense  the  proposition 

is  no  truism,  but  an  important  fact,  of  which  we  shall  hereafter  trace  the  consequences. 

7^  When  the  sun    shines    through    a    small    hole,    and    is    received    on    a   white    screen    behind    at    a    con- 

siderable distance,  we  see  a  round  luminous  spot,  which  enlarges  as  the  screen  recedes  from  the  hole.  If 
we  measure  the  diameter  of  this  image  at  different  distances  from  the  hole,  it  will  be  found  that  (laying 
out  of  the  question  certain  small  causes  of  difference  not  now  in  contemplation)  the  angle  subtended  by 
the  spot  at  the  centre  of  the  hole  is  constant,  and  equal  to  the  apparent  angular  diameter  of  the  sun.  The 
reason  of  this  is  obvious  ;  the  light  from  every  point  in  the  sun's  disc  passes  through  the  hole,  and  continues 
its  course  in  a  right  line  beyond  it  till  it  reaches  the  screen.  Thus  every  point  in  the  sun's  disc  has  a 
point  corresponding  to  it  in  the  screen  ;  and  the  whole  circular  spot  on  the  screen  is,  in  fact,  an  image  or 
representation  of  the  face  of  the  sun.  That  this  is  really  the  case,  is  evidently  seen  by  making  the  expe- 
riment in  the  time  of  a  solar  eclipse,  when  the  image  on  the  screen,  instead  of  appearing  round,  appears 
horned,  like  the  sun  itself.*  In  like  manner,  if  a  pin-hole  in  a  card  be  held  between  a  candle  and  a  piece 
of  white  paper  in  a  dark  room,  an  exact  representation  of  the  flame,  but  inverted,  will  be  seen  depicted  on 
the  paper,  which  enlarges  as  the  paper  recedes  from  the  hole  ;  and  if  in  a  dark  room  a  white  screen  be 
extended  at  a  few  feet  from  a  small  round  hole,  an  exact  picture  of  all  external  objects,  of  their  natural 
colours  and  forms,  will  be  seen  traced  upon  the  screen  ;  moving  objects  being  represented  in  motion,  and 
Fig.  6.  quiescent  ones  at  rest.  (See  fig.  6.)  To  understand  this,  we  must  recollect  that  all  objects  exposed  to  light 
are  luminous  ;  that  from  every  physical  point  of  them  light  radiates  in  all  directions,  so  that  every  point  in 
the  screen  is  receiving  light  at  once  from  every  point  in  the  object.  The  same  may  be  said  of  the  hole  ; 
but  the  light  that  falls  on  the  hole  passes  through  it,  and  continues  its  course  in  straight  lines  behind. 
Thus  the  hole  becomes  the  vertex  of  a  conoidal  solid  prolonged  both  ways,  having  the  object  tor  its 
base  at  one  end,  and  the  screen  at  the  other.  The  section  of  this  solid  by  the  screen  is  the  picture  we  see 
projected  on  it,  which  must  manifestly  be  exactly  similar  to  the  object,  and  inverted,  according  to  the  simplest 
rules  of  Geometry. 

8.  Now  if  in  our  screen  receiving  (suppose)  the  image  of  the  sun  we  make  another  small  hole,  and  behind 

it  place  another  screen,  the  light  falling  on  the  space  occupied  by  this  hole  will  pass  beyond  it,  and  reach 
the  other  screen;  but  it  is  clear  that  it  will  no  longer  dilate  itself,  after  passing  through  the  second  hole, 
and  form  another  image  of  the  whole  sun,  but  only  an  image  of  that  very  minute  portion  of  the  sun  which 
corresponds  to  the  space  occupied  in  his  image  on  the  first  screen  by  the  hole  made  there.  The  lines 
bounding  the  conoidal  surface  will'in  this  case  have  much  less  divergency,  and,  if  the  holes  be  small 
enough,  and  very  distant  from  each  other,  will  approach  to  physical  lines,  and  that  the  nearer,  as  the  holes 
Fig.  7.  are  smaller  and  their  distance  greater.  (See  fig.  7.)  If  we  conceive  the  holes  reduced  to  mere  physical  points, 
these  lines  form  what  we  call  rays  of  light.  Mathematically  speaking,  a  ray  of  light  is  an  infinitesimal 
pyramid,  having  for  its  vertex  a  luminous  point,  and  for  its  base  an  infinitely  small  portion  of  any  surface 
illuminated  by  it,  and  supposed  to  be  filled  with  the  luminous  emanation,  whatever  that  may  be.  This 
pyramid,  in  homogeneous  media,  and  when  the  course  of  the  ray  is  not  interrupted,  has,  as  we  have  seen, 


*  In  the  eclipse  of  September  7,  1820,  this  horned  appearance  vas  very  striking  in  the  luminous   interstices  between  the  shadows  of 
small  irregular  objects,  as  the  leaves  of  trees.  &c.     It  was  noticed  by  those  who  had  no  idea  of  its  cause. 


LIGHT.  343 

Jjght.      its  sides  straight  line*      If  cases  should  occur  (as  they  will)  when  the  course  of  the  light  is  curved,  or  sud-      Part 
v— -V— ^  denly  broken,  we  may  still  conceive  such  a    pyramid  having  curved  or  broken  sides  to   correspond  ;    or  we  ^ ->,- 
may  (for  brevity's  sake)  substitute  for  it  a  mere  mathematical    line,  straight,  curved,  or  broken,  as  the  case 
may  be. 

9.  Light  requires  time  for  its  propagation.      Two    spectators    at    different    distances  from  a   luminous  object 

Velocity  of  suddenly  disclosed,  will  not  begin  to  see  it  at  the  same  mathematical  instant  of  time.     The  nearer  will  see 

it  sooner  than  the  more  remote  ;    in  the  same  way  as  two  persons  at  unequal    distances  from    a   gun   hear 

the  report  at  different  moments.     In  like  manner,  if  a  luminous  object  be  suddenly  extinguished,  a  spectator 

will  continue  to  see  it  for  a  certain  time  afterwards,  as  if  it  still  continued  luminous,  and  this  time  will  be 

greater  the  farther  he  is  from  it.     The  interval  in  question  is,  however,  so  excessively  small  in  such  distances 

as  occur  on  the  earth's  surface,  as  to  be  absolutely  insensible ;  but  in  the  immense  expanse  of  the  celestial 

regions  the  case  is  different.     The  eclipses  and  emersions  of  Jupiter's  satellites  become  visible  much  sooner 

(nearly  a  quarter  of  an  hour)  when  the  earth  is  at  its  least  distance  from  Jupiter  than  when  at  its  greatest. 

Light  then  takes  time  to  travel  over  space.     It  has  a  finite,  though    immense  velocity,  viz.   192500  miles  per 

second ;    and  this  important  conclusion,  deduced    by  calculation  from  the  phenomenon   just   mentioned,  and 

which,  if  it  stood  unsupported,  might  startle  us  with  its  vastness,  and  incline  us  to  look  out  for  some  other 

Aberration   mode  of    explanation,  receives  foil  confirmation   from  another  astronomical   phenomenon,    the    aberration    of 

of  light.       light,  which  (without  entering  into  any  close  examination  of   the  mode  in  which  vision  is  produced)  may  be 

explained  as  follows : 

10.  Let  a  ray  of  light  from  a  star  S,  at  such  a  distance  that  all  rays  from  it  maybe  regarded  as  parallel,  be 

received  on  a  small  screen  A,  having  an  extremely  minute  opening  A  in  its  centre ;  and  let  that  ray  which 
Fig  1.  passes  through  the  opening  be  received  at  any  distance  A  B,  on  a  screen  B  perpendicular  to  its  direction; 
and  let  B  be  the  point  on  which  it  falls,  the  whole  apparatus  being  supposed  at  rest.  If  then  we  join  the 
points  A,  B  by  an  imaginary  line,  that  line  will  be  the  direction  in  which  the  ray  has  really  travelled,  and 
will  indicate  to  us  the  direction  of  the  star ;  and  the  angle  between  that  line  and  any  fixed  direction  (that 
of  the  plumb-line,  for  instance)  will  determine  the  star's  place  as  referred  to  that  fixed  direction.  For  sim- 
plicity, we  will  suppose  this  angle  nothing,  or  the  star  directly  vertical ;  then  the  point  B  on  which  the  ray 
falls  will  be  precisely  that  marked  by  a  plumb-line  let  fall  from  A ;  and  the  direction  in  which  we  judge  the 
star  to  lie  will  coincide  precisely  with  the  direction  of  gravity.  Such  will  be  the  case,  supposing  the  earth, 
the  spectator,  and  the  whole  apparatus  at  rest ;  but  now  suppose  them  carried  along  in  a  horizontal  direction 
A  C,  B  D,  with  a  uniform  and  equal  velocity,  of  whose  existence  they  will  therefore  be  perfectly  insensible,  and 
the  purnb-line  will  hang  steadily  as  before,  and  coincide  with  the  same  point  of  the  screen.  At  the  moment 
when  the  ray  S  A  from  the  star  passes  through  the  orifice  A,  let  A,  B  be  the  respective  places  of  the  orifice, 
and  the  point  on  the  screen  vertically  below  it.  When  the  ray  has  passed  the  orifice,  it  will  pursue  its  course 
in  the  same  straight  line  S  A  B  as  before,  independent  of  the  motion  of  the  apparatus,  and  in  some  certain 

(distance  A  B  \ 

=  — ;         — ,  ,.   ,      =  t  I   will  reach  the  lower  screen.     But  in  this  time  the  aperture,  screens,  and 
velocity  of  light          / 

plumb-line  will  have  moved  away  through  a  space 

„     /  earth's  velocity  \ 

A  a  =  B  6  I  =  t  x  velocity  of  motion  =  A  B  x  • — 

V  velocity  of  light/ 

At  the  instant,  then,  that  the  ray  impinges  on  the  lower  screen,  the -plumb-line  will  hang,  not  from  A  on  B,  but 
from  a  on  b  ;  and  a  being  the  real  orifice,  and  B  the  real  point  of  incidence  of  the  light  on  the  screen,  the 
spectator,  judging  only  from  these  facts,  will  necessarily  be  led  to  regard  the  ray  as  having  deviated  from  its 
vertical  direction,  and  as  inclining  from  the  vertical,  in  the  direction  of  the  earth's  motion  through  an  angle  whose 

A  a  earth's  velocity 

tangent  is  —  -  —  or 


A  B  velocity  of  light  ' 

The  eye  is  such  an  apparatus.  Its  retina  is  the  screen  on  which  the  light  of  the  star  or  luminary  falls, 
and  we  judge  of  its  place  only  by  the  actual  point  on  this  screen  where  the  impression  is  made.  The 
pupil  is  the  orifice.  If,  the  eye  preserving  a  fixed  direction,  the  whole  body  be  carried  to  one  side  with  a 
velocity  commensurate  to  that  of  light,  before  the  rays  can  traverse  the  space  which  separates  the  pupil  from 
the  retina,  the  latter  will  have  shifted  its  place  ;  and  the  point  which  receives  the  impression  is  no  longer 
the  same  which  would  have  received  it  had  the  eye  and  spectator  remained  at  rest ;  and  this  deviation  is  the 
aberration  of  light. 

Every  spectator  on  the  earth  participates  in  the  general  motion  of  the  whole  earth,  which  in  its  annual 
orbit  about  the  sun  is  very  rapid,  and  though  far  from  equal  to  that  of  light,  is  by  no  means  insensible, 
compared  to  it.  Hence  the  stars,  the  sun,  and  planets,  all  appear  removed  from  their  true  places  in  the 
direction  in  which  the  earth  is  moving. 

13.  This  direction  is  varying  every  instant,  as  the  earth  describes  an  orbit  round  the  sun.     The  direction  therefore 

of  the  apparent  displacement  of  any  star  from  its  true  situation  continually  changes,  i.  e.  the  apparent  place 
describes  a  small  orbit  about  the  true.  This  phenomenon  is  that  alluded  to.  It  was  noticed  as  a  fact  by 
Bradley,  while  ignorant  of  its  cause,  that  the  stars  appear  to  describe  annually  small  ellipses  in  the  heavens 
of  about  40"  in  diameter.  The  discovery  of  the  velocity  of  light  by  the  eclipses  of  Jupiter's  satellites,  then 
recently  made  by  Roemer,  however,  soon  furnished  its  explanation.  Later  observations,  especially  those  of 
Brinkley  and  Struve,  have  enabled  us  to  assign,  with  great  precision,  the  numerical  amount  of  this  inequality, 
and  thence  to  deduce  the  velocity  of  light,  which  by  this  method  comes  out  191515  miles  per  second,  differing 


344  L  I  G  II  T. 

Light,      from  the  former  only  by  a  two  hundredth  part  of  its  whole  quantity.     This  determination  is  certainly  to  be     Part  I. 
v— v-^*'  preferred.  v— y«» 

14.  But  this  is  not  the  only  information  respecting  light  which  astronomical  observations  furnish.      We  learn 
Light  uni-  from  them  also,  "  That  the  light  of  the  sun,  the  planets,  and  all  the  fixed  stars,  travels  with  one  and  the  same 
form   in  its  velocity."     Now  as  we  know  these  bodies  to  be  at  different  and  variable  distances  from  us,  we  hence  conclude 

[on'        that  the  velocity  of  light  is  independent  of  the  particular  source  from  which  it  emanates,   and  the  distance 
over  which  it  has  travelled  before  reaching  our  eye. 

15.  The  velocity  of  light,  therefore,  in  that  free  and  perhaps  void  space  which  intervenes  between  us  and  the 
planets  and  fixed  stars,  cannot  be  supposed  other  than  uniform  ;  and  the  calculations  of  the  eclipses  of  Jupiter's 
satellites,  and  the  places  of  the  distant  planets  made  on  this  supposition  agreeing  with  observation,  prove  it  ti> 
be  so.     In  entering  such  media  as  it  traverses,  when  arrived  within  the  limits  of  the  atmospheres  of  the  earth 
and  other  planets,  we  shall  find  reason  hereafter  to  conclude  that  its   velocity  undergoes  a  change  ;    but,  at 
all  events,   we   have    no  reason  to  suppose  it  to  differ  in  different  parts  of  one  and  the  same  homogeneous 
medium. 

16.  The  enormous  velocity  here  assigned  to  light,  surprising  as  it  may  seem,  is  among  those  conclusions  which 
Velocity  of  rest  on  the  best  evidence  that  science  can  afford,  and  may  serve  to  prepare  us  for  other  yet  more    amazing 
light  illus-    numerical  estimates.     It  is  when  we  attempt  to  measure  the  vastness  of  the  phenomena  of   nature  with   our 
eum  ari^     fee^'e  scale  of  units,  such  as  we  are  conversant  with  on  this  our  planet,  that  we  become  sensible  of  its  insig- 
solls            nificance  in  the  system  of  the  universe.     Demonstrably  true  as  are  the  results,  they  fail  to  give  us   distinct  con- 
ceptions ;  we  are  lost  in  the  immensity  of  our  numbers,  and  must  have  recourse  to  other  ways  of   rendering 
them  sensible.     A  cannon  ball  would  require  seventeen  years  at  least  to  reach  the  sun,  supposing  its  velocity 
to  continue  uniform  from  the  moment  of  its  discharge.     Yet  light  travels  over  the  same  space  in  7J  minutes. 
The  swiftest  bird,  at  its  utmost  speed,  would  require  nearly  three  weeks  to  make  the  tour  of  the  earth.     Light 
performs  the  same  distance  in  much  less  time  than  is  required  for  a  single  stroke  of  his  wing ;    yet  its  rapidity 
is  but  commensurate   to  the  distances  it  has  to  travel.     It  is  demonstrable  that  light  cannot  possibly  arrive  at 
our  system  from  the  nearest  of  the  fixed   stars  in  less  than  five  years,  and  telescopes  disclose  to  us  objects 
probably  many  thousand  times  more  remote. 

But  these  are  considerations   which   belong  rather  to  astronomy  than  to  the   present  subject ;  and  we  will, 
therefore,  return  to  the  consideration  of  the  phenomena  of  emitted  light. 

§  II.     Of  Photometry. 

,.  ;.  .  Of  these,  one  of  the  most  striking  is  certainly  the  diminution  of  the  illuminating  power  of  any  source  of  light, 
nishes  asth'e  arising  from  an  increase  of  its  distance.  We  see  very  well  to  read  by  the  light  of  a  candle  at  a  certain  distance : 
distance  of  remove  the  candle  twice,  or  ten  times  as  far,  and  we  can  see  to  read  no  longer. 

us  source  The  numerical  estimation  of  the  degrees  of  intensity  of  light  constitutes  that  branch  of  optics  which  is  termed 
increases.  Photometry.  ($>««,  fitTpw.) 

If  light  be   a  material  emanation,  a  something  scattered  in  minute   particles  in   all   directions,  it  is  obvious 
!nverseinS'as  tnat  tne  same  quantity  which  is  diffused  over  the  surface  of  a  sphere  concentric  with  the  luminous  points,  if  it 
the  square    continue  its  course,  will  successively  be  diffused  over  larger  and  larger  concentric  spherical  surfaces  ;  and  that  its 
of  the         intensity,  or  the  number  of  rays  which  fall  on  a  given  space,  in  each  will  be  inversely  as  the  whole  surfaces  over 
distance.      which  it  is  diffused ;  that  is,  inversely  as  the  squares  of  their  radii,  or  of  their  distances  from  the  source  of  light. 
Without  assuming  this  hypothesis,  the  same  thing  may  be  rendered  evident  as  follows.     Let  a  candle  be  placed 
behind  an  opaque  screen  full  of  small  equal  and  similar  holes ;   the  light  will  shine  through  these,  and  be  inter- 
cepted in  all  other  parts,  forming  a  pyramidal  bundle  of  rays,  having  the  candle  in  the  common  vertex.     If  a 
sheet  of  white  paper  be  placed  behind  this,  it  will   be  seen  dotted  over  with  small   luminous  specks,  disposed 
exactly  as   the  holes  in  the  screen.     Suppose  the  holes  so  small,  their  number  so  great,  and  the  eye  so  distant 
from  the  paper  that  it  cannot  distinguish  the  individual  specks,  it  will  still  receive  a  general  impression  of  bright- 
ness ;    the  paper  will  appear  illuminated,  and  present  a  mottled  appearance,  which,  however,  will   grow   more 
uniform   as  the  holes  are  smaller,  and  closer,  and   the   eye  more  distant ;  and  if  extremely  so,  the  paper  will 
appear  uniformly  bright.     Now,  if  every  alternate   hole  be  stopped,  the  paper  will  manifestly  receive  only  half 
the  light,  and  will  therefore  be  only  half  as  much  illuminated,  and  creteris  paribits,  the  degree  of  illumination 
is  proportional  to  the  number  of  the  holes  in  the  screen,  or  to  the  number  of  equally  illuminated  specks  on 
its  surface,  i.  e.  if  the  specks  be  infinitely  diminished  in    size,  and    infinitely    increased    in    number,  to   the 
number  of  rays  which  fall  on  it  from  the  original  source  of  light. 

'9  Let  a  screen,  so  pierced  with  innumerable  equal  and  very  small  holes  in  the  manner  described,  be  placed 

at  a  given  distance  (1  yard)  from  a  candle;  and  in  the  diverging  pyramid  of  rays  behind  it  place  a  small 
piece  of  white  paper  of  a  given  area,  (I  square  inch,  for  instance,)  so  as  to  be  entirely  included  in  the  pyramid. 
It  is  manifest  that  the  number  of  rays  which  fall  on  it  will  be  fewer  as  it  is  placed  farther  from  the  screen, 
because  the  whole  number  which  pass  the  screen  are  scattered  continually  over  a  larger  and  larger  space 
Thus  were  it  close  to  the  screen  it  would  receive  a  number  equal  to  that  of  the  holes  in  a  square  inch  of  the 
screen,  but  at  twice  the  distance  (2  yards)  from  the  candle  this  number  will  be  spread  over  four  square  inches 
by  their  divergence,  and  the  paper  can  therefore  receive  only  a  fourth  part  of  that  number.  If,  therefore, 

its  illumination  when  close  to  the  screen  be  represented  by  I,  it  will  at  twice  the  distance  be  only—,  and 


LIGHT.  345 

t_j—     -*_-  at  D  times  the  assumed  unit  of  distance,  its  illumination  will  be  —  ,  the    areas    of  sections  of  a   pyramid  s 

IJ 

being  as  the  squares  of  their  distances  from  the  vertex. 

20.  As  this  reasoning  is  independent  of  the  number  and  size  of  the  holes,  and  therefore  of  the  ratio  ot   the 
sum  of  their  areas  to  that  of  the  unperforated  part  of  the  screen,  we  may  conceive  this  ratio  increased  ad 
infinitum.     The  screen  then  disappears,  and  the  paper  is  freely  illuminated.     Hence  we  conclude  that  when 
a  small  plane  object  of  given  area  is  freely  and  perpendicularly  exposed  to  a  luminary  at  different  distances, 
the  quantity  of  light  it  receives,  or  the  degree  of  its  illumination,  is  inversely  as  the  squares  of   its  distance 
from  the  luminary,  cateris  paribus. 

21.  If  a  single  candle  be  placed  before  a  system  of  holes  in  a  screen,  as  before,  and  the  rays  received  on  a 
Illumination  screen  at  a  given  distance,  (1,)  the  degree  of  illumination  may  be  represented  by  a  given  quantity,  I.     Now 
proportional  jf  a  second  candle  be  placed  immediately  behind  the  other,  and  close  to  it,  so  as  to  shine  through  the  same 

D  the  num-  no]eS)  the  illumination  of   the  screen  is  perceived  to  be  increased,  though  the  number  and  size  of  the  illu- 

tensity  of'""  minated  points  on  it  be  the  same.     Each  point  is  then  said  to  be    more    intensely  illuminated.     Now,  (the 

the  rays;      eye  being  all  along  supposed  so  distant,  and  the  illuminated  points  so  small  as  to  produce  only  a  general 

sense  of  brightness,  without  distinguishing  the  individual  points,)  if  the  one  candle  be  shifted  a  little  sideways, 

without  altering  its  distance,  the  illumination  of  the  paper  will  not  be  altered.      In  this  ease  the  number  of 

illuminated  points  is  doubled,  but  each  is  illuminated  with  only  half  the  light  it  had  before.     The  same  holds 

for  any  number  of  candles.     Hence  we  conclude  that  the  illumination    of    a  surface    is    constant  when  the 

number  of  rays  it  receives  is  inversely  as  the  intensity  of  each,  and  that  consequently  the  degree  of  illumination 

is  proportional  to  the  number  and  intensity  of  the  rays  jointly. 

22.  Now  if  for  any  number  of  candles  placed  side  by  side  we  substitute  mere  physical  luminous  points,  each 
and   to   the  of  these  will  be  the  vertex  of  a  pyramid  of  rays,  and  the  number  of  equally  illuminated  points  in  the  paper, 
area  of  the.  anfj  therefore  illuminations  will  be  proportional  to  the  number  of  such  points.     If   we  conceive  the  number 

atln?  of  these  increased,  and  their  size  diminished  ad  infinitum,  so  as  to  form  a  continuous  luminous  surface,  their 
number  will  be  represented  by  its  area.  Hence  the  illumination  of  the  paper  will  be,  ctsteris  paribus,  as  the 
area  of  the  illuminating  surface,  (supposed  of  uniform  brightness.) 

2.3.  Uniting  all  these  circumstances,  we  see  that  when  a  given  object  is  enlightened  by  a  luminous  surface  of 

General  ex-  small  but  sensible  size,  the  degree  of  its  illumination  is  proportional  to  the 

area  of  the  luminous  surface  x  intensity  of  its  illuminating  power 
square  of  the  distance  of  the  surface  illuminated. 

2t.  The  foregoing  reasoning  applies  only  to  the  case  when  the  luminous  disc  is  a  small  portion  of  a  spherical 

Oblique  il-  surface  concentric  with  the  illuminated  object,  in  which  case  all  its  points  are  equidistant  from  it,  and  all  the 
lumination.  light  falls  perpendicularly  on  the  object.  When  the  object  is  exposed  obliquely,  conceive  its  surface  divided 
into  equal  infinitely  small  portions,  and  regard  each  of  them  as  the  base  of  an  oblique  pyramid,  having  its 
vertex  at  any  one  point  of  the  luminary  ;  then  will  the  perpendicular  section  of  this  pyramid  at  the  same 
distance  be  equal  to  the  base  x  sine  of  inclination  of  the  base  to  the  axis,  or  the  element  of  the  illuminated 
surface  x  by  the  sine  of  the  inclination  of  the  ray.  But  the  number  of  rays  which  falls  on  the  base  is  evidently 
equal  to  those  which  fall  on  the  section,  and  being  spread  over  a  larger  area  their  effect  will  be  to  illuminate  it 
less  intensely  in  the  proportion  of  the  area  of  the  section  to  that  of  the  base,  i.  e.  in  the  proportion  of  the  sine  of 
inclination  to  radius.  But  the  illumination  of  the  section  is  equal  to  the 

area  of  the  luminary  x  intrinsic  brightness 
(distance)5 

therefore  that  of  the  elementary  surface  equals   this  fraction  multiplied  by  the  sine  of  the  rays'   inclination ; 
or,   calling   A   the   area   of   the    luminary,    I    its   intrinsic    brightness,   D   its   distance,  and  0  the  inclination 

of  the  ray  to  the  illuminated  surface   — '—— will  represent  the  intensity  of  illumination 

25.  If  L  represent  the  absolute  quantity  of  light  emitted  by  the  luminary  in  a  given  direction,  which  may  be  called 
its  absolute  light,  we  have  L  =  A  x  I,  provided  the   surface  of   the  luminary  be  perpendicular   to   the   given 
direction.     If  not,  A  must  represent  the  area  of  the  section  of  a  cylindroidal  surface  bounded  by  the  outline  of 

the  luminary,  and  having  its  axis  parallel  to  the  given  direction  ;   consequently    — '— ;  —  represents  in  this  case 

the  intensity  of  illumination  of  the  elementary  surface. 

To  illustrate  the  application  of  these  principles  we  will  resolve  the  following 

PROBLEM. 

26.  A  small  white  surface  is  laid  horizontally  on  a  table,  and  illuminated  by  a  candle  placed  at  a  g-iven  (hori- 
zontal) distance :  What  ought  to  be  the  height  of  the  flame,  so  as  to  give  the  greatest  possible  illumination  to  the 
surface  ? 

g-  2  Let  A  be  the  surface,  B  C  the  candle.      Put  AB  =  a.AC  =  D;    BC=  A/DJ—  a4.      Then,  since  the 


sin  C  A  B                C  B            \/Ds  —  a* 
illumination  of  A  is,  ceetens  panbus,  as  —  ,  or  as  -  = — (=  F)  we  have  to  make  this 

VOL.  IV.  2   Z 


346  LIGHT. 

Light,      quantity  a  maximum  ;  consequently  d  F  =  o,  or  d  .  F  1J  =  o,  that  is, 

4  e 


/"a- 

D  =  a  .  \f    -  -    and    B  C   = 


«2  =  -     -  =  0.707  X  A  B. 
V  2 

27.  Definition.     The  apparent  superficial  magnitude,  or  the  apparent  magnitude  of  any  object,  is  a  portion  of  a 
Apparent  spherical  surface  described  about  the  eye  as  a  centre,  with  a  radius  equal  to   1,  and  bounded  by  an  outline 
nagnitude  being  the  intersection  of  this  spherical  surface  with  a  conoidal  surface,  having1  the  object  for  its  base  and  the 
defined.  eye  jor  ;ts  vertex. 

28.  Hence  the  apparent  superficial  magnitude  of  a  small  object  is  directly  as  the  area  of  a  section  (perpendi- 
cular to  the  line  of  sight)  of  this  conoidal  surface,  at  the  place  of  the  object,  and  inversely  as  the  square  of  its 
distance.     If  the  object  be  a  surface  perpendicular  to  the  line  of  sight,  this  ratio  reduces  itself  to  the  area 
of  the  object  divided  by  the  square  of  its  distance. 

29.  Definition.  The  real  intrinsic  brightness  of  a  luminous  object  is  the  intensity  of  the  light  of  each  physical 
Real  intrin-  point  in  its  surface,  or  the    numerical  measure  of    the  degree  in  which  such  a   point  (of  given  magnitude) 
sic  bright-    would  illuminate  a  given  object  at  a    given  distance,  referred  to  some  standard  degree  of  illumination  as  a 
ness  defined.  un;t      When  we  speak  simply  of  intrinsic  brightness,  real  intrinsic  brightness  is  meant. 

30.  Carol.  1.  Consequently  the  degree  of  illumination  of  an  object  exposed  perpendicularly  to  a  luminary  is  as 
the  apparent  magnitude  of  the  luminary  and  its  intrinsic  brightness  jointly. 

31.  Carol.  2.  Conversely,  if  these  remain  the  same,  the  degree  of  illumination  remains  the  same.     For  example, 
the  illumination  of  direct  sunshine  is  the  same  as  would  be  produced  by  a  circular    portion  of  the  surface 
of  the  sun  of  one  inch  in  diameter,  placed  at  about  10  feet  from  the  illuminated  object,  and  the  rest  of  the  sun 
annihilated  ;  for  such  a  circular  portion  would  have  the  same  apparent  superficial  magnitude  as  the  sun  itself 
This  will  serve  to  give  some  idea  of  the  intense  brightness  of  the  sun's  disc. 

32.  Definition.     The  apparent  intrinsic  brightness  of  any  object,  or  luminary,  is  the  degree  of  illumination  of 
Apparent  its  image  or  picture  at  the  bottom  of  the  eye.     It  is  this  illumination  only  by  which  we  judge  of  brightness. 
intrinsic  A.  luminary  may  in  reality  be  ever  so  much  brighter  than  another  ;  but  if  by  any  cause  the  illumination  of  its 
brightness.  jmage  jn  tne  eye  be  enfeebled,  it  will  appear  no  brighter  than  in  proportion  to  its  diminished  intensity.     Thus 

we  can  gaze  steadily  at  the  sun  through  a  dark  glass,  or  the  vapours  of  the  horizon. 

Definition.  The  absolute  light  of  a  luminary  is  the  sum  of  the  areas  of  its  elementary  portions,  each  multi- 
Absolute      plied  by  its  own  intrinsic  brightness  ;  or,  if  every  part  of  the  surface  be  equally  bright,  simply  the  area  multi- 
''  ()         plied  by  the  intrinsic  brightness.     It  is,  therefore,  the  same  quantity  as  that  above  represented  by  L. 

Definition.    The  apparent  light  of  an  object  is  the  total  quantity  of  light  which  enters  our  eyes  from  it, 
Apparent     however  distributed  on  the  retina. 

light  In  common  language,  when  we  speak  of  the  brightness  of  an  object  of  considerable  size,  we  often  mean  its 

defined.        apparent  intrinsic  brightness.     When,  however,  the  object  has  no  sensible  size,  as  a  star,  we  always  mean  its 

3*>-       apparent  light,  (or,  as  it   might  be  termed,  its  apparent  absolute  brightness,)  because,  as  we  cannot  distinguish 

such  an  object  into  parts,  we  can  only  be  affected  by  its  total  light  indiscriminately.     The  same  holds  good  with 

all  small  objects  which  require  attention  to  distinguish  them  into  parts.     Optical  writers  have  occasionally  fallen 

into  much  confusion  for  want  of  attending  to  these  distinctions. 

36.  As  we  recede  from  a  luminary,  its  apparent  light  diminishes,  from  two  causes  ;  first,  our  eyes,  being  of  a  given 
Diminution  gjze>  present  a  given  area  to  its  light,  and  therefore  receive  from  it  a  quantity  of  light  inversely  as  the  square  of 
"'  aPI'arent  tne  distance  ;  secondly,  in  passing  through  the  air,  a  portion  of  the  light  is   stopped,  and  lost  from  its  want  of 
distance       perfect  transparency.     This,  however,  we  will  not  now  consider.     In  virtue  of   the  first  cause  only,  then,  the 

apparent  light  of  a  luminary  is  inversely  as  the  square  of  its  distance,  and  directly  as  its  absolute  light. 

37.  The  apparent  intrinsic  brightness  is  equal  to  the  apparent  light  divided  by  the  area  of  the  picture  on  the  retina 
Objects  ap-  of  our  eye.     But  this  area  is  as  the  apparent  superficial  magnitude  of  the  luminary,  that  is,  as  its  real  area  A 
pear  equally  ^ 

bright  at  all  divided  by  the  square  of  its  distance  D,  or  as  —  —  .      Moreover,  the   apparent  light,  as  we  have  just  seen,  is  as 
distances.  D2 

A  I 

—  where  I  is  the  real  intrinsic  brightness.     Consequently  the  apparent  intrinsic  brightness  is  proportional  to 

-  -i-   —  —  ,  or  simply  to  I,  and  is  independent  on  A  or  D.     The  apparent  intrinsic  brightness  is,  therefore, 

the  same  at  all  distances,  and  is  simply  proportional  to  the  real  intrinsic  brightness  of  the  object.  This  con- 
In  what  elusion  is  usually  announced  by  optical  writers  by  saying,  that  objects  appear  equally  bright  at  all  distances, 
sense  to  be  which  must  be  understood  only  of  apparent  intrinsic  brightness,  and  the  truth  of  which  supposes  also  that  no  loss 
understood,  of  light  takes  place  in  the  media  traversed. 

38.  The  angle  of  emanation  of  a  ray  of  light  from  a  luminous  surface  is  the  inclination  of  the  ray  to  the  surface  at 

Angle  of      the  pOint,  from  which  it  emanates. 

A  question  has  been  agitated  among  optical  philosophers,  whether  the  intensity  ot'  the  light  of  luminous  bodies 

5  "39        be  the  same  in  all  directions  ;  or  whether,  on  the  other  hand,  it  be  not  dependent  on  the  angle  of  emanation. 

Euler,  in  his  Reflexions  sur  les  divers  degrts  de  la  lumiere  du  Soliel,  $c.  Berlin,  Me'm.  1750,  p.  280,  has  adopted 


LIGHT.  347 

Light,     the  former  principle.     Lambert,  on  the  other  hand,  Photometria,  p.  41,  contends  that  the  intensity  of  the  light,     fart  I. 

^—  -y—^  or  density  of  the  rays,  issuing-  from  a  luminous  surface  in  any  direction  is    proportional  to  the*me.of  the  v"^~\^'~-' 

Question      angle  of  emanation.     If  we  knew  the  intimate  nature  of  light,  and  the  real  mechanism  by  which  bodies  emit 

cians^-e's-'1  an<^  re^ect  it,  ll  might  be  possible  to  decide  this  question   a  priori.      If,  for  instance,  we  were  assured  that 

pecting  the  ''ght  emanated  strictly  and  solely  from  the  molecules  situated  on  the  external  surface  of  bodies,  and  that  the 

dependence  emanation  from  each  physical  point  of  the  surface  were  totally  uninfluenced  by  the  rest  of  the  molecules  of 

f  the  emis-  which  the  body  consists,  and  dispersed  itself  equally  in  all  directions,  then,  since  every  point  of  a  plane  surface 

o'n  "th°e  anMe  'S  v*s"3'e  to  an  eye  wherever  situated  above  it,  and  each  is  supposed  to  send  the  same  number  of  rays  to  the 

of  emaua-    eye  m  an  oblique  as  in  a  perpendicular  situation,  the  total  light  received  from  a  given  area  of  the  surface  in 

tion.  the  eye  ought  to  be  the  same  at  all  angles  of  emanation.     But  as  the  apparent  magnitude  of  this  area  is  as 

the  sine  of  its  inclination  to  the  line  of  sight,  i.  e.  of  its  angle  of  emanation,  this  light  is  distributed  over  a  less 

apparent  area  ;   and  therefore  its  intensity,  or  the  apparent  brightness  of  the  surface,  should  be  increased  in  the 

inverse  ratio  of  the  sine  of  the  angle  of  emanation.     On  the  other  hand,  if,  as  there  is  every  reason  to  suppose, 

light  emanates,  not  strictly  from  the  surfaces  of  bodies,  but  from   sensible  depths  within  their  substance  ;  if  the 

surfaces  themselves  be  not  true  mathematical  planes,  but  consist  of  a  series  of  physical  points  retained  in  their 

places  by  attractive  and  repulsive  forces,  and  if  the  intensity  of  emanation  of  each  of  these  points  depend  in  any 

way  on  its  relation  to  the  points  adjacent,  there    is    no    reason,  a  priori,  to  suppose  the  equal  emanation  of 

light  in  all  directions;    and  to  find  what  its  law  really  is,  we  must  have  recourse  to  direct  observation. 

Astronomy  teaches  us  that  the  sun  is  a  sphere.  Hence  the  several  par's  of  its  visible  disc  appear  to  us 
under  every  possible  angle  of  inclination.  Now  if  we  examine  the  surface  of  the  sun  with  a  telescope,  the 
circumference  certainly  does  not  appear  brighter  than  the  centre.  But  if  the  hypothesis  of  equal  emanation  were 
correct,  the  brightness  ought  to  increase  from  the  centre  outwards,  and  should  become  infinite  at  the  edges,  so 
that  the  disc  ought  to  appear  surrounded  by  an  annulus  of  infinitely  greater  splendour  than  the  central  parts.  To 
this  it  may,  however,  be  justly  objected,  that  as  the  surface  of  the  sun  is  obviously  though  generally  spherical, 
yet  full  of  local  irregularities,  every  minute  portion  of  it  may  be  regarded  as  presenting  every  possible  variety 
of  inclination  to  our  eye  ;  and  the  brightness  of  every  part  being  thus  an  average  of  all  the  gradations  of  which 
it  is  susceptible,  should  he  alike  throughout. 

40.  Bouguer,  in  his  Traite  d'Optique,  Paris,  1760,  p.  90,  states  himself  to  have  found,  by  direct  comparison,  that 

the  central  portions  of  the  disc  of  the  sun  are  really  much  more  luminous  than  the  borders.  A  result  so  extra- 
ordinary, however,  and  so  apparently  incompatible  with  all  we  know  of  the  constitution  of  the  sun  and  the  mode 
of  emission  of  light  from  its  surface,  would  require  to  be  verified  by  very  careful  and  delicate  reexamination.  If 
found  correct,  the  only  way  of  accounting  for  it  would  be  to  suppose  a  dense  and  imperfectly  transparent 
atmosphere  of  great  extent  floating  above  the  luminous  clouds  which  form  its  visible  surface.  This  is  certainly 
possible,  but  our  ignorance  on  the  subject  renders  it  unphilosophiral  to  resort  to  a  body  so  little  within  our  reach 
for  the  establishment  of  any  fundamental  law  of  emanation.  The  objection  above  advanced,  it  will  be  observed, 
applies  with  nearly  the  same  force  to  all  surfaces.  If  we  examine  a  piece  of  white  paper  with  a  magnifier, 
we  shall  find  its  texture  to  be  in  the  last  degree  rough  and  coarse,  presenting  no  approach  to  a  plane;  and 
so  of  all  surfaces  rough  enough  to  reflect  light  in  all  directions. 

41  However,  as  it  is  only  with  such  luminous  surfaces  as  occur  in  nature  that  we  have  any  concern,  we  must 

Surfaces      take  their  properties  as  we  find  them  ;  and,  waiving  all  consideration  of  what  would  be  the  law  of  emanation 
appear         from  a  mathematical  surface,  it  may  be  stated  as  a  result  of  observation,  that  luminous  surfaces  appear  equally 

b?[Uhtyat  all  br'ght  at  a!l  a"?/™  °f  inclinaiio>l  to  the  line  of  sight. 

angles  d  may  ^e  tried  with  a  surface  of  red-hot  iron  ;  its  apparent  intrinsic  brightness  is  not  sensibly  increased 

by  inclining  it  obliquely  to  the  eye. 

42.  If  we  take  a  smooth  square  bar  of  iron,  or  better,  of  silver,  or  a  polished  cylinder  of  either  metal,  heated 
Experimen-  to  redness,  into  a  dark  room,  the  cylinder  will  appear  equally  bright  in  the  middle  of  its  convexity  next  the 
tal  proof  of  eye,  and  at  the  edges,  and  cannot  be  distinguished  at  all  from  a  flat  bar  ;    and  the  square  bar,  when  so  pre- 

na^ion  sented  as  to  have  two  of  its  sides  at  very  different  angles  to  the  line  of  sight,  will  still  appear  of  perfectly 
equable  brightness,  nor  can  the  angle  separating  the  adjacent  sides  be  at  all  discerned  ;  and  if  the  whole 
bar  be  turned  round  on  its  axis,  the  motion  can  only  be  recognised  by  an  alternating  increase  and  decrease 
of  its  apparent  diameter,  according  as  it  is  seen  alternately  diagonally  and  laterally,  its  appearance  being 
always  that  of  a  flat  plate  perpendicularly  exposed  to  the  eye.  These  and  similar  experiments  with  surfaces 
artificially  illuminated,  which  the  reader  will  have  no  difficulty  in  imagining  and  making,  as  well  as  those 
recorded  by  Mr.  Ritchie  in  the  Edinburgh  Philosophical  Journal,  are  sufficient  to  establish  the  principle 
announced  in  Article  42,  to  which  (for  the  reasons  already  mentioned)  the  observation  of  Bou<nier  on  the 
unequal  brightness  of  the  sun's  disc  offers  no  conclusive  objection. 

43.  Hence  it  follows,  that  the  surfaces  of  luminous  bodies,  at  least  their  ultimate  molecules,  do  not  emit  light 
Law  of  trie  with  equal  copiousness  in  all  directions;  but  that,  on  the  contrary,  the  copiousness  of  emission,  in  any  direction, 
oblique        js  as  (fa  sjne  Of  ffa  a7(™/e  Of  emanation  from  the  surface. 

emanation  *  J  J 


emanation 
of  light. 


PROBLEM. 

To  determine  the  intensity  of  illumination  of  a  small  plane  surface  any  how  exposed  to  the  rays  from  a 
luminary  of  any  given  size,  figure,  and  distance  ;  the  luminary  being  supposed  uniformly  bright  in  every  part. 
Conceive  the  surface  of  the  luminary  divided  into  infinitesimal  elementary  portions,  of  which  let  each  be 
regarded  as  an  oblique  section  of  a  pyramid,  having  for  its  vertex  the  centre  of  the  infinitely  small  illuminated 

2z  2 


348  LIGHT. 

Light,      plane  B,  fig.  3.     Let  P  Q  be  any  such  portion,  and  let  the  pyramid  B  P  be  continued  till  it  meets  the  surface     Part  L 
v— - YT'  °f  the  heavens  in  p,  there  projecting  the  surface  PQ  into  the  areola  pq,  and  let  the  whole  luminary  C  D  E  F  ^ —  <•*•• 
Illumination  ke  jn  |j]je  ,,lanner  projected  into  the  disc  c  d  ef.     Let  T  Q  be  a  section  of  the  pyramid  A  P  Q,  perpendicular 
by  any*  *    to  'ts  ax's-     Then,  first,  the  plane  B  will  be  illuminated  by  the  element  PQ,  just  as  it  would  be  by  a  surface 
luminary       if  Q  equally  bright,   in  virtue  of  the  principle  just  established.     Hence   P  Q  is  equivalent   to  an  equally  bright 
investigated,  surface  IT  Q.     Again,  since  the  apparent  magnitude  of  TT  Q  seen  from  B  is  the  same  with  that  of  p  q,  the   area 
Fig- 3.          ,,-Q  is  equivalent  to  an  equally  bright  area  p  q  placed   at  p  q,  (Art.  29,  30,  31,  Cor.  1,  2.)     P  Q  is,  therefore, 
equivalent  to  pq.     And  since  the  same  holds  good  of  every  other  elementary  portion  of  the  surface,  and  the  total 
light  received  by  B  is  the  sum  o    the  lights  it  receives  from  all  the  elements   of    the    luminary,  the  whole 
surface  CDEF  must  be  equivalent  to  its  projection  cd ef. 

45-  Hence  the  illumination  of  B  depends,  not  at  all  on  the  real,  but  only  on  the  apparent  figure  and  magni- 

tude of  the  luminary;   and  whatever  the  luminary  be,  we  may  always  substitute  for  it  a  portion  of  the  visible 
heavens,  supposed  of  equal  intrinsic  brightness,  and  bounded  by  the  same  outline. 

46.  Thus,  instead  of  the  sun,  we  may  suppose  a  small  circle  equal  in  apparent  diameter  to  the  sun,  and  equally 
bright ;  instead  of   a  luminous  rectangle  perpendicular  to  the  illuminated  plane  B,  and  of  infinite  height,  as 
A  G  H  I,   fig.  3,  we    may  substitute    the    spherical    sector    Z  A  G,  bounded   by  the  two  vertical  circles    Z  A, 
Z  G,  and  so  on. 

47.  Let  then  p  q,  any  elementary  rectangle  infinitely  small  in  both  dimensions  of  the  spherical  surface,  be  repre- 
sented by  rf4A,  so  that     /  /  ds  A  shall  represent  the  surface  cdef  itself;    then  if  we   put  z  =  the  zenith 

distance  Z  p  of  this  portion,  its  illuminating  power  on  A  will  be  d*  A  .  cos  z,  and  the  total  illuminating  power 
of  the  whole  surface  A  will  be 


=ff 


d*  A  .  cosr. 


48.  Example  \ .  To  find  the  illuminating  power  of  the  sector  Z  A  G  confined  between  any  two  vertical  circles 

General  for-  and  the  horizon,  (fig.  3.)  Here,  putting  0  for  the  azimuth  of  the  element  d  2  A,  if  we  consider  it  as  terminated 
mula  for  il-  jjy  jwo  contiguous  verticals  and  two  contiguous  parallels  of  altitude,  we  have  d2  A  =  d  z  X  dd  .  sin  z.  Hence 
we  have 


of  a  small 


plane"  L  =    /  Id  G  d  z  .  sin  z  .  cos  z  =  £  Cid  Odz  .  sin  2  z  =  £  /  (0  +  C)  d  z  .  sin  2  z  ; 

and   extending   the   integral  from  6  =  o  to  0  =  A  G,  the   amplitude  of   the   sector,  (whicli  we  will    call  a,) 
we  get 


L  =  I dz.  sin  2z  =  -^—  (C  —  £  cos2z) 

&   »/  2 

which  extended  from  Z  =  o  to  z  =  90°  becomes  simply  L  =  — . 

49.  Carol.  1.  This  is  a  measure  of  the  illuminating  power  of  the  sector,  on  the  same  scale  that  that  of  an  infinitely 

small  area  (A)  placed  at  the  zenith  would  be  represented  by  A  itself.     Because  in  this  case 

cos  z  =  o,  and    /  /   d2  A  .  cos  z  =  A. 

50  Carol.  2.   On  the  same  scale  the  illuminating  power  of  the  whole  hemisphere  is  B-  where  v  =  3.14159535 

51.  Example  2.   Required  the  illuminating  power  of  a  circular  portion  of  the  heavens  whose  centre  is  the  zenith. 
Calling  z  the  zenith  distance  of  any  element,  and  0  its  azimuth,  we  shall  have,  as  before, 

d2  A  =  d  6  d  z  .  sin  z,  and  therefore  L  =  /  /  dtfds.sin  z.  cos  z  =  I  0  .  -  -  —  ir  I  d  z  .  sin  2  z 

extending  the  integral  from  0  —  o  to  0  =  2  v.     That  is  L  =  ir .  (const  —  i  cos  2  z)  which  being  made  to  vanish 
when  z  —  o  becomes  L  =  --  (1  —  cos  2  z)  =  JT  .  (sin  z)8 

52.  Carol.  3.  The  illuminating  power  of  a  circular  luminary,  whose  centre  is  in  the  zenith,  is  as  the  square  of 
the  sine  of  its  apparent  semidiameter. 

53.  Example  3.  Required  the  illuminating  power  of  any  circular  portion  of  the  heavens  whatever. 

Illumina-  Let  T  K L M  be  the  illuminating  circle;  conceive  it  composed  of  annuli  concentric  with  itself,  and  of  one 
ting  power  of  tnem>  X  Y  Z,  (fig.  4,)  let  X  x  be  an  infinitesimal  parallelogram  terminated  by  contiguous  radii  S  X  and 
cularVor-  S  z,  S  being  the  centre. 

t"on  0fP  an  Put  Z  S  =  «  ;    S  X  =  *,   Z  X  =  Z, 

e<tu*"y  Angle  ZSX  =  0,  S  T  =  r. 

bright 
.»ven.  Area  d4  A  =  Xar,  =  dx  x  d0.  sin  x 

.•.L  =  //d0da:.  sin  x  .  cos  z. 
but,  by  spherical  trigonometry,         cos  z  =  cos  a  .  cos  x  .  +  sin  a  .  sin  x  cos  0. 


LIGHT.  349 

V»Y»WBV'  Therefore  L  =    /   /  dx .  d<p .  sin  x  I  cos  a .  cos  x  +  sin  <z  .  sin  a;  .  cos  0.  >  ,_ 

The  first  integration  performed  relative  to  <p,  and  extended  from  0  =  o  to  0  =  360°,  or  2  w,  gives 

L  =    /  d  x  .  sin  x  x  2  IT  .  cos  a .  cos  x. 

After  which  integrating,  with  respect  to  x.  and  extending  the  integral  from  x  =  o  to  x  =  S  T  =  r,  we  find 

L  =  -  —  (1  —  cos  2  r)  =  n-  .  cos  a  (sin  r)a. 

This  lesult  is  particularly  elegant  and  remarkable.  It  shows,  that  to  obtain  the  illuminating  effect  of  a  circular 
luminary  (of  any  apparent  diameter)  at  any  altitude,  on  a  horizontal  plane,  we  have  only  to  reduce  its 
illuminating  effect  when  in  the  zenith,  in  the  ratio  of  radius  to  the  cosine  of  the  zenith  distance,  or  sine 
of  the  altitude.  For  other  examples,  the  reader  may  consult  Lambert's  Photometria,  cap.  ii.  from  which  this 
is  taken. 
54.  If  the  illuminating  surface  be  not  equally  intrinsically  bright  in  every  part,  if  we  call  I  the  intrinsic  brightness 

General  ex-  of  the  element  d  *  A,  we  shall  have 

pression  for  f*{* 

illumination  L=    //    Id'A.  COS  Z 

when  the  */»/ 

luminary  is  for  jne  general  formula  expressing  the  illuminating  power  of  the  surface  A.     The  moon,  Venus  and  Mercury  in 

brighT1"    ^  their  phases,  the  sky  during  twilight,  a  white  sphere  illuminated  by  the  sun,  &c.  afford  examples  of  this  when 

throughout,  themselves  regarded  as  luminaries. 

PROBLEM. 

55^  To  compare  the  illumination  of  a  horizontal  plane  by  the  sun  in  the  zenith  with  the  illumination  it  would 

have  were  the  whole  surface  of  the  heavens  of  equal  brightness  with  the  sun. 

By  Art.  53  we  have  L  :=  w  .  cos  a  .  (sin  r)2.      If,  therefore,  we  call  L  and  L'  the  two  illuminations  in 
question,  we  shall  have 

L  :  L'  :  :  TT  .  cos  o°  .  (sin  Q's  semidiam.)2  :  ir  .  cos  o°.  (sin  90°)* 

:  :  (sin  160"  :!::!:  46166. 

56.  The  illumination  of  a  plane  in  contact  with  the  sun's  surface  is  the  same  as  that  of  a  plane  on  the  earth's 
Illumination  surface  illuminated  by  a  whole  hemisphere  of  equal  brightness  with  the  sun  in  the    zenith.     Hence  we   see 

s  that  the  illumination  of  such  a  plane  at  the  sun's  surface  would  be  nearly  50,000  times  greater  than  that  of 
the  earth's  surface  at  noon  under  the  equator.  Such  would  be  the  effect  (in  point  of  light  alone)  of 
bringing  the  earth's  surface  in  contact  with  the  sun's ! 

57.  For  measuring  the  intensity  of  any  given  light,  various  instruments  called  Photometers  have  been  contrived, 
Photometers  many  of  which  have  little  to  recommend  them  on  the  score  of  exactness,  and  some  are  essentially  defective 

in  principle,  being  adapted  to  measure — not  the  illuminating — but  the  heating  power  of  the  rays  of   light  ; 
and,  therefore,  must  be  regarded  as  undeserving  the  name  of  photometers. 

58.  We  know  of  no  instrument,  no  contrivance,  as  yet,  by  which  light  alone  (as  such)  can  be  made  to  produce 
mechanical  motion,  so  as  to  mark  a  point  upon  a  scale,  or  in  any  way  to  give  a  direct  reading  off  of  its 
intensity,  or  quantity,  at  any  moment.     This  obliges  us  to  refer  all  our  estimations  of  the  degrees  of  bright- 
ness at  once  to  our  organs  of  vision,  and  to  judge  of  their  amount  by  the  impression  they  produce  imme- 
diately on  our  sense  of  sight.     But  the  eye,  though  sensible  to  an  astonishing  range  of  different  degrees  of 
illumination,  is  (partly  on  that  very  account)  but  little  capable  of  judging  of  their  relative  strength,  or  even 

The  eye  an  of  recognising  their  identity  when  presented  at  intervals  of  time,  especially  at  distant  intervals.     In  this  manner 
jiXe  of     ^e  Judgment  of  the  eye  is  as  little  to  be  depended  on  for  a  measure  of  light,  as  that  of  the  hand  would  be 
degrees  of    f°r  the  weight  of  a  body  casually  presented.     This  uncertainty,  too,  is  increased  by  the  nature  of  the  organ 
illumination  itself,  which    is  in  a  constant  state  of  fluctuation  ;    the  opening  of  the  pupil,  which  admits  the  light,  being 
continually  expanding  and  contracting  by  the  stimulus  of  the    light    itself,  and  the  sensibility  of  the  nerves 
which  feel  the  impression  varying  at  every  instant.     Let  any  one  call  to  mind  the  blinding  and  overpower- 
ing effect  of  a  flash  of  lightning  in  a  dark  night,  compared  with  the  sensation  an  equally  vivid  flash  pro- 
duces in  full  daylight.      In  the  one  case  the   eye  is   painfully  affected,  and  the  violent  agitation  into  which 
the  nerves  of  the  retina  are  thrown  is  sensible  for  many  seconds  afterwards,  in  a  series  of  imaginary  alter- 
nations of  light  and  darkness.      By  day  no    such  effect  is  produced,  and  we  trace  the  course  of  the  flash, 
and  the  zig-zags  of  its  motion  with  perfect  distinctness  and  tranquillity,  and  without  any  of  those  ideas  of 
overpowering  intensity  which  previous  and  subsequent  total  darkness  attach  to  it. 

59.  But   yet  more.     When  two   unequally  illuminated  objects  (surfaces  of  white  paper,   for  instance)  are  pre- 
sented at  once  to  the  sight,  though  we  pronounce  immediately  on  the  existence  of  a  difference,  and  see  that 
one  is  brighter  than  the  other,  we  are  quite  unable  to  say  what  is  the  proportion  between  them.     Illuminate 
half  a  sheet  of  paper  by  the  light  of  one  candle,  and  the  other  half  by  that  of  several  ;    the  difference  will 
be  evident.     But  if  ten  different  persons  are  desired,  from  their  appearance  only,  to  guess  at  the  number  of 
candles  shining  on  each,  the  probability  is  that  no  two  will  agree.      Nay,  even  the  same  person  at  different 
times  will  form  different  judgments.     This  throws  additional  difficulty  in  the  way  of  photometrical  estimations, 
and  would  seem  to  render  this  one  of  the  most  delicate  and  difficult  departments  of  optics. 


350  LIGHT 

Light  However,  the  eye,  under  favourable  circumstances,  is  a  tolerably  exact  judge  of  the  equality  of  two  degrees     Part  I. 

^— — v"""*'  °f  illumination  seen  at  once ;  and  availing  ourselves  of  this,  we  may  by  proper   management  obtain  correct  — »-^/-<- 

60.       information  as  to  the  relative  intensities  of  all  lights.     What  these  favourable  circumstances  are,  we  come  now 
The  eye       to  consider. 

;    .        1st.  The  degrees  of  illumination  compared  must  be  of  moderate   intensity.      If  so  bright  as  to  dazzle,  or 
theDequa°ity  so  ^a'nt  as  to  stram  tne  eYe>  no  correct  judgment  can  be  formed. 

•>f  two  de-  Hence,  it  is  rarely  adviseable  to  compare  two  luminaries  directly  with  each  other.  It  is  generally  better  to  let 
grees  of  il-  them  shine  on  a  smooth  white  surface,  and  judge  of  the  degree  in  which  they  illuminate  it ;  for  it  is  an  obvious 
immation,  axiom,  That  two  luminaries  are  equal  in  absolute  light  when,  being  placed  at  equal  distances  from,  and  in  similar 
what  cir-  situations  with  respect  to,  a  given  smooth  white  surface,  or  two  equal  and  precisely  similar  white  surfaces,  they 
cumstances.  illuminate  it  or  them  equally. 

Axiom  in          2nd.  The  luminaries,  or  illuminated  surfaces  compared,  must  be  of  equal  apparent  magnitude,  and    similar 
photometry,  figure,  and  of  such  small  dimensions  as  to  allow  of  the  illumination  in  every  part  of  each  being  regarded   as 
""•        the  same. 

64.  3rd.  They  must  be  brought  close  together,  in  apparent  contact ;   the  boundary  of  one  cutting  upon  that  of  the 
other  by  a  well-defined  straight  line. 

65.  4th.  They  should  be  viewed  at  once  by  the  same  eye,  and  not  one  by  one  eye,  and  the  other  by  the  other. 

66.  5th.    All  other  light  but  that  of  the  two  objects  whose  illumination  is  compared  should  be  most  carefully 
excluded. 

67.  6th.  The  lights  which   illuminate    both    surfaces    must  be    of  the  same   colour.      Between   very    differently 
coloured  illuminations  no  exact  equalization  can  ever  be  obtained,  and  in  proportion  as  they  differ  our  judgment 
is  uncertain. 

68.  When  all  these  conditions  obtain,  we  can  pronounce  very  certainly  on  the  equality  or  inequality  of  two  illu- 
minations.    When  the  limit  between  them   cannot  be  perceived,  on  passing  the  eye  backwards  and  forwards 
across  it,  we  may  be  sure  that  their  lights  are  equal. 

gg  Bouguer,  in  his   Traite  d'Optique,    1760,   p.  35,  has  applied  these  principles  to  the  measure    or  rather  the 

Bouguer's     comparison  of  different  degrees  of  illumination.     Two  surfaces  of  white  paper,  of  exactly  equal   size  and   re- 
principle  of  flective   power,  (cut  from  the  same  piece  in  contact,)  are  illuminated,  the  one  by  the  light  whose  illuminating 
compan    ve  power  is  to  be  measured,  the  other  by  a  light  whose  intensity  can  be  varied   at  pleasure  by  an   increase  of 
'etr^'  distance,  and  can  therefore  be  exactly  estimated.     The  variable  light  is  to  be  removed,  or  approached,  till  the 
two  surfaces  are  judged  to  be  equally  bright,  when,  the  distances  of  the  luminaries  being  measured,  or  otherwise 
allowed  for,  the  measure  required  is  ascertained. 

70.  Mr.  Ritchie  has  lately  made  a  very  elegant  and  simple  application  of  this  principle.     His  photometer  consists 
Ritchie's      of  a  rectangular  box,   about  an  inch  and  a  half  or  two  inches   square,  open  at   both  ends,  of   which  A  B  C  D 
photometer:  (^g  5)  js  a  sectjOn.      Jt  js  blackened  within,  to  absorb  extraneous  light.     Within,  inclined  at  angles  of  45°  to 

its  axis,  are  placed  two  rectangular  pieces  of  plane  looking-glass  F  C,  F  D,  cut  from  one  and  the  same  rectan- 
gular strip,  of  twice  the  length  of  either,  to  ensure  the  exact  equality  of  their  reflecting  powers,  and  fastened 
so  as  to  meet  at  F,  in  the  middle  of  a  narrow  slit  EFG  about  an  inch  long,  and  an  eighth  of  an  inch  broad, 
which  is  covered  with  a  slip  of  fine  tissue  or  oiled  paper.  The  rectangular  slit  should  have  a  slip  of  blackened 
card  at  F,  to  prevent  the  lights  reflected  from  the  looking-glasses  mingling  with  each  other. 

71.  Suppose  we  would  compare  the  illuminating  powers  of  two  sources  of  light  (two  flames,  for  instance)  PandQ. 
its  use.         They  must  be  placed  at  such  a  distance  from  each  other,  and  from  the  instrument  between  them,  that  the  light 

from  every  part  of  each  shall  fall  on  the  reflector  next  it,  and  be  reflected  to  the  corresponding  portion  of  the 
paper  E  F  or  F  G.  The  instrument  is  then  to  be  moved  nearer  to  the  one  or  the  other,  till  the  paper  on  either 
side  of  the  division  F  appears  equally  illuminated.  To  judge  of  this,  it  should  be  viewed  through  aprismoidal 
box  blackened  within,  one  end  resting  on  the  upper  part  A  B  of  the  photometer;  the  other  applied  quite  close 
to  the  eye.  When  the  lights  are  thus  exactly  equalized,  it  is  clear  that  the  total  illuminating  powers  of  the 
luminaries  are  directly  as  the  squares  of  their  distances  from  the  middle  of  the  instrument. 

72.  By  means  of  this  instrument  we  are  furnished  with  an  easy  experimental  proof  of  the  decrease  of  light  as  the 
Experimen-  inverse  squares  of  the  distances.     For  if  we  place  four  candles  at  P,  and  one  at  Q,  (as  nearly  equal  as  possible, 
ul  proof  of  an(j  burning  with  equal  flames,)  it  is  found  that  the  portions  E  F,   G  F  of  the  paper  will  be  equally  illuminated 
tion  onriit  wnen  tne  distances  PF,  QF  are  as  2  :  1,  and  so  for  any  number  of  candles  at  each  side. 

aTthe  To  render  the  comparison  of  the  lights  more  exact,  the  equalization  of   the   lights   should  be  performed 

squares  of   several  times,  turning  the  instrument  end  for  end  each  time.     The  mean  of  the  several  determinations  will  then 
thedistances  Jje  very  near  the  truth. 

In  some  cases  the  looking-glasses  are  better  dispensed  with,  and  a  slip  of  paper  pasted  over  them,  so  as  to 

^       present  two  oblique  surfaces  of  white   paper  inclined  at   equal   angles   to  the  incident  light.     In  this  case  the 

paper  stretched  over  the  slit  E  F  G  is  taken  away,  and  the  white  surfaces  below  examined  and  compared.     One 

tion.  advantage  of  this  disposition  is  the  avoiding  of  a  black  interval  between  the  two  halves  of  the  slit,  which  renders 

the  exact  comparison  of  their  illuminations  somewhat  precarious. 

75.  If  the  lights  compared  be  of   different  colours  (as  daylight,  or   moonlight,  and   candlelight,)  their  precise 
Comparison  equalization  is  impracticable,  (art.  67.)     The  best   way  of  employing  the  instrument,  in   this  case,  is  to  move 
of  lights  of  it  till  one  of  the  sides  of  the  slit  (in  spite  of  the  difference  of  colours)  is  judged  to  be  decidedly  the  brighter, 
different       anfj  tnea  to  move  ;t  the  other  way,  till  the  other  becomes  decidedly  the  brighter.     The  position  halfway  between 

these  points  is  to  be  taken  as  the  true  point  of  equal  illumination. 

76.  If  we  would  compare  the  degrees  of  illumination,  or  the  intrinsic  brightnesses  of  two  surfaces,  a  gi\'en  portion 
of  each  must  be  insulated  for  examination ;  this  may  be  best  done  by  the  adaptation  of  two  blackened  tubes  to 


LIGHT.  051 

the  openings  of  the  photometer,  of  equal  length,  and  terminated  by  orifices  of  equal  area,  or  subtending  equal      Part  I. 
angles  at  the  middle  of  the  instrument.     These,  of  course,  cut  off  equal   apparent  magnitudes  of  the  bright  ' 
Comparison  surfaceS)  the  light  of  which  is  then  to  be  equalized  on  the  oiled  papers   of  the  slit  E  F,  as  in  the  case  of 
rfbrifhT    candles,  &c.  Bouguer,  Traite,  p.  31. 

nessofillu-      Another  method  of  comparing  the  intensity  of  the  light  from  two  luminaries,  which  is  also  very  ready  and 
minated       convenient,  and  possesses  in  some  cases  considerable  advantages,  has  been  proposed  by  Count  Rumford.    (See 
surfaces.       Phil.  Trans.,  vol.  84,   p.  67.)      It   consists   in   the  equalization  of  the  shadows  cast  by  them  on  a  white  surface 
' '•        illuminated  by  them  both  at  once.     Suppose,  for  instance,  we  would  compare  the  illuminating  power  of  two 
'  flames  L  and  I  of  different  sizes,  or  from  different  combustibles,  as  of  wax  and  tallow.     Before  a  screen  C  D  of 

white  paper,  in  a  darkened  room,  place  a  blackened  cylindrical  stick  S,  and  let  the  flames  L  I  be  so  placed  as 
to  throw  the  shadows  AB  of  the  stick  on  the  screen,  side  by  side,  and  with  an  interval  between  them  about 
equal  in  breadth  to  either  shadow.  Moreover  the  inclination  of  the  rays  L  S  A  and  I S  B  to  the  surface  of  the 
screen  must  be  adjusted  to  exact  equality.  The  brighter  flame  must  then  be  removed,  or  the  feebler  brought 
nearer  to  the  screen,  till  the  two  shadows  appear  of  equal  intensity,  when  their  distances  (or  the  distance  of  the 
screen)  from  the  lights  must  be  measured,  and  their  total  illuminating  powers  will  be  in  the  direct  ratio  of  the 
squares  of  the  distances.  The  rationale  of  this  is  obvious,  the  shadow  thrown  by  each  flame  is  illuminated  by 
the  light  of  the  other.  The  screen  by  the  sum  of  the  lights.  The  eye  in  this  case  judges  of  the  degrees  of  defal- 
cation of  brightness  from  this  sum ;  and  if  these  degrees  be  alike,  it  is  clear  that  the  remaining  illuminations  must 
be  equal. 

78.  This  method  becomes  uncertain  when  the  lights  are  of  considerable  size  and  near  the  screen,  as  the  penum- 
brae  of  the  shadows  prevent  any  fair  comparison  of  the  relative  intensities  of  their  central  portions.     It  is  still 
more  so,  and  can  hardly  be  used  when  the  lights  differ  considerably  in  colour.     Its  convenience,  however,  as  an 
extemporaneous  method,  requiring  no  apparatus  but  what  is  always  at  hand,  (as  the  use  of  a  blackened  stick, 
though  preferable,  is  not  essential,)  renders  it  often  useful  in  the  absence  of  more  refined  means. 

79.  It  may  happen  that  the  lights  to  be  compared  are  not  movable,  or  not  conveniently  so.     In  this  case  the 
When  the     equalization  of  the  shadows  may  be  performed  by  inclining  the  screen  at  different  angles  to  the  directions  in 
lights  to  be  which  it  receives  the  light  of  each,  and  noting  the  angles  of  inclination  of  the  rays.     In  this  case  the  illumi- 
compared     nating  powers  of  the  luminaries  are  as  the  squares  of  their  distances  directly,  and  the  sines  of  the  respective 
movable        angles  of  inclination  of  their  rays  to  the  screen  inversely. 

80  When  a  ray  of  light  proceeds  in  empty  space,  or  in  a  perfectly  homogeneous  medium,  its  course,  as  we  have 

Modifies-      seen,  is  rectilinear,  and  its  velocity  uniform  ;    but  when  it  encounters  an  obstacle,  or  a   different  medium,  it 
tions  of  light  undergoes  changes  or  modifications  which  may  be  stated  as  follows  : 

enumerated.       jt  js  separate(i  into  several  parts,  which  pursue  different  courses,  or  are  otherwise  differently  modified.     One  of 

these  parts  is  regularly  reflected,  and  pursues,  after  reflexion,  a  course  wholly  exterior  to  the  new  medium,  or  obstacle. 

A  second  and  a    third  portion  are  regularly  refracted,  that  is,  they  enter  the  medium,   and   there   pursue 

g2         their  course   according  to  the    laws   of  refraction.      In  many  media  these   portions  follow  the  same  course 

Regular  re-  precisely,  and  perhaps  are  no  way  distinguishable  from   each  other.      In  such  media  (comprehending  most 

fraction         uncrystallized  substances  and  liquids)  the  refraction  is  said  to  be  single      In  numerous  others  (for  instance 

Single  and    ;n  mosi  crystallized    media)  they  follow  different  courses,  and  also  retain  different   physical  characters.      In 

fraction"5"    these  the  refraction  is  said  to  be  double. 

g3  A  fourth  portion  is  scattered  in  all  directions,  one  part  being  mtromitted  into  the  medium,  and  distributed 

Scattering  over  the  hemisphere  interior  to  it,  while  the  other  is  in  like  manner  scattered  over  the  exterior  hemi- 
sphere. These  two  portions  are  those  which  render  visible  the  surfaces  of  bodies  to  eyes  situated  any  how 
with  respect  to  them,  and  are  therefore  of  the  utmost  importance  to  vision. 

g4  Of  those  portions  which  enter  the  medium,  a  part  more  or    less  considerable  is  absorbed,  stifled,  or  lost, 

Absorption,  without  any  further  change  of  direction ;  and  that  not  at  once,  but  progressively,  as  they  penetrate  deeper 
and  deeper  into  its  substance.  In  perfectly  opaque  media,  such  as  the  metals,  this  absorption  is  total,  and 
takes  place  within  a  space  less  than  we  can  appreciate  ;  yet  even  here  we  have  good  reasons  for  believing 
that  it  does  not  take  place  per  sallum.  In  crystallized  bodies,  those  at  least  which  are  coloured,  this  absorp- 
tion takes  place  differently  on  the  two  portions  into  which  the  regularly  refracted  ray  is  divided,  according 
to  laws  to  be  explained  when  we  come  to  treat  of  the  absorption  of  light. 

85.  The  regularly  refracted  portions  of  a  ray  of  white  or    solar    light  are   (except  in    peculiar  circumstances) 
Separation    separated  into  a  multitude  of  rays  of  different    colours,  and  otherwise  differing  in  their   physical  properties, 
into  colours,  eacn  of  which  rays  pursues  its  course  afterwards,  independently  of  all  the  rest,  according  to  the  laws  of  re- 
or  dispersion  gu]ar  refraction  or   reflexion.      The  laws  of  this  separation,  or  dispersion,  of  the  coloured    rays,  and  their 

physical  and  sensible  properties,  form  the  subject  of  Chromatics. 

86.  All  those    portions  which  are    either    regularly  reflected,  or    regularly  refracted,  undergo,  more  or  less,  a 
Polarization,  modification    termed   polarization,  in  virtue  of   which  they  present,   on  their   encountering  another  medium, 

different  phenomena  of  reflexion  and  refraction  from  those  presented  by  unpolarized  light.  Generally  speaking, 
polarized  light  obeys  the  same  laws  of  reflexion  and  refraction  as  unpolarized,  as  to  the  directions  which 
the  several  portions,  into  which  it  is  divided  on  encountering  a  new  medium,  take ;  but  differs  from  it  in  the 
relative  intensities  of  those  portions,  which  vary  according  to  the  situation  in  which  the  surface  of  the  medium 
and  certain  imaginary  lines,  or  axes  within  it,  are  presented  to  the  polarized  ray. 

The  rays  of  light  under  certain  circumstances  exercise  a  mutual  influence  on  each  other,  increasing,  dimi- 
loterference  nishing,  or  modifying  each  other's  effects  according  to  peculiar  laws.      This    mutual    influence   is  called  the 
interference  of  the  rays  of  light.     We  shall  proceed  to  treat  of  these  several  modificatious  in  order ;  and  first 
of  the  regular  reflexion  of  light. 


352 


LIGHT. 


Light. 


Ptrt  I. 


88. 


89. 

Laws  of 
reflexion. 

90. 

91. 
92. 


93. 


94. 


95. 

Demon- 
strated by 
experiment. 


96 


97. 


Fig  9. 


98. 
Fig.  10 


§  3.   Of  the  regular  Reflexion  of  unpolarized  Light  from  Plane  Surfaces. 

When  a  ray  of  light  is  incident  on  a  smooth-polished  surface,  a  portion  of  it  is  regularly  reflected,  and 
pursues  its  course  after  reflexion  in  a  right  line  wholly  exterior  to  the  reflecting  medium.  The  direction 
and  intensity  of  this  portion  are  the  objects  of  inquiry  in  this  section  ;  the  physical  properties  acquired  by 
the  ray  in  the  act  of  reflexion  being  reserved  for  examination  at  a  more  advanced  period.  And  first,  with 
regard  to  the  direction  of  the  reflected  ray.  This  is  determined  by  the  following  laws : 

Laws  of  Reflexion. 

Law  1.  When  the  reflecting  surface  is  a  plane.  At  the  point  on  which  the  ray  is  incident  raise  a  perpendicular. 
The  reflected  ray  will  lie  in  the  same  plane  with  this  perpendicular,  and  with  the  incident  ray.  It  will  lie 
on  the  opposite  side  of  the  perpendicular,  and  will  make  an  angle  with  it  equal  to  that  made  by  the  in- 
cident ray. 

The  plane  in  which  the  perpendicular  to  any  surface  at  the  point  of  incidence,  and  the  incident  ray,  both 
lie,  is  called  the  plane  of  incidence. 

The  angle  included  between  the  incident  ray  and  the  perpendicular  is  called  the  angle  of  incidence. 

The  plane  in  which  the  reflected  ray  and  perpendicular  both  lie  is  called  the  plane  of  reflexion ;  and  the 
angle  included  between  the  perpendicular  and  reflected  ray  is,  in  like  manner,  termed  the  angle  of  re- 
flexion. 

Adopting  these  definitions,  the  law  of  reflexion  from  a  plane  surface  may  be  announced  by  saying,  that 
the  plane  of  reflexion  is  the  same  with  that  of  incidence,  and  the  angle  of  reflexion  equal  to  that  of  incidence, 
but  situated  on  the  contrary  side  of  the  perpendicular. 

Carol.  The  incident  and  reflected  rays  are  equally  inclined  to  the  surface  at  the  point  of  incidence. 

Law  2.  When  the  surface  is  a  curved  one,  the  course  of  a  ray  reflected  from  any  point  is  the  same  as  if 
it  were  reflected  at  the  same  point  from  a  plane,  a  tangent  to  the  curve  surface  at  that  point  ;  i.  e.  if  a 
perpendicular  be  raised  to  the  curve  surface  at  the  point  of  incidence,  the  reflected  ray  will  lie  in  the  plane 
of  incidence,  and  the  angle  of  reflexion  will  equal  that  of  incidence. 

The  demonstration  of  these  laws  is  a  matter  of  experiment.  If  we  admit  a  small  sunbeam  through  a 
hole  in  the  shutter  of  a  darkened  chamber,  and  receive  it  on  a  polished  surface  of  glass,  or  metal,  we  may 
easily  with  proper  instruments  measure  the  inclinations  of  the  incident  and  reflected  rays  to  the  surface, 
which  will  be  found  equal.  But  this  method  is  rude  and  coarse.  A  much  more  delicate  verification  of  this 
law  is  afforded  by  astronomical  observations.  It  is  the  practice  of  astronomers  to  observe  the  altitudes  of 
the  stars  above  the  horizon  by  direct  vision ;  and,  at  the  same  instant,  the  apparent  depression  below  the 
horizon  of  their  images  reflected  at  the  surface  of  Mercury,  (which  is  necessarily  exactly  horizontal,)  and  the 
depression  so  observed  is  always  found  precisely  equal  to  the  altitude,  whatever  the  latter  may  be,  whether  great 
or  small.  Now  as  these  observations,  when  made  with  large  instruments,  are  susceptible  of  almost  mathe- 
matical accuracy,  we  may  regard  the  law  of  reflexion,  or  plane  surfaces,  as  the  best  established  in  nature. 

Reflexion  at  a  curved  surface  may  be  considered  as  taking  place  at  that  infinitely  small  portion  of  the 
surface  which  is  common  to  it,  and  to  its  tangent  plane  at  the  point  of  incidence ;  so  that  if  a  perpendicular 
to  the  surface  be  erected  at  the  point  of  incidence,  the  incident  and  reflected  rays  will  make  equal  angles  with  it 
on  opposite  sides. 

Proposition.  To  find  the  direction  of  a  ray  of  light  after  reflexion  at  any  number  of  plane  surfaces,  given  in 
position. 

Construction.  Since  the  direction  of  the  ray  after  reflexion  is  the  same  whether  it  be  reflected  at  the  given 
surfaces,  or  at  surfaces  parallel  to  them,  conceive  surfaces  parallel  to  the  given  ones  to  pass  through  any 
point  C,  (fig.  9,)  and  from  C  draw  the  straight  lines  C  P,  C  P',  C  P'',  &c.  respectively  perpendicular  to  these 
respective  surfaces,  and  lying  wholly  exterior  to  the  reflecting  media.  Draw  S  C  parallel  to  the  ray  when 
incident  on  the  first  surface,  and  in  the  plane  S  C  P,  and  on  the  opposite  side  of  C  P,  from  the  incident  ray  S  C 
make  the  angle  PCs/=PCS,  then  will  C  </  be  the  direction  of  the  ray  after  reflexion  at  the  first  surface. 
Prolong  s'C  to  S',  then  S'C  will  represent  the  ray  at  the  moment  of  its  incidence  on  the  second  surface,  whose 
normal  is  C  P'.  Again,  make  the  angle  P'Cs"  in  the  plane  S'CP",  but  on  the  other  side  of  C  P',  equal  to 
the  angle  S'  C  P',  then  will  C  s1'  represent  the  ray  at  the  moment  of  its  reflexion  from  the  second  surface,  and, 
producing  s" C  to  S",  S"C  will  represent  it  at  the  moment  of  its  incidence  on  the  third  surfiice,  whose  normal 
is  C  P".  Similarly  in  the  plane  S"  C  P" ;  but  on  the  other  side  of  C  P"  make  the  angle  P"  C/"  =  P"  C  S", 
and  C  s'"  will  be  the  direction  of  the  ray  at  the  moment  of  its  quitting  the  third  surface,  and  so  on. 

Analysis.  About  C  as  a  centre  conceive  a  spherical  surface  described,  (fig.  10,)  then  will  the  plane 
P  S  s  intersect  it  in  a  great  circle  P  S  S'  p,  and  the  plane  in  which  C  P,  C  P"  lie,  or  the  plane  at  right  angles 
to  the  two  first  reflecting  planes  in  another  great  circle  PP'^,  and  the  planes  S'Cs"  and  S  Cs"  in  other  great 
circles  S'PV'and  Ska". 

Since  C  P  and  C  P'  are  given  directions,  the  angle  P  C  P',  or  the  arc  P  V  (which  is  equal  to  the  inclination 
of  the  two  first  surfaces  to  each  other)  is  given.  Call  this  I.  Again,  since  (he  direction  S  C  of  the  incident 
ray  is  given,  the  angle  of  incidence,  or  the  first  surface  P  C  S  (=  a)  and  the  angle  S  P  P7,  or  the  inclination  of 
the  plane  of  the  first  reflection  to  the  plane  P  P'  perpendicular  to  both  surfaces  (=  y/-)  are  given.  Hence  in 


LIGHT.  353 

Light,     the  spherical  triangle  P  F  S'  we  have  P  F  =  I ;  P  S'  =  1 80°  —  a, ;  and  the  angle  F  P  S'  =  ^- ;  consequently      Tart  I. 

— v~— '  S'F,  and  therefore  2  S'P'  =  SV  and  the  angle  S  S'  F  are  known,   as  also  the  angle  PFS',  and  therefore  >•—  v— 

its  supplement  PPV,  which  is  the  angle  made  by  the  second  reflexion  with  the  plane  P  F.     Again,  in  the 

spherical  triangle  SSV  we  have  given  S  S' =  180°  — 2a;  SV  =  2S'F   and   the  included   angle  S  S  V, 

whence  the  third  side  S  s"  may  be.  found,  which  is  the  angle  between  the  incident  and  twice  reflected  rays. 

Similarly,  if  a  third  reflexion  be  supposed,  we  have  given  P'  S"=  180°  —  S'F;  FP"  =  I',  and  the'anHe 
S"  F  P"  =  S'  F  P"  =  P  F  F'  -  P  F  S',  whence  we  may  compute  S"  P"  and  proceed  as  before,  and  so  on  to 
any  extent. 

Confining  ourselves  however  to  the  case  of  two  reflexions  we  have,  by  spherical  trigonometry,  putting  FS'=         99. 
a'  =  the  angle  of  incidence  on  the  second  reflecting  surface,  P  S'  P'  =  0 ;    P  F  S'  =  0,  and  180°  —  S  s"  =  D, 
the  deviation  of  the  ray  after  the  second  reflexion,  the  following  equations  : 

—  cos  a'  =  cos  a  .  cos  I  —  sin  o  .  sin  I  .  cos  •& 

BuUBUOm 


,  sin  I 

sin  9  =   —  -  ;-.  sin 


sin  0  = 


sin  «' 
sin  a 


sin  « 
cos  D  =  cos  2  a  .  cos  2  a!  —  sin  2  a  .  sin  2  a' .  cos  0 


of  reflexiou 
at  two 


(A)  Planes- 


From  these  equations,  any  three  of  the  seven  quantities  a,  a',  I,  0,  0,  \[r,  D  being  given,  the  other  four  may       JQQ 
be  found.     It  will  be  observed,  that  0  is  the  angle  between  the  plane  of  the  second  reflexion  and  the  principal  Values  of 
section  of  the  two  reflecting  planes,  and  0  the  angle  between  the  planes  of  the  first  and  second  reflexion.     If  0  the  symbols 
and  D  only  be  sought,  0  must  be  regarded  as  merely  an  auxiliary  angle  ;    but  this   may  not  be  the  case,  and 
cases  may  occur  in  which  0  alone  may  be  sought,  or  in  which  it  enters  as  a  given  quantity,  &c.      In  short, 
the  foregoing  equations  contain  in  themselves  all  the  conditions  which  can  arise  in  any  proposed  case  of  two 
reflexions. 

Carol.    If  Y*  =  o,  or  if  the  incident  ray  coincide  with  the  principal  section  PC  P,  i.  e.  if  the  two  reflexions        101 
both  take  place  in  the   plane  perpendicular  to  the  reflecting  surfaces,  these  formulae  take  a  very  simple  form, 
for  we  then  have 

0  =  o;  0  =  180°;  cos  «'  =  —  cos  (a  +  I) 

that  is  (a  +  a')  =  180°  —  I  ;  and  consequently  cos  (2  a  +  2  a')  =  cos  (360°  —  2  I)  =  cos  2  I,  or  2  a  +  2  a'  =  2  I. 
But  since  6  =  o,  we  have  by  the  last  of  the  equations  (A)  cos  D  =  cos  2  (a  +  «')  ;  consequently  D  =  2  a  +  2  a 
=  21.     That  is  to  say,  the  deviation  in  this  case,  after  two  reflexions,  is  equal  to  twice  the  inclination  of  the  „ 
reflecting  planes,  whatever  be  the  original  direction  of  the  ray.     This  elegant  property  is  the  foundation  of  the  bo"*  reflet™ 
common  sextant  and  of  the   reflecting  circle,  and  is  commonly  regarded   as  having  been  first  applied   to  the  ions   are  in 
measurement  of  angles  by  Hadley,  though  Newton    appears   also   to  have   proposed  it  for  the  same  object,  one  plane. 
See  the  explanation  of  these  instruments. 

In  other  cases,  however,  D,  the  deviation,  is  essentially  a  function  of  the  angles  expressing  the  position  of       ]02 
the  incident  ray,  and  can  only  be  obtained  from  the  equations  above  stated. 

Proposition.  Given  the  angles  of  incidence  on  the  two  planes,  and  the  angle  made  by  the  plane  of  the  first       103. 
reflexion  with  that  of  the  second  ;  required  the  positions  of  the  incident  and  twice  reflected  rays,  the  deviation 
of  the  ray  after  both  reflexions,  and  the  angle  included  between  the  reflecting  surfaces. 

Retaining  the  same  notation,  we  have  given,  a,  a'  0,  required  I,  D,  and  0,  y^. 

1st,  D  is  given  at  once,  by  the  last  of  the  general  equations,  (A.) 

2ndly,  To  find  the  rest,  put  x  =  sin  I  ;   y  =  sin  y-  ;  and  a  =  sin  o'  .  sin  0  ;  put  also  cos  a  =  c  ;  sin  a  =  s  ; 

cos  a'  =  cf  ;  sin  a'  =  /.     We  have  then  xy  =  a,  or  y  =  -  —  ;  and  the  first  of  the  equations  (A)  then  gives 


'    4 


-  <  —  c      i  _  x*  —  s      ^i  —  Oa 
which,  cleared  of  radicals  and  reduced,  gives 

o  =  x'+  x*{2J1(c*      s2)  -2c2  -2s2  a*}  +  (</2  -  c2)5  +  2  a2  s*  (c'2  4  c2)  +  a'  s 

and    this  equation,  which,   though  biquadratic,  is  of  a  quadratic  form,  contains   the   general  solution  of  the 
problem. 

Carol.  1.  If   0  =  90°,  or  if  the  planes  of  the  first  and  second  reflexions  be  at  right  angles  to  each   other,       JQ^ 

we  have  simply  sin  I  .  sin  •<!>•  =  sin  a'          and  a  =  sin  of  =  s'.  ?ase   when 

the  planes 
In  this  case  our  final  equation  becomes  of  the  two 


0  =  ,    - 


<which,  being  a  complete  square,  gives  x*  =  1  —  c2  </'. 

Now  i  =  sin  I,  therefore  *8  =  1  —  cos  I2,  consequently  we  have  the  following  simple  result, 


COS  I  (=  c  cO  =  COS  n  .  COS  a'. 
VOL.  IV,  3  A 


351  LIGHT. 

Light.      Or  the  cosine  of  the  Inclination  of  the   planes  to  each  other  is  equal  to  the  product  of   the  cosines  of   the      I'art  1. 
-  ^—  ^  angles  of  incidence  on   each.     And,  vice  versd,  if  this  relation  holds  good,  the  planes  of  the  two  reflexions  will  "——  v~" 
necessarily  be  at  right  angles  to  each  other  ;  for,  this  relation  being  supposed,  we  have  of  course  of  =  1  —  c*  c", 
and  therefore  1  —  e*  c"  being  put  for  x"  in  the  general  equation,  the  whole  must  vanish  ;   now  this  substitution 
gives  a  biquadratic  of  a  quadratic  form  for  determining  a,  which  must  evidently  be  satisfied  by  taking 

a  =  sin  a,  and  consequently  0  =  90°. 

This  elegant  property  will  be  useful  when  we  come  to  treat  of  the  polarization  of  light. 
in=S  Carol.  -2.  In  the  same  case  if  6  —  90,  the  deviation  D  is  given  by  the  equation 

cos  D  =  cos  2  a  .  cos  2  a', 

or,  the  cosine  of  the  deviation  is  equal  to  the  product  of  the  cosines  of  the  doubles  of  the  angles  of 
incidence. 

106.  Problem.   A  ray  of  light  is  reflected  from  each  of  two  planes  in  such  a  manner  that  all  the  angles  of  inci- 
dence and  reflexion  are  equal.     Given  the  inclination  of   the  planes,  and  the  angles  of  incidence  ;    required, 
first,  the  deviation  ;  secondly,  the  inclination  of  the  planes  of  the  first  and  second  reflexion  to  each  other,  and 
the  angles  made  by  each  of  these  planes  with  the  principal  section  of  the  reflecting  planes. 

Preserving  the  same  notation  we  have  a  =  a',  and  therefore  by  the  third  of  the  equations  (A)  ^  =  0,  so 
that  these  equations  become 

cos  a  (I  +  cos  I)  =  sin  a  .  sin  I  .  cos  ^-^ 

sin  a  .  sin  6  =  sin  I  .  sin  ty  \          (a) 

cos  D  =  (cos  2  a)*  —  (sin  2  a)8  .  cos  0 

107.  The  first  of  these  gives  (putting  for  1  +  cos  I  its  value  2  (cos  —  j    and  for  sin  I  its  equal  2  .  sin  —  •  .  cos  —  ) 

cos  Y^  =  cotan  a  .  cotan  —  ,  (6) 

whence  ^  is  immediately  known.     Hence  ty  is  had  by  the  equation 

sin  I 

sin  0  =   --  .  sin  \lr.  (c) 

sin  «. 

Lastly,  if  we   subtract  each  member  of  the  third  of  the  equations  (a)  from  1,  divide   both  sides  by  2,  and 
reduce,  we  transform  it  into  the  following 

D  0 

sin  —  =  sin  2  a  .  cos    —  .  (rf) 

These  equations  afford    ready  and   direct  means  of  computing  ^,  0,  and  D   in  succession,  from  the  known 
values  of  a  and  I  ;  the  formulae  are  adapted  to  logarithmic  evaluation,  and  are  in  themselves  not  inelegant. 

§  IV.    Of  Reflexion  from  Curved  Surfaces. 

108  '^''e  reflex'on  °f    a  ray  from  a  curved    surface  is  performed  as  if  it  took    place   at    a   reflecting   plane,  a 

tangent  to  the  point  of  incidence.  The  reflected  ray  will  therefore  lie  in  the  plane  which  contains  the 
incident  ray  and  the  normal  or  perpendicular  at  the  point  of  incidence.  The  general  expressions  for  the 
course  of  the  ray  after  reflexion  at  surfaces  of  double  curvature  being  considerably  complex,  and  not  likely 
to  be  of  great  service  to  us  in  the  sequel,  we  shall  confine  ourselves  to  the  particular  case  of  a  surface  of 
revolution  (comprehending  the  cases  of  a  plane,  and  conoidal  surfaces  of  all  kinds)  where  the  plane  of 
incidence  is  supposed  to  pass  through  the  axis  of  revolution. 

109.  Proposition.  -A  ray  being  incident  on  any  surface  of  revolution  in  a  plane  passing  through  the  axis,  to  find 

General  in-  the  direction  of  the  reflected  ray. 

>estigation         Qp  (n<r.  H)  being  a  section  of  the  surface  by  the  plane  of  incidence,  QN  the  axis,  QP  the  incident,  and 
ofthecourse  pr  tng  reflecte{j  raVi  which    produced  if  necessary  cuts  the  axis  in  q.      Draw  the  tangent  PT,  the  ordinate 
"  .  ancl  tne  normal  PN,  which  produce  to  O,  and  put  as  follows, 


any  nine  , 

*  =  Q  M  ;  y  =  MP;  p=  -^-  ;  0  =  the  angle  M  Q  P, 

nr  the  angle  made  by  the  incident  ray  with  the  axis  ;  then,  since  the  angle  of  reflexion  is  equal  to  that  of  inci- 
dence, we  have  /rPO  =  OPQ,  and  therefore  N  P  q  —  O  P  Q  ;  consequently  QPT  =  TPo.  Now  Qn  = 
QM-M</=QM  —  P  M  .  tan  M  P  9 


L  I  G  H  T. 

Light.  =_  x  —  y  .  tan  {  T  P  M  —  T  P  q  } 

=  i  -  y  .  tan  {TPM  -TPQ} 

=  *  -y  .  tan  {TPM-PTM  +  PQM} 

=  x  -  y  .Urn  {90°-iiPTM+PQM} 

d  y 
But  by  the  theory  of  curves  we  have  tan  PTM  =  — —  =  p,  consequently  PTM  =  arc  tan  p  =  tail  ~"1  />, 

CL  X 

denoting  by  tan~~  '  the  inverse  function  of  that  expressed  by  tan;  and  since  P  Q  M  =  0,  this  expression  becomes 
Q  q  =  x  —  y  .  cotan  {  2  .  tan  ~  '  p  —  0  } 

dy 


/  d  y  \  /  11  \ 

—  x  —  y  .  cotan  {  2  .  tan      '  I              1  —  tan  '  I  — -  1  ^a} 

\  d  x  /  \  x  / 

PM          y 


(' 
\ 


Because  tan  0  = 


QM 

This  then  is  the  general  expression  for  the  distance  between  the  points  in  which  the  incident  and  reflected  rays 
cut  the  axis. 

Now,  by  Trigonometry,  we  have  (A  and  B  being  any  two  quantities) 

{2  A  ) 

tan"' —  —  tan"1  B  [ 


.   /2A-  (1  -  A2) 
=  «*"•*"-'   ((1_A.)  +  ,A' 

that  is,  since   cotan  .  tan  ~ '  0  =  —  .  the  cotangent  and  tangent  being  reciprocals  of  each  other,  simply 

1  -  A" +  2  AB 
2A—  (1  -  Aa)  B 

d  y  y 

Applying  this  to  the  present  case,  A  =  — —  =  p  ;  B  =    — ,  and  therefore  the  expression  above  found  for  Q  q 

becomes 

(1  —  p*)  x  +  Zpy      -\ 

Q<1  ~         ~y  '      2PX-(\   -  p*)y~      I  -General  ex- 

>  (6)  pressions 

_  „        (J  +  Py)   (px-y)  for  the 

2  p  x  -  (  1  _  pi)  y      )  distance  of 

the  focus 

These  expressions  contain  the  whole  theory  of  the  foci  and  aberrations  of  reflecting  surfaces. 

Carol.  1.  To  find  the  angle  made  by  the  reflected  ray  with  the  axis,  which  we  will  call  ff.  poinTq 

This  is  the  angle  P  q  M,  which  is  the  complement  of  M  P  q.     Now  we  have  found  above  110 

M  P  q  =  90°  —  2  tan  - '  p  +  6.  An£le  maae 

by  the  re- 

Hence  0'  =  2  .  tan  - '  p  —  0  flected  ™v 

and  the  axis 

But  tan  0  =  —  ,  so  that  substituting  we  have 

2  p  x  —  (1  —  pj)v 

tan  ff  =  — i '     '  y    •  (C) 

(l-p*)jr  +  *py    ' 

Carol.  -2.  A9  =  a'  =  a  +  2     <*  + 


•2px  -(1  -  p*~)y  '  111. 

In  all  the  foregoing  formula;  we  have  supposed  the  origin  of  the  x  placed  at  Q  the  radiant  point.  If  we  112 
would  place  it  elsewhere,  as  at  A,  we  have  only  to  write  x  —  a,  for  x  throughout.  The  formulae  then  become  Formula- 
on  this  hypothesis,  when  the  ra- 

y  diant  point 

{tan  0  =  •  (e)  is  not  in  the 

origin  of  the 
tany-     2y(*-«)-(l-P*)y  coordinates 


^        a,  _.  2(J  +  py)  (px-y)  i-  { (i  —  p*)y  —  "r*  /  ^  . 

2  px  —  (1  —  p*')  y  ~  '2  pa 

3x2 


356  L  I  G  H  T. 

Light.           If  the  incident  ray  be  parallel  to  the  axis,  we  have  only  to  suppose  the  point  Q  infinitely  distant;  or  placing.      Part  I 
•——v—*'  as  in  the  last  article,  the  origin  of  the  x  at  a  point  A  at   a  finite  distance,  to  make  a  (=  AQ)  infinite.     The  '— -v— •» 
Formulas      ab°ve  expressions  then  give  Q  q  =  co 
when  the  in- 
cident rays  .          2  p 
ire  parallel  .   _      t 


tan  0'  = 
the  axis. 

1   - 


A  q     =  x  —  y . 


1 14.  Proposition.   To  represent  the  incident  and  reflected  rays  by  their  equations. 

The  equation  of  any  straight  line  is  necessarily  of  the  form  Y  =  a  X  +  /3.  Suppose  we  take  A  for  the 
common  origin  of  the  coordinates,  and,  retaining  the  foregoing  notation,  representing  by  T  and  y  the  coor- 
dinates of  the  point  P  in  the  curve,  let  X  and  Y  represent  those  of  any  point  in  the  incident  ray ;  and,  Q 
being  the  point  in  which  that  ray  cuts  the  axis,  and  A  Q  =  a,  it  is  evident,  first,  that  when  X  =  a,  Y  =  o  ; 
and  secondly,  since  the  ray  passes  through  P,  when  X  =  x,  Y  =  y.  Hence  we  have 

o  =  a  a  +  /3,         and         y  =  a  x  -(-  ft, 

y  ffy 

whence  we  get  a  =  — - —       ft  =  —  - — - —  ;       (1) 

x  —  a  i  —  a 

therefore,  the  equation  of  the  incident  ray  is 
or  which  is  the  same  in  a  different  form, 

y 

•r  —  a 

PM  y 

or,  since  tan  0  = 


M  Q  x  —  a 

Y=  (X-c).  tan<?;          (4) 

or,  again,  Y  —  u  =  (X  —  x)  .  tan  6.         (5) 

Similarly  for  the  reflected  ray,  it  is  obvious  that  if  we  represent  its  equation  by  Y  =  <J  X  +  /3',  we  shall  have 


;         (6) 


-       -,  -  __ 

x  —  a  x  —  a! 

and  consequently 

Y  =    (—M—\  .  (X  -  a')   =   (X  -  a')  .  tan  tf  ;       (7) 

*~~ 


Y-y=    —  ?—,   .  (X-j)   =  (X-  *).tan<X;       (8) 
x  —  a 

will  be  the  corresponding  forms  of  the  equation  of  the  reflected  ray,  in  which  a'  and  tan  ff  are  given  in  terms 

d  y 
of  x,  y,  a,,  and  p  =   —  —  by  the  equations  (g)  and  (A)  or  (i). 

(I   X 

1  1\  If  the  whole  figure  (fig.  11)  be  turned  about  the  axis  A  M,  and  Q  be  supposed  a  radiant  point,  the  rays  in  the 

Fig.  11.       whole  conical  surface  generated  by  the  revolution  of  QP  will  be  concentred  after  reflexion  in  one   and   tl.e 

same  point  q,  which  will  thus  become  infinitely  more   illuminated  than  by  any  single  ray  from  an  elementary 

molecule  of  the  surface.     The  point  P  will  generate  an  annulus,  having  M  P  for  its  radius  ;  and  q  is  called  the 
Focus.         focus  of  this  annulus,  and   the  distance  A  q  the  focal  distance  of  the  same  annulus.     This  last  expression  is 

commonly  understood   to  mean  the  distance  of  q  from   the  vertex,  or  point  where  the  curve  meets  the  axis, 

but  we  shall  use  it  at  present  in  the  more  general  sense. 
1  15  Generally  speaking,  then,  the  focus  varies  as  the  point  P  in  the   reflecting  annulus  varies,  unless  in  that 

particular  case    where,  by  the  nature  of  the  curve,  the  function  expressing  a    is  constant.     Let  us  examine 
117.      this  case. 

Investiga  Proposition  .  To  find  the  curve  which  will  have  the  same  focus  for  every  point  in  its  surface  of  revolution,  or 
tion  of  the  on  which  rays  diverging  from  or  converging  to  any  point  Q,  being  incident,  shall  all  after  reflexion  converge 
which  re  *°  or  Diverge  from  one  point  q. 

Sect  all  the      The  value  of  Q  q  assigned  in  Art.  109,  E  q,  (6)  being  made  constant,  affords  the  equation 
incident 


Zpx—  (1  —  p5)  y 


=  constant  =  c. 


LIGHT 


357 


Light.          This  equation,  cleared  of  fractions,  and  putting  x  for  x  —  c,  (which  is  merely  shifting  the  origin  ?f  the  co-     part  I. 
•— v-—''  ordinates  to  the  distance  c  from  their  former  origin)  becomes  ._,_      

p{x*-y°--c*}  =  (1-p*)  xy.        (a) 

To  integrate  this  equation,  assume  a  new  variable  z,  such  that  p  y  =  x  z,  and  (multiplying  the  original  equation 
by  y)  we  have  py  (x'-  —  y*  —  c2)  =  xy'1  —  x.  p*  y3, 

that  is  xz  (x2  —  y"1  —  c'2)  =  xy*  —  x3  z*, 

z  x2  —  zc*  +  z2  x2  z 


whence  we  find  y*  = 

Differentiating  this  equation  we  get 


1  -t-z 


=  x*  z 


1 


Zy  d  y  {  =  2  py  dx  =  zx  zdx 


because  p  =    -^-) 
d  x/ 


=  2xzdx 


that  is 


or 


—  ca  d 


1  +z 

f>Z  \ 

— \dz  =  o. 

(1  +  *r  j 


(6) 


This  equation  may  obviously  be  satisfied  in  two  ways ;  the  first  is,  by  putting  the  factor 

c 


x*  — 


=  o,    or  x  =  + 


(!+«)« 

which  gives  (restoring  the  value  of  z,  z  =  \  merely  x  +  py  =  c;  and,  eliminating  p  between  this  and 

x     / 

the  original  equation  (a)  we  find,  on  reduction, 

y*  +  (x  -  c)2  =  o. 

This  is,  however,  (as  is  clear  from  the  way  in  which  it  has  been  obtained,)  only  a  singular  solution  of  the 
differential  equation,  (see  DIFFERENTIAL  CALCULUS,  singular  solutions ;)  and  as  the  value  of  y  which  results 
from  it  is  always  imaginary,  it  affords  no  curve  satisfying  the  conditions  of  the  problem. 

The  other  way  in  which  the  equation  (6)  can  be  satisfied,  is  by  putting  d  z  =  o,  or  z  •=•  constant.     Let  The  curve  a 

in  all  cases 

this  constant  be  represented  by  —  h ;  then,  since  z  = 


-,  we  have 


a  conic 
section. 


py 
x 


ydy 

• 
x  d  x 


which,  integrated,  gives 


y*  —  h  (  a'2  —  x2), 


a  being  another  constant.  This  is  the  general  equation  to  the  conic  sections,  and  it  is  obvious,  from  the 
properties  of  these  curves,  that  they  satisfy  the  conditions  ;  because  two  lines  drawn  from  their  foci  to  any 
point  in  their  periphery  make  equal  angles  with  the  tangent  at  that  point,  and,  consequently,  a  ray  proceeding 
from,  or  converging  to,  one  focus,  and  reflected  at  the  curve,  must  necessarily  take  a  direction  to  or  from 
the  other.  But,  the  foregoing  analysis  being  direct,  shows  that  they  possess  this  property  in  common  with 
no  other  curves. 

Thus  in  the  case  of  the  ellipse,  all  rays,  (fig.  12,  )    S  P,  S  P',   &c.  diverging  from  the  focus    S  will  after       ng. 
reflexion  converge  to  the  other  focus  H,  the  interior  surface  of  the  ellipse  being  polished  ;  and  all  rays  Q  P,  Ellipse. 
Q  P',  &c.  converging  to  S,  will  after  reflexion  diverge  from  H.  Fi£-  I2- 

In  the  hyperbola,  (fig.  13,)  rays  Q  P,  Q'  P,  &c.  converging  to  one  focus  S,  and  incident  on  the  polished  Fig.  13. 
convex  surface  of  the  curve,  will  after  reflexion  converge  to  the  other  focus  H  ;  and  if  diverging  from  S,  1  19 
and  reflected  on  the  polished  concave  surface  P  P',  will  after  reflexion  diverge  from  H.  Hyperbola. 

In  the  case  of  the  parabola,  rays  parallel  to  the  axis,  incident  on  the  interior  or  concave  surface,  will  all  be  120. 
reflected  to  the  focus  S,  fig.  14;  and  if  reflected  at  the  exterior  or  convex  surfaces,  will  all  after  reflexion  diverge  Parabola. 
from  S.  Fig.  14. 

Rays  converging  to,  or  diverging  from,  the  centre  of  a  sphere  will  all  after  reflexion  diverge  from,  or  \-)\ 
converge  to,  the  same  centre.  Circle. 

Let  .us  now  apply  our  general  formula  (6)  (Art.  109)  to  some  particular  cases. 


358  L  I  G  H  T. 

Light.          Pmpntitinn.  Let  the  reflecting  surface  be  a  plane,  or  the  curve  PC  a  straight  line.     Required  the  focus  of     Part  I. 
v»v»*'  reflected  rays.  s_^v^-« 

122.  dy 

Focus  of  a        Here  we  have  x  =  constant  =  a  p  —  —  —  —  =   oc  ,  and  the  general  formula  becomes  simply 

plane  sur- 

face. 2  *  V 

Qq  =  a'  =  •  -  ?—  =  2  T  =  2  a. 

y 

So  that  the  focus  of  reflected  rays  is  a  point  on  the  opposite  side  of  the  reflecting  plane  equally  distant  from 
it  with  the  radiant  point  ;  and  as  this  is  independent  of  y,  or  of  the  situation  of  the  point  P,  we  see  that  all 
the  rays  after  reflexion  diverge  from  this  point,  see  fig.  15. 

123.  Proposition.  To  find  the  focus  of  any  annulus  of  a  spherical  reflector. 

Focus  of  a        Let  r  be  the  radius  of  the  sphere,  and,  if  we  fix  the  origin  of   the  coordinates  at  the  radiant    point,  t'.ie 

spheric?.!       equation  of  the  generating  circle  will  be 

annului. 

r*=  O-  a)"-  +  i/* 
This,  differentiated,  gives  (x  —  a)  d  x  +  ;/  d  y  =  o, 

d  y  x  —  a  2  ys  —  r°- 

consequently  p  =  —r^—    =  --  ;    1  —  p"  =  —=—  -  . 

dx  y  y2 

Hence,  substituting  in  the  general  expression  (6),  we  find  for  the  focal  distance  the  following  value, 

2  a  {  r*  +  a  (x  -  a)  } 
Qq=         r.48a(g-«)        '  (a) 

which  expresses  in  all  cases  the  distance  of  the  focus  of  reflected  rays  from  the  radiant  point. 

For  optical  purposes,  however,  it  is  more  convenient  to  know  its  distance  from  the  centre,  or  from  the 
surface. 

The  distance  from  the  centre  (E  q,  fig.  16,)  is 

=Qq-QE=     2«(«*-««  +  r«)        _a 

2  a  x  +  r*-  2  a* 


in  which  positive  values  of  E  q  lie  to  the  right  of  E,  or  the  same  way  with  those  of  x  or  of  Q  q. 
Focus  for          Carol.  1    If  we  would  find  the  focus  of  the  infinitely  small  annulus  immediately  adjoining  to  the  vertex  C, 
central  rays  or  C'  of  the  reflecting  spherical   surface,  or,   as  it  is  termed  in  Optics,  the  focus  of  central  rays,  we  must  put 
in  a  sphe-    jn  the  case  of  the  vertex  C  (when  the  reflexion  takes  place  on  a  concave  surface)  x  =  a  +  r,  and  in  the  other 

case,  viz.  that  where  the  rays  are  reflected  on  the  convex  surface  C',  x  =  a  —  r.     The  former  gives 


2  a  +  r  \  2  a  +  r 

the  latter  gives  the  same  results,  writing  only  —  r  for  r. 
124.  If  we  bisect  the  radii  C  E  and  C'  E  in  P  and  F",  and   suppose  q  and  of  to  be  the  foci  of  central  rays  reflected 


at  C  and  at  C',  we  shall  have  F  q  =  £  r  —  —  -  (d) 

&a  +  r  r 

a   +    -- 

which  gives  the  following  useful  analogy, 

Q  F  :  F  E  :  :  E  F  :  F  q.  (R) 

Similarly  we  have  QF*  :  F'E  :  :  E  F'  :  F'  q  ;  so  that  the  same  analogy  applies  to  both  cases,  and  may  be 
regarded  as  the  fundamental  proposition  in  the  theory  of  the  foci  for  central  rays.  For  it  is  obvious,  that  if  PC 
were  any  other  eurve  than  a  circle,  the  same  must  hold  good,  taking  only  E  the  centre  of  curvature  at  the 

vertex. 

125  Carol.  2.  If  a  be  infinite,  or  the  incident  rays  be  parallel,  we  have  F  q  =  o,  which   shows  that  the  fonts  of 

Principal      central  parallel  rays  bisects  the  radius.     This  focus,  for  distinction's  sake,  is  called  the  principal  J'OCM  of  the 
focus.  reflector. 

126.  Definition.     Q  and  q  are  termed  conjugate  foci.     It  is  evident  that  if  q  be  made  the  radiant  point,  Q  w  11 

Conjugate     be  its  focus  ;   for  the  rays  will  pursue  the  same  course  backwards. 

foci.  Carol.  3.  Regarding  only  central  ra\s:    the  conjugate  foci   move  in  opposite  directions,  and  coincide  at  the 

1  27'      centre  and  surface  of  the  reflector. 

For  let  a  vary  from  x  to  —  o>  ,  then  Fq  will  vary  as  follows  :  first,  while  a  varies  from  x    to,  --  ,  F,  q  is 


LIGHT.  359 

Light,      positive,  and  increases  from  o  to   co  ;  that  is,  as  Q  moves  up  to  F,  q  moves   through  C  to  infinity.     As  the      Part  I. 

motion  of  q  continues,  Fq  then  becomes  negative;  because  a  is  then  negative  and  greater  than  -— ,  and  a   in-  Conjugate 

foci  move 

creasing  Fq  diminishes  ;    therefore  q  moves  from  the  right  towards  F,  that  is  in  the  opposite  direction  to  Q's  '"  °PP°site 
motion ;   and  when  Q  is  at  an  infinite  distance  to  the  right,  q  is  again  at  F. 

When  Q  comes  to  E,  a  =  o  .  F  q  =  — ,  or  q  is  at  E  also. 
When  Q  comes  to  C,  a  =  —  r,  F  q  =  —  — -,  or  q  is  at  C  also. 

It  appears  by  the  value  of   E  9,  Equation  (6),  that  a  spherical    reflector  A  C  B,  fig.  17,  whose  chord  (or      128. 
aperture,  as  it  is  termed  in  Optics)  is  A  B,  causes  the  ray  reflected  at  its  exterior  annulus  A  to  converge  to,  or  Longitudi- 
di verge  from,  a  point  q,  different  from  the  focus  of  central  rays.     Let  f  be  this  latter  focus,  then  we  shall  have  nal  aberra- 
tion, for  an) 
"•  r  (a  +  r)  r  ar* a  r  aperture. 


2  a  +  r    ' 


2a(z  -  a)  +  r2  2  a  +  r 


This  quantity  fq  is  called  the  longitudinal  aberration  of  the  spherical  reflector.     If  the  rays  fall  on  the  convex 
portion,  we  need  only  write  —  r  for  r. 

Proposition.  To  express  approximately  the  longitudinal  aberration  of  a  spherical  reflector  whose  aperture  is      129. 
inconsiderable  with  respect  to  its  focal  length.  Longimdi- 

, «2  nal  aberrn- 

y  being  the  semi-aperture,  and  x  —  a  being  equal  to   *  r2  —  y  *  --  r         — —  ,  (neglecting  y4,  and  higher  tion  for 

"  r  small 

powers  of  y,)  we  have  apertures. 


fq  =  aberration   = 


2  a  r  4-  r4  - 


2  a  +r         r(2a  +  r)4' 


If  we  put  Cf  =;  f,  we  have  f=  —  — •,  and,  consequently,   we  may  eliminate  a,  the  distance  of  the      139. 

Another 
radiant  point,  and  express  the  aberration  in  terms  of  the  aperture,  radius  of  curvature,  and  distance  of  the  focus  ^P"*81011 

of  central  rays  from  C,  the  vertex  of  the  minor ;  for  this  gives  a  =    — j*j — ,  which,  substituted  for  a  in 

the  expression  (f)  gives 

aberration  =  — 


*.«*  E/2.  (semi-aperture)2 

-  P 


r3 

To  express  the  lateral  aberration,  or  the  quantity  by  which  the  reflected  ray  A  qg  deviates  from  the  axis,  at      131. 
the  focus  of  central  rays,  or  the  value  of  fg,  (fig.  17,)  we  have  Lateral 

aberration 


2  a  (x  —  a)  2  +  r*  (x  —  2  a) 

—~  ;  so  that 


2  a  (T  —  a)  + 


a  —  x  +  r 


When  the  aperture  is  very  small,  this  becomes  simply 

fff  = 


2  a(x  —  a)1 


(h) 


r«  .  (r  +  o)  (r  +  2  a) 
When  a  is  infinite,  or  the  incident  rays  are  parallel,  we  have  the  following, 

fq  =  longitudinal  aberration   =   — 


fg  =  lateral  aberration 


y3 


132. 

Lateral 
aberration 
for  smalt 
apertures. 

133. 

Aberrations 
for  paralle1 
rays  and 
small 
apertures. 


If  the  rays  fall  on  the  convex  side  of  the  sphere  we  must  make  r  negative,  which  only  changes  the  signs 
the  aberrations. 


of 


360  L  I  G  H  T. 


§  V.  Of  Caustics  by  Reflexion,  or  Catacaustia. 

134  If  rays  of  light  be  incident  on  a  medium  of  any  other  form  than  that  of  a  conic  Section,  having  the  radiant 

point  in  the  focus,  they  will  after  reflexion  no  longer  converge  to  or  diverge  from  any  one  point,  but  will  be 

dispersed  according  to  a  law  depending  on  the  nature  of  the  reflecting  curve  ;   the  inclination  of  each  reflected 

ray  to  the  axis  varying  according  to  the  point  of  the  curve  from  which  it  is  reflected,  and  not  being  the  same 

for  any  two  consecutive  rays.     Of  course  each  lay  will  intersect  that  immediately  consecutive  to  it  in  some  point 

or  other,  and  the  locus  of  these  points  of  continual  intersection  will  trace  out  a  curve  to  which  the  reflected  rays 

Caustics  by  will   all  necessarily  be  tangents,  and  which   is  called  a  caustic.     If  these  rays  fall  on  another  reflecting  curve, 

reflexion      they  WJH  De  again  dispersed,  and  another  caustic  will  originate  in  the  continual  intersections  of  the  consecutive 

rays  of  the  former,  and  so  on  to  infinity. 

135.  Let  Q  P,  Q'  P',  (fig.  1  8,)  be  any  two  contiguous  rays  incident  on  consecutive  points  P,  P'  of  a  reflecting  curve 
Fig.  18.       PP',  and  after  reflexion  let  them  pursue  the  courses  PR,  P'R';  and  since  they  are  not  necessarily  parallel, 

let  Y  be  their  point  of  intersection,  then  will  Y  be  the  point  in  the  caustic  Y  Y'  Y''  corresponding  to  the 
point  P  in  the  reflecting  curve  ;  and  if  we  determine  the  points  Y'  Y",  &c.  from  the  consecutive  points  P'  P'',  &c. 
in  the  same  manner,  the  locus  of  these,  or  the  curve  Y  Y'  ~Y"  will  be  the  whole  caustic. 

136.  Since  the  reflected  ray  passes  through  P,  whose  coordinates  are  xy,  its  equation,  as  we  have  already  seen 
Coordinates  (Art.  114),  is  necessarily  of  the  form 

of  the  -tT  -r,  fv          * 

caustic  in-  Y  -  y  =  P  (X  -  *) 

otfany  sup-  If  we  regard  x,  y,  P  as  variable,  this  will  represent  any  one  of  the  reflected  rays  P  R,  and  the  consecutive  ray 
position  of  P'  R'  will  be  represented  by 

Y  -  (3,  +  rf  y)  =  (P  +  d  P)   (X  -  (*  +  <*  *)  ) 


Now  since  the  point  Y  in  which  these  two  rays  intersect  is  common  to  both,  the  coordinates  X  and  Y  at  this 
point  are  the  same  for  both  ;  and  therefore  at  this  point  both  these  equations  coexist,  and  thereby  determine  the 
values  of  X  and  Y,  or  the  situation  of  the  point  Y.  Now  the  latter  of  these  equations  is  nothing  more  than 
the  former  plus  its  differential,  on  the  supposition  of  X  and  Y  remaining  constant.  Therefore,  we  have  to  find 
X  and  Y  from  the  two  equations, 

-dy  =  (X-;r)dP-Pd,r, 
which  gives  at  once 


In  these  equations  we  have  only  to  substitute  for  P  its  value  =  tan  &,  or  — ^—^ — — - r ^ j"  ;   and 

(1  —  p*)  (x  -a)  +  Zpy 

after  executing  all  the  differentiations  indicated,  or  implied,  to  eliminate  x  and  y  by  the  equations  of  the  curve 
and  the  other  conditions  to  which  the  quantity  a  may  be  subjected,  an  equation  between  X  and  Y  will 
result  which  will  be  the  equation  of  the  caustic. 

137.          Proposition.    To  determine  the  caustic  when  rays    diverge   from  one  fixed  point  in  the  axis  of  a  pven 
Caustic        reflecting  curve. 

when  rays        jn  tn;s  case  a  js  invariable,  and  the  differentiation  of  P  must  be  performed  on  this  hypothesis.     It  will, 

>m  therefore,  simplify  the  question  if  we  put  a  =  o ;  or  suppose  the  origin  of  the  coordinates  in  the  radiant  point, 

point.          'n  which  case 

p_    2px-  (\- 


d  P      _  (1  +  p*)  (y-px)+  2  q 


dx 

dp 
Where  <?  =  -/- 

(A    X 

(1  +  p*)  (px-y) 
-* 


Light.       which  substituted,  we  find 


=  2. 


LIGHT. 

p  (p  x  -  yy  -  q  x  (x*  +  y«) 

+p*)  (px-y)-2q(x*  + 

(px-yy  +  gy  (x*  +  y"-) 


36! 


Part  I. 


(m) 


Carol.  1.  If  the  incident  rays  be  parallel,  or  the  radiant  point  at  an  infinite  distance,  we  may  fix  the  origin       138. 
of  the  coordinates  where  we  please ;  and  since  in  this  case  the  equation  of  any  reflected  ray  is,  by  1 13  equation  Caustic  for 
(/)  and  114  equation  (8),  Parallel 


we  have 


P  = 


2p 


Y  -  y  =  (X  -  x)  . 

P(l  +P 


i  - 


putting  q  for 


dp 
d  x 


dx 


d'1  y 
~~ 


•,8) 

n  —  -!>-}•* 


These  substitutions  made,  we  get  the  following  values  for  the  coordinates  of  the  caustic, 

P  Pa 

X=x  +  ~    (1  —  ps);    Y  =  yr    -J-— .  (n) 


Carol.  2.  In  the  general  case,  if  we  put  /  =  the  line  Py,  or  the  distance  between  the  point  in  the  curve       j3g 


and  the  corresponding  point  in  the  caustic  we  have 


/=  V  (X-  jr)«  +  (Y-y)« 
Which,  if  we  write  for  X  —  x  and  Y  —  y,  their  values  above  found  become 

-3-TTF     P-J> 


Distance 
between 
correspond- 
ing points 
in  curve 
and  caustic 


/= 


or,  writing  for  P  its  value,  and  executing  the  operations, 

f_       -  (y—  px}  (i  -f- 

J  — 


+  y 


(y-px) 
Carol.  3.  In  the  case  of  parallel  rays,  when 


140. 


dx 


d  - 


.p'-L.  P_.    =   P(] 


1  - 


;     V  1  +  P«  = 


1  -  o" 


we  have 


f= 

*    ' 


Carol.  4.  Call  c  the  chord  of  the  circle  of  curvature  passing  through  the  origin  of  the  coordinates,  or  through       141 
the  radiant  point :    then,  by  the  theory  of  curves, 


so  that 


+  y«)  = 


and  substituting  this  for  q  (x*  +  y8),  in  the  general  expression  for  /,  we  eliminate  q,  and  get 

-  _          c  "Si*  +  yj  r  c 


4  AV 


putting 
Hence  we  have 


T  y    —  c 


4  r  —  c 


—  1  <" 


(r) 


which  gives 

Hence  the  following  general  property.     (Smith's  Optics,  ed.   1738,  p.  160.)  H* 

r  ?  SSSr^i*^  c,onJuSate  foci  of  an  elementary  pencil  of  rays  reflected  at  anv  curve  surface  at  P,  fig.  19.  fig.  19 
Let  V  P  W  be  the  circle  of  curvature ;    (if  the  curve  be  a  circle,  this  will  be  the  curve  itself.)     Let  the  chords 

VOL.  IV.  Q  „  ' 


369  LIGHT. 

Light.      PV,  PWin  the  direction  of  the  incident  and  reflected  rays  be  divided  in  F, /,  so  that  PF  and  Pf  shall  each     P"''  I- 
— v— — '  be  one  quarter  of  the  whole  chords,  and  the  relation  between  Q  and  q  will  be  expressed  by  the  proportion  ^— V* 

neral 


General 
[elation 

between 


elation  QF  •  FP  :  :  P/:  /,. 

J       J    I 


;  or 


'.  5.  Putting  •  d  x  =  M,  we  have 


----  o  _  _  u>  M/    —    -"*i    »T  v    uu-vc;    -    —     j.    ^* 

foci  of  re-  •  ™  d  .£  d 


fleeted  rays 

incident  on  d  Y  d  M  d  P  „/,  d  M 

«*™  ^  =  *  +  p-^r  +  M  77-  =  p  0  +  -jT 

j  V 

Hence  it  follows  that  P  =   ___  ; 

d  X 

P  therefore  is  to  the  caustic,  for  the  coordinates  X,  Y,  what  p  is  to  the  reflecting  curve  foi  the  corresponding 

point  whose  coordinates  are  x,  y. 

144.          Corel.  6.  If  we  put  S  for  the  length  of  the  caustic 

Length  of  _ 

the  caustic  =  the  arc  A  H  K  Y,  we  have  d  S  =  ^  d  X  £  +  d  Y  ! 

investigated.  _ 


dx     -/I  +  P" +  <*/•- M         Prfp 

A/r+~p* 

because  df=  d  .  M  .  •v/F+T4  +  M. — dl    —  ;  but  MdP  =  (P  -  p)  dx 

V  l  +  P' 


so  that  we  have 


=  d/1-hd*{  ^1  +  Ps  --  ^Ln^lH  1 

Vi  +  PJ  J 


that  is,  substituting  for  P  its  value  —  Px~  \  ~  p  '  y- 

2py  +  (1  -  p")  x 

dS  =  df+dx. 


ta 


and  integrating  S  =  constant  +  /  +  ^x*  +  y1. 

Caustics       Hence  it  follows,  that  the  caustic  is  always  a  rectifiable  curve,  and  its 

always  rec- 

t'fiabh.  Length  AK«  =  QP  +  Pw  +  constant  ) 

But  Arc        A  K  F  =  Q  C  +  C  F  +  constant  }  ^s^^'  .ubtn«Ung 

Arc        Fy       =  (QC  +  CF)  -  (QP  +  P  Y). 

Hence  it  appears,  that  the  caustic  is  necessarily  a   rectifiable   curve  when   the   reflecting  curve  is    not   itself 
transcendental. 

145.  If  the  rays  PR,  P'R',  P"  R",  &c.  after  reflexion  at  the  curve  PFP"fall  on  another  reflector  RR'R"  and 

Fig.  20.       are  reflected  in  the  directions  R  S,  R'  S',    R"  S",  &c.  (fig.  20)  their  continual  intersections  will  form  another 

caustic  Z  Z'  Z",   and  so   on  ad  infinitum,  which  may  be  determined  by  a  similar  analysis.     In  like  manner, 

whatever  be  the  law  according  to  which  the  rays  Q  P,  Q'  P',  &c.  are  dispersed,  we  may  conceive  each  to  be  a 

tangent  to  a  curve  which  may  be  regarded  as  the  caustic  of  another  reflecting  curve,  and  so  on.     Let  VV'V" 

be  this  curve.     Since  PVQ  is  a  tangent  to  it,  if  this  curve  and  the  curve  PP'P"  be  given,  the  point  Q  in 

the   axis  from  which  the  incident  ray  Q  P  may  be  regarded  as  radiating,   is  determined  in  terms   of  the  co- 

j^g       ordinates  of  P,  and   therefore   the  quantity   a  may  be   eliminated  altogether.     The  manner  of  doing  this  is 

General        shown  in  the  following 

relation  be-      Proposition.  To  determine  the  relations  between  any  two  consecutive,  or,  as  they  may  be  termed,  conjugate 
tween  two   caustics  VV'V",  YY'Y'',  and  the  intermediate  reflecting  curve  PP'P". 

Let  V  and  Y  be,  as  before,  any  two  conjugate  points  in  the  caustics,  P  the  reflecting  point  ;  then  if  we  put 

infeSate  ?  and  *  for  the  «*>ldinates  of  V 


x  and  y  for  those  of  P 

curve  inves- 

«ated-  X  and  Y  of  Y 


LIGHT. 

Light      Since  the  line  P  VQ  is  a  tangent  to  the  first  curve  at  V,  we  must  evidently  have 


and  this,  combined  with  the  equation  between  17  and  f,  which  represents  the  curve  V  V  V"  suffices  to  determine 
17  and  f  in  terms  of  x,  y,  or  vice  versd,  x  and  y  in  terms  of  f  and  r/. 
Again,  we  have  also  by  Art.  114,  equation  (2) 


and  consequently 


Thus  a  is  given  in  terms  either  of  x,  y,  or  of  17,  f,  whichever  we  may  prefer.     It  only  remains  to  substitute  this 
in  the  value  of  P. 


(1  —  p8)  (x  —  a)  +  2  py 
which  thus  becomes 

gpfr-P-d-y'Xy-,) 
(1  -  p1)  (*  -  f)  +  2  /»  (y  -  r/) 

and  this,  being  free  of  a,  may  be  substituted  in  the  equations  (k)  Art.  136,  when  X  and  Y  will  be  at  once  ob- 
tained in  terms  of  x,  y,  g,  i),  the  coordinates  of  the  reflecting  curve  and  the  preceding  caustic. 

We  shall  now  proceed  to  illustrate  the   theory  above  delivered  by  an  example  or  two. 

Required  the  caustic  when  the  reflecting  curve  is  a  cycloid,  and  the  incident  rays  are  parallel  to  each  other       147. 
and  to  the  axis  of  the  cycloid.  Caustic  of 

d  v  v~  a  cycloid- 

The  equation  of  the  cycloid  is  =  p  = 

I '      t 


x 


taking  unity  for  the  radius  of  the  generating  circle. 
From  this  we  get 

—  (2  -  *) 


and  therefore  —   =  2  x  —  £* ; 

9 

consequently,  by  the  equations  (k)  of  Art.  13o,  we  shall  have 


Y  =  y  +  p  .  --   =  y  +  x   A/2  x  —  x2 
whence 

_    ,.      i /o          o  f\    —   O  <v   ?     T   r2  —   ' 

— =:    p    -f-    :    ^«5  —  &  3,)   —   &    v    &i     J>  a.      —   , 

d  x  */2  _  x 

Now  we  have  also 

42L  =  2(i_,) 

But  since  X  =  %  x  —  x1,  we  have  1  —  x  =  ">/  I  —  X,         and  therefore 


dx 


=  2  -/l-X 


So  that  we  have,  finally,  =  \/  — - — — 

d  A.  '1   —  A. 

which  shows  that  the  caustic  is  itself  a  cycloid  of  half  the  linear  dimensions  of  the  reflecting  curve.  Is  itself 

To  take  one  other  example,  let  us  suppose  the  reflecting  curve  a  circle,  and  the    radiant   point  infinitely  cycloid, 
distant.     Here  we  have  (placing  the  origin  of  the  coordinates  in  the  centre) 

3  B  2 


364  LIGHT. 

Light.  x  r*  Pan  L 

— v~  *  +y  =  r;p=-v7^r;9=-(7^oV  ^~ 

Caustic  of  a  consequently,  by  the  equations  (k)  of  Art.  136 


2  q  2r* 


=  y  +  — 


Then  since  (supposing,  for  brevity,  r  =  1,  which  will  not  affect  the  result) 

4XS=9*3  —  12  T"  +  4  jr8 

4  Y2=  4-  12*'+  12  ,r4-4,r» 

Adding,  4  (Xs  +  Y8)  =  4  -  3  x2  ;  *"  =  —  (1  _  Xs  -  Y2) 

9 
So  that  we  get,  finally,  substituting  this  value  of  x'  in  that  of  Y,  and  reducing, 

(4X8+  4Y'-  1)3=27  Y';  (c) 

which  is  the  equation  of  the  caustic. 

This  equation  belongs  to  an  epicycloid  generated  by  the  revolution  of  a  circle  whose  radius  is  £  that  of  the 
reflecting  circle  on  another  concentric  with  the  latter,  and  whose  radius  is  J  that  of  the  reflecting  circle. 
Fi  21  ^'£'  ^  represents  the  caustic  in  this  case;  QP  being  the  incident  ray,  and  P  Y  the  reflected.  It  has  a  cusp 
at  F,  which  is  the  principal  focus  of  rays  reflected  at  the  concave  surface  BCD,  and  another  at  F',  which  is 
that  of  the  rays  reflected  from  the  convex  surface  BAD.  In  the  latter  case,  it  is  not  the  rays  themselves, 
but  their  prolongations  backwards  which  touch  the  caustic. 

149.  Carol.    When    y  is  very  small,  or  immediately  adjacent  to  the  cusp  F,  the  form  of  the  caustic  approaches 

indefinitely  to  that  of  a  semicubical  parabola.     For,  generally, 


X  =  1  -/I  +  3  Y  '  —  4  Y*, 
and  when  Y  is  very  small,  neglecting  Y8  in  comparison  with  Y 


It  is,  as  we  have  seen,  only  in  certain  very  particular  cases,  when  rays  proceeding  from  one  point  and  reflected 
at  a  curve  proceed  after  reflexion  all  to  or  from  one  point.  In  general  they  are  distributed  in  the  manner 
described  in  Art.  145,  146,  being  all  tangents  to  the  caustic.  The  density  of  the  rays  therefore  in  any  point  of  the 
caustic  is  infinitely  greater  than  in  the  space  surrounding  it,  and  in  the  space  between  the  caustic  and  the  re- 
flecting curve  (PC  F  Y,  fig.  18)  is  greater  than  in  the  space  without  the  caustic  Q  YF.  This  is  obvious,  for 
in  the  latter  space  only  the  incident  rays  occur,  while  in  the  former  are  included  all  the  reflected  rays  as  well 
as  the  incident  ones. 

jr.  This  may  be  easily  shown  experimentally,  in  a  very  satisfactory  manner  pointed  out  by  Dr.  Brewster,  by 

„  22'  bending  a  narrow  strip  of  polished  steel  into  any  concave  form,  as  in  fig.  22,  and  placing  it  upright  on  a  sheet 
of  white  paper.  If  in  this  state  it  be  exposed  to  the  rays  of  the  sun,  holding  the  plane  of  the  paper  so  as  to 
pass  nearly  but  not  quite  through  the  sun,  the  caustic  will  be  seen  traced  on  the  paper,  and  marked  by  a  very 
bright  well-defined  line  ;  the  part  within  being  brighter  than  that  without,  and  the  light  graduating  away  from 
the  caustic  inwards  by  rapid  gradations.  If  the  form  of  the  spring  be  varied,  all  the  varieties  of  catacaustics, 
with  their  singular  points,  cusps,  contrary  flexures,  &c.  will  be  seen  beautifully  developed.  The  experiment  is 
at  once  amusing  and  instructive. 

The  bright  line  seen  on  the  surface  of  a  drinking-glass  full  of  milk,  or,  better  still,  of  ink,  standing  in  sunshine, 
is  a  familiar  instance  of  the  caustic  of  a  circle  just  investigated. 

152  If  the  figure  18  be  turned  round  its  axis,  the  reflecting  curve  will  generate  a  surface  of  revolution,  which,  if 

supposed  polished  within  or  without,  as  the  case  may  be,  will  become  a  mirror.  The  caustic  will  also  generate 
a  conoidal  surface,  to  which  all  the  rays  reflected  by  the  mirror  will  be  tangents.  No  mirror,  therefore,  which 
is  not  formed  by  the  revolution  of  a  conic  section  having  the  radiant  point  in  its  focus,  can  converge  all  the 
reflected  rays  to  one  point  or  focus.  There  will,  however,  always  be  one  point  which  receives  the  reflected  rays 
in  a  more  dense  state  than  any  other.  This  point  is  the  cusp  F,  as  we  shall  presently  see.  The  deviation  ot 
any  reflected  ray  from  this  point  is  what  is  termed  its  aberration. 

Ug  The  concentration  and  dispersion  of  rays  by  reflecting  and  refracting  surfaces  forming  the  great  business  of 

practical  optics,  it  will  be  necessary  to  enter  at  large  into  this  subject  ;  and,  first,  it  will  be  proper  to  inquire 
how  far  any  given  reflector  will  enable  us  to  concentrate  the  rays  which  fall  on  it.  To  this  end  let  the  following 
problem  be  proposed. 

154  Proposition.  A  reflector  of  any  figure,  of  a  given  diameter  or  aperture  AB,  being  proposed,  to  find  the  circle 

of  least  aberration,  or  the  place  where  a  screen  must  be  placed  to  receive  all  the  rays  reflected  from  the  surface, 
within  the  least  possible  circular  space  (since  they  cannot  be  all  collected  in  one  point)  and  the  diameter  of  this 
circle. 


LIGHT.  365 

Light.          A  C  B  (fig.  23)  being1  the  mirror,  Q  the  radiant  point,   G  K.fkg  the  caustic,  /the  cusp  or  focus  for  central      P«t  I. 
— v-^-7  rays,  q  the  focus  of  the  extreme  rays  A  q,  B  q,  produce  these  lines  till  they  cut  the  caustic  in  Yy.     It  is  c.ear,  *-^-v— — 
then,  since  all  the  rays  reflected  from  the  portion  A  C  B  of  the  reflector  are  tangents  to  points  of  the  caustic     R'  2^- 
between  K,/and  k,f,  that  they  must  all  pass  through  the  line  Yy.    Retaining  the  notation  of  the  foregoing  pro- 

o  o  o  o 

positions,  (i.  e.  supposing  Q  x  =*  X  ;  X  y  =  Y.)     Let  us  put  Q  L  =  X,  L  K  =  Y,  QD  =  i;  D  A  =  y  ;  and  let 

o     o 

P,  p  represent  the  values  of  P  and  p  corresponding  to  the  points  K  and  A  of  the  caustic  and  reflecting  curves. 
The  equation  of  the  line  A  K  q  y  will  then  be 

Y-y 

Y  and  X  being  the  coordinates  of  any  point  in  it.  But  at  the  point  y,  where  it  cuts  the  other  branch  of  the 
caustic,  these  coordinates  are  common  to  the  straight  line,  and  to  the  caustic.  At  this  point,  therefore,  the  above 
equation,  and  those  expressing  the  nature  of  the  caustic,  must  subsist  together.  Now  these  are  the  equations 
(K)  Art.  136,  combined  with  the  original  equation  of  the  reflecting  curve.  Eliminating,  then,  x  and  y,  by  the 
aid  of  two  of  them,  and  determining  the  values  of  X,  Y  from  the  rest,  the  problem  is  resolved. 

Now  the  same  equation  which  gives  the  value  of  y,  or  xy,  must  also  give  that  of  L  K,  because  K  is  a  point       155. 
in  both  caustic  and  the  line  A  K  y,  as  well  as  y.     But,  moreover,  since  A  K  y  is  a  tangent,  the  point  K  is  a 
double  point ;  therefore  the  final  equation  in  Y  must  necessarily  have  two  equal  roots,  besides  the  value  of  Y 
sought ;  and  these  being  known,  the  other  may  be  found  from  a  depressed  equation. 

The  method  here  followed  is,  apparently,  different  from  that  usually  employed,  which  consists  in  making  the 
value  of  y  as  determined  by  the  intersection  of  the  extreme  reflected  ray  A  K  y,  and  any  other  reflected  ray  (from  P) 
a  maximum.  But  the  difference  is  only  apparent,  for  in  the  latter  method  we  have  to  make  Y  as  determined 
by  the  two  equations  (holding  good  jointly) 

Y  -  °y  =  P  (X  -  x),  and  Y  -  y  =  P  (X  -  x) 

a  maximum,  or  d  Y  =  o.  Now  in  this  case  the  former  equation  gives  dX  =  o  also ;  and  therefore,  differen- 
tiating the  latter,  we  have  —  dy  =  (X  —  x)dP  —  Pdx, 

whence  X  -  x  =  —  ~P     d  x 

p „ 

and  therefore  Y  —  y  =  P  .   -   — -    d  x. 

Now  these  are  nothing  more  than  the  equations  of  Art.  136,  expressing  the  general  properties  of  the  caustic  ; 
so  that  this  consideration  of  the  maximum  only  leads  by  a  more  circuitous  route  to  the  same  equations  as 
the  method  above  stated,  and  is  in  fact  nothing  more  than  a  different  mode  of  expressing  the  caustic. 

Let  us  apply  this    reasoning  to  the  case  when  the  reflector    is   spherical.      Resuming    the    equations  and       156. 
notation  of  Art.  148,  and  putting  a  for  the  extreme  value  of  y,  or  the  semi-aperture  of  the  mirror,  and   6  for  cirele  of 

the  corresponding  value  of  x,  that  of  P  will  be  least  aber 

ration   in 

•2  a  b  -2al>  sP]?eriraI 

reflector. 


_  

l-p'    "  /  b*-a*        '    l-2o' 

Hence  the  equation  (m,  2)  Art.  138,  of  the  extreme  reflected  ray  becomes 


'-• 


2 


whence  we  get  2  X  =  —  (l   + 

6     \  a 

Assume  z,  so  that  Y  =  a3  z3,  z  being  another  unknown  quantity,  then  we  have 

4  X*  =  —  —  .    {  1  +  (1  -  2  «!)  a2  z3  }  !. 

JL   ™  ~~  CL 

Substituting  this  for  4  Xs,  and  for  Y8  its  value  a"  z"  in  the  equation  of  the  caustic  (r)  Art.  148,  extracting  the 
cube  root,  and  reducing,  we  get  the  following  equation  for  finding  z, 

a*z°+  (2-4ae)  z=>  +  (3  a2  -  3)  z    +1=0. 

Now  this,  according  to  the  remark  in  Art.  155,  must  have  two  equal  roots,  viz.  when  x  =  b,  or  Y  =r  <z3, 
that  is,  when  z  =  1.  Hence  this  equation  must  necessarily  be  divisible  by  (z  —  I)2.  Performing  the  divi- 
sion we  find  it  is  so,  and  the  quotient  gives 

a'z4  +  2a2z3  +  3a°z*  +  2  z  +  1  =  o;  (y) 

for  determining  the  remaining  values  of  z. 


366  LIGHT. 

Light.          As  this  investigation  is  rigorous,  nothing  having  been  omitted  or  neglected  as  small,  we   have    here   the 
^—-V—*'  complete  solution  of   the  problem,  whatever  be  the  aperture  of  the  mirror.      If  this  be    supposed  small  in 

157.      comparison  with  the  radius,  an  approximation  to  the  value  of  z  will  be  had  by  t'-e  series  thence  derived, 
Case  when 
me  aperture  19  9  1395 

-      ~a     ~a 


and  of  course  since  Y  =  a'  ;3> 


27  675 

5r  B     -    164i    B    ~ 


158.          The  first  term  of  this  series  is  sufficient  for  most  cases  which  occur  in  practice,  and  gives 
Case  where 
tne  aperture  as 

is  small  Y  =  --  (a) 

-  G 

wnen  com- 

pared to 

radius.         or,  supposing  r  the  radius  of  curvature  of  the  reflector, 


The   lateral    aberration    corresponding   to  the  semi-aperture  a  is,  by  the  equation  (/),  Art.  133,  equal  to 
— ; ;  consequently,  in  the  case  of  small  apertures,  the  radius  of  the  least  circle  of  aberration  is  equal  to  \ 

of  the  lateral  aberration  (at  the  focus)  of  the  exterior  annulus. 

3  3 

159  Carol.  The  least  circle  of  aberration  is  nearer  the  mirror  than  its  principal  focus,  by  -—  f  gor  — -  the  lon- 

3  a 

gitudinal  aberration  =  — — —  .   -   — . 
16  r 

IgQ  To  complete  the  theory  of  caustics,  it  only  remains  to  examine  the  degree  of  concentration  of  the  reflected 

Density 'of   rays  at  any  assigned  point.     To  this  end,  let. S  (fig.  24)  be  any  point,  and  through  it  let  PSY<?  be  drawn 

reflected       touching  the  caustic  in  Y.     Then  S  may  be  regarded  as   lying  in  a  conical  surface  generated  by  the  revo- 

raysatany    lution  of  the  tangent  P  Ys^,  about  the  axis;  and  all  the  rays  in  the  annulus,  generated  by  the  revolution  of 

oomt  mves-  jne    e]ement    p  p7,  will  be    contained    in    the    hollow    conoidal  solid    formed  by  the  revolution  of  the  figure 

FTgtB24        PPYq'q  about  the  same  axis.     Hence  at  S  the  rays  will  be  concentrated:  first,  in  a  plane  parallel  to  that 

of  the  paper,  in  the  ratio  of  P  P'  to  S  S',  or  P  Y  to  S  Y  ;  and,  secondly,  in  a  plane  perpendicular  to  that  of 

the  paper,  or  in  the  ratio  of  the  circumferences  of  the  circles  generated  by  the  revolution  of  P  and  of   S, 

that  is,  in  the  ratio  of  these  radii  P M  :  ST.     On  both  accounts,  therefore,  the  concentration  at  S  will  be 

PM  PY  Po  PY 

represented  by  -       -     x   --^r>    or  ~  x      c  v  •      If-  therefore,  we  represent  by  1  the  density  of  the 

ol  ox  0*7  ^  * 

rays  immediately  on  their  reflexion  at  P,  their  density  at  S  corresponding,  will  be  represented  by ' — -, 

S  Y  .  S  q 
and  this  is  true,  whatever  be  the  situation  of  S. 

161.  But  there  are  now  several  cases  to  be  distinguished.     First,  when  S  is  situated  in  any  part  of  the  spaces 
1st  case.       K  H  V,  N  D  W.  no  such  tangent  can  be  drawn  to  cut  the  reflector  within  its  aperture  A  B  ;  therefore  these 

spaces  receive  no  rays  at  all,  and  the  density  =  o  in  every  point. 

162.  Secondly,  when    S  is   situated    anywhere  within  the  spaces  A  G  B,    V  H  F  E,    E  F  D  W,    only  one    such 
2nd  case,     tangent  can  be  drawn  to  cut  the  reflecting  curve  between  A  and  B.      So  that  in  these   spaces  the  density 

PY.P<? 

is  simply  represented  by  g  y — g — • 

Thirdly,  within  the  spaces  KGH  and  MGDtwo  tangents  can  be  drawn  from  any  point  S,  both  touching 
3rd  cue       tne  branch  F  k  on  the  same  side  of  the  axis  as  the  point  S.     If  we  suppose  PI  Y,  S  qt  and  P,  Y8  S  q,  to  be 
these  tangents,  S  will  receive  rays  belonging  to  both  these  converging  conoids,  and  the  density  will  therefore 
be  the  sum  of  those  belonging  to  either,  or 

D  =  2IL41L   +     **,** 


Fig.  25.       See  fig.  25. 

164.  Fourthly  and    lastly,  within  the    space  FHGD  there  maybe  drawn  three  tangents  9,  S  Y,  P,,  ^SY,?,, 

4th  case.      an(1  ,,3  s  y,  P3,  all  falling  within  A  B,  the  two  first  (fig.  26)  touching  the  branch  FA;  on  the  same  side  as  S,  the 


LIGHT.  ;>67 

Light,      third  on  the  opposite  side.     The  former  belong  to  cones  of  rays  converging  to  ql  qt,  the  latter  to  a  cone  con-     Part  I. 
""v"**'  verging  to  q,,  but  intercepted  by  S  after  meeting  at  q,  and  again  diverging.     Hence,  in  this  case,  the  density  will  •— ~ v»- 

Fig.  26. 

PY..P?,  PYa.P?a  PY3.Pg, 

be  expressed  by  D  =   —    + —    -f- ri— . 

It  would  lead  into  too  great  complication  to  attempt  developing  the  actual  value  of  these  fractions  in  terms  of  Application 

the  coordinates  of  S,  and  we  will  therefore  merely  apply  them  to  some  remarkable  positions  of  S  lar^ses" 

Case  1.  S  in  the  axis,  beyond  the  principal  focus,  or  between  the  mirror  and  its  focus  for  extreme  rays  G.    Here       jgj 

(P  F   \2  Case  1. 

-  „    },   which   shows    that   the 

density  is  inversely  as  the  square  of  the  distance  of  S  from  the  principal  focus. 

Case  2.  S  in  the  axis  between  the  principal  focus  and  the  focus  for  extreme  rays  G,  (i.  e.  in  the  line  GF.)  166. 
Here  S  ql  =:  o,  S  qt  =  o,  S  q3  =  o ;  so  that  here  all  the  three  several  component  portions  of  D  are  infinite,  and  Case  2. 
of  course  the  density  is  infinitely  greater  than  on  the  surface  of  the  reflector. 

Case  3.  S  at  F.  Here  not  only  89  =  0,  but  also  S  Y;  therefore  at  F  the  density  is  infinitely  greater  than  167 
in  the  last  case,  and  is  the  greatest  which  exists  anywhere.  Case  3. 

Case  4.  S  anywhere  in  the  caustic.  Here  S  Y  =  o,  therefore  in  this  case  also  D  is  infinite,  or  the  density  168. 
infinitely  greater  than  at  the  surface  of  the  reflector ;  and  as  S  approaches  F,  this  is  still  further  multiplied  by  Case  4. 
the  diminution  of  all  the  values  of  S  q. 

Case  5.     S  anywhere    in  H  zD,  the  circle  of  least  aberration.     At  the  centre  z  and  the  circumference  H  the       169. 
density  is  infinite.     Between  these  two  positions,  finite,  diminishing  to  a  minimum,  and  again  increasing  accord-  Case  5. 
ing  to   a  law  too  complicated  to  be  here  investigated.     It  will  be  observed,  that  the  relations  expressed   in 
these  articles  (160 — 169)  are  general,  and  not  restricted  to  the  case  where  the  reflecting  surface  is  spherical. 

In  all  the  foregoing  reasoning  the  point  S  is  supposed  to  receive  the  rays  perpendicularly.     The  density  of      170. 
the  rays  therefore  here  intended  must  be  understood  to  mean,  The  number  of  rays  not  incident  on  a  given  par-  Illumination 
ticular  plane  surface,  but  passing  through  a  given  infinitely  small  spherical  portion  of  space,  or  received  upon  of  a  screen 

an  infinitely  small  spherical  body  at  S.  f,xPosefld  '°  , 

J ,  J  -11  j.      i  .1  ..,,.,  -the  reHectet 

In  cases,  however,  where  the  aperture  is  small,  a  screen  perpendicular  to  the  axis  will  receive  the  rays  from  rays 

every  point  very  nearly  at  a  perpendicular  incidence ;  and  hence  the  above  expressions  will  in  this  case  represent 
the  intensity  of  illumination  of  the  several  points  in  such  a  surface,  the  screen  being,  however,  supposed  to  stop 
none  of  the  incident  light. 

For  further  information  respecting  caustics,  the  reader  is  referred  to  Tschirnaus,  Leipsic  acts  1682,  and  Hist, 
de  I'Acad.,  torn.  ii.  p.  54,  168S;  to  De  la  Hire's  Traite'  des  Epicycloides,  and  Me'm.  de  I'Acad.,  vol.  x. ;  to 
Smith's  Optics ;  Carre",  Mem.  de  I'Acad.,  1703  ;  J.  Bernoulli!,  Opera  Omnia,  vol.  iii.  p.  464  ;  VHSpital  Analyse 
des  Infiniment  Petits ;  Hayes's  Fluxions;  Petit,  Correspondence  de  I'Ecole  Polytechnique,  ii.  553;  Malus,  Journal 
de  I'Ecole  Polytech.,  vol.  vi. ;  Gergonne,  Annales  des  Mathematiques,  xi.  p.  229  ;  De  la  Rive,  Dissertation  sur 
les  Caustiques,  Sfc. ;  Sturm,  Annales  des  Math.,  xvi. ;  Gergonne,  ditto. 


OF  THE  REGULAR  REFRACTION  OF  LIGHT  BY  UNCRYSTALIJZED  MEDIA. 

§  VI.   Of  the  Refraction  of  Homogeneous  Light  at  Plane  Surfaces. 

When  a  ray  of  light  is  incident  on  the  surface  of  any  transparent  uncrystallized  medium,  a  portion  of  it  is       171. 
reflected ;  another  portion  is  dispersed  in  all  directions,  and  serves  to  render  the  surface  visible ;  and  the  remainder 
enters  the  medium  and  pursues  its  course  within  it. 

In  the  reflexion  of  light,  the  law  of  reflexion,  as  far  as  regards  the  direction  of  the  reflected  ray,  is  the  172 
same  for  all  reflecting  media ;  the  angle  of  reflexion  being  equal  to  that  of  incidence  for  all.  In  refraction, 
however,  the  case  is  otherwise,  and  each  different  medium  has  its  own  peculiar  Taw  of  action  on  light ;  some 
turning  a  ray  incident  at  a  given  angle  more  out  of  its  course  than  others.  Whatever  be  the  nature  of  the 
refracting  medium,  the  following  general  laws  are  found  to  hold  good,  and  suffices  (when  the  medium  is  known) 
to  determine  the  direction  of  the  refracted  ray. 

1st.  The  incident  ray,  the  perpendicular  to  the  surface  at  the  point  of  incidence,  and  the  refracted  ray,  all  lie       [73 
in  the  same  plane. 

2nd.  The  incident  and  refracted  rays  lie  on  opposite  sides  of  the  perpendicular.  j,-^ 

3rd.  Whatever  be  the  inclination  of  the  incident  ray  to  the  refracting  surface,  the  sine  of  the  angle  included       175 
between  the  incident  ray  and  the  perpendicular  is  to  the  sine  of  that  included  between  the  refracted  ray  and  the 
perpendicular  in  a  constant  ratio. 

These  laws  equally  hold  good  for  plane  and  for  curved  surfaces,  and  are  found  to  be  verified  with  perfect       17^ 
precision  by  the  most  delicate  experiments,  and  all  the  phenomena  of  refracted  light  to  take  place  in  exact  con- 
formity with  the  results  deduced  from  them  by  mathematical  reasoning. 


368  LIGHT. 

Light         Let  A  C  B  (fig.  23)  be  the  refracting  surface,  P  C  p  the  perpendicular  to  it  at  the  point  of  incidence  C,  S  C 
'  the  incident,  and  C  s  the  refracted  ray.     Then  we  shall  have 

sin  P  C  S  :  sin  p  C  *  :  :  p.  :  1, 

/a  being  a  constant  quantity  ;  that  is,  constant  for  the  same  medium  A  B,  though  its  value  is  different  for  different 
media. 

178.  It  is  usual,  for  brevity,  to  speak  of  the  sine  of  incidence,  and  the  sine  of  refraction,  instead  of  the  sines  of  the 
angle  of  incidence,  and  the  angle  of  refraction. 

179.  The  numerical  value  of  the  quantity  /t,  or  of  -  -  -  —  in  any  medium,  must  be  ascertained  before 

sin  of  refraction 

the  law  of  refraction  in  that  medium  can  be  regarded  as  perfectly  known.  This  may  be  done  experimentally 
by  actually  measuring  the  angle  of  refraction  corresponding  to  any  one  given  angle  of  incidence,  for  the  value 
of  the  above  fraction  being  thus  determined  for  one  incidence  holds  equally  for  every  other,  or  by  other  more 

Index  of      easy  or  more  renned  modes  to  be  described  hereafter.     This  quantity  u,  is  called  the  index  of  refraction  of  the 

retracuon. 


The  medium  in  which  the  ray  proceeds  previous  to  its  incidence  on  A  B  is  here  regarded  as  a  vacuum.     If 

the  medium  A  B  be  also  a  vacuum,  it  is  clear  that  the  ray  will  not  change  its  course  ;  so  that  the  angle  of  inci- 

dence will  equal  that  of  refraction,  and  the  value  of  fi  will  be  equal  to  1.     This  is  the  lowest  value  of  /t,  as  no 

medium  has  yet  been  discovered  which  refracts  rays  from  the  perpendicular  when  incident  from  a  vacuum.    The 

greatest  value  of  /t  yet  known  is  3,  when  the  refraction  is  made  into  chromate  of  lead  ;  and  between  these  limits 

almost  every  intermediate  gradation  has  been  found  to  belong  to  some  one  or  other  transparent  body.     Thus 

for  air  at  its  ordinary  density  /t=  1.00028,  while  for  water  it  is  1.336,  for  ordinary  crown  glass  1.535,  for  flint 

glass  1.60,  for  oil  of  cassia  1.641,  for  diamond  2.487,  and  for  the  greatest  refraction  of  chromate  of  lead  3.0. 

181.  It  is  a  general  law  in  Optics,  that  the  visibility  of  two  points  from  one  another  is  mutual,  whatever  be  the 

Refraction    course  pursued  by  the  rays  which  proceed  from  one  to  the  other.     In  other  words,  that  if  a  ray  of  light  pro 

ne^turSo  ceedin?  from  A  arrives   by  any  course  at  B,  however  often  reflected,  refracted,  &c.,  a  ray  can  also  arrive  at  A 

vacuum        from  B  by  retracing  precisely  the  same  course  in  a  contrary  direction.     It  follows  from  this,  that  if  the  ray  S  C 

incident  on  the  exterior  surface  of  a  medium  A  B,  (fig.  23,)  pursue  after  refraction  the  course  Cs,  then  will  a 

ray  sC,  incident  on  the  exterior  surface  of  the  medium,  be  refracted  out  of  it  in  the  direction  CS,  being  bent 

from  the  perpendicular.     Consequently,  since  in  this  case  the  angle  of  incidence  is  the  same  with  the  angle  of 

....  ,.  sin  incidence  1 

refraction  in  the  former  case,  and  vice  versd,  we  shall  have  here  -  -    =  —  .     Thus  we  see  that  the 

sin  refraction  /i 

index  of  refraction  out  of  any  medium  into  vacuum  is  the  reciprocal  of  the  index  of  refraction  into  the  medium 
from  the  vacuum. 

Hence  it  follows,  that  a  ray  can  be  intromitted  into  any  medium  from  a  vacuum  at  any  angle  of  incidence;  for 

since  sin  refr.  =  sin  p  ca  =  —  .  sin  P  C  S,  the  value  of  u  being  greater  than  1,  the  sine  of  pcs  will  neces- 

r* 

sarily  be  less  than  that  of  P  C  S,  and  of  course  less  than  unity  ;  so  that  the  angle  of  refraction  can  never  become 
imaginary.  Thus,  as  the  angle  of  incidence  PCS  increases  from  o,  or  as  the  ray  S  C  becomes  more  and  more 
oblique  to  the  surface  till  it  barely  grazes  it,  as  at  S"  C,  the  refracted  ray  becomes  also  more  oblique,  but  much 

less  rapidly,  and  never  attains  a  greater  obliquity  than  the  situation  C  /',  in  which  sin  p  C  s"  —  —  '-  —  =  •  —  . 

fi  [i 

Limit  of  the  This  limiting  angle,  then,  is  the  maximum  angle  of  refraction  from  vacuum  into  the  medium,  and  its  value  in  any 

angle  of      given  medium  is  found  by  computing  the  angle  whose  sine  is  the  reciprocal  of  the  index  of  refraction.     Thus  in 
refraction. 

water  the  angle  of  refraction  cannot  exceed  arc  sin  —  —          ,  or  48°  27'  40".     In  crown  glass  the  limit  is 

1  .33o 

40°  39',  in  flint  38°  41',  in  diamond  23°  42',  while  for  the  greatest  refraction  of  chromate  of  lead  the  limit  is  so 
low  as  19°  28'  20". 

183.  Conversely,  when  a  ray  is  incident  on  the  interior  surface  of  the  medium,  at  any  angle  less  than  the  limiting 

Limit  to  the  , 

possibility     an™]e  whose  sine  is  —  ,  it  will  be  refracted  and  emerge  according  to  the  law  laid  down  in  Art.  181.  being  bent 
of  a  ray's  p 

anye5me-0m  from  the  perpendicular.     But  as  the  angle  of  incidence  pCs  increases,  the  angle  of  refraction  PCS  increases 

ilium.          more  rapidly  ;  and  when  the  former  angle  has  reached  the  limiting  value  p  C  s",  the  transmitted   ray  emerges  in 

the  direction  C  S",  barely  grazing  the  external  surface.     If  the  angle  of  incidence  be  still  further  increased,  the 

angle  of  refraction  becomes  imaginary  :  for  we  have  sin  P  C  S  =  fi  x  sin  p  C  s,  and  if  sin  p  C  s  7  —  ,  the  sine 

When  the    of  P  C  S  must  be  greater  than  unity.     This  shows  that  the  ray  cannot  emerge  ;  but  it  does  not  inform  us  what 
ray  cannot  \jecornes  of  jt.     To  ascertain  this,  we  must  have  recourse  to  experiment  ;    from  which  we  learn,  that  after  this 
reflected*  "  ^'m't  's  Passed,  tne  ray>  instead  of  being  refracted  out  of  the  medium,  is  turned  back  and  totally  reflected  within 
it,  making  the  angle  of  reflexion  p  C  S'"  =  p  C  s"'. 


LIGHT.  369 

Lignt.          When  the  ray  is  incident  on  the  exterior  surface  of  the  medium,  a  portion  is  reflected  (R)  and  the  remainder      P"t  I. 
•v""''  (r)  refracted.     The  ratio  of  R  to  r  is  smallest  at  a  perpendicular  incidence,  and  increases  regularly  till  the  inci-  ^—  •—*'—•—' 
dence  becomes  90°  ;  but  even   at  extreme  obliquities,  and  when  the  incident  ray  just  grazes  the  surface,  the 
reflexion  is  never  total,  or  nearly  total,  a  very  considerable   portion   being  always  intromitted.     On   the  other  7*"s  Teftex~ 
hand,  when  the  ray  is  incident  on  the  interior  surface,  the  reflected  portion  (R)  increases  regularly,  with  a  very 

moderate  rate  of  increase,  till  the  angle  of  incidence  becomes  equal  to  the  critical  angle,  whose  sine  is  —  ;  when 

it  suddenly,  and,  as  it  were,  per  saltum,  attains  the  whole  amount  of  the  incident  light,  and  the  refracted  portion 
(r)  becomes  zero.  This  sudden  change  from  the  law  of  refraction  to  that  of  reflexion  —  this  breach  of  continuity, 
as  it  were,  is  one  of  the  most  curious  and  interesting  phenomena  in  Optics,  and  (as  we  shall  see  hereafter)  is 
connected  with  the  most  important  points  in  the  theory  of  light. 

The  reflexion  thus  obtained,  being  total,  far  surpasses  in  brilliancy  what  can  be  obtained  by  any  other  means  ;       185. 
from  quicksilver,  for  instance,  or  from  the  most  highly  polished  metals.     It  may  be  familiarly  shown  by  filling  a  Experiment 
glass  (a  common  drinking-glass)  with  water,  and  holding  it  above  the  level  of  the  eye,  (as  in  fig.  24,  No.  2.)    If  |jj^stratlns 
we  then  look  obliquely  upwards  in  the  direction  E  a  c,  we  shall  see  the  whole  surface  shining  like  polished  silver,  reflex|or 
with  a  strong  metallic  reflexion  ;    and  any  object,  as  a  spoon,  A  C  B,  for  instance,  immersed  in  it  will  have  its  Fig.24, 
immersed  part  C  B  reflected  on  the  surface  as  on  a  mirror,  but  with  a  brightness  far  superior  to  what  any  mirror  No.  2. 
would  afford.     This  property  of  internal   reflexion  is  employed   to  great  advantage  in  the  camera  lucida,  and 
might  be  turned  to  important  uses  in  other  optical  instruments,  especially  in  the  Newtonian  telescope,  to  obviate 
the  loss  of  light  in  the  second  reflexion,  of  which  more  hereafter. 

Some  curious  consequences  follow  from  this,  as  to  vision  under  water.     An  eye   placed  under  perfectly  still       186. 
water   (that  of  a  fish,  or  of  a  diver)  will   see  external  objects  only  through  a  circular  aperture  (as  it  were)  of  Appear- 
96°  55'  20"  in  diameter  overhead.     But  all  objects  down  to  the  horizon  will  be  visible  in  this  space  ;   and  those  *c 
near  the  horizon  much  distorted  and  contracted  in  dimensions,  especially  in  height.     Beyond  the  limits  of  this  :ects  to  a 
circle  will  be  seen  the  bottom  of  the  water,  and  all  subaqueous  objects,  reflected,  and  as  vividly  depicted  as  by  spectator 
direct  vision.     In   addition  to  these  peculiarities,  the  circular  space   above-mentioned    will   appear  surrounded  underwater 
with   a  perpetual  rainbow,   of  faint  but  delicate  colours,  the  cause  of  which  we  shall  take  occasion  to  explain 
further  on.     But  we  need  not  immerse   ourselves  in  water  to  see,  at  least,  a  part  of  these   phenomena.     We 
actually  live  under  an  ocean  of  air,  a  feebly  refracting  medium,  it  is  true,  in  comparison   with  water  ;  and  our 
vision  of  external  objects  near  the  horizon  is  modified  accordingly.     They  are  seen  distorted  from  their  true 
form,  and  contracted  in  their  vertical  dimensions  ;  thus  the  sun  at  setting,  instead  of  appearing  circular,  assumes  Elliptical 
an  elliptical,  or  rather  compressed  figure,  the  lower  half  being  more  flattened  than  the  upper,  and  this  change 
of  figure  is  considerable  enough  to  be  very  evident  to  even  a  careless  spectator.     The  spherical  form  of  the 
atmosphere,  and  its  decrease  of  density  in  the  higher  regions,  however,  prevent  the  rest  of  the  appearances  above 
described  from  being  seen  in  it. 

If  a  medium  be  bounded  by  parallel  plane  surfaces,  a  ray  refracted  through  it  will  have  its  final   direction       187. 
after  both  refractions  the  same  as  before  entering  the  medium.  Refraction 

Let  A  B,  D  F  be  the  parallel  surfaces  of  the  medium,  and  S  C  E  T  a  ray  refracted  through  it,  P  C  p,  Q  Eg,  thro»8l' 
perpendiculars  to  the  surfaces  at  C  and  E,  then  we  have  surfaces 

sin  S  C  P  :  sinp  C  E  (=  sin  C  EQ)  :  :  p  :  1  )  N^25' 

and,  compounding  these  proportions, 
sin  CEQ:  sing  CT  ::!:/.) 

sin  S  C  P  :  sin  q  E  T,  and  therefore  S  C  P  =  q  E  T,  and  the  ray  E  T  is  parallel  to  S  C. 

This  proposition  may  be  proved  experimentally,  by  placing  the  plane  glass  of  a  sextant  (unsilvered)  before  the  Experimen- 
object-glass  of  a  telescope   directed  to  a  distant  object,  or  before  the  naked  eye,  and  inclining  it  at  any  angle  to  •"proe* 
the  visual  ray.     The  apparent  place  of  the  object  will  be  unchanged. 

Experiment.     Let  a  plate  of  glass,  or  any  other  transparent  medium,  be  placed  parallel  to  the  horizon,  and  on       188. 
it  let  any  transparent  fluid  be  poured,  so  as  to  form  a  compound   medium   consisting  of  two  media  of  different  Refraction 
refractive  indices,  in  contact,  and  bounded  by  parallel  planes  ;  and  let  an  object  above  this  combination,  a  star,  *tthe  COIT>- 
for  instance,  be  viewed  by  an  eye  placed  below  it,  or  through  a  telescope.     It  will  be  found  to  appear  precisely  morlsurface 
in  the  same  situation  as  if  the  media  were  removed,  whatever  be  the  altitude  of  the  object,  or  star.     It  follows  medra°in 
from  this,  that  a  ray  S  B  (fig.  26,  No.  2)  incident  on  such  a  combination  of  media,  A  F  and  D  I,  as  described,  contact. 
will  emerge  in  a  direction  H  T  parallel  to  the  incident  ray  S  B.  Fig.  26, 

Proposition.     Let  there  be  any  two   media  (No.  1   and  2)  whose  respective  indices  of  refraction  from  a  No'2' 
vacuum  into  each  are  /t  and  p!.     Then  if  these  media  are  brought  into  perfect  contact,  (such  as  that  of  a  fluid  .        *?• 
with  a  solid,  or  of  two  fluids  with  one  another,)  the  refraction  from  either  of  them  (No.  1)  into  the  other  (No.  2) 


u!  from  one 

will  be  the  same  as  that  from  a  vacuum  into  a  medium,  whose  index  of  refraction  is  —  ,   the    index   of  refrac-  medium  int 

f-  another. 

tion  of  the  second  medium  divided  by  that  of  the  first. 

Let  D  E  F  (fig.  2fJ,  No.  2)  be  the  common  surface  of  the  two  media,  and  let  them  be  formed  into  parallel 
plates  A  F,  D  I,  as  in  the  experiment  last  described  ;  then  any  ray  S  B  incident  at  any  jingle  on  the  surface  A  C 
will  emerge  at  G  I  in  a  direction  H  T  parallel  to  S  B.  Let  B  E  H  be  its  path  within  the  media,  and  draw  the 
perpendiculars  P  B  p,  Q  E  q,  R  H  r,  then 

VOL.  tv  3  c 


370  LIGHT. 

sin  S  B  P  :  sin  E  B  p  =  sin  B  E  Q  :  :  /»  :   1 
sin  R  H  E  =  sin  9  E  H  :  sin  r  H  T  =  sin  P  B  S  : :  1  :  ,u , 
and,  compounding  these  proportional 

sin  H  E  q  :  sin  B  E  Q  : :  »  :  V; 


sin  H  E  q  u    ' 

Absolute          But  B  E  Q  is  the  angle  of  incidence,  and  H  E  9  that  of  refraction,  at  the  common  surface  of  the  media,  con- 
and  relative  sequently  the  relative  index,   or  index   of  refraction  from   the  first   into  the  second,  is  equal  to  the  quotient 

-  of  the  absolute  indices  fi',  p.,  of  the  second  and  first,  or  their  indices  of  refraction  from  vacuum. 

190.  This  demonstration,  it  is  true,  holds  good  only  for  the  case  when  the  angles  of  incidence  and  refraction  at  the 
common  surface  are  both   less  than  the  limits  of  the  angles  of  refraction  from  vacuum  into  each   medium.     If 
they  exceed  these  limits,  the  proposition  however  still  holds  good,  as  may  be  shown  by  direct  measures  of  the 
angles  of  incidence  and  retraction  in  any  proposed  case.     At  present,  therefore,  we  must  receive  it  as  an  experi- 
mental truth. 

191.  Example.  Required  the  ratio  of  the  sine  of  incidence  to  that  of  refraction  out  of  water  into  flint  glass.     The 
refractive  index  of  flint  glass  is  1.60,  and  that  of  water  1.336,  therefore  the  refractive  ratio  required  is 

1.60 
-L336-    -  L197' 

192.  If  the  index  /*  =  —!,  the  general  law  of  refraction  coincides  with   that  of  reflexion.     Thus  all  the  cases  of 
reflexion,  as  far  as  the  direction  of  the  reflected  ray  is  concerned,  are  included  in  those  of  refraction. 

Of  the  Ordinary  Refraction  of  Light  through  a  System  of  Plane  Surfaces,  and  of  Refraction  through  Prisms. 

193.  Definition.    In  Optics,  any  medium  having  two   plane  surfaces,   through  which   light  may  be  transmitted, 
inclined  to  each  other  at  any  angle,  is  called  a  prism. 

194.  Definition.  The  edge  of  the  prism  is  the  line,  real  or  imaginary,  in  which  the  two  plane  surfaces  meet,  or  would 
meet  if  produced. 

195.  Definition.  The  refracting  angle  of  the  prism  is  the  angle  on  which  its  two  plane  surfaces  are  inclined  to  each 
other. 

196.  Definition.  The  faces  of  a  prism  are  the  two  plane  surfaces. 

197.  Definition.    The  plane  perpendicular  to  both  surfaces,  and  therefore  to  the  edge  of  a  prism,  is  called  the 
principal  section  of  the  prism,  or  of  the  two  surfaces.     This  expression   has  been  used  in  this  general  sense 
already,  under  the  head  of  reflexion. 

To  determine  the  direction  of  a  Ray  after  Refraction  through  any  System  of  Plane  Surfaces. 

198.  Construction.  Since  the  direction  of  the  ray  is  the  same  whether  refracted  at  the  given  surfaces,  or  at  others 
General        parallel  to  them,  conceive  surfaces  parallel  to  the  given  ones,  all  passing  through  one  point,  and  from  this  point, 
problem  of   but  wj,oily  exterior  to   the  refracting   media,  let  perpendiculars  C  P,  C  P',  C  P",  &c.  be  drawn  to  each  of  the 
tioen  tf^gh  surfaces,  (fig.  '27.)     Let  S  C  be  the  direction  of  the  incident  ray.     Between  C  P  and  C  S'  draw  C  S'  in  the  plane 


any  system  I 

of  plane        S  C  P,  so  that  sill  P  C  S'  =  —  .  sin  P  C  S,  fi  being  the  index  of  refraction  of  the  first  medium  from  the  medium 

surfaces.  /<• 

Fig  in  which  the  ray  originally  moved,  which  we  will  at  present  suppose  a  vacuum,  then  will  S'C  be  the  direction  of 

the  ray  after  the  first  refraction.     Again,  let  ft'  =   the  relative  refractive  index  of  the  second  medium  out  of  the 
first,  or  ft.  fi!  =  its  absolute  refractive  index  from  a  vacuum  ;    draw  C  S"  in   the  plane   S'C  P'  so  as  to   make 

sin  P'  C  S"  =  —  r  .  sin  F  C  S',  then  will  S"  C  be  the  direction  of  the  ray  after  the  second  refraction,  and  so  on. 

(* 

199.  General  analysis.    Let  a  =  S  C  P  the  first  angle  of  incidence,  a!  =.  S'  C  P'  the  angle  of  incidence  on   the 

second  surface,  I  =  P  C  P"  the  inclination  of  the  two  first  surfaces  to  each  other,  and  putting,  moreover, 

0  =   p  S'  Pf  =   the  angle  which  the  planes  of  the  first  and  second  refraction  make  with  each  other. 

Y'  =   S  P  f  =  the  angle  made  by  the  plane  of  the  first  refraction  with  the  principal  section  of  the  two  first 

refracting  surfaces. 

0  =   S'  P*  P  =   the  angle  made  by  the  plane  of  the  second  refraction  with  the  same  principal  section. 
p   =  PCS'         the  first,  and  />'  =  P'  C  S"  the  second  angle  of  refraction. 
D  =   S  C  S"         the  deviation  after  the  second  refraction. 


LIGHT.  371 

Lignt       We  have  (conceiving  S  S' S"  P  P'  to  be  a  portion  of  a  spherical  surface  having  C  for  its  centre)  in  the  spherical     Part  I. 

— v^—'  triangle  S'  P  P7  given  P  S',  P  V,  and  the  included  angle,  required  S'P'andthe  angles  PS'P',  PP'S';  and,  v — v ' 

again,  in  the  triangle  S  S'  S"  given  S  S',  S  S"  and  the  angle  S  S'  S",  required  S  S''  the  deviation.  Or,  in 
symbols,  since  p  and  /•'  arc  the  angles  of  refraction  corresponding  to  the  angles  of  incidence  a,  a1,  and  the  indices 
of  refraction  ft,  ft', 

-  sin  a.  =  n  .  sin  /> 

cos  a'  =  cos  p  .  cos  I  +  sin  p  .  sin  I  .  cos  Y" 

sin  a'  =  fif .  sin  p' 

sin  a! .  sin  0  =  sin  I  .  sin  Y" 

sin  a!  .  sin  0  =  sin  p  .  sin  Y" 

cos  D  =  cos  (a  -  p)  .  cos  (a'  —  /)  —  sin  (a  —  p)  .  sin  (<*'  —  />')  .  cos  0. 

From  these  equations,  which,  however,  are  rather  more  involved  than  in  the  case  of  reflexion,  (Art.  99,  200. 
equation  A,)  we  may  determine  in  all  circumstances  the  course  of  the  ray  after  the  second  refraction  ;  and,  in  like 
manner,  as  in  the  case  of  reflexion,  of  any  of  the  eleven  quantities  a,  a',  p,  p',  fi,  [if,  I,  6,  <j>,  Y*,  D,  any  five  being 
given  the  remaining  six  may  be  found,  -md  we  may  then  go  on  to  the  next  refraction,  and  so  on  as  far  as  we 
please.  It  is  needless  to  observe,  however,  that,  except  in  particular  cases,  the  complication  of  the  formula 
becomes  exceedingly  embarrassing  when  more  than  two  refractions  are  considered.  Such  is  the  general  analysis 
of  the  problem ;  but  the  importance  of  it  in  optical  researches  requires  an  examination  in  some  detail  of  a  variety 
of  particular  cases. 

Case  1.  When  two  plane  surfaces  only  are  concerned,  at  both  of  which  the  refractions  are  made  in  one  plane,       201. 
viz.  that  of  the  principal  section  of  the  two  planes,  or  of  the  prism  which  they  include.  Case  1. 

Let  the  ray  S  C  (fig.  28)  be  incident  from   vacuum  on  any  refracting  surface  A  C  of  a  prism   CAD,  in  the  w!len  .both 
plane  of  its  principal   section  ;    draw  PC  perpendicular  to  that  surface,  and  draw  CS'  so  that  sin  P  C  S'  :  sin  aremade^n 
PCS  :  :  1  :  «,  then  will  S'C  be  the  direction  of  the  refracted  ray  CD.     Again,   draw   C  P'  perpendicular  to  one  piane. 
A  D,   and  take  the  angle  S"  C  F,  such  that  sin  P'  C  S''  :  sin  P'  C  S'  :  :  1  :  /*',  /  being  the  relative   index  of  Fig.  28.  ' 
refraction  from  the  medium  A  C  D  into  the  medium  A  D  E,  then  will  S"  C  be  parallel  to  the  ray  after  the  second 
refraciion;  draw,  therefore,  DE  parallel  to  Sr  C,  and  DE  will  be  the  twice  refracted  ray.     As  in  the  general 
case,  calling  S  C  P,  a. ;  S'  C  P,  p ;  S'  C  P',  * ;   S"  C  P',  p1 ;  and  P  C  P',  I,  &c. 

we  have 
and 

The  first  of  these  equations  gives  p  when  ft-  and  a  are  known  ;  the  second  gives  the  value  of  <*'  when  p  is  found ; 
the  third  gives  /•/  when  «.'•  and  p'  are  known  ;  and  the  last  exhibits  the  deviation  D. 

The  sign  of  D  is  ambiguous.     If  we  regard  a  deviation  from  the  original  direction  towards  the  thicker  part  of      202. 
the  prism,  or  from  its  edge  as  positive,  which  for  future  use  will  be  most  convenient,  we  must  use  the  lower  sign 

or  take  D  =  p'  —  I  —  a. ;  (6) 

but  if  vice  versa,  then  the  upper  sign  must  be  used.     We  shall  adhere  to  the  former  notation. 

Case  2.  If,  in  case  1,  we  suppose  the  medium  into  which  the  ray  emerges  to  be  the  same  as  that  from  which       203. 

j  Case  2. 

it  originally  entered  the  prism,  (a  vacuum,  for  example,)  we  have  fi'  =  .      This    is    the   case    of   refraction  Bothrefrae 

fi  tions  in  one 

through  an  ordinary  prism  of  glass,  or  any  transparent  substance.     In  this  case,  I  is  the  refracting  angle  of  the  tne  faces  of 
prism,  fj.  its  refractive  index,  (its  absolute  refractive  index  if  the  prism  be  placed  in  vacuo,  its  relative,  if  in  any  a  prism  in 
other  medium,)  and  the  system   of  equations   representing  the  deviation   and  direction   of  the    refracted  ray  vacuo. 
becomes 


sin  a.  =  i*  .  sin  p ;    a.'  =  1  +  p;    sin  a!  =  fi  .  sin  p' \ 

_  //_  t  -  -  (ff) 


sin  a.  =  ft,  .  sin  p 

sin  u1  =  I  +  p 

t  •      '    }••  <c> 

PHI  //  =  /LI  .  sin  a      I 

tin  D  =  f'-  a  -  I  / 


i 


Corol.  1.    The  deviation  may  be  expressed  in  another  form,  which  it  will  be  convenient  hereafter  to  refer  to        204. 
For  we  have 

sin  (I  +  D  4    -i)  =  sin  p •  =  /*  .  sin  a1  =  fi .  sin  (I  4-  f) 

=  »  {  sin  p  .  cos  I  +  cos  p  .  sin  I  } 

=  yu  |  sin  p  —  2  sin  p  .    (  sin  —  J   +  2  .  cos  p .  cos  —  .sin  —  j 

3  c  <2 


L  I  G  H  T. 

,    /   .         I   \ s  I  I  Pirt  I 

cos  1  =  1  —  2   I  sin   — 1     and  sin  I  =  2  .  sin  —  .  cos  — .  \— ,,-. 

sin  a.  l>y  the  first  of  the  equations  (c),  hence  we  get  (equation  d) 

sin  (I  +  D  +  a)  =  sin  a  +•  2  /i .  sin  —  .  cos  /  -  -  +  f  J  :  (d) 

whence,  I  and  a  being  given,  and  p  calculated  from  the  equation  sin  />  =    —  sin  a,  D  is  easily  had. 

205.  Carol.  2.  If  o  =  o,  or  if  the  ray  be  intromitted  perpendicularly  into  the  first  surface,  we  have  also  p  —  o,  ;mfl 
the  expression  (d)  becomes  simply 

sin  (I  +  D)  =  fi  .  sin  I ,  (e) 

whence  also  a  =     sl"  <'  +  D>-  {  (f) 

sin  I 

Thus  we  see  that  if /u  .  sin  I  7  I,  or  if  I,  the  angle  of  the  prism,  be  greater  than  sin"1  ,*  the  critical  angle,  or 

the  least  angle  of  total  internal  reflexion,  the  deviation  becomes  imaginary,  and  the  ray  cannot  be  transmitted 
at  such  an  incidence. 

206.  Carol.  3.  The  equation  (/")  affords  a  direct  method  of  determining  by  experiment  the  refractive  index  of  any 
1st  mode  of  medium    which  can  be  formed  into  a  prism.     We  have  only  to  measure  the  angle  of  the  prism,  and  the  deviation 
determining  of  a  rav  jntromitted  perpendicularly  to  one  of  its  faces.     Thus  I  and  D  being  given  by  observation,  u,  is  known. 
re'fracti     °  This  's  not>  however,  the  most  convenient  way  ;  a  better  will  soon  appear. 

i>v  experi-         Definitions.   One  medium  in  Optics  is  said  to  be  denser  or  rarer  than  another,  according  as  a  ray  in  passing 
ment.  from  the  former  into   the  latter  is  bent  towards  or  from  the  perpendicular.     When  we  speak  of  the  refractive 

207.  density  of  a  medium,  we  mean  that  quality  by  which  it  turns  the  ray  more  or  less  from   its  course  towards  the 
perpendicular  (from  a  vacuum,)  and  whose  numerical  measure  is  the  quantity  ft  the  index  of  refraction. 

208.  Proposition.  Given  the  index  of  refraction  of  a  prism,  to  find  the  limit  of  its  refracting  angle,  or  that  which 
Limit  ot"  the  if  exceeded,  no  ray  can  be  directly  transmitted  through  both  its  faces. 

refracting          This  limit  is  evidently  that  value  of  I  which  just  renders  the  angle  of  refraction  p  imaginary  for  all  angles  of 

angle  of  a    incidence  on  the  first  surface,  or  for  all  values  of  a,  that  is,  which  renders  in  all  cases 
prism. 

/t  .  sin  {  I  +/>}  —  1  positive, 
or,  sin  (i  +  p) positive  ;    that  is,  (since  I  +  p  cannot  exceed  90°)  which  renders  in  all  cases  I  +  p  - 

sin"1  (-—  )  positive.     Now  p  =  sin"1  —     — ,  and  consequently  the  value  of  a  least  favourable  to  a  positive 

value  of  the  function  under  consideration  is  —  90°,  which  makes  />  =  —  sin  -'    ( j,  its  greatest    negative 

value.     Consequently,  in   order  that  no   second  refraction  shall  take   place,  I  must  at   least  be  such  that  I  - 
2  sin-'  f )  shall  be  positive;  that  is,  I,  the  angle  of  inclination  of  the  faces  of  the  prism  to  each   other, 

Angle  of  a  or  as  it  is  briefly  expressed,  the  angle  of  the  prism,  must  be  at  least  twice  the  maximum  angle  of  internal 
prism.          incidence. 

209.  For  example,  if  fi  =  2, 1  must  be  at  least  60°.     In  this  case  no  ray  can  be  transmitted  directly  through  an 
equilateral  prism  of  the  medium  in  question. 

JIQ  Carol.  4.    If  p.  7  1,  or  if  the  prism   be  denser  than  the  surrounding  medium,  »  .  sin  I  is  7  sin  I  and  r,in~  ' 

(jt, .  sin  I)  7  I,  so  that  the  value  of  D  (equation  (d),  Art.  204)  is  positive,  or  the  ray  is  bent  towards  the  thicker 
part  of  the  prism,  (see  fig.  29.)  If  /t  ^  1,  or  the  prism  be  rarer  than  the  medium,  the  contrary  is  the  case, 
(see  fig.  30.) 

2ii  Problem.    The  same  case  being  supposed,  (that  of  a  prism  in  vacuo,  or  in  a  medium  of  equal   density  on 

Case  of'      both  sides,)  required  to  find  in  what  direction  a  ray  must  be  incident  on  its  first  surface  so  as  to  undergo  the  least 
minimum     possible  deviation. 
deviation.         Since  D  =  p  —  a.  —  I ;  (c)  Art.  203,  and  by  the  condition  of  the  minimum,  d  D  =  o,  we  must  have 

d  p  =  d  a. 
Now  the  equations  (c)  give  by  differentiation 

d  «  .  cos  a  =  ft.  d  p  .  cos  p ;      d  a'  =  d  p  ;     dp',  cos  p  =  ft  d  a  .  cos  a, 


that  is  dp.  cos  />'  =  /*  d  p  .  cos  of  =  d  a  . 


cos  p 


*  The  reader  will  observe,  that  by  the  eipression  sin  -' is  meant  what  in  most  books  would  be  expreised  by  arc  sin  =  — . 


LIGHT. 


373 


,  or 

That  is,  squaring, 


j.  _ 

d  a 


cos  «  .  cos  a 


;    or  cos  a  .  cos  a  =  cos  a  .  cos  p. 


Part 


cos  p  .  cos  p 

(1  —  sin  a2)  (1  —  sin  a'2)  =  (1  -  sin  p")  (1  —  sin  p'1 
in  which,  for  sin  a  and  sin  p'  writing  their  equals,  /a1 .  sin  p  and  ft  .  sin  a',  we  get 

1  —  ft2  .  sin  />2  1  —  ft ?  .  sin  a' 


1  —  sin  p1 


1  —  sin  a'2 


which  gives,  on  reduction,  simply  sin  />2   =  sin   re"2,   and  therefore  /j  =    +  a.',   that  is   I  +  p  =  I  +  a',  or 
a'  =  I  +  a'.     The  upper  sign  is  unsatisfactory,  as  it  would  give  1  =  0.     The  lower  therefore  must  be"  taken, 

which  gives  «'  =  —  ,  whioh  satisfies  the  conditions  of  the  question.     We  therefore  have 


—  —  4  *  ?    sin  <z  =   —  ,*  .  sin   f-ii  sin  p'  = 


.  sin  (  —  j  . 


This  state  of  things  is  represented  in  fig.  31,  for  the  case  where  ft  7  1,  or  where  the  prism  is  denser  than  the 
surrounding  medium,  and  in  fig.  32,  for  that  in  which  it  is  rarer,  or  ft  /  1  .  In  both  cases,  a,  being  negative, 
indicates  that  the  incident  ray  must  fall  on  the  side  of  the  perpendicular  C  P,  from  the  edge  A  of  the  prism  (as 
S  C).  In  both  cases,  the  equations  p  (—  P  C  S')  =  —  £  I  (=  —  \  P  C  P')  and  a!  =  P'  C  S'.  =  +  |  P  C  P', 
indicate  that  the  once  refracted  ray  S'  C  D  bisects  the  angle  P  C  P',  and  therefore  that  the  portion  of  it  C  D 
within  the  prism  makes  equal  angles  with  both  its  faces.  In  both  cases,  also,  the  equality  of  the  angles  a  and  «/ 
(without  reference  to  their  signs)  shows  that  the  incident  and  emergent  rays  make  equal  angles  with  the  faces 
of  the  prism,  and  therefore  that  it  is  of  no  consequence  on  which  face  the  ray  is  first  received. 
Carol.  5.  In  this  case,  also,  we  have  the  actual  amount  of  the  deviation 


(/) 


H  i 

Hence  also 


Fig.  31. 
Fig.  32. 


D  =  (f  —  o  -  I  =  2  sin  -  '    (p.  .  sin   —  ^  -  I. 


212. 
Expression 
for  the 
minimum 
deviation. 


D 


=  /i  .  sin  -  —  . 


Carol.  6.  In  the  same  case,  I  being  given  by  direct  measurement,  and  D  by  observation,  of  the  actual  213. 
minimum  deviation  of  a  ray  refracted  through  any  prism,  the  value  of  ft.  its  index  of  refraction,  is  given  at  Another 

once,  for  we  have  ™ode  °f  . 

,     ,    n  determining 

s;n    (__     \  the  index  of 

\       2       '  refraction 

ft  =  —        -  y  -  .  (g)  of  a  prism 

s;n    (M  byexperi- 

\    2    I  ment. 

And   this  affords  the  easiest  and  most  exact  means  of   ascertaining    the  refractive    index     of  any  substance 
capable  of  being  formed  into  a  prism. 

Example.  A  prism  of  silicate  of  lead,  consisting  of  silica  and  oxide  of  lead,  atom  to  atom,  had  its  refracting  214. 
angle  21°  12'.  It  produced  a  deviation  of  2-3°  46'  at  the  minimum  in  a  ray  of  homogeneous  extreme  red  light  :  Examp.e. 
what  was  the  refractive  index  for  that  ray  ? 


I  =  21°  12',  ~  =  10°  36',  D  =  24°  46',  -5_  = 


9.59158 


9.26470 


23' 


.      /  I             D  v 

sin  (T  +  T) 

=  sin  22°  59' 

i 

sin      —  - 

=  sin  10°  36' 

2 

/*=  2.123 

0.32688 

Case  3.  Let  us  now  take  a  somewhat  more  general  case,  viz.  to  find  the  final  direction  and  total  deviation 
of  a  ray,  after  any  number  of  refractions  at  plane  surfaces,  all  the  refractions  being  performed  in  one  plane, 
and,  of  course,  all  the  common  sections  of  the  surfaces  being  supposed  parallel. 

Supposing  (as  above)  I  to  represent  the  inclination  of  the  first  surface  to  the  second  ;  I'  that  of  the  second 
to  the  third,  &c.  ;    and  I,  I',  &c.  to  be  negative  when  the  surfaces  incline  the  contrary  way  from  one   certain 
side  assumed  as  positive,  taking  also  i,  £',  ff',&c  ......  {(«—')  to  represent  the  several  partial  bendings  of  the  rays 

at  the  first,  second,  third,  ?tth  surface  respectively,  and  the  ther  symbols  remaining  as  before,  we  have  the 
total  deviation,  D  =  a  +  £'  -)-  ----  «(«-').  Now  we  have,  s  nee  in  each  case  0  =  ISO0, 


215. 

Deviation  of 
a  ray  af'fr 

fractions^'is 
one  plane. 


374  LIGHT. 

Light.  sin  a  =  fj.  .  sin  p  ;     a'  —  p  +  I  ;      p?  .  sin  />'  ±:  sin  a'  ;     S  =z  a  —  p  ; 

sin  a  =  ft',  sin  p'  ;  a"  ==  p'  4-  I'  ;  ;*".  sin  /'  =  sin  a";     &  s  a  —  />'  ,-  &c.  <Skc. 
Hence  we  get  (supposing  n  to  represent  the  number  of  surfaces) 

1 

sin  p  =  —  ,  sin  n 
^ 

sin  p'  =  —  f  .  sin  (I  +  p) 

P 

sin  /'=  —  f   .  sin  (I'  +  p1) 


sin  ^  <»-»  =  -jj  .  sin  (I  ("-2:  +  />  ("~2  ) 

whence  the  series  of  values  p,  f,  &c.  may  be  continued  to  the  end.     Those  delei  mined,  we  get  a,  a',  &c.  by  the 
equations  a  =  a;      a'  =r  p  +  I  ;     o"  =  /+!';    ____  u  «-D  =  ^  (»-2)  +  I  '"-2J  T 

and  finally        •  D  =  {  a  +  a'  +  ____  «  (»-D  }  —  {,>  +  /  +  ____  ^  i»-ij  } 

=  o  +  {I  +  I'  +  ____  I  ("-2)  }—p  l"~}}  . 

Now  I  +  I'  +  ...  I  <"-2>  is  the  inclination  of  the  first  to  the  last  surface,  or  the  angle  (A)  of  the  compound 
prism,  formed  of  the  assemblage  of  them  all,  so  that  we  have  in  general 

D  =  a  +  A  -/.("-''  (A) 

216.  Let  us  now  inquire,  how  a  ray  must  be  incident  on  such  a  system  of  surfaces  so  that  its  total  deviation  shall 

Ca«e  of        be  a  minimum. 
minimum          Since  d  D  =  o  and  I,  I',  &c.  are  constant,  we  must  have 

deviation 

•to'™*  ddx=dpt*-», 

number  ot 

refractions.    i_    ,  • 

out  fi   .  sin  p   =  sin  a  ^         (  p  d  p  .  cos  p  =  d  a  .  cos  a 

fi   .  sin  p  =  sin  (p  4-  I)  V  •  <  p'dp'.  cos  />'  z=  d  p  .  cos  (p  +  I) 

&c. 


and  multiplying  all  these  equations  together 

HF  .      .  pi"-'),  cos.  pcosp' cosoC1-').  -^-^ — -   =   cos  a  .  cos  (j>  +  I) cosO>>B-2>+  I C'-2)) 

d  a 
or  simply 

ft  ft' .  .  .  .  f*(°~  ''.  COS  />  .  COS  />'....  COS  pC1"1'  =  COS  a  .  COS  a' COS  a'"      ');  (j) 

this  equation,  combined  with  the  relations  already  stated,  between  the  successive  values  of  p  and  those  of  a, 
afford  a  solution  of  the  problem ;  but  the  final  equations  to  which  it  leads  are  of  great  complexity  and  high 
dimensions.  Thus,  in  the  case  of  only  three  refractions,  the  final  equation  in  sin  p  or  sin  p',  &c.  rises  to  the 
sixteenth  degree ;  and  though  its  form  is  only  that  of  an  equation  of  the  eighth,  yet  there  appears  no  obvious 
substitution  by  which  it  can  be  brought  lower.  The  only  case  where  it  assumes  a  tractable  form  is  that  of  two 
surfaces,  when  the  equation  (;')  which  in  general  may  be  put  under  the  form 

p*^* fiW*  (1  —  sin  ?")  (1  -  sin  f2),  &c.  =  (1  -  fS .  sin  p")  (1  —  «'    .  sin  p"),  &c.  (;') 

reduces  itself  by  putting  sin  ?2  =  x,    and  sin  p'*  =  y, 

to  (p. 4/t'2  —  1)  —  ft-  (/i'2  -  1)  x  —  ,«'a  (*ta  —  l)y  =  0, 

which,  combined  with  the  equation  ft' .  sin  p'  =  sin  (j  +  I) 

or  (>/s  y  +  x  —  sin  I8)5  =  4  (*'  - .  cos  I2  .  x  y, 

gives  a  final  equation  of  a  quadratic  form  for  determining  x  or  y,  and  which  in  the  particular  case  of  /»/*  =  1, 
or  when  the  second  refraction  is  made  into  the  same  medium  in  which  the  ray  originally  moved  before  its  first 
incidence,  gives  the  same  result  we  have  already  found  for  that  case  by  a  similar  process.  Meanwhile,  though 
we  may  not  be  able  to  resolve  the  final  equations  in  the  general  case,  the  equation  (j)  affords  a  criterion  of  the 
state  of  minimum  deviation  which  may  prove  useful  in  a  variety  of  cases. 


L  I  G  H  T.  375 

Light.          Case  4.  When  the  planes  of  the  first  and  second  refraction  are  at  right  angles  to  each  other,  required  the  rela       Pan  I. 
^— ^    »  '  tions  arising  from  this  condition.  '  "^  v  "*" 

In  this  case  we  have  6  =  90°,  cos  0  =  0,  sin  0  =  1,  so  that  the  general  equation  (B,  199)  becomes  217. 

Case  when 
sin  a  =  /j.  .  sin  p          ~\  the  plane" 

of  the  firs* 
sin  a  =:  p  .  sin  p'          >and  cos  a'  =  cos  j  .  cos  I  +  sin  p  .  sin  I  .  cos  ty.  and  second 

retraction 
sin  a'  •=.  sin  1  .  sin  ty}  are  at  right 


The  last  of  these  equations,  by  transposition  and  squaring,  becomes 

cos  a'2  —  2  .  cos  a' .  cos  p  .  cos  I  +  cos  £  .  cos  1 2  =  sin  p*  .  sin  I*  (1  —  sin  ^) 

in  which,  substituting  for  sin  ^  its  value  -  — —   deduced  from  the  third  equation,  and  reducing  as  much  as  pos 

sible,  we  obtain 

cos  a'2 .  cos  p2  —  2  .  cos  a' .  cos  p  .  cos  I  +  cos  1 2  =  0, 

which,  being  a  complete  square,  gives  simply 

cos  p  .  cos  a'  =  cos  I.  (k) 

This  answers  to  the  equation  cos  a  .  cos  a!  =  cos  I,  obtained,  on  the  same  hypothesis,  in  the  case  of  reflexion 
(104)  ;  for  since  the  latter  case  is  included  in  the  case  of  refraction,  by  putting  p,  =  —  1  (Art.  192)  we  have 
then  a  =  —  p  and  cos  p  =  cos  a. 

Corol.  1.  If  i  and  i'  be  the  inclinations  to  the  first  and  second  surfaces  respectively  of  that  part  of  the  ray       218. 
which  lies  between  the  surfaces,  we  have 

i  =  90°  -  f        and  i'  =  90°  -  a, 

so  that  the  equation  above  found,  gives 

sin  i  .  sin  i'  =  cos  I, 

or  the  product  of  the  sines  of  the  inclination  of  the  ray  between  the  surfaces  to  either  surface  is  equal  to  the 
cosine  of  the  inclination  of  the  two  surfaces.  The  same  relation  may  be  expressed  otherw.ise,  thus  :  if  we 
suppose  the  ray  to  pass  both  ways  from  within,  out  of  the  prism,  the  product  of  the  cosines  of  its  interior 
incidences  on  the  two  surfaces  is  equal  to  the  cosine  of  their  inclination  to  each  other.  In  this  way  of  stating 
it,  the  case  of  reflexion  is  included. 

Corol.  2.  We  have  also  in  the  present  case  gio 


.    Ai*  .  sin  I2  —  sin  «2  1   .   / V  •  sin  I*  -  sin  oa 

S'n  °  =  V   ^_sira»         '  *  /=  2TV    '      ^-sina2~ 


1 

sin  p  =  .  sin  a  ; 

f>,  v  /*»—  sin 


and  cos  D  =  cos  (a  —  5)  .  cos  (a'  —  p') 

so  that  a  being  given,  all  the  rest  become  known.     The  last  equation  corresponds  to  the  equation  cos  D  =  cos 
2  a  .  cos  2  a  in  the  case  of  reflexion. 


§  VII.     Of  Ordinary  Refraction  at  Curried  Surfaces,  and  of  Diacaustics,  or  Caustics  by  Refraction. 

The  refraction  at  a  curved  surface  being  the  same  as  at  a  plane,  a  tangent  at  the  point  of  incidence,  if  we       220. 
know  the  nature  of  the  surface,  we  may  investigate,  by  the  rules  of  refraction  at  plane  surfaces,  combined  with 
the  relations  expressed  by  the  equation  of  the  surface,  in  all  cases,  the  course  of  the  refracted  ray.     We  shall 
confine  ourselves  to  the  simple  case  of  a  surface  of  revolution,  having  the  radiant  point  in  the  axis. 

Proposition.  Given  a  radiant  point  in  the  axis  of  any  refracting  surface  of  revolution,  required  the  focus  of       221. 
any  annulus  of  the  surface.  General  in- 

Let  C  P  be  the  curve,  Q  the  radiant  point,  Q  q  N  the  axis,  P  M  any  ordinate,  P  N  a  normal,  and  P  q  or  9  P  vestigation 
the  direction  of  the  refracted  ray,  and  therefore  q  the  focus  of  the  annulus  described  by  the  revolution  of  P.  of  the  focus 
Then  if  we  put  p  for  the  re  ractive  index,  and,  assuming  Q  for  the  origin  of  the  coordinates,  put  QM  =  x,  "urf^e"^8 
d  y  revolution. 

M  P  =  y,  r  —  "/i*  +  y\   p—  —?-,  we  have  Fi8-  33- 

d  I 

sin  QPM  =  — ;  cc 


376  LIGHT. 

1  Par  I. 


sin  N  P  M  = ;          cos  N  P  M  = 


V 


consequently  sin  N  P  Q  =  sin  Q  P  M  .  cos  N  P  M  +  sin  N  P  M  .  cos  Q  P  M 

x  +  py 
r.  v'l  -rpa> 

and  therefore  sin  N  P  9  =  .  sin  N  P  Q  = 


v  l  +  p* 

g 

and  cosNP9  =  --  ,    if  we  put  Z  =   J~*  rs  (1  +  »8)  —  (x  +  py)  »;  (a) 

ft  r  v  1  +  .p* 


consequently  since  MP9  =  NP9  +  NPM,          we  get 

x+py+pZ  -p(x  +  py)+Z 

sin  M  P  q  =  -  Vr  -  f—r-  ;         and  cos  M  P  q  =  -  -  -        —  - 
/*r(l  +  p*)  nr(\+p*) 

sin  M  P  9  x  +  py  +  p  Z 

whence  tan  M  Po  =  -        *      -   -  —  . 

cosMP9  -p(,r 


Now  we  have  M  q  =  P  M  .  tan  M  P  q  =  y  .  tan  M  Pg  =  -J^j    +/'>++f'?))    i 

consequently  Q9  =  jr+y.tanMP9  =  (x+Py).  Y(x'+  pii)  -~Z  '  (c) 

222.          Carol.  1.  If  we  put  s  for  the  arc  C  P  of  the  curve,  we  have,  since  rdr  =  xdx  +  y  dy  •=.  dx  (.r  + 


•223.          Carol.  2.    If  p  =  —  1,  in  which  case  the  refraction  becomes  a  reflexion,  we  have 

Z  —  "J  r*  (1  +  p*)  —  (x  +  py)1*  =•  y  —  px,  writing  for  r5  its  value  zj  +  yc  ;  so  that  the  general  value  above 
found  for  Q  9  reduces  itself  to 

Q9  =  2. 


which  is  the  same  as  that  found  in  (6)  Art.  109,  in  the  case  of  reflexion. 
22 1.          Carol.  3.   If  we  put  P  =  tan  M  9  P  =  cotan  M  P  q  =  . 


we  have  P=    ~  .  W 

f  +  f9+P* 

and  the  equation  of  the  refracted  ray,  if  X  and  Y  be  its  coordinates,  (Q  being  their  origin)  will  be  (since  Y  lies 
on  the  opposite  side  of  the  curve  from  Q) 


225  In  the  case  of  parallel  rays  these  expressions  become  (by  putting  first  x  +  a  for  x,  and  then  making  a  infinite) 

(g) 


-!   -p 


LIGHT. 


377 


VIII.    Of  Cawtics  by  Refraction,  or  Diacaustics. 


The  theory  of  Diacausti  ,3  is  in  all  respects  analogous  to  that  of  Catacaustics  already  explained.     To  find  the       226. 
coordinates  X  and  Y  of  the  point  in  the  diacaustic  which  corresponds  to  the  point  P  in  the  refracting  curve,  we  FlS-  34- 
have  only  to  regard  the  equation  (/)  and  its  differential  with  respect  to  x,  y,  and  p  alone,  as  subsisting  together, 
and  we  get  the  necessary  equations  for  determining  X  and  Y  in  terms  of  x,  y,  as  in  the  case  of  reflexion,  and 


these  are 


X  =  x   + 


dx; 


Y  =  y  -  P  . 


d  x  ; 


(i) 


the  only  difference  is  in  the  signs  and  in  the  value  of  P,  which,  instead  of  the  formula  (e,  Art.  110,)  is  here 
expressed  by  the  more  complicated  function  (e,  Art.  223,)  and  the  equation  of  the  diacaustic  will  be  obtained  as 
before  by  eliminating  all  but  X,  Y  from  these. 

It  is,  evident,  moreover,  that  if  we  suppose,  as  in  the  theory  of  Catacaustics,  M  =  —  -T-J;  —  d  x  ;    and   put       227. 
S  for  the  length  of  the  caustic,  and  /for  the  line  P  y,  we  shall  have,  exactly  as  in  that  theory, 


See  Art.  139,  143,  144. 

Now  we  have,  substituting  for  P  its  value  (e), 


x  +  py  +  pZ' 
and  consequently  the  value  of  d  S  becomes 


andrfS  =  df+ 


r 


V  i  + 


py+pZ 


dS  =  df 


=df+ 


,  because  (x  +  py)dx  = 


and  integrating 

so  that  we  have,  finally,  (tig.  34,) 


S  =  const  4-  f  +  -  ; 


Fig.  34. 


arc  F  y  —  (C  F  -  P  y)  +  —  (Q  C  —  Q  P). 


(/) 


In  the  case  of  reflexion,  /*  =  —!,  but  at  the  same  time  the  sign  of  f  is  negative,  because  in  this  case       228 
the  reflected  ray  lies  on  the  same  side  of  the  point  of  incidence  with  the  incident  one  ;  thus  both  terms  of 
the  formula  change  their  sign,  and  this  expression  coincides  with  that  found  Art.  144. 

In    the    case  of  parallel  rays,  we  must   use    the  value  of  P  found    in    Art.  225,   equation  (g).      Putting      229. 


9  = 


dp 


'  '    T 


and  executing  the  operations,  we  find,  then, 


Y  l 

x  =  *--p- 


S  (1  +  p')  -  1 

fflq 

^(1  +y2)-  1 

^•g 


(m) 


230. 


Carol.  If  we  suppose  /*  =  x  ,  or  the  refractive  power  infinite,  the  refracted  ray  will  coincide  with  the 
normal,  and  the  caustic  will  be  identical  with  the  evolute  ;  and  it  is  evident  that  the  expressions  (m),  when 
p  =  CD  ,  resolve  themselves  into  the  well-known  values  of  the  coordinates  of  the  evolute. 

If  the  rays  incident  on  the  refracting  curve  do  not  diverge  from  one  point,  but  be  all  tangents  to  a  curve      231. 
V  V  V",  (fig.  35,)  we  must  put  x  —  a  for  x  in  the  value  of  P,  (eq.  (e)  Art.  224  ;)  and  fix  the  origin  of  the  Fig.  35. 
coordinates  at  A,  putting  A  Q  =  a  ;  and  if,  then,  we  regard  a  as  variable  according  to  any  given  law,  (or 
regard  x  —  a  at  once  as  a   given  function   of  x,)  and  take  the    differential  of  P  on  this    supposition,  the 
equations  (i)  still  hold  good,  and  suffice  to  define  the  caustic. 

Problem.    The  radiant  point  and  refractive  index  of  a   medium  being   given,  to  determine  the    naUre   of      .^^ 
the  curve  surface  which  shall  refract  all  the  rays  to  one  point. 

Here  we  are  required  to  find  the  relation  between  *  and  y,  so  as  to  make  Q  q  invariable.  Let  Q  q  s:  c, 
and  we  have 

c  =  (x  +  py)  .  ~ 

This  equation  gives 


where  Z  =       ft  (*  +  y')  (1  +  ?')  -  (x  +  p  y*). 


VOL.  IV. 


(x  +  p  y)  (p  (x  -  c)  -  y)  =  Z  (x  -  c  +  p  y). 

3  D 


378  LIGHT. 

Light.      Squaring  both  sides,  and  substituting  for  Z  its  value,  we  get  Far.  I. 

(x  +  py)*{  (p  (r  -  c)  -  y)»  +  (x  -  c  +  py)*  }  =  (x  -  c 


which,  on  executing  the  operations  indicated  on  the  left  hand  side,  becomes  totally  divisible  by  1  +  p  *,  and 
reduces  itself  to 

(x  +  py)*  (y*  +  (x  -  c)2)  =  ,*»  (*  -  c 


that  is,  putting  for  p  its  value  ,  multiplying  by  d  a;2,  and  extracting  the  square  root, 

(I  X 

(x  —  c)  dx  +  y  d  y 

-  -  - 


V  (x  —  c)2  +  y* 
and  integrating  (each  side  being  a  complete  differential) 

V  a;2  -f  y  *  =  b  +  ft .  V  (x  —  e)*  +  y  2,  (n) 

which  is  the  equation  of  the  curve  required,  and  belongs,  generally,  to  a  curve  of  the  fourth  order. 
•233  Carol.  1.  About  Q,  (fig.  36,^  ••  ith  any  radius  Q  A,  arbitrarily  assumed,  describe  a  circle  A  B  D  E,  then  if  C  P 

Fig.  36.       be  the  refracting  curve,  and  we  put  Q  A  =  6,  we   have  Q  P  =  V  x*  +  y*,   P  q  —  "/(x  —  c)s  +  y4,  and  the 
nature  of  the  curve  is  expressed  by  the  property 

B  P  =  ft .  P  q,         or,  B  P  :  P  q  :  :  p  :  I. 

234.          Carol.  2.  If  6  =  o,  or  the  circle  A  B  E  be  infinitely  small,  we  have  Q  P  :  P  q  :  :  ft  :  1,  which  is  a  well  known 
property  of  the  circle.     In  fact,  in  this  case  we  have  simply 

In  this,  if  we  change  the  origin  of  the  coordinates  by  writing  x  +    — g  ^        c  for  x,  we  find 


M 

The  radius  of  the  circle,  therefore,  is  equal  to  — .  _  -i      x  Q  q,  and  the  distance  of  its  centre  from  the  radiant 

»2 
pio.  3-        point  is  — - —  X  Q  9.     Take  therefore  any  circle  H  P  C  whose  centre  is  E,  (fig.  37,)  and  two  points  Q,  q, 

such  that  Q  E  =  /*  X  E  C  and   Q  C  :  C  q  :  :  fi  :  1 .     Then  if  rays  diverge  from  Q,  and  fall  on  the  surface  P  H 
beyond  the  centre,  they  will,  after  refraction  into  the  medium  M,  all  diverge  from  q. 

9  5  Coral.  3.  If  p,  =  —  I,  the  equation  (»),  when  freed  from  radicals,  is  only  of  the  second  degree  between  x  and 

y,  and  therefore  belongs  to  a  conic  section.     On  executing  the  reduction  we  get 


*•=('-  -£)  ((4)' ='*-!)')• 


which  shows  that  the  radiant  point  Q  is  in  one  focus  and  q  in  the  o^rer,  which  is  the  same  result  as  that  before 

found  by  a  different  mode  of  integration. 

236.  Carol.  4.  When  Q  is  infinitely  distant,  and  the  rays  are  parallel,  we  iiust  shift  the  origin  of  the  coordinates 

For  parallel  from  Q  to  q,  by  putting  c  —  x  for  x,  and  afterwards  supposing  c  infinite.     This  gives 
rays  the 


curve  is  a  V  o2  —  2  CX  +  X*  +  y*  — 


conic 


Developing  the  first  term  in  a  descending  series,  we  find 
Fig.  38. 

(C  _  b)  -  X  +     *\+cf~     +  &C.   =  /.  . 

Let  c— 6  =  h,  which,  since  6  is  arbitrary,  is  equally  general,  and  may  represent  any  finite  quantity,  then,  as  c 
increases  and  at  length  becomes  infinite,  this  equation  becomes  ultimately 


h  —  x  =  ft  A/  x*  +  y*. 

Let  C  P  be  a  conic  section,  q  its  focus,  and  A  B  its  directrix,  q  M  =  x,  and  P  M  =  y,  then  will  Q  P  =  k  —  x 
if  we  take  q  \  —  h,  and  the  above  equation  we  see  expresses  that  well  known  property  of  a  conic  section,  in 
virtue  of  which  QP  :  Pq  in  a  constant  ratio,  (fi  :  1.) 

237  Corol.  5.  The  curve  is  an  ellipse  when  Q  P  7  P  </,  or  when  the  ray  is  incident  from  a  rarer  on  a  denser 

medium,  and  an  hyperbola  in  the  contrary  case.     If  Q  P  =  P  q,  the  curve  is  a  parabola  ;  in  this  case  /t  =  1,  and 
the  rays  converge  to  the  focus  at  an  infinite  distance,  i.  e.  remain  parallel. 

To  "take  a  single  example  of  the  investigation  of  the  diacaustic  curve,  fiom  the  general  expressions  above 


LIGHT.  379 

Light,      delivered, — let  the  refracting  surface  be  a  plane,  and  we  shall  have,  fixing  the  origin  of  the  coordinates  at  the      Part  I- 

— v— '  radiant  point,  and  supposing  the  axis  of  the  x  perpendicular  to  the  refracting  plane  A  C  B,  >— — v~~" 

Caustic  of  a 

x  =  constant  =  Q  C  =  a,     p  =  =  CD  .         Thus  we  get  refracting 

surface. 


p  —  


V  (^  -  1) 


Fig.  39  and 
and  therefore  by  the  equations  (f)  we  get,  substituting  these  values,  40. 


Y_ 


a 
Eliminating  y  from  these,  we  have  the  equation  of  the  caustic 


"  +  ,.*a*}M 

y3  ( 

~^~-  J 


fia 


/'/I-/.'     _Y  y  _ 
\       /t  a  J 


This  is  the  equation  of  the  evolute  of  a  conic  section  whose  centre  is  C,  and  focus  the  radiant  point  Q.  If  ft, 
be  greater  than  unity,  or  the  refraction  be  made  into  a  rarer  medium  from  a  denser,  the  conic  section  is  an 
ellipse,  (see  fig.  39,)  and  in  the  contrary  case  an  hyperbola,  (fig.  40.) 

§  IX     Of  the  Foci  of  Spherical  Surfaces  for  Central  Rays. 

Definitions.  The  curvature  of  any  spherical  surface  is  the  reciprocal  of  its  radius,  or  a  fraction  whose  nume-       239. 
rator  is  unity    and  denominator  the  number  of  units  of   any  scale  of  linear  measure  to  which  the  radius  is  Curvature 

,  J  defined, 

equal. 

The  proximity  of  one  point  to  another  is  the  reciprocal  of  their  mutual  distance,  or  the  quotient  of  unity  by      24C 
the  number  of  units  of  linear  measure  in  that  distance. 

The  focal  distance  of  a  spherical  surface  is  the  distance  from  the  vertex,  of  the  point  to  which  rays  converge,   ^241. 
or  from  which  rays  diverge  after  refraction  or  reflexion. 

The  principal  focal  distance,  or  focal  length,  is  the  distance  from  the  vertex  of  the  point  to  which  parallel    K™£% 
and  central  rays  converge,  or  from  which  they  diverge  after  refraction  or  reflexion.  Foca]    '  {h 

The  power  of  a  surface  is  the  reciprocal  of  its  principal  focal  distance,  or  focal  length,  estimated  as  in  the      343 
definitions  of  curvature  and  proximity.  Power. 

Problem.  To  find  the  focus  of  a  spherical  refracting  surface  after  one  refraction,  for  central  rays.  244. 

Here,  putting  a  for  the  distance  of  the  focus  of  incident  rays  Q,  (fig.  41,)  from  the  centre  E,  we  have  General  ei- 

pressions  foi 


any  annulus 

ind  these  substituted  in  the  general  expressions  Art.  221,  give  of  a  sphe- 

rical refract- 
y  Z  =  J  ^  r2  x*  +  (p*  r2  -  as)  y2        ^  ing  surface 

(  r*  | 

(J  0   "~7    ff,  <   J,   ^—   '  J 

ft  (ct  ^~  x)  —~  y  f*  I 


These  values  of  Q  q  and  C  q  contain  the  rigorous  solution  of  the  problem,  whatever  be  the  amplitude  (y)  of  the  Focus  for 
annulus  whose  focus  q  is,  and  we  shall  accordingly  again  have  recourse  to  them.  At  present,  however,  our  central  ra 
concern  being  only  with  central  rays,  we  must  put  y  =  0,  when  we  find  x  =  a  —  r;  yZ  —  ftrx  =  fir(a  —  r) 


a  — 


Carol.  1.  This  latter  is  the  focal  distance  for  central  rays.     Now,  since  a  —  r  =  Q  C,  this  gives  the  following      245. 
proportion, 

/..QC-QE:/i.QC::CE:C9.  (c) 

3D2 


380  LIGHT. 

L.ght.          Carol.  2.  If  we  suppose  the  focus  of  incident  rays  infinitely  distant,  or  a  ~  x,  and  take  P  the  place  of  q  for     Pa"  I- 
^- v-— '  central  rays,  on  that  supposition,  F  will  be  the  principal  focus,  and  we  shall  have  '*•"•  '™> 

246. 

Focus  for  C  F  = 

parallel  rays  /*  —  1  '  \ .  (d) 

urliAnstA    \va    alert   ^in/^  I  ^     ' 


whence  we  also  find 

CE 


?  =-•!—;     that  is,  C  E  :  C  F  :  :  /i  -  1  :  p  } 
:  E  F  :  :  /•  —  1  :  1,      and  C  F  :  F  E  :  :  /. :  1   ) 


•>47  These  results  will  be  expressed  more  conveniently  for  our  future  reference  by  adopting  a  different  notation. 

Let,  then, 

R  =  -  =  curvature  of  the  surface,  and  let  positive  values  of  r  and  R  correspond   to  the  case 

where  the  centre  E  lies  to  the  right  of  the  vertex  C,  or  in  the  direction  in  which  the 
rays  proceed. 

D  =  (fig.  42)  =   proximity  of  the  focus  of  incident  rays  to  the  surface,  D  being  regarded  as 

V  C 

positive  when  Q  lies  to  the  right  of  C,  as  in  fig.  42,  and  as  negative  when  to  the  left, 
as  in  fig.  41.  Then,  since  Q  E  =  a,  and  since  in  the  foregoing  analysis  a  is  regarded 
as  positive  when  Q  is  to  the  left  of  E,  we  must  have  (fig.  42)  Q  E  =  —  a,  and 
QC=QE-rEC  =  r  —  a,  so  that 

D  =  -  ;  a  =  —  --   -=-  .     Let  also  m  =  -  : 

r  —  a  R  D  it 

F  =  =  power  of  the  surface  : 

C  Jp 

/  =  —  -  —   =  proximity  of  the  focus  of  refracted  rays  to  the  surface. 
C  q 

Positive  values  of  F  and  /  as  well  as  of  D  and  R,  being  supposed  to  indicate  situations  of  the  points  F,  f,  Q,  E, 
respectively,  to  the  right  of  C,  or  in  the  direction  towards  which  the  light  travels.  This  is,  in  fact,  assuming  for 
our  positive  case  that  of  converging  rays  incident  on  a  convex  surface  of  a  denser  medium.  We  shall  have,  then, 


1  1111 

T;  :" 

—        -  tLL  —  JLl 


:"D;  :1T        -D'  '- 


Fundamen-   But  equation  (6)  gives  —  —  =  -  t      —     l    and  substituting  we  shall  get 

tal  equation  C  q  fir  (r  -  a) 

for  the  foci 

of  central  /=  (I  —  m)  R  +  mD.  (e) 

This  equation   comprises  the  whole  doctrine  of  the  foci  of  spherical  surfaces  for  central   rays,  and  may  be 
regarded  as  the  fundamental  equation  in  their  theory. 

In  the  case  of  parallel  rays,  we  have  D  =  0,  whether  the  rays  be  incident  from  left  to  right,  or  from  right  to 
^.x"  left.     In  -either  case,  then,  /has  the  same  value,  viz.  (1  —  m)  .  R,  and  the  principal  focal  distance  F  in  either 
:he  power    case  's  ^e  same>  being  given  by  the  equation 

of  <>ny  F  =  (1  -ni)  .  R,  (f) 

surface. 

which  shows,  moreover,  that  the  power  of  any  spherical  surface  is  in  the  direct  ratio  of  its  curvature. 

249.          Hence  also  we  have  /=  F  +  m  D.  (g) 

250  ^n  *ne  case  "^  reflex'on-  where  /»  =  —  1,  or  m  =  —  1,  these  equations  become  respectively 

Kundamen-  F  =  2R;          /=  2  R  -  D  ;          /=  F  -  D.  (A) 

tal  expres- 

sions for  the      Such  are  the  expressions  for  the  central  foci  in  the  case  of  a  single  surface.     Let  us  now  consider  that  of  any 

""caTe  fofC'  system  of  spherical  surfaces. 

reflexion"          Problem.  To  find  the  central  focus  of  any  system  of  spherical  surfaces. 

251.  Let  C',  C'',  C"',  &c.  be  the  surfaces.     Q'  the  focus  of  rays  incident  on  C',  Q"  that  of  refracted  rays,  or  the 

Central        focus  of  rays  incident  on  C",  and  so  on.     Call  also  R',  R",  &c.  the  radii  of  the  first,  second,  &c.  surfaces  «'.  u", 
focus  of  a  . 

system  of     ^     their  refractive  indices,  or  —  -  —  into  each  medium  from  that  immediately  preceding,  m'  =  -  ,  TO"=  —  , 
spherical  sin  ref  p  u.' 

surfaces  in- 

&c.     Also  let  D  =  -v  ,  D"  =  -  &c.  and  moreover  let  C'C"  =  f,  C"  C'"  =  I",  &c.  f,  I",  &c.  being 


regarded  as  positive  when  C",  C"',  &c.  respectively  lie  to  the  right  of  C',  C'',  &c.  or  in  the  direction  in  which 
the  nght  travels  ;  and  if  we  put    C/Q</    =  /,    (-v,1Q,,,   =  /",  &c.  F'  =  (1  -  m')  R',  F"  =  (1  -  m")  R"',  &c. 

we  shall  have  by  (249) 

/  =  F'  +  m'  D'  :        /'  =  F'  +  m''  D",  &c.  ;  (t) 


LIGHT.  38i 

but  we  have  also  ^art  L 

c'Q-  =  ^7  ;      C"Q"  =  —  =  c'  Q"  -  c'C",  =  ±  -r: 

and  so  on  ;   so  that  we  have,  besides,  the  following  relations, 

ry  _  rv  .  r»"  —         $         •          D'"  =  __  *—      -    &c  •  (  j") 

LI         -        L>       ,  LI  -        |        _         P,     ,.     ,  ,       _      f"t"         '  '  •" 

and  substituting  these  values  of  D",  D'",  &c.  in  the  equations  (i),  and  in  each  subsequent  one,  introducing  the 
values  of/',  f,  &c.  obtained  from  those  preceding,  we  shall  obtain  explicit  values  off,  f",  &c.  to  the  end. 

The  systems  of  equations  (i)  and  (./)  contain  the  general  solution  of  the  problem,  whatever  be  the  intervals      353 
between"  the   surfaces.     On  executing  the  operations,   however,  for  general  values  of  f,   t",  &c.  the  resulting 
expressions  are  found  to  become  exceedingly  complex,  nor  is  there  any  way  of  simplifying  them,  the  complication 
beino-  in  the  subject,  not  in  the  method  of  treating  it.     For  further  information  on  this  point,  consult  Lagrange, 
(Sur  la  TMorie  des  Lunettes,  Berlin,  Acad.  1778.)     We  shall  here  only  examine  the  principal  cases. 

Problem.   To  find  the  focal  distance  of  any  system  of  spherical  surfaces  placed  close  together.  253. 

Here  t''  t",  &c.  all  vanish,  and  the  equations  (i)  and  (/)  become  simply  Foci  of  a 

system  of 
D'  =  D'  ;         D"  =  /'  ;         D"'  —  f",  &c.  ;  spheric^ 

f  =  F  +  m!  D'  ;        /"  =  F"  +  m"  D",  &c.  ; 

•  placeo  clrst 

whence  by  substitution  we  obtain  :ogether. 

/"  =  F"  +  TO"  F  +  m  m"  D' 
j-ii>  —  F'"  ^  m'f  F"  +  m"i  m"  p'  +  m">  m"  m?  D', 

which  it  is  easy  to  continue  as  far  as  we  please. 

Carol.   1.    Let  the  number  of  surfaces  be   n,  and  let  M'  represent  /»',  or  the  absolute  refractive  index  out  of 
vacuum  into  the  first  medium;    M"  =  p'  p",  or  the  absolute  refractive   index  from  vacuum  into  the  second 
medium,  and  so  on  ;  /»',  ft'1,  &c.  representing  only  the  relative   refractive  indices  from  each  medium  into  that 
succeeding  it.     Thus  we  shall  have 

M(n)fM  -  D'  +  M'F'  +  M"F"+  ......  M<")  F«.  (fc) 

Cor.  2.  For  parallel  rays,  in  whichever  direction  incident,  we  have  D1  =  0  ;  and  the  principal  focal  length  of      255. 
the  system,  which  we  will  call  -^},  is  given  by  the  equation 

M<">  0(">=M'F'  +  M"F''  +  ____  MC»>  F<"'.  (0 

Cor.  3.   Hence  it  appears  that  0<B>,  the  power  of  the  system,  or  its  reciprocal  focal  length  for  parallel  rays,  being      256. 
found  by  the  last  equation,  the  focus  for  any  converging  or  diverging  rays  is  had  at  once  by  the  equation 

M  w  y;«)  =  M("J  0(n)  +  D'. 

For  brevity  and  convenience,  let  us,  however,  modify  our  notation  as  follows  :  confining  the  accented  letters      257. 
to  the  several  individual  surfaces  of  which  the  system  consists,  let  the  unaccented  ones  be  conceived  to  relate  Fundamen- 
to  their  combined  action  as  a  system.      Thus,  F',  F",  ....   Fw  representing  the  individual  powers  of  the  ^  *xPre! 
respective  surfaces;   let  F,  without  an  accent,  denote  the  resulting  power  of  the  system.     In   this   view  D'  may  cel"^"^! 
be  used  indifferently  ;  accented,  as  relating  to  the  incidence  on  the  first  surface  ;   or  unaccented,  as  expressing  Of  any 
the   proximity  of  the  focus  of  incident  rays  to  the  vertex  of  the  whole  system.     Similarly,  M(n)  may  be  used  system  of 
without  an  accent,  if  we  regard  the  total  refractive  index  of  the  system  as  that  of  a  ray  passing  at  one  refraction  spheric 
into  the  last  medium.     This  supposed,  the  equations  (k)  and  (/)  become 

MF  =  M'F'  +  M"F"  +  ......  M<">  F«  ;  (m) 

M/=MF+D;       M(F-/)  +  D  =  0.  («) 

If  the  whole  system  be  placed  in  vacuo,  or  if  the  last  refraction  be  made  into  vacuum,  we  have  M  =  1  =  M'"',      29b. 
and  the  equations  become 

F  =  M'F'  +  M"F"  +  ....M«  FWJ  (o) 

/=F  +  D 

Definitions.  A  lens  in  Optics  is  a  portion  of  a  refracting  medium  included  between  two  surfaces  of  revolution      259 
whose  axes  coincide.     If  the  surfaces  do  not  meet,  and  therefore  do  not  include  space,  an  additional  boundary  is  Lenses  de- 
required,  and  this  is  a  cylindrical  surface,  having  its  axis  coincident  with  that  of  the  surfaces.  '5" 


The  axis  of  the  lens  is  the  common  axis  of  all  the  bounding  surfaces.  int0  species, 

Lenses  are  distinguished  (after  the  nature  of  their  surfaces)  into  double-convex,  with  both  surfaces  convex, 
(fig.  44  ;)  plano-convex,  with  one  surface  plane,  the  other  convex,  (fig.  45  ;)  concavo-convex,  (fig.  46  ;)  double- 
concave,  (fig.  47  ;)  plano-concave,  (fig.  48  ;)  and  meniscus,  (fig.  49,)  in  which  the  concave  surface  is  less  curved 
than  the  convex.  Also  into  spherical,  (when  the  surfaces  are  segments  of  spheres  ;)  conoidal,  when  portions 
of  ellipsoids,  hyperboloids,  &o 


382  L  I  G  H  T. 

Light.          These  different  species  are  distinguished,  algebraically,  by  the  equations  of  the  surfaces,  and  by  the  signs  of    Part  I. 
•  _-v  _;  (heir  radii  of  curvature.     In  the  case  of  spherical  lenses,  to  which  our  attention  will  be  chiefly  confined,  if  we  <——>,-— 

260.      suppose  a  positive  value  of  the  radius  of  curvature  to  correspond  to  a  surface  whose  convexity  is  turned  towards 
Species  of    the  left,  or  towards  the  incident  rays,  and  a  negative  to  that  whose  convexity  is  turned  to  the  right,  or  from 

lenses  how   them,  ^e  shall  have  the  following  varieties  of  denomination  : 

distinguish- 

menisous  ~\    ("both  radii  +  ,  as  fig.  46,  49,  a,  or 

concavo-convex  j    (.both  radii  — ,  as  fig.  46,  49,  b, 

f  radius  of  first  surface  +,  of  second  infinite,  fia:.  45,  b, 
plano-convex   -J 

(.radius  of  first  surface  infinite,  of  second  — ,  fig.  45,  a, 

f  radius  of  first  surface  — ,  of  second  GO  ,  fig.  48,  b, 
plano-concave  -i 

(.radius  of  first  surface  oo  ,  of  second  +,  fig.  48,  a, 

double-convex  :   radius  of  first  surface  +,  of  second  — ,  fig.  44, 
double-concave:  radius  of  first  surface  — ,  of  second  +,  fig.  47, 

the  rays  being  supposed  in  all  cases  to  pass  from  left  to  right. 

A  compound  lens  is  a  lens  consisting  of  several  lenses  placed  close  together. 
An  aplanatic  lens  is  one  which  refracts  all  the  rays  incident  on  it  to  one  and  the  same  focus. 
2gi  Problem.   To  find  the  power  and  foci  of  a  single  thin  lent  in  vacua. 

Focus  of  a       Let  R'  and  R"  be  the  curvatures  of  its  first  and  second  surfaces  respectively,  ft  the  refractive  index  of  the 
single  lens.  \ 

medium  of  which  it  consists,  m  =  —  ;  F  its  power :  then  we  have,  since  the  last  refraction  is  made  into  vacuum, 

but,  F'=  (1  —  TO')  R',  and  F"  =  (1  —  m")  R" ;  and  as  ft1  = and  TO"  =  /j,  these  become  respectively 

—  (p  —  i)  ft*  and  —  (ft  —  1)  R'1,  so  that  the  foci  of  the  lens  are  finally  determined  by  the  equations 

F  =  (ft  -  1)  (R1  -  R")" 

Fundamen-  * 

talequations.  f  =  F  +  1) 

262.  Carol.  1.  The  power  of  a  lens  is  proportional  to  the  difference  of  the  curvatures  of  the  surfaces  in  a  meniscus 
Power  of  a  or  concavo-convex  lens  ;  and  to  their  sum,  in  a  double-convex  or  double-concave. 

leus.  In  plano-convex,  or  plano-concave  lenses,  the  power  is  simply  as  the  curvature  of  the  convex  or  concave 

surface. 

263.  Corol.  2.  In  double-convex  lenses  R'  is  positive  and  R"  negative,  so  that  when  ft  >  1,  F  is  positive,  or  the 
rays   converge  to  a  focus  behind  the  lens.     In  plano-convex,   R"  =  0  and  R'  is  +  ;  or   R'  =  0  and   R"  is 
negative,  (260)  ;    hence  in  both  cases  F  is  positive  and  the  rays   also  converge.     In  meniscus  lenses  also, 
R'is  +,  and  R", though  +,  is  less  than  R',  (fig.  49  ;)  therefore  in  these,  also,  the  same  holds  good.     In  all  these 

19        cases  the  focus  is  said  to  be  real,  because  the  rays  actually  meet  there.     In  double-concave,  plano-concave,  or 
virtuar"oci    concavo-concave  lenses,  the  reverse  holds  good  ;    the  focus  lies  on  the  opposite  side,  or  towards  the  incident 
rays,  and  parallel  rays,  after  refraction,  diverge  from  it.     In  this  case,  therefore,  they  never  meet,  and  the  focus 
is  called  a  virtual  focus. 

i>54  Corol.  3.  If  ft  be  <  1,  or  the  lens  be  formed  of  a  medium  rarer  than  the  ambient  medium  (which  need  not  be 

vacuum,  provided  the  whole  system  be  immersed  in  it,)  ft  —  1  is  negative,  and  all  the  above  cases  are  reversed. 
In  this  case  convex  lenses  give  virtual,  and  concave,  real  foci. 

-  Corol.  4.     For  lenses  of  denser  media,  the  powers  of  double-convex,  plano-convex,  and  menisci  are  positive  ; 

posjt;ve  an(j  and  those  of  double  plano-concave  and  concavo-convex  lenses,  negative  ;  vice  versa  for  rarer  media, 
negative    '       Corol.  5.    The  focus  of  parallel  rays  is  at  the  same  distance,  on  whichever  side  of  the  lens  the  rays  fall.     For 
powers.        if  the  lens  be  turned  above,  R'  becomes  R'r,  and  vice  versa;  but,  since  they  also  change  their  signs,  F  remains 
266.      unaltered. 

267  Corol.  6.    The  equation/^  F  +  D  gives  df=  d  D.     This  shows  that  the  foci  of  incident  and  refracted  rays 

Conjugate    move  always  in  the  same  direction,  if  the  former  be  supposed  to  shift  its  place  along  the  axis ;  and,  moreover, 
foci  move  in  that  their  proximities  to  the  lens  vary  by  equal  increments  or  decrements  for  each. 
the  same          Problem.  To  determine  the  central  foci  of  any  system  of  lenies  placed  close  together,  the  lenses  being  supposed 

f!'°n      infinitely  thin. 

r  tral  'foci  The  general  problem  of  a  system  of  spherical  surfaces  contains  this  as  a  particular  case  ;  for  we  may  regard 
ofa  system  the  posterior  surface  of  the  first  lens,  and  the  anterior  of  the  second,  as  forming  a  lens  of  vacuum  interposed 
of  thin  lenses  between  the  two  lenses,  and  so  for  the  rest.  Thus  the  system  of  lenses  is  resolved  into  a  system  of  spherical 
in  contact,  surfaces  in  contact  throughout  their  whole  extent ;  the  alternate  media  having  their  refractive  indices,  or  the 
alternate  values  of  M,  unity.  If  then  we  call  ft',  ft",  ft'",  &c.  the  refractive  indices  of  the  lenses,  we  shall  have 

M=l;        M'  =  ft';        M"=l;        M"'=V';        M"  =  1,  &c. 


LIGHT. 


The  compound  power  F  then  will  (258,  o)  be  represented  by 

F  =  /t'  F1  +  F"  +  n"  F'"  +  Fiv  +  /i1"  F"  +  Fvi  +,  &c. 

But  F'  =  (1  -  m>)  R'  =  4-  0*'  -  1)  R' 


because  m1  =  — ;—  and  m"  =  /t.'.     Consequently, 

,,'  F'  +  F"  =  (/«'  -  1)  (R(  -  R") 
and  similarly 

pii  fin  +  F"  =  GU"  -  1)  (R"'  -  •  Riv),  &c. 

so  that  we  get,  finally, 

F  =  0*'  -  1)  (R'  -  R'')  +  (/*"  -  i)  (R'"  -  Riv)  +  &c. 

Now,  the  several  terms  of  which  this  consists  are  (by  Art.  261)  the  respective  powers  of  the  individual  lenses 
of  which  the  system  consists,  so  that  if  we  put  (according  to  the  same  principle  of  notation)  L',  L",  L"',  &c. 
for  the  powers  of  the  single  lenses,  and  L  for  their  joint  power  as  a  system,  we  have 


Part  I. 


Superposi- 
tion of 
powers. 
Power  of  a 
system  of 
lenses  is  the 
sum  of  the 
powers  of 
the  compo- 
nent indivi- 
duals. 


L  =  L'  +  L"  +  L'"  +,  &c. 


(9) 


an  equation  which  shows  that  the  power  of  any  system  of  lenses  is  the  sum  of  the  powers  of  the  individual  lenses 
which  compose  it ;  the  word  sum  being  taken  in  its  algebraic  sense,  when  any  of  the  lenses  has  a  negative  power. 
Moreover  it  is  easy  to  see  that  we  also  have/=  L  +  D,  as  in  the  case  of  a  single  lens. 

Reciprocally,  we  may  regard  a  system  of  spherical  surfaces  forming  the  boundaries  of  contiguous  media  (as 
in  the  instance  of  a  hollow  lens  of  glass  enclosing  water)  as  cons'sting  of  distinct  lenses,  by  imagining  the 
concavity  of  one  medium  and  the  convexity  of  that  in  immediate  contact  with  it  separated  by  an  infinitely  thin 
film  of  vacuum,  or  of  any  medium  having  its  surfaces  equicurve,  as  in  fig.  50  ;  and  thus  a  system  of  any  number 
(«)  of  media,  whose  surfaces  are  in  contact  throughout  their  whole  extent,  may  be  conceived  replaced  by  an  equi- 
valent system  of  2  n —  1  lenses,  the  alternate  ones  being  vacuum,  or  void  of  power.  This  way  of  considering 
the  subject  has  often  its  use.  It,  moreover,  leads  to  the  result,  that  the  power  of  any  system  of  spherical 
surfaces  placed  in  vacua  is  the  sum  of  the  powers  of  the  several  lenses  into  which  it  can  be  resolved,  each  placed  in 
vacua  and  acting  alone. 

Let  us  now  return  to  the  case  of  surfaces  separated  by  finite  intervals  ;  and,  first,  let  us  inquire  the  foci  of  a 
system  of  surfaces  separated  by  intervals  so  small  that  their  squares  may  be  neglected.  In  this  case  the  equa- 
tions (j),  Art.  251,  become  simply 

D'=D;         D"  =  /'  +  f* 11  i        D'"  =  /"+/"2i",  &c.; 


269 


Fig.  50. 

Power  of  a 
° 


and  substituting  these  values  in  the  equations  (i),  and  retaining  the  notation  of  Art.  257,  we  find 


surfaces 
expressed. 
270. 
Foci  of  a 
system  of 
surfaces  se- 
parated by 
small  finite 
intervals. 


M"F"+  ____ 


+D 


Now  in  this  we  are  to  consider  that 

/'  =  F1  +  m'D,  /"  =  F"  +  m"F'  +  m'm"D',  &c. 
and  the  values  of  f',f",  &c.  so  expressed,  being  substituted  in  the  foregoing  equation,  we  find 

M/=  M1  F'  +  M"  F"  +  M'"  F"'  +  &c.  .  .  +  D  (r) 

+  M'  (F1  +  m'  D)a  if  +  M"  (F"  +  m"  F'  +  m"  m'  D)a  t"  +,  &c. 
Carol.  In  the  case  of  two  surfaces,  supposing  M  =  1,  or  in  the  case  of  a  single  lens  in  vacuo,  this  gives 

CO 


1 


/=  0*  -1)  (R'  -  R")  +  D  +  —  {  0»  -  1)  R'  +  D  }  4  1. 

For  parallel  rays,  this  becomes 


271. 

Case  of  a 

single  lens, 
of  small  Uii 
finite  thick- 


F =  G*  -  1)  (R'  -  R")  + 


R'a.  t; 


(0 


t  being  here  put  for  t',  the  interval  between  the  surfaces  or  total  thickness  of  the  lens. 

Problem.  To  determine  the  foci  of  a  lens,  whose  thickness  t  is  too  considerable  to  allow  of  any  of  its  powers 
being  neglected. 

Here  we  must  take  the  strict  formulae 

D'=D;         D"=_Z_;        /'=  (1  -mOR'  +  m'D;         and  /*  =  '1  -  m")  R"  +  m"  D" 


373 

Focus  O'f ,. 

lens  of  anj 
thickness. 


The  latter  equation  gives,  on  substitution,  and  recollecting  that  m'  =:  —  =  m  and  m"  =  /*, 


* 


LIGHT. 

u.  —  ]  Part  I. 

(/.  -  1)  (R'  -  R")  +  D  +  -          -  {  0*  -  1)  R'  +  D  }  R"  t  ^— v— 

f  —  f"   —    r i  (M) 

1-  _{0,_1)R'  +  D}« 

and  for  parallel  rays 

F  ==  — — — — -. ..  „, (») 

273.          Example  1.  To  determine  the  foci  of  a  sphere. 
Foci  of  a  _ 

sphere  jjere  R"  =  _  R'  —  _  R;         (  =  -_-  ;         and  the  equations  («)  and  (c)  become 

R 


F__^ 

(2  -  /.)  R  -  2  D~  -2=7"  ("° 

274.  Carol.  1.  If  /*  =  2,  for  instance,  these  values  become 

f-     ^    •         F- 
-P~' 

In  this  case,  then,  since  /  and  F  express  the  proximities  of  the  foci  to  the  posterior  surface  of  the  sphere,   w« 
see  that  the  focus  for  parallel  rays  falls  on  this  surface,  and  that  in  any  other  case  (as  in  fig.  51  and  52)  q  is  given 

by  the  proportion  Q  C  :  C  E  :  :  E  H  :  H  q, 

275.  Carol.  2.  Whatever  be  the  value  of  ft,  the  focus  for  parallel  rays  after  the  second  refraction  bisects  the  distance 
between  the  posterior  surface  of  the  sphere,  and  the  focus  after  the  first  refraction. 

276.  Example  2.  To  determine  the  foci  of  a  hemisphere,  in  the  two  cases  ;  first,  when  the  convex,  secondly,  when 
Foci  of  a     the  plane  surface  receives  the  incident  light. 

hemisphere.  i 

In  the  first  case,  R'  =  R  ;          R"  =  0  ;  t  —  —  :  therefore  we  find 

H 

fr-DR+D 
/  =  -  R—  '  D  --  '"  ~    ' 

277  In  the  other  case,  when  the  rays  fall  first  on  the  plane  side,  R1  =  0,  R"  =  —  R,  and  t  =  -—•  ,  so  that 

nfr-l)R  +  D 
^R-D 

If  the  thickness  of  a  spherical  segment  exposed  with  its  convex  side  to  the  incident  rays  be  to  the  radius  as 

*7o 

,  and  R"  =  0,  the  expressions  (M)  and  (u)  become 


to  u  -  1,  or  if  t  =  —   -j  .  -=-  =  -  -  r-5, 
ft  —  1     R        (1  —  m)  K 


In  this  case  the  focus  for  parallel  rays  falls  on  the  posterior  surface  of  the  segment. 
279.          In  general,  for  any  spherical  segment,  if  exposed  with  its  convex  side  to  the  rays,  R"  =  0,  and 

FOCUS  Of  /•1\T>|T»  /1\T> 

anyspheri-  Q.  -  1)  R  +  P  f.  (f.  -  1)  R 

cal  segmant,  •7~~^'u+{(>-l)R+D}<'  /.  +  (/*—  1)  R  > 

convex  side 

first.  if  the  plane  side  be  exposed  to  the  rays 

Plane  side  _ 


280.  If  R1  =  R     iv  if  the  lens  be  a  spherical  lamina  of  equal  curvatures,  the  one  convex,  the  other  concav*, 

Focus  of  a 
spherical  p  D  +  Q  -  1)  {  Qt  -  1)  R  +  D  }  R  t  Q.  -  1)«  R*  t 

-,-1)R+P  '*->-»  R<  ' 


curvatures. 


LIGHT. 


385 


Light. 


Part  I. 


§  X.    Of  the  Aberration  of  a  System  of  Spherical  Surfaces. 


Problem.  To  determine  the  focus  of  any  annulus  of  a  spherical  refracting  or  reflecting  surface  281. 

The  equations  (a)  of  Art.  244,  of  the  last  section,  in  fact,  contain  a  general  solution  of  this  problem  ;  but  Focus  of  a 

the  applications  of  practical  Optics  require  an  approximate  solution  for  annuli  of  small  diameter,  or  in  which  y  small  annu- 

is  small  compared  with  r.     Conceiving  y,  then,  so  small  that  its  fourth  and  higher  powers  may  be  neglected,  the  SpheriCa] 

expressions  in  the  article  cited  give  surface  in- 


and  substituting  these  in  the  value  of  C  q,  found  in  the  same  article,  we  get  for  the  distance  of  the  focus   of 
refracted  rays  from  the  vertex 

p,  -  1  a*  (a  +  ftr) 

2 


a  —  ft  a  +  p  r  2  fi          (a  —  r)  (a  —  /*  a  +  p  r)J    '     r 

In  conformity,  however,  with  the  system  of  notation  adopted  in  the  last  section,  instead  of  expressing  directly 
C  q,  we  will  take  its  reciprocal.  As  we  have  hitherto  represented  the  value  of  this  reciprocal  for  central  rays 
by  f,  we  will  continue  to  do  so  ;  and  for  rays  incident  at  the  distance  y  from  the  vertex,  we  will  represent  the 
same  reciprocal  byf  +  A  f;  A  f  then  will  be  vhat  part  of  /"due  to  the  deviation  of  the  point  of  incidence  from 
the  vertex.  Now,  neglecting  y*,  we  have 


1 


Cq 


a  —  ft  a  +  ft  r 
pr(r  —  a) 


-  1 


a"  (a 


2 


r3  (a  -  r)3 


" 


\°l 


Now    if  we    put,    as    we    have    hitherto    done,    /»   =   -  ,  r  =  —  ,  a  =  —  ---  —  ,    and   substitute   these 
in  the  above,  we  shall  get  the  value  of  —  -  —  ,  or  of/+  A  f,  in  terms  of  TO,  R,  and  D;   and  from  this,  subtracting 


the  term  independent  of  y*,  which  is  the  value  of  f,  we  shall  get  A  /as  follows, 
A/  =    m(l  ~*  ^-(R-D)' 


(c) 


Definition.  The  longitudinal  aberration,  is  the  distance  between  the  focus  for  central  rays  and  the  focus  q  of 
the  annulus,  whose  semidiameter,  or  aperture,  is  y  =  M  P. 

The  lateral  aberration  at  the  focus,  is  the  deviation  from  the  axis  of  the  refracted  ray,  or  the  portion  fk, 
intercepted  by  the  extreme  ray,  of  a  perpendicular  to  the  axis  drawn  through  the  central  focus. 

Carol.  These  aberrations  are  readily  found  from  the  value   of   A  f  above  given  ;    for  since  09  =  —7-,  we 

j 

___  I  A    f 

have  A  C  9  (=  longitudinal  aberration)  =  A  —f-  =  --  ^—  ;  or,  calling  to  this  aberration, 


283. 
Longitudi- 
nal  and 

aberration 

"' 
(&o4. 

Relation 
between 
them  and 


Cq  :  qk  :  :  y  :fk,  or  —   :  ui  :  :  y.fk, 

lateral  aberration  =  /.  y  .  o>  =  —         *    .  y; 
/=  (1  —  m)  R  + wiD. 


(e) 


and  since 

we  have  fk,  or  the 

where 

Thus  the  whole  theory  of  aberration  is  made  to  depend  on  the  value  of  A  /•  and  we  come  therefore  to  con- 
sider the  various  cases  of  this  which  present  themselves. 

Case  1.  For  parallel  rays  D  =  0  ;  and,  therefore,  285 

Case  of 
parallel  r.w. 


»OL.  iv. 


lateral  aberration  =  --  — 


386 


LIGHT. 


Light. 


Case  of 
reflectors. 


287. 
Aplanatic 
foci  defined 
and  inves- 
tigated. 


COM  2.  In  reflectors,  m  —  p  =  —  1,  and 


Kurt  L 


R  (R- 


(-2R-D)*    ' 

lateral  aberration  =  —  £  (R  —  D)2  y3, 


Of) 


lateral  aberration  =  —      R-     -. 


(A) 


which,  for  parallel  rays,  become 

In  the  general  case,  if  we  put  either  D  =  R,  or 

771 

m  R  —  (1  +  w)  D  =  o,  which  gives  D  =  „ ,  _  v_  ,   _,  . 

the  value  of  /^  ft  al>d  therefore  of  the  aberration,  vanishes.  The  former  case  is  that  of  rays  converging  to 
the  centre  of  curvature,  in  which,  of  course,  they  undergo  no  refraction.  In  the  latter,  the  point  is  the 
same  with  that  already  determined,  Art.  234.  It  is  evident,  from  what  was  there  demonstrated,  that  every 
spherical  surface,  C  P,  has  two  points  Q,  q  in  its  axis,  so  related,  that  all  rays  converging  to  or  diverging  from 
one  of  them,  shall  after  refraction  rigorously  converge  to  or  diverge  from  the  other.  These  points  may  be  called 
the  aplanatic  foci  of  the  surface;  and,  to  distinguish  them,  Q  may  be  called  the  aplanatic  focus  for  incident,  and 
q  for  refracted  rays.  To  find  them  in  any  proposed  case,  in  the  axis  of  any  proposed  surface  C,  and  on  the 


concave  side  of  the  surface,  take 


X  radius. 


288. 

Aberration 
shortens  the 
focus  for 
parallel  rays 

289.  ' 
Effect  of 
aberration 
in  other 
cases. 
Fig.  54. 

290. 


291. 

Aoerration 
of  any 
system  of 
spherical 
surfaces  in 
contact. 


=  (/i+l)X  radius  C  E  of  the  surface,  and  Cer  =  {  —  +  1  1 

V  f-         ' 

Then  will  Q  and  q  be  the  aplanatic  foci  required.     In  the  case  of  reflexion,  when  /*  =  —  1,CQ  =  C7==0,  and 
both  the  aplanatic  foci  coincide  with  the  vertex  of  the  reflector. 

Let  us  next  trace  the  effect  of  aberration  in  lengthening  or  shortening  the  focus,  for  all  the  varieties  of  position 
of  the  focus  of  incident  rays;  and,  first,  when  D  =  0,  or  for  parallel  rays,  A  fis  of  the  same,  and  therefore  u> 
of  the  contrary  sign  with  R,  and  therefore  with  F,  which  is  equal  to  (1  —  m)  .  R.  Hence  it  is  evident,  that  the 
effect  of  aberration  in  this  case  must  be  to  shorten  the  focus  of  exterior  rays. 

Q  in  this  case  is  infinitely  distant.  As  it  approaches  the  surface,  or  as  the  rays  from  being  parallel  become 
more  and  more  convergent,  or  divergent,  the  aberration  diminishes  ;  but  the  focus  of  exterior  rays  is  still  always 
nearer  the  surface  than  that  of  central,  till  Q  comes  up  to  the  aplanatic  focus  A.  for  incident  rays  on  the  concave, 
or  to  the  focus  F  of  parallel  rays  on  the  convex  side.  When  Q  is  at  the  former  of  these  points,  the  aberration  is 
0  ;  at  the  latter,  infinite. 

When  Q  is  situated  anywhere  between  these  points,  however,  the  reverse  is  the  case,  and  the  effect  of  aberra- 
tion is  to  throw  the  focus  for  exterior  rays  farther  from  the  surface  than  that  for  central  ones.  These  results  are 
easily  deduced  from  the  consideration  of  all  the  particular  cases,  and  hold  good  for  all  varieties  of  curvature,  and 
for  refracting  media  of  all  kinds.  In  reflectors,  the  aplanatic  foci  coincide  with  the  vertex.  In  these,  the  focus 
for  exterior  rays  is  shorter  than  for  interior  in  every  case,  except  when  the  radiant  point  is  situated  between  the 
surface  and  the  principal  focus  on  the  concave  side  of  the  reflecting  surface  ;  but  between  these  points,  longer. 

Problem.    To  determine  the  aberrations  of  any  system  of  spherical  refracting  surfaces  placed  close  together. 

Retaining  the  notation  of  Art.  257,  let  us  suppose  the  ray,  after  passing  through  the  first  surface,  to  be  incident 
on  the  second.  Its  aberration  at  this  will  arise  from  two  distinct  causes  :  first,  that  after  traversing  the  first 
surface,  instead  of  converging  to  or  diverging  from  the  focus  for  central  rays,  its  direction  was  really  to  or  from 
a  point  in  the  axis  distant  from  that  focus  by  the  total  aberration  of  the  first  surface  ;  and.  secondly,  that  being 
incident  at  a  distance  from  the  vertex  of  the  second  surface,  a  new  aberration  will  be  produced  here,  which  (being, 
as  well  as  the  other,  of  small  amount)  the  principles  of  the  differential  calculus  allow  us  to  regard  as  independent 
of  it,  and  which  being  computed  separately,  and  added  to  it,  gives  the  whole  aberration  of  the  two  surfaces 
regarded  as  a  system.  The  same  is  true  of  the  small  alterations  in  the  values  of  f,  f",  &c.  produced  by  the 
aberrations.  If  then  we  denote  by  f>  f"  the  change  in  the  value  of  /",  produced  by  the  action  of  the  first 
surface,  and  by  i1  f",  that  arising  immediately  from  the  action  of  the  second,  and  by  A  f",  the  total  alteration 
produced  by  both  causes,  we  shall  have 

A/''=  £f"+  c'f" 

Now,  first,  to  investigate  the  partial  alteration  if"  arising  from  the  total  alteration  A/'  in  the  value  of/',  or 
from  the.  aberration  of  the  first  surface,  we  have 


since,  in  this  case, 


/"  =  (1  -  m)  .  R"  +  m"  /',  and  therefore  c  /"  =  m"  A  /', 
D'  =  D,     D"  =  /',     D"'  =  /",  &c. 


Again,  to  discover  the  partial  variation  S'f"  in  /",  arising  immediately  from  the  action  of  the  second  surface, 
we  have,  by  the  equation  (c)  at  once,  putting/'  for  D",  and  neglecting  y4,  &c. 


*'/"  = 


fcut  we  have,  by  the  same  equation,  also 
£f"  =  m"  A  /'  - 


(R" 


{  m"  R"  -  (1  +  m")f  }  y*  ; 


-  _  D)*  {  m1  R'  -  (1  +  m')  D  }  y« 


LIGHT. 


387 


Light.      Consequently,  uniting  the  two,  we  have  the  value  of  ^  f".     Similarly,  the  value  of  A  f"  may  be  derived  from     Part  !• 
—  v~~^  that  of  A  f",  by  a  process  exactly  the  same,  and  which  gives  ~-"~v~~ 

lll~\ 

;  (R'"  -  /")2  {  7n'"R'"  -  (1  +  m'")  /"  }  y\ 


mhl  (\ 

*• 


and  so  on.     Calling,  then,  as  in  Art.  257,  M',  M  ',  M"'.  .  .  .  M(n)  the  absolute  refracting  indices  of  the  several  media 
into  which  the  successive  refractions  are  made,  and  putting  M("'  =  M,  we  shall  have  no  difficulty  in  arriving  at 
the  following  general  expression,  where  A  /denotes  the  total  effect  of  aberration  on  the  value  of  f,  the  reciprocal  General 
focal  distance  of  the  system, 


M' .    m'  (l~  m>)  <R'  -  D)2  {  TO'  R'  -  (I  +  m')  D  } 


expression 

for  A/ 


M.  A/=< 


+  M"  .  7""(1 


(R"  -/')*  {  m"  R"  -  (I  +  IB")/'  } 


+  M"'  .  m"'  (12    OT'")   (R"1  -/")*  {  m"1  R"'  -  (1  +  m"')/"} 
+   &c. 


Successive 
values  of/. 


in  which  it  will  be  recollected  that 

/'=(!-  m')    R'    +  m'  D 

/"  =  (1  -  m")  R"  +  m"  (1  -  m')  R1  +  m'  m"  D 

/'"  =  (1  -  m'")  R'"  +  m1"  (1  -  m")  R"  +  m"1  «i"  (1  -  O  R'  +  m1"  m'1  m1  D 
&c. 

and  these  values  being  substituted  give,  if  required,  an  explicit  resulting  value  of  A  /  in  terms  of  the  radii  and 
refractive  indices,  or  their  reciprocals,  of  the  surfaces. 

If  the  system  of  surfaces  be  placed  in  vacuo,  or  the   last   refraction  be  made  into  vacuum,  M  =  1,  and  the      292. 
second  member  of  the  equation  (i)  exhibits  simply  the  value  of  A  f.     In  all  cases,  the  aberration  <u  is  given  as 
before  by  the  equation 


"f 


>  and  the  lateral  aberration  is 


A/ 
/ 


y- 


To  express  the  aberration  of  any  infinitely  thin  lens  in  vacuo,  let  the  terms  of  the  general  equation  be  denoted       293 
respectively  by  Q',  Q",  &c.,  so  as  to  make  Aberration 


M  .  A  /  =  {  Q1  +  Q"  +  Q"1  +  ,  &c.  }  y  ». 


(A) 


.  . 

Then,  for  the   case   of  a   single   lens   in   vacuo,   when  TO"  =  —  -,  M'  =  —  -r,  M"  =  1,  M  =  1,  we  have 

m  m 

A  /=  Q1  +  Q"  ;  and  putting,  for  a  moment,  R'  -  D  =  B,  R'  —  R"  =  C,  we  find 

i  _  m> 
' 


thin  lens. 


Q"=- 


2  m 


whence 


Q"= 


2m'3 


(C  -  m' 


The  expression  in  brackets,  putting  for  B  and  C  their  values,  and  -  for  m1,  will  become 


--  {  ((2  -  ft  R'  +  f.  R"  -  2  D)  (R'  -  (1+  /.)  D)  +  f.  (  0*-  1)  R'  -/,  R"  +  D)'}. 
If  now  we  multiply  out,  arranging  according  to  powers  of  D,  and  substitute  the  result,  as  also  the  value  of  m!, 
(=  —  ,)  and  of  C,  (  =  R'  -  R",)  in  Q'  +  Q",  or  A  /,-  we  get 


where 


a  =  (-2  -  2 


R'2  +  (/i  +  2  /t2  -  2  ^3)  R1  R"  +  n3  R"* 


General 
expi 
for  it. 


B  =  (4  +  3  M  -  3  /*2)  R'  +  (/»  +  3 
'i  —  2  J-  3  ft 


R" 


(0 


3E2 


388  LIGHT. 

Light.      Now  it  has  been  shown,  (Art.  261,)  that  (^  —  1)  (R1  —  R")  expresses  the  power  of  the  lens,  so  that,  putting  L      Part  I. 
v—  -v—  "''  for  this,  we  have  v—  ^*—  • 

A/=-^-(a-/3D  +  -/B"-)3/2.  (m) 

Such  then  is  the  general  expression  for  A  f,  the  fundamental  quantity,  from  which  the  aberration  to  may  be  had 
in  any  lens  by  the  equation  «>  =  —         "   . 

294  Corol.  1.  The  aberration  of  a  lens  vanishes  when  D  is  so  related  to  R',  R"  and  p,  as  to  give 

Cases  in 


of  a  single 

lens  can  be  Now  we  find,  by  substitution  and  reduction, 

made  to 

vanish.  P*  -  4  a  7  =  p*  {  (R1  +  R   )  -  -  (2  p.  +  3  /i2)  (R1  -  R")  '  } 

and  unless  this  quantity  be  positive,  that  is,  unless 


the  focus  of  incident  rays  cannot  be  so  situated  as  to  render  the  aberration  nothing.  But,  if  the  curvatures  R' 
and  R"  of  the  surfaces  be  such  as  to  satisfy  this  condition,  the  value  of  D  may  be  calculated  at  once  from  the 
equation  (&.) 

295.  Carol.  2.  Whenever,  in  meniscus  or  concavo-convex  lenses,  the  difference  of  the  curvatures  of  the  surfaces  is 

small  in  comparison  with  their  sum,  that  is,  whenever  a  moderate  focal  length  is  produced  by  great  curvatures, 
the  aberration  admits  of  being  rendered  evanescent  by  properly  placing  the  focus  of  incident  rays.  In  a  lens  of 

crown  glass  where  fi  =  1.52,  we  have  ^2/4  +  3/t2  =  3.16;  therefore  the  sum  of  the  curvatures  must  be  at  least 
3.16  times  their  difference,  to  satisfy  the  condition  of  possibility.  In  double-convex  or  double-concave  lenses,  R' 
and  R''  having  opposite  signs,  the  condition  can  never  be  satisfied. 

ggg  Carol.  3.   If  a  =  0,  the  aberration  vanishes  for  parallel  rays.     This  condition  is,  however,  only  to  be  satisfied 

No  known    °y  rea'  values  of  R'  and  R1'  when  p.  is  equal  to  or  less  than  \,  and  no  such  media  are  known  to  exist, 
medium  can      Carol.  4.    The  effect  of  aberration  will  be  to  shorten  or  lengthen  the  focus  for   exterior  rays,  according  ns  -the 
render  the    sign  of  A  /is  the  same  as,  or  the  opposite  to,  that  of/.     In  particular  cases  it  will,  of  course,  however,  depend 

aberration     on  ^  va]ues  of  „   n    j^/   ancj  j)  which  shall  take  place.     The  principal  case  is  that  of  parallel  rays,  in  which 
nothing  for    --.  J 

parallel  rays  D  =  0,  and 

Ci2J£  A  /  =  -|1  .  L  {  (2  -  2  p*  +  p*)  R'*  +  (,u  +  «/.*-  9  &  R;  R"  +  ^  R"*  } 

which  the 

ation  an(j  t|,e  focus  Of  external  rays  will  be  shorter  or  longer  than  that  of  central  ones,  according  as  this  quantity  has 

lengthens"^  t'le  same'  or  opposite  sign  with  L,  that  is,  according  as 
the  focus'  (2  -  2  n*  +  /,')  R'*  +  GB  +  2  n*  -  2  /.')  R'  R"  +  /**  R'"- 

is  positive  or  negative.  Now,  from  what  we  have  already  seen  in  the  last  corollary,  this  quantity  never  can  be 
rendered  negative  by  any  real  values  of  R'  and  R",  unless  /»  be  less  than  £.  For  all  other  media,  therefore, 
(comprehending  all  yet  known  to  exist  in  nature,)  every  lens,  whatever  be  the  curvatures  of  its  surfaces,  has  the 
exterior  focal  length  for  parallel  rays  shorter  than  the  central. 

298.  Carol,  b.    In  a  glass  meniscus,  when  the  radiant  point  is  on  the  convex  side,  and  the  rays  diverge,  we  have 

Case  of  a     4  +  3  ft  —  3  ft-  a  positive  quantity ;  and,  R'  and  R"  being  both  positive,  ft  is  so  ;  hence  (D  being  negative  in 

glass  this  case)  the  term  —  ft  D,  and  therefore  the  whole  factor  a  —  /3  D  +-/D2   is  positive;    and  L  being  also 

meniscus,     positive,  A  /  is  so ;    and,  therefore,  w,  the  aberration,   negative.      Hence,   when   Q    is    beyond    F,  the  focus 

for  parallel   rays  incident  the  other  way,  the  exterior  focus  is  the  shorter  ;    but  when  between    F   and   C,  the 

longer. 

/  R1  -I-  R'1  \- 

299  Corol.  6.    Unless    (  —  -; —  — j-  )     >  2  p  +  3  /»-,  no  real  value  of  D  can  render  o  —  /3D  +  -/D2  negative. 
Rule,  for  a  V  R   -     R     ^ 

oVfense'to      ^  aPPears'  therefore,  that  in  all  double-convex  or  concave  lenses,  as  well  as  in  all  meniscus  and  concavo-convex 

effect'hof      ones,  in  which  the  sum  of  the  curvatures  of  the  surfaces  is  greater  than  J  '2  ft  +  3  u*  times  their  difference,  the 

aberration     factor  a  —  ft  D  +  f  D8  is  positive  for  all  values  of  D,  and  therefore  the  aberration  u>  has  in  all  such  lenses  the 

in  lengthen-  siffn  opposite  to  that  of  L.     Hence,  for  all  such  lenses,  we  have  the  following  simple  and  general  rule :  the  effect 

ingorsliort-  of  aberration  will  be  to  throw  the  focus  of  exterior  rays  more  TOWARDS  the  incident  light  than  that  of  central 

' lne     ones,  when  the  lens  is  of  a  positive  character,  or  makes  parallel  rays  CONVERGE,  but  more  FROM  the  incident  light 

if  of  a  negative,  or  if  it  cause  parallel  rays  to  DIVERGE. 

300  Corol.  7.  All  other  lenses  have,  as  in  the  case  of  single  surfaces,  aplanatic  foci,  corresponding  to  the  roots  of 
the  equation  a  —  ft  D  +  7  D*  =  0.     In  general  there  are  two  such  foci  of  incident  and  two  of  refracted  rays  :  and 


LIGHT.  389 

Light,      rules  might  easily  be  laid  down  for  determining  in  what  positions  of  the  radiant  point,  with  respect  to  these  foci      Part  I. 
•— ,/-»^  and  the  lens,  the  aberration  tends  to   shorten   or  lengthen  the  exterior  focus ;  but  it  is  simpler  and  readier  to  ^^—^^^ 
have  recourse  at  once  to  the  algebraic  expressions.  "ow  to 

Carol.  S.  In  the  case  of  reflexion,  as  when  rays  are  reflected  between  the  surfaces  of  thin  lenses  of  transparent  ^J^^," 
media,  we  have  m  =  m"  =  &c.  =  //  =  u"  =  &c.  =  —  1  ;  M'  =  —  1,  M'1  =  4  1,  &c.,  and  M  =  4  1,  accord-  °  g^*8" 
ing  as  the  number  of  reflexions  is  even  or  odd  ;  therefore  for  n  reflexions  we  have  C^  Of  le_ 

flexion  be- 

f    =  2  R'     —  D  ")  tween  any 

f»—  9n«          9  R'    4.   T,  f  system  of 

/  '     .  (    A  transparent 

f"  =  2  R"1  —   2  R"  4   2  R'  —  D    (    '  surfaces. 

&c.  ) 

and 

[  R'(R'-  D)2 

-R"  (R"  -  2  R'  4  D)2 

4  R'"  (R'"  -  2  R"  4  2  R'  -  D)s 

-&c. 

which  formula?  serve  to  determine,  in  all  cases  of  internal  reflexion  between  spherical  surfaces,  both  the  places 
of  the  successive  foci  and  the  aberrations. 

Carol.  9.    If  the  reflexions  take  place  between  equicurve  surfaces,  having  their  concavities  turned  opposite      3T>2. 
ways,  f,  f",  &c.  are  in  arithmetical,  and   therefore  their   reciprocals,  or  the  focal  distances,  in  harmonic  pro- 
gression. 

Problem.  To  construct  an  aplanatic  lens,  or  one  which  shall  refract  all  rays,  for  a  given  refractive  index,  and      303. 
converging  to  or  diverging  from  any  one  given  point,  to  or  from  any  other.  General 

Let  Q  and  q  be  the  points,  the  former  being  the  focus  of  incident,  the  latter  of  refracted  rays.     Let  /«  =  index  construction 
of  refraction  ;  and  putting  Q  q  =  2  f,  and  assuming  6  any  arbitrary  quantity,  construct  the  curve  whose  equation      an  a*'la~ 
is  (n),  Art.  232.     Let  H  PC,  (fig.  36,)  be  this  curve;  and  with   centre  q,  and  any  radius  gN  less  than  q  P,  any  p-tg^se"5 
one  of  the  refracted  rays  describe  the  circle  H  N  K.     Then  since  the  ray  Q  P,  by  the  nature  of  the  curve  H  P  C, 
is  after  refraction  directed  to  or  from  17,   and,  being  incident  perpendicularly   on   the   second   surface,   suffers 
there  no  flexure,  it  will,  if  supposed  to  emerge  from  the  medium,  here  continue  its  course  to  or  from  q.     If  then 
we  suppose  the  figure  C  P  H  N  K.  to  revolve  round  Qq,  it  will  generate  a  solid,  which,  being  composed  of  the 
proposed  medium,  is  the  lens  required.     If  the  rays  be  parallel,  as  in  fig.  38,  the  curve  H  PC,  as  we  have  seen,  is  Fig.  38. 
a  conic  section,  which,  if  the  lens  be  denser  than  the  ambient  medium,  is  an  ellipse.     Thus,  a  glass  meniscus  lens, 
whose  anterior  convex  surface  is  elliptic,  and  posterior  spherical,  having  its  centre  in  the  focus  of  rays  refracted 
by  the  first  surface,  is  aplanatic. 

But,  without  having  recourse  to  the  conic  sections,  the  same  thing  may,  in  certain  cases,  be  accomplished  by      304. 
spherical   surfaces  only.     For  if  Q   and  q  (fig.  53)  be  the  aplanatic  foci  of  the  spherical   refracting  surface,  (j'ase  v 
and  if  with  the  centre  q  and  any  radius  greater  than  qC,  when  the  incident  rays  diverge  from  Q,  as  in  the  lower  of'^"^*.8 
portion  of  the  figure,  but  less  if  they  converge  to  Q  as  in  the  upper,  we  describe  a  circle  K  L,  or  k  I,  and  turn  the  natic  lens 
whole  figure  about  Q  q  as  an  axis,  the  surfaces  C  P  K  L,  or  cp  k  I,  will  generate  the  aplanatic  lens  in  question,  are  all 
This  also  follows  evidently  from  the  general  formula,  (z,  Art.  29 1,)  for  if  R"  =  /',  the  expression  of  A  /  for  the  spherical, 
lens  becomes  simply 

1 **|  ' 

(R1  -  D)»  {  m'  R'  -  (1  4  m')  D  }  y*, 

which  vanishes  when  D  =  R',  or  when  Q  is  the  aplanatic  focus  of  incident  rays  for  the  first  surface. 

147/1'     . 

More  generally,  however,  the  equation  a  —  /3  D  4  7  D  *  =  0,  assigns  the  universal  relation  between  ft,  D,  R', 
R",  which  constitutes  the  lens  aplanatic.     See  Cor.  1,  Art.  294. 

Problem.  To  assign  the  most  advantageous  form  for  a  single  lens,  or  that  which,  with  a  given  power,  has  the      306. 
least  possible  aberration  for  parallel  rays.  Most  ad- 

Since  the  aberration  cannot  be  rigorously  made  to  vanish  for  parallel  rays,  when  u  >  4  (Art.  296)  we  have  to  "ntageous 

form  for  a 

A    f  A  f  sin-rle  lens 

make  it  a  minimum.     Now  ui  = -•*-  =  -         ^—  for  parallel  rays,  or  '"'  parallel 

f-  L8  rays  deter- 

mined. 

to  = —  .    —  ;  and,  in  general,  d  M  =  -    {  L  rf  «  —  a  rf  L  } 

2  «         L 


In  the  present  case  L  is  given,  therefore  we  must  put  d  a  =  0,  which  gives 

0=2  (2-  2fi*  4/1')  R'dR'4  (/»+  2/^-2 /*3)  (R'd  R"  4  R"d  R1)  4  2 /«'  R'-'rfR''. 
But  the  condition  d  L  =  0  g-ives  dR'  =  dR"  ;  so  that  our  equation  becomes,  on  substitution  and  reduction, 

Or=  (44  ,u-2/.«)R'  4  Gtt42/.')R"; 


390  L  I  G  II  T. 

Light.      that  is  to  say 


R"         2  /»*-/•  -4  ,  > 

* 


R'        a  p*  +  ^ 

In  the  case  of  a  glass  lens,  taking  p.  =  1.5,  this  fraction  becomes  equal  to  —  —  ,  which  shows  that  the  lens 

o 

must  be  double-convex,  having  the  curvature  of  the  posterior  surface  only  —  -  that  of  the  anterior,  or  its  radius 

six  times  as  great.     Artists  sometimes  call  such  a  lens  a  "  crossed  lens." 

306.  Carol.  1.     If  ft  =  1.6861,  as  is  nearly  the  case  with  several  of  the-  precious  stones  and  the  more  refractive 

?ase  in       .glasses,  R"  =  0  ;   and  the  most  advantageous  figure  for  collecting  all  the  light  in  one  place  is  plano-convex, 
formes*"    Caving  'ts  convex  side  turned  to  the  incident  rays. 

convex  Corol.  2.  Calling  the  aberration  of  a  lens  of  the  best  figure  ui,  we  shall  have  «.•  =  --  -  —  y2  .  L,  for  glass 

307. 

Aberrations  whose  refractive  index  is  1.5,  and  the  proportional  aberrations  of  other  forms  will  be  as  follows: 
of  various 

species  of  Plano-convex,  plane  side  first  (or  towards  the  light)    ....    4.2  x  u> 

mined  for  Plano-convex,  curved  surface  first  ..................    1.081   x  <u 

parallel  rays  Double  equi-convex,  or  concave    ..................    1.567  x  <a 

Problem,  To  investigate  a  general  expression  fir  the  aberration  of  any  system  of  infinitely  thin  lenses  placed 

Aberration     dogg  togdher  in  vacuo_ 

of  Tenses*"       I""6  general  expression  for  MA/,  or,  since  M  =  1  in  the  case  before  us,  of  A  /,  is 

(Q'  +  Q"  +  Q"+  Q'v  -K,  &c,)  y\ 

which  divides  itself  into  terms  originating  with  the  successive  lenses  in  the  following  manner, 

A/=  (Q'  +  Q")y*  +  (Q'"  +  Qiv)  y2  +,  &c. 

The  first  of  these  quantities  we  have  already  considered  ;  let  us  now,  therefore,  examine  the  constitution  of  the 
rest.  Let  then  ft'  be  the  refractive  index  of  the  first  lens,  /*"  of  the  second,  /if''  of  the  third  ;  and  let  of,  /3',  7' 
represent  the  values  of  a,  ft,  7  for  the  first  lens,  or  the  expressions  in  (I,  292,)  writing  only  ft'  for/*;  also  let 
a",  ft",  -/"  represent  their  values  for  the  second  lens,  or  what  the  same  expressions  become  when  ft"  is  put  for  ft, 
and  R'"  and  Riv  respectively  for  R'  and  R",  and  so  on  for  the  rest  of  the  lenses. 

309.  Now  if  we  consider  the  values  of  Q'''  and  Qiv,  it  will  be  seen  that  they  are  composed  of  the  quantities  m'", 
m",  M'",  Miv,  R'",  R1',/"  and/'",  precisely  in  the  same  manner  that  Q'  and  Q"  are  of  m',  m",  M1.  M",  R',  R", 
Dand/'. 

Moreover,  since  by  Art.  251  we  have 

/'  =  (1  —  m')  R'  +  m'D 
f"  =  (1  -  m")  R"  +  m'1  f 

=  (1  -m")  R"  f  m"  (!-»?')  R'  +  m"  m'  D 

=  (ft  —  1)  (Rr  -  R")  +  D,  since  m  =  —  ,  m"  =  p. 

=  L  +  D;  call  this  D";  (L  is  the  power  of  the  first  lens) 
/"'=  (1  -  m'")  R'"  +  m'"  D" 
./"  =  (1  —  «ii7)  RiT  +  m"f"  =  L"  -f-  D"  as  before  ;  (L"  is  the  power  of  the  second  lens) 

=  L  +  I/  +  D  ;  and  so  on. 

And  it  is  clear  that  Q'"  +  Qiv  will  be  the  same  function  of,  i.  e.  similarly  composed  of,  the  refractive  index  and 
curvatures  of  the  surfaces  of  the  second  lens,  and  of  the  quantities  D"  and  /"',  that  Q'  +  Q"  is  of  the  re- 
fractive index  and  curvatures  of  the  first  lens,  and  of  D  and  /'.  It  follows,  therefore,  that  the  very  same 
system  of  reductions  which  led  to  the  equation 

Q1  +  Q"  =  -  —  („  _  /J  D  + 
• 


•2  ft 
being  pursued  in  the  case  of  Q'  '  +  Q",  must  lead  to  the  precisely  similar  equation 

Q"'  +  Q»  =  -i^r  («»  -  ft"  D"  +  7"  D"s) 

Genera!        and  so  on  f°r  the  remaining  lenses  ;  so  that  we  shall  have,  ultimately,  for  the  whole  system  (writing  L',  D'. 
expression      for  L,  D,  pi) 

A  f  =  _|!  J  Jll  (a'  _  e1  D'  +  7'  D")  +    -L,','  («"  -  ft"  D"  +  7"  D"2)  +  &c.  |  ;  (*) 

in  which  there  are  as  many  terms  as  lenses. 


LIGHT.  3<jl 

Cored.   For  parallel  rays,          D;  =  0  ;         D"  =  L' ;         D'"  =  L'  +  L",  &c.  P»«  I. 

tlietefore 

-V  «'  +  -^r  (""  -  ft"  L'  +  7"  L'J) 

~*~^  +  -^77T  («"'  -  ft1"  (L'  +  L")  +  V"  (L'  +  L")')  f  ' 

+   &c. 

Although  the  aberration  of  a  single  lens  for  parallel  rays  admits  of  being  destroyed  only  on  a  certain  hypo-  311 
thesis  of  the  refractive  index,  which  has  no  place  in  nature,  yet,  by  combining  two  or  more  lenses,  it  may  be 
destroyed  in  a  variety  of  ways.  Thus,  in  the  case  of  two  lenses,  the  expression  (t)  being  put  equal  to  zero, 
gives  an  equation  involving  ft1,  /*",  L',  L",  R',  R",  R'",  Rlv  ;  or  (since  L'  and  L"  are  given  in  terms  of  ft,  u,1 
and  R',  R",  &c.  and  since  /»',  p"  are  given  quantities)  only  the  four  unknown  quantities  R',  R",  R'",  R'T.  Now 
as  there  are  four  of  these,  and  only  one  equation,  it  may  be  satisfied  in  an  infinite  variety  of  ways,  and  the 
problem  of  the  destruction  of  the  spherical  aberration  (as  it  is  termed)  becomes  indeterminate. 

The  equation  in  the  case  of  two  lenses  for  parallel  rays  is  312. 

,  General 

0  =         -V  |  (2  -  2  /•-  +  X3)  R'-  +  (ft  +  2  /»  -  2  n")  R'  R"  +  f'3  R  "  I  ;  («)          ZdTnlc 

tion  of  aber 
L '       r  ration  in  a 

-f- <  (2  —  2  ft" -  +  /a,"3)  R"''2  +  (/*''  +  2  it"-  —  2  fi"3)  R'"  Riv   +   u"s  Riv*  <•  double  lens 

fif/       (  (  for  parallel 

rays. 

R'"  +  U"  +  3  M";)  R"  ]  +  ^^'  !  2  +  3  u"] 


This  equation,  if  L'  and  L",  the  powers  of  the  separate  lenses,  be  assigned,  is  of  a  quadratic  form  in  either      313. 
R',  R'1,  R'",  or  R"  ;  it  will  therefore  depend  on  the  supposition  adopted  to  limit  the  problem,  whether  these  Another 
quantities   admit   real    corresponding    values.      Now   the    equations    L'  =    («'  —   1)   (R'  —  R")    and   L"  =  torm  of  the 
(ft"  —  1)  (R'"  —  Riv)  afford  the  means  of  eliminating  two  of  them,  and  the  resulting  equation  (in   R'  and  R"'  $.ame  e1ua 
for  instance)  is 

0=       L'A±^R"    -L'R'  '  W 


and,  as  the  unknown  quantities  R',  R'"  are  not  combined  by  multiplication,  the  equation  when  L'  and  L"  are 
given  is  of  an  ordinary  quadratic  form  with  respect  to  each.  This  equation  will  be  of  use  to  us  hereafter, 
when  we  come  to  treat  of  the  theory  of  refracting  telescopes. 

If  L'  and  L/'  be  not  given,  since  either  of  them  is  of  the  first  degree  in  terms  of  R',  R",  &c.,  the  equation      3^ 
(«)  is  of  the  third  degree  in  either  of  the  quantities  R',  R",  &c.,  or  in  L',  L",  if  either  R"  or  Riv  be  elimi- 
nated.     Now  as  an  equation  of   the  third  degree  must  necessarily  have  at  least  one  real  root,  we  conclude. 
first,  that   in  a   double  lent,  if  the  curvatures  of  three  of  the  surfaces  be  given,  that  of  the  fourth  may    be, 
found,  so  as  to   destroy  the  spherical  aberration. 

Secondly.    That  if  the  curvature  of  one  surface  of  each   lens,  and  the  power  of  either,  or  that  of  the   two       313 
combined,  be  given,  the  power  of  the  other  may  be  found  so  as  to  destroy  the  spherical  aberration.      This  is 
evident;    for,  supposing  R'  and  R"'  given,   and  either  L'  or  L",  or  L'  +  L",  also  given,  the  equation  (D) 
becomes    an    ordinary  cubic    in    which    L'  or  L'',  as  the  case  may  be,  is    the  only  unknown    quantity,    and 
therefore  necessarily  admits  a  real  value. 

As  examples  of  aplanatic  combinations,  we  may  set  down  the  following  cases,  in  which  a  lens  of  glass  of      315 
the  refraction  1.50,  and  of  the  best  form,  having  the  radii  of  its  surfaces  respectively  +  5.833  and  —  35.000 
inches,  and  its  focal  length   10.000    inches,   has  its    aberration  corrected  by  applying  behind  it  another  lens 
of  similar  glass,  as  in  fig.  55.     This  lens  is  a  meniscus.     If  its  curvatures  be  determined  by  the  condition  of  pjg.  55. 
giving  the    maximum  of  power  to  the  combination,  the  radii  of  its  surfaces  and  its  focal  length  will  be  as 
follows:  radius  of  first  surface,  =  +  2.054  inches  ;  radius  of  second  surface,  =  +  8.128;  focal  length  of  cor- 
recting lens,  =  +  5.497  ;   focal  length  of  the  two  combined,  =  +  3.474.      On  the  other  hand,  if  we"  deter- 
mine the  second  lens  by  the  condition  of  the  resulting  combination,  having  a  focal  length  as  nearly  10.000 
as  is  consistent  with  perfect  aplanaticity,  we  shall  find  radius  of  first  surface,  =  +  3.688  ;  radius  of  second, 
=  +  6.291  ;  focal  length  of  correcting  lens,  =  +  17.829;  focal  length  of  the  combination,  =  +  6.407. 

The  effect  of  aberration   may  be  very  prettily  exhibited  by  covering    a    large    convex    lens  with    a    paper      317 


392 


L  I  G  H  T. 


Light.  screen  full  ot  small  round  holes,  regularly  disposed,  and,  exposing  it  to  the  sun,  receiving  the  converged  rays 
*^s~^s  on  a  white  paper  behind  the  lens,  which  should  be  first  placed  very  near  it,  and  then  gradually  withdrawn.  The 
pencils  which  pass  through  the  holes  will  form  spots  on  the  screen,  and  their  disposition  will  become  more  and 
more  unequal  over  the  surface,  as  the  screen  is  further  removed  ;  those  at  the  circumference  becoming  crowded 
together  before  the  central  ones.  The  manner  in  which  the  several  spots  corresponding- to  central  rays  blend 
together  into  one  image  at  the  focus,  and  those  formed  by  the  exterior  ones  are  scattered  round  it,  gives  us  a 
very  good  idea  of  the  variation  of  density  of  the  rays  in  the  circle  of  aberration  at  or  near  the  principal  focus; 
and  if  the  white  screen  be  waved  rapidly  to  and  fro  in  the  cone  of  rays,  so  as  to  pass  over  the  focus  at  each 
oscillation,  the  whole  cone  will  be  seen  as  a  solid  figure  in  the  air,  and  the  place  of  the  circle  of  least  aberra- 
tion will  become  evident  to  the  eye,  forming  altogether  a  very  pleasing  and  instructive  experiment. 


§  XI.    Of  the  Foci  for  Oblique  Bays,  and  of  the  Formation  of  Images. 


318. 


Foci  of 
oblique 
pencils. 


We  have  hitherto  considered  rays  as  converging  to,  or  diverging  from,  a  single  point;  but  as  this  is  not 
the  case  with  luminous  bodies  of  a  sensible  diameter,  we  now  proceed  to  examine  the  cases  of  refraction  at 
spherical  .surfaces,  where  more  than  one  radiant  point  is  concerned,  or  where  several  pencils  are  incident  at 
once  on  the  surface.  We  shall  take  for  our  positive,  or  fundamentiil  case,  as  we  have  done  all  along,  that  of 
converging  rays  incident  on  the  convex  side  of  a  more  refractive  medium  than  the  ambient  one,  and  derive  all 
others  from  it  by  the  changes  in  the  sign  and  relative  magnitudes  of  R,  D,  &c. 

In  fig.  56,  then,  let  Q  and  Q'  be  the  foci  of  two  pencils  of  convergent  rays  incident  on  the  spherical  surface 
C  C',  whose  centre  is  E.  Draw  Q  E  C,  Q'  E  C',  cutting  the  surface  in  C  and  C',  and,  regarding  C  E  Q  as  the 

axis  of  the  pencil  R  Q,  S  Q,  T  Q,  the  focus  of  refracted  rays  will  be  found  by  taking  o,  such  as  that  ,    or 

Cq 

f,  shall  be  equal  to  (I  —  m)  R  +  m  D,  (247,  e.)     Similarly,  regarding  C'E  Q'  as  the  axis  of  the  pencil  con- 
verging to  Q',  the  focus  q'  will  be  had  by  the  equation 

-— -  =f'—(\—m)R  +  m  D'. 

Thus  when  C'Q'  =  C  Q,  C  q'  will  also  equal  C  q,  and,  in  general,  when  the  locus  of  the  point  Q  is  given,  that 
of  q  may  be  found. 

319.  Definition.  The   image  of  an  object,  in  Optics,  is  the  locus  of  the  focus  of  a  pencil  of  rays  diverging  from, 

Images  in     or  converging  to,  every  point  of  it,  and  received  on  a  refracting  surface.     Thus,  supposing  C  Q'  to  be  a   line, 
d  fin"         or  surface>  every  point  of  which  may  be  regarded  as  a  focus  of  incident  rays,  qqf  is  its  image. 

320  Problem.    To  find  the  form  of  the  image  of  a  straight  line  formed  by  a  spherical  refracting  or  reflecting 

Form  of  the  surface. 


image  of  a 
straight  line 


Put  C  E  =  r  ; 

Then  we  have 

and  therefore 

we  have,  consequently, 


I  —  m 


C'q> 


+ 


ra 


(1  —  m)  a'  +  m  r 
ra 

,  m  r  (of  —  r) 


(1  —  TO)  a'  +  m  r  ' 


(1  —  wi)  a'  +  m  r 


'    {(I  -  m)a'  +  mr 
But,  by  similar  triangles,  E  q' :  E  M  :  :  E  Q7  :  E  Q,  or 


equating  these  two  values  we  get 

a  (1  —  m)  a'  +  m  r 


a' 


r(a-  x) 


x  m  r  1  —  m  x 

so  that  eliminating  a',  by  substituting  this  value  for  it,  we  get  for  a  final  equation  between  .  and  y,  or  for  the 
«*c«on°mC    e(luat'on  °f  tne  'mage 

(1  —  wi)s  (T2  +  Vs)  =  (  —  I    .(ma—  x)* 
\  a  / 

which  belongs  to  a  conic  section. 

321  Problem.    When   an   oblique  pencil   is   incident  on   any   system    of  spherical  surfaces,  to  find  the  focus  of 

rtfracted  rays. 


LIGHT.  393 

Take  E',  (fig.  57,)  the  centre  of  the  first  surface,  and  let  Q'  be  the  focus  of  incident  rays.     Join  Q'  E'  and     P«n  I. 

produce  it  to  C',  then  will  C'  be  the  vertex  of  the  surface   corresponding  to   the  pencil  whose  focus  is  Q'  ;  and  <^-~>r~-' 
r  ,  .  Foci  of 

taklnff  .  ,  oblique 

pencils  ir  - 

C'  Q"    "       C'  E'  C'Q'  cident  on  a 

system  of 

Q"  will   be  the  focus  of  retracted  ravs.      Airain,  join  Q"  and  E",   the  centre  of   the  second  surface,  produce  spherical 

"    ,„.  °  J  surfaces. 

to  C  ',  and  take  v:™  *•, 

JjI^L         _m^_ 
~  "    "       '      "     " 


__ 
C"  Q'"  ~        C"  E"       '   C"  Q 

and  Q'"  will  be  the  focus  after  refraction  at  the  second  surface,  and  so  on. 

Carol.  In  the  case  of  an  infinitely  thin  lens,  when  the  obliquity  is  small,  it  is  evident,  from  this  construction,      322. 
that  the  focus  of  oblique  rays  will  lie  at  the   same  distance  from  the  lens  with  that  of  rays  convergent  to,  or 
divergent  from,  a  point  in  the  axis  at  the  same  distance  with  the  focus  of  incident  rays,  but  instead  of  lying  in  the 
axis,  will  deviate  from  it. 

Definition.  The  centre  of  a  lens  is  a  point  in  its  axis  where  a  line  joining  the  extremities  of  two  parallel  radii       323. 
of  its  two  surfaces  cuts  the  axis.     Thus,  in  the  various  lenses  represented  in  fig.  58,  59,  60,  and  61,  E'  A.  and  E"  B  £"' 
being  two  parallel  radii  ;  join  B  A,  and  produce,  if  necessary,  till  it  meets  the  axis  in  X,  and  X  is  the  centre. 

Carol.  1.    The  centre  is  a  fixed  point  ;  for,  since  AE'  and  B  E''  are  parallel,  we  have  E'X  :  E'  E"  :  :  A  E'  :       324. 
B  E"—  A  E',  in  which  proportion  three  terms  being  invariable,  the  other  is  so  also. 

Carol.  2.  If  C'C",  the  interval  of  the  surfaces  or  thickness  of  the  lens,  be  put  equal  to  t  (t  being  always       325. 
positive)  and  the  curvatures  be  respectively  R'  and  R",  we  have,  for  the  distance   of  the  centre  from  the   first 
surface   or  for  C'  X,  the  following  value. 

R" 

C'  \  _  _      / 

R'  -  R" 

Carol.  3.    If  a  ray  be  so  incident  on  a  lens  that  its  direction  after  the  first  refraction  shall   pass  through  its      326. 
centre,  it  will  suffer  no  deviation.     This  is  evident,  because  its  course  within  the  lens  will  be  A  B,  and  the  radii  Rays 
E'A  and  E''B  being  parallel,  the  internal   angles  of  incidence  on  the   surfaces  are  equal,  and,  therefore,  the  throngh  the 
angles  of  refraction  both  ways  out  of  the  lens  ;  consequently  the  two  portions  of  the  ray  without  the  lens  are  undeviatel? 
parallel 

Carol  4     If  the  thickness  of  a  lens  be  very  small,  the  ray  passing  through  its  centre  may  be  regarded  as       337. 
undergoing  no  refraction  whatever  ;   for  the  portion  A  B  within   the   lens  being  very  small,   the  two  portions 
exterior  to  the  lens  (being  parallel)  may  be  regarded  as  one  ray.     This  is,  a  fortiori,  still  nearer  the  truth  when 
the  obliquity  of  the  ray  to  the  axis  is  small  ;  because  then  the  portion  A  B  is  very  nearly  coincident  in  direction 
with  either  of  the  two  exterior  portions. 

Carol.  5.   Hence,  to  find  the  focus  of  refracted  rays  in  the  case  of  a  very  thin  lens  and  for  a  pencil  of  small      328. 
obliquity,  take  X,  the  centre  of  the  lens,  and  the  focus  will  lie  in  the  line  Q  X,  at  the  same  distance  from  the  lens  Focus  of  a 
as  if  the  axis  of  the  incident  pencil  were  coincident  with  that  of  the  lens.  tt'ue'^n'cli 

Proposition.  When  a  luminary,  or  illuminated  object,  is  placed  before  a  double  or  plano-convex,  or  meniscus  throughTa0' 
lens,  at  a  distance  from  it  greater  than  its  focal  length,  there  will  be  formed  behind  the  lens  an  image,  similar  thin  lens. 
to  the  object,  but  inverted  ;  and  the  object  and  image  subtend  the  same  angle  at  the  centre  of  the  lens.  329. 

For  the  pencil  of  rays  which  emanates  (either  by  direct  radiation  or  by  reflexion)  from  any  point,  as  P,  of  tne  Fig.  62. 
object,  will  after  refraction  be  all  made  to  converge  to  a  point  p  behind  the  lens,  or  at  least  very  nearly  so.  ^  inverted 
Were  the  aberration  of  the  lens  evanescent,  the  convergence  would  be  mathematically  exact  ;  and  since,  when-  j^^.'  °s  al 
ever  thv  aperture  of  the  lens  and  the  obliquity  of  the  pencil  are  small,  the  aberration  is  so  very  minute,  that  the  formed 
space  over  which  the  rays  are  spread  may  be  regarded  as  a  physical  point,  and  every  physical  point  in  the  object  behind  a 
will  have  a  corresponding  point  in  the  image.     Now,  C  being  the  centre  of  the  lens,  the  line  joining  Pp  passes  convex  lena 
through  C  ;  and  the  same  being  true  of  the  line  joining  any  other  corresponding  points  of  the  object  and  image, 
it   follows,  by  similar  triangles,  that  the  object  and  image  are  similar  in  figure  ;  and  as  the  rays  cross  at  C,  the 
image  is  inverted,  and  subtends  the  same  angle  p  C  q  at  C  that  the  object  does  on  the  other  side. 

If  a  screen  of  white  paper  be  placed  at  qp,  this  image  will  be  rendered  visible  as  a  picture  of  the  object.    The      330. 
experiment  may  be  tried  with  any  magnifier  or  spectacle-glass  at  a  window,  when  the  forms  of  external  objects,  Camera 
the  houses,  trees,  landscape,  &c.  will  be  painted  on  the  paper  screen  with  perfect  fidelity,  forming  a  miniature  of  obscura 
the  utmost  delicacy  and  beauty.     This  is  the  principle  of  the  common  camera  obscura,  in  which  the  rays  from  e*Plalne<i- 
external  objects  are  thrown  by  an  inclined  looking-glass  downwards,  and  being  received  on  a  convex  lens,  are 
brought  to   their  focus  on  a  white  horizontal  table,  in  a  room  where  no  other  light  is  admitted.     On  this  table 
a  moving  picture  of  all  external  objects,  in  their  proper  forms,  colours,  and  motions,  is  seen,  infinitely  more  correct 
Hiid  beautiful  than  the  most  elaborate  painting.     See  fig.  63,  in  which  P  is  the  object,  AB  the  reflector,  B  C  the 
lens,  and  ;;  the  image  on  the  table  D. 

If  the  rays,  instead  of  being  received  on  white  paper,  be  received  on  a  plate  of  glass  emeried  on  one  side,  331. 
ihe  picture  may  be  seen  by  an  eye  placed  at  the  other  side  of  the  glass,  as  well  as  by  one  in  front  of  it  ;  for  it  is 
a  property  of  such  roughened  transparent  surfaces  to  scatter  the  rays  which  fall  on  them,  not  only  by  reflexion 
outwards,  but  by  refraction  inwards.  If  the  surface  be  but  slightly  roughened,  however,  the  picture  will  appear 
much  less  vivid  when  looked  at  obliquely  than  when  the  eye  is  placed  immediately  behind  it;  and  in  this 
VOL.  iv.  3  F 


394  LIGHT. 

Light.      latter  situation  the  emeried  glass  may  even  be  removed  altogether,  and  the  image  will  still  be  seen,  and  even  more 
*Cr*»1'  distinctly   as  if  a  real  object  stood  in  the  place  in  all  respects  similar  to  the  picture. 

332.  We  may  examine  the  image  on  the  roughened  glass  with  a  magnifying  glass,  or  microscope.     It  will   then 

appear  as  a  delicate  painting,  accommodating  itself  to  all  the  inequalities  of  the  surface.     But  if,  in  the 
so  examiniiiT  it,  the  rough  glass  be  removed,  the  painting  remains  as  if  suspended  m   air,  and  the   objects 
represents  are  seen  brought  nearer  to  the  eye,  and  enlarged  in  their  dimensions.     In  short,  we  have  formed 

tC  If  Te^lens  used  to  form  the  image  be  a  concave  one,  or  if  a  convex  reflector  be  used,  as  in  fig .64  and  65, 
the  rays,  after  refraction  or  reflexion,  diverge,   not  from  any  actual  points  in  which  they  cross,  but  from  poi 
in  which  they  would  cross  if  produced  backwards.     There  is  in  this  case,  then,  no  real  image  formed  capab 
of  beino-  received  en  a  screen,  but  what  is  called  a  virtual  one,  visible  to  the  eye  if  properly  situated,  either 
assisted  or  aided  by  a  magnifier,  and  situated  on  the  same  side  of  the  lens,  or  on  the  contrary  side  of  the  refl 
with  the  object,  and  therefore  erect. 
oo.  The  perfection  of  the  image  formed  by  a  lens  or  reflector,  its  exact  re  ambiance  to  the  object,  and  the  , 

ness  of  its  parts,  will  depend  on  the  exact  convergence  of  all  the  rays  of  pencils  emanating  from  every  physical 
point  of  the  object  in  strict  mathematical  points,  or  in  as  near  an  approach  to  such  points  a-  may  be 
therefore,  a  lens  of  considerable  diameter  be  used,  especially  if  the  curvatures  of  its  surfaces  be  improperly  chosen 
so  as  to  produce  much  aberration,  the  image  will  be  confused;   for  each   point  of  the  object  will  form,  not  < 
point,  but  a  small  circular  spot  in  the  image,  over  which  the  rays  are  diffused  ;  and  as  these  spots  overlap  a 
encroach  on  each  other,  distinctness  is  destroyed.     For  the  formation,  therefore,  of  perfect  images,  the  destruc 
tion  of  aberration  is  the  essential  condition ;  and  whatever  imperfections,  either  in  the  figures  of  the  reflecting 
refractin^  surfaces  used,  or  in  the  materials  of  which  they  are  composed,  tends  to  throw  the  rays  aside  from  their 
strict  geometrical  direction,  must,  of  course,  confound  the  images.     Hence,  in  the  formation  of  optical  images, 
there  are  three  great  points  to  be  attended  to  :  first,  perfect  polish  of  the  surfaces ;  secondly,  perfect  homogeneity 
in  the  material  employed  ;    thirdly,  strict  conformity  in  the  figures  of  the  reflecting  and  refracting   surfaces  t( 
geometrical  rules,  and  the  results  of  analysis. 

335  There  is  one  case  where  the  aberrations  of  all  kinds  are  rigorously  destroyed,  and  HI  which  the  image  is  perfect. 
It  is  when  the  rays  are  reflected  at  a  plane  surface.     For  (fig.  66)  if  P  Q  be  an   object   placed   before   a  plane 
reflector  AB,  and  if  perpendiculars  be  let  fall  from  every  point  of  the  object  to  the  surface,  and  on  the  other 
points  in  these  be  taken  at  the  same  distances  respectively  behind  the  surface  as  p  q,  these  points  will  form  It 
image.     Now  we  have  seen,  that  all  rays  from  any  point  P.  retlected  at  A  B,  will  after  reflexion  diverge  strictly 
from  p  its  image.     Thus,  the  image  is  as  perfect  and  free  from  aberration  as  the  object ;  and  will  appear,  to  an 
eye  placed  so  as  to  receive  the  rays,  like  a  real  object  placed  behind  the  reflector. 

336  Corol.  The  image  formed  by  a  plane  reflecting  surface  is  similar  and  equal  to  the  object,  and  any  correspond- 
ing lines  in  both  are  equally  inclined  to  the  reflecting  surface.     A  common  lookmg-glass  is  the  best  illustratic 

,,7         3   ProposMon.    To  determine  the  image  of  any  object  formed  by  a  plane  refracting  surface      Let  B  C  be  the 
surface,  PQ  the  object.     From  any  point  Q  draw  Q  C  perpendicular  to  the  surface,  and,  /.  being  the  i 
refraction,  if  we  regard  the  surface  as  a  sphere  of  infinite  radius,  we  have  R  its  curvature  =  0,  and  the  equati. 

f  =  (I  -m)R  +  mD  becomes  simply  f  =  mD.     Now  /  =  —  ;    D  =  -^Q- ;    and  m  =   —. 

this  equation,  translated  into  geometrical  language,  gives  C  q  ~  fi  X  C  Q. 
338.  In  the  case  represented  in  the  figure,  the  refraction  is  made  out  of  a  denser  medium   into  a^ rarer,   the   object 


experiment  where  a  shilling  is  laid  in   an  empty  vessel,  and  the  eye  withdrawn  till  the  shilling  is  hidden  by  the 
edge,  but  reappears  again,  as  if  raised  up,  when  the  vessel   is  filled  with  water.      On  the   other  hand,  to  an 
eve  placed  under  water,  external  objects  would  appear  farther  removed  by  the  effect  of  refraction. 
,00          "  Corol.  1.  The  image  of  a  straight  line  PQ   in  the  object  is  a  straight  line  pqin  the  image,  less  inclined 
the  surface  if  the  refraction  be  made  from  a  denser  into   a  rarer  medium.     Thus,  if  a  stick 
plunged  into  water,  the  immersed  portion  A  Q  forms  the  image  Ag  less  inclined;  so  that  to  a  spectator  in  air, 
the  stick  appears  broken  and  bent  upwards  at  A.     The  appearance  is  familiar  to  every  one. 

340  In   refraction  at  a  plane  surface,  however,  the  rays  do  not  rigorously  diverge  from,  or  converge  to,  a  single 
point.     Therefore  the  above  result  is  only  approximately  correct,  and  supposes  the  rays  to  be  incident  nearly  at 
right  angles  to  the  surface.     And  this  leads  us  to  the  consideration  of  obl-que  vision  through  refracting  surfaces, 
or  in  reflectors  of  any  figure. 

341  The  eye  sees  by  the  rays  which  enter  it,  and  judges  of  the  existence  of  an  object,  by  the  fact  of  rays  diverging 
Oblique'       sensibly  from  some  point  in  space.     If,  then,  rays  diverge  rigorously  from  a  point,  the  eye  which  receives 
vision           is  irresistibly  led  to  the  belief  (unless   corrected  by  experience  and  judgment)  of  an  object  being  there  ;  tf 
through  re-  inusjon  js  complete,  and  vision  perfect.     But  if  such  divergence  be  only  approximate,  as  when  the  density  of  i 
fracting  or  ^.^  ^^  the         |n  Qne  directi(ln  is  very  Inuch  greater  than  in  directions  adjacent  on  either  bide, 

Surface"  of    vision   is  still  produced,  only  less  distinct,  in  proportion  to   the  degree  of  deviation  from  strict  mathematical 

»ny  figure     divergence  of  the  rays  which  produce  it.     Suppose,  now,  Q  to  be  a  radiant  point  placed  anywhere  with  respect 

Fig-  68.        to  the  refracting  or  reflecting  surface  A  C  B,  (fig.  68,)  and  let  A  ?  F  B  be  the  caustic  formed  by  the  intersection  ol 

al!  the  refracted  or  reflected  rays.     Let  us  suppose  an  eye  placed  at  E,  and  from  thence  draw  K  q  a   tangent 


LIGHT.  395 

Light      to  the  caustic,  which  continue  to  the  surface  C,  and  join  Q  C.     Then  it  is  obvious,  that  any  small  pencil  Q  C,  Q  C       Part  I- 
— ^— -^  diverging  from  Q,  will  form  a  focus  at  q  (Art.  134,  &c.)  from  which  it  will  afterwards  diverge,  and  fall  on  the  eye  ^-~ "V— ^ 
at  E,  nearly  as  if  the  rays  came  from  a  mathematical  point;    and  from  what  was  said  in  Art.  161  and  162,  it 
appears  that  the  density  of  rays  in  the  cone  q  E  is  infinitely  greater  than  in  any  adjacent  cone   having  the  eye  for 
its  base ;   so  that  q  will  appear  as  an  image  of  Q,  more  or  less  confused,  in  proportion  to  the  degree  of  curvature 
of  the  caustic  at  q;  for  it  is  evident,  that  if  the  curvature  be  great,  the  assumed  concentration  of  any  small  finite 
pencil  Q  C  C'  in  one  mathematical  point  q,  will  deviate  more  from  truth  than  if  the  caustic  approach  nearly  to  a 
straight  line. 

Carol.    As  the  eye  shifts  its  place,  the  apparent  position  of  an  object  seen  in  a  reflecting  or  refracting  surface       342. 
shifts  also,  for  as  E  varies,  the  tangent  ~Eq  shifts  its  place  on  the  caustic,  and  the  point  of  contact  q,  or  the 
place  of  the  image  shifts. 

This  doctrine  may  be  illustrated  by  a  very  familiar  instance.     If  we  look  through  a  surface  of  still  water,  not       343. 
very  deep,  but  having  a  level  horizontal  bottom,  the  bottom  will  not  appear  a  plane,  but  will  seem  to  rise  on  all  Apparent 
sides,  and  approach  nearer  the   surface  the   more  obliquely  we  look.     To  explain  this,  let  Q  be  a  point  in  the  figure  of  the 
bottom,  and  let  Q  P  e  be  the  course  of  the  pencil  of  rays  by  which  an  eye  at  e  sees  it  (fig.  39)  on  the  visual   ray.  jj' 
The  point  in  the  caustic  to  which  e  P  produced  is  a  tangent,  is  Y ;  and  from  the  form  of  the  caustic  D  Y  B  (see  Stjii0wat°er. 
Art.  238)  it  is  obvious,  that  Y  is  nearer  the  surface  the  more  oblique  eP  is  to  it.     The  apparent  figure  of  the  Hg.  39. 
bottom  will  therefore  be  thus  determined.     From  the  eye  E  (fig.  69)  draw  any  line  E  g-  to  the  point  G  of  the  Fig.  69. 
surface;   and  having  drawn  P  Y  parallel  to  E  G,  touching  the  branch  D  Y  B  of  the  caustic  having  Q,  vertically 
below  E  for  a  radiant  point  in  Y,  prolong  E  G  to  H,  making  G II  =  P  Y,  then  will  H  be  the  image  of  the  point 
Q'  in  the  bottom,  belonging  to  the  caustic  D'  H  B' ;  and  the  locus  of  H,  or  the  apparent  form  of  the  bottom, 
will   be  the  curve  D  F  H,  having  a  basin-shaped  curvature  at  D,  a  point  of  contrary  flexure   at  F,  and  an 
asymptote  C  G  K  coinciding  with  the  surface. 

But,  to  return  to  the  case  of  images  formed  by  rays  incident  at  very  small  obliquities   and  nearly  central,       344. 
the  following  rules  for  determining   their  places,  magnitudes,  and  apparent  situations  in  all  cases  of  spherical  Rules  for 
surfaces,  will  be  convenient  to  bear  in  memory,  and  will  need  no  express  demonstration  to  the  reader  of  the  fore-  Aiding  the 

going  pages.  Sw'faiw 

Rule  \.  Any  image  formed,  or  about  to  be  formed,  by  converging  rays,  or  from  which  rays  diverge,  may  be       345 

regarded  as  an  object. 

Rule  2.  In  spherical  reflectors  the  object  and  its  image  lie  on  the  same  side  of  tV  principal  focus.    They  move       346. 

in  contrary  directions,  and  meet  at  the  centre  and  surface  of  the  reflector.     The  J'rstance  of  the   image  from  the  Rule  foi 

principal  focus  and  centre  is  had  by  the  proportion  reflectors. 

QF:FE  ::EF:Fg:  :  QE  :  E  q, 

and  the  image  is  erect  when  the  object  and  surface  lie  on  the  same  side  of  the  principal  focus  ;  but  inverted  when 
on  contrary  sides.  The  relative  magnitudes  of  the  object  and  image  (being  as  their  distances  from  the  centre) 
are  given  by  the  proportion 

object  :  image  :  :  Q  F  :  FE  :  :  distance  of  the  object  from  the  principal  focus  :  focal  length  of  reflector. 

Ride  3.  In  thin  lenses,  of  all  species,  if  Q  be  the  place  of  the  object,  q  of  its  image,  E  the  centre  of  the  lens,       34". 
F  the  principal  focus  of  rays  incident  in  a  contrary  direction,  then  will  the  object  and  image  lie  on  the  same,  or  Rule  for 
opposite  side  of  the  lens,  according  as  the  object  and  lens  lie  on  the  same  or  opposite  sides  of  the  principal  'enses- 
focus  F.     In  the  former  case  the  image  is  erect,  in  the  latter  inverted,  with  respect  to  the  object.     The  distance 
of  the  image  from  the  lens,  or  from  the  object,  is  had  by  the  proportions 

QF  :  FE  :  :  QE  :  Eq;         Q  F  :  Q  E  :  :  Q  E  :  Q  q; 

and  the  magnitude  of  the  object  is  to  that  of  the  image  as  the  distance  of  the  object  from  F  is  to  the  focal  length, 
or  as  Q  F  :  F  E. 

Rule  4.  In  all  combinations  of  reflectors  and  lenses,  the  image  formed  by  one  is  to  be  regarded  as  the  object,       34   . 
whose  image  is  to  be  formed  by  the  next,  and  so  on,  till  we  come  to  the  last. 

It  has  been  already  remarked  (Art.  6)  that  visible  objects  are  distinguished  from  optical  images  by  this,  that  349. 
from  the  former  light  emanates  in  all  directions,  whereas  in  the  latter  it  emanates  only  in  certain  directions. 
This  is  an  important  limitation  in  practical  optics.  A  real  object  can  be  seen  whenever  nothing  opaque  is 
interposed  between  it  and  the  eye.  An  image  can  only  be  seen  when  the  eye  is  placed  in  the  pencil  of  rays 
which  goes  to  form  it,  or  diverges  from  it.  Thus  in  the  case  represented  in  fig.  62,  except  the  eye  be  placed 
somewhere  in  the  space  Dq  pll,  it  will  see  no  part  of  the  image,  B  9  D  and  A/)H  being  the  extreme  rays 
refracted  by  the  lens  from  the  extremities  of  the  object. 

The  brightness  of  an  image  is,  of  course,  proportional  to  the  quantity  of  light  which  is  concentrated  in  each  Brightness 
point  of  it ;  and,  therefore,  supposing  no  aberration,  as  the  apparent  magnitude  of  the  lens  or  mirror  which  forms  of  images. 

it,  seen  from  the  object  x  - — .     Or,  since  the  area  of  the  object  :  that  of  the  imaare  •  •  (distanced 

area  of  image 

of  object  from  lens  :  (distance)5  of  image  ;  and  since  the  apparent  magnitude  of  the  lens  seen  from  the  object 

(diameter  \ 2 

; I  ,  the  brightness  or  degree  or  illumination  of  the  imafre  is  as  the  anmrpnr 
distance  from  object  / 

3F2 


390  LIGHT. 

Light       magnitude  of  the  lens  seen  from  the  image,  alone,  whatever  he  the  distance  of  the   object.     Now  the  apparent 

>— — V— ^  magnitude  of  the  lens  seen  from  the  image  is  always  much  less  than  a  hemisphere.     Therefore  (even  supposing 

no  light  lost  by  reflection  or  refraction)  the  illumination  of  the  image  is  always  much  less  than  that  of  the  object. 

This  is  the  case  when  the  image  is  received  on  a  screen  which  reflects  all  the  rays,  or  when  viewed  by  an 

eye  behind  it  having  a  pupil   large  enough  to  receive  all  the  rays  which  have  crossed  at  the  image,  a  fortiori, 

then,  when  the  eye  does  not  receive  all  the  rays,  must  the  apparent  intrinsic  brightness  be  less  than  that  of  the 

object.     This  supposes  the  object  to  have  a  sensible  magnitude ;  but  when  both  the  object  and  its  image  are 

Images  are    physical  points,  the  eye  judges  only  of  absolute  light ;  and  the  light  of  the  image  is  therefore  proportional  to  the 

brUn^is       apparent  magnitude  of  the  lens,  as  seen  from  the  object.     In  the  case  of  a  star,  for  instance,  whose  distance  is 

their  objects  constant,  the  absolute  light  of  the  image  is  simply  as  the  square  of  the  aperture,  and  this  is  the  reason  why  stars 

can  be  seen  in  large  telescopes  which  are  too  faint  to  be  seen  in  small  ones. 


§  XII.  Of  the  Structure  of  the  Eye,  and  of  Vision. 

350.  It  is  by  means  of  optical  images  that  vision  is  performed,  that  we  see.     The  eye  is  an  assemblage  of  lenses 
which  concentrate  the  rays  emanating  from  each  point  of  external  objects  on  a  delicate  tissue  of  nerves,  called 
the  retina,  there  forming  an  image,  or  exact  representation  of  every  object,  which  is  the  thing  immediately  per- 
ceived or  felt  by  the  retina. 

Description       Fig.  70  is  a  section  of  the  human  eye  through  its  axis  in  a  horizontal  plane.     Its  figure  is,  generally  speaking, 

of  the  eye.    spherical,  but  considerably  more  prominent  in  front.     It  consists  of  three  principal  chambers,  filled  with  media 

Fig.  70.       of  perfect  transparency  and  of  refractive  powers,  diHering  sensibly  into-  se,  but  none  of  them  greatly  different  from 

Aqueous       that  of  pure  water.     The  first  of  these  media,  A,  occupying  the  anterior  chamber,  is  called  the  aqueous  humour, 

humour.        and  consists,  in  fact,  chiefly  of  pure  water,  holding  a  little  muriate  of  soda  and  gelatine  in  solution,  with  a  trace 

!uion>mp0"    °*  albumen  ;  the  whole  not  exceeding  eight  per  cent.*     Its  refractive  index,  according  to  the  experiments  of 

Refractive     ^.  Chossat.t  and  those  of  Dr.  Brewster  and  Dr.  Gordon,!  is  almost  precisely  that  of  water,  viz.  1.337,  that  of 

power.          water  being  1.336.     The  cell  in  which  it  is  contained  is  bounded,  on  its  anterior  side,  by  a  strong,  horny,  and 

Cornea.         delicately  transparent  coat  o,  called  the  cornea,  the  figure  of  which,  according  to  the  delicate  experiments  and 

Its  figure       measures  of  M.  Chossat,  §  is  an  ellipsoid  of  revolution  about  the  major  axis  ;   this  axis,  of  course,  determines  the 

>"  revoTu"     ax^s  °f  the  eye;   but  it  is  remarkable,  that  in  the  eyes  of  oxen,  measured  by  M.  Chossat,  its  vertex  was  never 

rton  found  to    be    coincident  with  the  central    point  of  the  aperture  of  the  cornea,  but  to  lie  always   about  10° 

(reckoned  on  the  surface)  inwardly,  or  towards  the  nose,  in  a  horizontal  plane.     The  ratio  of  the  semi-axis 

of  this  ellipse  to  the  excentricity,  he  determines  at  1.3  ;  and  this  being  nearly  the  same  with  1.337,  the  index 

of  refraction,  it  is  evident,  from  what  was  demonstrated  in  Art.  236,  that  parallel  rays  incident  on  the  cornea  in 

the  direction  of  its  axis,  will  be  made  to  converge  to  a  focus  situated   behind   it,  almost  with   mathematical 

exactness,  the  aberration  which  would  have  subsisted,  had  the  external  surface  a  spherical  figure,  being  almost 

completely  destroyed. 

351.  The  posterior  surface  of  the  chamber  A  of  the  aqueous  humour  is  limited  by  the  iris  /3  7,  which  is  a  kind  of 
Iris.              circular  opaque  screen,  or  diaphragm,  consisting  of  muscular  fibres,  by  whose  contraction  or  expansion   an 

aperture  in  its  centre,  called  the  pupil,  is  diminished  or  dilated,  according  to  the  intensity  of  the  light.  In  very 
strong  lights  the  opening  of  the  pupil  is  greatly  contracted,  so  as  not  to  exceed  twelve  hundredths  of  an  inch  in 
the  human  eye,  while  in  feebler  illuminations  it  dilates  to  an  opening  not  exceeding  twenty-five  hundredths,||  or 
double  its  former  diameter.  The  use  of  this  is  evidently  to  moderate  and  equalize  the  illumination  of  the 
image  on  the  retina,  which  might  otherwise  injure  its  sensibility.  In  animals  (as  the  cat)  which  see  well  in 
the  dark,  the  pupil  is  almost  totally  closed  in  the  daytime,  and  reduced  to  a  very  narrow  line  ;  but  in  the  human 
eye,  the  form  of  the  aperture  is  always  circular.  The  contraction  of  the  pupil  is  involuntary,  and  takes  place 
by  the  effect  of  the  stimulus  of  the  light  itself;  a  beautiful  piece  of  self-adjusting  mechanism,  the  play  of  which 
may  be  easily  seen  by  approaching  a  candle  to  the  eye  while  directed  to  its  own  image  in  a  looking-glass. 

352.  Immediately  behind  the  opening  of  the  iris  lies  the  crystalline  lens,  B,  enclosed  in  its  capsule,  which  forms  the 
Crystalline,   posterior  boundary  of  the  chamber  A.     Its  figure  is  a  solid  of  revolution,  having  its  anterior  surface  much  less 
Its  figure,     curved  than  the  posterior.     Both  surfaces,  according  to  M.  Chossat,  are  ellipsoids  of  revolution   about   their 

letifr  axes  ;  but  it  would  seem  from  his  measures,  that  the  axes  of  the  two  surfaces  are  neither  exactly  coincident 

in  direction  with  each  other,  nor  with  that  of  the  cornea.     This  deviation  would  be  fatal  to  distinct  vision 

were  the  crystalline  lens  very  much  denser  than  the  others,  or  were  the  whole  refraction  performed  by  it.     This, 

Refraction,    however,  is  not  the  case;    for  the  mean  refractive  index  of  this  lens   is  only  1.384,  while  that  of  the  aqueous 

Non-coinci-  humour,  as  we  have  seen,  is  1.337  ;   and  that  of  the  vitreous  C,  which   occupies  the  third  chamber,  is  1.339  ;  so 

mxes^of  its°  ^at   t'le  w'lo'e  amount  of  bending  which  the  rays  undergo  at  the  surface  of  the  crystalline  is  small,  in  compa- 

turfaces.       risen  with  the  inclination  of  the  surface  at  the  point  where  the  bending  takes  place,  and,  since  near  the  vertex,  a 


*  Chenevix,  Philosophical  Tratunctioni,  vol.  xciii.  p.  195. 

t  Bulletin  tie  la  Soc.  Philomatique,  1818,  p.  94. 

I  Edinburgh  Philutophical  Journal,  vol.  i.  p.  42. 

§  Sur  la  Courbiire  des  Milieux  Rtfriiiffens  de  I'CEil  chrz  le  Banif.    dnnalcs  df  Chim.  x.  p.  337. 

|;  Dr.  Young's  Lectures  on  the  Mechanism  of  the  Eye,  Philotophical  Traaiactions,  vol.  xci. 


LIGHT.  397 

Light.      material  deviation  in  the  direction  of  the  axis  can  produce  but  a  very  minute  change  in  the   inclination   of  the      Part  I. 
•— Y"""  raY  to  the    surface,  this  cause   of  error  is  so  weakened  in   its  effect,  as,  probably,  to  produce  no  appreciable  v— ~v"~^ 
aberration.  Wrioutto 

The  crystalline  is  composed  of  a  much  larg-er  proportion  of  albumen  and  gelatine  than  the  other  humours  of  vj,jon 
the  eye,  so  much  so  as  to  be  entirely  coagulable  by  the  heat  of  boiling1  water.     It  is   somewhat  denser  towards       353 
the  centre  than  at  the  outside.     According  to  Dr.  Brewster  and  Dr.  Gordon,  the  refractive  indices  of  its  centre   Composi- 
middle  of  its  thickness,  from  the  centre  to  the  outside,  and  the  outside   itself,  are  respectively  1.3999,  1.3786,  tionofcrys- 
and  1.3767,  that  of  pure  water  being  1.3358.     This  increase   of  density  is  obviously  useful  in   correcting  the  '»"'lne- 
aberration,  by  shortening  the  focus  of   rays  near  the  centre,  according  to  the  rule  laid  down  in  Art.  299  for  (owarc|s 
finding  the  effect  of  aberration.     The  effect  of  the  elliptic  figure  of  the  surfaces  is,  however,  a  matter  of  pretty  centre. 
complex  calculation,  and  cannot  be  entered  upon  in  the  limits  of  this  essay.     Its  use  is,  probably,  to  correct  the 
aberration  of  oblique  pencils. 

The  posterior  chamber  C  of  the  eye  is  filled  with  the  vitreous  humour,  a  fluid  differing  (according  to  Chenevix)      354. 
neither  in  specific  gravity  nor  in  chemical  composition  in  any  sensible  respect  from  the  aqueous ;  and,  as  we  hav« 
already  seen,  having  a  refractive  index  but  very  little  superior  to  it. 

The  refractive  density  of  the  crystalline  being  superior  to  that  of  either  the  aqueous  or  vitreous  humour,  the      355_ 
rays  which  are  incident  on  it  in  a  state  of  convergence  from  the  cornea,  are  made  to  converge  more,  and  exactly  Retina, 
in  their  final  focus  is  the  posterior  surface  of  the  cell  of  the  vitreous  humour  covered  by  the  retina  d,  a  network 
(as  its  name  imports)  of  inconceivably  delicate  nerves,  all  branching  from  one  great  nerve  O,  called  the  optic 
nerve,  which  enters  the  eye  obliquely  at  the  inner  side  of  the  orbit,  next  the  nose.     The  retina  lines  the  whole 
of  the  cavity  C  up  to  i,  where  the  capsule  of  the  crystalline  commences.      Its  nerves  are  in  contact  with,  or 
immersed  in,   the  pigmenttim  nigrum,   a  very  black  velvety  matter,  which  covers  the  choroid  membrane  g,  and 
whose  office  is  to  absorb  and  stifle  all  the  light  which  enters  the  eye  as  soon  as  it  has  done  its  office  of  exciting 
the  retina ;  thus  preventing  internal  reflexions,  and  consequent  confusion  of  vision.     The  whole  of  these  humours 
and  membranes  are  contained  in  a  thick  tough  coat,  called  the  sclerotica,  which  unites  with  the  cornea,  and  forms  Sclerotica. 
what  is  commonly  called  the  while  of  the  eye. 

Such  is  the  structure  by  which  parallel  rays,  or  those  emanating  from  very  distant  objects,  are  brought  to  a  356. 
focus  on  the  retina.  But  as  we  require  to  see  objects  near,  as  well  as  at  a  distance,  and  as  the  focus  of  a  lens  C-han"e  °' 
or  system  of  lenses  for  near  objects  is  longer  than  for  distant  ones,  it  is  evident  that  a  power  of  adjustment  must  forUEear 
reside  somewhere  in  the  eye ;  by  which  either  the  retina  can  be  removed  farther  from  the  cornea,  and  the  eye  objects. 
lengthened  in  the  direction  of  its  axis,  or  the  curvature  of  the  lenses  themselves  altered  so  as  to  give  greater 
convergency  to  the  rays.  We  know  that  such  a  power  exists,  and  can  be  called  into  action  by  a  voluntary  effort; 
and,  evidently,  by  a  muscular  action,  producing  fatigue  if  long  continued,  and  not  capable  of  being  strained 
beyond  a  certain  point.  Anatomists,  however,  as  well  as  theoretical  opticians,  differ  as  to  the  mechanism  by 
which  this  is  effected.  Some  assert,  that  the  action  of  the  muscles  which  move  the  eye  in  its  orbit,  called 
the  recli,  or  straight1  muscles,  when  all  contracted  at  once,  producing  a  pressure  on  the  fluids  within,  forces 
out  the  cornea,  rendering  it  at  once  more  convex,  and  more  distant  from  the  retina.  This  opinion,  however, 
which  has  been  advocated  by  Dr.  Olbers,  and  even  attempted  to  be  made  a  matter  of  ocular  demonstration  by 
Ramsden  and  Sir  E.  Home,  has  been  combated  by  Dr.  Young,  by  experiments  which  show,  at  least,  very 
decisively,  that  the  increase  of  convexity  in  the  cornea  has  little  if  any  share  in  producing  the  effect.  An  elon- 
gation of  the  whole  eye,  spherical  as  it  is  and  full  of  fluid,  to  the  considerable  extent  required,  is  difficult  to 
conceive  as  the  result  of  any  pressure  which  could  be  safely  applied,  as  to  give  distinct  vision  at  the  distance  of 
three  inches  from  the  eye,  (the  nearest  at  which  ordinary  eyes  can  see  well,)  the  sphere  must  be  reduced  to  an 
ellipsoid,  having  its  axis  nearly  one-seventh  longer  than  in  its  natural  state  ;  and  the  extension  of  the 
sclerotica  thus  produced,  would  hardly  seem  compatible  with  its  great  strength  and  toughness.  Another  opinion, 
which  has  been  defended  with  considerable  success  by  the  excellent  philosopher  last  named,  is,  that  the  crystalline 
itself  is  susceptible  of  a  change  of  figure,  and  becomes  more  convex  when  the  eye  adapts  itself  to  near  distances. 
His  experiments,  on  persons  deprived  of  this  lens,  go  far  to  prove  the  total  want  of  a  power  to  change  the  focus 
of  the  eye  in  such  cases,  though  a  certain  degree  of  adaptation  is  obtained  by  the  contraction  of  the  iris,  which, 
limiting  the  diameter  of  the  pencil,  diminishes  the  space  on  the  retina  over  which  imperfectly  converged  rays  are 
diffused,  and  thus,  in  some  measure,  obviates  the  effect  of  their  insufficient  convergence.  When  we  consider 
that  the  crystalline  lens  has  actually  a  regular  fibrous  structure,  (as  may  be  seen  familiarly  on  tearing  to 
pieces  the  lens  of  a  boiled  fish's  eye,)  being  composed  of  layers  laid  over  each  other  like  the  coats  of  an 
onion,  and  each  layer  consisting  of  an  assemblage  of  fibres  proceeding  from  two  poles,  like  the  meridians 
of  a  globe,  the  axis  being  that  of  the  eye  itself;  we  have,  so  far  at  least,  satisfactory  evidence  of  a  muscular 
structure ;  and  were  it  not  so,  the  analogy  of  pellucid  animals,  in  which  no  muscular  fibres  can  be  discerned, 
and  which  yet  possess  the  power  of  motion  and  obedience  to  the  nervous  stimulus,  though  nerves  no  more 
than  muscles  can  be  seen  in  them,  would  render  the  idea  of  a  muscular  power  resident  in  the  crystalline 
easily  admissible,  though  nerves  have  as  yet  not  been  traced  into  it.  On  the  whole,  it  must  be  allowed,  that  the 
presumption  is  strongly  in  favour  of  this  mechanism,  though  the  other  causes  already  mentioned  may,  perhaps, 
conspire  to  a  certain  extent  in  producing  the  effect,  and  though  the  subject  must  be  regarded  as  still  open 
to  fuller  demonstration.  It  is  the  boast  of  science  to  have  been  able  to  trace  so  far  the  refined  contrivances 
of  this  most  admirable  organ ;  not  its  shame  to  find  something  still  concealed  from  its  scrutiny ;  for,  how- 
ever anatomists  may  differ  on  points  of  structure,  or  physiologists  dispute  on  modes  of  action,  there  is  that 
in  what  we  do  understand  of  the  formation  of  the  eye  so  similar,  and  yet  so  infinitely  superior,  to  a  product 
of  human  ingenuity, — such  thought,  such  care,  such  refinement,  such  advantage  taken  of  the  properties  of 
natural  agents  used  as  mere  instruments,  for  accomplishing  a  given  end,  as  force  upon  us  a  conviction  of 


398  L  I  G  H  T. 

Light,      deliberate   choice  and  premeditated  design,  more  strongly,  perhaps,  than  any  single  contrivance  to  be  found,     Part 
>-"~v~™"/  whether  in  art  or  nature,  and  render  its  study  an  object  of  the  deepest  interest.  ^~Y 

357.  The  images  of  external  objects  are  of  course  formed  inverted  on  the  retina,  and  may  be  seen  there,  by  dissect- 

Image  on     jngp  off  the  posterior  coats  of  the  eye  of  a  newly-killed  animal,  and   exposing  live  retina  and  choroid  membrane 
the  retina     from  behind,  like  the  image  on  a  screen  of  rough  glass,  mentioned  in  Art.  331.     It  is  this  image,  and  this  only, 
diate'obirct  wn'ch  's  fe^  by  the  nerves  of  the   retina,  on  which  the  rays  of  light  act  as  a  stimulus  ;    and  the  impressions 
•f  vision.      therein  produced  are  thence  conveyed  along  the  optic  nerves  to  the  sensorium,  in  a  manner  which  we  must  rank  at 
present  among  the  profounder  mysteries  of  physiology,  but  which   appears  to   diti'er  in  no  respect  from  that  in 
which  the  impressions  of  the  other  senses  are  transmitted.     Thus,  a  paralysis  of  the  optic  nerve  produces,  while 
i  it  lasts,  total  blindness,  though  the  eye  remains  open,  and  the  lenses  retain  their  transparency ;  and  some  very 

curious  cases  of  half  blindness  have  been  successfully  referred  to  an  affection  of  one  of  the  nerves  without  the 
other.*  On  the  other  hand,  while  the  nerves  retain  their  sensibility,  the  degree  of  perfection  of  vision  is  exactly 
commensurate  to  that  of  the  image  formed  on  the  retina.  In  cases  of  cataract,  where  the  crystalline  lens  loses 
its  transparency,  the  light  is  prevented  from  reaching  the  retina,  or  from  reaching  it  in  a  proper  state  of  regular 
concentration,  being  stopped,  confused,  and  scattered  by  the  opaque  or  semi-opaque  portions  it  encounters  in  its 
passage.  The  image,  in  consequence,  is  either  altogether  obliterated,  or  rendered  dim  and  indistinct ;  and  the 
progress  of  blindness  is  accordingly.  If  the  opaque  lens  be  extracted,  the  full  perception  of  light  returns  ;  but 
one  principal  instrument  for  producing  the  convergence  of  the  rays  being  removed,  the  image,  instead  of  being 
formed  on  the  retina,  is  formed  considerably  behind  it,  and  the  rays  being  received  in  their  unconverged  state  on 
it,  produce  no  regular  picture,  and  therefore  no  distinct  vision.  But  if  we  give  to  the  rays,  before  their  entry  into 
the  eye,  a  certain  proper  degree  of  convergence,  by  the  application  of  a  convex  lens,  so  as  to  render  the  remain- 
ing lenses  capable  of  finally  effecting  their  exact  convergence  on  the  retina,  restoration  of  distinct  vision  is  the 
immediate  result.  This  is  the  reason  why  persons  who  have  undergone  the  operation  for  the  cataract  (which 
consists  either  in  totally  removing,  or  in  putting  out  of  the  way  an  opaque  crystalline)  wear  spectacles  of 
comparatively  very  short  focus.  Such  glasses  perform  the  office  of  an  artificial  crystalline.  A  similar  imper- 
fection of  vision  to  that  produced  by  the  removal  of  the  crystalline,  is  the  ordinary  effect  of  old  age,  and  its 
remedy  is  the  same.  In  aged  persons  the  exterior  transparent  surface  of  the  eye,  called  the  cornea,  loses  some- 
what of  its  convexity,  and  becomes  flatter.  The  power  of  the  eye  is  therefore  diminished,  (Art.  248  and  255,) 
and  a  perfect  image  can  no  longer  be  formed  on  the  retina.  The  deficient  power  is  however  supplied  by  a 
convex  lens,  or  spectacle-glass,  (Art.  268,)  and  vision  rendered  perfect  or  materially  improved. 

358  Short-sighted  persons  have  their  eyes  too  convex,  and  this  defect  is,  like  the  other,   remediable  by  the  use  of 

proper  lenses  of  an  opposite  character.  There  are  cases,  however,  though  rare,  in  which  the  cornea  becomes  so 
very  prominent  as  to  render  it  impossible  to  apply  conveniently  a  lens  sufficiently  concave  to  counteract  its  action. 
Such  cases  would  be  accompanied  with  irremediable  blindness,  but  for  that  happy  boldness,  justifiable  only  by  the 
certainty  of  our  knowledge  of  the  true  nature  and  laws  of  vision,  which  in  such  a  case  has  suggested  the 
opening  of  the  eye  and  removal  of  the  crystalline  lens,  though  in  a  perfectly  sound  state. 

359.  But  these  are  not  the  only  cases  of  defective  vision  arising  from  the  structure  of  the  organ,  which  are  suscep- 

Malconfor-    tible  of  remedy.     Malconformations  of  the  cornea  are  much  more  common  than  is  generally  supposed,  and  few 
nutions  of    eves  arCi  m  facit  free  from  them.     They  may  be  detected  by  closing  one  eye,  and  directing  the  other  to  a  very 
the  cornea.    narroWi  well-defined  luminous  object,  not  too  bright,  (the  horns  of  the  moon,  when  a  slender  crescent,  only  two 
or  three  days  old,  are  very  proper  for  the  purpose,)  and  turning  the  head  about  in  yarious  directions.     The  line 
will  be  doubled,  tripled,  or  multiplied,  or  variously  distorted  ;   and  careful  observation  of  its  appearances,  under 
different  circumstances,  will   lead  to  a  knowledge  of  the  peculiar  conformation  of  the  refracting  surfaces  of  the 
Remarkable  eye  which  causes  them,  and  may  suggest  their  proper  remedy.     A  remarkable  and  instructive  instance  of  the 
case,  sue-     kind  has  recently  been  adduced  by  Mr.  G.  B.  Airy,   (Transactions  of  the   Cambridge  Philosophical  Society,) 
cessfully       ;n  tne  case  of  one  of  n;s  own  eves  .  wnich,  from  a  certain  defect  in  the  figure  of  its   lenses,  he  ascertained  to 
refract  the  rays  to  a  nearer  focus  in  a  vertical  than  in  a  horizontal  plane,  so  as  to  render  the  eye  utterly  useless. 
This,  it  is  obvious,  would  take  place  if  the  cornea,  instead  of  being  a  surface  of  revolution,  (in  which  the  curvature 
of  all  its  sections  through  the  axis  must  be  equal,)  were  of  some  other  form,  in  which  the  curvature  in  a  vertical 
plane  is  greater  than  in  a  horizontal.     It  is  obvious,  that  the  correction  of  such  a  defect  could   never  be  accom- 
plished by  the  use  of  spherical  lenses.     The  strict  method,  applicable  in  all  such  cases,  would  be  to  adapt  a  lens 
to  the   eye,  of  nearly  the  same  refractive  power,  and  having  its  surface  next  the  eye  an  exact  intaglio  fac-simile 
of  the  irregular  cornea,  while  the  external  should  be  exactly  spherical  of  the  same  general  convexity  as  the  cornea 
itself;  for  it  is  clear,  that  all  the  distortions  of  the  rays  at  the  posterior  surface  of  such  a  lens  would  be  exactly 
counteracted  by  the  equal  and  opposite  distortions  at  the  cornea  itself.f     But  the  necessity  of  limiting  the  cor- 
recting lens  to  such  surfaces  as  can  be  truly  ground  in  glass,  to   render  it  of  any  real   and   everyday  use,   and 
which  surfaces  are  only  spheres,  planes,  and  cylinders,  suggested   to   Mr.  Airy  the   ingenious  idea  of  a  double 
concave  lens,  in  which  one   surface  should  be  spherical,  the  other  cylindrical.     The  use  of  the   spherical  surface 
was   to  correct   the   general  defect  of  a  too  convex  cornea.     That  of  the  cylindrical    may  be   thus    explained. 
Suppose  parallel  rays  incident  on  a  concave  cylindrical  surface,  A  B  C  D,  in  a  direction  perpendicular  to  its  axis, 
Fig  71.        as  in  fig.  71,  and  let  S  S'  P  I"  Q  Q'  T  T',  be  any  laminar  pencil  of  them  contained  in  a  parallelepiped  infinitely 

*  Wollaslon,  on  Semi -decussation  of  the  Optic  Nerves,  Philosophical  Transiictioits,  1824. 

t  Should  any  very  bad  cases  of  irregular  cornea  be  found,  it  is  worthy  of  consideration,  whether  at  least  a  temporary  distinct  vision  coniil 
not  be  procured,  by  applying  in  contact  with  the  surface  of  the  eye  some  transparent  animal  jelly  contained  in  a  spherical  capsule  of  glass  ;  or 
whether  an  actual  mould  of  the  cornea  might  not  be  Ukeu.  r.nd  impressed  on  some  transparent  medium.  The  operation  would,  of  course,  be 
delicate,  but  certainly  less  so  than  that  of  cutting  open  a  living  eye,  and  taking  out  its  contents. 


LIGHT.  399 

I  >fl  '•  thin,  and  having  its  sides  parallel  to  the  axis.  Any  of  the  rays  S  P,  S'  P',  of  this  pencil  lying-  in  a  plane  APS  Part  I. 
•~v~~-'  perpendicular  to  the  axis,  will  after  refraction  converge  to,  or  diverge  from,  a  point  X,  also  in  this  plane  ;  and,  v^-^^-^ 
therefore,  all  the  rays  incident  on  P  Q,  P'  Q',  will  after  refraction  have  for  their  focus  the  line  X  Y,  in  the  caustic 
surface  A  F  G  D,  and  the  principal  focus  of  the  cylinder  will  be  the  line  F  G,  whose  distance  from  the  vertex 
C  C'  of  the  surface,  or  F  C,  is  the  same  with  the  focal  length  of  a  spherical  surface,  formed  by  the  revolution  of 
A  B  about  the  axis  F  C.  Thus  we  see  that  a  cylindrical  lens  produces  no  convergency  or  divergency  in  parallel 
rays,  incidental  in  the  plane  of  its  axis ;  while  it  converges  or  diverges  rays  in  a  plane  at  right  angles  to  the 
axis,  as  a  spherical  surface  of  equal  curvature  would  do  If  then  such  a  cylindrical  surface  he  conjoined  with 
a  spherical  one,  the  focus  of  the  spherical  surface  will  remain  unaltered  in  one  plane,  but  in  the  other  will  be 
changed  to  that  of  a  lens  formed  by  it,  and  a  spherical  surface  of  equal  curvature  with  the  cylinder.  Hence  by 
properly  placing  such  a  cyliudro-spheric  lens  across  the  defective  eye,  its  detect  will  be  (approximately,  at  least) 
counteracted.  It  would  be  wrong  to  conclude  our  account  of  this  interesting  application  of  mathematical 
knowledge  to  the  increase  of  the  comforts  and  improvement  of  the  faculties  of  its  possessor,  in  other  than  his 
own  words.  "  After  some  ineffectual  applications  to  different  workmen,  I  at  last  procured  a  lens  to  these 
dimensions,*  from  an  artist  named  Fuller,  at  Ipswich.  It  satisfies  my  wishes  in  every  respect.  I  can  now  read 
the  smallest  print  at  a  considerable  distance  with  the  left"  (the  defective)  "  eye  as  well  as  with  the  right.  I 
have  found  that  vision  is  most  distinct  when  the  cylindrical  surface  is  turned  from  the  eye  :  and  as,  when  the 
lens  is  distant  from  the  eye,  it  alters  the  apparent  figure  of  objects  by  refracting  differently  the  rays  in  different 
planes,  I  judged  it  proper  to  have  the  frame  of  my  spectacles  made  so  as  to  bring  the  glass  pretty  close  to 
the  eye.  With  these  precautions,  I  find  that  the  eye  which  I  once  feared  would  become  quite  useless,  can  be 
used  in  almost  every  respect  as  well  as  the  other." 

Blindness,  partial  or  total,  may  be  caused,  not  only  by  the  opacity  of  the  crystalline  lens,  but  of  any  other  360. 
part,  or  by  anything  extraneous  to  the  materials  of  which  they  consist,  interposed  between  the  external  trans- 
parent surface  of  the  cornea  and  the  retina.  In  all  such  cases,  if  the  sensibility  of  the  nerve  be  uninjured,  the 
restoration  of  sight  is  never  to  be  despaired  of.  In  a  recent  most  remarkable  case,  operated  by  Mr.  Wardrop, 
and  by  him  recorded  in  the  Philosophical  Transactions  for  1826.  blindness  from  infancy,  accompanied  with 
complete  obliteration  of  the  pupil,  by  a  contraction  of  the  iris,  owing  to  an  unskilful  operation,  performed  at 
six  months  of  age,  was  removed,  and  perfect  sight  restored  after  a  lapse  of  forty-six  years,  by  a  simple  removal 
of  the  obstruction,  by  breaking  a  hole  through  the  closed  membrane.  The  details  of  this  case  are  in  the 
highest  degree  interesting,  but  we  must  refer  the  reader  to  the  volume  of  the  Philosophical  Transactions  cited  for 
the  account. 

As  we  have  two  eyes,  and  a  separate  image  of  every  external  object  is  formed  in  each,  it  may  be  asked,  why  do       361. 
we  not  see,  double  ?  and  to  some,  the  question  has  appeared  to  present,  much  difficulty.     To  us  it  appears,  that  we  Single 
might  with  equal  reason  ask,  why — having  two  hands,  and  five  fingers  on  each,  all  endowed  with  equal  sensi-  *lslon  WItn 
bility  of  touch   and  equal  aptitude  to  discern  objects  by  that  sense — we  do  not  feel  decuple?     The  answer  is  the 
same  in  both  cases  :  it  is  a  matter  of  habit.     Habit  alone  teaches  us  that  the   sensations  of  sight  correspond  to 
any  thing  external,  and  to  what  they  correspond.     An  object  (a  small  globe  or  wafer  suppose)  is  before  us  on  a 
table  ;  we  direct  our  eyes  to  it,  i.  e.  we  bring  its  images  on  both  retinae  to  those  parts  which  habit  has  ascer- 
tained to  be  the  most  sensible  and  best  situated  for  seeing  distinctly  ;  and  having  always  found  that  in  such 
circumstances  the  object  producing  the  sensation  is  one  and  the  same,  the  idea  of  unity  in   the   object  becomes 
irresistibly  associated  with  the  impression.     But  while  looking  at  the  globe,  squeeze  the  upper  part  of  one  eye  Double 
downwards,  by  pressing  on  the  eyelid  with  the  finger,  and  thereby  forcibly  throw  the  image  on  another  part  of  vision 
the  retina  of  that   eye,  and  double  vision  is  immediately   produced,  two   globes  or  two  wafers  being   distinctly  art|ficially 
seen,  which  appear  to  recede  from  each  other  as  the  pressure  is  stronger,  and  approach,  and  finally  blend  into  pro  u 
one  as   it  is  relieved.     The  same  effect  may  be  produced  without  pressure,  by   directing   the   eyes  to  a  point  Another 
nearer  to,  or  farther  from  them  than  the  wafer  ;  the  optic  axes  in  this  case  being  both  directed  away  from  the  method. 
object  seen.     When  the  eyes  are  in  a  state  of  perfect  rest,  their  axes  are  usually  parallel,  or  a  little  diverging. 
In  this  state  all  near  objects  are  seen  double ;  but  the  slightest  effort  of  attention  causes  their  images  to  coalesce 
immediately.     Those  who  have  one  eye  distorted  by  a  blow,  see  double,  till  habit  has  taught  them  anew  to  see 
single,  though  the  distortion  of  the  optic  axis  subsists. 

The  case  is  exactly  the  same  with  the  sense  of  touch.     Lay  hands  on  the  globe,  and  handle  it.     It  is  one,       362. 
nothing  can  be  more  irresistible  than  this  conviction.     Place  it  between  the  first  and  second  fingers  of  the  right  Single 
hand  in  their  natural  position.     The  right  side  of  the  first  and  left  of  the  second  finger  feel  opposite  convexities  ;  ^^  felt 
but  as  habit  has  always  taught  us  that  two  convexities  so  felt  belong  to  one  and  the  same  spherical  surface,  we  j^JJain  cLej 
never  hesitate  or  question  the  identity  of  the  globe,  or  the  unity  of  the  sensation.     Now  cross  the  two  fingers, 
bringing  the  second  over  the  first,  and  place  the  globe  on  the  table  in  the  fork  between  them,  so  as  to  feel  the  left 
side  of  the  globe  with  the  right  side  of  the  second  finger,  and  the  right  with  the  left  of  the  first.     In  this  state  of 
things  the  impression  is  equally  irresistible,  that  we  have  two  globes  in  contact  with  the  fingers,  especially  if  the 
the  eyes  be  shut,  and  the  fingers  placed  on  it  by  another  person.     A  pea  is  a  very  proper  object  for  this  experi- 
ment.    The  illusion  is  equally  strong  when  the  two  fore  fingers  of  both  hands  are  crossed,  and  the  pea  placed 
between  them. 

So  forcible  is  the  power  of  habit  in  producing  single  vision,  that  it  will  bring  the  two  images  to  apparent  353. 
coalescence,  when  the  rays  which  form  one  of  them  are  really  turned  far  aside  from  their  natural  course.  To  Force  of 
show  this,  place  a  candle  at  a  distance,  and  look  at  it  with  one  eye  (the  left  suppose)  naked,  the  other  having  hal)it '" 

producing 

single  vision 

•  »»diu*  of  the  sphencai  surface  3$  inches,  of  the  cylindrical  4£.  illustrated  by 

experiment. 


LIGHT 

* 

Light,  before  it  a  prism,  with  a  variable  refracting  angle,  (an  instrument  to  be  described  hereafter,  see  INDRX,)  and,  first,  P.irt  I 
V—V »/  let  the  angle  be  adjusted  to  zero,  then  will  the  prism  produce  no  deviation,  and  the  object  will  appear  single,  v— -%— 
Now  vary  the  prism,  so  as  to  produce  a  deviation  of  2°  or  3°  of  the  rays  in  a  horizontal  plane  to  the  right.  The 
candle  will  immediately  be  seen  double,  the  image  deviated  by  the  prism  being  seen  to  the  left  of  the  other ;  but 
the  slightest  motion,  such  as  winking  with  the  ejelids,  blends  them  immediately  into  one.  Again,  vary  the  prism 
a  few  degrees  more  in  the  same  direction  ;  the  candle  will  again  be  doubled,  and  again  rendered  single  by  winking, 
and  directing  the  attention  more  strongly  to  it  ;  and  thus  may  the  optic  axes  be,  as  it  were,  inveigled  to  an 
inclination  of  20°  or  30°  to  each  other.  In  this  state  of  things,  if  a  second  candle  be  placed  exactly  in  the 
direction  of  the  deviated  image  of  the  first,  but  so  screened,  that  its  rays  shall  not  fall  on  the  left  eye,  and  the 
pri«m  be  then  suddenly  removed  in  the  act  of  winking,  the  two  candles  appear  as  one.  If  the  deviation  of  the 
image  seen  with  the  right  eye  be  made  to  the  apparent  right,  the  range  within  which  it  is  possible  to  bring  them 
to  coalesce  is  much  more  limited,  as  it  is  much  more  usual  for  us  to  direct  by  an  effort  the  optic  axes  towards, 
than  from  each  other.  If  the  deviation  be  made  but  a  very  little  out  of  the  horizontal  plane,  no  effort  will 
enable  us  to  correct  it.  It  is  probable  that  s,,.ae  cases  of  squinting  might  be  cured  by  some  such  exercise  in 
the  art  of  directing  the  optic  axes,  if  continued  perseveringly. 

304.  Such  is,  undoubtedly,  a  sufficient  explanation  of  single  vision  with  two -eyes;  yet  Dr.  Wollaston  lias  rendered 

A  further  jt  probable  that  a  physiological  cause  has  also  some  share  in  producing  the  effect,  and  that  a  semi-decussation  of 
l]na\e°  ^e  °Pt'c  nerves  takes  place  immediately  on  their  quitting  the  brain,  half  of  each  nerve  going  to  each  eye,  the 
vision  right  half  of  each  retina  consisting  wholly  of  fibres  of  one  nerve,  and  the  left  wholly  of  the  other,  so  that  all 

Nervous       images  of  objects  out  of  the  optic  axis  are  perceived  by  one  and  the  same  nerve  in  both  eyes,  and  thus  a  power- 
sympathy,     ful  sympathy  and  perfect  unison  kept  up  between  them,  independent  of  the  mere  influence  of  habit.     Immediately 
in  the  optic  axis,  it  is  probable,  that  the  fibres  of  both   nerves  are   commingled,  and  this  may  account  for  the 
greater  acuteness  and  certainty  of  vision  in  this  part  of  the  eye. 

365.  Another  point,  on  which  much  more  discussion  has  been  expended  than  it  deserves,  is  the  fact  of  our  seeing 
Erect  vision  objects  erect  when  their  images  on  the  retina  are  inverted.     Erect,  means  nothing  else  than  having   the  he^d 

tr"  •  ""  fi""tner  from  the  ground,  and  the  feet  nearer,  than  any  other  part.     Now,  the  earth,  and  the  objects  which  ' 

on  it,  preserve  the  same  relative  situation  in  the  picture  on  the  retina  that  they  do  in  nature.  In  th»*''  ,.ar*- 
men,  it  is  true,  stand  with  their  heads  downwards  ;  hut  then,  at  the  same  time,  heavy  bodies  fall  upv,  a.rus  ;  and 
the  mind,  or  its  deputy,  the  nerve,  which  is  present  in  every  part  of  the  picture,  judges  only  of  the  relations  of  its 
parts  to  one  another.  How  these  parts  are  related  to  external  objects,  is  known  only  by  experience,  and  judged 
of  at  the  instant  only  by  habit. 

366.  There  is  one  remarkable  fact  which  ought  not  to  escape  mention,  even  in  so  brief  an  abstract  of  the  doctrine 
Puuctuin      of  vision  as  the  present,  it  is,  that  the  spot  Q,  at  which  the  optic  nerve  enters  the  eye,  is  totally  insensible  to  the 
cacum.        stimulus  of  light,  for  which  reason  it  is  called  the  punctum  ceeciim.     The  reason  is  obvious :  at  this   point  the 

nerve  is  not  yet  divided  into  those  almost  infinitely  minute  fibres,  which  are  fine   enough   to  be  either  thrown 

into  tremors,  or  otherwise  changed  in  their  mechanical,  chemical,  or  other  state,  by  a  stimulus  so  delicate  as  the 

Experiment  rays  of  light.     The  effect,  however,  is  curious  and  striking.     On  a  sheet  of  black  paper,  or  other  dark  ground, 

proving  its    place  two  white  wafers,  having  their  centres   three  inches  distant.     Vertically   above  that  to  the  le/l,  hold  the 

Ke'     right  eye,  at  12  inches  from  it,  and  so   that  when   looking  down   on   it,  the   line  joining  the  two  eyes  shall  be 

parallel  to  that  joining  the  centre  of  the  wafers.     In  this  situation  closing  the  left  eye,  and  looking  full  with  the 

right   at  the  wafer  perpendicularly  below  it,  this   only   is  seen,   the   other  being  completely    invisible.     But  if 

remove-'l  ever  so  little  from  its  place,  either  to  the  right  or  left,  above  or  below,  it   becomes  immediately  visible, 

and  starts,  as  it  were,  into  existence.     The  distances  here  set  down  may  perhaps  vary  slightly  in  different  eyes. 

367.  I*  w'l'  cease  to  be  thought  singular,  that  this  fact,  of  the  absolute  invisibility  of  objects   in  a  certain  point  of 
the  field  of  view   of  each  eye,  should  be  one  of  which  not  one  person  in  ten  thousand   is  apprized,  when  we 
learn,  that  it  is  not  extremely  uncommon  to  find  persons  who  have  for   some  time  been   totally  blind  with   one 
eye   without    being  aware  of    the  fact.     One  instance  has   i'allen  under  the   knowledge  of  the  writer  of  these 
pages. 

368.  In  the  eyes  of  fishes,  the  humours  being  nearly  of  the  refractive  density  of  the  medium  in  which  they  live,  the 
Eyes  of       refraction  at  the  cornea  is  small,  and  the  work  of  bringing  the  rays  to  a  focus   on  the  retina   is  almost  wholly 
fishes.           performed   by  the  crystalline.     This  lens,  therefore,  in  fishes  is  almost  spherical,  and  of  small  radius,  in  compa- 
rison with  the  whole  diameter  of  the  eye.     Moreover,  the  destruction  of  spherical  aberration  not  being  producible 
in  this  case  by  mere  refraction  at  the  cornea,  the  crystalline  itself  is  adapted  to  execute  this  necessary  part  of  the 
process,  which    it    does    by  a  very    great    increase    of  density    towards    the    centre.     (Brewster,   Treatite  on 
New  Philosophical  Instruments,  p.  2G8.)     The  fibrous  and  coated  structure  of  the  crystalline  lens  is  beautifully 
shown  in  the  eye  of  a  fish  coagulated  by  boiling. 

369  The  same  scientific   principles  which  enable  us  to   assist  natural  imperfections  of  sight,  can  be   employed  in 

giving  additional  power  to  this  sense,  even  in  individuals  who  enjoy  it  naturally  in  the  greatest  perfection.  It 
being  once  understood,  that  the  image  on  the  retina  is  that  which  we  really  see,  it  follows,  that  if  by  any  means 
we  can  render  this  image  brighter,  larger,  more  distinct  than  in  the  natural  state  of  the  organ,  we  shall  see  objects 
brighter  than  in  their  natural  state,  enlarged  in  dimension,  and,  therefore,  capable  of  being  examined  more  in 
detail,  or  more  sharply  defined  and  clearly  outlined.  The  means  which  the  principles  already  detailed  put  in  em- 
power, for  the  accomplishment  of  such  ends,  are  the  concentration  of  more  rays  than  enter  the  natural  eye  by 
lenses;  the  enlargement  of  the  image  on  the  retina,  by  substituting  for  the  object  seen  an  image  of  it,  either 
larger  than  the  object  itself,  or  capable  of  being  brought  nearer  to  us  ;  and  the  destruction  of  aberration,  Im- 
properly adapting  the  figure  and  materials  of  our  instruments  to  the  end  proposed. 

Proposition.     The  apparent  magnitude  of  a  rectilinear  object  is  measured   by  the  angle  subtended  by    it   at 


LIGHT.  401 

Light      the  centre  of  the   eye,  or  by  the  linear  magnitude   of  its  image   on  the  retina,  and  is   therefore  proportional     Part  I- 
—  V"  "^         linear  magnitude  of  object  ^—p-y-^' 

its  distance  from  the  eye 

The  centre  of  the  eye,  in  its  optical  sense,  is  a  point  nearly  in  the  centre  of  the  pupil  in  the  plane  of  the  iris, 
and  the  image  of  any  ex  ernal  object  P  Q,  being  formed  at  the  bottom  of  the  eye  at  p  q,  by  rays  crossing  there,  F'?-  "2. 

pE 
must  subtend  the  same  angle  ;  so  that  p  q  =  P  Q  .  =;•=• 

X     III 

Carol.    If  the  object  be  so  distant  that  the  rays  from  each  point  of  it  may  be  regarded  as  parallel,  the  angular      *•*• 
diameter  of  the  object  is  measured  by  the  inclination  of  rays  of  its  extreme  pencils  to  each  other.     Whenever, 
therefore,  the  eye  sees  by  parallel,  or  very  nearly  parallel,  rays,  the  apparent  magnitude  of  the  object  seen,  is 
measured  by  the  inclination  of  its  extreme  pencils,  and  the  object  itself  is  referred  to  an  infinite  distance,  or  to  the 
concave  surface  of  the  heavens. 

Prop.     When  a  convex  lens  is  placed   between   the  eye  and  any  object,  so  as  to  have  the  object  at  a  distance      372. 
from  the  lens  equal  to  its  focal  length,  it  will  be  distinctly  seen  by  an  eye  capable  of  converging  parallel  rays,  and 
will  appear  enlarged  beyond  its  natural  size. 

Let  P  Q  be  the  object,  C  the  lens,  and  E  the  centre  of  the  eye.  Since  the  object  is  ;.n  the  focus  of  the  lens,  Fig.  73. 
the  rays  of  a  pencil  diverging  from  any  point  P  in  it,  will  emerge  parallel  to  P  C,  and  to  each  other  ;  they  will, 
therefore,  after  refraction  in  the  eye,  be  brought  to  converge  on  the  retina  to  a  point  p,  such  that  E  p  is  parallel 
to  P  C.  Similarly,  rays  from  Q  will,  after  refraction  through  the  lens  and  eye,  converge  to  tj;  such  that  E  q  is 
parallel  to  Q  C.  Thus,  a  distinct  image  will  be  formed  at  p  q  on  the  retina,  and  the  apparent  angular  magnitude 
of  the  object  seen  through  the  lens  will  be  the  angle  q  E  p.  Now  this  is  equal  to  P  C  Q,  or  the  angle  subtended 
by  the  object  at  the  centre  of  the  lens,  and  is,  therefore,  greater  than  P  E  Q,  or  that  subtended  by  it  at  the  centre 
of  the  eye,  because  the  lens  is  between  the  eye  and  object. 

Hence,  the  nearer  the  eye  is  to  the  lens,  the  less  will  be  the  difference  between  the  apparent  magnitudes  of  the      373. 
object,  as  seen  with  and  without  the  lens  interposed.    But  if  the  lens  be  of  shorter  focus  than  the  least  distance  at 
which  the  eye  can  see  distinctly,  there  will  be  this  essential  difference  between  vision  with  and  without  the  lens, 
that  in  the  former  case  the  object  is  seen  distinctly,  and  well-defined  ;  while  in  the  latter,  or  with  the  naked  eye, 
it  will  be  indistinct  and  confused,  and  the  more  so  the  nearer  it  is  brought. 

Hence,  by  the  use  of  a  convex  lens  of  short  focus,  objects  may  be  seen  distinct,  and  magnified  to  any  extent  we      374 
please  :  for  let  L  be  the  power,  or  reciprocal  focal  length  of  the  lens,  and  D  the  greatest  proximity  of  the  object  By  a  con- 
to  the  centre  of  the  eye  at  which  it  can  be  seen   distinctly  without  a  lens.     Then  we  shall  have  L  :  D  :  :  angle  vex  lens  of 

p  E  q  :  angle  subtended  by   the  object    at  the  proximity  D  ;    and,  therefore,  :  :  apparent  linear  magnitude  of  sllort  focus 

objects  are 

object  seen  through  the  lens  :  apparent  linear  magnitude  at  proximity  D,  with  the  naked  eye.     Therefore  -  is  magml 

the  ratio  of  these  magnitudes,  or,  as  it  is  called,  the  magnifying  power  of  the  lens,  beyond  that  of  the  naked  eye,  Magnifying 
at  its  greatest  proximity.  power. 

Carol.     D  being  given,  the  magnifying  power  is  as  L,  or  as  (/»  —  1)  (R'  —  R").     This  explains  the  use  of  the      375. 
word  power  in  the  foregoing  sections.     Whatever  we  have  demonstrated  of  the  powers  of  lenses  in  the  foregoing  Magnifying 
pages,  is  true  of  magnifying  powers.     Thus  the  sum   of  the   magnifying  powers  of  two  convex  lenses  is  the  po   er  of,  a 
magnifying  power  of  the  two  combined.     If  one  be  concave,  its  magnifying  power  is  to  be  regarded  as  negative,  lenses 
and  instead  of  their  sum  we  must  take  their  difference. 

Prop.     To  express,  generally,  the  visual  angle  under  which  a  small  object  placed  at  any  distance  from  a  lens,      376. 
and  seen  by  an  eye  any  how  situated,  appears,  supposing  it  seen  distinctly. 

Let  P  Q,  fig.  74,  75,  76,  77,  be  the  object,  E  the  lens,  O  the  eye,  and  p  q  the  image.     Put  —  ^—  =  D,  —  —  f/'J74'  7o> 

h,  Q  E  q       > 

j  Visual 

=  f;         _    =  e;  e  being  reckoned  in  the  same  direction  from  the  centre  of  the  lens  that   D  and/"  are.     Then  angle. 

O  1) 

the  visual  angle  under  which  the  image  is  seen  is  q  O  p,  and  we  have,  therefore,  visual  angle  (=  A)  =  ~—  = 

**f 

•     But,  qp  =  Q  P  .  —  -^  =  Q  P  .  —  -  =  O  .  ~r-  putting  O  for  Q  P  the  linear  magnitude  of  the  Vision 


.-.r.T.      •  ,  .  —  -  .  —  -  .  ~r-  e 

U  ti  -  •  tj  q  t,   (J  J  J  through 

1  1  f—e  convex 

object  ;  and,  moreover,  O  E  —  E  q  =  —    --  —  —J——  —  ,  therefore  we  have,  lenses. 

/  J  e 


A=0  . 


•  f-e  ~        -L  +  D-e 

when  L,  as  all  along,  represents  the  power  of  the  lens.     Now  O  .  D  is  the  visual  angle  of  the  object,  as  seen 

Q  P 

from  the  centre  of  the  lens  ;  therefore,  putting  O  .  D,  or  —  —  =  (A)  we  get 

Q  h, 


VOL.  IV. 


-102 


I.  I  G  H  T. 


377. 
Through 

concave. 

378. 

Inreflectors. 


In  concave  lenses,  the  images  of  distant  objects  are  formed  erect,  and  on  the  same  side  of  the  lens  with  the  object. 
'  If,  therefore,   such  a  lens  be  held  between  the  eye  and   distant  objects  at  a  sufficient  distance  from  the  eye  for  1 
distinct  vision,  the  objects  will  be  seen  erect,  and  diminished  in  magnitude.     In  this  case,  e  is  positive,  and  L/  and 
D  both  negative;  therefore   L  +  D  —  e  is  a  negative  quantity,  greater  (without  regard  to  the  sign)  than  e,  and, 
consequently,  A  is  negative,  and  less  than  (A). 


In  reflectors,  /  =  2  R  —  D,  and,  therefore, 


379. 
General 
principles 
of  tele- 
scopes. 


380. 

Astronomi- 
cal tele- 
icope. 

Fig.  80, 81 
381. 


Field  of 
view. 


382. 


A  =  (A)  . 


2R-D-e' 


In  a  convex  reflector,  e  is  necessarily  negative,  at  least  if  the  mirror  be  made  of  metal,  because  the  eye  must  be 
on   the  side  of  the  surface  towards  the  incident  light ;    and,  therefore,  2  R  —  e  is  positive,  and 


R  -  D  -  « 

will  be  greater  or  less  than  unity,  according  to  the  value  of  2  R  —  D  —  e.  In  concave  reflectors,  R  is 
negative,  and  e  is  also  negative  for  the  same  reason  as  in  concave  ;  therefore  the  sign  and  magnitude  of 
A  in  this,  as  well  as  the  former  case,  may  vary  indefinitely,  according  to  the  place  of  the  eye,  the  image, 
and  the  object.  The  varieties  of  these  cases  are  represented  in  fig.  78  and  79. 

If  the  image,  instead  of  being  seen  directly  by  the  naked  eye,  be  seen  through  the  medium  of  another 
lens  or  reflector,  so  plactd  as  to  cause  the  pencils  diverging  primarily  from  each  point  of  the  object,  to 
emerge  finally,  either  exactly  parallel,  or  within  such  limits  of  convergence  or  divergence  as  the  eye  can 
accommodate  itself  to,  the  object  will  be  seen  distinctly,  and  either  larger  or  smaller  than  it  would  be  seen  by  the 
unassisted  eye,  according  to  the  magnitude  of  the  image,  and  the  power  of  the  lens  or  reflector  used  to  view 
it.  This  is  the  principle  of  all  telescopes  and  microscopes.  As  most  eyes  can  see  with  parallel  rays,  they  are 
so  constructed  as  to  make  parallel  pencils  emerge  parallel  ;  and  a  mechanical  adjustment  allows  such  a  quantity 
of  motion  of  the  lenses  or  reflectors  with  respect  to  each  other,  as  to  give  the  rays  a  sufficient  degree  of  conver- 
gence or  divergence  as  may  be  required. 

In  the  common  refracting,  or,  as  it  is  sometimes  called,  the  astronomical  telescope,  the  image  is  first 
formed  by  a  convex  lens,  and  is  viewed  through  a  convex  lens,  placed  at  a  distance  from  the  other  nearly 
equal  to  the  sum  of  their  focal  lengths.  The  lens  which  forms  the  image  is  called  the  object-glass,  and  that 
through  which  it  is  viewed,  the  eye-glass  of  the  telescope.  If  the  latter  be  concave,  the  telescope  is  said  to 
be  of  the  Galilaean  construction,  such  having  been  the  original  arrangement  of  Galiheo's  instruments.  The 
situation  of  the  lenses,  and  the  course  of  the  rays  in  these  two  constructions,  are  represented  in  fig.  80  and  81. 

In  the  former  construction,  let  P  Q  be  the  object.  Draw  Q  O  G  through  the  centres  of  the  object  and 
eye-glass,  and  this  line  will  be  the  axis  of  the  telescope.  From  R  any  point  in  the  object  draw  P  O  p  through 
the  centre  O  of  the  object-glass,  and  meeting  p  q,  a  line  through  q,  the  focus  of  the  point  Q,  perpendicular 
to  the  axis  in  p,  then  will  p  q  be  the  image  of  P  Q.  Let  P  A,  P  B  be  tne  extreme  rays  of  the  pencil  diverging 
from  P,  and  incident  on  the  object-glass,  and  they  will  be  refracted  to  and  cross  at  p.  Hence,  unless 
the  diameter  of  the  eye-glass  6  Go  be  such,  that  the  ray  Ap  a  shall  be  received  on  it,  the  point  p  will  be 
seen  less  illuminated  than  the  point  Q  in  the  centre  of  the  object,  and  if  it  be  so  small  that  the  line 
Bp  produced  does  not  meet  it,  then  none  of  the  rays  from  P  can  reach  the  eye  at  all.  Thus,  the  field  of 
view,  or  angular  dimensions  of  the  object  seen,  is  limited  by  the  aperture  of  the  eye-glass.  To  find  its  extent, 
then,  join  B  6,  A  a,  opposite  extremities  of  the  object  and  eye-glass,  meeting  the  image  in  r  and  p,  and  the 
axis  in  X,  then  r  p  is  the  whole  extent  of  the  image  which  is  seen  at  all,  and  the  angle  p  O  r,  which  is 
equal  to  P  O  R,  is  the  angular  extent  of  the  field  of  view.  Now  we  have  AB:a6::OX:GX,  and, 


therefore,  AB  +  o6:AB::OG:OX,  whence  we  get  O  X  = 


B 


a  b 
'          '  ~ 


O  G.     But  we  have,  moreover,  X  q  =  O  q  —  OX;  p  r  =  a  b  .  ^-^>  and  angle  r  O  p  =  — — .     To    express 

1 1  ,\  ^9 

this  algebraically,  put 

Diameter  of  object-glass  =  a, ;  Power  of  object-glass  =  L 
Diameter  of  eye-glass  ==  /3 ;       Power  of  eye-glass  =  I. 


Then 


OX  = 
QX  = 


a+p 

I 


ftl-aL, 


This  last  is  the  linear  magnitude  of  the  visible  portion  of  the  image ;  and  it  is,  as  we  see,  symmetrical 
both  with  respect  to  the  eye-glass  and  object-glass. 

Now  from  this  it  is  easy  to  deduce  both  the  field  of  view  and  magnifying  power  of  the  telescope  ;  for  the 
former  is  equal  to  the  angle  subtended  by  p  r,  at  the  centre  of  the  object-glass,  and  the  latter  is  obtained  from 
the  former,  when  the  angle  r  Gp  subtended  at  the  centre  of  the  eye-glass  is  obtained.  But  we  have 


LIGHT.  403 

Light.  /3/  —  oL  fil       aL>  Parti. 


/  jv  Formulae 

rGp  I      l"  for  field  of 

therefore  magnifying  power  =  —     '      =  —  —  I  view  and 

r  O  p  LA     J  magnifying 

Hence  we  see,  that  the  greater  the  power  of  the  eye-glass  is,  compared  with  that  of  the  object-glass,  the  greater 
the  magnifying  power  of  the  telescope  ;  or,  in  other  words,  the  greater  the  focal  length  of  the  object  glass  com- 
pared with  that  of  the  eye-glass. 

The  pencils  of  rays  after  refraction  at  the  eye-glass  will   emerge  parallel,  and  therefore   proper  for  distinct      383. 
vision  to  an  eye  properly  placed  to  receive  them.     Now  the  eye  will  receive  both  the  extreme  rays  b  R'  and  a  P"  Distance 
of  the  pencils  diverging  from  r  and  p,  if  it  be  placed  at  their  point  of  concourse  E  ;   but  since  6  E  is  parallel  to  of  eve- 
f  G,  and  a  E  to  p  G,  we  have 


(e) 


,  . 

pr  ftl  —aL, 

If  the  eye  be  placed  nearer  to,  or  farther  off  from,  the  eye-glass  than  this  distance,  it  will   not  receive  the      384. 
extreme  rays,  and  Ihefeld  of  view,  or  visible  area  of  the  object,  will  be  lessened.     In  the  construction  of  convex 
single  eye-pieces,  therefore,  care  must  be  taken  to  prolong  the  tube  which  carries  them,  (as  in  the  figure,)  so  that 
when  the  eye  is  applied  close  to  its  end,  it  shall  still  be  at  this  precise  distance  from  the  glass. 

If  the  telescope  be  inverted,  and  the  eye  applied  behind  the  object-glass,  it  is   evident  that  it  will  remain  a      385. 

•  Inversion  of 

telescope,  but  its  magnifying  power  will  be  changed  to  —  —  ;  so  that,  if  it  magnified  before,  it  will  diminish  objects  telesc°Pes- 

now,  and  the  field  of  view  will  be  proportionally  increased.  In  this  way,  beautiful  miniature  pictures  of  distant 
objects  may  be  seen. 

If  the  telescope,  instead  of  being  turned  on  objects  so  distant  as  that  the  pencils  flowing  from   them  may  be      386. 
regarded  as  parallel,  be  directed   to  near   objects,  the  distance   between  the  object-glass  and  eye-glass  must  be  Adjust- 
lengthened  so  as  to  bring  the  image  exactly  into  the  focus  of  the  latter.     To  accomplish  this,  the   eye-glass  is  ments- 
generally  set  in  a  sliding  tube  movable  by  a  rack-work,  or  by  hand.     The  same  mechanism  serves  also  to  adjust 
the  telescope  for  long  or  short-sighted  persons.     The  former  require  parallel  or  slightly  divergent  rays,  the  latter 
very  divergent  ;  and  to  obtain  the  necessary  divergence  for  the  latter,  the  eye-glass  must  be  brought  nearer  the 
object-glass. 

The  same  theory  and  formula  apply  to  the  second,  or  Galilaean,  construction,  only  recollecting  that  in  this  case  L,      387. 
the  power  of  the  eye-glass,  is  negative.     In  this  case,  therefore,  the  value  of  G  E  is  negative,  or  the  eye  should  Galil!e»n 
be  placed  between  the  object-glass  and  eye-glass  ;  but,  as  that  is  incompatible  with  the  other  conditions,  in  order  telesc°Pe- 
to  get  as  great  a  field  of  view  as  possible,  the  eye  must  be  brought  as  near  to  jts  proper  place  as  possible,  and 
therefore  close  to  the  eye-glass. 

In  the  astronomical  telescope  objects  are  seen  inverted,  in  the  Galilaean,  erect  ;  for,  in  the  former,  the  rays      388. 
from  the  extremities  of  the  object  have  crossed  before  entering  the  eye,  in  the  latter,  not. 

It  the  object  be  brought  nearer  the  object-glass,  the  magnifying  power  is  increased  ;    because  in  this  case      339. 

I  Micro- 

(calling  D  the  proximity  of  the  object)   -  -  —   expresses  the  magnifying  power,  as  is  easily  seen  from  what  has  scopes. 

been  said  Art.  382.  Thus  a  telescope  used  for  viewing  very  near  objects  becomes  a  microscope.  The  ordinary 
construction  of  the  compound  microscope  is  nothing  more  than  that  of  the  astronomical  telescope  modified  for 
the  use  it  is  intended  for.  The  object-glass  has  in  this  instrument  a  much  greater  power  than  the  eye-glass,  so 
that,  when  employed  for  viewing  distant  objects,  it  acts  as  a  telescope  inverted,  and  requires  to  be  greatly 

shortened.  But  for  near  objects,  as  D  increases,  /  —  D  diminishes,  and  the  fraction  --  may  be  increased 
to  any  amount,  by  bringing  the  object  nearer  to  the  object-glass,  and  at  the  same  time  lengthening  the  interval 
between  the  lenses,  which  is  expressed  by  -  -  —  -  +  -  —  -.  But  as  this  requires  two  operations,  it  is 

LJ  —  D         / 

usual  to  leave  the  latter  distance  unaltered,  and  vary,  by  a  screw  or  rack-work,  only  the  former.  Fig.  82  is  a  Fig.  82. 
section  of  such  an  instrument.  It  is,  however,  convenient  to  have  the  power  of  lengthening  and  shortening  the 
distance  between  the  glasses,  as  by  this  means  any  magnifying  power  between  the  limits  corresponding  to  the 
extreme  distances  may  be  obtained  ;  and  if  a  series  of  object-glasses  be  so  selected,  that  the  greatest  power 
attainable  by  one  within  the  limits  of  the  adjustment  in  question,  shall  just  surpass  the  least  obtainable  by  the 
next,  and  so  on,  we  may  command  any  power  we  please.  Such  a  series  is  usually  comprised  in  a  small  revolving 
plate  containing  cells,  each  of  which  can  be  brought  in  succession  into  the  axis  of  the  microscope  by  a  simple 
mechanism. 

In  the  reflecting  telescope,  of  the  most  simple  construction,  the  image  is  formed  by  a  concave  mirror,  and      390 
viewed  by  a  convex  or  concave  eye-glass,  as  in  refracting  telescopes  ;  but  since  the  head  of  the  observer  would  Reflecting 
intercept  the  whole  of  the  incident  light  in  small  telescopes,  and  a  great  part  of  it  in  large  ones,  the  axis  of  the  telescope. 
reflector  itself  is  turned  a  little  obliquely,  so  as  to  throw  the  image  aside,  by  which  it  can  be  viewed  with  little  or 
no  loss  of  light.     The  inconvenience  of  this  is  a  little  distortion  of  the  image,  caused  by  the  obliquity  of  the  rays; 


404 


LIGHT. 


Light 


392. 

Gregorian 
telescope. 
Fig.  84. 


393. 

Catsegrain- 
ian. 


but  as  such  telescopes  are  only  used  of  a  great  size,  and  for  the  purpose  of  viewing  very  faint  celestial  objects, 
in  which  the  light  diffused  by  aberration  is  insensible,  little  or  no  inconvenience  is  found  to  arise  from  this  cause. 
Such  is  the  construction  of  the  telescopes  used  by  Sir  William  Herschel  in  his  sweeps  of  the  heavens. 

To  obviate  the  inconvenience  of  the  stoppage  of  rays  by  the  head,  Newton,  the  inventor  of  reflecting  tele- 
scopes, employed  a  small  mirror,  placed  obliquely,  as  in  fig.  83,  opposite  the  centre  of  the  large  one.  Thus 
parallel  rays  PA,  P  B,  emanating  from  a  point  in  the  axis  of  the  telescope,  are  received,  before  their  meeting,  on 
a  plane  mirror  C  D  inclined  at  45°  to  the  axis,  and  thence  reflected  through  a  tube  projecting  from  the  side  of 
the  telescope  to  the  lens  G,  and  by  it  refracted  to  the  eye  E.  It  is  manifest,  that  if  the  image  formed  by  the 
mirror  A  B  behind  C  D  be  regarded  as  an  object,  an  image  equal  and  similar  to  it  (Art.  335)  will  be  formed 
at  F,  at  an  equal  distance  from  the  plane  mirror ;  and  this  image  will  be  seen  through  the  glass  G,  just  as  if  it 
were  formed  by  an  object-glass  of  the  same  focal  length  placed  in  the  prolongation  of  the  axis  of  the  eye-tube, 
beyond  the  small  mirror,  (supposed  away.)  Hence  the  same  propositions  and  formulae  will  hold  good  in  the 
Newtonian  telescope,  as  in  the  astronomical  and  Galitean,  for  the  magnifying  power,  field  of  vievv,  and  position 
of  the  eye,  substituting  only  2  R  for  L,  and  2  R  —  D  for  L  —  D,  and  recollecting  that  R  is  negative,  as  the 
mirror  has  its  concavity  turned  towards  the  light 

The  Gregorian  telescope,  instead  of  a  small  plain  mirror  turned  obliquely,  has  a  small  convex  mirror  with  its 
concavity  turned  towards  that  of  the  large  one,  as  in  fig.  84 ;  but  instead  of  being  placed  at  a  distance  from  the 
large  one  equal  to  the  sum  of  the  focal  lengths,  the  distance  is  somewhat  greater ;  hence  the  image  p  q,  formed 
in  the  focus  of  the  great  mirror,  being  at  a  distance  from  the  vertex  of  the  small  one  greater  than  its  focal  length, 
another  image  is  formed  at  a  distance,  viz.  at  or  near  the  surface  of  the  great  mirror,  at  r  s.  In  the  centre  of  the 
large  m'rror  there  is  a  hole  which  lets  pass  the  rays  to  an  eye-lens  g-.  The  adjustment  to  parallel  or  diverging 
rays,  or  for  imperfect  eyes,  is  performed  by  an  alteration  of  the  distance  between  the  mirrors  made  by  a  screw. 

The  Cassegrainian  construction  differs  in  no  respect  from  the  Gregorian,  except  that  the  small  mirror  is  convex 
and  receives  the  rays  before  their  convergence  to  form  an  image.  The  magnitude  of  the  field,  the  distance  of  the 
eye  and  of  the  mirrors  from  each  other,  are  easily  expressed  in  these  constructions  ;  the  latter  being  derived  from 
the  former  by  a  mere  change  of  sign  in  the  curvature  of  the  small  mirror.  Let  then  R'  and  R"  be  the  curvatures 
of  the  two  mirrors,  then  in  the  Gregorian  telescope  R'  is  negative  and  R''  positive  ;  and  if  we  put  t  for  the 
distance  between  their  surfaces,  (t  being  negative,  because  the  second  reflecting  surface  lies  towards  the  incident 
light)  we  shall  have  for  an  object  whose  proximity  is  D 

D'=D;        /'=  2R/-D  =2R'-D;        /"=2R"— "".         TV  _.    /' 


D"; 

adopting  the  formulae  and  notation  of  Art.  251.     Now  these  give,  by  substitution, 

2  R'  -  D  2  R'  -  D 


D"- 


l-ff 


D'  = 


1  -  t  (2  R'  -  D) 


/"  =  2  R"  - 


1  -  t  (2  R'  -  D) 


2  R"  -  2  R'  +  D  -  2  t  (2  R'  -  D)  .  R" 


1  -  t  (2  R'  -  D) 

This  is  the  reciprocal  distance  of  the  second  image  from  the  second  reflecting  surface.     If  we  wish  that  the  image 
to  be  viewed  by  the  eye-lens   should   fall  just  on   the  surface  of  the  large  mirror,  we  have  only  to  put  f"  = 

(because  /"  is  positive,  and  t  negative.)     For  parallel  rays  this  gives 


—  t 


R'  R"  .  <*  +  (4  R'  -  2  R")  t  -  1  =  o ; 


394. 


whence  t  may  be  found  when  R'  and  R"  are  given,  or  vice  versa. 

The  description  of  other  optical  instruments,  and  of  the  more  refined  construction  of  telescopes,  &c.  must  be 
deterred  till  we  are  farther  advanced  in  our  account  of  the  physical  properties  of  light,  and  especially  of  the 
different  refrangibility  of  its  rays  and  their  colours,  which  will  form  the  object  of  the  next  part. 


L  I  G  H  T.  405 

Light.  JTL 

PART  II. 
CHROMATICS. 

§  I.     Of  the  Dispersion  of  Light. 

HITHERTO  we  have  regarded  the  refractive  index  of  a  medium  as  a  quantity  absolutely  given  and  the  same  for      395 
all  rays  refracted  by  the  medium.     In  nature,  however,  the  case  is  otherwise.     When  a  ray  of  light  falls  obliquely  General 
on  the  surface  of  a  refracting  medium,  it  is  not  refracted  entirely  in  one  direction,  but  undergoes  a  separation  phenome- 
into  several  rays,  and  is  dispersed  over  an  angle  more  or  less  considerable,  according  to  the  nature  of  the  medium  "°t"0°  ^^' 
and  the  obliquity  of  incidence.     Thus   if  a   sunbeam   S  C   be  incident   on  the  refracting  surface  A  B,  and  be  ray  jnto 
afterwards  received  on  a  screen  R  V,  (fig.  85,)  it  will,  instead  of  a  single  point  on  the  screen  as  R,  illuminate  colours. 
a  space  R  V  of  a  greater  extent  the  greater  is  the  angle  of  incidence.     The  ray  S  C,  then,  which,  before  refraction  *''g-  85. 
was  single,  is  separated  into  an   infinite  number   of  rays   C  R,    CO,    C  Y,  &c.   each   of  which   is   refracted 
differently  from  all  the  rest. 

The  several  rays  of  which  the  dispersed  beam  consists,  are  found  to  differ  essentially  from  each  other,  and  from  396. 
the  incident  beam,  in  a  most  important  physical  character.  They  are  of  different  colours.  The  light  of  the  sun 
is  white.  If  a  sunbeam  be  received  directly  on  a  piece  of  paper,  it  makes  on  it  a  white  spot ;  but  if  a  piece  of 
white  paper  (that  is,  such  as  by  ordinary  daylight  appears  white)  be  held  in  the  dispersed  beam,  as  RV,  the 
illuminated  portion  will  be  seen  to  be  differently  coloured  in  different  parts,  according  to  a  regular  succession  of 
lints,  which  is  always  the  same,  whatever  be  the  refracting  medium  employed. 

To  make  the  experiment  in  the  most  striking  and  satisfactory  manner,  procure  a  triangular  prism  of  good  397. 
flint-glass,  and  having  darkened  a  room,  admit  a  sunbeam  through  a  small  round  hole  O  P  in  the  window  Fig.  8fi 
shutter.  If  this  be  received  on  a  white  screen  D  at  a  distance,  there  will  be  formed  a  round  white  spot,  or 
image  of  the  sun,  which  will  be  larger  as  the  paper  is  farther  removed.  New  in  the  beam  before  the  screen 
place  the  prism  ABC,  having  one  of  its  angles  C  downwards  and  parallel  to  the  horizon,  and  at  right  angles 
to  the  direction  of  the  sunbeam,  and  let  the  beam  fall  on  one  of  its  sides  B  C  obliquely.  It  will  be  refracted 
and  turned  out  of  its  course,  and  thrown  upwards,  pursuing  the  course  F  G  R,  and  may  be  received  on  a  screen 
E  properly  placed.  But  on  this  screen  there  will  no  longer  be  seen  a  white  round  spot,  but  a  long  streak,  or, 
as  it  is  called  in  Optics,  a  spectrum  R  V  of  most  vivid  colours,  (provided  the  admitted  sunbeam  be  not  too  large, 
and  tha  distance  of  the  screen  from  the  prism  considerable.)  The  tint  of  the  lower  or  least  refracted  extremity 
R  is  a  brilliant  red,  more  full  and  vivid  than  can  be  produced  by  any  other  means,  or  than  the  colour  of  any 
natural  substance.  This  dies  away  first  into  an  orange,  and  this  passes  by  imperceptible  gradations  into  a  fine 
pale  straw-yellow,  which  is  quickly  succeeded  by  a  pure  and  very  intense  green,  which  again  passes  into  a  blue, 
at  first  of  less  purity,  being  mixed  with  green,  but  afterwards,  as  we  trace  it  upwards,  deepening  to  the  purest 
indigo.  Meanwhile,  the  intensity  of  the  illumination  is  diminishing,  and  in  the  upper  portion  of  the  indigo  tint 
is  very  feeble,  but  it  is  continued  still  beyond,  and  the  blue  acquires  a  pallid  cast  of  purplish  red,  a  livid  hue 
more  easily  seen  than  described,  and  which,  though  not  to  be  exactly  matched  by  any  natural  colour,  approaches 
most  nearly  to  that  of  a  fading  violet :  "  tinctux  viola  pallor," 

If  the  screen  on  which  the  spectrum  be  received  have  a  small  hole  in  it,  too  small  to  allow  the  whole  of  the      398. 
spectrum  to  pass,  but  only  a  very  narrow  portion  of  it,  as  X,  (fig.  87,)  the  portion  of  the  beam  which  goes  to  Insulation 
form  that  particular  spot  X  may  be  received  on  another  screen  at  any  distance  behind  it,  and  will  there  form  a  °'  eaoh 
spot  d  of  the  very  same  colour  as  the  part  X  of  the  spectrum.     Thus  if  X  be  placed  in  the  red  part  of  the  co 
spectrum  the  spot  d  will  be  red  ;  if  in  the  green,  green ;    and  in  the  blue,  blue.     If  the  eye  be  placed  at  d,  it 
will  see  through  the  hole  an  image  of  the  sun  of  dazzling  brightness ;  not,  as  usually,  white,  but  of  the  colour 
which  goes  to  form  the  spot  X  of  the  spectrum.     Thus  we  see,  that  the  joint  action  of  all   the  rays   is  not 
essential  to  the  production  of  the  coloured  appearance  of  the  spectrum,  but  that  one  colour  may  be  insulated 
from  the  rest,  and  examined  separately. 

If,  instead  of  receiving  the  ray  X  d,  transmitted  through  the  hole  X,  on  a  screen  immediately  behind  it,  it  be      399. 
intercepted  by  another  prism  acb,  it  will  be  refracted,  and  bent  from  its  course,  as  in  Xfgz  ;  and  after  this  Second 
second  refraction  may  be  received  on  a  screen  e.     But  it  is   now  observed  to  be   no  longer  separated  into  a  refracii°n 
coloured  spectrum  like  the  original  one  R  V,  of  which  it  formed   a  part.     A  single  spot  x  only  is  seen  on  the  n0°fuUrT- 
screen,  the  colour  of  which  is  uniform,  and  precisely  that  which  th<>  part  X  of  the  s]>ectruin  would  have  had,  change  of 
were  it  intercepted  on  the  first  screen.     It  appears,  then,  that  the  ray  which  goes  to  form  any  single  point  of  the  colour 
spectrum  is  not  only  independent  of  all   the  rest,  but   having  been  once  insulated  from  them,  is  no  longer 
capable  of  further  separation  into  different  colours  by  a  second  refraction. 

This  simple,  but  instructive  experiment,  then,  makes  us  acquainted  with  the  following  properties  of  light : 


406  LIGHT. 

1.  A  beam  of  white  light  consists  of  a  great  and  almost  infinite  variety  of  rays  differing  from  each  other  in     Part  II, 
colour  and  refrangibility.  v^"\~"~l 

For  the  ray  S  F  from  any  one  point  of  the  sun's  disc,  which   if  received  immediately  on   the  screen  would 
hf'refran'gi-  nave  occuP'ed  on'y  a  s'ns'e  point  on  it,  or  (supposing  the  hole  in  the  screen  to  have  a  sensible  diameter)  only  a 
bility.      '    space   equal  to   its  area,  is  dilated  into  a  line  V  R  of  considerable  length,   every  point  of  which  (speaking 
loosely)  is  illuminated.     Now   the   rays  which   go  to  V  must  necessarily  have  been  more  refracted  than  those 
which  go  to  R,  which  can  only  have   been  in  virtue  of  a  peculiar  quality  in  the  rays  themselves,  since  the 
refracting  medium  is  the  same  for  all. 

401.  2.  White  light  may  be  decomposed,  analyzed,  or  separated  into  its  elementary  coloured  rays  by  refraction.    The 
act  of  such  separation  is  called  the  dispersion  of  the  coloured  rays. 

402.  3.  Each  elementary  ray  once  separated  and  insulated  from  the  rest,  is  incapable  of  further  decomposition  or 
analysis  by  the  same  means.     For  we  may  place  a  third,  and  a  fourth,  prism  in  the  way  of  the  twice  refracted 
ray  g  x,   and  refract  it  in  any  way,  or  in   any  plane  ;    it  remains  undispersed,  and  preserves  its  colour  quite 
unaltered. 

403.  4.  The  dispersion  of  the  coloured  rays  takes  place  in   the  plane  of  the  refraction  ;  for  it  is  found  that  the 
spectrum  VR  is  always  elongated  in  this  plane.     Its  breadth  is  found,  on  the  other  hand,  by  measurement,  to  be 
precisely  the  same  as  that  of  the  white  image  D,  (fig.  86,)  of  the  sun,  received  on  a  screen   at  a  distance  O  D 
from  the  hole,  equal  to  O  F  +  F  G  4-  G  R,  the  whole  course  of  the  refracted  light,  which  shows  that  the  beam 
has  undergone  no   contraction  or  dilation  by  the  effect  of  refraction  in  a  plane  perpendicular  to  the  plane  of 
refraction. 

404.  To   explain  all  the  phenomena  of  the  colours  produced  by  prismatic  dispersion,  or  of  the  prismatic  colours, 
Index  of      as  they  are  called,  we  need  only  suppose,  with  Newton,  that  each  particular  ray  of  light,  in  undergoing  refraction 
refraction     at  the  surface  of  a  given  medium,  has  the  sine  of  its  angle  of  incidence  to  that  of  refraction  in  a  constant  ratio, 
regarded  as  so  |ong  as  t|,e  me(lium  and  the  ray  are  the  same  ;  but  that  this  ratio  varies  not  only,  as  we  have  hitherto  all  along 

e'  assumed,  with  the  nature  of  the  medium,  but  also  with  that  of  the  ray.  In  other  words,  that  there  are  as  many 
distinct  species,  or  at  least  varieties  of  light,  as  there  are  distinct  illuminated  points  in  the  spectrum  into  which 
a  single  ray  of  white  light  is  dispersed.  This  amounts  to  regarding  the  quantity  /*,  for  any  medium,  not  as  one 
and  invariable,  but  as  susceptible  of  all  degrees  of  magnitude  between  certain  limits  :  one,  the  least  of  which, 
corresponds  to  the  extreme,  or  least  refracted  red  ray  ;  the  other,  the  greatest  value  of  p.,  to  the  extreme  or 
most  refracted  violet.  Each  of  these  varieties  separately  conforms  to  the  laws  of  reflexion  and  refraction  we 
have  already  laid  down.  As  in  Geometry  we  may  regard  a  whole  family  of  curves  as  comprehended  under  one 
equation,  by  the  variation  of  a  constant  parameter  ;  so  in  Optics  we  may  include  under  one  analysis  all  the 
doctrine  of  the  reflexions,  refractions,  and  other  modifications  of  a  ray  of  white  or  compound  light,  by  regarding 
the  refractive  index  ft  as  a  variable  parameter.  • 

405  To  apply  this,  for  instance,  to  the  experiment  of  the  prism   just  related  :    A  single  ray  of  white  light  being 

supposed  incident  on  the  first  surface,  must  be  regarded  as  consisting  of  an  infinite  number  of  coincident  rays, 
of  all  possible  degrees  of  refrangibility  between  certain  limits,  any  one  of  which  may  be  indifferently  expressed 
by  the  refractive  index  fi.  Supposing  the  prism  placed  so  as  to  receive  the  incident  ray  perpendicularly  on  one 
surface,  then  the  deviation  will  be  given  by  the  equation 

/»  .  sin  I  =  siu  (I  +  D) 

I  being  the  refracting  angle  of  the  prism.  D  therefore  is  a  function  of  /t,  and  if  fi  vary  by  the  infinitely  small 
increment  t>  fi,  i.  e.  if  we  pass  from  any  one  ray  in  the  spectrum  to  the  consecutive  ray,  D  will  vary  by  £  D, 
and  the  relation  between  these  simultaneous  changes  will  be  given  by  the  equation  resulting  from  the  differen- 
tiation of  the  above  with  the  characteristic  &  :  thus  we  get 


=  «D.cos(I  +  D);         5  D  =  K  fi 


_ 


It  is  evident,  then,  that  as  fi  varies,  D  also  varies  ;  and,  therefore,  that  no  two  of  the  refracted  and  coloured  rays 
will  coincide,  but  will  be  spread  over  an  angle,  in  the  plane  of  refraction,  the  greater,  the  greater  is   the  total 
variation  of  fi  from  one  extreme  to  the  other. 
406.  In  order  to  justify  the  term  analysis,  or  decomposition,  as  applied  to  the  separation  of  a  beam  of  white  light 

Analysis       jnto  coloured  rays,  we  must  show  by  experiment  that  white  light  may  be  again  produced  by  the  synthesis  of  these 
I^of^wh'"  eUmentary  rays-     The  experiment  is  easy.     Take  two  prisms  A  B  C,  a  b  c  of  the  same  medium,  and  having 

light  *     *  eo,ua'  refracting  angles,  and  lay  them  very  near  together,  having  their  edges  turned  opposite   ways,  as  in  fig.  87. 

Fig.  87.  With  this  disposition,  a  parallel  beam  of  white  light  intromitted  into  the  face  A  C  of  the  first  prism,  will  emerge 
from  the  face  b  c  of  the  last,  undeviated,  and  colourless,  as  if  no  prisms  were  in  the  way.  Now  the  dispersion 
having  been  fully  completed  by  the  prism  ABC,  the  rays  in  passing  through  the  thin  lamina  of  air  B  C  a  c  must 
have  existed  in  their  coloured  and  independent  state,  and  been  dispersed  in  their  directions  ;  but  being  refracted 
by  the  second  prism  so  as  to  emerge  parallel,  the  colour  is  destroyed  by  the  mixture  and  confusion  of  the  rays. 

Fig.  88.  fo  see  more  clearly  how  this  tnkes  place  in  fig.  88,  let  S  R  and  S  V  be  two  parallel  white  rays,  incident  on  the 
first  prism,  and  separated  by  refraction  ;  the  former  into  the  coloured  pencil  11  c»,  the  latter  into  a  pencil  exactly 
similar  to  V  re.  Let  Re  be  the  least  retracted  ray  of  the  former  pencil,  and  Vc  the  most  refracted  of  the 
latter.  These,  of  course,  must  meet  :  let  them  meet  in  c,  and  precisely  at  c  apply  the  vertex  of  the  second 
prism,  having  its  side  ca  parallel  to  C  B,  but  its  edge  turned  in  the  opposite  direction  ;  then  will  the  rays  R  C 
and  V  c,  each  for  itself,  and  independent  of  the  other,  be  refracted  so  as  to  emerge  parallel  to  its  original  direction 


LIGHT.  407 

Light.      S  R,  S  V,  and  the  emergent  rays  will  therefore  be  coincident  and  superimposed  on  each  other  as  c».     Thus  the     Part  II. 

— ~v~— '  emergent  ray  ex  will  contain  an  extreme  red  and  an  extreme  violet  ray.     But  it  will  also  contain  every  inter-  • v— — ' 

mediate  variety;  for  draw  c/anywhere  between  cR  and  cV.  Then,  since  the  angle  which  of  makes  with  the 
surface  B  C  is  greater  than  that  made  by  the  extreme  violet  ray  C  B,  but  less  than  that  made  by  the  extreme 
red,  there  must  exist  some  value  of  it,  intermediate  between  its  extreme  values,  which  will  give  a  deviation  equal 
to  the  angle  between  cf  and  S  Y  parallel  to  S  R.  Consequently,  if  S  Y  be  a  white  ray,  separated  into  the 
pencil  Y  v'  r  by  refraction,  the  coloured  ray  Y_/"c  of  that  particular  refrangibility  will  fall  on  c,  and  be  refracted 
along  cs.  Every  point  then  of  the  surface  gfh  will  send  to  c  a  ray  of  different  refrangibility,  comprehending  all 
the  values  of  /i  from  the  greatest  to  the  least.  Tims  all  the  coloured  elements,  though  all  belonging  originally 
to  different  white  rays,  will,  after  the  second  refraction,  coincide  in  the  ray  cs,  and  experience  proves  that  so 
reunited  they  form  white  light.  White  light,  then,  is  re-composed  when  all  the  coloured  elements,  even  though 
originally  belonging  to  separate  white  rays,  are  united  in  place  and  direction. 

In  the  reflexion  of  light,  regarded  as  a  case  of  refraction,  /i  has  a  specific  numerical  value,  and  cannot  vary      407. 
without  subverting  the  fundamental  law  of  reflexion.     Thus,  there  is  no  dispersion  into  colours  produced  by 
reflexion,  because  all  the  coloured  rays  after  reflexion  pursue  one  and  the  same  course.     There  is  one  exception 
to  this,  more  apparent  than  real,  when  light  is  reflected  from  the  base  of  a  prism  internally,   of  which  more 
hereafter. 

The  recomposition  of  white  from  coloured  light  may  be  otherwise  shown,  by  passing  a  small  circular  beam  of     408. 
solar  light  through  a  prism  ABC,  (fig.  89,)  and  receiving  the  dispersed  beam  on  a  lens  E  D  at  some  distance.  Synthesis 
If  a  white  screen  be  held  behind  the  lens,  and  removed  to  a  proper  distance,  the  whole   spectrum  will  be  °.'  white 
reunited  in  a  spot  of  white  light.     The  way  in  which  this  happens  will  be  evident  by  considering  the  figure,  in  :'* 
which  TE  and  TD  represent  the  parallel  pencils  of  rays  of  any  two  colours  (red  and  violet,  for  instance)  into 
which  the  incident  white  beam  S  T  is  dispersed.     These  will  be  collected  after  refraction,  each  in  its  own  proper 
focus  ;  the  former  at  F,  the  latter  at  G ;  after  which  each  pencil  diverges  again,  the  former  in  the  cone  F  H,  the 
latter  in  G  H.     If  the  screen  then  be  held  at  H,  each  of  these  pencils  will  paint  on  it  a  circle  of  its  own  colour, 
and  so  of  course  will  all  the  intermediate  ones ;  but  these  circles  all  coinciding,  the  circle  H  will  contain  all  the 
rays  of  the  spectrum  confounded  together,  and  it  is  found  (with  the  exception  of  a  trifling  coloured  fringe  about 
the  edges,  arising  from  a  slight  overlapping  of  the  several  coloured  images)  to  be  of  a  pure  whiteness. 

That  the  reunion  of  all  the  coloured  rays  is  necessary  to  produce  whiteness,  may  be  shown  by  intercepting  a      409. 
portion  of  the  spectrum  before  it  falls  on  the  lens.     Thus,  if  the  violet  be  intercepted,  the  white  will  acquire  a  A"  the  rav> 
tinge  of  yellow ;   if  the  blue  and  green  be  successively  stopped,  this  yellow  tinge  will  grow  more  and  more  ruddy,  !" 
and  pass  through  orange  to  scarlet  and  blood  red.     If,  on   the  other  hand,  the  red  end  of  the  spectrum  be  white. 
stopped,  and  more  and  more  of  the  less  refrangible  portion  thus  successively  abstracted  from  the  beam,  the  white 
will  pass  first  into   pale  and   then    to  vivid  green,  blue-green,  blue,  and  finally   into  violet.      If  the    middle 
portion  of  the  spectrum  be   intercepted,  the   remaining  rays,  concentrated,  j  reduce  various  shades  of  purple,  All  natural 
crimson,  or  plum-colour,  according  to  the  portion  by  which  it  is  thus  rendered  deficient  from  white  light;    and  J^"?  'm'~ 
by  varying  the  intercepted  rays,  any  variety  of  colours  may  be  produced  ;   nor  is  there  any  shade  of  colour  in  Combina- 
nature  which  may  not  thus  be  exactly  imitated,  with  a  brilliancy  and  richness  surpassing  that  of  any  artificial  tions  of  the 
colouring.  prismatic. 

Now,  if  we  consider  that  all  these  shades  are  produced  on  white  paper,  which  receives  and  reflects  to  our  eyes 
whatever  light  happens  to  fall  on  it ;  and  that  the  same  paper  placed  successively  in  the  red.  green,  and  blue 
portion  of  the  spectrum,  will  appear  indifferently  red,  or  green,  or  blue,  we  are  naturally  enough  led  to  conclude, 
that 

The  colours  of  natural  bodies  are  not  qualities  inherent  in  the  bodies  themselves,  by  which  they  immediately  affect      410. 
our  sense,  but  are  mere  cotuequences  of  that  peculiar  disposition  of  the  particles  of  each  body,  by  which  it  is  Colours  not 
enabled  more  copiously  to  reflect  the  rays  of  one  particular  colour,  and  to  transmit,  or  stifle,  or,  as  it  is  called  in  inherent  in 
Optics,  absorb  the  others.  bodies- 

Such  is  the  Newtonian  doctrine  of  the  origin  of    olours.     Every  phenomenon  of  optics  conspires  to  prove  its      4jj 
justice.     Perhaps  the  most  direct  and  satisfactory  (>roof  of  it  is  to  be  found  in  the  simple  fact,  that  every  body,  proved  by 
indifferently,  whatever  be  its  colour  in  white  light,  when  exposed  in  the  prismatic  spectrum,  appears  of  the  colour  experiment 
appropriate  to  that  part  of  the  spectrum  in  which  it  is  placed  ;  but  that  its  tint  is  incomparably  more  vivid  and 
full  when  laid  in  a  ray  of  a  tint  analogous  to  its  hue  in  white  light,  than  in  any  other.     For  example,  vermillion 
placed  in  the  red  rays  appears  of  the  most  vivid  red  ;  in  the  orange,  orange  ;  in  the  yellow,  yellow,  but  less  bright 
In  the  green  rays,  it  is  green  ;  but  from  the  great  inaptitude  of  vermillion  to  reflect  green  light,  it  appears  dark 
and  dull ;  still  more  so  in  the  blue ;  and  in  the  indigo  and  violet  it  is  almost  completely  black.     On  the  other 
hand,  a  piece  of  dark  blue  paper,  or  Prussian  blue,  in  the  indigo  rays  has  an  extraordinary  richness  and  depth  of 
blue  colour.     In  the  green  its  hue  is  green,  but  much  less  intense ;  while  in  the  red  rays  it  is  almost  entirely 
black.     Such  are  the  phenomena  of  pure  and  intense  colours;    but  bodies  of  mixed  tints,  as  pink  or  yellow 
paper,  or  any  of  the  lighter  shades  of  blue,  green,  brown,  &c.,  when  placed  in  any  of  the  prismatic  rays,  reflect 
them  in  abundance,  and  appear,  for  the  time,  of  the  colour  of  the  ray  in  which  they  are  placed. 

Refraction  by  a  prism  affords  us  the  means  of  separating  a  ray  of  white  light  into  the  rays  of  different  refran-       412. 
gibility  of  which  it  consists,  or  of  analyzing  it.     But  to  make  the  analysis  complete,  and  to  insulate  a  ray  of  any  Precautions 
particular  refrangibility  in  a  state  of  perfect  purity,  several  precautions  are  required,  the  chief  of  which  are  as  '"'"su™11" 
follows:    1st.  The  beam  of  light  to  be  analyzed  must  be  very  small,  as  nearly  as  possible  approaching  to   a  m"  eneit°~ 
mathematical  ray;  for  if  A  B,  aft  be  a  beam  of  parallel  rays  of  any  sensible  breadth  (fig.  89)  incident   on  the  of  any; 
prism  P,  the  extreme  rays  A  B,  a  6  will  each  be  separated  by  refraction  into  spectra  G  B  H  and  g  b  h :  B  G,  b  g  Fig.  89. 
being  the  violet,  and  B  H,  bh  the  red  rays  of  each  respectively ;  and  since  A  B,  a  b  are  parallel,  therefore  C  G 


408  L  I  G  H  T. 

Light.      and  eg  will  be  so,  and  also  D  H  and  d  h.     Hence  the  red  ray  D  H  from  B  will  intersect  the  violet  eg  from  ft,      Parl  "• 
^~^s-~>  in  some  point  F  behind  the  prism  ;  and  a  screen  E  F/  placed  at  F  will  have  the  point  F  illuminated  by  a  red  >>—"v-"~< 

Small-  ray  from  ftt  antj  a  violet  one  from  b ;  and  therefore  (as  is  easily  seen)  by  all  the  rays  intermediate  between  the 
ictdent'  6  an<^  v'o'e*>  from  points  between  B  and  b.  F  therefore  will  be  white.  If  the  screen  be  placed  nearer  the 
pencil  prism  than  F,  as  at  K  L  k  I,  it  is  clear  that  from  any  point  between  L  and  k  lines  drawn  parallel  to  K  C,  D  L,  to 
any  intermediate  direction,  will  fall  between  C  and  c,  D  and  d,  &c.,  respectively;  and  therefore  that  every  point 
between  L  and  k  will  receive  from  some  point  or  other  of  the  surface  C  d  of  the  prism  a  ray  of  each  colour, 
and  will  therefore  be  white.  Again,  any  point  as  x  between  k  and  I  can  receive  no  violet  ray,  nor  any  ray  of  the 
spectrum  whose  angle  of  deviation  is  greater  than  180°  —  a  b  x;  for  such  ray  to  reach  x  must  come  from  a  part 
of  the  prism  below  6,  which  is  contrary  to  the  supposition  of  a  limited  beam  A  B,  a  b ;  but  all  rays  whose 
angle  of  deviation  is  less  than  180° —  abx,  will  reach  x  from  some  part  or  other  of  the  surface  B  D.  Hence 
the  colour  of  the  portion  kl  of  the  image  on  the  screen  will  be  white  at  k,  pure  red  at  I,  and  intermediate 
between  white  and  red,  or  a  mixture  of  the  least  refrangible  rays  of  the  spectrum  at  any  intermediate  point  ; 
and,  in  the  same  manner,  the  portion  K  L  will  be  white  at  L,  violet  at  K,  and  at  any  intermediate  point  will  have 
a  colour  formed  by  a  mixture  of  a  greater  or  less  portion  of  the  more  refrangible  end  of  the  spectrum.  If  the 
screen  be  removed  beyond  F,  as  into  the  situation  G  s  H  h,  the  white  portion  will  disappear,  no  point  between  g 
and  H  being  capable  of  receiving  any  ray  whose  angle  of  deviation  is  between  180° —  abg  and  180  —  a  b  H. 
We  may  regard  the  whole  image  G  h  as  consisting  of  an  infinite  number  of  spectra  formed  by  every  elementary 
ray  of  which  the  beam  A  B  a  b  is  composed,  overlapping  each  other,  so  that  the  end  of  each  in  succession  projects 
beyond  that  of  the  foregoing.  The  fewer,  therefore,  there  are  of  these  overlapping  spectra,  or  the  smaller  the 
breadth  of  the  incident  beam,  the  less  will  be  the  mixture  of  rays  so  arising,  and  the  purer  the  colours.  Removal 
of  the  screen  to  a  greater  distance  from  the  prism,  evidently  produces  the  same  effect  as  diminution  of  the  size 
of  the  beam  ;  for  while  each  colour  occupies  constantly  the  same  space  on  the  screen  (for  G  g  =  K  k)  the  whole 
spectrum  is  diffused  over  a  larger  space  as  the  screen  is  removed,  by  the  divergence  of  its  component  rays  of 
different  colours,  and  therefore  the  individual  colours  must  of  necessity  be  continually  more  and  more  separated 
from  each  other. 

413.  2ndly.  Another  source  of  confusion  and  want  of  perfect  homogeneity  in  the  colours  of  the  spectrum   is  the 
2nd.  Small  angular  diameter  of  the  sun  or  other  luminary,  even  when  the  aperture  through  which  the  beam  is  admitted  is 
veivenre  of  ever  so  muc^  diminished.     For  let  S  T  (fig.  90)  be  the  sun,  whose  rays  are  admitted  to  the  prism  ABC  through 
the  pencil.    a  verY  sma"  h°'e  O  in  a  screen  placed  close  to  it.     The  beam  will  be  dilated  by  refraction  into  the  spectrum  v  r. 
Fig.  90.       Now,  if  we  consider  only  the  rays  of  one  particular  kind,  as  the  red,  and  regard  all  the  rest  as  suppressed,  it  is 

clear  that  a  red  image  r  of  the  sun  will  be  formed  by  them  alone  on  the  screen  ;  the  rays  from  every  point  of 
the  sun's  disc  crossing  at  O,  and  pursuing  (after  refraction)  different  courses.  If  the  prism  be  placed  in  its 
situation  of  minimum  deviation,  which  at  present  we  will  suppose,  this  image  will  be  a  circle,  and  it  and  the  sun 
will  subtend  equal  angles  at  O.  In  like  manner,  the  violet  rays  (considered  apart  from  the  red)  will  form  a 
circular  violet  image  of  the  sun,  at  r,  by  reason  of  their  greater  refrangibility;  and  every  species  of  ray,  of 
intermediate  refrangibility,  will  form,  in  like  manner,  a  circular  image  between  r  and  v.  The  constitution  of  the 
spectrum  so  arising  will  therefore  be  as  in  fig.  91,  a,  being  an  assemblage  of  images  of  every  possible  refrangi- 
bility superposed  on  and  overlapping  each  other.  Now,  if  we  diminish  the  angular  diameter  of  the  sun  or 
luminary,  each  of  these  images  will  be  proportionally  diminished  in  size  ;  but  their  number,  and  the  whole 
extent  over  which  they  are  spread,  will  remain  the  same.  They  will  therefore  overlap  less  and  less,  (as  in 
Fig.  91.  fig.  9l(  b,  c,-)  and  if  the  luminary  be  conceived  reduced  to  a  mere  point  (as  a  star)  the  spectrum  will  consist  of 
a  line  d  composed  of  an  infinite  number  of  mathematical  points,  each  of  a  perfectly  pure  homogeneous  light. 

414.  There  are  several   ways  by  which  the  angular  diameter,  or  the  degree  of  divergence  of  the  incident  beam  may 
Experimen-  be  diminished.     Thus,  first,  we  may  admit  a  sunbeam  through  a  small  hole,  as  A,  in  a  screen,  and  receive  the 
tal  methods  divergent  cone  of  rays  behind  it  on  another  screen  B,  (fig.  7,)  at  a  considerable  distance,  having  another  small 
homoee-n'ng  no'e  ^  to  'et  Pass>  not  tne  whole,  but  only  a  small  portion  of  the  sun's  image.     The  beam  B  C,  so  transmitted, 
neous  pris-  w'"  manifestly  have  a  degree  of  divergence  less  than  that  of  the  beam  immediately  transmitted  from  A  in  the 
matic  rays,    proportion  of  the  diameter  of  the  aperture  B  to  the  diameter  of  the  sun's  image  on  the  screen  B. 

F'?-  7  Another  and  much  more  commodious  method  is  to  substitute  for  the    sun  its  image  formed  in  the  focus 

*'*•      of  a  convex  lens  of  short  focus.     This  image  is  of  very  small  dimensions,  its  diameter  being  equal  to  focal 

Fig.  92.  length  of  the  lens  x  sine  of  sun's  angular  diameter,  (or  sine  of  30',  which  is  about  one  114th  part  of  radius,) 
so  that  a  lens  of  an  inch  focus  concentrates  all  the  rays  which  fall  on  it  within  a  circle  of  about  the  114th 
of  an  inch  in  diameter,  which,  for  this  purpose,  may  be  regarded  as  a  physical  point.  The  disposition  of  the 
apparatus  is  as  represented  in  fig.  92.  The  rays  converged  by  the  lens  L  to  F,  afterwards  diverge  as  if  they 
emanated  from  an  intensely  bright  luminous  point  placed  at  F,  and  a  screen  with  a  small  aperture  O  being 
placed  at  a  distance  from  it,  and  close  behind  it  the  prism  ABC,  the  spectrum  r  v  may  be  received  on  a  screen 
again  placed  at  a  considerable  distance  behind  the  prism,  each  of  whose  points  will  be  illuminated  by  rays  of  a 
very  high  degree  of  purity  and  homogeneity,  and  by  diminishing  the  focal  length  of  the  lens,  and  the  aperture 
O,  and  increasing  the  distance  F  O,  or  O  r,  this  may  be  carried  to  any  extent  we  please.  It  should,  however, 
be  remarked,  that  the  intensity  of  the  purified  ray,  and  the  quantity  of  homogeneous  light  so  obtained,  are 
diminished  in  the  same  ratio  as  the  purity  of  the  ray  is  increased. 
416.  A  third  method  of  obtaining  a  homogeneous  beam  is  to  repeat  the  process  of  analysis  on  a  ray  as  nearly 

Tig  93  pure  as  can  be  conveniently  obtained  by  refraction  through  a  single  prism.  Thus,  in  fig.  93,  V  R,  the 
spectrum  formed  by  a  first  refraction  at  the  prism  A,  is  received  on  a  screen  which  intercepts  the  whole  of 
it,  except  that  particular  colour  we  wish  to  insulate  and  purify,  which  is  allowed  to  pass  through  an  aperture 
M  N ;  behind  this  is  placed  another  prism  B,  so  as  to  refract  this  beam  a  second  time.  If  then  the  portion 


LIGHT.  409 

Light.      M  N  were  already  perfectly  pure,  it  would  pass  the  second  pristn  without  undergoing  any  further  separation  ;     Part  II. 
~-<v-~~>  but  if  there  be  (as  there  always  will)  other  rays  mixed  with  it,  these  will  be  dilated  by  the  subsequent  refraction  *— v-~-- 
into  a  second  spectrum  vr  of  faint  light,  with  a  much  brighter  portion  mn  in  the  midst;  and  if  the  rest  of  the 
rays  be  intercepted,  and  this  portion  only  allowed  to  pass  through  an  aperture,  the  emergent  beam  mp  will  be 
much  more  homogeneous  than  before  its  incidence  on  the  second  prism, — and  in  proportion  as  the  distance  be- 
tween the  second  prism  and  the  screen  is  increased,  the  purity  of  the  ray  obtained  will  be  greater. 

Another  source  of  impurity  in  the  prismatic  rays  is  the  imperfection  of  the  materials  of  our  ordinary  prisms,       417 
which  are  full  of  strise  and  veins,  which  disperse  the  light  irregularly,  and  thus  confound  together  in  the  spectrum  Imperfec- 
rays  which  properly  belong  to  different  parts  of  it.     Those  who  are  not  fortunate  enough  to  possess  glass  prisms  tion  of 
free  from  this  defect  (which  are  very  rare,  and  indeed  hardly  to  be  procured  for  any  price)  may  obviate  the  in-  Pr  ims 
convenience  by  employing  hollow  prisms  full  of  water,  or,  rather,  any  of  the  more  dispersive  oils.     A  great  part  i 
of  the  inconvenience  arising  from  a  bad  prism  may,  however,  be  avoided  by  transmitting  the  rays  as  near  the 
edge  of  it  as  possible,  so   as  to  diminish  the  quantity  of  the  material  they  have  to  pass  through,  and  therefore 
their  chance  of  encountering  veins  and  strife  in  their  passage. 

When  every  care  is  taken  to  obtain  a  pure  spectrum ;  when  the  divergence  of  the  incident  beam  is  extremely  418. 
small,  and  its  dimensions  also  greatly  reduced  ;  when  the  prism  is  perfect,  and  the  spectrum  sufficiently  elon-  Fixed  line* 
gated  to  allow  of  a  minute  examination  of  its  several  parts,  some  very  extraordinary  facts  have  been  observed  '"  the 
respecting  its  constitution.  They  were  first  noticed  by  Dr.  Wollaston,  in  a  Paper  published  by  him  in  the  Phil.  sPectmm- 
Trans.,  1802  ;  and  have  since  been  examined  in  full  detail,  and  with  every  delicacy  and  refinement  which 
the  highest  talents  and  the  most  unlimited  command  of  instrumental  aids  could  afford,  by  the  admirable  and 
ever-to-be-lamented  Fraunhofer.  It  does  not  appear  that  the  latter  had  any  knowledge  of  Dr.  Wollaston's 
previous  discovery,  so  that  he  has,  in  this  respect,  the  full  merit  of  an  independent  inventor.  The  facts  are 
these  :  The  solar  spectrum,  in  its  utmost  possible  state  of  purity  and  tenuity,  when  received  on  a  white  screen, 
or  when  viewed  by  admitting  it  at  once  into  the  eye,  is  not  an  uninterrupted  line  of  light,  red  at  one  end  and 
violet  at  the  other,  and  shading  away  by  insensible  gradations  through  every  intermediate  tint  from  one  to  the 
other,  as  Newton  conceived  it  to  be,  and  as  a  cursory  view  shows  it.  It  is  interrupted  by  intervals  absolutely 
dark ;  and  in  those  parts  where  it  is  luminous,  the  intensity  of  the  light  is  extremely  irregular  and  capricious, 
and  apparently  subject  to  no  law,  or  lo  one  of  the  utmost  complexity.  In  consequence,  if  we  view  a  spectrum 
formed  by  a  narrow  line  of  light  parallel  to  the  refracting  edge  of  the  prism,  (which  affords  a  considerable 
breadth  of  spectrum  without  impairing  the  purity  of  the  colours,  being,  in  fact,  an  assemblage  of  infinitely  narrow 
linear  spectra  arranged  side  by  side,)  instead  of  a  luminous  fascia  of  equable  light  and  graduating  colours,  it 
presents  the  appearance  of  a  striped  riband,  being  crossed  in  the  direction  of  its  breadth  by  an  infinite  multi- 
tude of  dark,  and  by  some  totally  black  bands,  distributed  irregularly  throughout  its  whole  extent.  This  irregu- 
larity, however,  is  not  a  consequence  of  any  casual  circumstances.  The  bauds  are  constantly  in  the  same  parts 
of  the  spectrum,  and  preserve  the  same  order  and  relations  to  each  other ;  the  same  proportional  breadth  and 
degree  of  obscurity,  whenever  and  however  they  are  examined,  provided  solar  light  be  used,  and  provided  the 
prisms  employed  be  composed  of  the  same  material  :  for  a  difference  in  the  latter  particular,  though  it  causes  no 
change  in  the  number,  order,  or  intensity  of  the  bands,  or  their  places  in  the  spectrum,  as  referred  to  the  several 
colours  of  which  it  consists,  yet  causes  a  variation  in  their  proportional  distances  inter  se,  of  which  more  here- 
after. By  solar  light  must  be  understood,  not  merely  the  direct  rays  of  the  sun,  but  any  rays  which  have  the 
sun  for  their  ultimate  origin  ;  the  light  of  the  clouds,  or  sky,  for  instance ;  of  the  rainbow  ;  of  the  moon,  or  of 
the  planets.  All  these  lights,  when  analyzed  by  the  prism,  are  found  deficient  in  the  identical  rays  which  are 
wanting  in  the  solar  spectrum  ;  and  the  deficiency  is  marked  by  the  same  phenomenon,  viz.  by  the  occurrence  of 
the  same  dark  bands  in  the  same  situations  in  spectra  formed  by  these  several  lights.  In  the  light  of  the  stars, 
on  the  other  hand,  in  electric  light,  and  that  of  flames,  though  similar  bands  are  observed  in  their  spectra, 
yet  they  are  differently  disposed ;  and  the  spectrum  of  each  several  star,  and  each  flame,  has  a  system  of  bands 
peculiar  to  itself,  and  characteristic  of  its  light,  which  it  preserves  unalterably  at  all  times,  and  under  all 
circumstances. 

Fig.  94  is  a  representation  of  the  solar  spectrum  as  laid  down  minutely  by  Fraunhofer,  from  micrometrical      419. 
measurement,  and  as  formed  by  a  prism  of  his  own  incomparable  flint  glass.     Only  the  great  number  of  small  Fij.  94. 
bands  observed  by  him  (upwards  of  500  in  number)  have  been  omitted,  to  avoid  confusing  the  figure.     Of  these 
bands,  or,  as  he  terms  them,  "  fixed  lines"  in  the  spectrum,  he  'has  selected  seven,  (those  marked  B,  C,  D,  E,  F, 
G,  II,)  as  terms  of  comparison,  or  as  standard  points  of  reference  in  the  spectrum,  on  account  of  their  distinct- 
ness, and  the  facility  with  which  they  may  be  recognised.     Of  these,  B  lies  in  the  red  portion  of  the  spectrum, 
near  the  end  ;  C  is  farther  advanced  in  the  red ;  D  lies  in  the  orange,  and  is  a  strong  double  line  easily  recog- 
nised ;  E  is  in  the  green  ;  F  in  the  blue  ;    G   in   the  indigo  ;   and  H  in  the  violet.     Besides  these,  there  are 
others  very  remarkable ;  thus  6  is  a  triple  line  in  the  green,  between  E  and  F,  consisting  of  three  strong  lines, 
of  which  two  are  nearer  each  other  than  the  third,  &c. 

The  definiteness  of  these  lines,  and  their  fixed  position,  with  respect  to  the  colours  of  the  spectrum, — in       420. 
other  words,  the  precision  of  the  limits  of  those  degrees  of  refrangibility  which   belong  to  the  deficient  rays  Utility  of 
of  solar  light, — renders  them  invaluable  in  optical  inquiries,  and  enables  us  to  give  a  precision  hitherto  unheard  'I16  &?*& 
of  to  optical  measurements,  and  to  place  the  determination  of  the  refractive  powers  of  media  on  the  several  rays  t'j"!j  a"t<£-T 
almost  on  the  same  footing,  with  respect  to  exactness,  with  astronomical   observations.      Fraunhofer,  in  his  minati0ns. 
various  essays,  has  made  excellent  use  of  them  in  this  respect,  as  we  shall  soon  have  occasion  to  see. 

To  see  these  phenomena,  we  must  place  the  refracting  angle  of  a  very  perfect  prism  parallel  to  a  very  small       421 
linear  opening  through  which  a  sunbeam  is  admitted  ;  or,  in  place  of  an  opening,  we  may  employ  a  glass 
cylinder,  or  semi-cylinder  of  small  radius,  to  bring  the  rays  to  a  linear  focus  behind  and  parallel  to  it,  from 
VOL.  iv.  3  H 


410 


LIGHT. 


light. 


First  me- 
thod of  ex 


422. 

Second 
method. 


Fig.  95. 


423. 

Third 
method. 


Fig.  96. 
424. 

Colours  of 
the  spec 
trum. 


which  the  rays  diverge,  as  from  a  fine  luminous  line,  in  the  manner  described  in  Art.  41b  for  a  lens.  If  now  the 
eye  be  applied  close  behind  the  prism,  the  line  will  be  seen  dilated  into  a  broad  coloured  band,  consisting  of  the  - 
prismatic  colours  in  their  order ;  and  if  the  prism  be  good,  and  carefully  placed  in  its  situation  of  minimum 
deviation,  and  of  sufficiently  large  refracting  angle  to  give  a  broad  spectrum,  some  of  the  more  remarkable  of 
e  the  fixed  lines  will  be  seen  arranged  parallel  to  the  edges  of  the  spectrum,  especially  the  lines  D  and  F,  the 
former  of  which  appears,  in  this  way  of  viewing  it,  to  form  a  separation  between  the  red  and  the  yellow.  If 
the  light  of  the  sun  be  too  bright,  so  as  to  dazzle  the  eye,  any  narrow  line  of  common  daylight  (as  the  slit 
between  two  nearly  closed  window-shutters)  may  be  substituted.  This  was  the  mode  in  which  the  fixed  lines 
were  first  discovered  by  Dr.  Wollaston. 

But  it  is  difficult  and  requires  acute  sight  to  perceive,  in  this  manner,  any  but  the  most  conspicuous  lines. 
The  reason  is,  their  very  small  angular  breadth  ;  which,  in  the  largest  of  them,  can  scarcely,  under  any  circum- 
stances, exceed  half  a  minute,  and  in  the  smaller  not  more  than  a  few  seconds.  They  require,  therefore,  to  be 
magnified.  This  may  be  done  by  a  telescope  interposed  between  the  eye  and  the  prism,  in  the  manner  repre- 
sented in  fig.  95,  in  which  L  /  is  the  line  of  light,  from  which  rays,  diverging  in  all  directions,  fall  on  the  prism 
ABC,  are  refracted  by  it,  and  after  refraction  are  received  on  the  object-glass  D  of  the  telescope.  This  object- 
glass,  it  should  be  observed,  must  be  of  that  kind  denominated  achromatic,  to  be  presently  described,  (see  Index,) 
and  of  which  it  need  only  be  here  said,  that  it  is  so  constructed  as  to  be  capable  of  bringing  rays  of  all  colours  to  foci 
at  one  and  the  same  distance  from  the  glass.  Now,  if  we  consider  only  rays  of  any  one  degree  of  refrangibility 
(the  extreme  red,  for  instance)  the  pencils  diverging  from  every  point  of  L,l  will,  after  refraction  at  the  two 
surfaces  of  the  prism,  diverge  from  corresponding  points  of  an  image  L'  I1  situated  in  the  direction  from  the 
base  towards  the  vertex  of  the  prism.  Rays  of  any  greater  refrangibility  will,  after  refraction  at  the  prism,  diverge 
from  a  linear  image  L"/"  parallel  to  L'/',  but  farther  from  the  original  line  L/.  Thus  the  white  line  L  I  will, 
after  refraction  at  the  prism,  have  for  its  image  the  coloured  rectangle  L'L"/'/",  which  will  be  viewed  through 
the  telescope  as  if  it  were  a  real  object.  Now  every  vertical  line  of  this  parallelogram  will  form  in  the  focus  of 
the  object-glass  a  corresponding  vertical  image  of  its  own  colour  ;  and  the  object-glass  being  achromatic,  all 
these  images  are  equidistant  from  it,  so  that  the  whole  image  of  the  parallelogram  L'  I"  will  be  a  similar  coloured 
parallelogram,  having  its  plane  perpendicular  to  the  axis  of  the  telescope.  This  will  be  viewed  as  a  real  object 
through  the  eye-glass,  and  the  spectrum  will  thus  be  magnified  as  any  other  object  would  be,  according  to  the 
power  of  the  telescope,  (Art.  382.)  With  this  disposition  of  the  apparatus  (which  is  that  employed  by  Fraun- 
hofer) the  fixed  lines  are  beautifully  exhibited,  and  (if  the  prism  be  perfect)  may  be  magnified  to  any  extent. 
The  slightest  defect  of  homogeneity  in  the  prism,  however,  as  may  be  readily  imagined,  is  fatal.  With  glass 
prisms  of  our  manufacture  it  would  be  quite  useless  to  attempt  the  experiment ;  and  those  who  would  repeat  it 
in  this  country  should  employ  prisms  of  highly  refractive  liquids,  enclosed  in  hollow  prisms  of  good  plate  glass 
The  eye-pieces  of  telescopes,  not  being  usually  achromatic,  a  slight  change  of  focus  is  still  required,  when  the 
lines  in  the  red  and  violet  portions  of  the  spectrum  are  to  be  viewed.  This  (if  an  inconvenience)  might  be 
obviated  by  the  use  of  an  achromatic  eye-piece. 

That  an  actual  image  of  the  spectrum,  with  its  fixed  lines,  is  really  formed  in  the  focus  of  the  object-glass, 
as  described,  may  be  easily  shown,  by  dismounting  the  telescope,  and  receiving  the  rays  refracted  by  the  object- 
glass  on  a  screen  in  its  focus.  This,  indeed,  affords  a  peculiarly  elegant  and  satisfactory  mode  of  exhibiting  the 
phenomena  to  several  persons  at  once.  An  achromatic  object-glass  of  considerable  focal  length  (6  feet,  for 
instance)  should  be  placed  at  about  twice  its  focal  length  from  the  line  of  light,  and  (the  prism  being  placed 
immediately  before  the  glass)  the  image  will  be  formed  at  about  the  same  distance,  12  feet  behind  it,  (f=  L  + 
D  ;  L  =  £ ;  D  =  —  7V ;  f—  IT  —  -rV  =  +  -rV)  and  being  received  on  a  screen  of  white  paper  or  emeried  glass 
may  be  examined  at  leisure,  and  the  distances  of  the  lines  from  each  other,  &c.  measured  on  a  scale.  But  by 
far  the  best  methods  of  performing  these  measurements  are  those  practised  by  Fraunhofer,  viz.  the  adaptation 
of  a  micrometer  to  the  eye-end  of  the  telescope,  (see  Micrometer,  in  a  subsequent  part  of  this  Article,)  for  ascer- 
taining the  distances  of  the  closer  lines ;  and  the  giving  the  axis  of  the  telescope,  together  with  the  prism  which 
is  connected  with  it,  a  motion  of  rotation  in  a  horizontal  plane,  the  extent  of  which  is  read  off  by  verniers  and 
microscopes  on  an  accurately  graduated  circle,  in  the  same  way  as  in  astronomical  observations.  The  apparatus 
employed  by  him  for  this  purpose,  and  which  is  applicable  to  a  variety  of  useful  purposes  in  optical  researches, 
is  represented  in  fig.  96. 

The  fixed  lines  in  the  spectrum  do  not  mark  any  precise  limits  between  the  different  colours  of  which  it 
consists.  According  to  Dr.  Wollaston,  (Phil.  Trans.,  1802,)  the  spectrum  consists  of  only  four  colours,  red, 
green,  blue,  and  violet ;  and  he  considers  the  narrow  line  of  yellow  visible  in  it  in  his  mode  of  examination 
already  described  (looking  through  a  prism  at  a  narrow  line  of  light  with  the  naked  eye)  as  arising  from  a 
mixture  of  red  and  green.  These  colours,  too,  he  conceives  to  be  well  defined  in  the  spaces  they  occupy,  not 
graduating  insensibly  into  each  other,  and  of,  sensibly,  the  same  tint  throughout  their  whole  extent.  We  confess 
we  have  never  been  able  quite  satisfactorily  to  verify  this  last  observation,  and  in  the  experiments  of  Fraunhofer, 
(which  we  had  the  good  fortune  to  witness,  as  exhibited  by  himself  at  Munich,)  where,  from  the  perfect  distinctness 
of  the  finest  lines  in  the  spectrum,  all  idea  of  confusion  of  vision,  or  intermixture  of  rays  is  precluded,  the  tints 
are  seen  to  pass  into  each  other  by  a  perfectly  insensible  gradation  ;  and  the  same  thing  may  be  noticed  in  the 
coloured  representations  of  the  spectrum  published  in  the  first  essay  of  that  eminent  artist,  and  executed  by 
himself  with  extraordinary  pains  and  fidelity.  The  existence  of  a  pale  straw  yellow,  not  of  mere  linear  breadth, 
but  occupying  a  very  sensible  space  in  the  spectrum,  is  there  very  conspicuous,  and  may  also  be  satisfactorily 
shown  by  other  experiments  to  be  hereafter  described,  when  we  come  to  speak  of  the  absorption  of  light.  In 
short,  (with  the  exception  of  the  fixed  lines,  which  Newton's  instrumental  means  did  not  enable  him  to  see,)  the 
spectrum  is,  what  that  illustrious  philosopher  originally  described  it,  a  graduated  succession  of  tints,  in  which  all 


Part  II. 


LIGHT.  411 

I^ight.  jne  seven  colours  he  enumerates  can  be  distinctly  recognised,  but  shading  so  far  insensibly  into  each  other  that  a  Part  II. 
~ "V "•'  positive  limit  between  them  can  be  nowhere  fixed  upon.  Whether  these  colours  be  really  compound  or  not,  whether  v»v<"'' 
some  other  mode  of  analysis  may  not  effect  a  separation  depending  on  some  other  fundamental  difference  between 
the  rays  than  that  of  the  degree  of  their  refrangibility,  is  quite  another  question,  and  will  be  considered  more  at 
large  hereafter.  At  present  it  may  be  enough  to  remark,  that  all  probability,  drawn  from  everyday  experience, 
is  in  favour  of  this  idea,  and  leads  us  to  believe  that  orange,  green,  and  violet  are  mixed  colours  ;  and  red, 
yellow,  and  blue,  original  ones ;  the  former  we  everyday  see  imitated  by  mixtures  of  the  latter,  but  never  vice 
versd.  This  doctrine  has  been  accordingly  maintained  by  Mayer,  in  a  curious  Tract  published  among  his  works. 
(See  the  Catalogue  of  Optical  Writers  at  the  end  of  this  Article.)  A  very  different  doctrine  has,  however,  been 
advanced  by  Dr.  Young,  (Lectures  on  Natural  Philosophy,  i.  441,)  in  which  he  assumes  red,  green,  and  violet, 
as  the  fundamental  colours.  The  respective  merits  of  these  systems  will  be  considered  more  at  large  hereafter. 
(See  Index,  Composition  of  Colours.) 

Media,  as  we  have  seen,  differ  very  greatly  in  their  refractive  power,  or  in  the  degree  in  which  prisms  of  one  and  the      425. 
same  refracting  Dingle  composed  of  different  substances,  deflect  the  rays  of  light.    This  was  known  to  the  optical  phi-  Media 
losophers  who  preceded  Newton.    This  great  man,  on  establishing  the  general  fact,  that  one  and  the  same  medium  jj? 
refracts  differently  the  differently  coloured  rays,  might  naturally  have  been  led  to  inquire  experimentally  whether  power?'" 
the  amount  of  this  difference  of  action  were  the  same  for  all  media.     He  appears  to  have  been  misled  by  an  acci- 
dental circumstance  in  the  conduct  of  an  experiment,  in  which  the  varieties  of  media  in  this  respect  ought  to  have 
struck  him,*  and  in  consequence  adopted  the  mistaken  idea  of  a.  proportional  action  of  all  media  on  the  several  homo- 
geneous rays.     Mr.  Hall,  a  gentleman  of  Worcestershire,  was  the  first  to  discover  Newton's  mistake ;  and  having 
ascertained  the  fact,  of  the  different  dispersive  powers  of  different  kinds  of  glass,  applied  his  discovery  successfully 
to  the  construction  of  an  achromatic  telescope.     His  invention,  however,  was  unaccountably  suffered  to  fall  into 
oblivion,  (though   it  is   said  that  he  made  several  such  telescopes,  some  of  which  still  exist,)  and  the  fact  was 
re-discovered  and  re-applied   to  the  same  great  purpose  by  Mr.  Dollond,  a  celebrated  optician  in  London,  on 
the  occasion  of  a  discussion  raised  on  the  subject  by  some  a  priori  and  paradoxical  opinions  broached  by 
Euler. 

If  a  prism  of  flint  glass  and  one  of  crown,  of  equal  refracting  angles,  be  presented  to  two  rays  of  white      426. 
light,  as  A  B  C,  a  be,  (fig.  97  ;)  S  C  and  sc  being  the  incident  rays,  C  R,  CV  the  red  and  violet  rays  refracted  Differencet 
by  the  flint,  and  cr,  cv  those  refracted  by  the  crown  ;  it  is  observed,  first,  that  the  deviation  produced  in  either  of  disper- 
the  red  or  violet  ray  by  the  flint  glass,  >s  much  greater  than  that  produced  by  the  crown  ;  secondly,  that  the  angle  si,°"  eV 
R  C  V,  over  which  the  coloured  rays  are  dispersed  by  the  flint  prism,  is  also  much  greater  than  the  angle  rev,  Fjt'"!/ 
over  which  they  are  dispersed  by  the  crown  ;  and,  thirdly,  that  the  angles  R  C  V,  rev,  or  the  angles  of  disper- 
sion, are  not  to  each   other  as  Newton  supposed  them  to  be,  in  the  same  ratio  with  the  angles  of  deviation 
T  C  R,  tcr,  but  in  a  much  higher  ratio ;  the  dispersion  of  the  flint  prism  being  much  more  than  in  proportion 
to.the  deviation  produced  by  it.    And  if,  instead  of  taking  the  angles  of  the  prism  equal,  the  refracting  angle  of  the 
crown  prism  be  so  increased  as  to  make  the  deviation  of  the  red  ray  equal   to  that  produced  by  the  flint  prism, 
the  deviation   of  the  violet  will  fall   considerably  short  of  such  equality.     In  consequence  of  this,   if  the  two 
prisms  be  placed  close  together,  with  their  edges  turned  opposite  ways,  as  in  fig.  98,  so  ^as  to  oppose  each  other's  Fi?-  98> 
action,  the  red  ray,  being  equally  refracted  in  opposite  directions,  will  suffer  no  deviation  ;  but  the  violet  ray, 
being  more  refracted  by  the  Hint  than  by  the  crown  prism,  will,  on  the  whole,  be  bent  towards  the  thicker  part 
of  the  flint  prism,  and   thus  an  uncorrected  colour  will  subsist,  though  the   refraction  (for  one  ray,  at  least)  is 
corrected.     Vice  versd,  if  the  dispersion  be  corrected,  that  is,  if  the  refracting  angle  of  the  crown  prism,  acting 
in  opposition  to  the  flint,  be  so  further  increased  as  to  make  the  difference  of  the  deviations  of  the  red  and  violet 
rays  produced  by  it  equal  to  the  difference  of  their  deviations  produced  by  the  flint,  the   deviation  produced  by 
it  will  now  be  greater  than  that  produced  by  the  flint ;   and  the  total  deviation,  produced  by  both  prisms  acting 
together,  will  now  be  in  favour  of  the  crown. 

By  such  a  combination  of  two  prisms  of  different  media  a  ray  of  white  light  may  therefore  be  turned  aside      427. 
considerably  from  its  course,  without  being  separated  into  its  elementary  coloured  rays.     It  is  manifest,  that  (sup-  R?'ract">n 
posing  the  angles  of  the  prisms  small,  and  that  both  are  placed  in  their  positions  of  minimum   deviation)  the  Wl'  ?ut  s< 
deviations  to  produce  this  effect  must  be  in  the  inverse  ratio  of  the  dispersive  powers  of  the  two  media ;    for  into  colours, 
supposing  p,  fi  to  be  the  refractive  indices  of  the  prisms  for  extreme  red  rays,  and/i  +  S  ft,  p!  +  8/»'  for  extreme 
violet,  A  and  A'  their  refracting  angles,  and  D  and  D'  their  deviations,  we  have,  generally,  in  the  position  of 
minimum  deviation 

A  A+D  A  A+D 

ft  .  sin  — -  =  sin ,  whence  o  /*  .  sin  —  =  \  6  D  .  cos 

A'              A'+  D'                           A'                        A'  +  D' 
ft' .  sin  — - —  =  sin o  ft .  sin  —  =  £  o  D  .  cos —  - 

whence,  since  the  prisms  oppose  each  other, 

'  He  counteracted  the  refraction  of  a  glass,  by  a  water  prism.  There  ought  to  have  been  a  residuum  of  uncorrected  colour;  but 
unluckily,  he  had  mixed  sugar  of  lead  with  the  water  to  increase  its  refraction,  and  the  high  dispersive  power  of  the  salts  of  lead  (of  which' 
of  course,  he  could  not  have  the  least  suspicion)  thus  robbed  him  of  one  of  the  greatest  discoveries  in  physical  optics. 

3  H2 


412  LIGHT. 

Light.  A  .A' 

c  ft  .  sin  —  c  p..  sin  - 

4«(D-DO=    - 


cos 

Putting  this  equal  to  zero,  we  have 

&  ft       sin  4  A  cos  J  (A  +  D)_ 

f>'     '  sin  \  A'  =      cosi  (\r+W)' 

and,  eliminating  sin  £  A  and  sin  ~  A'  from  this,  by  means  of  the  two  original  equations  from  which  we  set  out. 
we  get 

S/t  /          cos  i  (A  +  D)  sin  J  (A'  +  PQ  tan  4  (A7  +  DO 

T/    :     ~/T  :  :    cos  4  (A'  +  DO    : :    sin  i  (A  +  D)  tan  i  (A  +  D) 

Now  if  we  call  p,  p'  the  dispersive  powers  of  the  media,  or  the  proportional  parts  of  the  whole  refractions  of  the 
extreme  red  ray,  to  which  the  dispersion  is  equal,  we  shall  have 


so  that 


3/i  Bfi'  p  *£_  ,, 

P  =  -    i       P  —       i       i     and  ~^T  ~  ~T~'~   x 
ju  —  1  ft  '  —  1  p  Iff 


'  -  1  tan  ^  (A'  -f  DQ    _    ft1-  I         sin  &  A'  /I  -  ft*  .  (sin  4  A)* 

tani(A-fD)  u—  1          sin  i  A     '    V    1  -  Xs  .  (sin  4  AO4' 


p'     -   ^    '    ^-1  tan£(A+D) 

Such  is  the  strict  formula,  which,  when  A  and  A'  are  verv  small,  becomes 

--    =  ;     <».  -ce  (,-l)A=D,  and  0,'-1 


428.  The  formula  just  obtained,  furnishes  us  with  an  experimental  method  of  determining  the  ratio  of  the  dispersive 
Dispersive    powers  of  two  media.     For  if  we  can  by  any  means  succeed  in  forming  them  into  two  prisms  of  such  refracting 

>werscom-  angies>  that,  when  placed  in  their  respective  positions  of  minimum  deviation,  a  well  defined  bright  object,  viewed 
experiment    through  both,  shall  appear  well  defined  and  free  from  colour  at  its  edges  ;  then,  by  measuring  their  angles,  and 
knowing  also  from  other  experiments  their  refractive  indices,  the  equation  (a)  gives  us  immediately  the  ratio  in 
question. 

429.  When  we  view  through  a  prism  any  well  defined  object,  either  much  darker  or  much  lighter  than  the  ground 
Coloured      against  which  it  is  seen  projected,   as,  for  instance,  a  window  bar  seen  against  the  sky,  its  edges  appear  fringed 
fringes  bor-  wjt^  coiours  and  ill  defined.     The  reason  of  this  may  be  explained  as  follows: 

i'ecrtsnfeen~  Let  A  B,  fig.  99,  be  the  section  of  a  horizontal  bar  seen  through  the  prism  P  held  with  its  refracting  edge 
through  downwards,  and  first  lefc  us  consider  what  will  be  the  appearance  of  the  upper  edge  B  of  the  object.  Since  we 
prisms  ex-  see  by  light,  and  not  by  darkness,  the  thing  really  seen  is  not  the  dark  object,  but  the  bright  ground  on  which  it 
plained.  stands,  or  the  bright  spaces  B  C,  A  D  above  and  below.  Now  the  bright  space  B  C  above  the  object  being 
g  illuminated  with  white  light,  will,  after  refraction  at  the  prism,  form  a  succession  of  coloured  images  b  c,  b'  c', 

b"  (/',  &c.,  superposed  on  and  overlapping  each  other.  They  are  represented  in  the  figure  as  at  different  dis- 
tances from  P,  but  this  is  only  to  keep  '.hem  distinct.  In  reality,  they  must  be  supposed  to  lie  upon  and  interfere 
with  each  other.  The  least  refracted  6  c  of  these  is  red,  and  the  most  refracted  b"  c'"  violet,  and  any  intermediate 
one  (as  b'  </)  of  some  intermediate  colour,  as  yellow  for  instance.  Beyond  b"  no  image  exists,  so  that  the  whole 
space  below  b"  will  appear  dark  to  an  eye  situated  behind  the  prism.  On  the  other  hand,  above  6  the  images  of 
every  colour  in  the  spectrum  coexist,  the  bright  space  b  c  being  supposed  to  extend  indefinitely  above  B.  There- 
fore the  space  above  6  in  the  refracted  image  will  appear  perfectly  white.  Between  6  and  6"  there  will  be  seen, 
first,  a  general  diminution  of  light,  as  we  proceed  from  6  towards  h",  because  the  number  of  superposed  luminous 
images  continually  decreases  ;  secondly,  an  excess  in  all  this  part,  of  the  more  refrangible  rays  in  the  spectrum 
above  what  is  necessary  to  form  white  light,  for  beyond  b  no  red  image  exists,  beyond  6'  no  yellow,  and  so  on  ; 
the  last  which  projects  beyond  all,  at  b",  being  a  pure  unmixed  violet.  Thus  the  light  will  not  only  decrease  in 
intensity,  but  by  the  successive  subtraction  of  more  and  more  of  the  less  refrangible  end  of  the  spectrum  will 
acquire  a  bluer  and  bluer  tint,  deepening  to  a  pure  violet,  so  that  the  upper  edge  of  the  dark  object  will  appear 
fringed  with  a  blue  border,  becoming  paler  and  paler  till  it  dies  away  into  whiteness.  The  reverse  will  happen 
at  the  lower  edge  A.  The  bright  space  A  D  forms,  in  like  manner,  a  succession  of  coloured  images,  a  d,  a'  d', 
a''  d',  of  which  the  least  deviated  a  d  is  red.  the  most  a"  d"  violet,  and  the  intermediate  ones  of  the  intermediate 
colours.  Therefore  the  point  a,  which  contains  only  the  extreme  red,  will  appear  of  a  sombre  red  ;  a',  which 
contains  all  the  rays  from  red  to  yellow  (suppose),  of  a  lively  orange  red  ;  and  in  proportion  as  the  other  images 
belonging  to  the  more  refrangible  end  of  the  spectrum  come  in,  this  tendency  to  an  excess  of  red  will  be  neutralized, 
and  the  portion  beyond  a'',  containing  all  the  colours  in  their  natural  proportions,  will  be  purely  white.  Hence, 
the  lower  edge  of  the  dark  object  will  appear  bordered  with  a  red  fringe,  whose  tint  fades  away  into  whiteness, 
in  the  same  way  as  the  blue  fringe  which  borders  the  upper  edge.  These  fringes,  of  course,  destroy  the  dis- 
tinctness of  the  outlines  of  objects,  and  render  vision  through  a  prism  confused.  The  confusion  ceases,  and 
objects  resume  their  natural  well  defined  outlines,  if  illuminated  with  homogeneous  light,  or  if  viewed  through 
coloured  glasses  which  transmit  only  homogeneous  rays. 


LIGHT.  413 

Light.          The  eye  can  judge  pretty  well,  by  practice,  of  the  destruction  of  colour,  and  indistinctness  in  the  edges  of     I""*  K- 
•— ^-»'  objects,  when  prisms  are  made  to  act  in  opposition  to  one  another,  as  above  described ;  but  (owing1  to  causes  V>^-V'~ ^ 
presently  to  be  considered)  the  compensation  is  never  perfect,  and  there  always  remains  a  small  fringe  of  uncor-       430. 
rected  purple  on  one  side,  and  green  on  the  other,  when  the  eye  is  best  satisfied  ;  so  that  observations  of  dispersive 
powers  by  this  method  are  liable  to  a  certain  extent  of  error,   and,   indeed,  precision  in   this  department  of 
optical  science  is  very  difficult  to  obtain. 

To  determine  the  dispersive  power  of  a  medium,  having  formed  it  into  a  prism,  and  measured  by  the  goniometer,  431 . 
or  otherwise,  its  refracting  angle,  and  ascertained  its  refractive  index,  the  next  step  is  to  find  the  refracting  To  deter- 
angle  of  a  prism  of  some  standard  medium,  which  shall  exactly  compensate  its  dispersion,  so  as  to  produce  ™'ne  '*)* 

a  refraction  as  nearly  as  possible  Iree  from    colour.     But  as  it  is    impossible    to  have  a  series    of   standard  d'sPers'°n 

<•        •  u'   t-  i-  •  •.       -i   u  i  f  ofamedium 

pnsms  with  every  refracting  angle  which  may  be    requisite,  it  becomes    necessary  to  devise  some  means  of 

varying  the  refracting  angle  of  one  and  the  same  prism  by  insensible    gradations.      Many  contrivances  may 
be  had  recourse  to    for  this.     Thus,  first,    we  may    use  a  prism    composed   of  two  plates  of  parallel  glass.  Prisms  with 
united  by  a  hinge,  or  otherwise,  and  enclosing  between  them  a  fluid,  which  may  be  prevented  from  escaping  variable  re- 
either  by  capillary  attraction,  if  in  very  small  quantity,  or  by  close-fitting  metallic  cheeks,  forming  a  wedge-  fracling 
shaped  vessel,  if  in  larger.     This    contrivance,  however,  is  liable  to  a  thousand  inconveniencies  in  practice,  described 
Secondly,  we  may  use  two  prisms  of  the  same  kind  of  glass,  one  of  which  has  one  of  its  faces  ground  into 
a  convex,  and  the    other  into  a    concave    cylinder,  of  equal    curvatures,  having  their    axes   parallel    to    the 
refracting  edges.     These  being  applied  to  each  other,    and  one  of  them  being    made  to    revolve  round  the  Another 
common  axes  of  the  two  cylindric  surfaces  upon  the  other,  the  plane  faces  will  evidently  be  inclined  to  each  construct'°n 
other  in    every  possible  angle  within  the  limits  of  the  motion,  (see  fig.   100,   a,  6,  exhibiting  two  varieties     l?' 
of  this  construction.)     The  idea,  due,  we  believe,  to  Boscovich,  is  ingenious,  but  the  execution  difficult,  and 
liable  to  great  inaccuracies. 

The  following  method  succeeds  perfectly  well,  and  we  have  found  it  very  convenient  in  practice.     Take  a       432. 
prism  of  good  flint  glass,  whose  section  is  a  right  angled  triangle,  ABC,  having  the  angle  A  about  30°  Third  con- 
or  35°,  C  being  the  right  angle,  and  whose  length  is  twice  the  breadth  of  the  side  A  C  ;   and,  having  ground  y™0/^"' 
and  polished  the  side  A  C,  and  the  hypothenuse  of  the  prism  to  true  planes,  cut  it  in  half,  so  as  to  form  102. 
two  equal  prisms  with  one  face  in  each  a  square,   and  whose    refracting  angles  (A,  A')  cannot,  of  course, 
be  otherwise  than  exactly  equal.     Cement  the  square  faces  together  very  carefully  with  mastic,  so  that   the 
edges  A,  A',  shall  be  on  opposite  sides  of  the  square  surface,  which  is  common  to  both ;  and  then,  making 
the  whole  solid  to  revolve  round  an  axis  perpendicular  to  the  common  surface,  and  passing  through  its  centre, 
grind  off  all  the  angles  of  the    squares  in  the  lathe,  and  the  whole  will  be  formed  into  a  cylindrical  solid, 
with  oblique,  parallel,  elliptical,  plane  ends,  as  in    fig.   101.     Then    separate    the   prisms,   (by  warming  the 
cement,)  and  set  each  of  them  in  a  separate  brass  mounting,  as  in  fig.  102,  so  as  to  have  their  circular  faces 
in  contact,  and  capable  of  revolving  freely  upon  each  other  about  their  common  centre.     The  lower  one  is  fixed 
in  the  centre  of  the  divided  circle  D  E,  while  the  mounting  of  the  upper  or  moveable  one  carries  an  arm  with  an 
adjustable  vernier  reading  off  to  tenths  of  degrees,  or,  if  necessary,  to  minutes.     The  whole  apparatus  is  set  in  a 
swing  frame  between  plates,  which  grasp  the  divided  plate  by  a  groove  in  its  edge,  allowing  a  motion  in  its  own 
plane,  and  a  capability  of  adjusting  it  to  any  required  position,  so  as  to  admit  of  the  compound  prism  deviating 
an  incident  ray  in  every  possible  plane,  and  under  every  possible  situation,  with  respect  to   the  faces  of  the 
prisms.     It  is  evident,  that  in  the  position  here  represented,  where  the  prisms  oppose  each  other,  (und  at  which 
the  vernier  must  be  set  to  read  off  zero,)  the  refracting  angle  is  rigorously  nothing ;  and  when  turned  round  ISO3 
since  the  prisms  then  conspire,  their  combined  angle  must  be  double  that  «f  each.     In   intermediate  situations, 
the  angle  between  the  planes  of  their  exterior  faces  must,  of  course,  pass  through  every  intermediate  state,  and 
(by  spherical  trigonometry)  it  is  readily  shown,  that  ifO  be  the  reading  off  of  the  vernier,  or  the  angle  of  rotation 
of  the  prisms  on  each  other  from  the  true  zero,  the  angle  of  the  compound  prism  will  be  had  by  the  equation 

sin  — — -  =  sin  —  .  sin  (A)  (b) 

where  (A)  is  the  refracting  angle  of  each  of  the  simple  prisms,  and  A  the  angle  of  the  compound  one. 

To  use  this  instrument,  place  the  prism  A',  whose  dispersive  power  is  to  be  compared  with  the  medium  of  433. 
which  the  standard  prism  (A)  is  formed,  with  its  edge  downwards  and  horizontal,  before  a  window,  and,  selecting  How  used, 
one  of  the  horizontal  bars  properly  situated,  fix  it  so  that  the  refraction  of  this  bar  shall  be  a  minimum,  or  till, 
on  slightly  inclining  the  prism  backwards  and  forwards,  the  image  of  the  bar  appears  stationary.  Then  take 
the  standard  compound  p.-ism,  adjust  it  to  zero,  and  set  it  vertically  on  its  frame  behind  the  first  prism.  Move  its 
index  a  few  degrees  from  zero,  and  turn  the  divided  circle  in  its  own  plane,  till  the  refraction  so  produced  by  the 
second  prism  is  contrary  to  that  produced  by  the  first.  The  colour  will  be  found  less  than  before ,  continue 
this  till  the  colour  is  nearly  compensated,  then,  by  means  of  the  swing  motion,  and  of  the  motion  round  the 
vertical  axis,  adjust  the  apparatus  so  that  two  of  the  window  bars,  a  horizontal  and  a  vertical  one,  seen  through 
both  prisms,  shall  appear  to  make  a  right  angle  with  each  other,  (an  adjustment,  at  first,  rather  puzzling,  but 
which  a  little  practice  renders  very  easy.)  Then  complete  the  compensation  of  the  colour ;  verify  the  position  of 
the  standard  prism,  (by  the  same  test,)  and  finally  read  off  the  vernier,  and  the  required  angle  A  of  the  com- 
pensating prism  is  easily  calculated  by  the  equation  (6).  This  calculation  may  be  saved  by  tabulating  the  values 
of  A  corresponding  to  those  of  0,  (the  value  of  (A)  being  supposed  known  by  previous  exact  measures,)  or,  by 
graduating  the  divided  circle  at  once,  not  into  equal  parts  ol  6,  but  according  to  such  computed  values  of  A,  so 
as  to  read  off  at  once  the  value  of  the  angle  required. 


414  L  I  G  H  T. 

Light.          A  simpler,  perhaps,  on   the  whole,  a  better,  method  of   comparing  the  dispersions  of  two  prisms,  is  one     Part  II. 
^—  -Y~™P'  proposed  and  applied  extensively  by  Dr.  Brewster,  in  his  ingenious  Treatise  On  New  Philosophical  Instruments,  ^—  -v—  • 

43  1.       a  work  abounding  with  curious  contrivances  and  happy  adaptations.     It  consists  in  varying,  not  the  refracting 

Another       angle  of  the  standard  prism,  but  the  direction  in  which   its  dispersion  is  performed.     It  is  manifest,  that  if  we 

pro  osedb    can  Pro^uce  fr°m  a  u'ne  °f  white  light,  by  means  of  a  standard  prism  any  how  disposed,  a  coloured  fringe,  in 

Dr  Brew/  which  the  colours  occupy  the  same  angular  breadth  as  in   that  produced  by  a  prism  of  unknown  dispersion  ; 

ster  then,  the  latter,  being  made  to  refract  this  fringe  in.  a  direction  perpendicular  to  its  breadth,  and  opposite  to  the 

order  of  its  colours,  must  destroy  all  colour  and  produce  a  compensated  refraction  ;  and  therefore  if  the  position 

of  the  standard  prism  which   produces  such  a  fringe  be  known,  the  dispersion  of  the  other  may  be  calculated. 

To  accomplish  this,  let  A  B  be  a  horizontal  luminous  line  of  considerable  length,  and  let  it  be  refracted  downwards, 

Fig.  103.     but  obliquely  in  the  direction  A  a,  B  6,  by  a  standard  prism  whose  dispersion  is  greater  than  that  of  the  prism  to  be 

measured.    Then  it  will  form  an  oblique  spectrum  abb'  a',  ab  being  the  red,  and  a'  6'  the  violet;  and  the  angular 

breadth  of  this  coloured  fringe  will  beam  =  a  a'  x  sin  inclination  of  the  plane  of  refraction  to  the  horizon.     Now, 

let  the  prism  whose  dispersion  is  to  be  measured  be  made  to  refract  this  coloured  band  vertically  upwards  ;  then,  if 

the  plane  of  the  first  refraction  be  so  inclined  to  the  horizon  that  the  angle  subtended  by  a  m  at  the  eye  shall  be  just 

equal  to  the  angle  of  dispersion  of  the  other  prism,  all  the  colours  of  the  rectangular  portion  b  caf  d  will  be  made  to 

coalesce  in  the  horizontal  line  A'  B',  which  will  appear  therefore  free  from  colour,  except  at  its  extremities  A'  B', 

where   the  coloured  triangles  aca',   bdb'vritt  produce  a  red  termination  A'  A"  and   a  blue  one  B'B"at  the 

respective  ends  of  the  line  to  which  they  correspond.     Hence,  if,  the  second  prism  remaining  fixed,  with  its  edge 

downwards  and  parallel  to  the  horizon,  the  other  or  standard  prism  be  turned  gradually  round  in  the  plane  perpen- 

dicular to  its  principal  section,  a  position  must  necessarily  be  found  where  the  twice  refracted  line  A'  B'  will  appear 

free  from  colour  both  above  and  below.    In  this  position  let  it  be  arrested,  and  the  angle  of  inclination  of  its  edge 

to  the  horizon  read  off,  its   complement  is  the  angle  a  a'  in,  which  we  will  call  0.     Let  us  now  suppose  each 

prism  adjusted  to  its  position  of  minimum  deviation,  and  (as  it  is  a  matter  of  indifference  which  is  placed  first) 

let  the  prism  to  be  examined  or  the  fixed  prism  be  placed  next  the  object.*     Then,  D'  and  D  being  the  total 

deviations  produced  by  the  fixed  and  revolving  prisms  on  the  extreme  red  ray,  we  must  have 

A'  A'  +  D'  A  A  +  D 

&  D  —  6  D  .  sin  0  =  o  ,-         or  K  ft'  .  sin  —  j—  .  sec  -  -  -  =  $  fn  .  sm  —  —  .  sec  -  -  -  .  sin  0, 

£  £  8  •  8 


whence  we  obtain 


p'  &  /         /.-I  f!          p-l         tan  J(A  +  D) 

p    '  '    If,    •     /.'-I    '        ,.        ~7^\    •    tani(A'+D')   ' 


, 
where  the  angles  £  (A  -f-  D)  and  £  (A'  +  D')  are  given  by  the  equations 

sin  \  (A  +  D)  =  p  .  sin  x  J  A  ;         sin  £  (A'  +  D')  =  /.'  .  sin  \  A'  ; 

from  which  formula,  0  being  known,  and  also  the  angles  anu    efractive  indices  of  the  two  prisms,  the  ratio  ot 
their  dispersions  is  found. 

435.  By  tnese,  or  other  similar  methods,  may  the  dispersions  of  any  media  be  compared  with   those  of  any  other 

Absolute      taken  as  a  standard.     If  the  media  be  solid,  they  must  be  formed  into  prisms  ;  if  fluid,  they  must  be  enclosed  in 

dispersive     hollow  prisms  of  truly  parallel  plates  of  glass,  whose  angles  must  be  accurately  determined,  (and  one  of  which 

powers,how  wjj]  serve  for  anv  number  of  fluids.)     But  to  ascertain  directly  the  dispersion  of  that  standard  prism,  we  must 

IstBvmea-  Pursue  a  different  course.     The  first  method  which  obviously  presents  itself,  is  to  measure  the  actual  length  of 

suring  the     the  solar  spectrum  cast  by  a  prism  of  given  refracting  angle  ;  but  the  light  of  the  spectrum  dies  away  so  inde 

spectrum  on  finitely  at  both  ends,  and  its  visible  extent  varies  so  enormously  with  the  brightness  of  the  sun,  and  the  more 

a  screen.      or  ]ess  perfect  exclusion  of  extraneous  light,  that  nothing  certain  can  be  concluded  from  such  measures.     Yet, 

if  the  brighter  rays  of  the  spectrum  be  destroyed,  and  the  eye  defended  from  all  offensive  light  by  a  glass  which 

permits  only  the  extreme  red  and  violet  rays  to  pass,  (see  Index,  Absorption,)  some  degree  of  accuracy  may  be 

obtained  by  this  means.     A  method  founded  on  this  principle  has  been  described  by  the  writer  of  these  pages 

Fig.  104.     in  the  Transactions  of  the  Royal  Society  of  Edinburgh,  vol.  ix.  as  follows  :  Let  A  and  B  be  two  vertical  rect- 

tnd.Another  angular  slits  in  a  screen  placed  before  an  open  window,  the  one  being  half  the  length  of  the  other,  and  at  a 

known  distance  from  each  other.     The  eye  being  guarded  as  above  described,  let  the  slits  be  refracted  by  the 

prism  (in  its  minimum  position)  from   the  longer  towards  the  shorter.     Then  will  a  red  and  violet  image  of 

each  a,  b,  and  a',  b'  be  seen.     Now  let  the  prism  be  removed  from  the  slits,  (or  vice  versd,)  still  preserving  its 

position  of  minimum  deviation,  till  the  violet   image  of  the  longer  slit  exactly  falls  upon  and  covers  the  red 

image  of  the  shorter,  as  in  the  position  a:  b  of  the  figure.     Then  it  is  obvious,  that  the  distance  between  the 

slits,  divided  by  their  distance  from  the  prism,  is  the  sine  of  the  total  angle  of  dispersion,  or  is  equal  to  2  D, 

and  this  being  known 

3D       cos  &  (A  +  D) 

2  sin  i  A       ' 

&  u. 
and   therefore   —  —  ,  or  p,  the  dispersive  power,  is  obtained. 

*  Dr.  Brewster  has  chosen  a  somewhat  different  position,  (Trratite,  Sfc.  p.  296,)  with  a  view  to  simplify  the  formula;  j  but  it  doe*  n<* 
ippear  to  us  that  any  advantage  is  gained  in  that  respect  by  his  arrangement. 


LIGHT. 


415 


Light.  But  all  these  methods  are  only  rude  approximations,  as  the  great  discrepancies  of  the  results  hitherto  obtained 
wv-»'  by  them  abundantly  prove  ;  thus,  the  dispersions  of  various  specimens  of  flint  glass,  obtained  by  the  method  last 
described,  come  out  no  less  than  one-sixth  larger  than  those  previously  given  by  Dr.  Brewster.  The  only  method 
which  can  really  be  relied  on  is  that  practised  by  Fraunhofer,  (where  the  media  can  be  procured  in  a  state  of  sum- 
cient  purity  and  quantity  for  its  application  ;)  and  consists  in  determining,  with  astronomical  precision,  by  direct 
measures,  the  values  of  it.  for  the  several  points  of  definite  refrangibility  in  the  spectrum,  marked,  either  by  the 
fixed  lines,  or  by  the  phenomena  of  coloured  flames  or  absorbent  media.  (See  Index,  Flames—  Absorption.) 
By  taking  advantage  of  the  properties  of  the  latter,  a  red  ray,  of  a  refrangibility  strictly  definite,  may  be 
insulated  with  great  facility  ;  and  as  it  lies  so  near  the  extremity  of  the  spectrum  as  not  to  be  perceptible  till  all 
the  brighter  rays  are  extinguished,  it  is  invaluable  as  a  fixed  term  in  optical  researches,  arid  will  always  be  un- 
derstood by  us  in  future,  when  speaking  of  the  commencement  of  the  spectrum,  or  the  extreme  red,  even  though 
a  red  ray  still  less  refrangible  should  be  capable  of  being  discerned  by  careful  management,  and  in  favourable 
circumstances.  In  like  manner,  by  the  simple  artifice  of  putting  a  little  salt  into  a  flame,  a  yellow  ray  of  a 
character  perfectly  definite  is  obtained,  which,  it  is  very  remarkable,  occupies  precisely  the  place  in  the  scale  of 
refrangibility  where  in  the  solar  spectrum  the  dark  line  D  occurs,  (Art.  4LS,  419.)  These,  and  the  fixed  lines 
there  mentioned,  leave  us  at  no  loss  for  rays  identifiable  at  all  times  and  in  all  circumstances,  (with  a  good  appa- 
ratus,)  and  enable  us  to  place  the  doctrine  of  refractive  and  dispersive  powers  on  the  footing  of  the  most  accu- 
rate branches  of  science. 

The  following  table,  extracted  from  Fraunhofer's  Essay  on  the  Determination  of  Refractive  and  Dispersive 
Powers,  8fc.  contains  the  absolute  values  of  the  index  of  refraction  fi  for  the  several  rays  whose  places  in  the 
spectrum  correspond  to  the  seven  lines  B,  C,  D,  E,  F,  G,  H,  assumed  by  him  as  standards  (see  Art.  419,  &c.) 
for  several  different  specimens  of  glass  of  his  own  manufacture,  and  for  certain  liquids.  These  values,  for  dis- 
tinction's sake,  we  may  designate  by  the  signs  n  (B),  /t  (C),  p  (D),  &c. 


Part  II. 

^"v™- 
436. 


Fraunhofer 


Use  of  the 


437 


Table  of  the  refractive  indices  of  various  glasses  and  liquids  for  seven  standard  rays. 


Specific 

Values  of 

Refracting  medium. 

gravity. 

*(B) 

,<o 

MD) 

/•(E) 

MF) 

MG) 

/•(H) 

Flint  glass  No.  13  

3.723 

1  627749 

1  629681 

1  635036 

1.642024 

I  648260 

1.660285 

1.671062 

Crown  glass  No  9  . 

2.535 

1  525832 

1  526849 

1  529587 

1  533005 

1  536052 

1  541657 

1  546566 

Water  

1.000 

1.330935 

1.331712 

1.333577 

1.335851 

1.337818 

1.341293 

1.344177 

Water,  another  experiment 

1.000 

1.330977 

1.331709 

1.333577 

1.335849 

1.337788 

1.341261 

1.344162 

Solution  of  potash  

1.416 

1  399629 

1  400515 

1.402805 

1.405632 

1.408082 

1.412579 

1.416368 

Oil  of  turpentine  

0.885 

1.470496 

1.471530 

1.474434 

1.478353 

1.481736 

1.488198 

1.493874 

Flint  glass,  No.  3   

3.512 

1.602042 

1.603800 

1.608494 

1.614532 

1.620042 

1.630772 

1.640373 

Flint  glass,  No.  30  

3.695 

1.623570 

1.625477 

1.630585 

1.637356 

1.643466 

1.655406 

1.666072 

Crown  glass,  No.  13  .... 

2.535 

1.524312 

1.525299 

1.527982 

1.531372 

1.534337 

1.539908 

1.544684 

Crown  glass,  letter  M.  .  .  . 

2.756 

1.554774 

1.555933 

1.559075 

1.563150 

1.566741 

1.5*3535 

1.579470 

Flint  glass,  No.  23  ....  ~» 
Prism  of  60°  15'  42"  J 

3.724 

1.626596 

1.628469 

1  .633667 

1.640495 

1.646756 

1.658848 

1.669686 

Flint  glass,  No.  23  ~) 

Prism  of  45°  23'  14") 

3.724 

1.626564 

1.628451 

1.633666 

1.640544 

1.646780 

1.658849 

1.669680 

The  above  table  renders  very  evident  a  circumstance  which  has  long  been  recognised  by  experimental  opticians, 
and  which  is  of  great  importance  in  the  construction  of  telescopes,  viz.  the  irrationality,  (as  it  has  been  termed,)  or 
want  of  proportionality  of  the  spaces  occupied  in  spectra  formed  by  different  media  by  the  several  coloured  rays, 
or  by  those  whose  refrangibilities,  by  any  one  standard  medium,  lie  between  given  limits.  If  we  fix  upon 
water,  for  example,  as  a  standard  medium,  (and  we  see  no  reason  why  it  should  not  be  generally  adopted  as  a 
term  of  reference  in  this,  as  in  other  physical  inquiries — of  course  at  a  given  temperature — that  of  its  maximum 
density,  for  instance,)  it  is  obvious,  that  any  ray  may  be  identified  by  stating  its  index  of  refrangibility  by  water ; 
thus,  a  scale  of  refrangibilities,  which,  for  brevity,  we  shall  term  the  water  scale,  is  established ;  and  so  soon  as  we 
know  the  refractive  index  of  a  ray  from  vacuum  into  water,  we  have  its  place  in  the  water  spectrum,  its  colour, 


438. 

Identifica- 
tion  of  a  raj 
by  its  place 
in  a  water 
spectrum. 


416  LIGHT. 

Light      and  its  other  physical  properties  (so  far  as  they  depend  on  the  refrangibility  of  the  ray)  determined.     Thus     Part  II. 
v—  -v—  '  1.333577  being  known  to  be  the  retractive  index  for  a  ray  in  water,  that  ray  can  be  no  other  than  the  particular  *—  —  v» 
ray  D,  whose  colour  is  pale  orange-yellow,  and  which  is  totally  deficient  in  solar  light,  and  peculiarly  abundant 
in  the  light  of  certain  flames.     Now  let  x  be  the  refractive  index  of  any  ray  whatever  for  water,  or  its  place  in 
the  water  scale.     Then  it  is  evident,  that  its  refractive  index  for  any  other  medium  must  of  necessity  be  a  function 
of  x,  because  the  value  of  x  determines  this  and  all  the  other  properties  of  the  ray.     Hence  we  must  have 
between   /i    and  x  some    equation  which  may   be    generally  represented    by  /*  =  F  (x)  ;  F  (x)  denoting   a 
function  of  x. 

*39.          To  determine  the  form  of  this  function,  we  must  consider,  that  if  A  be  the  very  small   angle  of  a  prism, 
Function  of  A  A  +  D 

refrangibi-    and  D  the  deviation  produced  by  it  at  the  minimum,  we  have  p  .  ——  =     —  -  -  ,  or  D  =  (/»  —  1)  A.     Hence, 

lity.  •"  *• 

supposing  A  the  redacting  angle  constant,  the  deviation  is  proportional  to  fi  —  1.  Now,  since  in  all  media,  as 
well  as  in  water,  the  deviations  observe,  at  least,  the  same  order,  being  always  least  for  the  red  and  greatest  for 
the  violet,  it  follows,  that  in  all  media  fi  —  1  increases  as  x  increases  ;  so  that,  supposing  x0  to  be  the  index  of 
refraction  in  the  water  scale  for  the  first  visible  red  ray,  or  the  commencing  value  of  x,  and  p0  the  index  for  the 
same  ray  in  the  other  medium,  (u,  —  1)  —  (f*0—  1),  or  /t  —  ft0  must  increase  with  x  —  xa  ;  and  since  they 
vanish  together,  we  may  represent  the  one  in  a  series  with  indeterminate  coefficients,  and  powers  of  the  other, 
thus 

/*  -  PO  =  A  Cr  -  xa)  4-  B  (x  -  xj*  +C(x-ior  +  &c.  ; 

or,  which  comes  to  the  same  thing,  a  b,  c,  &c.,  representing  other  indeterminate  coefficients,  (x0  —  1  being 
constant,) 

4  .          Y+  &c.        «f) 


440.          The  simplest  hypothesis  we  can  form  respecting  the  values  of  a,  b,  &c.  is  that  which  makes  a  =  1,  and  6, 
Hypothesis 

ion'   and  all  the  other  coefficients  vanish.     This  gives   ^>    = 


. 

in  all  media.  °  *° 

We  have  before  used  «  /i  to  denote  what  is  here  signified  by  p  —  n0,  viz.  the  difference  between  the  refractive 

6  ft 
indices  of  any  ray  in  the  spectrum,  and  that  at  its  commencement;  and  we  have  denoted  by  —  —  —  the  same 

quantity  which  is  here  expressed  by  —  ^£_.     This  then  is  the  expression,  in  our  present  notation,  of  the 

f'o         1 

Not  the  law  dispersive  power  of  the  medium;  and  the  equation  now  under  consideration  therefore  indicates,  lhat,  on  the 
of  nature,     hypothesis  made,  the  dispersive  power  of  the  medium  must  necessarily  be  the  same  with  that  of  water;    and 
••  of  course  (supposing  this  hypothesis  to  be  founded  in  the  nature  of  light)  all  media  must  have  the  same  dis- 

persive power.     This,  as  we  have  already  seen,  is  not  the  case. 

Nor  that  of      The  next  simplest  hypothesis  is  that  which  admits  a  as  an  arbitrary  constant  determined  by  the  nature  of  the 
proportional  medium,  but  still  makes  b,  c,  &c.  =  o.     This  reduces  the  equation  to 
dispersions. 


'o 


consequently  (if  fi'  and  j;1  be  any  other  corresponding  values  of  ft  and  x)  we  mu:  t  have  also 

u'  —  u,  x'  —  x  /»'  —  !*  x'  —  x  p'  —  P  fo  ~  * 

-£ a.  =  a  .  — —2,  and  therefore   r  _       =  a  .  — —  ;  whence  we  have  ^  _  ^  =  a  .  ^   _  }. 


Hence,  if  this  hypothesis  be  correct,  and  ^  x  and  /,  *'  be  any  two  pairs  of  corresponding  refractive  indices 
for  rays  however  situated,  the  fraction  £-H£  must  be  invariable.     The  foregoing  table,  however,  shows  very 

X    ~~  X 

distinctly  that  this  is  far  from  being  the  case.     Thus,  if  we  take  the  flint  glass,  No.  13,  the  comparison  of  the 
two  rays  B  and  C  gives  for  the  value  of  the  fraction  in  question  2.562 ;  and  if  we  compare  in  like  manner  the 
rays  C  and  D,  D  and  E,  E  and  F,  F  and  G,  G  and  H  respectively,  we  obtain  the  values  2.871,   3  0,3,  3.1H 
3.460,  3  726  ;  the  great  deviation  of  which  from  equality,  and  their  regular  progression,  leaves  no  doul 
incompatibility  of  the  hypothesis  in  question,  as  a  general  law,  with  nature.     If  we  institute  the  same  compare 
for  the  other  media  in  the  table,  we  shall  find  the  greatest  diversity  prevail ;  and  if,  instead  of  water  we  a 
any  other  as  a  standard,  the  8ame  incompatibility  will  be  found.     Thus  if  the  flint  Bh-.No.18,  be  compai 
with  oil  of  turpentine,  we  find  for  the  values  of  the  series  of  fractions  in  question,  1.868,  1.844,  1. 
1.861,   1.899,  which  first  diminish  to  a  minimum  and  then  increase  again,  &c. 
441  It  follows  from  this,  that  the  proportion  which  the  several  coloured  spaces  (or  the  interval 

&c.)  bear  to  each  other  in  spectra  formed  by  different  media,  is  not  the  same  in  all       Ihus  taking  the 
ray  E  for  the  middle  colour,  ano*  calling  all  that  part  of  the  spectrum  which  lies  on  the  red  side 


LIGHT. 


417 


Light,      and  all  on  the  other  side  the  blue  portions,  the  ratio  of  the  spaces  occupied  by  the  red  and  blue  in  any  spectrum 

will  be  represented  by  the  fraction  -^ — 

fj.  (E)  —  ^  (B) 

foregoing  table  are  set  down  in  the  following  list : 


Now  the  values  of  this  in  the  several  media  of  the 


Flint,  No.  23 

2.0922 

Crown,  M  

1.9484 

Flint,  No.  30  

2.0830 

Crown,  No.  9   .... 

1.8905 

Flint,  No.  3  

2.0689 

Crown,  No.  13.  ... 

1.8855 

Flint,  No.  13  

2  0342 

Solution  of  potash. 

1.7884 

Oil  of  turpentine   .  . 

1.9754 

Water  

1.6936 

Part  IL 


Incommen- 
surability of 
tbe  coloured 
spaces  in 
.pectra  of 
different 
media. 


Here  we  see  that  the  same  coloured  spaces  which  in  the  flint  No.  23  are  in  the  ratio  of  21  :  10,  in  the  water 
spectrum  are  only  in  the  ratio  of  17  :  10  (nearly,)  so  that  the  blue  portion  of  the  spectrum  is  considerably  more 
extended  in  proportion  to  the  red  in  the  flint  glass  than  in  the  water  spectrum. 

From  this  it  follows,  that  if  two  prisms  be  formed  of  different  media  (such  as  flint  glass  and  water)  of  such      442. 
refracting  angles  as  to  give  spectra  of  equal  total  lengths,  and  these  be  made  to  refract  in  opposition  to  each  Secondary 
other,  although  the  red  and  violet  rays  will,  of  course,  be  united  in  the  emergent  beam,  yet  the  intermediate  8Pectra- 
rays  will  still  be  somewhat  dispersed,  the  water  prism  refracting  the  green,  or  middle  rays  more  than  in  pro- 
portion to  the  extremes ;  consequently,  if  a  white  luminous  line  be  the  object  examined  through  such  a  combi- 
nation, instead  of  being  seen  after  refraction  colourless,  it  will  form  a  coloured   spectrum  of  small  breadth 
compared  with  what  either  prism  separately  would  form,  and  having  one  side  of  a  purple  and  the  other  of  a 
green  tint.     Any  dark  object  viewed  against  the  sky  (as  a  window  bar)  will  be  seen  fringed  with  purple  and 
green  borders,  the  green  lying  on  the  same  side  of  the  bar  with  the  vertex  of  the  flint  prism ;  because  in  such 
a  combination,  green  must  be  considered  as  the  most,  and  purple  as  the  least,  refrangible  tint ;  and  the  flint 
prism,  of  necessity,  having  the  least  refraction  in  this  case,  the  most  refrangible  fringe  will  lie  towards  its  vertex, 
that  being  the  least  refracted  side  of  the  bar ;  for  the  same  reason  that,  when  seen  through  a  single  prism,  a 
dark  object  on  a  white  ground  appears  fringed  with  blue  on  its  least  refracted  edge.   (Art.  429.) 

This  result  accords  perfectly  with  observation.  Clairaut,  and,  after  him,  Boscovieh,  Dr.  Blair,  and  Dr  443. 
Brewster,  have  severally  drawn  the  attention  of  opticians  to  these  coloured  fringes,  or,  as  they  may  be  termed, 
secondary  spectra,  and  demonstrated  their  existence  in  the  most  satisfactory  manner.  Dr.  Brewster,  in  parti- 
cular, has  entered  into  a  very  extensive  and  highly  valuable  series  of  experiments,  described  in  his  Treatise  on 
new  philosophical  instruments,  and  in  his  paper  on  the  subject  in  the  Edinburgh  Transactions ;  from  which  it 
follows,  that  when  a  compound  prism,  consisting  of  any  of  the  media  in  the  following  list  refracting  in  oppo- 
sition to  each  other,  unites  the  red  and  violet  rays,  the  green  will  be  deviated  from  their  united  course  by 
the  combination,  in  the  direction  of  the  refraction  of  that  medium  which  stands  before  the  other  in  order : 


1.  SULPHURIC  ACID. 
2.  Phosphoric  acid. 
3.  Sulphurous  acid. 
4.  Phosphorous  acid. 
5.  Super-sulphuretted  hydrogen. 
6.  WATER. 
7.  Ice. 
8.  White  of  egg. 
9.  Rock  crystal. 
10.  Nitric  acid. 
11.  Prussic  acid. 
12.  Muriatic  acid. 
13.  Nitrous  acid. 
14.  Acetic  acid. 
15.  Malic  acid. 
16.  Citric  acid. 
17.  Fluor  spar. 
18.  Topaz,  (blue.; 
19.  Beryl. 
20.  Selenite. 
21.  Leucite. 
22.  Tourmaline. 
23.  Borax. 
24.  Borax,  (glass  of.) 
25.  Ether. 
26.  Alcohol. 
27.  Gum  Arabic. 
28.  CROWN  GLASS. 
29.  Oil  of  almonds. 
30  Tartrate  of  potash  and  soda. 

31.  Gum  juniper. 
32.  Rock  salt. 
33.  Calcareous  spar. 
34.  Oil  of  ambergris. 

61.  Oil  of  nutmegs. 

64.  Amber. 
65.  Oil  of  spearmint. 

39.  Zircon. 
40.  FLINT  GLASS. 
41.  Oil  of  rhodium. 
42.  —     -  rosemary. 

71.  Canada  balsam. 
72.  Oil  of  lavender. 
73.  Muriate  of  antimony. 
74.  Oil  of  cloves. 

44.  Balsam  of  capivi. 
45.  Nut  oil. 
46.  Oil  of  savine. 

76.  Red-coloured  glass. 
77.  Orange-coloured  glass. 
78.  Opal-coloured  glass. 
79.  Acetate  of  lead,  (melted.) 
80.  Oil  of  amber. 

49.  Nitrate  of  potash. 
50.  Diamond. 
51.  Resin. 
52.  Gum  copal. 
53.  Castor  oil. 
54.  Oil  of  chamomyl*. 

83.  anise  seeds. 
84.  Essential  oil  of  bitter  almonds. 
85.  Carbonate  of  lead. 
86.  Balsam  of  Tolu. 
87.  Sulphuret  of  carbon. 
86.  Sulphur. 
89   Oil  of  cassia. 

56.  wormwood. 

59.  peppermint. 

3  i 


Dr.  Brew. 
ster's  table 
of  media 
according 
to  action  on 
green  light. 


418 


LIGHT. 


Light. 

444. 
445. 

Achromatic 
refraction. 
446. 

Dispersive 

powers  of 

higher 

orders. 

Tertiary 

spectra. 

447. 

Computa- 
tion of  their 
coefficients. 


It  is  evident  from  this  table,  that  (generally  speaking)  the  more  refractive  a  medium  is,  the  greater  is  the 
extent  of  the  blue  portion  of  its  spectrum  compared  with  the  red. 

If  two  prisms  of  the  proper  refracting  angles,  composed  of  media  not  very  remote  from  each  other  in 
this  list,  be  made  to  oppose  each  other,  the  secondary  spectrum  will  be  small,  and  the  refraction  almost  perfectly 
colourless.  Such  a  combination  is  said  to  be  achromatic,  (a-\(iLifi.a.) 

The  existence  of  the  secondary  spectrum,  while  it  renders  the  attainment  of  perfect  achromaticity  impossible, 
by  the  use  of  two  media  only,  shows,  also,  that  in  a  theoretical  point  of  view  we  are  not  entitled  to  neglect  the 
coefficients  6,  c,  &c.  of  the  equation  (d),  Art.  439.  The  law  of  nature  probably  requires  the  series  to  be  continued 
to  infinity  ;  and  if,  by  way  of  uniting  three  rays,  we  employ  prisms  of  three  media,  tertiary  spectra,  and  after 
them  still  others  in  succession,  would  doubtless  be  found  to  arise.  These,  however,  will  be  small  in  comparison 
of  each  other. 

The  table  (Art.  437)  gives  us  the  means  of   computing  the   coefficients  on  which  they  depend  for   the 


particular  media  there   stated.     If    we   put 


=  P,  and 


=  p,   and   suppose  P,  P',  P//, 


P,  p',  p'1,  &c.  to  be  the  values  of  P  and  p  corresponding  to  the  several  values  of  /*  and  x  set  down  in  the  table, 
we  shall  have,  for  determining  a,  b,  c,  &c.  in  any  one  of  those  media,  the  equations 


-f  cp 


&c.     F'  = 


=  ap 


cp 


"3 


&c. 


and  as  many  such  equations    must  be  used  as  there  are  coefficients  to   determine.     Confining   ourselves    at 
present  to  two,  we  find  P  =  a  p  +  bp3;     P'  =  a  p'  +  b  p'*,  whence 


PP  (P'  - 


b=  - 


Pp'-  P'p 
p  p'  (p'  —  p) 


and,  since  it  is  desirable  to  select  rays  as  far  removed  from  each  other  in  the  spectrum  as  possible,  we  shall 
take  /i0  and  X0  from  the  column  n  (B)  ;  and  determine  P  and  p  by  the  values  in  the  column  /»  (E),  and  I",  p' 
by  those  under  ft  (H).  The  results  will  be  as  follows  : 


Refracting  media. 

Dispersive  powers 
of  the  first  order, 
that  of  water  being 
1.000. 

Dispersive  powers 
of  the  second  order, 
that  of  water  being 
0.000. 

Flint  glass,  No.  13 
Crown  glass.  No.  9  .  . 
Water  

a=  +  1.42580 
0.88419 
1.00000 

6  =  +  7.57705 
2.34915 
0.00000 

Solution  of  potash    .  . 
Oil  of  turpentine  .... 
Flint  glass,  No.  3..  .. 
Flint  glass,  No.  30    .  . 
Crown  glass,  No.  13.  . 
Crown  glass,  letter  M  . 
Flint  glass,  No.  23   .  . 

0.99626 
1.06149 
1.29013 
1.37026 
0.87374 
0.90131 
1.37578 

1.13262 
4.58639 
7.63048 
8.44095 
2.49199 
3.49000 
8.66904 

Problem.  To  determine  the  analytical  relation  which  must  hold  good  in  order  that  two  prisms  may  form  an 


448. 

achromatic  combination  ;    that  is,  may  refract  a  white  ray  without  separating  the  extreme  colours, 
of  achro-         Resuming  the  equations  and  notation  of  Art.  215,  since  the  prisms  are  placed  in  vacuo,  we  have  to  substi- 

maticity.  .  \  \ 

tute  p,  —f,  ft'  and    — j-  for  ft,  ft',  ft",  ft'",  in  those  equations  respectively,  and  we  shall  have 


/t .  sin  p  =  sin  a      j  /»'  •  sin  a'"  =  sin  p'") 

<*•'  =  I  +  P  f  (1)  ;  p"  =  —  I"  +  «'"      /  (2) ; 

sin  p'  =  ft  .  sin  a'  J  sin  a"  =  /»' .  sin  />"    J 


and 


D  =  *  +  I  +  I'  +  I"  -  p"'. 


Now,  since  by  hypothesis  the  incident  and  emergent  rays  are  both  colourless,  we  must  have  £  a  =  0,  and 
B  D  =  0,  that  is  $  p"'  =  0,  the  sign  8  being  supposed  to  refer  to  the  variation  of  the  place  of  the  ray  in  the 
spectrum.  Hence  the  two  systems  of  equations  (1)  and  (2)  are  exactly  similar,  in  their  form ;  the  former  a» 
relates  to  p,  a,,  a!,  p',  and  the  latter  as  to  a'",  p"1,  p'',  a.''.  Now,  the  first  system  gives 

8  ft .  sin  p  +  ft  5  p  ,  cos  p  =  0  ;         £  a'  =  &  p;        &  p'  cos  p'  =  $  ft .  sin  o'  +  /» 5  a',  cos  tf ; 

whence,  by  elimination  and  reduction,  we  find 

sin  I  ,  . 

V= /  */•;  (?) 

COS  p  .  COS  fl 


LIGHT. 

Light,      and,  consequently,  by  reason  of  the  analogy  of  the  two  systems  of  equations  pointed  out  above, 

«.'=  -          Si"  l"         Kft'  (/) 

But,  since  a1'  =  !'  +  /,  we  have  S  p'  =  S  a",  so  that  we  finally  get 

cos  p  .  cos  p'  sin  I        S  fi  . 

cos  a'" .  cos  a"  sin  1"    '    c  /»' 

The  property  expressed  by  this  equation  may  be  thus  stated.  Conceive  the  ray  to  pass  both  ways  outwards 
from  a  point  in  its  course  between  the  two  prisms;  then,  in  order  that  the  combination  maybe  achromatic, 
the  products  of  the  cosijies  of  its  incidences  on  the  surfaces  of  each  prism  must  be  to  each  other  in  the  ratio  com- 
pounded of  that  of  the  sines  of  their  respective  refracting  angles,  and  the  differences  of  their  refractive  indices 
for  red  and  violet  rays ;  besides  which,  they  must  refract  in  opposition  to  each  other,  or  I  and  I"  their  refracting 
angles  must  have  opposite  signs. 

The  combination  of  this  equation  with  the  system  of  equations  above  stated,  expressing  the  conditions  of       449. 
refraction  by  the  prism,  and  their  relative  position  with  regard  to  each  other  (which  is  included  in  the  equation  Progress  of 
a"  =  I'  +  p')  suffice,  algebraically  speaking,  to  resolve  every  problem  which  can   occur,  of  this  kind ;  but  the  JJ'S Pe™°"e 
final  equations  are  for  the  most  part  too   involved  to  allow  of  direct  solution.     Nevertheless,  the  results  we  pnsm  traced 
have  arrived  at  will  furnish  occasion  for  remarks  of  moment ;  and,  first,  since  pf  is  the  angle  of  refraction  from 
the  second  surface  of  the  first  prism,  3  p'  is  the  angular  breadth  of  the  spectrum  produced  by  it ;  this  is,  there- 
fore, proportional,  cizteris  paribus,  to  the  product  of  the  secants  of  the  angles  of  refraction  at  its  two  surfaces. 
Let  us  trace  the  progress  of  the  variation  of  this,   as  the  incident  ray  changes  its  inclination  to  the  first 
surface,  beginning  with  the  case  when  it  just  grazes  the  surface  from  the  back  towards  the  edge.     In  this  case 

a  =  90°,  sin  p  =  — ,  consequently  p,  and  therefore  I  +  p  or  a!,  and  therefore  /  are  all  finite,  and  at  their 

maximum.     Hence  cos  p .  cos  p'  is  finite,  and  at  its  minimum  ;  and  therefore  1  5',  or  the  breadth  of  the  spectrum, 

is  also  finite,  but  a  maximum.     As  the  incident  ray  becomes  more  inclined  to  the  surface  p,  and  therefore  a1  and 

j'  diminish,   and  the  denominator  of    1  p'  increases,    so    that    the  breadth   of   the  spectrum  diminishes,   and 

reaches  a  minimum  when  cos  p  .  cos  pf  attains  its  maximum ;  that  is,  when  d  p .  tan  />  +  dp' .  tan  p'  =  0.     Now  Position  of 

this  equation,  substituting  and  reducing  gives,  for  determining  the  value  of  p,  and  therefore  of  a,  or  the  inci-  least  disper- 

dence  when  the  spectrum  is  a  minimum,  s'?n  <jeter- 

mmed. 

p.* .  sin  (I  -f-  p)  .  cos  (I  +  2  p)  +  sin  p  =  0.  (A) 

Hence  we  see  that  the  position  which  gives  a  minimum  of  breadth  to  the  spectrum  is  very  different  from  that 
which  gives  a  minimum  of  deviation,  being  given  by  the  above  equation,  which  is  easily  resolved  by  a  table  of 

logarithms,  and  which   shows  at  once  that  p  must  be  greater  than  45°  —  — — . 

m 

After  attaining  the  position  so  determined,  the  breadth  of  the  spectrum  again  increases,  and  continues  to  do 
so  till  the  rays  can  be  no  longer  transmitted  through  the  prism.  At  this  limit  the  emergent  ray  just  grazes  the 
posterior  face  of  the  prism  from  its  thinner  towards  its  thicker  part  g'  =  90°,  cos  p'  =  0.  At  this  limit,  therefore, 
the  dispersion  becomes  infinite.  All  these  stages  are  easily  traced  by  turning  a  prism  round  its  edge  between 
the  eye  and  a  candle  ;  or,  better,  between  the  eye  and  the  narrow  slit  between  two  nearly  closed  window-shutters. 

Hence,  as  the  incident  ray  varies  from  the  position  S  E  (fig.  105)  to   S'  E,  and  therefore  the  refracted  from      * 
F  G  to  F'  G',  the  breadth  of  the  spectrum  commences  at  a  maximum,  but  finite  value,  diminishes  to  a  minimum  Of'sg™t°um 
and  then  increases  to  infinity.     The  distribution  of  the  colours  in  the  spectrum,  or  the  breadths  of  the  several  at  extreme 
coloured  spaces  in  any  state  of  the  data,  will  moreover  differ  according  to  the  values  of  p,  p1  and  sin  I;  for  the  incidences, 
equation  (e),  by  assigning  to  S  11  the  values  which  correspond  in  succession  to  the  intervals  between  red  and  Fig.  105. 
orange,  orange  and  yellow,  yellow  and  green,  &c.  will  give  the  corresponding  values  of  2  p',  or  the  apparent 
breadths  of  these  spaces.     Now  the  denominator  cos  p .  cos  p'  is  an  implicit  function  of  ft,  and  therefore  varies 
when  the  initial  ray  is  taken  in  different  parts  of  the  spectrum.     The  variation  is  trifling  when  the  angles  p,  p' 
are  considerable  ;  but  near  the  limit,  when  the  ray  can  barely  be  transmitted,  it  becomes  very  great,  the  spectrum 
is  violently  distorted,  and  the  violet  extremity  greatly  lengthened  in  proportion  to  the  red.     The  effect  is  the 
same  as  if  the  nature  of   the  medium  changed  and  descended  lower  in  the  order  of  substances  in  the  table 
Art.  443. 

From  what  has  just  been  said,  we  see  the  possibility  of  achromatising  any  prism,  however  large  its  refracting  451. 
angle,  by  any  other  of  the  same  medium,  however  small  may  be  its  angle ;  for  since,  by  properly  presenting  a  Achromatic 
prism  to  the  incident  ray,  its  dispersion  may  be  increased  to  infinity ;  if  made  to  refract  in  opposition  to  another  combina- 
whose  dispersion  has  any  magnitude,  however  great,  it  may  be  made  to  counteract,  or  even  overcome  it.  Thus  medium' 
in  fig.  106  the  dispersion  of  the  second  prism  a,  of  small  refracting  angle,  being  increased  by  the  effect  of  its  Fig.  106. 
inclined  position,  is  rendered  equal  and  opposite  to  that  of  the  prism  A,  whose  refracting  angle  is  large. 

When  the  prisms  differ  greatly  in  their  angles,  however,  the  second  must  be  very  much  inclined,  so  as  to      453. 
bring  it  near  to  the  limit  of  transmission.     In  this  case,  its  law  of  dispersion,  as  just  shown,  will   be  greatly  Subordinate 
disturbed,  and  rendered  totally  different  from  what  obtains  in  the  other  prisms  ;   so  that  perfect  achromaticity  spectra. 

3i2 


420  LIGHT. 

Light.      cannot  be  produced  ;  but  when  the  extreme  red  and  violet  rays  are  united,  the  green  will  be  too  little  refracted  by     Part  H. 

^—  "-v—  •  the  second  prism,  and  a  purple   and  green  spectrum  will  arise,  as  in  the  case  of  prisms  of  different  media.     To  N—  -v^« 

this  spectrum  Dr.  Brewster  (who  was  the  first  to  place  it  in  evidence)  has  given  the  name  of  a  tertiary  spectrum  ; 

but  it  appears  to  us,  that  this  term  had  better  be  reserved  for  the  spectra  mentioned  in  Art.  446,  and  those  now 

in  question  may  be  called  subordinate  spectra. 

If  a  small  rectangular  object  be  viewed  through  such  a  combination  as  above  described,  in  which  the  prism  A 
is  placed  in  its  position  of  minimum  deviation,  and  achromatised  by  a  second  a,  whose  angle  is  less  than  that 
of  A,  but  not  so  small  as  to  introduce  this  cause  of  colour,  it  will  appear  distorted  in  figure  ;  for  the  sides 
parallel  to  the  edges  of  the  prisms  will  undergo  no  change  in  their  apparent  length,  while  the  breadth  of  the 
rectangle  will  appear  magnified.  For  the  first  prism,  by  reason  of  its  position,  does  not  alter  the  angular 
dimensions  of  objects  seen  through  it  ;  but  the  second  changes  their  angular  breadth  in  the  ratio  of  d  p1"  to 

d  a",  that  is  (by  differentiation)  in  the  ratio  of  -  '•  --  -  to  unity,  a  ratio  which   increases  rapidly  as  the 

COS  f  .  COS  p' 

inclination  of  the  prism  increases,  and  p'  approaches  a  right  angle. 

453.  M.  Amici  has  taken  advantage  of  these  properties  to  construct  a  species  of  achromatic  telescope,  which,  at 

Amici's  first  sight,  appears  very  paradoxical,  being  composed  merely  of  four  prisms  of  the  same  kind  of  glass,  with 
plane  surfaces.  To  understand  its  construction,  conceive  a  small  square  object  op  placed  with  the  side  o  parallel 
to  the  refracting  edges  of  a  pair  of  prisms  so  adjusted,  and  perpendicular  to  their  principal  sections,  i.  e.  to  the 
plane  of  the  paper.  Then,  after  refraction  through  both,  it  will  be  seen  by  an  eye  at  E,  as  a  real  object  o'  p', 
having  its  length  o  unaltered,  but  magnified  in  breadth.  Now,  if  we  add  a  second  pair  of  prisms,  similar  to  the 
first,  and  similarly  disposed  with  respect  to  each  other,  so  as  to  form  a  second  achromatic  combination,  but 
having  the  plane  of  their  principal  sections  at  right  angles  to  the  former,  producing  a  refraction  perpendicular  to 
the  plane  of  the  paper,  or  parallel  to  the  length  of  the  distorted  square,  this  will  be  in  like  manner  seen  as  a  real 
and  colourless  object,  but  again  distorted,  its  side  o1  p'  remaining  unaltered,  but  o'  being  magnified.  Thus,  by 
the  effect  of  the  first  distortion,  the  breadth  of  the  square  is  magnified,  and,  by  that  of  the  second,  its  length, 
and  in  the  same  ratio  ;  and  therefore  the  final  result  will  be  an  image  undistorted,  achromatic,  and  magnified. 
The  writer  of  these  pages  had  the  pleasure  of  witnessing  the  very  good  performance  of  one  of  these  singular 
telescopes,  magnifying  about  four  times  in  the  hands  of  its  inventor,  at  Modena,  in  1826.  It  is  evident,  that,  by 
superposing  several  such  telescopes  on  each  other,  the  magnifying  power  may  be  increased  in  geometrical  pro- 
gression. It  is  equally  clear,  that,  by  using  prisms  of  two  different  media  to  form  the  several  binary  combina- 
tions, the  tubordinate  spectra  may  be  made  to  counteract  the  secondary  spectra,  arising  from  the  difference  in 
the  scales  of  dispersion  in  the  two  media  ;  and  thus  an  achromaticity,  almost  mathematically  perfect,  might  be 
obtained.  It  is  worthy  of  consideration,  whether,  for  the  purpose  of  viewing  very  bright  objects,  as  the  sun, 
for  instance,  this  species  of  telescope  might  not  prove  of  considerable  service.  It  would  have  the  advantage  of 
being  its  own  darkening  glass,  of  not  bringing  the  rays  to  a  focus,  and  therefore  of  requiring  no  extraordinary 
care  in  the  figuring  of  the  surfaces  ;  and,  in  short,  of  being  exempt  from  all  those  inconveniencies  which  oppose 
the  perfection  of  telescopes  of  the  usual  constructions,  as  applied  to  this  particular  object. 

.  t  j  Proposition.  To  find  the  conditions  of  achromaticity  when  several  prisms  of  different  media  refract  a  ray  of 

Conditions    white  light,  supposing  all  their  refracting  angles  very  small,  and  the  ray  to  pass  nearly  at  right  angles  to  the 
of  achrmna-  principal  section  of  each. 

ticity  for          The  refracting  angles  being  A,  A',  A",  &c.,  and  the  refractive  indices  ft,  /»',  &c.,  the  several  partial  deviations 
•eyeral         wij|  be  D  =  0*  —  1)  A  ;   D'  =(/»'  —  1)  A',  &c.;  and  their  sum,  or  the  total  deviation,  will  be  (/*  —  1)  A  + 
•man  'angles  (/•'  —  1)  A'  +  (/'  —  1)  A"  +  &c.     In  order  that  the  emergent  ray  may  be  colourless,  this  must  be  the  same 
for  rays  of  all  colours;  and  its  variation,  when  /*,  /,  &c.  are  made  to  vary,  must  vanish,  or 


+  &c.  =0. 
Now,  by  equation  (rf)  of  Art.  439,  we  have  3  p,  (or,  in  the  notation  of  that  article,  ^  —  /»0) 


Therefore  the  above  equation  gives,  when  arranged  according  to  powers  of  3  x, 


0  =  { 
(. 


A  0*0  -  1)  a  +  A'  (jl0  -  1)  a'  +  A"  (jJ'0  -  1)  a"  +  &c. 


+  I A  (ft,  -  1)  6  +  A'  (X0  -  1)  6'  +  A"  (/0  -  1)   6"  +  &c. : 

+   &c. 

taking  a',  V,  Sue.  to  represent  the  dispersive  powers  of  the  various  orders  for  the  second  prism,  a",  6",  &c.  for 
the  third,  and  so  on.  Hence,  in  order  that  this  may  vanish  for  all  the  rays  in  the  spectrum,  we  must  have 
(putting,  for  brevity,  /*  for  p.a,  /i!  for  /0,  &c.) 


LIGHT.  421 

light.  0«-1).  A«+  (/-I)  AV+  (X'-  1)  A"a"+  &c.  =  <n  P'rtll. 


+  &c.  =  0  "I 

G*  -  1)  .  A  6  +  (/  -  1)  A'  6'  +  (/'  -  1)  A"  6*  +  &c.  =  0  I 
G*  -  1)  .  Ac  +  0*'  -  1)  A'  c1  +  C""  -  n  A"  c"  +  &c.  =  0  f 

&C.  &C.  &C.  &C. 


:ia>      w 


and  so  on.  Generally  speaking,  the  number  of  these  equations  being  infinite,  no  finite  number  of  prisms  can 
satisfy  them  all ;  but  if  we  attempt  only  to  unite  as  many  rays  in  the  spectrum  as  there  are  prisms,  which  is  the 
greatest  approach  to  achromaticity  we  can  attain,  we  have  as  many  equations  as  unknown  quantities,  minus  one, 
and  the  ratios  of  the  angles  to  each  other  become  known.  Thus,  to  unite  two  rays  two  media  suffice,  and  we 
can  only  take  into  consideration  the  first  order  of  dispersions,  which  give 

0.- 1)  An +(^-1)  A' of  =0;   -£-=  -y^4  •  -£-•  O') 

To  unite  three  rays  we  have 

C«  -  1)  A  a  +  (/  -  1)  A'  «'  +  0*"  -  1)  A"  a"  =  0 

GU  -  1)  A6  +  O'  -  1)  A'  6'  +  GI"  -  1)  A"  6"  =  0 
whence  by  elimination 

JV  p-  1         abf'-ba'       A"  »-l          a  b'  -  b  a! 

~A~  :  X-l    '    a'b"-ba"      A  /•"-!'    a"6'-6'V; 

and  so  on  for  any  number. 

In  the  case  of  two  media,  if  the  quantities  6,  c,  &c.  be  not  known,  the  dispersive  powers  of  the  first  order,      453. 
a,  a',  should  be  determined,  not  by  comparison  of  the  extreme  red  and  violet  rays,  which  are  too  little  luminous  Case  of  two 
to  render  their  strict  union  a  matter  of  importance  ;    we  should  rather  endeavour  to  unite  those  rays  which  are  me"'a- 
at  once  powerfully  illuminating,  and  differing  much  in  colour,  such  as  the  rays  D  and  F.     The  exact  union  of  Best  rays 
these  will  insure  the  approximate  union  of  all  the  rest  better,  on  the  whole,  than  if  we  aimed  at  uniting  the  to  unite. 
extremes  of  the  spectrum,  and  a  far  greater  concentration  of  light  will  be  produced.     This  should  be  carefully 
borne  in  mind  in  all  experiments  on  the  dispersions  of  glass  to  be  used  in  the  construction  of  telescopes. 

If  we  would  produce  the   greatest  possible  achromaticity  by  three  prisms,  the  rays  to  be  selected  for  deter-      454. 
mining  the  values  of  a,  b,  a',  b1,  should  be  C,  E,  and  G ;  or,  which  would,  perhaps,  be  still  better,  C,  P,  and  a  Best  rays  to 
ray  half  way  between  D  and  E  ;  but  the  want  of  a  sufficiently  well  marked  line  in  that  part  of  the  spectrum  u" 
throws  some  slight  difficulty  in  the  way  of  this  latter  combination,  when  solar  light  is  used,  and  would  oblige  us  ^dia. * 
to  have  recourse  to  some  other  method  of  measurement,  of  which  a  variety  might  be  suggested. 

In  the  case  of  three  media,  if  the  numerators  and  denominators  of  the  expressions  (It)  vanish,  or  nearly  so,  the      455. 
solutions  become  illusory,  or  at  least  inapplicable  in  practice.     This  happens  whenever  either  of  the  fractions  Cases  in 

which  the 

a      a       a'  b         b  V  ,    formula 

~T>  ~~ii>  — ir  becomes  equal  to  either  of  the  corresponding  fractions  — -T-,  -TTT,  or  — — -.      Hence,   to  obtain  become  in- 
a       a         a  b'       b'  applicable 

practicable  combinations,  it  is  necessary  to  employ  media  which  differ  as  much  as  possible  in  their  scales  of  dis- 
persive powers,  i.  e.  in  which  the  coloured  spaces  differ  as  far  as  possible  from  proportionality ;  such,  for 
instance,  as  flint  glass,  crown  glass,  and  muriatic  acid ;  or,  still  better,  oil  of  cassia,  crown  glass,  and  sulphuric 
acid,  &c. 

§  II.  Of  the  Achromatic  Telescope. 

In  the  refracting  telescopes  described  in  Art.  380,  &c.  the  different  refrangibility  of  the  differently  coloured  rays      456. 
presents  an  obstacle  to  the  extension  of  their  power  beyond  very  moderate  limits.     The  focus  of  a  lens  being  Chromatic 
shorter  as  the  refractive  index  is  greater,  it  follows,  that  one  and  the  same  lens  refracts  violet  rays  to  a  focus  a°e"'ati°n 
nearer  to  its  surface  than  red.     This  is  easily  seen  by  exposing  a  lens  to  the  sun's  rays,  and  receiving  the  con-  e" 
verging  cone  of  rays  on  a  paper  placed  successively  at  different  distances  behind  it.     At  any  distance  nearer  to 
the  lens  than  its  focus  for  mean  rays,  the  circle  on  the  paper  will  have  a  red  border,  but  beyond  it  a  blue  one ; 
for  the  cone  of  red  rays  whose  base  is  the  lens,  envelopes  that  of  violet  within  the  focus,  its  vertex  lying  beyond 
the  other,  but  is  enveloped  by  it  without,  for  the  converse  reason.     Hence,  if  the  paper  be  held  in  the  focus  for 
mean  rays,  or  between  the  vertices  of  the  red  and  violet  cones,  these  will  then  form  a  distinct  image,  being  col- 
lected in  a  point :  but  the  extreme,  and  all  the  other  intermediate  rays,  will  be  diffused  over  circles  of  a  sensible 
magnitude,  and  form  coloured  borders,  rendering  the  image  indistinct  and  hazy.     This  deviation  of  the  several 
coloured  rays  from  one  focus  is  called  the  "  chromatic  aberration." 

The  diameter  of  the  least  circle  within  which  all  the  coloured  rays  are  concentrated  by  a  lens  supposed  free      457. 
from  spherical  aberration  is  easily  found.     Thus,  in  fig.  107,  if  v  be  the  focus  for  violet,  and  r  for  red  rays,  n  mo  Least  circle 

of  chromatic 


mv  mr 


aberration. 


will  be  the  diameter  of  this  circle.    Now,  by  similar  triangles,  n  o  r:  A  B  .  —= — ,  and  also  n  o  =*  A  B  .  ;  i. rr.a'. 


422  LIGHT 

Light.  mv 

therefore  equating  these    —  —  s 

" 


Cr  Cr  re 

sequently  m  r  =  r  o  .  —  -  —  —  =  r  p  .  —  —  q  --  =  -  •  very  nearly,  since  the  dispersion  is  small  in 

V>  T  —|~   t/  U  "-  \_/  7*  ~~   7"  17  >& 

comparison  with  the  whole  refraction.     Therefore  n  o  =  —  -  —  .  -—  —  .      Now,    /  being-    the   reciprocal   focal 

*&  O  T 

distance  (=  L  +  D  =  (/»  -  1)  (R1  -  R")  +  D)  we  have  r  »  =  -  8  ~  =  *  ^  (R  ~  R  ) 


J          J  J 

u  L  1 

and  C  r  =    —  ,  supposing  /t  to  represent  the  index  of  refraction  for  extreme  red  rays. 


,-!/•>/  L 

Hence  we  get  diameter  of  least  circle  of  chromatic  aberration  =  semi-aperture  X  —.- 


f    '/—I 
=  semi-aperture  x  dispersive  index  x   — ^-  ; 

and  for  parallel  rays,  when  L  =  f,  simply  semi-aperture  X  dispersive  index. 

458.  Carol.  Hence  the  circle  of  least  colour  has  the  same  absolute  linear  magnitude  whatever  be  the  focal  length  of 
Use  of  very  the  lens,  provided  the  aperture  be  the  same.     Now,  in  the  telescope,  the  magnifying  power,  or  the  absolute  linear 
scope's'6"      magnitude  of  the  image  viewed  by  a  given  eye-glass,  increases  in  the  ratio  of  the  focal  length  of  the  object-glass, 

(382.)  Therefore,  by  increasing  the  focal  length  of  an  object-glass  without  increasing  its  aperture,  the  breadth 
of  the  coloured  border  round  the  image  of  any  object  diminishes  in  proportion  to  the  image  itself,  and  thus  the 
confusion  of  vision  is  diminished,  and  the  telescope  will  possess  a  proportionally  higher  magnifying  power.  In 
consequence  of  this  property,  before  the  invention  of  the  achromatic  telescope,  astronomers  were  in  the  habit  of 
using  refracting  telescopes  of  enormous  length,  even  so  far  as  100  or  150  feet;  and  Huyghens,  in  particular, 
distinguished  himself  by  the  magnitude  and  excellent  workmanship  of  his  glasses,  and  by  the  important  astrono- 
mical discoveries  made  with  them. 

459.  The  achromatic  object-glass,  however,  by  enabling  us  to  reduce  the  length  of  the  telescope  within  more  reason- 
Principle  of  able  bounds,  has  rendered  it  a  vastly  more  manageable  and  useful  instrument.     To  conceive  its  principle,  we  have 
•Ae  achro-    only  to  recur  to  what  has  already  been  said  in  Art.  451 — 454,  respecting  achromatic  prisms.     A  lens  is  nothing 
""j1'10           more  than  a  system  of  infinitely  small  prisms  arranged  in  circular  zones  round  a  centre,  with  refracting  angles 

increasing  as  their  distance  from  the  centre  increases,  so  as  to  refract  all  the  rays  to  one  point ;  and  if  we  can 
achromatise  each  elementary  prism,  the  whole  system  is  achromatic.  The  equations  (i)  apply  at  once  to  this 
view  of  the  structure  of  a  lens.  For,  suppose  R',  R"  to  be  the  curvatures  of  the  two  surfaces  of  the  first  lens, 
£«'  its  power,  and  f!  its  refractive  index,  then,  for  a  given  aperture,  or  at  a  given  distance  from  the  centre, 
R'  —  R",  the  difference  of  the  curvatures,  expresses  the  angle  made  by  tangents  to  the  surfaces,  or  the  refracting 
angle  of  the  elementary  prism  ;  or  R'  —  R"  =  A1;  and  similarly  for  the  other  lenses,  A''  =  R'"  —  R'T,  and  so 
on,  so  that  the  equations  become 

(ft-  1)  (R1  -  R")  .  a'  +  (X-l)  (R'"--  Rlv)  a"  +  &c.  =  0  &c. ; 

General         that  is  simply 

e?ua'ions  L' .  a'  +  L"  .  a"  +  L'" .  a'"  +  &c.  =  0  ' 

of  achroma- 

u<%-  L' .  V  +  L"  .  b"  +  L'" .  b'"  +  &c.  =  0  ,  .  j 

L' .  d  +  L"  .  c"  +  L'" .  d"  +  &c.  =  0 
&c. 

460.  These  equations  afford  all  the  relations  necessary  to  insure  achromaticity  ;  and  when  satisfied,  since  they  do 
Otherwise     not  contain  D,  they  show  that  an  object-glass  which  is  achromatic  for  any  one  distance  of  the  object  is  so  for 
deduced.       aji  distances.     It  is  evident,  that  the  same  system  of  equations  may  be  obtained  directly  from  the  expression  in 

Art.  265  for  the  joint  power  of  a  system  of  lenses  whose  individual  powers  are  L',  L",  &c.  For  the  condition 
of  achromaticity  gives  J  L  =  0,  that  is 

3  L'  +  J  L"  +  JL"'  +  &c.  =  0. 


But  since  L'  =  (jl  —  1)  (R'  —  R")  &c.  (according  to  the  system  of  notation  there  adopted) 

J  L'  =  (R'  -  R")  V  =  L'  .      ,  ^     .. 

But  in  the  equation  (d)  if  we  put  in  succession  for/u0  the  values  /,  /',  &c.,  for/i  —  /»0  respectively,  V,  J/i",&c., 
and  for  a,  b,  &c.  the  systems  of  coefficients  a',  b',  &c.  ;  a",  b",  &c.  ;  and  suppose  —  --  y-  =  p,  we  shall  have 


— 
—  1 


LIGHT.  423 

and  therefore  Part  ". 

0  =  L'  {  a!  p  +  b'p*  +  &c.  }  +  L"  {  a" p  +  b"  p*  +  &c.  }  +  &c. 

which,  being  made  to  vanish  independently  of  p,  gives  the  very  same  system  of  equations  as  (a.) 

To  satisfy  all  these  equations  at  once  with  any  finite  number  of  lenses  being  impossible,  we  must  rest  content      461. 
with  satisfying  as  many  of  the  most  important  as  the  number  of  lenses  will  permit.     Thus,  if  we  have  two  lenses  Object  glast 
of  different  media,  such  as  flint  and  crown  glass,  for  instance,  one  only  of  them  can  be  satisfied,  and  this  must      J,"0 
of  course  be  the  first,  viz. 

L'a'  +  L"a"=0,        or  ¥-  =  -  -£-  •  (b) 

which  shows  that  the  powers  of  the  lenses  must  oppose  each  other,  and  be  to  each  other  inversely  (and  of  course 
their  focal  lengths  directly)  as  the  dispersive  powers.  In  such  a  combination,  the  values  of  a',  a",  the  dispersive 
powers,  however,  ought  not  to  be  obtained  from  the  relative  refractions  for  the  extreme  red  and  violet  rays  of  the 
spectrum,  (according  to  the  remark  in  Art.  453,)  but  rather  from  the  strongest  and  brightest  rays  whose  colours 
are  in  decided  contrast ;  such,  for  instance,  as  the  rays  C  and  F  in  Fraunhofer's  scale. 

With  three  lenses  of  different  media,  two  of  the  equations  of  achromaticity  can  be  satisfied,  and  the  secondary      462. 
spectrum  corrected,  thus  we  have  Object  glass 


0  =  L'  a1  +  L"  a"  +  L'"  a!"  I    I     L'  «"  *'"  -  6"  «'"  > 

0  =  L'  b'  +  L"  6"  +  L'"  6'"  [    )     L'"  a'b"  -b'a"        ' 

J    (."IT          '    a'"  b"  -  6"V7 

and  in  determining  the  values  of  a',  b',  &c.  the  rays  to  be  employed  should  be  the  brightest  yellow  for  a  middle 
ray,  and  a  pretty  strong  red  and  blue  for  the  extremes.  The  rays  B,  E,  H  are  perhaps  inferior  to  C,  E,  G  for 
this  purpose. 

Hence  in  a  double  object-glass  having  a  positive  focus  the  least  dispersive  lens  must  be  of  a  convex  or  positive,       463. 
and  the  most  so  of  a  negative,  or  concave  character.     The  order  in  which  they  are  placed  is  of  no  consequence, 
as  far  as  achromaticity  is  concerned. 

A  single  lens,  as  we  have  seen,  neither  admits  of  the  destruction  of  the  spherical,  nor  chromatic  aberration,      464. 
(Art.  296  and  457  ;)  but  if  we  combine  two  or  more  lenses  of  different  media,  the  equations  s,  t,  u,  v  of  Art.  Simulta- 
309,  310,  312,  and  313,  combined  with  the  equations  just  derived  (a),  Art.  459,  or  so  many  of  them  as  are  not  nteou*d| 
incompatible,  afford  us  the  means  of  annihilating  both  species  of  aberration  at  once ;  and  what  is  curious,  and  bot),  aberra- 
must  be  regarded  as  singularly  fortunate,  the  relations  afforded  by  the  destruction  of  the  chromatic  aberration,  tions. 
which,  at  first  sight,  would   appear  likely  greatly  to  complicate  the  inquiry,  tend,  on  the  contrary,  remarkably 
to  simplify  it,  being  in  fact  the  very  relations  the  analyst  would  fix  upon  to  limit  his  symbols,  and  give  his  final 
equations  the  greatest  simplicity  their  nature  admits,  if  left  at  his  disposal.     For,  it  will  be  remarked,  that  in  the 
general  equation  for  the  destruction  of  the  spherical  aberration,  A  /  =  0,  or 

0  =  ^-  («'  -  ft1  D'  +  V  D'2)  +  ~  (a'"  +  ft"  D"  +  7"  D"«)  +  &c. ;  (d) 

the  expressions  within  the  parentheses  are  all  of  the  second  degree  when  expressed  in  terms  of  the  curvatures  of 
the  surfaces,  and  of  D'  =  D  the  proximity  of  the  radiant  point  to  the  first  lens ;  and  as  L',  L",  &c.  are  respec- 
tively of  the  first  degree,  in  terms  of  the  curvatures,  the  whole  is,  in  its  general  form,  of  the  third  degree,  and 
the  equation  of  a  cubic  form.  But  the  conditions  of  achromaticity,  which  assign  relations  only  between  L',  L", 
&c.  without  involving  R',  R",  &c.  enable  us  to  eliminate  these  quantities  and  replace  them  in  the  above  equation, 
by  giving  combinations  of  a',  a",  b',  b",  &c.,  so  that  it  becomes  reduced  to  a  quadratic  form,  and  its  treatment 
simplified  accordingly. 

Let  us  proceed  now  to  develope  the  equation  (d),  in  which,  according  to  the  foregoing  remark,  when  the  con-       465. 
ditions  of  achromaticity  are  introduced,  L',  L",  &c.  may  be  regarded  as  given  quantities ;  for,  taking  L  =  L'  +  Determina- 
L"  +  &c.  =  the  power  of  the  compound  lens,  (which  we  may  suppose  given,  or,  if  we  please,  assume  equal  to  tion  of  tne 
unity,)  this,  combined  with  the  equations  (a),  determines  the  values  of  L',  &c.     Thus,  in  the  case  of  two  lenses,  JJJ'J1^  of 

d  L  wL  lenses. 

if  we  put  JT  for  the  ratio  of  the  dispersive  powers,  or  TS  =.  -—j-  we  have  L'  =  ,  L"  =  —  -     —  ;    and 

a'  l  —  «,  1  —  zj 

similarly  for  three  or  more  lenses.  Suppose  then  we  represent  by  /,  r",  r"',  &c.  the  respective  curvatures  of  the 
first,  or  anterior  surfaces  of  the  first,  second,  third,  &c.  lens,  in  order ;  the  first  being  that  on  which  the  rays  first 

T  i  Develope- 

fall.    Then  we  have  L'  =  0*'  -  1)  (R'  -  R")  =  (jj  -  1)  (r1  -  R",)  so  that  R"=  r1 ^ — ;  and  similarly  ment  »f  «"• 

pi  —  1  J  general 

L"  equation. 

R"  =  r"  —  — „_     ,  &c.     We  must  therefore  put  in  the  foregoing  expressions 

T '  L/ 

Ri  —  -/  .         Rtf  —  ~;  _  _  .          pw  —  ,Ji  .         Riv  —  r"  —        &c 

—  r  ,          a.    —  r  —  •    ; —  ,          iv     —  r  ,          s\    —  r  ..          ,  U.L. 


424  LIGHT. 

Ligh^      Hence  by  substitution  of  these  in  the  values  of  o,  ft,  &c.  (Art.  293)  we  get 

a'  =  (2  4-  X)  r'2  —  (2  /+  1)  .     /*       .U  r1  +  n' . 


ft>=  (4  +  4  /,')  /  -  (3  /•'  +  1)  .  -£— t  L' 

y  =  2  +  3  /, 

and  similarly  for  a.",  ft",  </',  &c.  So  that,  substituting  again  these  expressions,  and  putting  for  D"  its  equal 
L'  +  D',  for  D'"  its  equal  L'  +  L"  +  D',  and  so  on,  we  have,  finally,  for  the  general  equation  A  f  =  0,  aa 
follows : 

0  =   j(y  +  l)  LV*  +   (-4-  +  1  )  L"r"s  +  (-^  +  l)  L"V"2  +  &c.J 


-  4  I  (l  +  4- 


+ 


+ 


'  +  L")  L"V"  +  &c.  J 


D.< 


+   {(4-    +  3) 
-  4  {0  +7r)  L'  r 


+  3 


-1 


L"V"  +  &, 


D8 


{  (-7-  +  3) 


-    +3   L 


)  L/" 


&c- 


466. 


For  brevity,  let  us  represent  by  X,  the  terms  of  this  expression,  independent  of  the  quantity  D  ;    by  Y,  the 
assemblage  of  terms  multiplied  by  D';  and  by  Z,  those  multiplied  by  D'2,  and  we  shall  have 

*    f          y         f  "V    i    "V      T\     i     TJ      T\  e  > 
A  /  =  {X  +  Y.U  +  Z.D8}; 


467. 

The  distinc- 
tion of  aber- 
ration an  in- 
determinate 
problem. 

Conditions 
limiting  it 
Clairaut's. 


and  if  this  vanish  the  aberration  is  destroyed.  Now,  first,  if  we  regard  only  parallel  rays,  or  suppose  D  =  0- 
this  reduces  itself  to  X  =  0,  so  that  the  condition  X  =  0  being  satisfied,  the  telescope  will  be  perfect  when  used 
for  astronomical  purposes,  or  for  viewing  objects  so  distant  that  D'  may  be  disregarded. 

The  equation  X  =  0  is  of  the  second  degree  in  each  of  the  quantities  r1 1",  &c.,  whose  number  is  that  of  the 
lenses.  Consequently,  this  condition  alone  is  not  sufficient  to  fix  their  values ;  and,  without  assuming  some 
further  relations  between  them,  or  some  other  limitations,  the  problem  is  indeterminate,  and  the  aberration  may 
be  destroyed  in  an  infinite  variety  of  ways.  Confining  ourselves  at  present  to  the  consideration  of  two  lenses 
only,  since  X  =  0  contains  only  two  unknown  quantities,  one  other  equation  only  is  required,  and  we  have  only 
to  consider  what  other  condition  will  be  attended  with  the  greatest  practical  advantages.  Clairaut  has  proposed 
to  adjust  the  two  lenses  so  as  to  have  their  adjacent  surfaces  in  contact  throughout  their  whole  extent,  to  allow 
of  their  being  cemented  together,  and  thus  avoid  the  loss  of  light  by  reflection  at  these  surfaces.  This  certainly 
would  be  a  great  advantage  were  it  possible  so  to  cement  two  glasses  of  large  size  together,  as  to  bring  neither 
of  them  into  a  state  of  strain  as  the  cement  cools,  or  otherwise  fixes ;  and  were  it  not  for  the  further  incon- 
venience, that  the  media  being  of  course  differently  expansible  by  heat,  every  subsequent  change  of  temperature 
would  necessarily  distort  their  figure,  as  well  as  strain  their  parts,  when  thus  forcibly  held  together,  just  as  we 
see  a  compound  lamina  of  two  differently  expansible  metals  assume  a  greater  or  less  curvature,  according  to 
the  temperature  it  is  exposed  to.  Meanwhile  the  condition  in  question  is  algebraically  expressed  by  L'=  (/i1 — 1) 
(r>  -.  /')  ;  for  jn  this  case  R'  =  r1,  and  R"  =  R'"  =  r",  and  this  being  of  the  first  degree  only  in  /,  r",  affords  a 
final  equation  of  a  quadratic  form  by  elimination  with  X  =  0,  which  latter,  in  the  case  before  us  of  two  lenses, 
is  the  same  as  the  equation  (»),  Art.  312,  writing  only  r'  for  R',  and  i"  for  R'". 


Part  II 


LIGHT.  4'J5 

But  this  condition  of  Cluiraut's  has  another  and  much  greater  inconvenience,  which   is,  that   the  resulting     ft*  N. 
quadratic  has  its  roots  imaginary,  when  the  refractive  and  dispersive  powers  of  the  glasses  are  such  as  are  by  no  s^>^~" 
means  unlikely  to  occur  in  practice;  and  without  the  limits  of  refraction  and  dispersion,  for  which  they  are  real,         " 


the  resulting  curvatures  change  so  rapidly  on  slight  variations  of  the  data,  as  to  make  their  computation  delicate,  bert.gem 
and  interpolation  between  them,  so  as  to  form  a  table,  very  troublesome.     D'Alembert,  in  his  Opuscules,  torn,  iii., 
has  proposed  a  variety  of  other  limitations,  such,  for  instance,  as  annihilating  the  spherical  aberration  for  rays 

?  X  S  X 

of  all  colours,  (which  comes  to  the  same  as  supposing  at  once  X  =r  0  and       '  ,     3//  H  --  %-  ^  /*''    =  ^>   an<^ 

which  leads  to  biquadratic  equations,  and  affords  no  practical  advantage,)  &c.     But,  without  going  into  useless 
refinements  of  this  kind,  the  very  form  of  the  general  equation  X  +  Y.D'  +Z.D/i  =  0  points  out  a  condition 
combining  every  advantage  the  case  is  susceptible  of.     This  consists  in  putting  Y  =  0.     By  this  supposition,  the 
term  depending  on  D'  is  destroyed,  without  assuming  D'=  0  ;  so  that  the  telescope  is  not  only  perfect  for  parallel  Another 
rays,  but  admits  of  as  considerable  a  proximity  of  the  object  without  losing  its  aplanatic   character,  as   the  proposed. 
nature  of  the  case  will  allow.     The  term  Z  .  D"2  indeed,  or 


cannot  vanish  when  two  lenses  only  are  used,  being  composed  wholly  of  given  functions  of  the  refractive  and 
dispersive  powers,  unless  by  D'  itself  vanishing,  or  by  an  accidental  adjustment  of  the  values  of  ft',  /t",  L',  &c. 
But  except  the  object  be  brought  within  a  comparatively  small  distance  from  the  telescope,  (such  as  ten  times  its 
own  length,)  the  square  of  D'  is  always  so  small  sis  to  allow  of  our  disregarding  this  term,  and  considering  the 
instrument  as  perfectly  aplanatic  when  Y  =  0.  Now  this  equation,  being  of  the  first  degree  in  /,  r",  adds  no 
new  algebraic  difficulty  to  the  problem,  but  leads  by  elimination  to  a  final  quadratic  ;  and,  what  is  of  most  con- 
sequence, for  such  values  of  /uf,  p!1,  and  the  dispersive  ratio  CT  as  occur  in  practice,  the  roots  of  this  quadratic 
are  always  real,  and  the  resulting  curvatures  of  all  the  surfaces  are  moderate,  and  well  adapted  for  practice  ;  more 
so,  indeed,  than  in  any  construction  hitherto  proposed.  They  are,  moreover,  such  as  to  afford  remarkable  and 
peculiar  facilities  for  interpolation,  as  we  shall  presently  see.  These  reasons  seem  to  leave  no  room  for  hesita- 
tion in  fixing  on  the  condition  Y  =  0,  as  tha»  which  ought  to  be  introduced  to  limit  the  problem  of  the  con- 
struction of  a  double  object-glass,  and  to  render  it,  so  far  as  it  can  be  rendered,  aplanatic. 

This  equation,  in  the  case  in  question,  is  469 


which  is  to  be  combined  with  (L-),  Art.  412,  in  which  R'  =  /  and  R'"=:  r1' .  To  reduce  these  to  numbers,  p' ,  ft"  ^70 
and  the  dispersive  ratio  CT  must  first  be  known.  The  readiest  and  most  certain  way  in  practice,  for  the  use  of 
the  optician,  is  to  form  small  object-glasses  from  specimens  of  the  glasses  intended  to  be  employed,  and  by  trial 
work  them  till  the  combination  is  as  free  from  colour  as  possible,  by  the  test  usually  had  recourse  to  in  practice. 
This  is,  to  examine  with  a  high  magnifying  power  the  image  of  a  well  defined  white  circle,  or  circular  annulus  on 
a  black  ground.  If  its  edges  are  totally  free  from  colour,  the  adjustment  is  perfect,  but  (owing  to  the  secon- 
dary spectrum)  this  will  seldom  be  the  case  ;  and  there  will  generally  be  seen  on  the  interior  edge  of  the  annulus 
a  faint  green,  and  on  the  exterior  a  purplish  border,  when  the  telescope  is  thrown  a  little  out  of  focus  by  bringing 
the  eye-glass  too  near  the  object-glass,  and  vice  rend.  The  reason  is,  that  while  the  great  mass  of  orange  and 
blue  rays  is  collected  in  one  focus,  the  red  and  violet  are  converged  to  a  focus  farther  from,  and  the  green  to 
one  nearer  to  the  object-glass;  the  refraction  of  the  green  rays  being  in  favour  of  the  convex  or  crown  glass,  and 
of  the  red  and  violet  (which  united  form  purple)  in  favour  of  the  flint  (see  table,  Art.  443)  or  concave  lens.  The 
focal  lengths  of  the  lenses  are  then  to  be  accurately  determined,  and  the  ratio  of  the  dispersions  (w)  will  then 
be  known,  being  the  same  with  that  of  the  focal  lengths  (454).  The  refractive  indices  will  be  best  ascertained 
by  direct  observation,  forming  portions  of  each  medium  into  small  prisms.  Now,  CT  being  known,  if  we  take 

unity  for  the  power  of  the  compound  lens,  we  have  L'  =.  and  L"  =  —    ,   so   that  L'  and  L"  are 

]  —  OT  1  —  •as 

known,  and  we  have  therefore  only  to  substitute  their  values  and  those  of  ft',  p,",  in  the  algebraic  expressions, 
and  proceed  to  eliminate  by  the  usual   rules.     The  following   compendious  table  contains  the  result  of  such  Dimensions 
calculations  for  the  values  of  p},  ft"  and  cr  therein  stated,  together  with  the  amount  of  variation  produced   by  of  an  ap'a- 
varying  either  of  the  refractive  indices  independently  of  the  other,  for  the  sake  of  interpolation  by  proportional  naticobjec'- 
parts.     Fig.  108  is  a  representation  of  the  resulting  object-glass. 


VOL.    IV.  3  K 


426 


LIGHT. 


UgUL 


Table  for  folding  the  Dimensions  of  an  Aplanatw  Object-glass. 

Refractive  index  of  crown,  or  convex  lens  =  /*'  =  1.524. 

Refractive  index  of  flint,  or  concave  lens  =  /*"  =  1.585. 

Compound  focal  length  =  10.000. 


PirtH. 


CROWN  LENS. 

FLINT  LENS. 

Second 

Third 

Fourth  surface,  convex. 

First  surface,  convex. 

surface, 

Surface, 

convex. 

Concave. 

Variation  of 

Variation  of 

Variation  of 

Variation  of 

Dis- 
per- 
sive 
ratio 

for  the 
above  re- 
fractive 

radius  for  a 
change  of 
+  0.010  in 
ref.  index  of 

radius  for  a 
change  of 
+  0.010  in 
ref.  index  of 

Radius 
of  con- 
vexity. 

Focal 
length 
of 
crown 

Radius 
of  con- 
cavity. 

Radius  for 
the  above 
refractive 
indices. 

radius  for  a 
change  of 
+  0.010  in 
ref.  index  of 

radius  for  a 
change  of 
+  0.010  in 
ref.  index  of 

Focal 
length  of 
flint  lens. 

TT  =. 

crown  glass. 

flint  glass. 

lens. 

crown  glass. 

flint  glass. 

0.50 

6.7485 

+  0.0500 

-  0.00304.2827 

5.0 

4.1575 

14.3697 

+  0.9921 

-  0.3962 

10.0000 

0.55 

6.7J  84 

+  0.0740 

-  0.0011 

3.6332 

4.5 

3.6006 

14.5353 

+  1.0080 

—  0.5033 

8.1818 

0.60 

6.7069 

+  0.0676 

+  0.0037 

3.0488 

4.0 

3.0640 

14.2937 

+  1.1049 

—  0.5659 

6.6667 

0.65 

6.7316 

+  0.0563 

+  0.0125 

2.5208 

3.5 

2.5566 

13.5709 

+  1.1614 

-  0.6323 

5.3846 

0.70 

6.8279 

+  0.0335 

+  0.0312 

2.0422 

3.0 

2.0831 

12.3154 

+  1.1613 

—  0.7570 

4.2858 

0.75 

7.0816 

-  0.0174 

+  0.0568 

1.6073 

2.5 

1.6450 

10.5186 

+  1.0847 

-0.7207J   3.3333 

the  table. 


To  apply  this  table  to  any  other  proposed  state  of  the  data,  we  have  only  to  consider  that  to  compute  the  radius 
of  any  one  of  the  surfaces,  as  the  first  or  fourth,  we  have  only  to  regard  each  element  as  varying  separately,  and 
471.       take  proportional  parts  for  each.     The  following  example  will   elucidate  the  process  :    Required  the  dimensions 
Example  of  for  an  object-glass  of  30  inches  focus,  the  refractive  index  of  the  crown  glass  being  1.519,  and  that  of  the  flint 
the  use  of    1.5Q9;    the  dispersive  powers  being   as  0.567  :  1,   or    0.567  being  the   dispersive   ratio.      Here  /»' =  1.519, 
/>!'  =  1.589,  and  -m  —  0.567.    The  computation  must  first  be  instituted  for  a  compound  focus  =  10.000,  as  in  the 
table,  and  we  proceed  thus: 

1st.  Subtract  the  decimal  (0.567)  representing  the  dispersive  ratio  from  1.000,  and  10  times  the  remainder 
(=  10  x  0.433  =  4.330)  is  the  focal  length  of  the  crown  lens. 

2nd.  Divide  unity  by  the  decimal  above  mentioned,  (0.567,)  subtract  1.000  from  the  quotient  (  — 

U.OO/ 

1.7635,  minas  1  =  0.7635)  and  the  remainder  multiplied  by  10  (or  7.635)  is  the  focal  length  of  the  flint  lens. 
We  must  next  determine  by  the  tables  the  radii  of  the  first  and  fourth  surfaces  for  the  dispersive  ratios  there  set 
down  (0.55  and  0.60)  next  less  and  next  greater  than  the  given  one.  For  this  purpose  we  have 


Refractive  powers  given.  . . 
Refractive  powers  in  table 


1.519  and  1.589 
1.524    .       1.5S5 


Differences      —  0.005      +  0.004 

The  given  refraction  of  the  crown  being  less,  and  of  the  flint  greater,  than  their  average  values  on  which  the 
tabre  is  founded.  Looking  out  now  opposite  to  0.55  in  the  first  column  for  the  variations  in  the  two  radii 
corresponding  to  a  change  of  +  0.010  in  the  two  refractions,  we  find  as  follows: 

First  surface.       Fourth  surface. 

For  a  change  =  +  0.010  in  the  crown  +  0.0740       +  1.0080 
For  a  change  =  +  0.010  in  the  flint      -  0.0011        -  0.5033 

But  the  actual  variation  in  the  crown  instead  of  +  0.010  being  -  0.005,  and  of  the  flint  +  0  004,  we  must  take 
the  proportional  parts  of  these,  changing  the  sign  in  the  former  case  ;  thus  we  find  the  variations  in  the  first  and 
last  radii  to  be 


L  1  G  H  T.  427 

First  surface. 
i  in  the  crown       —  0.0370 

For  +  0.004  variation  in  the  flint    .  .    —  0.0004 


Ught.  First  surface.         Fourth  surface.  Put  II. 

— v—  For  —  0.005  variation  in  the  crown       —  0.0370          —  0.5040  — v  — 


Total  variation  from  both  causes  ....    —  0.0374          —  0.7053 
But  the  radii  in  the  table  are    6.7184  14.5353 


Hence  the  radii  interpolated  are  ....          6.6810  13.8300 

If  we  interpolate,  by  a  process  exactly  similar,  the  same  two  radii   for  a  dispersive  ratio  0.60,  we  shall  find, 
respectively, 

First  surface.         Fourth  surface. 
For  a  variation  of  —  0.005  in  the  crown  —  0.0338          —  0.5524 

For  a  variation  of  +  0.004  in  the  flint    +  0.0015          —  0.2264 


Total  variation   —  0.0323          —  0.7788 

Radii  in  table 6.7069  14.2937 


Interpolated  radii 6.6746  13.5149 

Having  thus  got  the  radii  corresponding;  to  the  actual  refractions  for  the  two  dispersive  ratios  0.55   and  0.60, 
it  only  remains  to  determine  their  values  for  the  intermediate  ratios  0.567  by  proportional  parts ;  thus 
First  radius.         Fourth  radius. 

For 0.600  6.6746  13.5149 

For 0.550  6.6810  13.8300  0.050  :  0.567  -  0.050  =  0.017  ::- 0.0064  :- 0.0022 

— 0.050  0.017::- 0.3151: -0.1071 

Differences    +0.050         -0.0064          -0.3151 

So  that  6.6810  —  0.0022  =  6.6788,  and  13.8300  —  0. 1071  =  13.7229,  are  the  true  radii  corresponding  to  the 

given  data.     Thus  we  have,  for  the  crown  lens,  focal  length   =  4.330  =  -=-r,  radius  of  first  surface  =  6.6788 

L 

=  -f^ ,  index  of  refraction  =  1.519  =  p.',  whence  by  the  formula  L'  =  (/«'  —  1)  (R'  —  R")  -=j^  radius  of  the 
other  surface  is  —  3.3868.     Again,  for  the  flint  lens,  the  focal  length  =    •     ;/      =    —    7.635,    radius  of    the 

posterior   surface  =  -    -  =  —  13.7729,  index  of  refraction  ft,"  =  1.589,  whence  we  find         f.   =  —  3.3871 
iv  '  i 

for  the  radius  of  the  other  surface.     The  four  radii  are  thus  obtained  for  a  focal  length  of  10  inches,  and  multi 
plying  by  3  we  have  for  the  telescope  proposed 

in.  in.  in.  in. 

radius  of  first  surface  =  +  20.0364;     of  second,  —  10.1604;     ofthird,  -  10.1613;     offourth,  —41.1687. 

Here,  then,  we  see  that  the  radii  of  the  two  interior  surfaces  of  the  double  lens  (fig.  108)  differ  by  scarcely  472. 
more  than  a  thousandth  part  of  an  inch  ;  so  that,  should  it  be  thought  desirable,  they  may  be  cemented  together. 
This  is  not  merely  a  casual  coincidence,  for  the  particular  state  of  the  data  ;  if  we  cast  our  eyes  down  the 
table  we  shall  find  this  approximate  equality  of  the  interior  curvatures  (those  of  the  second  and  third  surfaces) 
maintained  in  a  singular  manner  throughout  the  whole  extent  of  the  variation  of  ra.  Thus  the  construction, 
here  proposed  in  reality  for  glasses  of  the  ordinary  materials,  approaches  considerably  to  that  of  Clairaut  already 
mentioned. 

In  order  to  put  these  results  to  the  test  of  experience,  Mr.  South  procured  an  achromatic  telescope  to  be  473. 
executed  on  this  construction  by  Mr.  Tulley,  one  of  the  most  eminent  of  our  British  artists,  which  is  now 
in  the  possession  of  J.  Moore,  Esq.  of  Lincoln.  Its  focal  length  was  45  inches,  and  aperture  3J,  and  its  per- 
formance was  found  to  be  fully  adequate  to  the  expectation  entertained  of  it,  bearing  a  magnifying  power  of 
300  with  perfect  distinctness,  and  separating  easily  a  variety  of  double  stars,  &c.  A  more  minute  account 
of  its  performance  will  be  found  in  the  Journal  of  the  Royal  Institution,  No.  26.  Should  the  splendid  example 
set  by  Fraunhofer  be  followed  up,  and  the  practice  of  the  optician  be  in  future  directed  by  a  rigorous  adherence 
to  theory,  grounded  on  exact  measurements  of  the  refractive  powers  of  his  glasses  on  the  several  coloured  rays, 
it  will  become  necessary  to  develope  the  above  table  more  in  detail. 

When  three  media  are  employed  in  the  construction  of  object-glasses,  it  should  be  our  object  to  obtain  as       474 
great  a  difference  as  possible  in  their  scales  of  action  on  the  differently  coloured  rays.     Dr.  Blair,  to  whom  we  Object- 
are  indebted  for  the  first  extensive  examination  of  the  dispersive  powers  of  media  as  a  physical  character,  and  glasses  of 
who  first  perceived  the  necessity  of  destroying  the  secondary  spectrum,  and  pointed  out  the  means  of  doing  it,  ;hree  media 
is  the  only  one  hitherto  who  has  bestowed  much  pains  on  this  important  part  of  practical    optics ;    which, 
considering  the  extraordinary  success  he  obtained,  and  the  perfection  of  the  telescopes  constructed  on  his  prin- 
ciples, is  to  be  regretted.     We  have  no  idea,  indeed,  for  the  reasons  already  mentioned,  that  very  large  object- 

3  K  2 


428  LIGHT. 

Light.       glasses,  enclosing  fluids,  can  ever  be  rendered  available  ;   but  to  render  glasses  of  moderate  dimensions  more 
>— — v^-»'  perfect,  and  capable  of  bearing  a  higher  degree  of  magnifying  power,  is   hardly  less  important  as  an  object  of 
practical   utility.     His  experiments  are  to  be  found   in    the   Transactions  of  the  Royal  Society  of  Edinburgh, 
1791.     We  can  here  do  little  more  than  present  a  brief  abstract  of  them. 

475.  Dr.  Blair  having  first  discovered  that  the  secondary  fringes  are  of  unequal  breadths,  when  binary  achromatic 

Dr.  Blair  a  combinations,  having  equal  total  refractions,  are  formed  of  different  dispersive  media,  was  immediately  led  to 
construction  consider,  that  by  employing  two  such  different  combinations  to  act  in  opposition  to  each  other,  if  the  total 
'"  refractions  were  equal,  the  ray  would  emerge  of  course  undeviated,  and  with  its  primary  spectrum  destroyed  ; 
three  media.  but  a  secondary  spectrum  would  remain,  equal  to  the  difference  of  the  secondary  spectra  in  the  two  combina- 
tions. Therefore,  by  a  reasoning  precisely  similar  to  that  which  led  to  the  correction  of  the  primary  spectrum 
itself,  (Art.  426  and  427,)  if  we  increase  the  total  refraction  of  that  combination  A  which,  ceeteris  paribus,  gives 
(he  leant  secondary  spectrum,  its  secondary  colour  will  be  increased  accordingly,  till  it  becomes  equal  to  that  of 
the  other  B  ;  so  that  the  emergent  beam  will  be  free  from  the  secondary  spectra  altogether,  and  will  be  deviated 
on  the  whole  in  favour  of  the  combination  A.  Reasoning  on  these  grounds,  Dr.  Blair  formed  a  compound,  or 
binary  achromatic  convex  lens  A,  (fig.  109,)  of  two  fluids  a  and  b,  (two  essential  oils,  such  as  naphtha  and  oil 
of  turpentine,  differing  considerably  in  dispersion,)  which,  when  examined  alone,  was  found  to  have  a  greater 
refractive  power  on  the  green  rays  than  on  the  united  red  and  violet.  He  also  formed  a  second  binary  lens  B, 
of  a  concave  character,  and  also  achromatic,  (i.  e.  having  the  primary  spectrum  destroyed,)  consisting  of  the  more 
dispersive  oil  (6)  and  glass,  and  in  which  the  green  rays  are  also  more  refracted  than  the  united  red  and  violet, 
but  in  a  greater  degree  in  proportion  to  the  whole  deviation,  than  in  the  other  combination  ;  and  in  precisely 
the  same  degree  was  the  focal  length  of  this  lens  increased  or  its  refraction  diminished,  when  compared  with 
that  of  the  combination  A.  When,  therefore,  these  two  lenses  were  placed  together,  as  in  fig.  109,  an  excess  of 
refraction  remained  in  favour  of  the  convex  combination  ;  but  the  secondary  spectra  of  each  being  equal  and 
opposite  (by  reason  of  the  opposite  character  of  the  lenses)  were  totally  destroyed.  In  fact,  he  states,  that  in 
a  compound  lens  so  constructed,  he  could  discover  no  colour  by  the  most  rigid  test ;  and  thence  concluded, 
not  only  the  red,  violet,  and  green  to  be  united,  but  also  all  the  rest  of  the  rays,  no  outstanding  colour  of  blue 
or  yellow  being  discernible.  In  placing  the  lenses  together,  the  intermediate  plane  glasses  may  be  suppressed 
altogether,  as  in  fig.  110. 

476  It  was  in  the   course  of  these  researches  that  Dr.  Blair  was  led  to  the  knowledge  of  the  possibility  of  forming 

Remarkable  binary  combinations,  having  secondary  spectra  of  opposite  characters  ;    that  is,  in  which  (the  total  refraction 
property  ot'  lying  the  same  way)  the  order  of  the  colours  in  the  secondary  spectra  should  be  inverted.     In  other  words,  that 
the  muriatic  wnj]e  jn  SOme  combinations  the  green  rays  are  more  refracted  than  the  united  red  and  violet,  in  others  they  are 
less  so.     He  found,  for  instance,  that  while  in  most  of  the  highly  dispersive  media,  including  metallic  solutions, 
the  green  lay  among  the  less  refrangible  rays  of  the  spectrum,  there  yet  exist  media  considerably  dispersive,  in 
which   the  reverse  holds  good.     The   muriatic  acid,  among  others,  is  in  this   predicament.     Hence,  in  binary 
combinations  of  glass  with  this  acid,  the  secondary  spectrum  consists  of  colours  oppositely  disposed  from  that 
formed  by  glass  and  the  oils,  or  by  crown  and  flint  glass.     In  consequence  of  this,  to  form  an  object-glass  of 
two  binary  combinations,  as  described  in  the  last  article,  they  must  both  be  of  convex  characters.     But  this  affords 
Dr.  Blair's     no  particular  advantage.     Dr.  Blair,  however,  considered  the  matter  in  another  and  much  more  important  light, 
discovery  of  a.s  offering  the  means  of  dispensing  with  a  third  medium  altogether,  and  producing  by  a  single  binary  combina- 
"l1^          tion  a  refraction  absolutely  free  from  secondary  colour.     To  this  end  he  considered,  that   it  appears   to  depend 
•ame^scale    entirely  <">  the  chemical  nature  of  the  refracting  medium,  what  shall  be  the  order  and  distribution  of  the  colours 
ofdispersion  in  the  spectrum,  as  well  as  what  shall  be  the  total  refraction  and  dispersive  powers  of  the  medium  ;    and  that 
a«  glass.        therefore  by  varying  properly  the  ingredients  of  a  medium,  it  may  be  practicable,  without  greatly  varying  the 
total  refraction  and  dispersion,  still  to  produce  a  considerable  change  in  the  internal  arrangement  (if  we  may 
use  the  phrase)  of  the  spectrum  ;    and  therefore,    perhaps,  to   form  a  compound  medium  in  which  the  seven 
colours  shall  occupy  spaces  regulated  by  any  proposed  law,  (within  certain  limits.)     Now  if  a  medium  could  be 
so  compounded  as  to  have  the  same  scale  of  dispersions,  or  the  same  law  of  distribution  of  the  colours  as  crown 
glass  with  a  different  absolute  dispersion,  as  we  have  already  seen,  nothing  more  would  be  required  for  the  per- 
fection of  the  double  object-glass.     The  property  of  the  muriatic  acid  just  mentioned  puts  this  in  our  power. 
It  is  observed,  that  the  presence  of  a  metal  (antimony,  for  instance)  in  a  fluid,  while  it  gives  it  a  high  refrac- 
tive and  dispersive  power,  at  the  same  time  tends  to  dilate  the  more  refrangible   part  of  the  spectrum   beyond 
its  due  proportion   to  the  less.     On  the  other  hand,  the  presence  of  muriatic  acid  tends  to  produce  a  contrary 
effect,  contracting  the  more  refrangible  part  and  dilating  the  less,  beyond  that  proportion  which  they  have  in 
glass.     Hence,  Dr.  Blair  was  led  to  conclude,  that  by  mixing  muriatic  acid  with  metallic  solutions,  in  proportions 
to  be  determined  by  experience,  a  fluid  might  be  obtained  with  the  wished  for  property  ;  and  this  on  trial  he 
found  to  be  the  case.     The  metals  he  used  were  antimony  and  mercury ;  and  to  ensure  the  presence  of  a  sufli- 
cient  quantity  of  muriatic  acid,  he  employed  them  in  the  state  of  muriates,  in  aqueous  solution  ;  or,  in  the  case 
of  mercury,  in  a  solution  of  sal  ammoniac,  which  is  a  compound  of  ammonia  and  muriatic  acid,  and  which   is 
capable  of  dissolving  a  considerably  greater  quantity  of  corrosive  sublimate  (muriate,  or  chloride  of  mercury) 
His  double   than  water  alone.     By  adding  liquid  muriatic  acid  to  the  compound  known  by  the  name  of  butter  of  antimony, 
object-          (chloride  of   antimony,)  or  sal   ammoniac  to   the   mercurial   solution,  he  succeeded  completely  in  obtaining  a 
|          spectrum  in  which  the  rays  followed  the  same  law  of  dispersion  as  in  crown  glass,  and  even  in  over-correcting 
w™  meaia  the  secondary  spectrum,  so  as  to  place  its  exact  destruction  completely  in  his  power.     It  only  remained  to  form 
an  object-glass  on  these  principles.     Fig.  Ill  is  such  an  one,  in  which,  though  there  are  two  refractions  at  the 
confines  of  the  glass  and  fluid,  yet  the  chromatic  aberration,  as  Dr.  Blair  assures  us,  was  totally  destroyed,  and 
the  rays  of  different  colours  were  bent  from  their  rectilinear  course  with  the  same  equality  as  in  reflexion. 


LIGHT.  429 

f  0  sucj,  an  extent  has  Dr.  Blair  carried  these  interesting  experiments,  that  he  assures  us  he  has  found  it  prac- 
ticable  to  construct  an  object-glass  of  nine  inches  focal  length,  capable  of  bearing  an  aperture  of  three  inches,  a  v—  p^""" 
thing  which  assuredly  no  artist  would  ever  dream  of  attempting  with  glass  lenses  ;  and  we  cannot  close  this 
account  of  his  labours  without  joining  in  a  wish  expressed  on  a  similar  occasion  by  Dr.  Brewster,  whose 
researches  on  dispersive  powers  have  so  worthily  filled  up  the  outline  sketched  by  his  predecessor,  that  this 
branch  of  practical  optics  may  be  resumed  with  the  attention  it  deserves,  by  artists  who  have  the  ready  means 
of  executing  the  experiments  it  would  require.  Could  solid  media  of  such  properties  be  discovered,  the  telescope 
would  become  a  new  instrument. 

These  experiments  of  Dr.  Blair  lead  to  the  remarkable  conclusion,  that  at  the  common  surface  of  two  media      478. 

a  white  rav  may  be  refracted  without  separation  into  its  coloured  elements.     In  fact,  u,  and  uf  being  the  refrac-  Case  of 
1         *  colourless 


live  indices  of  the  media  for  any  ray  as  the  extreme  red,  -—  will  be  their  relative  refractive  index  for  that  ray, 


common 


an(]     P      _  ?^_  will  be  the   relative  index  for  any  other  ray.     If,  then,  the  refractive  and  dispersive  powers  of  *^ 
the  media  be  such  that  j"   +     ^     =   —  ,  or  /»  5/  =  /  s/*.  that  is-  if  -*~J  =  ~T  5  and  if>  moreover,  this 


/*  -r- 

relation  hold  good  throughout  the  spectrum,  i.  e.  if  the  increments  of  the  refractive  indices,  in  proceeding  from 
the  red  to  the  violet  end  of  the  spectrum,  be  proportional  to  the  refractive  indices  themselves,  then  the  relative 
index  is  the  same  for  all  rays,  and  no  dispersion  will  take  place.  Now  this  gives  a  relation  between  the  disper- 

/  /  i  ^ 

P  u  u  —  1  u, 

sive  and  refractive  indices  of  the  two  media,  viz.  —  =  -  .  —  -.  -  '•-•  =   -  -  ;  and,  in  addition  to  this 

P  /*          F  —  * 


condition,  the  scale  of  dispersions  must  be  the  same  in  both  media.  According  as  the  dispersions  differ  one  way 
or  the  other  from  this  precise  adjustment,  the  violet  ray  may  be  either  more  or  less  refracted  than  the  red  at  the 
common  surface  of  the  two  media. 

We  shall  terminate  the  theory  of  achromatic  object-glasses  with  a  problem  of  considerable  practical  import-  479. 
ance,  as  it  puts  it  in  our  power,  having  obtained  an  approximate  degree  of  achromaticity  in  an  object-glass,  to  Achromatic 
complete  the  destruction  of  the  colour  without  making  any  alteration  in  the  focal  lengths  or  curvatures  of  the  object-glass 
lenses,  by  merely  placing  them  at  a  greater  or  less  distance  from  one  another.  ™ 

Problem.  To  express  the  condition  of  achromaticity,  when  the  two  lenses  of  a  double  object-glass  are  placed 
at  a  distance  from  each  other,  (=  t.) 

Resuming  the  notation  of  Art.  251  and  268,  we  have 

/"=L'  +  D;  /"  =      '  f"  " 


]-/    t 
and 


Now,  that  the  combination  may  be  achromatic,  we  must  have  if"  ==  0  ;  and,  since  t  and  D  are  constant,  and 
L'  and  I/'  only  vary  by  the  variations  of  /t',  fi"  the  refractive  indices,  we  have   3  L'  =  (R'  —  R")  3/i'  = 

— ; — —  I/  =  p'L',  and  similarly  3  L"  =  p"  L",  so  that  substituting  we  get 


Such  is  the  condition  of  achromaticity.     Since  it  depends  on  D,  it  appears  that  if  the  lenses  of  an  object-      480. 
glass  be  not  close  together,  it  will  cease  to  be  achromatic  for  near  objects,  however  perfectly  the  colour  be  cor- 
rected for  distant  ones.     The  eye  therefore  cannot  be  achromatic  for  objects  at  all  distances,  its  lenses  being 
of  great  thickness  compared  to  their  focal  lengths;  and,  therefore,  although  in  contact  at  their  adjacent  surfaces, 
yet  having  considerable  intervals  between  others. 

For  parallel  rays  the  equation  becomes  481. 

j/'L"(l-<L')8  =  -p'L'; 
hence,  the  dispersions  and  powers  of  the  lenses  being  given  their  interval  t  may  be  found  by  the  expression 

1    d     x/_  PL  J 
"17  (. i      V       p"  •  L" 

The  condition  of  achromaticity,  were  the  lenses  placed  close  together,  would  be,  as  we  have  already  shown, 


430  L  I  G  H  T. 

L.gnt  p:  J/ 

-_>v^-. —rf-  .   -j-rr  =  !•     Hence,  whenever  this  fraction  is  less  than  unity,  that  is  whenever  L",  the  power  of  the 

concave  or  flint  lens  (which  we  here  suppose  to  be  the  second)  is  too  great ;  or  when,  as  the  opticians  call  it,  the 
colour  is  over-corrected,  the  object-glass  may  be  made  achromatic,  or  their  over-correction  remedied,  without  re- 
grinding1  the  glasses,  merely  by  separating  the  lenses ;  for  in  this  case  the  quantity  under  the  radical  is  less  than 
unity,  a'nd  therefore  t  is  positive,  a  condition  without  which  the  rays  could  not  be  refracted  as  we  have  supposed 
them. 

4gg  Moreover,  this  affords  a  practical  and  very  easy  means  of  ascertaining,  with  the  greatest  precision,  the  dis- 

persive ratio  of  the  two  media.  Let  a  convex  lens  of  crown  be  purposely  a  little  over-corrected  by  a 
concave  of  flint,  and  then  let  the  colour  be  destroyed  by  separating  the  lenses.  Measure  their  focal  lengths 

(  -=-7-  and  ,,  J  and  the  interval  t  between  them  in  this  state,  and  we  have  at  once  for  the  value  of  -a  the 
dispersive  ratio, 


§  III.  Of  the  Absorption  or  Extinction  of  Light  by  uncrystallized  Media. 

Transparency  is  the  quality  by  which  media  allow  rays  of  light  freely  to  pass  through  their  substance,  or,  it 
All  m*<j|a  may  be,  between  their  molecules  ;  and  is  said  to  be  more  or  less  perfect,  according  as  a  more  or  less  consider- 
"able  part  of  the  whole  light  which  enters  them  finds  its  way  through.  Among  media,  consisting  of  ponderable 
matter,  we  know  of  none  whose  transparency  is  perfect.  Whether  it  be  that  some  of  the  rays  in  their  passage 
encounter  bodily  the  molecules  of  the  media,  and  are  thereby  reflected ;  or,  if  this  supposition  be  thought  too 
coarse  and  unrefined  for  the  present  state  of  science,  be  stopped  or  turned  aside  by  the  forces  which  reside  in 
the  ultimate  atoms  of  bodies,  without  actual  encounter,  or  otherwise  detained  or  neutralized  by  them  ;  certain  it 
is,  that  even  in  the  most  rare  and  transparent  media,  such  as  air,  water,  and  glass,  a  beam  of  light  intromitted, 
is  gradually  extinguished,  and  becomes  more  and  more  feeble  as  it  penetrates  to  a  greater  depth  within  them, 
and  ultimately  becomes  too  faint  to  affect  our  organs.  Thus,  at  the  tops  of  very  high  mountains,  a  much 
greater  multitude  of  stars  is  visible  to  the  naked  eye  than  on  the  plains  at  their  feet ;  the  weak  light  of  the 
smallest  of  them  being  too  much  reduced  in  its  passage  through  the  lower  atmospheric  strata  to  affect  the  sight. 
Thus,  too,  objects  cease  to  be  visible  at  great  depths  below  wate*  however  free  from  visible  impurities,  &c.  Dr. 
Olbers  has  even  supposed  the  same  to  hold  good  with  the  imponderable  media  (if  any)  of  the  celestial  spaces, 
and  conceives  this  to  be  the  cause  why  so  few  stars  (not  more  than  about  five  or  ten  millions)  can  be  seen  with 
the  most  powerful  telescopes.  It  is  probable  that  we  shall  be  long  without  means  of  confirming  or  refuting 
this  singular  doctrine. 

485.  On  the  other  hand,  though  no  body  in  nature  be  perfectly,  all  are  to  a  certain  degree,  transparent.     One  of 
the  densest  of  metals,  gold,  may  actually  be  beaten  so  thin  as  to  allow  light  to  pass  through  it;  and  that  it  passes 
through  the  substance  of  the  metal,  not  through  cracks  or  holes  too  small  to  be  detected  by  the  eye,  is  evident 
from  the  colour  of  the  transmitted   light,  which  is  green,  even  when  the  incident  light  is  white.     The  most 
opaque  of  bodies,  charcoal,  in  a  different  state  of  aggregation,  (as  diamond,)  is  one  of  the  most  perfectly  trans- 
parent ;  and  all  coloured  bodies,  however  deep  their  hues,  and  however  seemingly  opaque,  must  necessarily  be 
rendered  visible  by  rays  which   have  entered  their  substance  ;    for  if  reflected  at  their  surfaces,  they  would  all 
appear  white  alike.     Were  the  colours  of  bodies  strictly  superficial,  no  variation  in  their  thickness  could  affect 
their  hue  ;  but,  so  far  is  this  from  being  the  case,  that  all  coloured  bodies,  however  intense  their  tint,  become 
paler  by  diminution  of  thickness.     Thus  the  powders  of  all  coloured  bodies,   or  the   streak  they  leave  when 
rubbed  on  substances  harder  than  themselves,  have  much  paler  colours  than  the  same  bodies  in  mass. 

486.  This  gradual  diminution  in  the  intensity  of  a  transmitted  ray  in  its  progress  through  imperfectly  transparent 
And  all        media,  is  termed  its  absorption.     It  is  never  found  to  affect  equally  rays  of  all  colours,  some  being  always  absorbed 
absorb  the    jn  preference  to  others;  and  it  is  on  this  preference  that  the  colours  of  all  such  media,  as  seen  by  transmitted 

light,  depend.  A  white  ray  transmitted  through  a  perfectly  transparent  medium,  ought  to  contain  at  its  emer- 
imequally.  ?ence  the  same  proportional  quantity  of  all  the  coloured  rays,  because  the  part  reflected  at  its  anterior  and 
posterior  surfaces  is  colourless ;  but,  in  point  of  fact,  such  perfect  want  of  colour  in  the  transmitted  beam  is 
never  observed.  Media,  then,  are  unequally  transparent  for  the  differently  coloured  rays.  Each  ray  of  the 
spectrum  has,  for  every  different  medium  in  nature,  its  own  peculiar  index  of  transparency,  just  as  the  index  of 
refraction  differs  for  different  rays  and  different  media. 

The  most  striking  way  in  which  this  different  absorptive  power  of  one  and  the  same  medium  on  differently 
Experiment,  coloured  rays  can  be  exhibited,  is  to  look  through  a  plain  and  polished  piece  of  smalt-blue  glass,  (a  rich  deep  blue, 
very  common  in  the  arts — such  as  sugar-basins,  finger-glasses,  &c.  are  often  made  of,)  at  the  image  of  any  narrow 
line  of  light  (as  the  crack  in  a  window-shutter  of  a  darkened  room)  refracted  through  a  prism  whose  edge  is 
parallel  to  the  line,  and  placed  in  its  situation  of  minimum  deviation.  If  the  glass  be  extremely  thin,  all  the 
colours  are  seen  ;  but  if  of  moderate  thickness  (as  TV  inch)  the  spectrum  will  put  on  a  very  singular  and  striking 
appearance.  It  will  appear  composed  of  several  detached  portions  separated  by  broad  and  perfectly  black 


L  I  G  H  T  431 

intervals,  the  rays  which  correspond  to  those  points  in  the  perfect  spectrum  being  entirely  extinguished.    If  a  less     Part  II. 
'  thickness  be  employed,  the  intervals,  instead  of  being  perfectly  dark,  are  feebly  and  irregularly  illuminated,  some  ^-—v—  —  - 
parts  of  them  being  less  enfeebled  than  others.     If  the  thickness,  on  the  other  hand,  be  increased,  the   black 
spaces  become  broader,  till  at  length  all  the  colours  intermediate  between  the  extreme  red  and  extreme  violet  are 
totally  destroyed. 

The  simplest  hypothesis  we  can  form  of  the  extinction  of  a  beam  of  homogeneous  light  in  passing  through  a  488. 
homogeneous  medium,  is,  that  for  every  equal  thickness  of  the  medium  passed  through,  an  equal  aliquot  part  of  f'aw 
the  rays,  which,  up  to  that  depth  had  escaped  absorption,  is  extinguished.  Thus,  if  1000  red  rays  fall  on  and 
enter  into  a  certain  green  glass,  and  if  100  be  extinguished  in  traversing  the  first  tenth  of  an  inch,  there  will- 
remain  900  which  have  penetrated  so  far;  and  of  these  one-tenth,  or  90,  will  be  extinguished  in  the  next  tenth 
of  an  inch,  leaving  810,  out  of  which  again  a  tenth,  or  81,  will  be  extinguished  in  traversing  the  third  tenth, 
leaving  729,  and  so  on.  In  other  wdYds,  the  quantity  unubsorhed,  after  the  beam  has  traversed  any  thickness  of 
the  medium,  will  diminish  in  geometrical  progression,  as  t  increases  in  arithmetical.  So  that  if  1  be  taken  for 
the  whole  number  of  intromitted  rays,  and  y  for  the  number  that  escape  absorption  in  traversing  an  unit  of 
thickness,  y'  will  represent  the  number  escaping,  after  traversing  any  other  thickness,  =  t.  This  only  supposes 
that  the  rays  in  the  act  of  traversing  one_  stratum  of  a  medium  acquire  no  additional  facility  to  penetrate  the 
remainder.  In  this  doctrine,  y  is  necessarily  a  fraction  smaller  than  unity,  and  depending  on  the  nature  both  of 
the  ray  and  the  medium.  Hence,  if  C  represent  the  number  of  equally  illuminating  rays  of  the  extreme  red  in 
a  beam  of  white  light,  C'  that  of  the  next  degree  of  refranjibilitv,  and  so  on  ;  the  beam  of  white  light  will  be 
represented  by  C  +  C'  +  C"  -f-  &c.  ;  and  the  transmitted  be^in,  after  traversing  the  thickness  t,  will  be  properly 
expressed  by 

C.y<  +  C'.y"  +  C".y'"  +  &c. 

Each  term  representing  the  intensity  of  the  particular  ray  to  which  it  corresponds,  or  its  ratio  to  what  it  is  in  the 
original  white  beam. 

It  is  evident  from  this,  that,  strictly  speaking,  total  extinction  can  never  take  place  by  any  finite  thickness  of      489. 
the  medium  ;  but  if  the  fraction  y  for  any  ray  be  at  all  small,  a  moderate  increase  in  the  thickness,  (which  enters 
as  an  exponent,)  will  reduce  the  fraction  y  '  to  a  quantity  perfectly  insensible.     Thus,  in  the  case  taken  above, 
where  a  tenth  of  an  inch  of  green  glass  destroys  one-tenth  only  of  the  red  rays,  a  whole  inch  will  allow  to  pass 

/9V* 

only  l-r^r)    ,  or  304  rays  out  of  a   thousand,   while  ten    times  that  thickness,  or  10  inches,  will  suffer  only 


9  V00 
-r-r-l      =  0.0000266,  or  less  than  three  rays  out  of  100,000  to  pass,   which   amounts  to    almost   absolute 

opacity. 

If  x  be  the  index  of  refraction  of  any  ray  in  the  water  spectrum,  we  may  regard  y  as  a  function  of  x  ;  and  if  on       490. 
the  line  RV,  (fig.  11  2,)  representing  the  whole  length  of  the  water  spectrum,  we  erect  ordinates,  Rr,  MN,VVequal  L»w  of  ab- 
to  unity  and  to  each  other  ;  and  also  other  ordinates  R  r,  M  P,  V  v  representing  the  values  of  y  for  the  rays  at  ^T^"^^ 
the  corresponding  points;  the  curve  rP  v,  the  locus  of  P,  will  be,  as  it  were,  a  type,  or  geometrical  picture  of  expressed 
the   action  of   the  medium  on   the  spectrum,  and  the  straight  line  R  N  V  will  be  a  similat  type  of  a  perfectly  by  a  curve. 
transparent   medium.     Now   if  this  be  supposed   the  case  when  the  thickness  of  the  medium  is  1,  if  we  take  Fig.  112. 
always  M  F  :  M  P  :  :  M  P  :  M  N,  and  M  P"  :  M  P'  :  :  M  P'  :  M  P,  &c.  and  so  on,  the  loci  of  P'  P",  &c.  will  be 
curves  representing  the  quantities  of  the  rays  transmitted  by  the  thicknesses  2,  3,  &c.  of  the  medium,  and  so  for 
intermediate  thicknesses,  or  for  a  thickness  less  than  1,  as  in  the  curve  %  w. 

Hence,  whatever  be  the  colour  of  a  medium,  if  its  thickness  be  infinitely  diminished,  it  will  transmit  all  the      491. 
rays  indifferently  ;  for  when  t  =  0,  y  '  =  1  ,  whatever  be  y  ;  and  the  curve  p  TT  v  approaches  infinitely  near  to  the 
line  R'N  V.     Thus  all  coloured  glasses  blown  into  excessively  thin  bubbles  are  colourless,  and  so  is  the  foam 
of  coloured  liquids. 

Again,  if  there  be  any,  the  least,  preference  given  by  the  medium  to  the  transmission  of  certain  rays  beyond      492. 
others,  the  thickness  of  the  medium  may  be  so  far  increased  as  to  give  it  any  assignable  depth  of  tint  ;  for  if  y 
be  ever  so  little  less  than  unity,  and  if  between  the  values  of  y  for  different  rays  there  be  ever  so  little  difference, 
t  may  be  so  increased  as  to  make  y  '  as  small  as  we  please,  and  the  ratio  of  y  '  to  y"  '  as  different  from  unity  as 
we  please. 

In  very  deep  coloured  media  all  the  values  of  y  are  small.     If  they  were   equal,  the  medium  would  merely       493. 
stop  light,  without  colouring  the  transmitted  beam,  but  no  such  media  are  at  present  known. 

If  the  curve  rPv,  or  the  type  of  an  absorbent  medium  have  a  maximum  in  any  part  of  the  spectrum,  as  in  the    i   494. 
green,  for  instance,  (fig.  113  ;)  then,  whatever  be  the  proportion  in  which  the  other  rays  enter,  by  a  sufficient  ^J1™*  '®n 
increase  of  thickness,  that  colour  will  be  rendered  predominant  ;  and  the  ultimate  tint  of  the  medium,  or  the  a'bsoJLj™ 
last  ray  it  is  capable  of  transmitting,  will  be  a  pure  homogeneous  light  of  that  particular  refrangibility  to  which  medium. 
the  maximum  ordinate  corresponds.      Thus  green  glasses,  by  an  increase  of  thickness,  become  greener  and  Fig.  113. 
greener,  their  type  being  as  in  fig.  113;  while  yellow  ones,  whose  type  is  as   in  fig.  114,  change  their  tint  by 
reduplication,  and  pass  through  brown  to  red. 

This  change  of  tint  by  increase  of  thickness  is  no  uncommon  phenomenon;  and  though  at  first  sight  para-      49"). 
doxical,  yet  is  a  necessary  consequence  of  the  doctrine  here  laid  down.     If  we  enclose  a  pretty  strong  solution  Tint 
of  sap-green,  or,  still  better,  of  muriate  of  chromium  in  a  thin  hollow  glass  wedge,  and  if  we  look  through  the  Changes  by 
edge  where  it  is  thinnest,  at  white  paper,  or  at  the  white  light  of  the  clouds,  it  appears  of  a  fine  green;  but  if*' 
we  slide  the  wedge  before  the  eye  gradually  so  as  to  look  successively  through  a  greater  and  greater  thickness 


432  LIGHT. 

Light,      of  the  liquid,  the  green  tint  grows  livid,  and  passes  through  a  sort  of  neutral,  brownish  hue,  to  a  deep  blood-     Part  II. 
<— -v— — '  red.     To  understand  this,  we  must  observe,   that  the  curves  expressing  the  types  of  different  absorbent  media  v—  v^1 
Case  of  a    admit  the  most  capricious  variety  of  form,  and  very  frequently  have  several  maxima  and  minima  corresponding 
green-red     to  as  manv  different  colours.     The  green  liquids  in  question  have  two  distinct  maxima,  as  in  fig.  1 15  ;  the  one 
Fig  'llS       corresponding  to  the  extreme  red,  the  other  to  the  green,  but  the  absolute  lengths  of  the  maximum  ordinates 
are  unequal,  the  red  being  the  greater.     But  as  the  extreme  red  is  a  very  feebly  illuminating  ray,  while  on  the 
other  hand  the  green  is  vivid,  and  affects  the  eye  powerfully,  the  latter  at  first  predominates  over  the  former,  and 
entirely  prevents  its  becoming  sensible  ;  and  it  is  not  till  the  thickness  is  so  far  increased  as  to  leave  a  very  great 
preponderance  of  those  obscure  red  rays,  and  subdue  their  rivals,  as  in  the  case  represented  by  the  lowest  of 
the  dotted  curves  in  the  figure,  that  we  become  sensible  of  their  influence  on  the  tint.     Suppose,  for  instance. 
Numerical    to  illustrate  this  by  a  numerical  example,  the  index  of  transparency,  or  value  of  y,  in  muriate  of  chromium,  to 
illustration,  be  for  extreme  red  rays,  0.9  ;  for  the  mean  red,  orange,  and  yellow,  0.1  :  for  green,  0.5  ;  and  for  blue,  indigo, 
and  violet,  0.1  each  ;  and  suppose,  moreover,  in  a  beam  of  white  light,  consisting  of  10,000  rays,  all  equally 
illuminative,  the  proportions  corresponding  to  the  different  colours  to  be  as  follows  : 


Extreme  red 
200 

Red  and  orange. 
1300 

Yellow. 

3000 

Green. 
2800 

Blue. 
1200 

Indigo. 
1000 

Violet. 
500. 

Then,  after  passing  through  a  thickness 
would  be 

equal  to  1  of 

the  medium, 

the  proportions  in 

the  transmitted  beam 

Extreme  red. 
180 

Red  and  orange. 
130 

Yellow. 
300 

Green. 
1400 

Blue. 

120 

Indigo. 
100 

Violet. 
50. 

After  traversing  a  second  unit  of  thickness, 

they  would  be 

Extreme  red. 
162 

Red  and  orange. 
13 

Yellow. 
30 

Green. 
700 

Blue 
12 

Indigo. 

10 

Violet. 
5. 

and 

after  a  third,  a 

fourth,  a  fifth,  and  sixth 

respectively, 

Extreme  red. 
146 
131 

Red  and  orange. 
1 
0 

Yellow. 
3 
0 

Green. 
350 
175 

Blue. 

J. 

0 

Indigo. 

i 

0 

Violet 
0 
0 

118 

0 

0 

87 

0 

0 

(1 

106 

0 

0 

43 

0 

0 

0. 

Thus  we  see,  that  in  the  first  of  these  transmitted  beams  the  green  greatly  preponderates  ,  after  the  second 
transmission,  it  is  still  the  distinguishing  colour ;  but  after  the  third,  the  red  bears  a  proportion  to  it  large 
enough  to  impair  materially  the  purity  of  its  tint.  The  fourth  transmission  may  be  regarded  as  totally  extin- 
guishing all  the  other  colours,  and  leaving  a  neutral  tint  between  red  and  green  ;  while,  in  all  the  tints 
produced  by  further  successive  transmissions,  the  red  preponderates  continually  more  and  more,  till  at  length 
the  tint  becomes  no  way  distinguishable  from  the  homogeneous  red  of  the  extremity  of  the  spectrum. 
496.  Whether  we  suppose  the  obscurer  parts  of  the  spectrum  to  consist  of  fewer  rays  equally  illuminative,  or  of 

Relative  il-  the  same  number  of  rays  of  less  intrinsic  illuminating  power  with  the  brighter,  obviously  makes  no  difference  in 
luminative  the  conclusion,  but  the  former  supposition  has  the  advantage  of  iiffording  a  hold  to  numerical  estimation  which 
power  of  the  latter  does  not.  In  the  instance  here  taken,  the  numbers  are  assumed  at  random.  But  Fraunhofer  has  made  a 
rim""  ser'es  of  experiments  expressly  to  determine  numerically  the  illuminating  power  of  the  different  rays  of  the  spectrum, 
ravs.  According  to  which,  he  has  constructed  the  curve  fig.  116,  whose  ordinate  represents  the  illuminative  power  of 

Fig.  116.  the  ray  in  that  part  of  the  spectrum  on  which  it  is  svipposed  erected,  or  the  proportional  number  of  equally 
illuminative  rays  of  that  refrangibility  in  white  light.  If  we  would  take  this  into  consideration  in  our  geome- 
trical construction,  we  must  suppose  the  type  of  white  light,  instead,  of  being  a  straight  line,  as  in  fig.  112.  ... 
114,  to  be  a  curve  similar  to  fig.  116,  ajid  the  other  derivative  curves  to  be  derived  from  it  by  the  same  rules 
as  above.  But  as  the  only  use  of  such  representations  is  to  express  concisely  to  the  eye  the  general  scale  of 
action  of  a  medium  on  the  spectrum,  this  is  rather  a  disadvantageous  than  a  useful  refinement. 

49~  To  take  another  instance.     If  we  examine  various  thicknesses  of  the  smalt-blue  glass  above  noticed,  it  will 

be  found  to  appear  purely  blue  in  small  thicknesses.  As  the  thickness  increases,  a  purple  tinge  comes  on,  which 
becomes  more  and  more  ruddy,  and  finally  passes  to  a  deep  red  ;  a  great  thickness  being,  however,  required 
to  produce  this  effect.  If  we  examine  the  tints  by  a  prism,  we  shall  find  the  type  of  this  medium  to  be  as  in 
Fig.  117.  fig- 117,  having  four  maximum  ordinates,  thp  greatest  corresponding  to  a  ray  at  the  very  farthest  extremity  of  the 
red,  and  diminishing  with  such  rapidity  as  to  cause  an  almost  perfect  insulation  of  this  ray ;  the  next  corresponds 
to  a  red  of  mean  refrangibility,  the  next  to  the  mean  yellow,  and  the  last  to  the  violet,  the  ordinate  increasing 
continually  to  the  end  of  the  spectrum.  Thus,  when  apiece  of  such  glass  of  the  thickness  0.042  inch  was  used, 
the  red  portion  of  the  spectrum  was  separated  into  two,  the  least  refracted  being  a  well  defined  band  of  per 
fectly  homogeneous  and  purely  red  light,  separated  from  the  other  red  by  a  band  of  considerable  breadth,  and 
totally  black.  This  red  was  nearly  homogeneous ;  its  tint,  however,  differing  in  no  respect  from  the  former, 
and  being  free  from  the  slightest  shade  of  orange.  Its  most  refracted  limit  came  very  nearly  up  to  the  dark  line 
D  in  the  spectrum.  A  small,  sharp,  black  line  separated  this  red  from  the  yellow,  which  was  a  pretty  well  defined 
band  of  great  brilliancy  and  purity  of  colour,  of  a  breadth  exceeding  that  of  the  first  red,  and  bounded  on  the 


LIGHT.  433 

Ligot     preen  side  by  an  obscure  but  not  quite  black  interval.     The  green  was  dull  and  ill  defined,  but  the  violet  was     Hart  II. 

•—>,-— •>  transmitted  with  very  little  loss.     A  double  thickness  (0.084  inch)  obliterated  the  second  red,  greatly  enfeebled  ^— ~v— — 

the  yellow,  leaving  it  now  sharply  divided  from  the  green,  which  was  also  extremely  enfeebled.     The  extreme 

red,  however,   retained   nearly  its  whole  light,  and  the  violet  was  very  little  weakened.     When  a  great  many 

thicknesses  were  laid  together,  the  extreme  red  and  extreme  violet  only  passed. 

Among  transparent  media  of  most  ordinary  occurrence,  we  may  distinguish,  first,  those  whose  type  has  its       498. 
ordinate  decreasing  regularly,  with  more  or  less  rapidity  from  the  red  to  the  violet  end  of  the  spectrum,  or  Red  media, 
which  absorb  the  rays  with  an  energy  more  or  less  nearly  in  some  direct  ratio  of  their  refrangibility.     In  red 
and  scarlet  media  the  absorbent  power  increases  very  rapidly,  as  we  proceed  from  the  red  to  the  violet.     In 
yellow,  orange,  and  brown  ones,  less  so  ;  but  all  of  them  act  with  great  energy  on  the  violet  rays,  and  produce 
a  total  obliteration  of  them.     In  consequence  of  this,  by  an  increase  of  thickness,  all  these  media  finally  become 
red.     Examples:    red,  scarlet,  brown,  and  yellow  glasses;   port  wine,  infusion  of  saffron,  permuriate  of  iron, 
muriate  of  gold,  brandy,  India  soy,  &c. 

Among  green  media,  the  generality  have  a  single  maximum  of  transmission  corresponding  to  some  part  of      499. 
the  green  rays,  and  their  hue  in   consequence  becomes  more  purely  green  by  increase  of  thickness.     Of  this  Simple 
kind  are  green  glasses,  green  solutions  of  copper,  nickel,  &c.     They  absorb  both  ends  of  the  spectrum  with  greenmedia. 
great  energy  ;  the  red,  however,  more  so,  if  the  tint  verges  to  blue  ;    the  violet,  if  to  yellow.     Besides  these, 
however,  are  to  be  remarked  media  in  which  the  type  has  two  maxima ;    such  may  be   termed  dichromatic,  Dichromatic 
having  really  two  distinct  colours.     In  most  of  these,  the  green  maximum  is  less  than  the  red ;  and  the  green  media. 
tint,  in  consequence,  loses  purity  by  increase  of  thickness,  and  passes  through  a  livid  neutral  hue  to  red,  though 
this  is  not  always  the  case.     Examples :    muriate  of  chrome,  solution  of  sap-green,  manganesiate  of  potash, 
alkaline  infusion  of  the  petals  of  the  peonia  officinalis  and  many  other  red  flowers,  and  mixtures  of  red  and 
blue  or  green  media. 

Blue  media  admit  of  great  variety,  and  are  generally  dichromatic,  having  two  or  even  a  great  many  maxima       500 
and  minima  in  their  types ;  but  their  distinguishing  character  is  a  powerful  absorption  of  the  more  luminous  Blue  media, 
red  rays  and  the  green,  and  a  feeble  action  on  the  more  refrangible  part  of  the   spectrum.     Among  those  whose 
energy  of  absorption  appears  to  increase  regularly  and  rapidly  from  the  violet  to  the  red  end  of  the  spectrum, 
we  may  place  the  blue  solutions  of  copper.     The  best  example  is  the  magnificent  blue  liquid  formed  by  super- 
saturating sulphate  of  copper  with  carbonate  of  ammonia.     The  extreme  violet  ray  seems  capable  of  passing 
through  almost  any  thickness  of  this  medium  ;  and  this  property,  joined  to  the  unalterable  nature  of  the  solution, 
and  the  facility  of  its  preparation,  render  it  of  great  value  in  optical  researches.     A  vessel,  or  tube,  of  some  Insulationof 
inches  in  length,  closed  at  two  ends  with  glass  plates,  and  filled  with  this  liquid,  is  the  best  resource  for  experi-  '^e  extreme 
ments  on  the  violet  rays.     Ammonio-oxalate  of  nickel  transmits  the  blue  and  extreme  red,  but  stops  the  violet.      V1°  et> 

Purple  media  act  by  absorbing  the  middle  of  the  spectrum,  and  are  therefore  necessarily  always  dichromatic,       591 
some  of  them  having  red  and  others  violet  for  their  ultimate  or  terminal  tint.     Example:    solution  of  archil ;  purpie  ' 
purple,  plum-coloured,  and  crimson  glasses ;  acid  and  alkaline  solutions  of  cobalt,  &c.     They  may  be  termed  red-  media, 
purple  and  violet-purple,  according  to  their  terminal  tint. 

In  combinations  of  media,  the  ray  finally  transmitted  is  the  residuum  of  the  action  of  each.     If  x,  y,  z  be      502. 
the  indices  of  transmissibility  of  a  given  ray  C  in  the  spectrum  for  the  several  media,  and  r,  s,  t  their  thicknesses,  Combina- 
the  transmitted  portion  of  this  ray  will  be  C  .  jf  y'  z' ;  and  the  residuum  of  a  beam  of  white  light  (supposing  'ions  of 
none  lost  by  reflexion  at  the  surfaces)  after  undergoing  the  absorptive  action  of  all  the  media,  will  be  media. 

C  .xry'z'  +  C'.xlry"  z"  +  &c. 

An  expression  which  shows  that  it  is  indifferent  in  what  order  the  media  are  placed.  They  may  therefore  be 
mixed,  unless  a  chemical  action  take  place.  Thus  also,  by  the  same  construction  as  that  by  which  the  type  I 
of  the  first  medium  is  derived  from  the  straight  line  representing  white  light,  may  another  type  2  be  derived  from 
1,  and  so  on ;  and  thus  an  endless  variety  of  types  will  originate,  having  so  many  tints  corresponding  to  them. 

This   circumstance  enables  us  to  insulate,  in  a  state  of  considerable  homogeneity,  various  rays.     Thus,  by      o03. 
combining  with  the  smalt-blue  glass,  already  mentioned,  any  brown  or  red  glass  of  tolerable  fulness  and  purity  Insulation 
of  colour,  a  combination  will   be  formed  absolutely  impermeable  to   any  but  the  extreme  red   ray,  and  the  ofanex- 
refrangibility  of  this  is  so  strictly  definite  as  to  allow  of  its  being  used  as  a  standard  ray  in  all  optical  inquiries,  ^aelus"^ 
which  is  the  more  valuable,  as  the  coloured  glasses  by  which  it  is  insulated  are  the  most  common  of  any  which  ray. 
occur  in  the  shops,  and  may  be  had  at  any  glazier's.     If  to  such  a  combination  a  green  glass  be  added,  a  total 
stoppage  of  all  light  takes  place.     The  same  kind  of  glass,  too,  enables  us  to  insulate  the  yellow  ray,  corres-  Insuiatioa 
ponding  to  the  maximum  Y  in  the  type  fig.  117,  by  combining  it  with  a  brown  glass  to  stop  out  the  more,  and  of  tlle 
a  green  to  destroy  the  less,  refrangible  rays,  and  by  their  means  the  existence   of  a  considerable  breadth  of  yellow  ra7»- 
yellow  light,  evidently  not  depending  on  a  mixture,  or  mutual  encroachment  of  red  and  green,  may  be  exhibited 
in  the  solar  spectrum. 

It  has  been  found  by  Dr.  Brewster,  that  the  proportions  of  the  different  coloured  rays   absorbed  by  media       5°4. 
depend  on  their  temperature.     The  tints  of  bodies  generally  deepen  by  the  application  of  heat,  as  is  known  to  Alteration ol 
all  who  are  familiar  with  the  use  of  the  blow-pipe ;  thus  minium  and  red  oxide  of  mercury  deepen  in  their  hues  *  ^  'bv* 
by  heat  till  they  become  almost  black,  but  recover  their  red  colours  on  cooling.     Dr.  Brewster  has,  however,  heat, 
produced  instances,  not  merely  among  artificial  glasses,  but  among  transparent  minerals,  where  a  transition  takes 
place  from  red  to  green  on  the  application  of  a  high  temperature ;    the  original  tint  being,  however,  restored  on 
cooling,  and  no  chemical  alteration  having  been  produced  in  the  medium. 

The  analysis  of  the  spectrum  by  coloured  media  presents  several  circumstances  worthy  of  remark.     First,  the       50i> 
irregular  and  singular  distribution  in  the  dark  bands  which  cross  the  spectrum,  when  viewed  through    such 
VOL.  iv.  3  t 


434  LIGHT. 

L'ght-      media  as  have  several  maxima  of  transmission,  obviously  leads  us  to  refer  Fraunhofer's  Fixed  lines,  and  the     Pari 
S""V»''  analogous  phenomena  to  be  noticed  in  the  light  from   other  sources,  to  the  same  cause,  whatever  it  may  be,  v""v 
which  determines  the  absorption  of  some  ray  in  preference  to  others.     It  is  no  impossible  supposition,  that  the 
deficient  rays  in  the  light  of  the  sun  and  stars  may  be  absorbed  in  passing  through  their  own  atmospheres,  or,  to 
approach  still  nearer  to  the  origin  of  the  light,  we  may  conceive  a  ray  stifled  in  the  very  act  of  emanation  from 
a  luminous  molecule  by  an  intense  absorbent  power  residing  in  the  molecule  itself;    or,  in  a  word,  the  same 
indisposition  in  the  molecules  of  an  absorbent  body  to  permit  the  propagation  of  any  particular  coloured  ray 
through,  or  near  them,  may  constitute  an  obstacle  in  limine  to  the  production  of  the  ray  from    them.     At  all 
events,  the  phenomena  are  obviously  related,  though  we  may  not  yet  be  able  to  trace  the  particular  nature    of 
their  connection. 

506.  The  next  circumstance  to  be  observed  is,  that  when  examined  through  absorbent  media  all  idea  of  regular 
gradation  of  colour  from  one  end  to  the  other  of  the  spectrum  is  destroyed.     Rays  of  widely  different  refrangi- 
bility,  as  the  two  reds  noticed  in  Art.  497,  have  absolutely  the  same  colour,  and  cannot  be  distinguished.     On 
the  other  hand,  the  transition  from  pure  red  to  pure  yellow,  in  the  case  there  described,  is  quite  sudden,  and  the 
contrast  of  colours  most  striking,  while  the  dark  interval  which  separates  them,  by  properly  adjusting  the 
thickness  of  the  glass,  may  be  rendered  very  small  without  any  tinge  of  orange  becoming  perceptible.     What 
then,  we  may  ask,  is  become  of  the  orange ;  and  how  is  it,  that  its  place  is  partly  supplied  with  red  on  one  side, 
and  yellow  on  the  other  ?    These  phenomena  certainly  lead  us  very  strongly  to  believe  that  the  analysis  of  white 
light  by  the  prism  is  not  the  only  analysis  of  which  it  admits,  and  that  the  connection  between  the  refrangibility 
and  colour  of  a  ray  is  not  so  absolute  as  Newton  supposed.     Colour  is  a  sensation  excited  by  the  rays  of  light, 
and  since  two  rays  of  different  refrangibilities  are  found  to  excite  absolutely  the  same  sensation  of  colour,  there 
is  no  primd  facie  absurdity  in  supposing  the  converse, — that  two  rays  capable  of  exciting  sensations  of  different 
colours  may  have  identical  indices  of  refraction.     It  is  evident,  that  if  this  be  the  case,  no  mere  change  of 
direction  by  refractions  through  prisms,  &c.  could  ever  separate  them ;  but  should  they  be  differently  absorbable 
by  a  medium  through  which  they  pass,  an  analysis  of  the  compound  ray  would  take  place  by  the  destruction  of 
one  of  its  parts.     This  idea  has   been   advocated  by  Dr.  Brewster,  in  a  Paper   published  in  tile  Edinburgh 
Philosophical  Transactions,  vol.   ix.,  and  the  same  consequence  appears  to  follow  from  other  experiments,  pub- 
lished in  the  same  volume  of  that  collection.     According  to  this  doctrine,  the  spectrum  would  consist  of  at  least 
three  distinct  spectra  of  different  colours,  red,  yellow,  and  blue,  over-lapping  each    other,  and  each  having  a 
maximum  of  intensity  at  those  points  where  the  compound  spectrum  has  the  strongest  and  brightest  tint  of 
that  colour. 

507.  It  must  be  confessed,  however,  that  this  doctrine  is  not  without  its  objections ;  one  of  the  most  formidable  of 
Cases  of       which  may  be  drawn  from  the  curious  affection  of  vision  occasionally  (and  not  very  rarely)  met  with  in  certain 
persons  who  individuals,  who  distinguish  only  two  colours,  which  (when  carefully  questioned  and  examined  by  presenting  to 

two  them,  not  the  ordinary  compound  colours  of  painters,  but  optical  tints  of  known  composition)  are  generally 
found  to  be  yellow  and  blue.  We  have  examined  with  some  attention  a  very  eminent  optician,  whose  eyes  (or 
rather  eye,  having  lost  the  sight  of  one  by  an  accident)  have  this  curious  peculiarity,  and  have  satisfied  ourselves, 
contrary  to  the  received  opinion,  that  all  the  prismatic  rays  have  the  power  of  exciting  and  affecting  them  with 
the  sensation  of  light,  and  producing  distinct  vision,  so  that  the  defect  arises  from  no  insensibility  of  the  retina 
to  rays  of  any  particular  refrangibility,  nor  to  any  colouring  matter  in  the  humours  of  the  eye,  preventing 
certain  rays  from  reaching  the  retina,  (as  has  been  ingeniously  supposed,)  but  from  a  defect  in  the  sensorium, 
by  which  it  is  rendered  incapable  of  appreciating  exactly  those  differences  between  rays  on  which  their  colour 
depends.  The  following  is  the  result  of  a  series  of  trials,  in  which  a  succession  of  optical  tints  produced  by 
polarized  light,  passing  through  an  inclined  plate  of  mica,  in  a  manner  hereafter  to  be  described,  was  submitted 
to  his  judgment.  In  each  case,  two  uniformly  coloured  circular  spaces  placed  side  by  side,  and  having  comple- 
mentary tints  (i.  e.  such  that  the  sum  of  their  light  shall  be  white)  were  presented,  and  the  result  of  his  judguieut 
is  here  given  in  his  own  words. 


LIGHT. 


435 


Light. 


Colours  according  to  the  judgment  of  an  ordinary  eye. 

Colours  as  named  by  the  individual  in  question. 

Inclination 
of  the 
plate  of 
mica  to  eye. 

Circle  to  the  left. 

Circle  to  the  right. 

Circle  to  the  left. 

Circle  to  the  right. 

Pale  green. 

Pale  pink. 

Both  alike,  no  more  colour 

in  them  than  in  the  cloudy 

89.5 

sky  out  of  window. 

Dirty  white. 

Ditto,  both  alike. 

Both  darker  than  before,  but 

no  colour. 

85.0 

Fine  bright  pink. 

Fine  green,  a  little  verging 
on  bluish. 

Very  pale  tinge  of  blue. 

Very  pale  tinge  of  blue. 

81.1 

White. 

White. 

Yellow. 

Blue. 

76.3 

The  limit  of 

pink  and  red. 

Both  more  coloured 

than  before 

Rich  grass  green. 

Rich  crimson. 

Yellow. 

Blue. 

74.9 

Better,  but  neither 

full  colours. 

Dull  greenish  blue. 

Pale  brick  red. 

Blue. 

Yellow. 

79.8 

Neither  so  rich 

colours  as  the  last. 

Purple  (rather  pale.) 

Pale  yellow. 

Blue. 

Yellow. 

717 

Coming  up  to  good  colours, 

the   yellow    a  better  colour 

than  a  gilt  picture-frame. 

Fine  pink. 

Fine  green. 

Yellow,  but  has  got  a  good 

Blue,  but  has  a  good  deal  of 

69.7 

deal  of  blue  in  it. 

yellow  in  it. 

Fine  yellow. 

Purple. 

Good  yellow. 

Good  blue. 

68.2 

Better  colours  than 

any  yet  seen. 

Yellowish  green. 

Fine  crimson. 

Yellow,  but  has  a  good  deal 

Blue,  but  has  a  good  deal  of 

67.0 

of  blue. 

yellow. 

Good  blue,  verging  to  in- 

Yellow, verging  to  orange. 

Blue. 

Yellow. 

65.5 

digo. 
Red,  or  very  ruddy  pink. 

Very    pale    greenish   blue, 

Both  gay  colours,  particularly 
Yellow. 

the  yellow  to  the  right. 
Blue. 

63.8 

- 

almost  white. 

Rich  yellow. 

Full  blue. 

Fine  bright  yellow. 

Pretty  good  blue. 

62.7 

White. 

Fiery  orange. 

Has  very  little  colour. 

Yellow,  but  a  different  yel- 

61.2 

low,  it  is  a  blood-looking 

yellow. 

Dark  purple. 

White. 

A  dim  blue,  wants  light. 

White,  with  a  dash  of  yel- 

59.5 

low  and  blue. 

Dull  orange  red. 

White. 

Yellow 

White,  with  blue  and  yel- 

59.0 

low  iri  it. 

White. 

Dull  dirty  olive. 

White. 

Dark. 

57.1 

Very  dark  purple. 

White. 

Dark. 

White. 

55.0 

Part  II. 


Instead  of  presenting  the  colours  for  his  judgment,  he  was  now  desired  to  arrange  the  apparatus  so  as  to       508 
make  the  strongest  possible  succession  of  contrasts  of  colour  in  the  two  circles.     The  results  were  ;  s  follow  : 


Colours  according  to  the  judgment  of 

Colours  as  named  by  the  individual 

Inclination 

an  ordinary  eye. 

in  question. 

of  the 

Circle  to  the  left. 

Circle  to  the  right. 

Circle  to  the  left. 

Circle  to  the  right. 

mica  to  eye. 

Pale  ruddy  pink. 

Blue  green. 

Yellow. 

Blue. 

59.1° 

Blue  green. 

Pale  ruddy  pink. 

Blue. 

Yellow. 

65.3 

Yellow. 

Blue. 

Yellow. 

Blue. 

63.1 

White. 

Fiery  orange. 

Blue. 

Yellow. 

61.1 

Pale  brick-red. 

White. 

Yellow. 

Blue. 

58.5 

Indigo. 

Pale  yellow. 

Blue. 

Yellow. 

54.2 

Yellow. 

Indigo. 

Yellow. 

Blue. 

52.1 

It  appears  by  this,  that  the  eyes  of  the  individual  in  question  are  only  capable  of  fully  appreciating  blue  and 
yellow  tints,  and  that  these  names  uniformly  correspond,  in  his  nomenclature,  to  the  more  and  less  refrangible 
rays,  generally  ;  all  which  belong  to  the  former,  indifferently,  exciting  a  sense  of  "  blueness,"  and  to  the  latter 
of  "  yellowness."  Mention  has  been  made  of  individuals  seeing  well  in  other  respects,  but  devoid  altogether 
of  the  sense  of  colour,  distinguishing  different  tints  only  as  brighter  or  darker  one  than  another;  but  the  case 
is,  probably,  one  of  extremely  rare  occurrence. 

Mayer,  in  an  Essay  De  Affinitate  Colorum,  (Opera  inedita,  1775,)  regards  all  colours  as  arising  from  three 
primary  ones,  red,  yellow,  and  blue ;  regarding  white  as  a  neutral  mixture  of  rays  of  all  colours,  and  black  as  a 
mere  negation  of  light.  According  to  this  idea,  were  we  acquainted  with  any  mode  of  mixing  colours  in 
simple  numerical  ratios,  a  scale  might  be  formed  to  which  any  proposed  colour  might  be  at  once  referred.  He 
proposes  to  establish  such  a  scale  in  which  the  degrees  of  intensity  of  each  simple  colour  shall  be  represented 
by  the  natural  numbers  1,2,  3.  ...  12 ;  1  denoting  the  lowest  degree  of  it  capable  of  sensibly  affecting  a  tint, 
and  12  the  full  intensity  of  which  the  colour  is  capable,  or  the  total  amount  of  it  existing  in  white  light.  Thus 
r1*  denotes  a  full  red  of  the  brightest  and  purest  tint,  yu  the  brightest  yellow,  and  612  the  brightest  blue.  To 
represent  mixed  tints,  he  combines  the  symbols  of  the  separate  ingredients.  Thus  r14  y*,  or,  more  conveniently. 
12  r  +  4  y,  represents  a  red  verging  strongly  to  orange,  such  as  that  of  a  coal  fire. 

The  scale  proposed  is  convenient  and  complete,  so  far  as  regards  what  he  calls  perfect  colours,  which  arise 
from  white  light  by  the  subtraction  of  one  or  more  proportions  of  its  elementary  rays ;  but  a  very  slight  modifi- 


509. 

Mayer's 
hypothesis 
of  three 
primary 
colours. 


510 

Modification 
of  Mayer's 
scale. 


436 


L  I  G  H  T. 


Light. 


511. 

Whites, 
greys,  and 
neutral 
tints. 


512. 

Reds,  yel- 
lows, and 
allies. 


513. 


514. 
Brawns. 


515. 

Purples. 


516. 
Gretas. 


517. 
The  same 
colour  pro- 
duced by 
different 
orismatic 
eombina 
tion. 


cation  of  his  system  will  render  it  equally  applicable  to  all,  and  it  may  be  presented  as  follows.  Suppose  we 
fix  on  100  as  a  standard  intensity  of  each  primary  colour;  or  the  number  of  rays  of  that  colour  (all  supposed  v 
equally  effective)  which  falling1  on  a  sheet  of  white  paper,  or  other  surface  perfectly  neutral,  (i.  e.  equally 
disposed  to  reflect  all  rays)  shall  produce  a  full  tint  of  that  particular  kind,  and  let  us  denote  by  such  an 
expression  as  .rR  +  yY  +  iB,  the  tint  produced  by  the  incidence  of  x  such  rays  of  primary  red,  y  such  rays 
of  yellow,  and  2  such  rays  of  blue  on  the  same  surface  together.  It  is  obvious  then,  that  the  different  numerical 
values  assigned  to  x.  y,  2,  from  1  to  100,  will  give  different  symbols  of  tints,  whose  number  will  be  100  X 
100  x  100  =  1000000,  and  therefore  quite  sufficient  in  point  of  extent  to  embrace  all  the  variety  of  colours 
the  eye  can  distinguish.  The  number  of  tints  recognised  as  distinct  by  the  Roman  artists  in  Mosaic  is  said 
to  exceed  30,000 ;  but  if  we  suppose  ten  times  this  amount  to  occur  in  nature  (and  it  is  obvious  that  these 
must  be  greatly  more  numerous  than  the  purposes  of  the  painter  admit)  we  are  still  much  within  the  limits  of 
our  scale.  It  only  remains  to  examine  how  far  the  tints  themselves  are  expressible  by  the  members  of  the  scale 
proposed. 

And  first,  then,  of  whites,  greys,  and  neutral  tints.  The  most  perfectly  neutral  tints,  which  are,  in  fact,  only 
greater  and  less  intensities  of  whiteness,  are  those  we  observe  in  the  clouds  in  an  ordinary  cloudy  day,  with 
occasional  gleams  of  sunshine.  From  the  most  sombre  shadows  to  the  snowy  whiteness  of  those  cumulus- 
shaped  clouds  on  which  the  sun  immediately  shines,  we  have  nothing  but  a  series  of  whites,  or  greys,  repre- 
sented by  such  combinations  as  R  +  Y  +  B,  2R+2  Y  +  2  B,  &c.  ;  orn  (R  +  Y  +  B)  which,  for  brevity,  we  may 
represent  by  n  W.  To  be  satisfied  of  this  we  need  only  look  through  a  tube  blackened  on  the  inside  to  prevent 
surrounding  objects  influencing  our  judgments  ;  and  any  small  portion  thus  insulated  of  the  darkest  clouds 
will  appear  to  differ  in  no  respect  from  a  portion  similarly  insulated  of  a  sheet  of  white  paper  more  or  less 
shaded. 

The  various  intensities  of  pure  reds,  yellows,  and  blues  are  represented  by  n  R,  n  Y,  and  n  B  respectively. 
They  are  rare  in  nature ;  but  blood,  fresh  gilding,  or  gamboge  moistened,  and  ultramarine  may  be  cited  as 
examples  of  them.  Scarlets  and  vivid  reds,  such  as  vermilion  and  minium,  are  not  free  from  a  mixture  of 
yellow,  and  even  of  blue ;  for  all  the  primary  colours  are  greatly  increased  in  splendour  by  a  certain  mixture 
of  white,  and  whenever  any  primary  colour  is  peculiarly  glaring  and  vivid,  we  may  be  sure  that  it  is  in  some 
degree  diluted  with  white.  The  blue  of  the  sky  is  white,  with  a  very  moderate  addition  of  blue. 

The  mixture  of  red  and  yellow  produces  all  the  shades  of  scarlet,  orange,  and  the  deeper  browns,  when  of 
feeble  intensity.  When  diluted  with  white,  we  have  lemon  colour,  straw  colour,  clay  colour,  and  all  the  brighter 
browns  ;  the  last-mentioned  tints  growing  duskier  and  dingier  as  the  coefficients  are  smaller. 

The  browns,  however,  are  essentially  sombre  tints,  and  produce  their  effects  chiefly  by  contrast  with  other 
brighter  hues  in  their  neighbourhood.  To  produce  a  brown,  the  painter  mixes  black  and  yellow,  or  black  and 
red,  (that  is,  such  impure  reds  as  the  generality  of  red  pigments,)  or  all  three ;  his  object  is  to  stifle  light, 
and  leave  only  a  residuum  of  colour.  There  u  a  brown  glass  very  common  in  modern  ornamental  windows. 
If  examined  with  a  prism,  it  is  found  to  transmit  the  red,  orange,  and  yellow  rays  abundantly,  little  green,  and 
no  pure  blue.  The  small  quantity  of  blue,  then,  that  its  tint  does  involve,  must  be  that  which  enters  as  a 
component  part  of  its  green,  (in  this  view  of  the  composition  of  colours,)  and  its  characteristic  symbol  may 
thus  be,  perhaps,  of  some  such  form  as  1 0  R  +  9  Y  +  1  B  ;  that  is  to  say,  (9  R  +  8  Y)  +  1  (R  +  Y  +  B ) ,  or  an 
orange  of  the  character  9  R  +  8  Y  diluted  with  one  ray  of  white.  It  must  be  confessed,  however,  that  the 
composition  of  brown  tints  is  the  least  satisfactory  of  all  the  applications  of  Mayer's  doctrine.  He  himself  has 
passed  it  unnoticed. 

Combinations  of  red  and  blue,  and  their  dilutions  with  white,  form  all  the  varieties  of  crimson,  purple,  violet, 
rose  colour,  pink,  &c.  The  richer  purples  are  entirely  free  from  yellow.  The  prismatic  violet,  when  compared 
with  the  indigo,  produces  a  sensible  impression  of  redness,  and  must  therefore  be  regarded  on  th;,s  hypothesis 
as  consisting  of  a  mixture  of  blue  and  red  rays. 

Blue  and  yellow,  combined,  produce  green.  The  green  thus  arising  is  vivid  and  rich ;  and,  when  proper 
proportions  of  the  elementary  colours  are  used,  no  way  to  be  distinguished  from  the  prismatic  green.  Nothing 
can  be  more  striking,  and  even  surprising,  than  the  effect  of  mixing  together  a  blue  and  a  yellow  powder,  or 
of  covering  a  paper  with  blue  and  yellow  lines,  drawn  close  together,  and  alternating  with  each  other.  The 
elementary  tints  totally  disappear,  and  cannot  even  be  recalled  by  the  imagination.  One  of  the  most  marked 
facts  in  favour  of  the  idea  of  the  existence  of  three  primary  colours,  and  of  the  possibility  of  an  analysis  of 
white  light  distinct  from  that  afforded  by  the  prism,  is  to  see  the  prismatic  green  thus  completely  imitated  by 
a  mixture  of  adjacent  rays  totally  distinct  from  it,  both  in  refrangibility  and  colour. 

The  hypothesis  of  three  primary  colours,  of  which,  in  different  proportions,  all  the  colours  of  the  spectrum 
are  composed,  affords  an  easy  explanation  of  a  phenomenon  observed  by  Newton,  viz.  that  tints  no  way 
distinguishable  from  each  other  may  be  compounded  by  very  different  mixtures  of  the  seven  colours  into  which 
he  divided  it.  Thus  we  may  regard  white  light,  indifferently,  as  composed  of 

b  rays  of  pure  red  =  R1 

c  +  d  rays  of  orange  (c  red  +  d  yellow)  =  O 

e  rays  of  pure  yellow  =  Y' 

/  +  h  rays  of  green  (/yellow  +  h  blue)  =  G' 

g  +  i  rays  of  prismatic  blue  (§•  yellow  +  i  blue)  =  B 

k  rays  of  indigo,  or  pure  blue  =  I1 

t  +  a  rays  of  violet  (I  blue  +  a  red)  ==  V 


R  =  a  +  b  +  c  rays  of  pure  red 

rays  of  pure  yellow 


"i 

>  or  of  •< 
rays  of  pure  blue     J 


LIGHT.  437 

Light,     and  any  tint  capable  of  being'  represented  by  x  .  R  +  y  .  Y  +  z  B,  may  be  represented  equally  well  by 

m  .  R'  +  n .  O'  +  p  .  Y'  +  q  .  G'  +  r  .  B'  +  s  .  I'  +  t .  V, 
provided  we  assume  m,  n,  p,  &c.,  such  as  to  satisfy  the  equations 

mb+nc+ta  =  x;         n  d  +  p  e  +  q  f  +  rg  =  y  ;        q  h  +  r  i  +  s  k  +  tl  =  z. 

From  what  has  been  said  we  shall  now  proceed  to  show,  that,  without  departing  from  Mayer's  doctrine,  any      518. 
other  three  prismatic  rays  may  still  be  equally  assumed  as  fundamental  colours,  and  all  the  rest  compounded  Dr.  ^™s"Xf 
from  them,  provided  we  attend  only  to  the  predominant  tint  resulting,  and  disregard  its  dilution  with  white.  J^regS's 
For  instance,  Dr.  Young  has  assumed  red,  green,  and  violet  as  his  fundamental  colours  ;    and  states,  as  an  other  prima- 
experimental  fact  in  support  of  this  doctrine,  that  the  perfect  sensations  of  yellow  and  blue  may  be  produced,  ry  colours, 
the  former  by  a  mixture  of  red  and  green,  and  the  latter  by  green  and  violet.  (Lectures  on  Natural  Philosophy, 
p.  439.)     Now,  if  we  mix  together  yellow  and  white  in  the  proportion  of  m  yellow  +  n  white,  the  compound 
will  produce  a  perfect  sensation  of  yellow,  unless  m  be  small   compared  to  n ;    but,  assuming  white  to  be 
composed  as  above,  this  compound  is  equivalent  to 

n  R  red  +  (m  +  n)  Y  yellow  +  n  B  blue. 

On  the  other  hand,  if  we  mix  together  P  such  red  rays  (each  of  the  intensity  6)  and  Q  such  green  rays  (each 
consisting  of  yellow,  of  the  intensity/,  and  blue  of  the  intensity  A)  as  are  supposed  in  the  foregoing  article  U» 
exist  in  the  spectrum,  we  have  a  compound  of 

P .  6  red  +  Q  .  /yellow  +  Q  .  A  blue, 
and  these  will  be  identical  with  the  former,  if  we  take 

7iR=P6,-        (m  +  «)  Y=Q/j        nB  =  QA. 
Eliminating  Q  from  the  two  last  of  these,  we  get 

JL,   /    JL 

n       '    h    '     Y 

for  the  relation  between  M  and  N.  Now  the  only  conditions  to  be  satisfied  are  that  M  shall  be  positive,  and 
not  much  less  than  N  ;  and  it  is  evident  that  these  conditions  may  be  fulfilled  an  infinite  number  of  ways  by  a 
proper  assumption  of  the  ratio  of /to  A.  In  the  same  manner,  if  we  suppose  a  mixture  of  M  rays  primary 
blue  =  B  with  N  rays  of  white  (=  R  +  Y  +  B)  to  be  equivalent  to  P  rays  of  prismatic  green  mixed  with  Q 
of  violet,  we  get  the  equation 

_^L    _L    JL      h     Y 

n    '''   a         B    "    /         B 

Suppose,  for  example,  we  regard  white  light  as  consisting  of  20  rays  of  primary  red,  30  of  yellow,  and  50      519. 
of  blue,  and  the  several  prismatic  rays  to  consist  as  follows :  Numerictl 

'Iliutration. 

Red         8  rays  primary  red  =  A. 

Orange   7  red  +  7  primary  yellow  =  c  +  d. 

Yellow    8 yellow  =  e. 

Green    10 yellow  +  10  primary  blue  =  /-f  A. 

Blue        6 yellow  +  12  primary  blue  =  §•+»'. 

Indigo  12 blue  =  k. 

Violet    16  blue  +  5  primary  red.  =  /  +  a. 

Then  will  the  union  of  15  rays  of  such  red  with  30  of  such  green,  produce  a  compound  ray  containing 
15  X  8  =  120  of  primary  red,  30 x  10  =  300  of  primary  yellow,  and  30  X  10  =  300  of  primary  blue;  which 
are  the  same  as  exist  in  a  yellow,  consisting  of  6  rays  of  white  combined  with  4  of  primary  yellow.  In  like 
manner,  if  75  such  green  rays  be  combined  with  100  such  violet,  the  result  will  be  100  x  5  =  500  rays  of 
primary  red,  +  75  X  10  =  750  of  primary  yellow,  +  75  x  10  -f  100  x  16  =  2350  of  primary  blue,  which 
together  compose  a  tint  identical  with  that  which  would  result  from  the  union  of  25  rays  of  white  with  22  of 
primary  blue ;  that  is  to  say,  a  fine  lively  blue.  The  numbers  assumed  above,  it  must  be  understood,  are 
merely  taken  for  the  sake  of  illustration,  and  are  no  way  intended  to  represent  the  true  ratios  of  the  differently 
coloured  rays  in  the  spectrum. 

The  analogy  of  the  fixed  lines  in  the  solar  spectrum  might  lead  us  to  look  for  similar  phenomena  in  other 
sources  of  light.     Accordingly,  Fraunhofer  has  found,  that  each  fixed  star  has  its  own  particular  system  of  dark      ^® 
and  bright  spaces  in  its  spectrum  ;  but  the  most  curious  phenomena  are  those  presented  by  coloured  flames,    J16"01"^™* 
which  produce  spectra  (when  transmitted  through  a  colourless  prism)  hardly  less  capricious  than  those  afforded  flames  " 
oy  solar  light  transmitted  through  coloured  glasses.     Dr.  Brewster,  Mr.  Talbot,  and  others,  have  examined  these 


438  LIGHT. 

Light,      phenomena  with  attention  ;    but  the  subject  is   not  exhausted,  and  promises  a  wide  field  of  curious  research.      Part  !!. 
**— - v— • *  The  following  facts  may  be  easily  verified :  ^^^^«^ 

521.  t-    Most  combustible  bodies  consisting  of  hydrogen  and  carbon,  as  tallow,  oil,  paper,  alcohol,  &c.  when 

Flames  of    first  lighted  and  in  a  state  of  feeble  and  imperfect  combustion,  give  blue  flames.     These,  when  examined  bv 

combusti-     the  prism,  by  letting  them  shine  through  very  narrow  slits  parallel  to  its  edge,  as  described  in  Art.  487,  all  give 

Mes  burning  jnterrupted  spectra,  consisting,  for  the  most  part,  of  narrow  lines  of  very  definite  refrangibility,  either  separated 

by  broad  spaces  entirely  dark,  or  much  more  obscure  than  the  rest.     The  more  prominent  rays  are,  a  very  narrow 

definite  yellow,  a  yellowish  green,  a  vivid  emerald  green,  a  faint  blue,  and  a  strong  and  copious  violet. 

g22  2.  In  certain  cases  when  the  combustion  is  violent,  as  in  the  case  of  an  oil  lamp  urged  by  a  blow-pipe, 

Burning       (according  to  Fraunhofer,)  or  in  the  upper  part  of  the  flame  of  a  spirit  lamp,  or  when  sulphur  is  thrown  into 

strongly.       a  white-hot  crucible,  a  very  large  quantity  of  a  definite  and  purely  homogeneous  yellow  light  is  produced ;  and 

in  the  latter  case  forms  nearly  the  whole  of  the  light.     Dr.  Brewster  has  also  found  the  same  yellow  light  to  be 

produced  when  spirit  of  wine,  diluted  with  water  and  heated,  is  set  on  fire ;  and  has  proposed  this  as  a  means 

of  obtaining  a  supply  of  homogeneous  yellow  light  for  optical  experiments. 

523.          3.   Most   saline  bodies  have  the  power  of  imparting  a  peculiar  colour  to  flames  in  which  they  are  present, 
Flames        either  in  a  solid  or  vaporous  state.     This  may  be  shown  in  a  manner  at  once  the  most  familiar  and  most  effi- 
cac'ous>  by  tne  following  simple  process :  Take  a  piece  of  packthread,  or  a  cotton  thread,  which  (to  free  it  from 
saline  particles  should  have  been  boiled  in  clean  water,)  and  having  wetted  it,  take  up  on  it  a  little  of  the  salt 
to  be  examined  in  fine  powder,  or  in  solution.     Then  dip  the  wetted  end  of  it  into  the  cup  of  a  burning  wax 
candle,  and  apply  it  to  the  exterior  of  the  flame,  not  quite  in  contact  with  the  luminous  part,  but  so  as  to  be 
immersed  in  the  cone  of  invisible  but  intensely-heated  air  which  envelopes  it.     Immediately  an  irregular  sput- 
tering combustion  of  the   wax  on  the  thread  will  take  place,  and  the  invisible  cone  of  heat  will   be  rendered 
luminous,  with  that  particular  coloured  light,  which  characterises  the  saline  matter  employed. 
524  Thus  it  will  be  found  that,  in  general, 

The  colour  Salts  of  soda  give  a  copious  and  purely  homogeneous  yellow. 

depends  gajts  Qf  potasn  nrive  a  beautiful  pale  violet. 

chiefly  on  £-,..,.,  ,1.1  ,. 

the  base  -Salts  of  lime  give  a  brick  red,  in  whose  spectrum  a  yellow  and  a  bright  green  line  are  seen. 

Salts  of  strontia  give  a  magnificent  crimson.     If  analyzed  by  the  prism  two  definite  yellows  are  seen,  one 

of  which  verges  strongly  to  orange, 
Salts  of  magnesia  give  no  colour. 

Salts  of  lithia  give  a  red,  (on  the  authority  of  Dr.  Turner's  experiments  with  the  blow-pipe.) 
Salts   of  baryta  give  a  fine  pale  apple-green.     This  contrast  between  the  flames  of  baryta  and   strontia  is 

extremely  remarkable. 

Salts  of  copper  give  a  superb  green,  or  blue  green. 
Salt  of  iron  (protoxide)  gave  while,  where  the  sulphate  was  used. 

Of  all  salts,  the  muriates  succeed  best,  from  their  volatility.  The  same  colours  are  exhibited  also  when  any  of 
the  salts  in  question  are  put  (in  powder)  into  the  wick  of  a  spirit  lamp.  If  common  salt  be  used,  Mr.  Talbot 
has  shown  that  the  light  of  the  flame  is  an  absolutely  homogeneous  yellow ;  and,  being  at  the  same  time  very 
copious,  this  property  affords  an  invaluable  resource  in  optical  experiments,  from  the  great  ease  with  which  it 
is  obtained,  and  its  identity  at  all  times.  The  colours  thus  communicated  by  the  different  bases  to  flame,  afford 
in  many  cases  a  ready  and  neat  way  of  detecting  extremely  minute  quantities  of  them ;  but  this  rather  belongs 
to  Chemistry  than  to  our  present  subject.  The  pure  earths,  when  violently  heated,  as  has  recently  been  prac- 
tised by  Lieutenant  Drummond,  by  directing  on  small  spheres  of  them  the  flames  of  several  spirit  lamps  urged 
by  oxygen  gas,  yield  from  their  surfaces  lights  of  extraordinary  splendour,  which,  when  examined  by  prismatic 
analysis,  are  found  to  possess  the  peculiar  definite  rays  in  excess,  which  characterise  the  tints  of  flames  coloured 
by  them  ;  so  that  there  can  be  no  doubt  that  these  tints  arise  from  the  molecules  of  the  colouring  matter  reduced 
to  vapour,  and  held  in  a  state  of  violent  ignition. 


L  I  G  H  T  439 


PART  III. 

OF  THE  THEORIES  OF  LIGHT. 

Light.  AMONG  the  theories  which  philosophers  have  imagined  to  account  for  the  phenomena  of  light,  two  principally  part  ni. 
•v-^^  have  commanded  attention ;  the  one  conceived  by  Newton,  and  called  from  his  illustrious  name,  in  which  light  \^. -^~m. 
is  conceived  to  consist  of  excessively  minute  molecules  of  matter  projected  from  luminous  bodies  with  the  525. 
immense  velocity  due  to  light,  and  acted  on  by  attractive  and  repulsive  forces  residing  in  the  bodies  on  which 
they  impinge,  which  turn  them  aside  from  their  rectilinear  course,  and  reflect  and  refract  them  according  to 
the  laws  observed.  The  other  hypothesis  is  that  of  Huygens,  and  also  called  after  his  name ;  which  supposes 
light  to  consist,  like  sound,  in  undulations,  or  pulses,  propagated  through  an  elastic  medium.  This  medium  is 
conceived  to  be  of  extreme  elasticity  and  tenuity  ;  such,  indeed,  that  though  filling  all  space,  it  shall  offer  no 
appreciable  resistance  to  the  motions  of  the  planets,  comets,  &c.  capable  of  disturbing  them  in  their  orbits.  It 
is,  moreover,  imagined  to  penetrate  all  bodies  ;  but  in  their  interior  to  exist  in  a  different  state  of  density  and 
elasticity  from  those  which  belong  to  it  in  a  disengaged  state,  and  hence  the  refraction  and  reflexion  of  light. 
These  are  the  only  mechanical  theories  which  have  been  advanced.  Others,  indeed,  have  not  been  wanting  ; 
such  as  Professor  Oersted's,  who,  in  one  of  his  works,  considers  light  as  a  succession  of  electric  sparks,  or  a 
series  of  decompositions  and  recompositions  of  an  electric  fluid  filling  all  space  in  a  neutral  or  balanced  state, 
&c.  &c.  In  this  part,  however,  we  propose  only  to  give  an  account  of  the  Newtonian  and  Huygenian  theories, 
so  far  as  they  apply  to  the  phenomena  already  described ;  and  thus  prepare  ourselves  for  the  remaining  more 
complex  branches  of  the  History  of  the  Properties  of  Light,  which  can  hardly  be  understood,  or  even  described, 
without  a  reference  to  some  theoretical  views. 


§  I.    Of  the  Newtonian  or  Corpuscular  Theory  of  Light. 

Postulata.  1.  That  light  consists  of  particles  of  matter  possessed  of  inertia  and  endowed  with  attrac- 
tive  and  repulsive  forces,  and  projected  or  emitted  from  all  luminous  bodies  with  nearly  the  same  velocity, 
about  200,000  miles  per  second. 

•2.  That  these  particles  differ  from  each  other  in  the  intensity  of  the  attractive  and  repulsive  forces  which 
reside  in  them,  and  in  their  relations  to  the  other  bodies  of  the  material  world,  and  also  in  their  actual  masses, 
or  inertia. 

3.  That  these   particles,  impinging  on  the  retina,   stimulate  it  and  excite  vision.      The    particles   whose 
inertia  is  greatest  producing  the  sensation  of  red,  those  of  least  inertia  of  violet,  and  those  in  which  it  is  inter- 
mediate the  intermediate  colours. 

4.  That  the  molecules  of  material  bodies,  and  those  of  light,  exert  a  mutual  action  on  each  other,  which 
consists  in  attraction  and  repulsion,  according  to  some  law  or  function  of  the  distance  between  them  ;  that  this 
law  is  such  as  to  admit,  perhaps,  of  several  alternations,  or  changes  from   repulsive  to  attractive  force ;  but 
that  when  the  distance  is  below  a  certain  very  small  limit,  it  is  always  attractive  up  to  actual  contact ;  and  that 
beyond  this  limit  resides  at  least  one  sphere  of  repulsion.     This  repulsive  force  is  that  which  causes   the 
reflexion  of  light  at  the  external  surfaces  of  dense  media ;   and  the  interior  attraction  that  which  produces  the 
refraction  and  interior  reflexion  of  light. 

5.  That  these  forces  have  different  absolute  values,  or  intensities,  not  only  for  all  different  material  bodies, 
but  for  every  different  species  of  the  luminous  molecules,  being  of  a  nature  analogous  to  chemical  affinities,  or 
electric  attractions,  and  that  hence  arises  the  different  refrangibility  of  the  rays  of  light. 

6.  That  the  motion  of  a  particle  of  light  under  the  influence  of  these  forces  and  its  own  velocity  is  regu- 
lated by  the  same  mechanical  laws  which  govern  the  motions  of  ordinary  matter,  and  that  therefore  each  particle 
describes  a  trajectory  capable  of  strict  calculation  so  soon  as  the  forces  which  act  on  it  are  assigned. 

7.  That  the  distance  between  the  molecules  of  material  bodies  is  exceedingly  small  in  comparison  with  the 
extent  of  their  spheres  of  attraction  and  repulsion  on  the  particles  of  light.     And 

8.  That  the  forces  which  produce  the  reflexion  and  refraction  of  light  are,  nevertheless,  absolutely  insensible 
at  all  measurable  or  appreciable  distances  from  the  molecules  which  exert  them. 

9.  That  every  luminous  molecule,  during  the  whole  of  its  progress  through  space,  is  continually  passing 
through  certain  periodically  recurring  states,  called  by  Newton  fits  of  easy  reflexion  and  easy  transmission,  in 
virtue  of  which  (from  whatever  cause  arising,  whether  from  a  rotation  of   the  molecules  on  their  axes,  and  the 
consequent  alternate  presentation  of  attractive  and  repulsive  poles,  or  from  any  other  conceivable  cause)  they 
are  more  disposed,  when  in  the  former  states  01  phases  of' their  periods,  to  obey  the  influence  of  the  repulsive 
or  reflective  forces  of  the  molecules  of  a  medium  ;  and  when  in  the  latter,  of  the  attractive.     This  curious  and 
delicate  part  of  the  Newtonian  doctrine  will  be  developed  more  at  large  hereafter. 


440  LIGHT. 

Ijgr.t.  It  is  the  7th  and  8th  of  these  assumptions  only  which  render  the  course  pursued  by  a  luminous  molecule,    '>art 

^^-v— -^  under  the  influence  of  the  reflective  or  refractive  forces,  capable  of  being  reduced  to  mathematical  calculation  ;  ^• 

527.  for  it  follows  immediately  from  the  8th,  that,  up  to  the  very  moment  when  such  a  molecule  arrives  in  physical 
contact  with  the  surface  of  any  medium,  it  is  acted  on  by  no  sensible  force,  and  therefore  not  sensibly  deviated 
from  its  rectilinear  path ;  and,   on  the  other  hand,  as  soon  as  it  has  penetrated  to  any  sensible  depth  within  the 
surface,  or  among  the  molecules,  by  reason  of  the  7th  of  the  above  postulates,  it  must  be  equally  attracted  and 
repelled  by  them  in  all  directions,  and  therefore  will  continue  to  move  in  a  right  line,  as  if  under  the  influence 
of  no  force.     It  is  only,  therefore,  within  that  insensible  distance  on  either  side  the  surface,  which  is  measured 
by  the  diameter  of  the  sphere  of  action  of  each  molecule,  that  the  whole  flexure  of  the  ray  takes  place.     Its 
trajectory  then  may  be  regarded  as  a  kind  of  hyperbolic  curve,  in  which  the  right  lines  described  by  it,  previous 
and  subsequent  to  its  arrival  at  the  surface,  are  the  infinite  branches,  and  are  confounded  with  the  asymptotes, 
and  the  curvilinear  portion  is  concentered  as  it  were  in  a  physical  point.     Now,  in  explaining  the  phenomena 
of  reflexion  and  refraction,  it  is  not  the  nature  of  this  curve  that  we   are  called  on  to  investigate.     This  will 
depend  on  the  laws  of  corpuscular  action,  and  must  necessarily  be  of  great  complexity.     All  we  have  to  inquire, 
is  the  direction  the  ray  will  ultimately  take  after  incidence,  and  the  final  change,  if  any,  in  its  velocity. 

528.  Let  us,  then,  consider  the  motion  of  a  molecule  urged  to  or  from  the  surface  of  a  medium   by  the  united 
Motion  of  a  attractions  or  repulsions  of  all  its  particles  acting  according  to  any  conceivable  mathematical  law.     And,  first, 
luminous       jj  js  evident,  that  supposing  the  surface  mathematically  smooth,  and  the  number  of  attractive   or   repulsive 

iderUthe     Particles   of  which   it   consists,   infinite,  their   total  resultant   force  on  the   luminous  molecule   will  act  in  a 
influence  of  direction  perpendicular  to  the  surface ,  and  will  be  insensible  at  all  sensible  distances  from  the  surface,  provided 
any  forces,    the  elementary  forces  of  each  molecule  decrease  with  sufficiently  great  rapidity  as  the  distances  increase.     This 
condition  being  supposed,  let  x  and  y  be  the  coordinates  of  the  molecule  at  any  assigned  instant ;  the  plane  of 
the  x  and  y  being  supposed  to  coincide  with  that  of  its  trajectory,  out  of  which  plane  there  is  evidently  no  force 
to  turn  it,  and  which  must  of  course  be  perpendicular  to  the  surface  of  the  medium  in  which  x  is  supposed 
to  lie :    y  then  will  be  the  perpendicular  distance  of  the  luminous  molecule  from   this   surface,  and  Y  (some 
function  of  y  decreasing  with  extreme  rapidity)  will  represent  the  force  urging  it  inwards,  or  towards  the  surface 
when  the  molecule  is  without,  from  when  within  the  medium.     Therefore,  by  the  principles  of  Dynamics,  sup- 
posing d  t  to  denote  the  element  of  the  time,  we  shall  have  for  the  equations  of  the  motion 


dt*  dt* 

and  hence,  multiplying  the  first  by  dx,  the  second  by  dy,  adding  and  integrating,  we  get 

dx*  +  rfy*  /»,, 
-r— -      -  +2   /  Y  d  y  =  constant. 


Now,  c  being  the  velocity  of  the  molecule,  we  have  t>4  = — j ,  and  therefore  this  equation  becomes 

/» 

c*  =  constant  —  2    I  Y  d  y. 


It  is,  however,  only  with  the  terminal  velocity,  or  that  attained  by  the  light  after  undergoing  the  total  action  of 
the  medium,  that  we  are  concerned,  and  therefore  if  we  put  V  for  its  primitive,  or  initial,  and  V  for  its  terminal 
velocity,  we  shall  have,  by  extending  the  integral  from  the  value  of  y  at  the  commencement  of  the  ray's  motion 
(y0)  to  its  value  at  the  end  (y;), 


-  V«=  -  2  fa  dy.  (6) 


Since  y0  and  yt  are  supposed  infinite,  and  since  the  function  Y  decreases  by  hypothesis  with  such  rapidity  as  to 
become  absolutely  insensible  for  all  sensible  values  of  y,  it  is  clear  that  we  may  take  y0  =  +  CD  for  the  first 
limit  of  the  integral  in  all  cases.  With  regard  to  the  other,  we  have  now  to  distinguish  two  principal 
cases  : 

529.  The  first  is  that  of  reflexion,  where  the  ray,  no  matter  whether  before  its  arrival  at  the  surface,  or  at  reaching 

Case  of  re-  it,  or  even  after  passing  some  small  distance  into  the  medium,  is  turned  back  by  the  prevalence  of  the  repulsive 

tuion.         force,  and  pursues  the  whole  of  its  course  afterwards  without  the  medium.     Now  in  this  case  if  we  resolve  the 

integral  /"Y  dy  into  its  elements,  these,  in  the  approach  of  the  molecule  to  the  surface,  may  be  represented  as 

follows, 

&c.  +  Y'  x  -  d  y  +  Y"  x  -  d  y  +  Y'"  x  -  dy  +&c 

But  in  the  recess  of  the  molecule,  the  values  of  y  increase  again  by  the  same  steps  as  they  before  diminished 
and  become  identical  with  the  former  ones;  and  Y',  Y",  &c.,  the  values  of  Y  corresponding  to  the  successive 
values  of  y,  remain  therefore  the  same,  both  in  size  and  magnitude ;  the  corresponding  elements  of  the  integral 
generated  during  the  recess  of  the  molecule  will  be  then 

&c.  +  Y'  x  -f  d  y  +  Y"  x  +  d  y  +  Y'"  x  +  d  y  +  &c. 


LIGHT.  441 

L«1|L       So  that,  combining  both,  the  latter  exactly  destroy  the  former,  and  give  y  Y  d  y  —  0  when  extended  from  one  end     Par»  HI- 
"-V~™"/  to  the  other  of  the  trajectory.     Thus  we  have,  in  the  case  of  reflexion,  v— "V~~*' 

\'t  _  Vs  =  0,         or  V  =  V. 

The  second  case  is  that  in  which  the  whole  course  of  the  ray  after  incidence  lies  within  the  medium,  or  the  case        530. 
of  refraction.     Here  the  values  of  y  before  incidence  are  all  positive,  and  after,  all  negative;  and,  moreover,  the  Case  of 
change  of  sign  in  dy  which  happened  in  the  case  of  reflexion,  does  not  here  take  place.     Hencey' Ydy  must  refraction. 
be  extended  from  -\-  oo  to  —  oo ,  and  its  value  will  not  vanish,  but  (on  account  of  the  rapid  decrease  of  the 
function  Y)  will  have  some  finite  value.     Now  this  can  only  be  dependent  on  the  arbitrary  quantities  which 
enter  into  the  composition  of  Y ;  in  other  words,  on  the  nature  of  the  medium  and  the  ray,  and  not  at  all  on  the 
constants  which  determine  the  direction  of  the  ray  with  respect  to  the  surface,  (as  its  inclination  or  the  position 
of  the  plane  of  incidence.)     Hence  we  may  suppose  J"Y  dy  =  —  \  k  V2,  where  k  is  a  constant  independent  of 
the  direction  of  the  ray,  and  determined  only  by  its  nature  and  that  of  the  medium,  and  we  shall  have 

putting  v'l  -(-  k  =  fi. 

Hence  we  see  that  both  in  refraction  and  reflexion,  on  this  hypothesis,  the  velocity  of  the  ray  after  deviation       531. 
is  the  same  in  whatever  direction  the  ray  be  incident,  viz.  in  a  given  ratio  to  the  velocity  before  incidence,  this  Law  of 
'ratio  being  one  of  equality  in  the  case  of  reflexion.  velocities. 

Let  us  next  consider  the  direction  of  the  ray  after  flexure.     To  this  end  let  0  =  the  angle  made  by  its  path       55^. 

,  Direction  at 

at  any  moment  with  the  perpendicular  to  the  surface,  then  will  sin  0  =  — — ,  putting  ds  for   -v/dar4-)-  dy\  the  'llerayafte* 

element  of  the  arc.     Now  if  we  integrate  the  equation  —r-^-  =  0  once    we  get  — —  =  constant  =  c,  and 

d  t  tit 

d  x  =  c  d  t,  wherefore  sin  0  =  — ; — .     But  x  =  — — -,  therefore  sin  0  =  — .     Let  therefore  0n  and  0,  repre- 

a  s  \        a  t  r> 

sent  the  initial  and  terminal  values  of  6,  or  the  angles  of  incidence  and  reflexion,  or  refraction  of  the  rectilinear 
oortions  of  the  ray,  and  we  get  Coustancy 

of  ratio  of 

sin  00  =  — ,  and  sin  0,  =  -L  !^"  °f  '""A 

V  V  cidence  and 

.    ,.   ...  refraction. 

and  dividing  one  by  the  other 

sin60          V 


sin  0,  V 

That  is  to  say,  the  sines  of  ircidence  and  refraction,  or  reflexion,  are  to  each  Other  in  a  constant  ratio,  viz.  the 
inverse  ratio  of  the  velocities  of  the  ray  before  and  after  incidence. 

Thus  we  see  the  Newtonian  hypothesis  satisfies  the  fundamental  conditions  of  refraction  and  reflexion  without  533. 
entering  into  any  consideration  respecting  the  laws  of  the  refracting  and  reflecting  forces,  or  even  the  order  of 
their  superposition.  There  may  be  as  many  alternations  of  attraction  and  repulsion  as  we  please,  and  the 
reflected  or  refracted  ray  may  therefore,  prior  to  its  final  recess  from  the  surface,  make  any  variety  of  undulations; 
all  that  is  required  is  the  extremely  rapid  decrease  of  the  function  Y  expressing  the  total  force  before  the  distance 
attains  a  sensible  magnitude. 

Hence  also,  V  and  V  being  the  velocities  before  and  after  incidence,  and  p  the  index  of  refraction,  we  have       534. 

which  shows,  that  when  a  ray  passes  from  a  rarer  medium  to  a  denser,  its  velocity  is  increased,  and  vice  vend. 

Moreover,  we  have  535_ 

ys  _  ys        /  V  \*  2  /*—  Y  d  ll  Refractive 

V*         ~  \  V~)   ~        ~^  ~« '  m°edfum.  * 

Now  if  we  suppose  the  form  of  the  function  Y  to  be  the  same  for  all  media,  and  that  they  differ  in  the  energy 
of  action  only  by  reason,  first,  of  a  greater  density,  owing  to  which  more  molecules  are  brought  within  the 
sphere  of  activity ;  and,  secondly,  by  reason  of  a  greater  or  less  affinity,  or  intensity  of  action  of  each  molecule, 
we  may  suppose  Y  to  be  represented  by  S  .  n .  0  (y),  where  S  is  the  specific  gravity,  or  density,  n  the  intrinsic 
refractive  energy  of  the  medium,  and  0  (y)  a  function  absolutely  independent  of  the  peculiarities  of  the  medium, 
and  the  same  for  all  natural  bodies.  Hence  f  —  \  dy=S  .n.f  —  <j>(y)  d  y  =  S  .n  .  constant  because 
f  —  0  (y)  ^y  taken  from  y  =  -j-cctoy=—  cc  •win  nOw  be  an  absolute  numerical  constant.  We  have  then, 
according  to  this  doctrine, 

/»*  —  !.  V* 

S          '2  .  constant ' 

If  ft  be  the  refractive  index  of  a  given  standard  ray  out  of  a  vacuum,  V  the  velocity  of  that  ray  in  vacuo  is  known, 
and  is  also  an  absolute  constant ;  so  that  n,  the  intrinsic  refractive  power  of  the'  medium  is  proportional  to 
voi.  iv.  3  M 


442  LIGHT. 

Light,      (refractive  index)  *  —  1 

_^         .  ..  —  -  -  .     Such  is  Newton  s  idea  of  the  refractive  power  of  a  medium  as  differing  from  its  • 

specific  gravity 

efractive  index.  It  rests,  however,  on  a  purely  hypothetical  assumption,  that  of  the  similarity  of  form  of  the 
law  of  force  for  all  media,  respecting  which  we  can  be  said  to  know  nothing  whatever.  For  a  table  of  its  values 
for  different  media,  see  the  Collection  of  Tables  at  the  end  of  this  Essay. 

536.  The  constancy  of  the  ratio  of  the  sines  of  incidence  and  refraction  has  here  been  derived  by  direct  integration 
Principle  of  of  the  fundamental  equations.     There  is,  however,  another  mode  of  deducing  this  important  law,  much  more 
least  action  cjrcujtOus,  it  is  true,  in  this  simple  case,  but  which  offers  peculiar  advantages  in  the  more  complicated  ones  of 

double  refraction  ;  and  which,  therefore  we  shall  here  explain,  to  familiarize  the  reader  beforehand  with  its 
principle  and  mode  of  application.  It  consists  in  the  employment  of  what  is  called,  in  Dynamics,  the  principle 
of  least  action,  in  virtue  of  which  the  sum  of  each  element  of  the  trajectory  described  by  any  moving  molecule 
multiplied  by  the  velocity  of  its  description  (or  the  integral  fv  d  s)  is  a  minimum  when  taken  between  any  two 
fixed  points  in  the  trajectory.  The  trajectory  described  by  any  luminous  molecule  may  be  regarded  as  consisting 
of  two  rectilineal  portions,  or  hyperbolic  branches,  confounded  with  their  asymptotes,  and  one  curvilinear  one 
concentrated  in  a  space  of  insensible  magnitude,  a  physical  point.  Within  this  point  the  whole  operation  of  the 
flexure  of  the  ray,  however  complicated,  is  performed  ;  and  here  the  velocity  is  variable.  In  the  branches  it  is 
uniform.  Suppose,  then,  A  and  B  to  be  any  two  fixed  points  in  these,  taken  as  points  of  departure  and  arrival 
of  a  ray,  and  let  C  be  the  point  in  the  surface  of  a  reflecting  or  refracting  medium  where  the  flexure  takes  place, 
and  suppose  A  C  =  S,  B  C  =  S'  and  let  a  be  the  excessively  minute  curvilinear  portion  of  the  ray  at  C,  and  v 
the  variable  velocity  with  which  it  is  described,  V  and  V  being  those  with  which  S  and  S'  are  described.  Then 
may  the  integral  fvd  s  be  resolved  into  the  three  portions  f\  dS  +  fv  da  +_/"V'd  S'.  Of  these  the  second 
is  utterly  insensible,  by  reason  of  the  minuteness  of  a,  and  the  other  two,  since  V  and  V  are  constant,  become 
merely  V.S+V.  S'. 

537.  The  position  of  C,  then,  with  respect  to  A  and  B,  will  be  determined  by  the  condition  V  .  S  +  V.  S1  =  a 
minimum,  A  and  B  being  supposed  fixed,  and  C  any  how  variable  on  the  surface.     Now,  in  the  case  before  us, 
V  the  velocity  of  the  light  before,  and  V  that  after  incidence,  are  both,  as  we  showed  in  Article  529  and  530, 
independent  of  the  direction   of  the  incident  and  reflected  or  refracted  rays,  or  of  the  position  of  C  ;    and, 
therefore,  are  to  be  considered  as  absolute  constants  in  this  problem  of  minima,  which  is  thus  reduced  to  a 
simple  geometrical  question.     Given  A  and  B  to  find  C,  a  point  in  a  given  plane,  such  that  V  (=r  constant)  x 
A  C  +  V  (==  constant)  x  B  C  shall  be  a  minimum.     Nothing  is  easier  than  the  solution.     Put  a,  b,  c,  a1,  b',  c1 
for  the  respective  coordinates  of  A  and  B,  and  x,  y,  o  for  that  of  C,  taking  the  given  plane  for  that  of  the  x,  y. 

Solution  of   Then  __     __  ____ 

thegeom*  V  .  S  +  V  .  S'  .  =  V  .  */(*  -«)»  +  (y  -  6)«  +  e*  +  V  .  V(x  —  of)*  +  (w_4')»  +  c« 

tried  pro- 

minlrnum     is  to  be  a  minimum  by  the  variation  of  x  and  y,  independent  of  each  other.    This  gives,  by  differentiation, 

'  -  *)  dx  +  (&'  -  y)  dy  1  =  0; 


and  this,  since  x  and  y  are  independent,  must  vanish,  whatever  values  are  assigned  to  d  x  and  d  y,  therefore  we 
must  have  separately 

Jl  («_*)+  X-  (rf-*)  =  0;  JL(6_y)+_|L(6'_y)  =0.  (d) 

These  give,  respectively, 

S'  V         a  -  x  S'  V        b1  -  y 


IT  V          a-x  '  S  'V          b  -y  ' 

by  equating  which  we  get 

or  multiplying  out  and  reducing 

6  —  b'          a  b'  —  5  a' 

a  ^~  QI  a  "™*  c£ 

and,  consequently, 


a  —  a 

This  equation  expresses,  that  the  two  portions  S  and  S'  of  the  ray  before  and  after  incidence  on  *.he  surface  at 
C  both  lie  in  one  plane,  and  that   this  plane  is  perpendicular  to  the  surface,  or  to  the  plane  of  the  coordi- 
nates x,  y. 
538  Again,  if  we  resume  the  equations  (d)  and  putting  them  under  the  form 

Constancy  V'  V 

of  the  ratio  S'  (a  -  x)  = rr-  S  (a1  -  *) ;  S'  (6  -  y)  =  -  -=-  (b'  -  y)  .  S. 

of  the  Mues  V 

deduced.       gquare  and  ad(J  thenj  we  gw 


LIGHT.  443 


Now  if  we  put  0  for  the  angle  made  by  the  portion  S  with  a  perpendicular  to  the  surface,  or  the  angle  of  inci- 
dence of  the  ray,  and  Q'  for  that  made  by  the  other  S'  with  the  same  perpendicular,  or  the  angle  of  emergence, 
we  shall  have  _ 

sin  e  =  ^^F-f  (6  -  ir)«    and  sin  ,,  =  ^  «  -  *)•  +  V  -jOj  m 


So  that  the  above  equation  is  equivalent  simply  to 

V 

sin  0  =  -rrr-  .  sin  fft 

which  is  the  same  with  the  result  before  obtained. 

The  principle  of  least  action,  then,  in  the  case  before  us,  has  enabled  us  to  dispense  with  one  integration  of     539. 
the  differential  equations  expressing  the  motion  of  the  luminous  molecule  ;   and  its  applicability  to  this  purpose  Advantages 
depends,  as  we  have  seen,  on  the  relation  between  V  and  V  ;   the  velocities  of  the  light,  before  and  after  inci-  afforded  by 
dence,  being  known.     This  relation  has  here  been   deduced  &  priori;  but  had  it  been  merely  known,  as  a  *  £  {J""ecal^t 
matter  of  fact,  a  conclusion  established  by  experiment,  it  would  not  be  on  that  account  the  less  applicable  to  acti0n. 
the   same  purpose,  and  the  laws  of  refraction  and  reflexion  would  be  derivable  from  it  by  the  same  process. 
There  would,  however,  be  this  main  difference  ;   that,  in  the  latter  case,  we  should  have  no  occasion  to  employ 
the  differential  equations  at  all,  and  therefore  none  to  enter  into  any  consideration  of  the  forces  acting  on  the 
luminous  molecule,  or  their  mode  of  action.     The  principle  of  least  action  establishes,  independent  of,  and 
anterior  to,  all  particular  suppositions  as  to  the  forces  which  operate  the  flexure  of  the  ray,  (further  than  that  they 
are  functions  of  the  distances  from  their  origins  or  centres,)  an  analytical  relation  between  the  velocities  before 
and  after  incidence,  and  the  directions  of  its  direct  and  deviated  branches  ;    a  relation  nearly  as  general  as 
the  laws  of  dynamics  themselves,  and  expressive,  in  fact,  of  only  the  one  condition  above  mentioned.     And  this 
relation,  from  its  form,  enables  us,  whenever  the  relation  of  the  velocities  is  known,  to  determine  that  of   the 
directions  of  the  two  portions  of  the  ray,  and  vice  versd,  without  having  recourse  to  the  differential  equations  at 
all.     In  the  simple  case  before  us  this  may  seem  a  needless  refinement,  the  equations  being  so  simple.     It  is  Applicable 
otherwise,  however,  in  the  theory  of  double  refraction.     There  the  forces  in  action  are  altogether  unknown,  not  to  other 
only  in  respect  of  their  intensity,  but  of  their  directions  ;  and  so  far,  therefore,  from  being  able  in  that  theory  to  case$- 
integrate  the  equations  of  the  ray's  motion,  we  cannot  even  express  them  analytically.     The  principle  we  are 
now  considering  is,  in  such  a  case,  all  the  ground  we  have  to  stand  upon  ;  and  has  been  ingeniously  and   ele- 
gantly applied  by  Laplace,  in  that  theory,  to   reduce  the  complicated  laws   of  double   refraction   under  the 
dominion  of  analysis. 

In  fact,  suppose  that  the  velocities  of  the  incident  and  deviated  portions  of  the  rays,  instead  of  being  the  same      540. 
in  every  direction,  varied  with  the  positions  of  these  portions  with  respect  to  the  surface  of  the  medium,  or  to  Mode  of  in 
any  fixed  lines  or  axes  in  space.     Then  will  V  and  V,  instead  of  being  invariable,  be  represented  by  functions  application 
of  the  three  coordinates  of  the  point  C,  either  rectangular,  as  x,  y,  z  ;   or  polar,  as  0,  0,  and  7  ;  and  the  portions  ln  gcneral- 
S  and  S'  of  the  rays  intercepted  between  A  and  B  respectively,  and  the  surface  at  C,  will,  in   like  manner,  be 
functions  of  the  same  coordinates.     So  that  the  condition 

V  .  S  -f  V  .  S1  =  a  minimum 

will  afford,  by  differentiation  and  putting  the  differential  equal  to  zero,  an  equation  of  the  form  iidx  +  M<£y 
4-  X  d  z  =  0,  or  Ld0  +  Md0  +  Nd<y=±:0,  as  the  case  may  be.  The  equation  of  the  surface  also  being 
differentiated  affords  another  relation  of  the  same  kind  ;  and  these  being  the  only  conditions  to  which  the  diffe- 
rentials dx,  dy,  dz  are  subject,  we  may  eliminate  one,  and  put  the  coefficients  of  the  remaining  ones  separately 
equal  to  zero.  Thus  we  get  two  equations  between  the  coordinates,  which,  combined  with  that  of  the  surface, 
suffice  to  determine  them,  i.  e.  to  fix  the  point  C  at  which  the  ray  A  C  must  meet  the  surface,  in  order  that,  being 
there  deviated  by  the  action  of  the  medium,  it  may,  after  flexure,  proceed  to  B  ;  and  thus  the  problem  of 
reflexion  or  refraction  may  be  resolved  in  all  its  generality,  so  soon  as  the  nature  of  the  functions  V,  V  is 
known.  But  to  return  to  the  case  of  ordinary  reflexion  and  refraction,  from  which  this  is  a  digression. 

Let  us  consider,  a  little  more  in  detail,  what  may  be  conceived  to  happen  to  a  ray  at  the  confines  of  the  surface      541. 
of  a  medium.     We  may  suppose,  then,  that  there  exist  a  series  of  laminar  spaces,  or  strata,  within  which  the  Course  of  a 
attractive  and  repulsive  action  of  the  molecules  of  the  medium  alternately  predominate.     Of  these  there  may  rayjn  tl)e, 
be  any  number,  and  either  may  be  exterior  to  the  rest.     It  is,  in  fact,  the  assemblage  of  these  laminae  which  is  ref)ectjr? 
to  be  regarded  as  the  surface  of  the  medium.     Suppose  now  a  ray  A  a  (fig.  119)  to  be  moving  towards  the  and  retract- 
medium.     Its  course  will  be  rectilinear  up  to  a,  where  it  first  comes  within  the  action  of  the  medium.     If  the  i»?  medium 
first  stratum  into  which  it  enters  be  one  of  attraction,  its  course  will  be  bent  as  a  b  into  a  curve  concave  towards  'race(l 
the  surface,  and  its  velocity  in  the  direction  perpendicular  to  the  surface  will  be  increased.     Arrived  at  ft  let  the    >s' 
force  change  to  repulsive  ;  the  trajectory  will  have  at  6  a  point  of  contrary  flexure,  the  portion  b  c  within  this 
lamina  will  be  convex  to  the   surface,  and  the  velocity  towards  the  surface  will  be  diminished    in   the  whole 
progress  of  the  ray  througli  it,  and  so  for  any  number  of  alternations.     Let  us  now  suppose,  that   in  passing 
through  any  repulsive  lamina,  as  C,  the  repulsion  should  be  so  strong,  or  the  original  velocity  of  approach  to  the 
surface  so  small,  as  that  the  whole  of  it  shall  be  destroyed.     In  this  case  the  ray  for  a  moment  will  be  moving 
as   at  C,  parallel  to  the  surface,  but  the  repulsive  force  continuing  its  action  will  turn  it  back  ;  and  the  forces 

3  M  2 


444 


L  I  G  H 


Light. 


542. 

Motion  of  a 
ray  at  com- 
mon surface 
of  two 
media. 

543. 

Newtonian 
idea  of  a 
ray  of  light 
as  composed 
of  a  succes- 
sion of 
molecules. 


Their 
distance 
niter  $e. 


Their  ex- 
treme 
tenuity  il- 
lustrated. 


544. 

Partial  re- 
flexion ex- 
plained on 
Aewton's 

pi  inciple  J. 


545. 

Reflexion 
more  co- 
pious at 
gre»t  obli- 
quitiet. 


now  being  all  equal  to  what  they  were  before,  but  acting  in  a  contrary  direction  with  respect  to  the  motion  of    Part  III. 
the  molecule,  it  will  describe  a  portion  Cd'  c'  b'  a'  B  similar,  and  equal  to  the  portion  on  the  other  side  of  C.   Wv"** 
This  is  the  case  of  reflexion.     But  suppose,  as  in  fig.  120,  the  ray  to  have  such  an  initial  velocity  of  approach, 
or  the  repulsive  forces  to  be  so  feeble,  compared  to  the  attractive,  that  before  its  whole  velocity  perpendicular  ta 
the  surface  is  destroyed,  it  shall  have  passed  through   all  the  strata  of  attraction  and  repulsion,  and  entered 
the  region  where  the  forces  of  all  the  molecules  are  in  equilibrium,  as  at  e.     In  this  case   the  remainder  of  its 
course  will  be  rectilinear,  and  wholly  within  the  medium.     This  is  the  case  of  refraction.     In  both  cases,  it  is 
the  final  course  it  takes,  or  the  direction  of  the  asymptotic  branches  a'  B  or  e  B,  about  which  only  we  have  any 
knowledge ;    of  the  number  and  nature   of  the  undulations  of  its  course  between  a  and  a',  or  e,  we  know 
nothing. 

The  whole  of  this  reasoning  applies  equally  to  the  motion  of  a  luminous  molecule  at  the  confines  of  two 
media,  as  at  the  surface  separating  one  medium  from  a  vacuum.  The  molecules  of  either  medium  being  sup- 
posed uniformly  distributed,  and  acting  equally  in  all  directions  around  them,  the  resultant  of  all  their  forces 
on  the  luminous  particle  must  be  perpendicular  to  the  common  surface,  which  is  all  that  is  required  in  the 
foregoing  theory. 

In  the  Corpuscular  doctrine,  a  ray  of  light  is  understood  to  mean  a  continued  succession  or  stream  of  mole- 
cules, all  moving  with  the  same  velocity  along  one  right  line,  and  following  each  other  close  enough  to  keep  the 
retina  in  a  constant  state  of  stimulus,  i.  e.  so  fast,  that  before  the  impression  produced  by  one  can  have  time  to 
subside  another  shall  arrive.  It  appears,  by  experiment,  that  to  produce  a  continued  sensation  of  light,  it  is 
sufficient  to  repeat  a  momentary  flash  about  8  or  10  times  in  a  second.  If  a  red-hot  coal  on  the  point  of  a 
burning  stick  be  whirled  round,  so  as  to  describe  a  circle,  and  the  velocity  of  rotation  be  greater  than  8  or  10 
circumferences  per  second,  the  eye  can  no  longer  distinguish  the  place  of  the  luminous  point  at  any  instant,  and 
the  whole  circle  appears  equally  bright  and  entire.  This  shows,  evidently,  that  the  sensation  excited  by  the  light 
falling  on  any  one  point  of  the  retina,  must  remain  almost  without  diminution  till  the  impression  is  repeated 
during  the  subsequent  revolution  of  the  luminary.  Now,  if  uninterrupted  vision  can  be  produced  by  momen- 
tary impressions,  repeated  at  intervals  so  distant  as  a  tenth  of  a  second,  it  is  easy  to  conceive  that  the  indivi- 
dual molecules  of  light  in  a  ray  will  not  require  to  follow  close  on  each  other  to  affect  our  organs  with  a 
continued  sense  of  light.  As  their  velocity  is  nearly  200,000  miles  per  second,  if  they  follow  each  other  at 
intervals  of  1000  miles  apart,  200  of  them  would  still  reach  our  retina  per  second,  in  every  ray.  This  conside- 
ration frees  us  from  all  difficulties  on  the  score  of  their  jostling,  or  disturbing  each  other  in  space,  and  allows  of 
infinite  rays  crossing  at  once  through  the  same  point  of  space  without  at  all  interfering  witli  each  other,  espe- 
cially when  we  consider  the  minuteness  which  must  be  attributed  to  them,  that  (moving  with  such  swiftness) 
they  should  not  injure  our  organs.  If  a  molecule  of  light  weighed  but  a  single  grain,  its  inertia  would  equal 
that  of  a  cannon  ball  of  upwards  of  150  pounds  weight,  moving  at  the  rate  of  1000  feet  per  second.  What 
then  must  be  their  tenuity,  when  the  concentration  of  millions  upon  millions  of  them,  by  lenses  or  mirrors,  has 
never  been  found  to  produce  the  slightest  mechanical  effect  on  the  most  delicately  contrived  mechanism,  in 
experiments  made  expressly  to  detect  it.  (See  Mr.  Bennet's  Experiments,  Phil  Tram.  1792,  vol.  Ixxxii.  p.  87.) 

When  a  ray  of  light  falls  on  a  reflecting  or  refracting  surface,  since  all  its  molecules  move  with  equal  velocity 
and  are  incident  in  the  same  line,  it  would  seem  that  whatever  took  place  with  one  should  equally  happen  to 
all ;  and  that,  if  the  first  underwent  reflexion,  all  should  do  so ;  while,  on  the  other  hand,  if  one  could  penetrate 
the  surface,  and  pursue  its  course  entirely  within  the  medium,  all  ought  to  do  the  same.  This,  however,  is 
contrary  to  experience  ;  as  whenever  a  ray  of  light  is  incident  on  the  exterior  surface  of  any  medium,  a  part 
only  is  reflected,  and  the  rest  enters  the  medium.  No  theory  can  be  satisfactory  which  does  not  render  a  good 
account  of  so  principal  a  fact.  The  Newtonian  doctrine  accounts  for  it  by  the  fits  of  easy  reflexion  and  trans- 
mission. To  understand  this  explanation  we  must  recur  to  the  ninth  postulate,  (Art.  526,)  and  suppose  two 
molecules  to  arrive  at  the  surface  under  the  same  incidence,  the  one  in  a  fit  of  easy  reflexion,  the  other  in  one 
of  easy  transmission.  The  former  will  then  be  more  affected  by  the  repulsive  forces,  the  latter  by  the  attractive 
of  the  molecules  of  the  medium ;  and  hence  it  is  evident,  that  (he  one  may  be  reflected  under  circumstances  of 
incidence,  &c.  in  which  the  other  will  penetrate  the  surface  and  be  refracted.  Now  it  will  depend  entirely  on 
the  nature  of  the  medium,  and  the  initial  velocity  of  a  luminous  molecule  towards  it,  (which  is  as  the  cosine  of 
the  angle  of  incidence,)  whether  it  will  require  the  whole  exertion  of  its  repulsive  forces,  in  their  most  energetic 
manner,  to  destroy  that  velocity  and  produce  reflexion,  or  only  a  part  of  them.  In  the  former  case  only  such 
molecules  as  arrive  in  the  most  favourable  disposition  to  be  reflected,  or  in  the  most  intense  phase  of  a  fit  of 
easy  reflexion,  can  be  reflected.  In  the  latter,  such  as  arrive  in  less  favourable  dispositions,  or  in  less  intense 
phases  of  fits  of  reflexion,  may  be  reflected ;  and  if  the  repulsive  forces  of  the  medium  be  very  intense,  in 
comparison  with  the  attractive  ones,  or  if  the  obliquity  of  incidence  be  so  great  as  to  give  the  molecule  a  very 
small  velocity  perpendicular  to  the  surface,  even  those  molecules  which  arrive  in  the  less  energetic  phases  of  fits 
of  easy  transmission  may  still  be  unable  to  penetrate  the  strata  of  repulsion. 

Hence,  then,  we  see  that  the  proportion  of  the  molecules  of  a  ray  falling  on  the  surface  of  a  medium  in  every 
possible  state  or  phase  of  their  fits,  which  undergo  reflexion,  will  depend,  first,  on  the  nature  of  the  medium  on 
whose  surface  they  fall,  or  if  it  be  the  common  surface  of  two,  then  on  both  ;  secondly,  on  the  angle  of  incidence. 
At  great  obliquities,  the  reflexion  will  be  more  copious ;  but  even  at  the  greatest,  when  the  incident  ray  just 
grazes  the  surface,  it  by  no  means  follows  that  every  molecule,  or  even  the  greater  part,  must  be  reflected.  Those 
which  arrive  in  the  most  favourable  phases  of  their  fits  of  transmission,  will  obey  the  influence  of  small  attrac- 
tive forces,  in  preference  to  strong  repulsive  ones ;  but  it  will  depend  entirely  on  the  nature  of  the  media  whether 
the  former  or  the  latter  shall  prevail,  the  fits  in  the  Newtonian  doctrine  being  conceived  only  to  dispose  the 
luminous  molecules,  other  circumstances  being  favourable,  to  reflexion  or  transmission  ;  to  exalt  the  forces  which 


LIGHT.  445 

Light,      tend  to  produce  the  one  and  to  depress  those  which  act  in  favour  of  the  other,  but  not  to  determine,  absolutely.    Part  III. 
*— v"1"1*  'ts  reflexion  or  transmission  under  all  circumstances.  ^^Tifi""""' 

These  conclusions  are  verified  by  experience.     It  is  observed,  that  the  reflexion  from  the  surfaces  of  transparent      Mo. 
(or  indeed  any)  media,  becomes  sensibly  more  copious  as  the  angle  of  incidence  increases ;  but  at  the  external  ^  °n™^_ 
surface  of  a  single  medium  is  never  total,  or  nearly  total.     In  glass,  for  instance,  even  at  extreme  obliquities,  a  ,,;ent 
Very  large  portion  of  the  light  still  enters  the  glass  and  undergoes  refraction.     In  opaque  media,  such  as  polished 
metals,  the  same  holds  good ;  the  reflexion  increases  in  vividness  as  the  incidence  increases,  but  never  becomes 
total,  or  nearly  so.   The  only  difference  is,  that  here  the  portion  which  penetrates  the  surface  is  instantly  absorbed 
and  stifled. 

The  phenomena  which  take  place  when  light  is  reflected  at  the  common  surface  of  two  media,  are  such  as  from 
the  above  theory  we  might  be  led  to  expect, — with  the  addition,  however,  of  some  circumstances  which  lead  us  to  ^^"^ 
J'mit  the  generality  of  our  assumptions,  and  tend  to  establish  a  relation  between  the  attractive  and  repulsive  surface  Of 
forces,  to  which  the  refraction  and  reflexion  of  light  are  supposed  to  be  owing.     For  it  is  found,  that  when  two  two  me(iia. 
media  are  placed  in  perfect  contact,  (such  as  that  of  a  fluid  with  a  solid,  or  of  two  fluids  with  one  another,)  the 
intensity  of  reflexion  at  their  common  surface  is  always  less,  the  nearer  the  refractive  indices  of  the  media  approach 
to  equality ;  and  when  they  are  exactly  equal,  reflexion  ceases  altogether,  and  the  ray  pursues  its  course  in  the 
second  medium,   unchanged  either  in  direction,  velocity,  or  intensity.     It  is   evident,  from   this  fact,  which  is 
general,  that  the  reflective  or  refractive  forces,  in  all  media  of  equal  refractive  densities,  follow  exactly  the  same 
laws,  and  are  similarly  related  to  one  another ;   and  that  in  media  unequally  refractive,  the  relation  between  the 
reflecting  and  refracting  forces  is  not  arbitrary,  but  that  the  one  is  dependent  on  the  other,  and  increases  and 
diminishes  with  it.     This  remarkable  circumstance  renders  the  supposition  made  in  Art.  535,  of  the  identity  of 
form  of  the  function  Y,  or  0  (y)  expressing  the  law  of  action  of  the  molecules  of  all  bodies  on  light  indif- 
ferently, less  improbable. 

To  show  experimentally  the  phenomena  in  question,  take  a  glass  prism,  or  thin  wedge  of  very  small  refracting      543. 
angle  (half  a  degree,  for  instance :  almost  any  fragment  of  plate  glass,  indeed,  will  do,  as  it  is  seldom  the  two  sides  Phenomem 
are  parallel,)  and  placing  it  conveniently  with  the  eye  close  to  it,  view  the  image  of  a  candle  reflected  from  the  exhibited 
exterior  of  the  face  next  the  eye.     This  will  be  seen  accompanied  at  a  little  distance  by  another  image,  reflected 
internally  from  the  other  face,  and  the  two  images  will  be  nearly  of  equal  brightness,  if  the  incidence  be  not 
very  great.     Now,  apply  a  little  water,  or  a  wet  finger,  or,  still  better,  any  black  substance  wetted,  to  the  pos- 
terior face,  at  the  spot  where  the  internal  reflexion  takes  place,  and  the  second  image  will  immediately  lose  great 
part  of  its  brightness.     If  olive  oil  be  applied  instead  of  water,  the  defalcation  of  light  will  be  much  greater, 
and  if  the  substance  applied  be  pitch,  softened  by  heat,  so  as  to  make  it  adhere,  the  second  image  will  be  totally 
obliterated.     On  the  other  hand,  if  we  apply  substances  of  a  higher  refractive  power  than  glass,  the  second  image 
again  appears.     Thus,  with  oil  of  cassia  it  is  considerably  bright ;  with  sulphur,  it  cannot  be  distinguished  from 
that  reflected  at  the  first  surface;   and  if  we  apply  mercury,  or  amalgam,  (as  in  a  silvered  looking-glass,)  the 
reflexion    at  the  common  surface  of  the  glass  and  metal  is  much  more  vivid  than  that  reflected  from  the  glass 
alone. 

The  destruction  of  leflexion  at  the  common  surface  of  two  media  of  equal  refractive  powers  explains  many       ^49. 
curious  phenomena.     If  we  immerse  an  irregular  fragment  of  a  colourless  transparent  body  (as  crown  glass)  in  | 
a  colourless  fluid  of  precisely  equal  refractive  power,  it  disappears  altogether.     In  fact,  the  surface  being  only  on  the    ' 
visible  by  the  rays  reflected  from  it ;  destroy  this  reflexion,  and  the  object  must  cease  to  he  seen,  unless  from  any  foregoing 
opacity  in  its  substance  reflecting  rays  from  its  interior,  which  is  not  here  contemplated.     Hence,  if  the  powder  principles. 
of  any  such  substance  De  moistened  with  a  fluid  of  the  same  refractive  density,  all  the  internal   and  external 
reflexions  at  the  surfaces  of  the  small  fragments  of  which  it  consists,  which,  blended  and  confused,  present  the 
general  appearance  of  a  white  opaque  mass,  will  be  destroyed,  and  the  powder  will  be  rendered  perfectly  trans-  xranspa. 
parent.     A  familiar  instance  of  this  nature  is  the  transparency  given  to  paper  by  moistening  it  with  water,  or,  rency  of 
still  better,  with  oil ;  paper  is  composed  of  an  infinity  of  minute  transparent,  or  nearly  transparent  fibres  of  a  oiled  paper 
ligneous  substance,  having  a  refractive  power  probably  not  very  different  from  some  of  the  more  refractive  oils. 
Its  whiteness  is  caused  by  the  confused  reflexion  of  the  incident  rays  at  all  possible  angles,  both  internally  and 
externally,  those  which  have  escaped  reflexion  at  one  fibre,  undergoing  it  among  those  beneath.     If  moistened 
with   any  liquid,  the  intensity  of  these  reflexions  is  weakened,  and  the  more  the  more  nearly  its  refractive  power 
approaches  to  that  of  the  paper  itself;    so  that  a  considerable  number  of  rays  find  their  way  through  it,  and 
emerge  at  the  posterior  surface.     The  transparency  acquired  by  the  hydrophane,  by  immersion  in  water,  is,  no 
doubt,  owing  to  this  cause ;  the  water  filling  up  the  minute  pores,  and  enfeebling  the  internal  reflexion;    and 
Dr.  Brewster,  in  a  very  curious  and  interesting  Paper  on  the  tabasheer,  (a  siliceous  concretion  found  in  sugar- 
canes,  and  the  lowest  in  the  scale  of  refracting  powers  among  solids,)  has  explained  on  this  principle  a  number 
of  extraordinary  phenomena  exhibited  on  moistening  that  substance  with  various  liquids,  (see  Philosophical 
Transactions,  1819.) 

The  reasoning  of  Art.  529  applies,  it  is  evident,  equally  to  the  case  when  a  ray  is  reflected  from  the  interior      550. 
surface  of  a  dense  medium  placed  in  air,  and  when  from  the  exterior.     The  only  difference  is,  that  in  the  latter  Total 
case  the  reflexion  is  performed  by  the  action  of  repulsive,  and  in  the  former  by  that  of  attractive  forces.     The  in'ern.af 
course  of  a  ray  internally  reflected  may  be  conceived,  as  in  fig.  121  and  122 ;  and  the  reflexion  may  take  place  re 
in  any  of  the  attractive  regions,  or  laminae,  whether  within  or  without  the  true  surface,  i.  e.  the  last  layer  of 
molecules  which  constitute  the  medium.     There  is  one  case  of  internal  reflexion,  however,  too  remarkable  to  be 
passed  without  more  particular  notice.     It  is,  that  when  the  interior  angle  of  incidence  exceeds  the  limiting 

angle  whose  sine  is  — ,  (see  Art.  183  d  seq. ;)  and  when,  as  we  there  stated,  as  a  result  of  experiment,  the 


446  LIGHT. 

Light.  internal  reflexion  is  total.  To  see  how  this  happens,  let  us  consider  a  ray  incident  exactly  at  this  angle,  and 
v— - -Nl— — '  in  the  most  intense  phase  of  its  fit  of  transmission.  Then  will  it  be  refracted  ;  and,  since  the  angle  of  refraction 
must  be  just  90°,  (by  reason  of  the  generality  of  the  demonstration  of  the  law  of  refraction  in  Art.  529,)  it 
will  emerge,  grazing  the  surface,  exactly  at  the  extreme  boundary  of  the  outermost  region  C  B,  (fig.  123,)  where 
all  sensible  action  ceases.  Its  initial  velocity  under  these  circumstances  in  the  direction  perpendicularly  to  the 
surface,  is  barely  sufficient  to  carry  it  up  to  this  extreme  limit,  where  it  is  quite  annihilated.  If,  then,  we 
conceive  another  ray,  also  incident  in  the  most  intense  phase  of  its  fit  of  transmission,  but  at  an  anffle  more 
oblique  by  an  infinitely  small  quantity,  then,  since  its  initial  velocity  at  right  angles  to  the  surface  is  less.it  will 
be  destroyed  before  it  has  quite  reached  this  limit,  and  the  ray  will  therefore  begin  to  move  parallel  to  the 
surface,  just  within  the  last  limit  to  the  sphere  of  its  action. 

551.  Now  the  last  action  which  the  surface  exerts,  or  that  force  which   extends  to  the  greatest  distance  from  it, 
The  outer-    cannot  be  otherwise  than  attractive;  for,  first,  were  it  repulsive,  it  is  evident  that  no  ray  incident  externally  at 
most  sphere  an  extreme  incidence,  (i.  e.  approaching  indefinitely  to  90°,)  could  by  possibility  escape  reflexion  ;  and,  secondly, 
necessarily    no  rav  on  tna'  supposition  could  emerge  from  within  the  medium,  without  having  at  its  emergence  an  obliquity 
attractive,     to  the  surface  greater  than  some  finite  angle,  the  last  action  of  the  medium  being  in  this  case  to  bend  it  outwards, 

both  which  consequences  are  contrary  to  fact.  Or  we  may  consider  the  point  thus,  Since  a  ray  incident  within* 
at  the  limiting  angle,  emerges,  if  it  emerge  at  all,  parallel  to  the  surface  ;  and  since  every  point  in  the  curve 
described  by  it  previous  to  the  instant  of  emergence  is  nearer  to  the  medium  than  the  line  of  its  ultimate 
direction,  it  is  geometrically  impossible  that  the  curvature  immediately  adjacent  to  the  point  of  emergence  should 
be  otherwise  than  concave  towards  the  medium  ;  and  must,  therefore,  of  necessity  be  produced  by  a  force  directed 
to  it,  i.  e.  an  attractive  one. 

552.  Hence,  the  luminous  molecule  we  have  been  considering,  will  be  within  the  attractive  region  at  the  moment 
when  its  perpendicular  motion  is  destroyed ;    it  will,  therefore,  be  turned  inwards,  as  at  the  dotted  line  fig. 
123,  and   be  reflected.     A  fortiori,  therefore,  win  every  molecule  incident  in   a  less   intense  phase  of  a  fit 
of  transmission,  or  in  one  of  reflexion,  as  well  as  every  one   incident  at  a  more  oblique  incidence,  i.  e.  with 
a  less  initial  perpendicular  velocity,  be  reflected.     Those  in  which   the  circumstances  are  most  favourable  to 
transmission  will  reach  the  exterior  attractive  region,  as  in  fig.  123.     Others  in  which  they  are  less  so  will  be 
reflected  in  some  intermediate  region,  as  in  fig.  122,  while  those  which  are  incident  at  extreme   internal  obli- 
quities, or  in  the  most  intense  phases  of  fits  of  reflexion,  will  have  their  courses  as  represented  in  fig.  121. 

553.  The  conclusion  at  which  we  have  arrived  in  the  last  Art.  that  the  attractive  force  of  a  medium  on  the  molecules, 
:l.re"      of  light  extends  to  a  greater  distance  than  the  repulsive,  is,  as  we  have  seen,  a  necessary  consequence  of  dyna- 

!JmiVr"fl°x-  m'cal  principles  ;  and  so  far  from  being  in  opposition  to  Newton's  doctrine  of  reflexion,  as  has  been  said,  is  in 

ion  from      perfect  accordance  with  it.     Dr.  Brewster  has  been  led   to  the  same  conclusion  by  peculiar  considerations 

water.          grounded  on  his  experiments  on  the  law  of  polarization,  (Phil.  Trans.,  1815,  p.  133,)  and  has  applied  it  to 

explain  a  curious  fact  noticed  by  Bouguer,  viz.  that  although  water  be  much  less  reflective  than  glass  at  small 

incidences,  yet  at  great  ones  (as  87°£)  it  is  much  more  so.     Now,  supposing  the  light  to  have  undergone  the 

whole  effect  of  the  refracting  forces,  in  both  cases  before  it  suffers  reflexion,  its  incidence,  when  it  reaches  the 

region  of  the  repulsive  forces,  will  have  been  diminished  in  the  case  of  glass,  to  57°  44',  but  in  that  of  water 

only  to  61°  5',  and  therefore  being  incident  more  obliquely  on  the  water  it  ought  to  be  more  copiously  reflected. 

Whatever  we  may  think  of  the  validity  of  this  explanation,  it  is  certainly  ingenious,   and  the  fact  extremely 

remarkable,  and  deserving  of  all  attention. 

554.  To  see  the  phenomena  of  total  reflexion   to  the  best  advantage,  lay  down  a  right-angled  glass  prism  on   a 
Experiment  black  substance  close  to  a  window,  with  its  base  horizontal,  as  in  fig.  124,  and  apply  the  eye  close  to  the  side, 
showing  the  looking  downwards.     The   base  will  be  seen  divided  into  two   portions,  by  a  beautiful  coloured  arch  like  a 

f  T  taf6"1  ramDOW  concave  to  the  eye,  the  portion  above  the  arch  being  extremely  brilliant  and  vivid,  and  giving  a  reflexion 
reflexion.  °f  a"  external  objects  no  way  to  be  distinguished  from  reality.  On  the  other  hand,  the  space  within  the 
concavity  of  the  bow  is  comparatively  sombre,  the  reflexion  of  the  clouds,  &c.  on  that  part  of  the  base  being 
much  less  vivid.  If,  instead  of  placing  it  on  a  black  body,  we  hold  it  in  the  hand,  and  place  a  candle  below  it, 
this  will  be  visible  ;  but  (wherever  placed)  will  always  appear  in  some  part  of  the  base  within  the  concavity  of 
the  bow.  Fig.  124  represents  the  course  of  the  rays  in  this  experiment,  E  being  the  eye,  N  G,  OF,  PD  rays 
incident  through  the  farther  side  at  various  angles  of  obliquity  on  the  base,  and  reflected  to  the  eye  at  E,  of 
which  O  F  is  incident  precisely  at  the  limiting  angle.  It  is  obvious,  that  all  the  rays  towards  N  incident  on  that 
part  of  the  base  beyond  F  being  too  oblique  for  transmission  will  be  totally  reflected,  while  those  incident 
between  F  and  A,  being  less  oblique  than  is  required  for  total  reflexion,  will  be  only  partially  so,  a  portion 
escaping  through  the  base  in  the  direction  D  Q.  Again,  if  we  place  a  luminary  at  any  point  as  L  below  the 
base,  it  is  manifest  that  to  reach  the  eye,  a  ray  from  it  must  fall  between  A  and  F,  as  L  D,  and  that  no  ray 
falling  on  any  part  of  the  base  between  B  and  F  can  be  refracted  to  E. 

555.  The    coloured  arch  separating  the  region  of  total  from  that  of  partial  reflexion,  is  thus  explained.     For, 
Reflected     simplicity,  let  us  suppose  the  eye  within  the  medium,  (to  avoid  considering  the  reflexion  at  the  inclined  surface 
msmatic      A  C  of  the  prism  ;)  and,  first,  considering  only  the  extreme  red  rays,  if  we  drop  a  perpendicular  from  the  eye  on 

the  base  of  the  prism,  and  make  this  the  axis  of  a  cone,  the  side  of  which  is  inclined  to  the  axis  at  the  angle 

whose  sine  is  — ,  (or  the  limiting  angle  for  extreme  red  rays;)  and  if  we  conceive  such  ravs  to  emanate  in 

all  directions  from  the  eye,  then  all  which  fall  without  the  circular  base  of  this  cone  will  be  totally,  but  those 
within  only  partially  reflected.  Thus,  were  there  no  other  than  such  red  rays  of  this  precise  refrangibility,  the 


LIGHT.  447 

lagbt      region  of  partial  reflexion  would  be  a  circle  whose  radius  =  height  of  the  eye  above  the  base  X  tangent  of  the    P"*  in. 

angle  whose  sine  is  -  =  —  =        —  .     In  like  manner,  the  radius  of  the  circular  space,  within  which  only  a 
p          V^-  1 

TT  TT 

partial  reflexion  of  violet  rays  takes  place,  is  —  —          -  ,   or  -  ,  being  less  than  the  value 


of  the  same  radius  for  the  red  rays.  Hence,  in  the  space  between  the  two  circles,  the  violet  rays  will  be  totally, 
and  the  red  only  partially  reflected  ;  and,  therefore,  the  whole  of  this  space  will  have  an  excess  of  violet  light. 
A  similar  reasoning  holds  good  for  the  intermediate  rays;  and  the  shading  away  from  the  bright  space  without, 
to  the  comparatively  dark  one  within,  will,  in  consequence,  be  performed  by  the  abstraction  first  of  the  red,  next 
of  the  orange  rays,  and  so  on  through  the  spectrum,  leaving  a  residual  light,  which  continually  deviates  more 
and  more  from  white,  and  verges  to  blue.  If  now  we  suppose  each  ray  to  be  incident  in  the  contrary  direction 
so  as  to  be  reflected  to  the  eye  instead  of  emanating  from  it,  every  thing  will  equally  hold  good,  and  the  eye 
will  see  a  bright  space  without  ;  separated  from  an  obscure  space  within  the  base  of  the  cone,  the  transition  from 
one  to  the  other  being  not  sudden,  but  marked  by  a  blue  border,  the  colour  of  which  is  more  lively  towards  the 
Ulterior.  Now  such  is  the  fact,  with  one  difference,  however,  that  the  coloured  arch  appears  slightly  tinged 
with  pink  on  its  convex  side.  This,  as  it  is  incompatible  with  theory,  can  be  owing,  it  should  seem,  to  no 
cause  but  contrast  ;  a  most  powerful  source  of  illusion  in  all  the  phenomena  of  colours,  and  of  which  this  is, 
perhaps,  one  of  the  most  striking  and  curious  instances.  Newton  (Optics,  part  ii.  exp.  16)  takes  no  notice  of 
this  part  of  the  phenomenon,  (which  was  first  observed  and  described  by  Sir  W.  Herschel,)  though  he  gives  the 
same  explanation  of  the  rest  with  that  here  set  down.  The  effect  of  refraction  at  the  side  B  A  of  the  prism 
will  somewhat  modify  the  figure  of  the  bow,  giving  it  a  tendency  to  a  conchoidal  form  at  great  obliquities  of 
the  emergent  rays. 

If  the  side  B  C  of  the  prism  be  covered  with  black  paper,  and  a  bright  scattered  light  be  thrown  on  the  base      555 
from  below,  (as  from  an  emeried  glass  applied  with  its  rough  side  close  to  the  base,)  the  converse  of  the  above  Transmitted 
described  phenomena  will  be  seen.     A  totally  black  space  will  be  seen  beyond  F,  and  a  bright  one  within  it.    The  prismatic 
separation  being  marked  by  a  bow  of  a  vivid  red  colour,  graduating  through  orange  and  pale  yellow  into  white,  the  l)OW- 
red  being  outwards.     It  is  evident  that  this  phenomenon  is,  in  all  its  parts,  complementary  to  that  of  the  blue 
bow  seen  by  reflexion,  and  therefore  requires  no  more  particular  explanation.     It  should  be  noticed,  however,  that 
in  this  bow  no  appearance  of  blue  or  violet  within  its  concavity  is  ever  seen  ;    so  that  the  effect  which  we 
have  above  attributed  to  contrast  in  the  reflected  bow  has  nothing  corresponding  to  it  in  the  transmitted  one. 

The  intensity  and  regularity  of  reflexion   at  the   external  surface  of  a  medium,  is  found   to    depend    not      5^7. 
merely  on  the  nature  of  the  medium,  but  very  essentially  on  the  degree  of  smoothness  and  polish  of  its  surface.  Reflexion 
But  it  may  reasonably  be  asked,  how  any  regular  reflexion  can  take  place  on  a  surface  polished  by  art,  when  we  ^(j^;*"* 
recollect  that  the  process  of  polishing  is,  in  fact,  nothing  more  than  grinding  down  large  asperities  into  smaller  polished 
ones  by  the  use  of  hard  gritty  powders,  which,  whatever  degree  of  mechanical  comminution  we  may  give  them,  explained. 
are  yet  vast  masses,  in  comparison  with  the  ultimate  molecules  of  matter,  and  their  action  can  only  be  considered 
as  an  irregular  tearing  up  by  the  roots  of  every  projection  which  may  occur  in  the  surface.     So  that,  in  fact,  a 
surface  artificially  polished  must  bear  somewhat  of  the  same  kind  of  relation  to  the  surface  of  a  liquid,  or  a 
crystal,  that  a  ploughed  field  does  to  the  most  delicately  polished  mirror,  the  work  of  human  hands.     Now  to 
this  question  the  Newtonian  doctrine  furnishes  an  answer  quite  satisfactory.     Were  the  reflexion  of  light  per- 
formed by  actual  impact  of  its  molecules  upon  those  of  the  reflecting-  medium,  no  regular  ordinary  reflexion 
could  ever  take  place  at  all,  as  it  would  depend  entirely  on  the   shape  of  the  molecules,  or  asperities  of  the  Light  net 
surface,  and  the  inclinations  of  their  surfaces  to  the  general  surface  of  the  medium  at  the  point  of  incidence,  U^,'*^1"* 
what  should  be  the  direction  ultimately  taken  by  each  particular  ray.     Now  these  must  vary  in  every  possible  pact  on 
manner  in  uncrystallized  bodies,  so  that  in  reflexion  from  the  surfaces  of  these  the  light  would  be  uniformly  scat-  bodies. 
tered  in  every  direction.     On   the  other   hand,  in  crystallized  media,  each  molecule  presenting  only  a  limited 
number  of  strictly  plane  surfaces,  and  the  corresponding  faces  of  all  being  mathematically  parallel,  reflexion 
would  indeed  be  regular  ;  but  its  direction  would  be  regulated  only  by  that  of  the  incident  ray  and  the  position 
of  certain  fixed  lines  within  the  crystal  ;    and  would  be  quite  independent  of  either  the  smoothness  or  the 
inclination  of  the  polished  surfaces  of  it,  whether  natural  or  artificial  ;  add  to  which,  that  instead  of  the  reflected 
pencil  of  rays  being  single,  it  would  in  most  cases  be  multiple.     All  these  consequences  are  so  contrary  to  fact,  But  bj 
that  it  is  evident  we  must  suppose  the  distance  to  which  the  forces  producing  reflexion  extend  much  greater  *.°rces  *** 
not  only  than  the  size  of,  or  interval  between  individual  molecules,  but  even  greater  than  the  minute  inequalities 
or  furrows  in  the  artificially  polished  surfaces  of  media.     Granting  this,  the  difficulty  vanishes  ;  for  the  average 
action  of  many  molecules,  or  many'  corrugations,  will   present  an  uniformity,  while  individually  they  may  offer 
the  greatest  diversity.     To  illustrate  this,  we  need  only  cast  our  eyes  on  fig.  125,  where  A  B   represents  the 
rough  surface  of  a  medium,  and  A  C  the  radius  of  one  of  the  spheres  of  attraction,  or  repulsive  activity  of  a 
single  molecule  A.     Conceiving  now  the  summits  of  all  the  elevations  a,  b,  c,  d  to  lie  in  a  plane,  let  spheres  be 
described  with  their  centres  equal  to  A  C.     Then  their  intersections  will  generate  a  kind  of  mamillated  surface 
a  ft  ff  S,  which,  however,  if  the  radii  of  the  spheres  be  at  all  considerable  with  respect  to  the  distances  of  their 
centres,  will  approach  exceedingly  near  to  a  mathematical  plane,  infinitely  more  so  than  the  surface  A  B  need  be 
supposed.     Hence,  a  ray  of  light  impinging  on  the  medium  will  come  within  the  sphere  of  its  action  not  at  an 
irregular  surface,  but  nearly  at  a  plane  one  ;  and  the  resultant  action  of  all  the  molecules  in  action,  supposing 
them  distributed  with  uniformity  over  A  B,  will  be  perpendicular  to  this  surface.     The  same  will  hold  good  of 
the  layer  of  molecules  (however  interrupted)  immediately  under  the  summits  6,  c,  d,  &c.,  and  ot  all  the  other 


448  .LIGHT. 

layers  into  which  the  whole  surface  can  be  divided.     So  that  the  essential  conditions  on  which  the  Newtonian    P»rt  AI. 
doctrine  of  reflexion  and  refraction  reposes,  (viz.  equality  of  force  at  equal  distances  from  the  general  level  of > 
the  surface,  and  the  perpendicularity  of  its  direction  to  that  level,)  still  obtain. 

559.  It  is  evident  that  the  inequalities  in  the  mamillary  surfaces  above  described  will  become  more  considerable  as 

Oblique       their  radii  are  diminished,  or  as  the  interval  of  their  centres  is  greater,  and  in  proportion  will  the  regularity  of 
regular  re-   reflexion  and  refraction  be  interrupted.     Hence  too  it  follows,  that  the  more  oblique  the  incidence  of  the  ray,  the 
"n  greater  may  be   the  roughness  of  the  surface  which  will  give  a   regular  reflexion ;    and  this  is  perfectly'con- 
tartaces.       sonant  to  fact,  as  may  be  easily  tried  with  a  piece  of  emeried  glass,  which,   although  so  rough  as  to  give  ntr 
regular  image  at  a  perpendicular  incidence,  will   yet    give  a  pretty  distinct   one  at  great    obliquities.     The 
reasons  are,  first,  that  a  very  oblique  ray  will  not  require  to  penetrate  so  far  within  the  sphere  of  repulsion,  to- 
have  its  motion  perpendicular  to  the  surface  destroyed  ;  and,  secondly,  that   it  cannot  pass  between   two  conti- 
guous elevations  or  depressions  of  the  imaginary  surface  a  ft  7  c,  but  by  reason  of  its  obliquity  must  traverse 
several  of  them,  and  thus  undergo  a  more  regular  average  exertion  of  the  forces  of  the  medium. 

559.  Thus  the  reflexion  of  light  is  explained  on  the  Newtonian  doctrine.     But  it  may  still  be  asked,  how  refraction 
Regular        at  a  surface  artificially  polished  can  ever  be  regular.     In  reflexion,  the  ray  never  reaches  the  asperities  of  the 
refraction      surface  ;   it  undergoes  their  average  action,  equalized  by  distance,  and  mutually  compensated.     In  refraction,  it 
aVfi     if*    's  otherwise.     Here  the  rays  must  actually  traverse  the  surface,  and  must  therefore  actually  pass  through  all 
polished.      'ts  inequalities  at  every  possible  angle  of  obliquity.     The  answer  to  this  is  equally  plain.     Neither  refraction  nor 

reflexion  are  performed  close  to  the  surface,  either  wholly,  or  in  great  part.  The  greater  part  by  far  of  the 
flexure  of  the  ray  is  performed  (either  internally  or  externally)  at  a  distance,  out  of  the  reach  of  these  irregu- 
larities, and  by  the  action  of  a  much  more  considerable  thickness  of  the  medium  than  is  occupied  by  them. 
Their  action  must  be  compared  to  the  effect  of  mountains  on  the  earth's  surface  in  disturbing  the  general  force 
of  gravity.  A  stone  let  fall  close  to  one  of  them,  from  a  moderate  height,  follows  not  the  true  vertical  but  the 
direction  of  the  deviated  plumbline,  which  is  sensibly  different.  Whereas,  if  let  fall  from  the  moon  to  the  earth's  . 
centre,  it  would  pass  among  them,  were  they  greater  a  thousand  fold  than  they  are,  without  experiencing  any 
sensible  perturbation  or  change  of  direction  in  their  neighbourhood. 

560.  In  fact,  however,  no  regular  refraction  can  be  obtained  from  surfaces  sensibly  rough,  at  all  comparable  to  the 
regularity  of  their  reflexion.     This  may  arise  from  the  impossibility  of  a  refracted  ray  penetrating  the  surface 
at  a  sufficient  degree  of  obliquity.     It  is,  however,  a  remarkable  fact,  that  the  regular  internal  reflexion  from  a 
roughened  surface,  even  at  extreme  obliquities,  is  scarcely  sensible,  even  in  cases  where  the  external  reflexion  at 
the  same  obliquities  is  perfectly  regular  and  copious.     This   would  seem  to  indicate,  that  the   forces  which 
operate  the  external  reflexion  of  a  ray  exert  their  energy  wholly  without  the  medium. 

b61.           Whatever  be  the  forces  by  which   bodies   reflect  and  refract  light,  one  thing  is   certain,  that  they  must  be 
Intensity  of  incomparably  more  energetic   than   the  force  of  gravity.     The   attraction   of  the   earth  on  a  particle  near  its 
forcei     surface  produces  a  deflexion  of  only  about  16  feet  in  a  secoiv  ;    and,  therefore,  in  a  molecule  moving  with  the 
-efractioif    ve'oc'ty  °f  "ght,  would  cause  a  curvature,  or  change  of  direction,  absolutely  insensible  in  that  time.     In  fact, 
we  must  consider,  first,  that  the  time  during  which  the  whole  action  of  the  medium  takes  place,  is  only  that 
within  which  light  traverses  the  diameter  of  the  sphere  of  sensible  action  of  its  molecules  at  the  surface.     To 
allow  so  much  as  a  thousandth  of  an  inch  for  this  space  is  beyond  all   probability,  and  this  interval  is  tra- 
versed by  light  in  the  — part  of  a  second.      Now,  if  we  suppose  the   deviation  produced 

1 2,672,000,000,000 

by  refraction  to  be  30°,  (a  case  which  frequently  happens,)  and  to  be  produced  by  a  uniform  force  acting 
during  a  whole  second;  since  this  is  equivalent  to  a  linear  deflexion  of  200,000  miles  X  sin  30°,  or  of  100,000 
miles  =  33,000,000  x  16  feet,  such  a  force  must  exceed  gravity  on  the  earth's  surface  33,000,000  times. 
But,  in  fact,  the  whole  effect  being  produced  not  in  one  second,  but  in  the  small  fraction  of  it  above  mentioned, 
the  intensity  of  the  force  operating  it  (see  MECHANICS)  must  be  greater  in  the  ratio  of  the  square  of  one 
second  to  the  square  of  that  fraction ;  so  that  the  least  improbable  supposition  we  can  make  gives  a  mean 
force  equal  to  4,969,126,272  X  10"  times  that  of  terrestrial  gravity.  But  in  addition  to  this  estimate  already 
so  enormous,  we  have  to  consider  that  gravity  on  the  earth's  surface  is  the  resultant  attraction  of  its  whole 
mass,  whereas  the  force  deflecting  light  is  that  of  only  those  molecules  immediately  adjoining  to  it,  and  within 
the  sphere  of  the  deflecting  forces.  Now  a  sphere  of  -rnW  of  an  inch  diameter,  and  of  the  mean  density  of 

the  earth,  would  exert  at  its  surface  a  gravitating  force  only  T,W  x  -r JT-T — r-  of  ordinary  gra- 
vity, so  that  the  actual  intensity  of  the  force  exerted  by  the  molecules  concerned  cannot  be  less  than 

1000  x  earth  s  diameter          46>352,000,000)  times  the   above  enormous   number,  or  upwards  of  2  x  10" 

1  inch 

when  compared  with  the  ordinary  intensity  of  the  gravitating  power  of  matter.  Such  are  the  energies  concerned 
in  the  phenomena  of  light  on  the  Newtonian  doctrine.  In  the  undulatory  hypothesis,  numbers  not  less  immense 
will  occur;  nor  is  there  any  mode  of  conceiving  the  subject  which  does  not  call  upon  us  to  admit  the  exertion 
of  mechanical  forces  which  may  well  be  termed  infinite. 

ten  Dr.  Wollaston  has  proposed  the  observation  of  the  angle  at  which  total  reflexion  first  takes  place  at  the 

common  surface  of  two  media,  the  index  of  refraction  of  one  of  which  is  known,  as  a  means  of  determining 
that  of  the  other;  and,  in  the  Philosophical  Transactions  for  1S02,  has  described  an  ingenious  apparatus  which 
gives  a  measure  of  the  index  required  almost  by  inspection.  If  we  lay  any  object  under  the  base  of  a  prisiu 


LIGHT.  449 

of  flint   glass  with  air  alone  interposed,  the  internal   angle  of  incidence  at  which  the  visual  ray  begins   to  be    Part  HI. 
totally  reflected,  and  at  which  of  course  the  object  ceases  to  be  seen  by  refraction  is  about  39°  10' ;  but  when  v— - v— - ^ 
the  object  has  been  dipped  in  water,  and  brought  into  contact  with  the  glass,  it  continues  visible  (while  the  eye  ^    ,w°"as- 
is  depressed)  by  means  of  the  greater  refractive  power  of  the  water,  as  far  as  57£°  of  incidence.     When  any  tJJ"ds  ^e" 
kind  of  oil,  or  any  resinous  cement,  is  interposed,  this  angle  is  still  greater,  according  to  the  refractive  power  of  determining 
the  medium  employed  ;   and  by  cements  that  refract  more  strongly  than  the  glass,  the  object  may  be  seen  through  refractive 
the  prism  at  whatever  angle  of  incidence  it  is  viewed.     All  that  is  requisite,  then,  to   determine  the  refractive  powers- 
index  of  any  body  less  refractive  than  glass,  is  to  bring  the  substance  to  be  examined  in  optical  contact  with 
the  base  of  a  prism,  and  to  depress  the  eye  (or  increase  the  angle  of  incidence)  till  it  ceases  to  be  seen  as  a 
dark  spot  on  the  silvery  reflexion  of  the  sky  on  the  rest  of  the  base.     With  fluids  and  soft  solids,  or  fusible 
ones,  the  requisite  contact  is  easily  obtained  ;  but  with  solids,  they  must  be  brought  to  smooth  surfaces,  and 
applied  to  the  base  by  the  intervention  of  some  fluid  or  cement  of  higher  refractive  power  than  the  glass,  which 
(since  the  surfaces  of  the  interposed  stratum  are  parallel)  will  produce  no  change  in  the  total  deviation  of  a 
ray  passing  through  it,  and  therefore  no  error  in  the  result.     By  this  method,  opaque  as  well  as  transparent 
substances  may  be  examined,  or  bodies  of  unhomogeneous  density,  as  the  crystalline  lens  of  the  eye.     It  may 
seem  paradoxical  to  speak  of  the  refractive  power  of  an  opaque  body ;  but  it  will  be  remembered,  that  opacity 
is  merely  a  consequence  of  intense  absorbent  power,  and  that  before  a  ray  can  be  absorbed,  it  must  enter  the 
medium,  and  of  course  obey  the  laws  of  refraction  at  its  surface.     By  this  method,  Dr.  Wollaston  has  determined 
the  refractions  of  a  great  variety  of  bodies  ;  but  Dr.  Brewster  remarks,  that  the  method  must  be  liable  to  some 
source  of  inaccuracy,  which  renders  it  unsafe  to  trust  entirely  to  it  in  practice.     Dr.  Young  has  remarked,  that 
the  index  of  refraction  given  by  it,  belongs  in  strictness  to  the  extreme  red  rays. 

§  II.   General  Statement  of  the  Undulatory  Theory  of  Light. 

The  undulatory  theory,  among  whose  chief  supporters  we  have  to  number  Huygens,  Descartes,  Hooke,  and       563. 
Euler,  and,   in  later  times,  the  illustrious  names  of  Young  and  Fresnel,  who  have  applied  it  with   singular 
success  and  ingenuity  to  the  explanation  of  those  classes  of  phenomena  which  present  the  greatest  difficulties 
to  the  Corpuscular  doctrine,  requires  the  admission  of  the  following  hypotheses  or  postulata : 

1.  That  an  excessively  rare,  subtle,  and  elastic  medium,  or  ether,  as  it  is  called,  fills  all  space,  and  pervades  Postulata 
all  material  bodies,  occupying  the  intervals  between  their  molecules;  and,  either  by  passing  freely  among  them,  in  'he 
or,  by  its  extreme  rarity,  offering  no  resistance  to  the  motions  of  the  earth,  the  planets,  or  comets  in  their  orbits,  system  pf 
appreciable  by  the  most  delicate  astronomical  observations ;  and  having  inertia,  but  not  gravity. 

2.  That  the  molecules  of  the  ether  are  susceptible  of  being  set  in  motion  by  the  agitation  of  the  particles  of 
ponderable  matter,  and  that  when  any  one  is  thus  set  in  motion  it  communicates  a  similar  motion  to  those 
adjacent  to  it ;    and  thus  the  motion  is  propagated  further  and  further  in  all  directions,  according  to  the  same 
mechanical  laws  which  regulate  the  propagation  of  undulations  in  other  elastic  media,  as  air,  water,  or  solids, 
according  to  their  respective  constitutions. 

3.  That  in   the  interior  of  refracting  media  the  ether  exists  in   a  state  of  less  elasticity,  compared  with  its 
density,  than  in  vacuo,  (i.  e.  in  space  empty  of  all  other  matter ;)  and  that  the  more  refractive  the  medium,  the 
less,  relatively  speaking,  is  the  elasticity  of  the  ether  in  its  interior. 

4.  That  vibrations  communicated  to  the  ether  in  free  space  are  propagated  through  refractive  media  by  means 
of  the  ether  in  their  interior,  but  with  a  velocity  corresponding  to  its  inferior  degree  of  elasticity. 

5.  That  when  regular  vibratory  motions  of  a  proper  kind  are  propagated   through  the  ether,  and,  passing 
through   our  eyes,  reach   and  agitate  the  nerves  of  our  retina,  they  produce  in  us  the  sensation  of  light,  in  a 
manner  bearing  a  more  or  less  close  analogy  to  that  in  which  the  vibrations  of  the  air  affect  our  auditory  nerves 
with  that  of  sound. 

6.  That  as,  in  the  doctrine  of  sound,  the  frequency  of  the  aerial  pulses,  or  the  number  of  excursions  to  and 
fro  from  its  point  of  rest  made  by  each  molecule  of  the  air,  determines  the  pitch,  or   note,  so,  in  the  theory  of 
light,  the  frequency  of  the  pulses,  or  number  of  impulses  made  on  our  nerves  in  a  given  time  by  the  ethereal 
molecules  next  in  contact  with  them,  determines  the  colour  of  the  light;  and  that  as  the  absolute  extent  of  the 
motion  to  and  fro  of  the  particles  of  air  determine  the  loudness  of  the  sound,  so  the  amplitude,  or  extent  of  the 
excursions  of  the  ethereal   molecules  from   their  points  of  rest,  determine  the  brightness  or  intensity  of  the 
light. 

The  application  of  these  postulates  to  the  explanation  of  the  phenomena  of  light,  presumes  an  acquaintance       564. 
with  the  theory  of  the  propagation  of  motion  through  elastic  media.     This  we  shall  assume,  referring  to  our  The  velo- 
article  on  sound  for  the  demonstration  of  all  the  properties  and  laws  of  motions   so  propagated,  as  we  shall  city  of  all 
have  occasion  to  employ.     One  of  the  principal  of  these  is,  that  supposing  the  elastic   medium  uniform   and  undulation» 
homogeneous,  all  motions  of  whatever  kind  are  propagated  through  it  in  all   directions  with  one  and  the  same  CI5 
uniform  velocity,  a  velocity  depending  solely  on  the  elasticity  of  the  medium  as  compared  with  its  inertia,  and 
bearing  no  relation  to  the  greatness  or  smallness,  regularity  or  irregularity  of  the  original  disturbance.     Thus, 
while  the  intensity  of  light,  like  that  of  sound,  diminishes  as  the  distance  from  its  origin  increases,  its  velocity 
remains  invariable ,  and  thus,  too,  as  sounds  of  every  pitch,  so  light  of  every  colour,  travels  with  one  and  the 
same  velocity,  either  in  vacuo,  or  in  a  homogeneous  medium. 

Now  here  arises,  in  limine,  a  great  difficulty;   and  it  must  not  be  dissembled,  that  it  is  impossible  to  look  on 
VOL.  iv.  3  N 


450  LIGHT. 

it  in  any  other  light  than  as  a  most  formidable  objection  to  the  undulatory  doctrine.  It  will  be  shown  presently 
that  the  deviation  of  light  by  refraction  is  a  consequence  of  the  difference  of  its  velocities  within  and  without  v 
oraie  the  refracting  medium,  and  that  when  these  velocities  are  given  the  amount  of  deviation  is  also  given.  Hence 
phenomena  li  would  appear  to  follow  unavoidably,  that  rays  of  all  colours  must  be  in  all  cases  equally  refracted ;  and  that, 
•f  disper-  therefore,  there  could  exist  no  such  phenomenon  as  dispersion.  Dr.  Young  has  attempted  to  gloss  over  this 
sion.  difficulty,  by  calling  in  to  his  assistance  the  vibrations  of  the  ponderable  matter  of  the  refracting  medium  itself, 

as  modifying  the  velocity  of  the  ethereal  undulations  within  it,  and  that  differently  according  to  their  frequency, 
and  thus  producing  a  difference  in  the  velocity  of  propagation  of  the  different  colours  ;  but  to  us  it  appears  with 
more  ingenuity  than  success.  We  hold  it  better  to  state  it  at  once  in  its  broadest  terms,  and  call  on  the  reader 
to  suspend  his  condemnation  of  the  doctrine  for  what  it  apparently  will  not  explain,  till  he  has  become 
acquainted  with  the  immense  variety  and  complication  of  the  phenomena  which  it  will.  The  fact  is,  that 
neither  the  corpuscular  nor  the  undulatory,  nor  any  other  system  which  has  yet  been  devised,  will  furnish  that 
complete  and  satisfactory  explanation  of  all  the  phenomena  of  light  which  is  desirable.  Certain  admissions 
must  be  made  at  every  step,  as  to  modes  of  mechanical  action,  where  we  are  in  total  ignorance  of  the  acting 
forces ;  and  we  are  called  on,  where  reasoning  fails  us,  occasionally  for  an  exercise  of  faith.  Still,  if  we  regard 
hypotheses  and  theories  as  no  other  way  valuable  than  as  means  of  classifying  and  grouping  together  pheno- 
mena, and  of  referring  facts  to  laws  which,  though  possibly  empirical,  are  yet,  so  far  as  they  are  so,  correct 
representations  of  nature,  and  as  such  must  be  deducible  from  real  primary  laws,  whenever  they  shall  be  disco- 
vered, we  cannot  but  admit  their  importance.  The  undulatory  system  especially  is  necessarily  liable  to  consi- 
derable obscurities  ;  as  the  doctrine  of  the  propagation  of  motion  through  elastic  media  is  one  of  the  most 
abstruse  and  difficult  branches  of  mathematical  inquiry,  and  we  are  therefore  perpetually  driven  to  indirect  and 
analogical  reasoning,  from  the  utter  hopelesness  of  overcoming  the  mere  mathematical  difficulties  inherent  in 
the  subject  when  attacked  directly. 

566.  It  is  thus  that  we  are  encountered  at  the  very  outset  of  its  application  with  another  objection,  which,  in  the 

Objection  eves  of  Newton,  appeared  decisive  against  its  admission,  but  which  has  since  been,  in  a  considerable  degree, 
jectrti  e  overcome.  How  is  it  that  shadows  exist.  Sounds  make  their  way  freely  round  a  corner, — why  does  not  light 
propagation  ^°  so'  A  vibration  propagated  from  a  centre  in  an  elastic  medium,  and  intercepted  by  an  immovable  obstacle 
of  light  having  a  small  orifice,  ought  to  spread  itself,  it  is  said,  from  this  orifice  beyond  the  screen  as  from  a  new  centre, 
answered,  and  fill  the  space  beyond  with  undulations  propagated  from  it  in  every  direction.  Thus,  as  in  Acoustics,  the 
orifice  is  heard  as  a  new  source  of  sound ;  so,  in  Optics,  it  ought  to  be  seen  in  all  directions  as  a  new  luminary. 
To  this  the  answer  is,  first,  that  it  is  not  demonstrable  that  a  vibratory  motion  communicated  to  one  particle  of 
an  elastic  medium  is  propagated  with  equal  intensity  to  every  surrounding  molecule  in  whatever  direction 
situated  with  respect  to  the  line  of  its  motion,  though  it  is  with  equal  rapidity ;  and  therefore  that  we  have  no 
reason  to  presume,  a  priori,  but  rather  the  contrary,  that  the  motions  of  the  vibrating  particles  at  the  orifice 
should  be  propagated  laterally  with  equal  intensity  in  all  directions ;  secondly,  that  it  is  not  true,  in  fact,  that 
sounds  are  propagated  round  the  corner  of  an  obstacle,  with  the  same  intensity  as  in  their  original  direction,  as 
any  one  may  convince  himself  by  the  following  simple  experiment.  Take  a  common  tuning  fork,  and,  holding 
it  (when  set  in  vibration)  about  three  or  four  inches  from  the  ear,  with  its  flat  side  towards  it,  when  its  sound 
is  distinctly  heard,  let  a  strip  of  card,  somewhat  longer  than  the  flat  of  the  tuning  fork,  be  interposed,  at  about 
half  an  inch  from  the  fork.  The  sound  will  be  almost  entirely  intercepted  by  it ;  and  if  the  card  be  alternately 
removed  and  replaced  in  pretty  quick  succession,  alternations  of  sound  and  silence  will  be  perceived ;  proving 
that  the  undulations  of  the  air  are  by  no  means  propagated  with  equal  intensity  by  the  circuitous  route  round 
the  edge  of  the  card,  as  by  the  direct  one.  Indeed  any  one  has  only,  to  be  convinced  of  the  fact,  to  attend  to 
the  sound  of  a  carriage  in  the  act  of  turning  a  corner  from  the  street  in  which  he  happens  to  be  to  an  adjoining 
one  ;  to  which  we  may  add,  that,  even  when  there  is  no  obstacle  in  the  way,  sounds  are  by  no  means  equally 
audible  in  all  directions  from  the  sounding  body,  as  any  one  may  convince  himself  by  holding  a  vibrating  tuning 
fork,  or  pitchpipe,  near  his  ear,  and  turning  it  quickly  on  its  axis.  This  last  phenomenon  was  first  noticed,  we 
believe,  by  Dr.  Young,  (Phil.  Trans.,  1802,  p.  25,)  and  since  more  fully  described  (in  Schweiggers  Jahrbuch, 
1826)  by  M.  Weber.  Now  if  there  be  any  inequality  at  all  in  the  intensity  of  the  direct  and  lateral  propa- 
gation of  undulations  in  a  medium,  it  must  arise  from  the  constitution  of  the  medium,  and  the  proportion  of 
the  amplitude  of  the  excursions  of  the  vibrating  particles  to  their  distance  from  each  other  ;  and  may  therefore 
easily  be  conceived  to  differ  in  any  imaginable  degree  in  different  media,  and  there  is,  at  least,  no  absurdity  in  sup- 
posing the  ether  so  constituted  as  to  admit  of  comparatively  very  feeble  lateral  propagation.  Now,  thirdly,  in  point 
of  fact,  light  does  spread  itself  in  a  certain  small  degree  into  the  shadows  of  bodies,  out  of  its  strict  rectilinear 
course,  giving  rise  to  the  phenomena  of  inflexion  or  diffraction,  of  which  more  presently,  and  which  are  com- 
pletely accountable  for  on  the  undulatory  doctrine,  and  form,  in  fact,  its  strongest  points.  For  further  informa- 
tion on  this  confessedly  abstruse  subject,  the  reader  must  consult  our  article  on  SOUND,  and  the  works  cited  at 
the  end  of  this  Essay.  It  is  enough  here  to  show,  that  the  objection  which  has  been  urged  by  Newton  and  his 
followers  with  such  force  against  the  doctrine  of  undulations,  is  really  not  conclusive  against  it,  but  founded 
rather  on  inadequate  conceptions  of  the  nature  of  elastic  fluids,  and  the  laws  of  their  undulations. 
867.  Although  any  kind  of  impulse,  or  motions  regulated  by  any  law,  may  be  transferred  from  molecule  to 

Mode  in       molecule  in  an  elastic  medium,  yet  in  the  theory  of  light  it  is  supposed  that  only  such  primary  impulses  as  recur 
which  the     according  to  regular  periodical  laws,  at  equal   intervals  of  time,  and  repeated  many  times  in  succession,  can 
it*d*b      a^ec^  our  organs  with  the  sensation  of  light.     To  put  in  motion  the  molecules  of  the  nerves  of  our  retina  with 
ribrmtbni     sufficient  efficacy,  it  is  necessary  that  the  almost  infinitely  minute  impulse  of  the  adjacent  ethereal  molecules 
of  ethe  .      should  be  often  and  regularly  repeated,  so  as  to  multiply,  and,  as  it  were,  concentrate  their  effect.     Thus,  as  a 
great  pendulum  may  be  set  in  swing  by  a  very  minute  force  often  applied  at  intervals  exactly  equal  to  its  time 


LIGHT.  451 

of  oscillation,  or  as  one  elastic  solid  body  can  be  set  in  vibration  by  the  vibration  of  another  at  a  distance,  Parc  '!«'• 
"""V^  propagated  through  the  air,  if  in  exact  unison,  even  so  may  we  conceive  the  gross  fibres  of  the  nerves  of  the  *— • ~V~~' 
retina  to  be  thrown  into  motion  by  the  continual  repetition  of  the  ethereal  pulses ;  and  such  only  will  be  thus 
agitated,  as  from  their  size,  shape,  or  elasticity  are  susceptible  of  vibrating  in  times  exactly  equal  to  those  at 
which  the  impulses  are  repeated.  Thus  it  is  easy  to  conceive  how  the  limits  of  visible  colour  may  be  established ; 
for  if  there  be  no  nervous  fibres  in  unison  with  vibrations  more  or  less  frequent  than  certain  limits,  such  vibra- 
tions, though  they  reach  the  retina,  will  produce  no  sensation.  Thus,  too,  a  single  impulse,  or  an  irregularly 
repeated  one,  produces  no  light ;  and  thus  also  may  the  vibrations  excited  in  the  retina  continue  a  sensible 
time  after  the  exciting  cause  has  ceased,  prolonging  the  sensation  of  light  (especially  of  a  vivid  one)  for  an 
instant  in  the  eye  in  the  manner  described,  (Art.  543.)  We  may  thus  conceive  the  possibility  of  other  animals, 
such  as  insects,  incapable  of  being  affected  with  any  of  our  colours,  and  receiving  their  whole  stock  of  luminous 
impressions  from  a  class  of  vibrations  altogether  beyond  our  limits,  as  Dr.  Wollaston  has  ingeniously  imagined 
(we  may  almost  say  proved)  to  be  the  case  with  their  perceptions  of  sound. 

The  law  of  motion  of  every  particle  of  the  ether  is  regulated  by  that  of  the  molecule  of  the  luminary  from      568. 
which  it  takes  its  origin  ;    and  will  be  regular  or  irregular,  periodical  or  not,  according  as  that  of  the  original  Motion  of  " 
molecule  is  so  or  otherwise.     But  it  is  only  with  motions  which  may  be  regarded  as  infinitely  small   that  we  J',',^',^ 
are    concerned   in  this    theory.     The   displacement  of  each  particle,  either  of  the  ether  or  of  the    luminary,  is  n,™ecuia 
supposed  to  be  so  minute  as  not  to  detach  it  from,  or  change  its  order  of  situation  among  the  neighbouring 
ones.     Now  when  we  consider  only  such   infinitesimal  displacements  from    the  position  of  equilibrium,  it  is 
evident,  that  the  tension  arising  from  them,  or  the  force  by  which   the  displaced  molecule  is  urged,  must  be 
proportional  in  quantity  to  its  distance  from  its  point  of  rest,  and  must  tend  directly  to  that  point,  provided  we 
suppose  the  medium  equally  elastic  in  all  directions.     Hence,  by  the  laws  of  Dynamics,  its  trajectory  must  be  an 
ellipse  described  in  one  plane  about  the  point  of  equilibrium  as  its  centre;   or,  if  one  of  the  axes  of  the  ellipse 
vanish,  a  straight  line  having  that  point  in  its  middle,  in  which  it  oscillates  to  and  fro,  performing  all  its  excur- 
sions in  the  latter  case,  or  its  revolutions  in  the  former,  whether  great  or  small,  in  equal  times,  and  following  the 
law  of  a  vibrating  pendulum.     We  will,  for  the  present,  consider  the  case  of  rectilinear  vibrations  as  the  most 
simple,  and  show  hereafter  how  the  more  general  one  may  be  reduced  to  it. 

Proposition.  To  define  the  motion  of  a  vibrating  molecule  of  a  luminary,  supposing  its  excursions  to  and  fro       559 
to  be  performed  in  straight  lines.  Lawg  Of 

Putting  x  for  its  distance  from  its  point  of  rest,  t  for  the  time  elapsed  since  a  given  epoch,  and  v  for  its  rectilinear 
velocity,  and  E  for  the  absolute  elastic  force,  the  force  urging  the  molecule  to  its  point  of  equilibrium  will  be  "brations- 
E  .  x,  and  will  tend  to  diminish  x;  hence  (supposing  gravity  to  be  represented  by  32 £  feet)  we  must  have 

dv  d*x  2  d* x  .  d x  d  x* 

Ex,  and   therefore  -     — -— =  -  2  E  x  d  x,  or,  integrating, or   c2    =  E 


j   j  j  jo  j  JQ  ~~  ™    •*-**   "*  **  **»    UI »    im-ctinitiiiti,  ^~ 

U    t  (It  Ct  6  U   t 

(a*  —  ,r2)  where   a  is  the    greatest   distance  of  excursion,  or  the   semiamplitude   of  the  vibration.      Hence, 

_  dx  fig 

v  —  v  E  .  v  a'  -  x"-  —  — — — ,   and  therefore  d  <  =  —   — ^ ;    or,  integrating,  t  -+-   C   = 

—  arc  .  cos  ,  that  is 

a 


x  =  a  .  cos  {  v'E  .  (t  +  C)  }  ;         v  =  a  .  ^E   .  sin  {  </E  (t  -f-  C)  } 

Such  are  the  velocity  and  distance  from  the  middle  point  of  its  vibration  of  the  molecule  at  any  instant.  If 
we  call  T  the  whole  period  in  which  the  molecule  has  performed  one  complete  evolution,  consisting  of  a 
complete  excursion  to  and  fro  on  both  sides  of  its  point  of  equilibrium,  we  shall  have  at  the  commencement  of 


the  motion  when  v  =  0,  or  x  —  a,  a  .  cos  {  VE  .  (t  -f-  C)  }  =  a,  or  (t  -f-  C)  VE  =  0  ;  and  when  one  quarter 
of  a  period  has  been  performed,  or  the  molecule  has  arrived  at  its  greatest  distance  —  a  on  the  opposite  side  of 
the  centre  -  a  =  a  .  cos  {  VE  (t  +  \  T  +  C)  }  .  or  Vff  .  (t  +  C  -f  £  T)  =  ir,  putting  *  for  the  semicircum- 
ference  of  a  circle  whose  diameter  is  1  .  Hence  we  get  by  subtraction 


"/  E 
Hence  we  may  eliminate  E,  and  introduce  T  instead  of  it,  which  will  give  the  equations  ^~E=  — — , 

t  +  C  . —  t  +  C 

x  —  a  .  cos  2  IT  ,  — i- —  ;  v  =  a  v  E   sin   2  IT  .  —^ — ; 

which  equations  express  the  laws  required,  and  which  if  the  time  t  be  supposed  to  commence  at  the  moment 
when  r>  =  0,  or  when  the  molecule  is  at  the  extremity  of  one  of  its  excursions,  become  simply 

x  =  a  .  cos  2  v .  -—-  •  v  =  a  V  E  sin  2  ir .  -—. 


3  N2 


452 


L  I  G  H  T. 


Light. 

— ~\— 
570. 


571. 

Laws  of 

rectilinear 

vibrations 

of  an 

ethereal 

molecule. 


573. 

Waves  of 

light 

defined. 


574. 

Undula- 
tions or 
pulses. 

575. 
Different 
colours 
have  dif- 
ferent 
lengths  of 
their  undu- 
lations. 


Carol.  Hence  the  excursions  of  the  molecule  to  and  fro  will  consist  of  four  principal  phases,  in  each  of  which     l'-rt  HI- 
•  its  motion  is  similar,  but  in   contrary  directions,  or  on  contrary  sides  of  the  centre.     In  the  first  phase  the  >—— -v— • 
molecule  is  to  the  right  of  the  centre  of  motion,  and  is  approaching  the  centre,  or  moving  from  right  to  left. 
In  the  second,  it  is  to  the  left  of  the  middle  point,  and  moving  from  it,  or  still  from  right  to  left.     These  two 
phases  we  shall  term  the  positive  phases.     In  the  third  phase  the  molecule  lies  on  the  left  side,  and  its  motion  is 
towards  the  centre,  and  from  left  to  right.     In  the  fourth,  it  is  to  the  right  again,  receding  from  the  centre,  and 
moving  still  from  left  to  right.     These  we  shall  term  the  negative  phases  of  its  vibration. 

Proposition.  To  define  the  rectilinear  vibrations  of  any  molecule  of  the  ether,  propagated  from  a  luminous 
particle  vibrating  as  in  the  last  proposition. 

In  the  propagation  of  motions  through  elastic,  uniform  media,  the  same  or  a  similar  motion  to  that  of  any 
one  molecule  is  communicated  to  every  other  in  succession ;  but  this  communication  occupies  time,  and  the 
motion  of  a  molecule  at  a  distance  from  the  origin  of  the  vibrations  does  not  commence  till  after  the  lapse  of  an 
interval  of  time  proportional  to  that  distance,  being  the  time  in  which  the  propagated  impulse,  whether  of  sound 
or  light,  &c.  runs  over  that  distance  with  a  certain  uniform  velocity  due  to  the  intrinsic  elasticity  of  the  medium, 
and  which  in  the  case  of  light  is  about  200,000  miles  per  second  ;  in  that  of  sound  about  1100  feet.  And  when 
the  vibration  of  the  original  source  of  motion  has  ceased,  that  of  the  ethereal  molecule  does  not  cease  on  the 
instant,  but  continues  for  a  time  equal  to  that  which  elapsed  before  its  commencement.  Hence,  if  we  call  V  the 

velocity  of  light,  and  D  the  distance  of  the  molecule  from  the  luminous  point,  will  be  the  interval  between 

the  commencement  of  the  motion  of  the  latter  and  of  the  former ;   hence  —  t  being  the  time  elapsed  at  any 
instant  since  the  commencement  of  the  first  positive  phase  of  the  vibration  of  the  luminous   point,  t  —   — 

will  be  the  corresponding  time  in  the  case  of  the  ethereal  molecule.     Thus  we  have,  for  the  equations  of  the 
motions  of  the  former, 

t  t 

x  =  a  .  cos  2  TT  .  -—  ;  c  =  b  .  sin  2  TT  •-  - ;  where  b  =  a   v    E 


and  in  that  of  the  latter 
x  =  a  .  cos  2  TT 


\.         T         / 


v  =  /3  .  sin  2  sr 


;   where  /3  =  a  V  E 


a  being  the  seraiamplitude  of  the  vibration,  or  the  extent  of  the  excursion  of  the  ethereal  molecule  from  its 
point  of  rest. 

Carol.  Hence  it  is  evident  that  the  actual  velocity  of  the  molecules  of  ether  may  be  less  in  any  proportion 
than  that  of  light ;  for  the  maximum  value  of  v  depends  for  its  numerical  magnitude  solely  on  a,  or  on  the 
amplitude  of  excursion,  and  on  E,  and  not  at  all  on  V  the  velocity  of  propagation  of  the  wave. 

Coral.  2.  If  we  suppose  the  luminous  molecule  to  have  made,  from  the  commencement  of  its  motion,  any 
number  of  vibrations  and  parts  of  a  vibration  in  the  time  t;  then  if  we  consider  an  ethereal  molecule  at  a 
distance  V  .  t  from  it  in  any  direction,  (i.  e.  situated  in  a  spherical  surface  whose  radius  is  V  .  t,)  this 
molecule  will  be  just  beginning  to  be  put  in  motion.  If  we  suppose  another  spherical  surface  concentric  with 
the  former,  but  having  its  radius  less  than  the  former  by  V  .  T,  which  in  future  we  shall  call  X,  every  particle 
situated  in  this  surface  will  have  just  completed  one  vibration,  and  be  commencing  its  second,  and  so  on. 
The  interval  between  these  surfaces  will  comprehend,  arranged  in  spherical,  concentric  shells,  molecules  in 
every  phase  of  their  vibrations, — those  in  each  shell  being  in  the  same  phase.  This  assemblage  of  molecules 
is  termed  a  wave,  and  as  the  impulse  continues  to  be  propagated  forwards  it  is  evident  that  the  wave  will 
continue  to  increase  in  radius,  and  will  comprehend  in  succession  all  the  molecules  of  the  medium  10 
infinity. 

Definition.  The  interval  between  the  internal  and  external  surface  of  a  luminous  wave  is  called  an  undulation, 
or  a  pulse,  and  its  length  is  evidently  =  V .  T  —  X,  or  the  space  run  over  by  light  in  the  time  T  of  one  complete 
period,  or  vibration  of  the  luminous  molecule.  It  is  therefore  proportional  to  that  time. 

Hence  the  lengths  of  the  undulations  of  differently  coloured  rays  differ  inter  se.  For,  by  Postulate  6,  the 
number  of  vibrations  made  in  any  given  time  by  the  ethereal  particles  determines  the  colour.  Now  the  more 
numerous  the  vibrations  are,  data  tempore,  the  shorter  their  duration ;  hence  T,  which  represents  this  duration, 
is  less  ;  and  therefore  X,  or  the  length  of  the  undulation  less  for  the  violet  than  for  the  red  rays.  From 
experiments  to  be  presently  described,  it  has  been  found,  that  the  lengths  of  the  undulations  in  air,  or 
the  values  of  X  for  the  different  rays,  as  also  the  number  of  times  they  are  repeated  in  one  second,  are  a* 
in  the  following'  table  . 


LIGHT. 


453 


Lignt. 


Colours, 

Length  of  an  undulation 
in   part;  of  an  inch  in 
air  X  =. 

Number  of  such  undu- 
lations in  an  inch  or  •  — 
>. 

Number  of  undulations  per  second. 

Extreme   

0-0000266 

37640 

458,000000,000000 

Red  

0-0000256 

39180 

477,000000,000000 

Intermediate 

0-0000246 

40720 

495,000000,000000 

Orange  

0-0000240 

41610 

506,000000,000000 

Intermediate  ... 

0-0000235 

42510 

517,000000,000000 

Yellow  .  . 

0-0000227 

44000 

535,000000,000000 

Intermediate  

0-0000219 

45600 

555,000000,000000 

Green   

0-0000211 

47460 

577,000000,000000 

Intermediate 

0-0000203 

49320 

600,000000,000000 

Blue  

0-0000196 

51110 

622,000000,000000 

Intermediate  
Indigo 

0-0000189 
0-0000185 

52910 
54070 

644,000000,000000 
658,000000,000000 

Intermediate  . 

0-0000181 

55240 

672,000000,000000 

Violet 

0-0000174 

57490 

699  000000,000000 

Extreme  

0-0000167 

59750 

727,000000,000000 

Taking   the  velocity   of  light  at 
192000  miles  per  second. 

Part  III. 


From  this  table  we  see,  that  the  sensibility  of  the  eye  is  confined  within  much  narrower  limits  than  that 
of  the  ear,  the  ratio  of  the  extreme  vibrations  being  nearly  1'58  :  1,  and  therefore  less  than  an  octave,  and  about 
equal  to  a  minor  sixth.  That  man  should  be  able  to  measure,  with  certainty,  such  minute  portions  of  space  and 
time,  is  not  a  little  wonderful  ;  for  it  may  be  observed,  whatever  theory  of  light  we  adopt,  these  periods  and  these 
spaces  have  a  real  existence,  being1,  in  fact,  deduced  by  Newton  from  direct  measurements,  and  involving  nothing 
hypothetical  but  the  names  here  given  them. 

The  direction  of  a  ray  in  the  undulatory  system  is  a  line  perpendicular  to  the  surface  of  the  wave  at  any 
point.  When,  therefore,  the  vibration  is  propagated  through  an  uniform  ether,  the  wave  being  bounded  by 
spherical  surfaces,  the  direction  of  the  ray  is  constant,  and  from  the  centre.  Thus  in  this  system  a  ray  of  light 
moves  in  a  right  line  in  an  uniform  medium. 

The  intensity  of  a  ray  is,  of  course,  in  some  certain  determinate  ratio  of  the  impulse  made  on  the  retina  data 
tempore  by  the  ethereal  molecules,  and  therefore  in  some  certain  ratio  of  their  amplitudes  of  excursion,  or  their 
absolute  velocities.  The  principle  of  the  conservation  of  living  forces  requires  that  the  amplitude  of  excursion 
of  a  molecule,  situated  at  any  distance  from  the  vibrating  centre,  should  be  as  the  distance  inversely,  (see 
ACOUSTICS.)  If  then  we  suppose  the  sensation  created  in  the  retina  to  be  as  the  simple  vis  inertia  of  the  mole- 
cules producing  it,  light  ought  to  decrease  inversely  as  the  distance  ;  if  as  the  vis  viva,  (which  is  as  the  square  of 
the  velocity,)  inversely  as  the  square  of  the  distance.  As  we  know  nothing  of  the  mode  in  which  the  immediate 
sensation  of  light  or  sound  is  produced  in  the  sensorium,  we  have  no  reason  to  prefer  one  of  these  ratios  to  the 
other  a  priori.  But  when  we  consider,  that  in  the  division  of  a  beam  of  light  by  partial  reflexion,  or  by  double 
refraction,  or  otherwise,  there  is  neither  gain  nor  loss  of  light,  (supposing  the  perfect  transparency  and  polish  of 
the  medium  which  operates  the  division)  so  that  the  sum  of  the  intensities  remains  constant,  however  the  absolute 
velocities  of  the  vibrating  molecules  may  change,  either  in  quantity,  or  (as  in  the  case  of  reflexion,  where  they 
must  be  conceived  to  rebound  from  each  other,  mediately  or  immediately)  in  sign,  the  agreement  of  this  law  in 
all  cases  with  that  of  the  conservation  of  the  vis  viva,  and  its  opposition  in  the  other  mentioned  case  to  that  of 
the  uniform  motion  of  the  centre  of  gravity,  (which  would  make  not  the  sum,  but  the  difference  of  the  intensities 
constant,  were  the  simple  ratio  of  their  velocities  assumed  for  their  measure,)  (see  DYNAMICS,)  leaves  us  no 
choice  in  preferring  the  square  of  the  absolute  velocity,  or  of  the  amplitude  of  excursion  of  a  vibrating  molecule, 
for  the  measure  of  the  intensity  of  the  ray  it  propagates  ;  and  thus  the  observed  law  of  the  diminution  of  light 
is  reconciled  to  the  undulatory  doctrine. 

When  the  medium  through  which  the  vibrations  are  transmitted  is  not  uniformly  elastic,  the  waves  will  make 
unequal  progress  in  different  directions,  according  to  the  law  of  elasticity.  In  this  case  the  figure  of  the  wave 
will  not  be  spherical.  If  we  suppose  the  elasticity  to  vary  by  insensible  gradations,  as  when  light  passes  through 
the  atmosphere,  whose  refracting  power  is  variable,  the  figure  of  the  wave  will  be  flattened  towards  that  part 
where  the  elasticity  is  less.  Thus,  in  fig.  126,  if  A  B  be  the  earth's  surface,  C  D,  E  F,  G  H,  &c.  the  atmo- 
spheric strata,  and  S  a  luminous  point,  the  waves  will  be  less  curved  as  they  approach  the  perpendicular  S  B  ;  and 
the  line  S,  1,2,  3,  4,  5,  &c.  drawn  so  as  to  intersect  them  all  at  right  angles,  will  be  a  curve  convex  downwards, 
so  that  a  ray  will  appear  to  be  continually  bent  downwards  towards  the  earth,  as  we  see  really  happens.  Let  us 
now  proceed  to  consider  the  explanation  of  the  phenomena  of  reflexion  and  refraction  on  the  undulatory  system. 

The  perpendicular  reflexion  of  light  may  be  conceived,  by  the  analogy  of  an  elastic  ball  in  motion  impinging 
directly  on  another  at  rest,  and  in  this  way  it  has  been  illustrated  by  Dr.  Young.  If  the  balls  be  equal,  the 
whole  motion  of  the  impinging  ball  will  be  transferred  to  the  other,  no  reflexion  taking  place  ;  and  thus  the 
impulse  may  be  propagated  undiminished  along  a  line  of  balls  as  far  as  we  please.  So  it  is  with  light  moving1 
in  a  uniform  medium,  or  passing  from  one  medium  to  another  of  equal  elasticity.  But  if  a  less  ball  impinge  on 


576 


577. 

Direction  of 
a  ray. 

578. 
Law  of  in- 
tensity of 
light 


579. 

Form  ot  the 
wave- 


PelPend'- 


454  L  I  G  H  T. 

Light,     a  greater  at  rest,  it  will  be  reflected,  and  with  a   momentum  which  is  greater  in  proportion   to  the  difference  in 
— V— -^  size  of  the  balls. 

But  to  render  an  account  of  oblique  reflexion  and  refraction,  and  the  other  phenomena  we  shall  have  to  speak 
Principles.    ojr   jt  w;]]  be  necessary  jo  ]ay  down  the  following  principles,  which  are  either  self-evident  or  follow  immediately 
from  the  elementary  principles  of  dynamics. 

582.  1.  When  any  number  of  very  minute  impulses  is  communicated   at  once  to  the  particles  of  any  medium,  or 
Superpo-     of  any  mechanical  system  under  the  influence  of  any  forces,  the  motion  of  each  particle  at  any  instant  will  be  the 
small  °'       SUm  °^  a"  t'le  mot'ons  wn'ch  it  would  have  at  that  instant,  had  each  of  the  impulses  been  communicated  to  the 
motions        system  alone,  (the  word  sum  being  understood  in  its  algebraical  sense.) 

583.  2.  Every  vibrating  molecule  in  an  elastic  medium,  whether  vibrating  by  an  original  impulse,  or  in  consequence 
Principle  of  of  an  impulse  propagated  to  it  from  others,  may  be   regarded  as  a  centre  of  vibration   from  which  a  system  of 
secondary    secondary  waves  emanates  in  all  directions,  according  to  the  laws  of  the  propagation  of  waves  in  the  medium. 

Proposition.  In  the  reflexion  of  light  on  the  undulatory  doctrine,  the  angle  of  incidence  is  equal  to  that  of 
reflexion. 

584.  Let  A  B  be  a  plane   surface  separating  the  two  media,  and  S   the  luminous   point  propagating  a  series  of 
Law  of  re-    spherical  waves,  of  which  let  A.  a  be  one.     So  soon  as  this  reaches  the  surface  at  A,  a  partial  reflexion  will  take 

LIOU  at  a  piace  ;  an(j  regarding  the  point  A  as  a  new  centre  of  vibration,  spherical  waves  will  begin  to  be  propagated 
from  it  as  a  centre,  one  of  which  proceeds  forwards  into  the  reflecting  medium,  with  a  velocity  greater  or  less 
than  that  of  the  incident  wave,  as  the  case  may  be  ;  the  other  backwards  into  the  medium  of  incidence,  with  a 
velocity  equal  to  that  of  the  incident  wave.  It  is  only  with  the  latter  we  are  at  present  concerned.  Conceive 
now  the  wave  A  a  to  move  forward  into  the  position  B  6 ;  then  in  the  time  that  it  has  run  over  the  space  P  B,  the 
wave  propagated  from  A  will  have  run  back  over  a  distance  A  d  =  P  B,  and  the  hemisphere  whose  radius  is  A  d 
will  represent  this  wave.  Between  A  and  B  take  any  point  X,  and  describe  the  hemispheric  surface  X  c.  Then 
regarding  X  as  a  centre  of  vibration,  its  vibrations  will  not  commence  till  the  wave  has  reached  it.  It  will,  there- 
fore, begin  to  vibrate  later  than  A,  by  the  whole  time  the  wave  A  a  takes  to  run  over  P  Q  ;  but  when  once  set 
in  vibration,  it  propagates  backwards  a  spherical  wave  with  the  same  velocity,  so  that  when  the  original  wave 
has  advanced  into  the  situation  B  6,  the  wave  from  X  will  have  expanded  into  a  hemisphere,  whose  radius  X  c 
is  equal  toPB,  —  PQ.orA  B.  Now  this  being  true  of  every  point  X,  if  we  conceive  a  surface  touching  all  these 
hemispheres  in  d,  c,  B,  this  surface  will  mark  the  points  at  which  the  reflected  impulse  has  just  arrived,  and 
which  just  begins  to  move  when  the  original  wave  has  reached  B,  and  will,  therefore,  be  the  surface  of  the 
reflected  wave.  Conceive  now  the  spherical  surface  6  B  prolonged  below  the  plane  A  B,  as  represented  by 
the  dotted  line  D  C  B,  and  the  same  of  the  spheres  about  A  and  X.  Then  the  spherical  surfaces  D  C  B  and 
C  c  being  both  perpendicular  to  S  X  C,  must  touch  each  other  in  C,  hence  the  surface  touching  all  the  hemi- 
spheres about  A,  X,  &c.  as  centres,  below  A  B  is  a  segment  of  a  sphere  having  S  for  a  centre,  and  therefore 
the  surface  B  c  d  or  the  reflected  wave  is  a  segment  of  a  sphere  having  its  centre  at  s  as  much  below  the  line 
A  B  as  S  is  above  it. 

Now  to  an  eye  placed  at  X,  the  luminous  point  S  will  appear  in  the  direction  S  X  perpendicular  to  the  incident 
wave,  and  the  eye  placed  in  c  will  perceive  the  reflected  image  of  S  at  *  in  the  direction  cs,  perpendicular  to 
the  reflected  wave  ;  but  cs  passes  through  X,  because  the  spheres  c  C  and  B  b  touch  at  c.  Therefore  the  ray  by 
which  s  is  seen  at  c  passes  through  X.  But  the  surfaces  B  D,  B  d  being  similar  and  equal,  the  angle  B  X  c  = 
B  X  C  =  A  X  S,  that  is,  the  angle  of  incidence  is  equal  to  that  of  reflexion.  Q.  E.  D. 

585.  Cor.     If  the  reflecting  surface  be  not  a  plane,  the  reflected  wave  will  not  be  spherical ;  its  form  is,  however, 
ReBexion  at  easily  determined  as  follows :  Suppose  the  direct  wave  to  have  assumed  the  position  B  6.     Take  any  point  X  in 
curved  sur-   the  reflecting  surface,  and  describe  the  sphere  X  Q,  and  with  the  centre  X  and  radius  =  B  Q,  describe  another 
pCes'.Qft       sphere.     Do  this  for  every  point  in  the  surface  A  B,  and  the   surface  which  is  a  common   tangent  (as  B  erf)  to 

'*>•  all  these  spheres,  is  the  surface  of  the  reflected  wave,  because  it  marks  the  farthest  limit  to  which  the  reflected 

impulse  has  reached  in  all  directions  at  the  instant  when  the  direct  impulse  has  reached  B.  Now  take  Y 
infinitely  near  to  X,  and,  making  the  same  construction  at  Y,  let  c,  e  be  the  points  in  the  reflected  wave  to  which 
X  c  and  Y  e  are  respectively  perpendicular.  Draw  X  r  perpendicular  to  Y  e,  and  X  q  to  S  Yg,  then,  since  Y  e  = 
S  B  -  S  Y,  and  X  c  =  S  B  -  S  X,  we  have  Ye-Xc,orYr  =  SX-SY=Yg,  andXY  being  common  to 
the  right  angled  triangles  X  Y  r,  X  Y  q,  the  angle  r  Y  X  must  be  equal  to  X  Y  q  or  to  S  Y  A,  so  that  the  same 
law  of  reflexion  holds  good  in  curve  as  in  plane  surfaces. 
586  Proposition.  To  demonstrate  the  law  of  refraction  in  the  undulatory  system. 

Let  S,  fig.  129,  be  a  luminous  point,  and  let  any  wave  propagated  from  it  reach  in  succession  the  points  Y, 
X,  B  of  any  curve  surface  Y  X  B  of  a  refracting  medium,  whereof  X  and  Y  are  supposed  infinitely  near  each 
other.  As  the  wave  strikes  Y,  X,  B,  each  of  these  points  will  become  centres  of  undulation,  which  will  be 
propagated  in  the  refracting  medium  with  a  velocity  different  from  that  of  light  in  the  medium  of  incidence,  by 
reason  of  their  different  elasticities,  (Postulate  3.)  Let  V  :  v  '  '.  velocity  in  the  first  medium  to  that  in  the 

second,  (a  constant  ratio  by  hypothesis,)  and,  describing  the  sphere  B  Q  R,  take  X  c  =  —  .  Q  X  and  Y  e  = 

-=r- .  V  R,  then  will   X  c  and   Ye  represent  the  spaces  run  over  by  the   refracted  secondary  waves  propagated 

from  X  and  Y  respectively,  when  the  direct  wave  has  reached  B.  Hence,  if  about  X  and  Y  as  centres,  and  with 
these  radii  we  describe  spheres,  and  suppose  e,  c  to  be  points  in  the  curve  surface  which  is  a  tangent  to  all  such 
spheres,  it  is  clear  that  X  cand  Yewill  be  perpendicular  to  this  surface,  that  is,  to  the  surface  of  the  refracted  primary 
wave  ;  hence,  X  c  and  Y  e  will  be  the  directions  of  the  refracted  rays  at  X  and  Y.  Draw  X  q,  Xr  perpendiculai 


LIGHT.  455 

respectively  to  YR  and  Ye,  then  will  Yg  =  SX-SYandYr  =  Ye-  Xc=  ^- .  YR  -  -^-  .XQ  = 

(YR-XQ)  =  ^  {(SR-  SY)-  (SQ   -  S  X)  }  =  ^- .  (SX  -  S  Y)  =  -^  .  Yg.     Hence  we  have  Y9  . 

Yr  "  V  :  v.     But   since  S  X,  SY  are  direct  rays,  and  X  c,  Y  e  the  corresponding  refracted  ones,  therefore 
S  X  Y  is  the  complement  of  the  angle  of  incidence  of  S  X,  and,  consequently,  Y  X  q  is  equal  to  the  angle  of 
incidence  itself,  and  X  Yr  will  be  the  complement  of  the  angle  of  refraction,  and   therefore   Y  X  r  (=  90°  - 
X  Y  r)  =  the  angle  of  refraction  of  S  Y,  or,  (since  the  points  Y,  X  are  infinitely  near  each  other,)  of  S  X,  hence 
we  have 

Y  q  :  X  Y  ;  ;  sin  incidence  :  1, 

X  Y  :  Y  r  '.  \  1  :  sin  refraction. 

And  compounding  Y^  :  Yr  '.'.  sin   incidence  :  sin  refraction.     But  we  proved  before,   that  Y<7  :  Yr  in  the 
constant  ratio  of  V  :  » ;  therefore  the  sine  of  incidence  :  that  of  refraction  in  the  same  constant  ratio.   Q.  E.  D. 

Corollary  1.  In  the  cases  both  of  reflexion  and  refraction,  the  undulation  is  propagated  from  the  luminous  point  587. 
to  any  other  point  in  the  least  possible  time.  For  the  surface  both  of  the  reflected  and  refracted  waves  mark  the 
extreme  limits  to  which  the  impulse  has  been  propagated  by  reflexion  or  refraction  in  a  given  time.  The  undu- 
lation propagated  from  X  (fig.  127)  in  any  other  direction  than  X  e,  as,  for  instance,  X  <y,  will  fall  short  of  the  surface 
B  erf,  and  the  point  <y  therefore  will  have  been  reached,  and  passed  by  the  reflected  or  refracted  primary  wave  in 
the  situation  /3  7  S,  before  it  can  be  reached  by  the  secondary  undulation  propagated  from  X  in  the  direction  X  7. 

Corollary  2.     This  property  in   the  undulatory  system  corresponds  to  the  principle   of  least  action  in  the      588. 
corpuscular  doctrine,  and  may  be  thus  stated  generally  :  Law  of 

A  reflected  or  refracted  ray  will  always  pursue  such  a  course  as  would  be  described  in  the  least  possible  time,  sw|ftest 
by  a  point  moving  from  the  point  of  its  departure  to  that  of  its  arrival,  with  the  velocities  corresponding  to  the  pro' 
media  in  which  it  moves,  and  the  direction  of  its  motion. 

It  is  evident  that  this  is  general,  and  applies  to  cases  where  the  medium  is  either  of  variable  elasticity,  or  has      589. 
different  elasticities  in  different  directions ;  for  the  ray  is  by  definition  a  perpendicular  to  the  surface  of  the  wave,  Applies 
or  to  a  surface,  the  locus  of  all  the  molecules  in  the  medium,  which  are  just  attained  by  the  undulation,  and  just  genera"y- 
commencing  their  vibration,  so  that  the  reasoning  of  Corol.  1,  applies  equally  to  all  cases. 

The  properties  of  foci  and  Caustics  flow  with  such  elegance  and  simplicity  from  this  doctrine,  that  it  would      590. 
be  unpardonable  not  to  instance  its  application  to  that  part  of  the  theory  of  Optics. 

Definition.     A  focus  is  a  point  at  which  the  same  wave  arrives  at  the  same  instant  from  more  than  one  point  svstem 
in  a  surface.  Defined. 

It  is  evident,  that  when  this  is  the  case,  the  ethereal  molecules  in  the  focus  will  be  agitated  by  the  united  force 
of  all  the  undulations  which  reach  them  in  the  same  phase  at  the  same  instant,  and  will  be  proportionally  more 
violent  as  the  focus  is  common  to  a  greater  number  of  points,  and  the  light  in  the  focus  will  be  proportionally 
more  intense. 

Proposition.     Required  to  determine  the   nature  of  the  surface  which  shall   refract  all  rays  from  one  point       59} 
rigorously  to  one  focus.     Let  F  (fig.  129)  be   the  focus,  then  will  every  part  of  a  wave  propagated  from  S  and 
refracted  at  the  surface  A  B,  reach  F  at  the  same  instant ;   therefore  time  of  describing  S  X  with  velocity  V  + 
time  of  describing  X  F  with  velocity  v  is  constant  for  every  point  in  the  surface.     Or, 

S  X          F  X 
— — f-  -    —  =  constant,  orSX-f-,u.FX=  constant,  /*  being  the  relative  index  of  refraction. 


This  equation  then  defines  the  nature  of  the  curve  sought,  and  it  is  easy  to  perceive  its  identity  with  that 
expressed  by  the  equation  (n)  Art.  232,  obtained  from  a  direct  consideration  of  the  law  of  refraction,  but  by  a 
much  more  intricate  process. 

The  intensity  of  the  reflected  or  refracted  ray  cannot  be  computed  generally  in  the  present  very  imperfect  state      592. 
of  our  knowledge  of  the  theory  of  waves.     M.  Poisson,  however,  in  the  case  of  perpendicular  incidence,  and  on  Intensity  of 
the  particular  hypothesis  of  the  luminous  vibrations  being  performed  in  the  direction  of  the  ray  itself,  has  ^  ray  re- 
succeeded  in  investigating  the  comparative  intensities  of  the  incident,  reflected,    and  transmitted  rays.     His  J^jici£e' 
results  are  as  follows :    Taking  p.,  ft'  for  the  absolute  refractive  indices  of  the   media,  he  finds  (on  the  supposi-  larly 
tion  that  the  intensity  of  light  is  as  the  square  of  the  absolute  velocity  of  the  vibrating  molecules)  : 

Intensity  of  reflected  ray  :  that  of  incident  ; ;  {p!  —  /t)4  :  (p!  +  /i)4.  Intensity  of  the  intromitted  ray  :  that 
of  the  incident  ; ;  4/ta  :  < p.  -\-  p.')*.  Intensity  of  the  ray  intromitted  from  a  medium  whose  refractive  index  —  /i 
into  a  parallel  plate  of  one  whose  refractive  index  =  p!,  in  contact  at  its  second  surface  with  a  third  whose  refractive 
index  =  p.",  reflected  at  their  common  surface,  and  again  emergent  at  the  first  surface  :  intensity  of  the  ray 
originally  incident  on  the  first  surface  ; ;  16  p?  ,u's  (/'  p.')°~  -.  (p.  +  /<,')<  .  (/  +  /u")2.  And,  lastly,  the  intensity 
of  the  ray  transmitted  through  the  parallel  plate  of  the  second  medium  into  the  third  :  that  of  the  original 
incident  ray  '.'.  16  ft?  /»'8 :  (ft,  +  /t1)8 .  (/*'  +  ,u.")2  which  (in  the  case  where  the  third  medium  is  the  same  as  the 
first,  becomes  16  ft"  p.1 2  :  (/«-)-  p!)4. 

These  results  of  M.  Poisson,  so  far  as  they  have  been  hitherto  satisfactorily  compared  with  experiment,  593 
manifest  at  least  a  general  accordance,  and  the  undulatory  doctrine  thus  furnishes  a  plausible  explanation  of  the 
connection  of  the  reflecting  power  of  a  medium  with  its  refractive  index,  and  of  the  diminished  reflection  at  the 
common  surfaces  of  media  in  contact. — They  have  been  in  great  measure  (it  should  be  observed)  anticipated  by 
Dr.  Young,  in  his  Paper  on  Chromatics,  (Encydop.  Brit.)  by  reasoning  which  M.  Poisson  terms  indirect,  but 
which,  we  confess,  appears  to  us  by  no  means  to  merit  the  epithet. 


456  LIGHT. 

If  photometrical  experiments  enable  us  to  determine  the  proportion  of  the  reflected  to  the   incident  light,  we    Tart  HI. 
— — s<~- '  may  thence  conclude  the  index  *f  refraction  of  the  reflecting  medium,  and  that  in   cases  where  no   other  mode  v-~"v" ••' 
594.      will  apply.     Thus,  M.  Arago  having  ascertained  that  about  half  the  incident  light  is  reflected  at  a  perpendicular 

;eh™ve     incidence  from  mercury,  we  have  in  this  case  (  ^   J    =  \  ;     —  =  5'829  for  the  refractive  index  of  mer- 

indices.  ^  /*+/*/  /» 

curyout  of  air;  and  this  is  perfectly  consonant  to  the  general  tenor  of  optico-chemical  facts,  which  assign  to  the 
heavy  and  especially  to  the  white  metals  (as  indicated  in  their  transparent  combinations)  enormous  refractive 
and  dispersive  powers.  This  curious  and  interesting  application  has  not  been  overlooked  by  Dr.  Young  in  the 
Paper  alluded  to. 

595  To  complete  the  theory  of  reflexion  and  refraction  on  the  undulatory  hypothesis,  it  will  be  necessary  to  show 

what  becomes  of  those  oblique  portions  of  the  secondary  waves,  diverging  in  all  directions  from  every  point  of 
the  reflecting  or  refracting  surfaces  (as  X  7,  fig.  127)  which  do  not  conspire  to  form  the  principal  wave.  But 
to  understand  this,  we  must  enter  on  the  doctrine  of  the  interference  of  the  rays  of  light, — a  doctrine  we  owe 
almost  entirely  to  the  ingenuity  of  Dr.  Young,  though  some  of  its  features  may  be  pretty  distinctly  traced  in  the 
writings  of  Hooke,  (the  most  ingenious  man,  perhaps,  of  his  age,)  and  though  Newton  himself  occasionally 
indulged  in  speculations  bearing  a  certain  relation  to  it.  But  the  unpursued  speculations  of  Newton,  and  the 
appercus  of  Hooke,  however  distinct,  must  not  be  put  in  competition,  and,  indeed,  ought  scarcely  to  be 
mentioned  with  the  elegant,  simple,  and  comprehensive  theory  of  Young, — a  theory  which,  if  not  founded  in 
nature,  is  certainly  one  of  the  happiest  fictions  that  the  genius  of  man  has  yet  invented  to  group  together  natural 
phenomena,  as  well  as  the  most  fortunate  in  the  support  it  has  unexpectedly  received  from  whole  classes  of  new 
phenomena,  which  at  their  first  discovery  seemed  in  irreconcileable  opposition  to  it.  It  is,  in  fact,  in  all  its 
applications  and  details  one  succession  of  felicities,  insomuch  that  we  may  almost  be  induced  to  say,  if  it  be  not 
true,  it  deserves  to  be  so.  The  limits  of  this  Essay,  we  fear,  will  hardly  allow  us  to  do  it  ju  tice. 


§  III.     Of  the  Interference  of  the  Rays  of  Light. 

596.  The  principle  on  which  this  part  of  the  theory  of  Light  depends,  is  a  consequence  of  that  of  the  "  Superposition 
General        of  small  motions"  laid  down  in  Art.  583.     If  two  waves  arrive  at  once  at  the  same  molecule  of  the  ether,  that 
pr'nc'p'esof  molecule  will  receive  at  once  both  the  motions  it  would  have  had  in  virtue  of  each  separately,  and  its  resultant 

•e  motion  will,  therefore,  be  the  diagonal  of  a  parallelogram  whose  sides  are  the  separate  ones.  If,  therefore,  the 
two  component  motions  agree  in  direction  or  very  nearly  so,  the  resultant  will  be  very  nearly  equal  to  their  sum, 
and  in  the  same  direction.  If  they  very  nearly  oppose  each  other,  then  to  their  diti'ertnce.  Suppose,  now,  two 
vibratory  motions  consisting  of  a  series  of  successive  undulations  in  an  elastic  medium,  all  similar  and  equal  to 
each  other,  and  indefinitely  repeated,  to  arrive  at  the  same  point  from  the  same  original  centre  of  vibration,  but 
by  different  routes  (owing  to  the  interposition  of  obstacles  or  other  causes)  exactly,  or  very  nearly  in  the  same 
final  direction  ;  and  suppose,  also,  that  owing  either  to  a  difference  in  the  lengths  of  the  routes,  or  to  a  differ- 
ence in  the  velocities  with  which  they  are  traversed,  the  time  occupied  by  a  wave  in  arriving  by  the  first  route 
(A)  is  less  than  that  of  its  arriving  by  the  other  (B).  It  is  clear,  then,  that  any  ethereal  molecule  placed  in  any 
point  common  to  the  two  routes  A,  B,  will  begin  to  vibrate  in  virtue  of  the  undulations  propagated  along  A. 
before  the  moment  when  the  first  wave  propagated  along  B  reached  it.  Up,  then,  to  this  moment  its  motions 
will  be  the  same  as  if  the  waves  along  B  had  no  existence.  But  after  this  moment,  its  motions  will  be  very 
nearly  the  sum  or  difference  of  the  motions  it  would  have  separately  in  virtue  of  the  two  undulations 
each  subsisting  alone,  and  the  more  nearly,  the  more  nearly  the  two  routes  of  arrival  agree  in  their  final 
direction. 

597.  Now  it  may  happen,  that  the  difference  of  the  lengths  of  the  routes  or   the  difference  of  velocities  is  such,  that 
Case  of        the  waves  propagated  along  B  shall  reach  the  intersection  exactly  one-half  an  undulation  behind  the  others,  t.  e. 
complete      later   by  exactly  half  the  time  of  a   wave  running  over  a  space  equal  to  a  complete  undulation.     In   that  case, 
discordance  tjje  moiecuie  which  in   virtue  of  the  vibrations  propagated   along  A  would  (at  any  future  instant)  be  in  one 

phase  of  its  excursions  from  its  point  of  rest,  would,  in  virtue  of  those  propagated  along  B,  if  subsisting 
alone,  be  at  the  same  instant  in  exactly  the  opposite  phase,  i.  e.  moving  with  equal  velocity  in  the  contrary 
direction.  (See  Art.  570.)  Hence,  when  both  systems  of  vibration  coexist  the  motions  will  constantly  destroy 
each  other,  and  the  molecule  will  remain  at  rest.  The  same  will  hold  good  if  the  difference  of  routes  or 
velocities  be  such,  that  the  vibrations  propagated  along  B  shall  reach  the  intersection  of  the  routes  exactly 
^,  £,  £,  &c.  of  a  complete  period  of  undulation  after  those  propagated  along  A;  for  the  similar  phases  of  vibra- 
tion recurring  periodically,  and  being  (by  hypothesis)  continually  repeated  for  an  indefinite  time,  it  is  no  matter 
whether  the  first  vibration  propagated  along  B  be  superimposed  on,  or  interfere  with  (as  it  is  called)  the  first,  or 
any  subsequent  one  propagated  along  A,  provided  the  difference  of  their  phases  be  the  same. 

598.  On  the  other  hand  it   may  happen,   that  the  waves  propagated  along  B  do  not  reach  the  intersection   till 
Case  of        exactly  one,  two,  or  more  whole  periods  after  the  corresponding  waves   propagated  along  A.     In  this  case,  the 
complete       molecule  at  the  intersection  will,  at  any  instant   subsequent  to  the  time  of  arrival   of  the  first  wave  along  B, 

>ce-  be  agitated  at  once  by  both  vibrations  in  the  same  phase,  and  therefore  the  velocity  and  amplitude  of  its  excur- 
sions will,  instead  of  being  destroyed,  be  doubled. 


LIGHT.  457 

I.u-ht.          Lastly,  it  may  happen,  that  the  difference  of  the  times  of  arrival  of  the  corresponding  waves  is  neither  an    Part  Hi. 
.— Y—^/  exact  even,  or   odd  multiple  of  half  a  complete  period  of  undulation.     In  that  case,  the  molecule  will  vibrate  *— — N.^— '' 
with  a  joint  motion,  less  than  double  what  it  would  have  in  virtue  of  either  separately.  599- 

An  apt  illustration  of  the  case  of  interference  here  described,  may  be  had  by  considering  the  analogous  case  in  goo 
the  interference  of  waves  on  the  surface  of  water.  Conceive,  for  instance,  two  equally  broad  canals  A  and  B  to  illustration 
enter  two  canals  at  right  angles  into  the  side  of  a  reservoir,  at  both  whose  apertures,  from  an  origin  at  a  great  dis-  from  waves 
tance,  a  wave  arrives  at  the  same  instant,  and  runs  along  the  two  canals  with  equal,  uniform  velocities.  Let  their  Pr°Pa»a 
sides  be  perfectly  smooth,  and  their  breadths  everywhere  equal,  but  let  them  be  led,  by  a  gentle  curvature,  to  meet 
in  a  point  at  some  distance,  and,  the  curvature  of  B  being  supposed  somewhat  greater  than  that  of  A,  let  the 
distance  from  their  intersection  to  the  reservoir,  measured  along  B,  be  greater  than  along  A.  It  is  obvious, 
that  (if  we  consider  only  a  single  wave)  the  portion  of  it  propagated  along  A  will  reach  the  intersection  first, 
and  after  it  that  propagated  along  B,  so  that  the  water  at  that  point  will  be  agitated  by  two  waves  in  succession. 
But,  let  the  original  cause  of  undulation  be  continually  repeated  so  as  to  produce  an  indefinite  series  of  equal 
and  similar  waves.  Then,  if  the  difference  of  lengths  of  the  two  canals  be  just  equal  to  half  the  interval  between 
the  summits  of  two  consecutive  waves,  it  is  evident  that  when  the  summit  of  any  wave  propagated  along  A  has 
reached  the  intersection,  the  depression  between  two  consecutive  summits  (viz.  that  corresponding  to  the  wave 
propagated  along  A,  and  that  of  the  wave  immediately  preceding  it)  will  arrive  at  the  intersection  by  the  course 
B.  Thus,  in  virtue  of  the  wave  along  A  the  water  will  be  raised  as  much  above  its  natural  level,  as  it  will  be 
depressed  below  it  by  that  along  B.  Its  level  will,  therefore,  be  unchanged. — Now  as  the  wave  propagated 
along  A  passes  the  intersection,  it  subsides,  from  its  maximum,  by  precisely  the  same  gradations  as  that  along 
B,  passing  it  with  equal  velocity,  rises,  from  its  minimum,  so  that  the  level  will  be  preserved  at  the  point  of 
intersection,  undisturbed  so  long  as  the  original  cause  of  undulation  continues  to  act  regularly.  So  soon  as  it 
ceases,  however,  the  last  half  wave  which  runs  along  B  will  have  no  corresponding  portion  of  a  wave  along 
A  to  interfere  with,  and  will,  therefore,  create  a  single  fluctuation  at  the  point  of  concourse. 

In  the  theory  of  the  interferences  of  light  we  may  disregard  these  commencing  and  terminal,  uncompensated      601. 
undulations,  and  parts  of  undulations,  as  being  so  few  "in  number  as  to   excite  no  impression  on  the  retina,  and  Initial  anj 
consider  the  interfering  rays  as  of  indefinite  duration,  or  as  destitute  of  either  beginning  or  end.  termmal  vi- 

According  to  the  foregoing  reasoning  then  it  appears,  that  if  two  rays  having  a  common  origin,  i.  e.  forming 
parts  of  one  and  the  same  system  of  luminous  waves  proceeding  from  a  common  centre,  be  conducted  by  different    °go2 
routes  to  one  point  which  we  will  suppose  to  be  situated  on  a  white  screen,  or  on  the  retina  of  the  eye,  they  Mutual  an- 
will  there  produce  a  bright  point,  or  the  sensation  of  light,  if  their  difference  of  routes  be  an  even  multiple  of  the  nihilation  of 
length  of  half  an  undulation  and  a  dark  one  ;    or  the  sense  of  darkness,  if  an  odd  multiple  of  it;   and  if  inter-  two  rajs  of 
mediate,  then  a  feebler  or  a  stronger  sense  of  light,  as  the  difference  of  routes  approximates  to  one  or  the  other  of  '' 
these  limits.     That  two  lights  should  in  any  case  annihilate  each  other,  and  produce  darkness,  appears  a  strange 
paradox,  yet  experiment  confirms  it ;  and  the  fact  was  observed,  and  broadly  stated  by  Grimaldi  long  before  any 
plausible  reason  could  be  given  of  it. 

Having  thus   obtained    a  general  idea  of  the  nature  of  interferences,  let  us  now  endeavour  to  subject  their      603. 
effects  to  a  more  strict  calculation.     To  this  end  it  will  be  necessary  to  fix  with  precision   the   sense  of  some 
words  hitherto  used  rather  loosely. 

Definition.     The  phase  of  an  undulation  affecting  any  given  molecule  of  ether  at  any  instant  of  time,   is      604. 
numerically  expressed  by  an  arc  of  a  circle  to  radius  unity,  increasing  proportionally  to  the  time — commencing  Definition*! 
at  0  when  the  molecule  is  at  rest  at  its  greatest  positive  distance  of  excursion,  and  becoming  equal  to   one  cir-         e' 
cumference  when  the  molecule,  after  completing  the  whole  of  a  vibration,  returns  again  to  the   same  state  of 

(/    |   {"*  \  /    i    {~* 

2  TT  .  — — —  I,  2  IT.  — — —  is  the  phase  of 

the  undulation  at  the  instant  t. 

Definition.     The  amplitude  of  vibration  of  a  ray  or  system  of  waves  is  the  coefficient  a,  or  the  maximum       605. 
excursion  from  rest,  of  each  molecule  of  the  ether  in  its  course.  Amplitude 

Carol.     The  intensity  of  a  ray  of  light  is  as  the  square  of  the  amplitude  of  the  vibrations  of  the  waves  of  which 
it  consists. 

Definition.     Similar  rays,  or  systems  of  luminous  waves,  are-  such  as  have  the  vibratory  motions  of  the       606. 
ethereal  molecules  which  compose  them  regulated  by  the  same  laws,  and  their  vibrations  performed  in  equal  Slmilalr'')'s- 
times,  and  the  curves  or  straight  lines  they  describe  in  virtue  of  them,  similar  and  similarly  situated  in  space,  so 
that  the  motions  of  any  two  corresponding  molecules  in  each,  shall  at  every  instant  of  time  be  parallel  to  each 
other. 

Carol.     Similar  rays  have  the  same  colour. 

Definition.     The  origin  of  a  ray,  or  a  system  of  waves,  is  the  vibrating  material  centre  from  which  the  waves       607. 
begin  to  be  propagated,  or  more  generally,  a  fixed  point  in  its  length,  at  which  an  ethereal    molecule,  at  an  Origin  of  a 
assumed  epoch,  was  in  the  phase  0  of  its  undulation.  ray- 

Carol.  Two  systems  of  interfering  waves  having  their  origins  distant  by  an  exact  number  of  undulations,  may       608. 
be  regarded  as  having  a  common  origin. 

Proposition.     To  find  the  origin  of  a  ray,  having  given  the  expression  for  the  velocity  of  one  of  its  vibrating       "09. 

molecules.  Tl! find  lhe 

origin  ot  a 

/          i  _j_  Q  \  ray- 

Let  a  =  a  .    "J  E,   and   let    v  =  a  .   sin  (  2  ir  .  — — — -  \   be    the    expression    given  for    the   velocity 
VOL.  iv.  3  o 


458  LIGHT. 

Light,     of  any  assumed  molecule    (M)   at  the  instant  1.     Let  V  represent  the  velocity  of  light,  and  X  the  length     Pirl  "I. 
v~^v"~-'  of  an   undulation,    and    &  the  distance  run  over  by   light   in   the  time  1.      Then  will  8  =  V  t  and  X  =  VT    ^•v"™*' 

and  consequently  --•  =  —  -.     Suppose  t>0  to  represent  the  velocity  of  a  vibrating  molecule  at  the  origin  of  the 

ray  at  the  instant  t,  then  will  »„  =  a  .  sin  (  2  v  .  —  j  =  a  .  sin  (  2  TT  —  \     But  the  molecule  M  moves  only 

by  an  impulse  communicated  to  it  from  the  origin,  and  therefore  all  its  motions  are  later  than  those  at  the  origin 
by  a  constant  interval  equal  to  the  time  required  for  light  to  run  over  the  distance  of  M  from  the  origin.     Call 

D  that  distance,  then  -=-  is  the  interval  in  question,  and  t  --  —  is  the  time  elapsed  at  the  instant  t,  since  the 

S        t-—\ 

molecule  commenced  its  periodic  motions  ;  therefore  its  velocity  0  must  =  a  .  sin  I  2  w  V     I,  and  con- 

sequently C  =  --  —  ,  or  D  =  -  V  C. 

Hence  we  see  that  the  distance  of  the  molecule  M  from  the  origin  of  the  ray,  is  equal  to  the  space  described  by 
Light,  in  a  time  represented  by  the  arbitrary  constant  C,  and  is  therefore  given  when  C  is  so,  and  vice  versa. 
610.          Carol,     Since  V  T  =  X  the  expression  for  the  velocity  becomes 

v=a.  sin  2  TT  .(  —  --  —  j  =  a  .  sin  2  v{  -  -  --  jand  similarly  x=  a.  cos  2  ir  f  —  '-  —  .  j 

Proposition.     To  determine  the  colour,  origin,  and  intensity  of  a  ray  resulting  from  the  interference  of  two 
Resu       of  sjmiiar  rayS)  differing  in  origin  and  intensity. 

fering  rays        Let  a8  and  a'*  be  the  intensities  of  the  rays,  or  a,  a1  their  amplitudes  of  vibration,  and  take  a  =  a  .  V  E, 
™  a1  =.  a'  .  */  E,  then,  if  we  put  0  for  the  phase  of  vibration  of  a  molecule  M  at  the  instant  t  which  it  would  be  in, 

k 
in  virtue  of  the  first  system  of  waves  (A),  and  0  -j-  k  for  its  phase,  in  virtue  of  the  other  (B),  —  .   T  will  repre- 

2  7T 

sent  the  time  taken  by  light  to  run   over  a  space    equal  to  the   interval    of  their  origin,   and  the  velocities 
and  distances  from  rest  which  M  would  have,  separately  at  the  instant  t,  in  virtue  of  the  two  rays,  will  be 

»  =  a  .  sin  0  ;  vf  =  a'  .  sin  (0  -f-  k),  and  x  =  a  .  cos  0  ;  x1—  a'  .  cos  (0  -f  k). 
Therefore,  in  virtue  of  the  resulting  ray,  it  will  have  the  velocity 

v  +  v1  =  a  .  sin  0  -f-  "'  •  sin  ("  +  &)•  and  x  +  d  =  o  •  cos  0  +  a'  .  cos  (0  -f-  k). 

Let  the  former  be  put  equal  to  A  .  sin   (0  +  B),  the  possibility  of  which   assumption  will  be  shown   by  our 
being  able  to  determine  A  and  B,  so  as  to  satisfy  this  condition.     Then  we  have 


(o  -j-  a'  .  cos  K)  sin  0  +  <*'  •  sin  k  .  cos  0  =  A  .  cos  B  .  sin  0  -(-  A  .  sin  B  .  cos  0, 
and  equating  like  terms, 

A  .  cos  B  =  n  -\-  a  .  cos  k  ;  A  .  sin  B  =  a'  .  sin  k, 
whence  we  get,  dividing  one  by  the  other, 


a' .  sin  k  a' ,  sin  k 


„  a    .  sill  K  «.   ,  am  K  / 

tan  B  =  — — ; ;  A  =  --        — •=  V  a?  -  2  a  a' .  cos  k  -f  a'» 

a-f-a  .  cosAr  sin  B 

and  these  values  being  determined,  A  and  B  are  known,  and,  therefore,  v  -f-  i/  =  A  .  sin  (0  -j-  B).  Similarly, 
if  we  put  x  -j-  x1  =  A' .  cos  (0  -j-  B')  we  obtain  values  of  A'  and  B'  precisely  similar,  writing  only  a  a'  for  a,  a' 
respectively. 

612.  Carol.  1.  Hence  we  conclude,  1st.  that  the  resultant  ray  is  similar  to  the  component  ones,  and  has  the  same 
period,  i.  e.  the  same  colour. 

613.  Carol.  2.  M.  Fresnel  has  given  the  following  elegant  rule  for  determining  the   amplitude  and  origin  of  the 
Fresnel's  / 

theorem.       resultant  ray,  which  follows  immediately  from  the  value  of  A  and  the  equation  sin  B  =  — — .  sin  K  above  found. 

A 

Construct  a  parallelogram,  having  its  adjacent  sides  proportional  to  the  amplitudes  a,  a'  of  the  component  rays, 
and  the  angle  between  them  measured  by  a  circular  arc  to  radius  unity,  equal  to  the  differences  of  their  phases, 
then  will  the  diagonal  of  this  parallelogram  represent  on  the  same  scale  the  amplitude  of  the  resulting  ray, 
and  the  angle  included  between  it,  and  either  side  will  represent  the  difference  of  phases  between  it  and 
the  ray  corresponding  ;  or,  which  comes  to  the  same  thing,  the  difference  of  their  origins  (when  reduced  to 
space.) 


L  1  G  H  T.  453 

Light.          Carol.  3.  Thus  in  the  case  of  complete  discordance,  the  diagonal  of  the  parallelogram  vanishes,  and  the  angle  Part  "f- 

•—•*-"—   becomes  180°,  or  half  a  circumference,  corresponding  to  a  difference  of  origins  of  half  an  undulation.     In  that  x— ' ^v^— ' 

of  complete  accordance,  the  angle  is  0,  or  360°,  and  the  origins  of  the  rays  coincide,  or  (which  comes  to  the  614. 

same  thing)  differ  by  an  exact  undulation,  and  the  diagonal  is  double  of  the  side,  so  that  the  intensity  of  the  Cases  ol 

compound  ray  is  four  times  that  of  either  ray  singly.  concord  and 

Carol.  4.     If  the  origins  of  two  equally  intense   rays  differ  by  one  quarter  of  an  undulation,  the  resultant  discord. 

ray  will  have  its  amplitude  to  that  of  either  component  one,  as  ^2  :  1,  and,  therefore,  its  intensity  double,  and  615. 

its  origin  will  differ  one-eighth  of  an  undulation  from  that  of  either.     Thus  in  this  particular  care,  the  brightness  Caseoi 

of  the  compound  ray  is  the  sum  of  the  brightnesses  of  the  components,  and  its  position  exactly  intermediate  a  c,uarter  of 

between  them.  an  unduU- 

Corol.  5.     Any  ray  may  be  resolved  into  two,  differing  in  origin  and  amplitude,  by  the  same  rules  as  govern  ''<»>• 

the  resolution  of  forces  in  Mechanics.  616. 

Carol.  6.     The  sum  of  the  intensities  of  the  component  rays  exceeds  that  of  the  resultant,  when  their  origins  Composi- 
differ  by  less  than  a  quarter  of  an  undulation,  falls  short  of  it  when  the   difference  is  between  £  and  J,  again 

exceeds  it  when  between  ^  and  f-,  and  so  on.     For  the  value  of  A',  above  found,  gives  ray's ' 

o4  -j-  a"1  -  A2  =  2  a  a',  cos  k  ;  617. 

now  cf,  a's,  and  A2,  represent  the  intensities  of  the  respective  rays  whose  momenta  are  a,  a1,  and  A.  intensities 

Carol.  7      In  the  same  manner  may  any  number  of  similar  rays  be  compounded,  and  the  resultant  ray  will  be  of  simple 
similar  to  the  elementary  rays,  and  vice  versa.  and  com 

Let  us  now  consider  the  interference  of  waves  having  the  same  period  (or  colour)  but  in  all   other  respects  pou"d  ra5" 
dissimilar.  Generd 

The  law  of  vibration  of  the  molecules  of  the  luminous  bodies  which  agitate  the  ether,  restricting  their  motions  problem  of 
to  ellipses  performed  in  planes,  the  same  will    hold  good  of  the  motions  of  each  molecule  of  the  ether.     Now  inter- 
every  elliptic  vibration,  or  rather  revolution,  performed  under  the  influence  of  a  force  directed  to  its  centre  and  ferences. 
proportional  to  the  distance,  is  decompostd  into  three   rectilinear  vibrations,  lying  in   any  three  planes  at  right 
angles  to  each  other,  each  of  which  separately  would  be  performed  by  the  action  of  the  same  force  in  the  same 
time,  and  according  to  the  same  laws  of  velocity,  time,  and   space.     Hence,  every  elliptic  vibration  may  be 
expressed  by  regarding  the  place  of  the  vibrating  molecule  at  any  instant  t  as  determined  by  three  coordinates 
T,  y,  z,  such  that,  0  being  an  arc  proportional  to  the  time,  we  shall  have 

dx 
x  =  a  .  cos  (0  -f-  p)  ;    -    — —  =  u  =  a  .  sin  (0  ~j-  p) 

(1.)  y  -  b  .  cos  (6  f  q)  •    -    -^-  =  v  =  /i  .  sin  (0  -j-  q)    }•   (2.) 

z  =  c  .  cos  (0  -4-  r)  ;    —  • =  w  —  </  .  sin  (f  -J-  r) 

d  t 

In  fact,  if  we  multiply  the  first  of  these  equations  by  an  indeterminate  I,  the  second  by  m,  and  the  third  by  n 
and  add,  we  get 

(3);  I  x  -j-  my  -\-nz  =  cos  0  {  I  a  .  cos  p  -f-  m  b  .  cos  q  -f-  n  c  .  cos  r  } 

—  sin  6  {  /  a  .  sin  p  -(-  m  b  .  sin  q  -f-  n  c .  sin  r  } 
and,  therefore,  if  we  determine  I,  m,  n,  so  that 

/  a  .  cos  p  -j-  m  b  .  cos  q  -f-  n  c  .  cos  r  =  0  ;  /  a  .  sin  p  -f-  in  b  .  sin  q  -j-  n  c  .  sin  r  =  0 
which  (being  equations  of  the  first  degree  only)  is  always  possible,  we  shall  have,  independently  of  0, 

lx-\-my-\-nz  =  0;          (4,) 

and  this,  being  the  equation  of  a  plane,  shows  that  the  whole  curve  represented  by  the  above  equations  lies  in 
one  plane.     Again,  if  we  eliminate  0  between  the  equations,  involving  x  and  y  only,  we  have 


-  1     x  -  1    y 

cos          --  cos 
a 


or,  taking  the  cosines  on  both  sides, 


ind  reducing,  we  get  the  equation 

2-ir  -f  •  COS(P  -  9)  =  sin  (p  ~  ?)!;    (5° 

which  is  the  equation  of  an  ellipse  having  the  origin  of  the  x  and  y  in  its  centre,  and  the  same  is  true  mutatis 
mutandis  of  the  equations  between  x  and  z,  and  between  y  and  z.  Thus  the  curve  represented  by  the  three 
equations  between  x,  y,  2,  0,  has  an  ellipse  about  the  centre  for  its  projection  on  each  of  the  planes  at  right 
angles  to  each  other,  and  is,  of  course,  itself  an  ellipse. 

3o2 


460  L  I  G  H  T. 

Light.          Suppose  now  two  systems  of  waves,  or  two  rays  coincident  in  direction,  to  interfere  with  each  other.     If  we    Part  HI. 
^— -v-" ~s  accent  the  letters  of  the  above  expressions  to  represent  corresponding  quantities  for  the   second  system,   we  ^ — y— 
619.       shall  have 

X  =  x  -f  a?  =  a  .  cos  (0  -f-  p)  -f  a' .  cos  (0  -f-  p')         -^ 

Y  =  y  +  y'  =  b  .  cos  (0  -j-  q)  -f-  6'  .  cos  (0  +  <]')  (6) 

Z  =  z  -f  z'  =  c  .  cos  (0  -J-  r)  -j-  c1  .  cos  (0  -f  /)         J 

and  similarly  for  the  velocities  u  -f  u',  v  -J-  t>',  jo  -f-  M/.     In  the  same  manner,  then,  as  we  proceeded  in  the  case 
of  two  similar  rays,  let  us  suppose 

a  .  cos  (0  +  f)  -j-  a'  .  cos  (0  -f  p')  =  A  .  cos  (6  -f-  P) 
and  developing 

(a  .  cos  p  -j-  «'  •  cos  p1)  cos  0  —  (a  .  sin  p  -f  a!  .  sin  p')  sin  6  =  A  .  cos  P  .  cos  0  —  A  .  sin  P  .  sin  0, 
whence  we  get 

—  _    ffl  .  sin  p  -f  a' .  sin  p'  a  .  sinp  -f-  a>  •  sinp'      \ 

a  .  cosp  -f-  a' .  cosp'  '  sin  P  >  '     (7) 

or,  A  =  J  of  -f-  2  a  a' .  cos  (p  -  p1)  +-a' «  ) 

Thus  we  have  X  =  A  .  cos  (0  -J-  P),  and,  similarly,  Y  =  B  .  cos  (0  -f-  Q),  and  Z  =  C  .  cos  (0  -f.  R),  and  a  process 

exactly  similar  gives  us  the  corresponding  expressions  for  the  velocities. 

620.  Thus  we  see  that  the  same  rules  of  composition  and  resolution  apply  to  dissimilar  as  to  similar  vibrations. 

Composi-  Each  vibration  must  first  be  resolved  into  three  rectilinear  vibrations  in  three  fixed  planes  at  right  angles  to  each 
resolution  ot'ler-  These  must  be  separately  compounded  to  produce  new  rectilinear  vibrations  in  the  coordinate  planes, 
of  vibra-  which  together  represent  the  resulting  elliptic  vibration,  and  will  have  the  same  period  as  the  component  ones, 
tions  gene-  By  inverting  the  process,  a  vibration  of  this  kind  may  be  resolved  into  any  number  of  others  we  please,  having 
rally.  the  same  period. 

A  great  variety  of  particular  cases  present  themselves,  of  which  we  shall  examine  some  of  the  principal.     And 
Case  of  in-   first,  when  the  interfering  vibrations  are  both  rectilinear. 

ofrectili-  Since  the  choice  of  our  coordinate  planes  is  arbitrary,  let  us  suppose  that  of  the  x,  y  to  be  that  in  which  both 
near  vibra-  *'le  vibrations  are  performed.  Of  course  the  resulting  one  will  be  performed  in  the  same.  Therefore  we  may 
tions  put  z  =i  0,  or  c  =  0,  d  —  0,  and  content  ourselves  with  making 


};       (8) 


x  =  a  .  cos  (0  -f-  p)  ;  y  =  b  .  cos  (0  -f-  p) 
x'=  a'.cos(0-f-p');  y'=&'.cos(0+y) 

;  in  this  case,  and  X,  Y,  A,  B,  P,  Q 

!  =  A.  cos(0  +  P);     Y=B.cos(0+Q); 


The  resul-    because  -  -   and  — r  are  constant  in  this  case,  and  X,  Y,  A,  B,  P,  Q,  denoting  as  in  the  general  case,  we  have 

tant  vibra-  y  y 


elliptic.         and,  by  elimination  of  0, 

\A/\B/  AB 

where  A,  B,  P,  Q,  are  determined  as  in  equations,  (7.)  In  the  general  case,  then,  the  resulting  vibration  in 
elliptic. 

622.          The  ellipse  degenerates  into  a  straight  line  by  the  diminution  of  its  minor  axis  when  P  =  Q.     Now  this  gives 
Case  when   tan  P  =  tan  Q,  or 

the  resul-                                                                  a  .  sin  p  +  a' .  sinp'  6  .  sin  fl  +  6' .  siny 

tentisrec-  ^  ,      , ^7=    , -hr- ^T 

tilinear.                                                                     a  .  cosp  +  a  ,  cosp'  b  .  cosp  -f-o  .  cosp 

which,  reduced,  takes  the  form 


There  are,  therefore,  two  cases,  and  two  only  in  which  the  resulting  vibration  is  rectilinear.     The  first,  when 
p  —  p1  =:  0,  or  when  the  component  vibrations  have  a  common  origin,  or  are  in  complete  accordance;  the 

/  L I 

Case  when   other,  when  —  =  — ,  that  is,  when  they  are  both  performed  in  one  plane,  and  in  the  same  direction.     For  if 

(I  O 

their  direc- 
tions coin-    we  call  mand  m1  the  amplitudes,  and  Y">  V'' tne  angles  tnev  make  with  the  axis  of  the  x,  we  have 

a  =  m  .  cos  ^  ;  b  =  m  .  sin  ^  ;  a'  =  m'  .  cos  y/ ;  b'  —  m' .  sin  ^', 

so  that  the  above  equation  is  equivalent  to  tan  ty  =  tan  \fr',  or  ^  =  Y^'. 
623.          The  latter  case  we  have  already  fully  considered.     In  the  former,  we  have  cos  (p  —  p')  =  0,  and,  therefore, 

\  =  a  +  a';  B  =  6-f  6';  P-p;  Q=p, 


LIGHT.  4G1 

Y  6+6'  Pail  III. 

'  and,  finally,  —  =  -        —  =  tan  0     ;  (10)  ^—v—^ 

X  a  +  °  Case  if 

which  is  the  tangent  of  the  angle  made  by  the  resulting  rectilinear  vibration  with  the  axis  of  the  x.  complete 

If  we  put  M  for  the  amplitude  of  the  resulting  vibration,  we  have  M  .  cos  0  =  A;  M  .  sin  0  =  B  ;  therefore,  accordance 
M'  .  (cos  0*  +  sin  0')  or  M'  =  A«  +  B'.  ^"^ 

Now,  A*  =  (a  -f-  a')8  =  (m  .  COS  Y"  +  m'  •  cos  YO*  vibrations. 

B8  =  (6  +  6')°  =  (rn  .  sin  y,  +  m'  .  sin  Y/)8  Aog^ 

and,  therefore,  adding  these  values  together,  and  reducing  and  positioo 

M'  =  m*  +  2  m  m'  .  cos  (Y"  -  f  )  +  m"  ;         (11)  *£$L. 

Now,  Y"  —  Y"'  is  the  angle  between  the  directions  of  the  component  vibrations,  so  that  this  equation  expresses  tlon  deter- 
that  the  amplitude  of  the  resultant  vibration  is  in  this  case  also  the  diagonal  of  a  parallelogram,  whose  sides  mmed- 

are  the  amplitudes  of  the  component  ones  ;  and  it  is  easily  shown,  by  substituting  in  tan  0  =  -     —  r  the  above 

Qi  —  f—  Or 

values  of  a  -f-  a',  6  -+-  &',  that  the  diagonal  has  also  the  position  of  the  resultant  line  of  vibration. 

Carol.  1.     Any  rectilinear  vibration  may  be  resolved  into  two  other  rectilinear  vibrations,  whose  amplitudes      625. 
are  the  sides  of  any  parallelogram,  of  which  the  amplitude  of  the  original  vibration  is  the  diagonal,  and  which 
are  in  complete  accordance,  or  have  a  common  origin  with  it. 

Carol.  2.     Hence  any  rectilinear  vibration   may  be  readily  reduced  to  the  directions  of  two    rectangular      626. 
coordinates,  or,  if  necessary,  into  those  of  three,  by  the   rules   of  the  resolution  of  forces,  and  the  component 
vibrations,  however  numerous,  will  be  in  complete  accordance  with  the  resultant. 

The  ellipse  degenerates  into  a  circle  when   cos  (P  —  Q)  =  0,  or  P  —  Q  =  90",  and,  also,  A  =  B.     Now  the      627. 
former  condition  gives  tan  P  -f-  cot  Q  =r  0,  that  is  Case  of 

a  .  sin  p  -f  a'  .  sin  p  b  .  cos  p  +  6'  .  cos  j/  circular 

_  £  —  !  -  1-  --  --  -  —  !  -  -  —  —  0  vibrations. 

a  .  cosp  -f-  ft  •  cosp'          6  .  sin  p  -j-  6'  .  sin  p' 

or  reducing 

ab  +  a'b'  m*  .  sin  2  Y"  +  ™'8  sin  2 

' 


"-*          ' 


. 
'~  '  ' 


a  V  -\-  a'  b  mm'  .  sin  (Y- 

The  condition  A  =  B,  or  A2  =  B8,  gives 

a3  -f  2  a  a'  .  cos  (p  -  p')  -f  a'»  =  6«  +  2  b  b'  .  cos  (p  -  p1)  -f  b" 
whence  we,  in  like  manner,  obtain 

(a8  +  a'8)  -  (68  +  ft'8)  wi«  .  cos  2  Y-  +  »"  •  cos  2  Y-' 

s(p-^}  2  a  a'  -  2  6  V  cos  (Y-  -  Y-') 

and,  equating  the  values  of  cos  (p  —  p'),  we  find  the  following  relation  between  a,  a,  b,  b',  which  must  subsist 
when  the  vibrations  are  circular, 

fjL.        JL\  (a*  -f  &•  -  a'8  -  ft'8)  =  0. 

The  vanishing  of  the  first  factor  gives  no  circular  vibration,  it  being  introduced  with  the  negative  root  of  the 
equation  A8  =  B8,  with  which  we  have  no  concern.  The  other  gives 

o«  -f-  68  =  o'»  -f-  6'8,  or  m  =  m', 

which  shows  that  the  component  vibrations  must  have  equal  amplitudes.  Now,  if  for  a  and  b  we  write  their 
values  m  .  cos  Y"  and  m  .  sin  Y".  and  f°r  a'  and  b',  respectively,  m  .  cos  Y*'  and  m  .  sin  Y1/,  in  either  of  the 
expressions  for  cos  (p  —  p'),  it  will  reduce  itself  to 

cos  (j)  —  p')  =  —  cos  (Y"  —  Y")  ?  or>  P  —  P'  —  180°  —  (^  —  Y"')- 

Hence  it  appears,  that  the  interference  of  two  equal  rectilinear  vibrations  will  produce  a  resultant  circular  one, 
provided  the  difference  of  their  phases  be  equal  to  the  supplement  of  the  angle  their  directions  make  with  each 
other,  so  that  when  the  molecule  is  just  commencing  its  motion  towards  its  centre,  in  virtue  of  one  vibration,  it 
shall  be  receding  from  it  at  an  obtuse  angle  with  this  motion,  in  virtue  of  the  other. 

Carol.      Hence,  if  two  vibrations  have  equal  amplitudes,  but  differ  in  their  phases  by  a  quarter  of  an  undula- 
tion, their  resultant  vibration  will  be  circular. 

We  are  now  in  a  condition  to  explain  what  becomes  of  the  portions  of  the  secondary  waves  which  diverge      625. 
obliquely  from  the  molecules  of  the  primary  ones,  as  alluded  to  in  Art.  595,  and  to  explain  the  mode  in  which  Fig.  130 
those  which  do  not  conspire  with  the  primary  wave  mutually  destroy  each  other.     To  this  end,  conceive  the  sur- 
face of  any  wave  A  B  C  to  consist  of  vibratory  molecules,  all  in  the  same  phase  of  their  vibrations.     Then  will  the 
motion  of  any  point  X  (fig.  130)  be  the  same,  whether  it  be  regarded  as  arising  from  the  original  motion  of  S,  Mutual 
or  as  the  resultant  of  all  the  motions  propagated  to  it  from  all  the  points  of  this  surface.     Conceive  the  surface  destruction 
ABC  divided  into  an  infinite  number  of  elementary  portions,  such  that  the  difference  of  distance  of  each  con-  J 
secutive  pair  from  X  shall  be  constant,  or  =  d  f,  putting  the  distance  of  any  one  from  that  point  =  f;  and  let 
A  B,  B  C,  CD,  &c.,  and  A  6,  b  c,  c  d,  &c.  be  finite  portions  of  the  surface  containing  each  the  same  number  of 


4G2  I.  I  G  H  T. 

'-i?"'  these  elements,  and  in  each  of  which  the  corresponding  values  of/  are  exactly  half  an  undulation  (^  X)  greater 
v— "v'"*''  than  in  the  preceding,  so  that  (for  instance)  BX^AX  +  ^X,  CX  =  BX"-fJX,  &c.  Then  it  is  evident,  ' 
that  the  vibrations  which  reach  X  simultaneously  from  the  corresponding  portions  of  any  two  consecutive  ones 
as  of  A  B  and  B  C,  will  be  in  exactly  opposite  phases  ;  and,  therefore,  were  they  of  equal  intensity,  and  in 
precisely  the  same  direction,  would  interfere  with,  and  destroy  each  other.  Now,  first,  with  regard  to  their 
intensity,  this  depends  on  the  magnitudes  of  the  elements  of  the  wave  A  B,  from  which  they  are  derived,  and  on 
the  law  of  lateral  propagation.  Of  the  latter,  we  know  little,  a  priori  ;  but  all  the  phenomena  of  light'indicate 
a  very  rapid  diminution  of  intensity,  as  the  direction  in  which  the  secondary  undulations  are  propagated  deviates 
from  that  of  the  primary.  With  respect  to  the  former,  it  is  evident  that  the  elements  in  the  immediate  vicinity 
of  the  perpendicular  A  X,  corresponding  to  a  given  increment  rf/of  the  distance  from  X,  are  much  larger  than 
those  remote  from  it ;  so  that  all  the  elements  of  the  portion  A  B  are  much  larger  than  those  in  B  C,  and  these 
again  than  in  those  of  C  D,  and  so  on.  Thus  the  motion  transmitted  to  X  from  any  element  in  A  B  will  be 
much  greater  than  that  from  the  corresponding  one  in  B  C,  and  that  again  greater  than  that  from  the  element  in 
C  D,  and  so  on.  Thus  the  motion  arriving  at  G,  from  the  whole  series  of  corresponding  elements,  will  be  repre- 
sented by  a  series  such  asA-B  +  C-D  +  E  —  F  +  &c.,  in  which  each  term  is  successively  greater  than 
that  which  follows.  Now  it  is  evident  that  the  terms  approach  with  great  rapidity  to  equality ;  for  if  we  consider 
any  two  corresponding  elements  as  M,  N  at  a  distance  from  A  at  all  considerable,  the  angles  X  M  and  X  N  make 
with  the  surface  approach  exceedingly  near  to  equality,  so  that  the  obliquity  of  the  secondary  wave  to  the  pri- 
mary, and  of  course  its  intensity,  compared  with  that  of  the  direct  wave,  is  very  nearly  alike  in  both  ;  and  the 
elements  M,  N  themselves,  at  a  distance  from  the  perpendicular,  approach  rapidly  to  equality,  for  the  elementary 
triangles  M  mo,  Mjip  are  in  this  case  very  nearly  similar,  and  have  their  sides  m  o,  np  equal  by  hypothesis. 
Finally,  the  lines  M  X,  N  X  approach  nearer  to  each  other  in  direction  so  as  to  produce  a  more  complete  inter- 
ference, as  their  distance  from  A  is  greater. 

629.  Thus  we  see  that  the  terms  of  the  series  A— B  +  C  —  D-(-  &c.,  at    a   distance  from   its    commencement, 
have   on     all    accounts    (viz.    their  smallness,    near    approach    to    equality,  and  disposition  to  interfere)    an 
extremely  small  influence  on  its  value  ;  and  as  the  same  is  true  of   every  set  of  corresponding  elements  into 
which  the  portions  A  B,  B  C,  &c.  are  divided,  it  is  so  of  their  joint  effect,  so  that  the  motion  of  the  molecule  X  is 
governed  entirely  by  that  of  the  portion  of  the  wave  ABC  immediately  contiguous  to  A,  the  secondary  vibrations 
propagated  from  parts  at  a  distance  mutually  interfering  and  destroying  each  others  effect. 

630.  It    is  obvious,    that    in    the    case    of    refraction    or    reflexion,  we    may  substitute  for   the  wave  A  M  the 
refracting  or  reflecting  surface ;  and  for  the  perpendicular  X  A  the  primary  refracted  ray,  when  the  same  things, 
mutatis  mutandis,  will   hold    good.     See  M.  Fresnel's  Paper   entitled  Explication  de  la  Refraction  dans   le 
Systeme  des  Ondes,  published  in  the  Bulletin  de  la  Societe  Philomatique,  October,  1821. 

g,jj  This  is  the  case  when  the  portion  of  the  wave  A  B  C  D  whose  vibrations  are  propagated  to  X  is   unlimited, 

Case  of  a     or  at  'east  so  considerable,  that  the  last  term  in  the  series  A  —  B  +  C  —  &c.  is  very  minute  compared  with  the 

wave  first.     But  if  this  be  not  the  case,  as,  if  the  whole  of  a  wave  except  a  small  part  about  A  be  intercepted  by  an 

transmitted  obstacle,  the  case  will  be  very  different.    It  is  easy  on  this  supposition  to  express  by  an  integral  the  intensity  of  the 

lirough  a     undulatory  motion  of  X,  compared  with  what  it  would  be  on  the  supposition  of  no  obstacle  existing.     For  this 

Jpe'rture       purpose,  let  d2  s  be  the  rr -o-nitude  of  any  vibrating  element  of  the  surface,  /  its  distance  from  X  =  M  X,  and  let 

<p  (<?)  be  the  function  of  the  a.^'-e  made  by  a  laterally-divergent  vibration  with  the  direct  one,  which  expresses  its 

relative  intensity,  and  which  is  unity  when  0  =  0,  and  diminishes  with  great  rapidity  as  0  increases.     Then  if  t 

be  the  time  since  a  given  epoch,  X  =   the  length  of  an  undulation,  S  A  =  a,  the  phase  of  a  vibration  arriving 

at  X  by  the  route  S  M  X  will  be  2  TT  I  — —  I,  and  the  velocity  produced  in  X  thereby  will  be  repre- 
sented by  a  .  d*  s  .  0  (0)  .  sin  2  ir  (  —  -        ^— -  j,   so  that  the  whole  motion  produced  will  be  represented  by 

f  t          a  +  f 
//  a  .  d*  »  .  0  (0)  .  sin  2  v    ' 

the  integral  being  extended  to  the  limits  of  the  aperture. 

632.  Carol.  1.    If  but  a  very  small  portion  of  the  wave  be  permitted  to  pass,  as  in  the  case  of  a  ray  transmitted 
through  a  very  small  hole,  and  received  on  a  distant  screen,  0  and  0  (0)  are  very  nearly  constant,  so  that  the 
motion  excited  in  X  is  in  this  case  represented  by 

We  shall  have  occasion  to  revert  to  these  expressions  hereafter. 

§  IV.  Of  the  Colours  of  Thin  Plates. 

633.  Every  one   is  familiar  with  the  brilliant  colours  which  appear  on  soap-bubbles ;   with   the   iridescent   hues 
General       produced  by  heat  on  polished  steel  or  copper  ;  with  those  fringes  of  beautiful  and  splendid  colours  which  appear 
iccount  of  ;„  the  cracks  of  broken  glass,  or  between  the   laminae  of  fissile  minerals,  as  Iceland  spar,  mica,  sulphate  of 

he  pheno-   |imei  &c      jn  a[1  tnese,  and  an  infinite  variety  of  cases  of  the  same  kind,   if  the  fringes  of  colour  be  examined 


LIGHT.  463 

Light,      with  care  they  will  be  found  to  consist  of  a  regular  succession  of  hues,  disposed  in  the  same  order,  and  deter-    Part  III. 
— •v*-'  mined,  obviously,  not  by  any  colour  in  the  medium  itself  in  which  they  are  formed,  or  on  whose  surfaces  they  ^— ^  — „ 
appear,  but  solely  by  its  greater  or  less  thickness.     Thus  a  soap-bubble  (defended  from  currents  of  air  by  being 
placed  under  a  glass)  at  first  appears  uniformly  white  when  exposed  to  the  dispersed  light  of  the  sky  at  an  open 
window ;  but,  as  it  grows  thinner   and  thinner  by  the  subsidence  of  its  particles,  colours  begin  to  appear  at  its 
top  where  thinnest,  which  grow  more  and  more  vivid,  and  (if  kept  perfectly  still)  arrange  themselves  in  beautiful 
horizontal  zones  about  the  highest  point  as  a  centre.     This  point,  when  reduced  to  extreme  tenuity,  becomes 
black,  or  loses  its  power  of  reflecting  light  almost  entirely.     After  which  the  bubble  speedily  bursts,  its  cohesion 
at  the  vertex  being  no  longer  sufficient  to  counteract  the  lateral  attraction  of  its  parts. 

But  as  it  is  a  matter  of  great  delicacy  to  make  regular  observations  on  a  thing  so  fluctuating  and  unmanage-      634. 
able  as  a  soap-bubble,  the  following  method  of  observing  and  studying  the  phenomena  is  far  preferable.     Let  a  Rinss 
convex  lens,  of  a  very  long  focus  and  a  good  polish,  be  laid  down  on  a  plane  glass,  or  on  a  concave  glass  lens  formel1  be" 
having  a  curvature  somewhat  less  than  the  convex  surface  resting  on  it ;  so  that  the  two  shall  touch  in  but  a  j,"^", 
single  point,  and   so  that  the  interval  separating  the  surfaces  in  the  surrounding  parts  shall  be  exceedingly  glasses. 
small.     If  the  surfaces  be  very  carefully  cleaned  from  dust  before  placing  them  together,  and  the  combination  be 
laid  down  before  an  open  window  in  full  daylight,  the  point  of  contact  will  be  seen  as  a  black  spot  in  the  general 
reflexion  of  the  sky  on  the  surfaces,  surrounded  with  rings  of  vivid  colours.     A  glass   of  10    or  12   feet  focus 
laid  on  a  plane  glass,  will  show  them  very  well.     If  one  of  shorter  focus  be  used,  the  eye  may  be  assisted  by  a 
magnifying  glass.     The  following  phenomena  are  now  to  be  attended  to  : 

Phenomenon  1.    The  colours,  whatever  glasses  be  used,  provided  the  incident  light  be  white,  always  succeed      635. 
each  other  in  the  very  same  order ;  that  is,  beginning  with  the  central  black  spot,  as  follows :  Order  of 

First  ring,  or  first  order  of  colours, — black,  very  faint  blue,  brilliant  white,  yellow,  orange,  red.  succession 

Second  ring,  or  second  order, — dark  purple  or  rather  violet,  blue,  green,  (very  imperfect,  a  yellow-green,)  of,th< 
vivid  yellow,  crimson  red. 

Third  ring,  or  third  order, — purple,  bine,  rich  grass  green,  Jine  yellow,  pink,  crimson. 
Fourth  ring,  or  fourth  order, — green,  (dull  and  bluish,)  pale  yellowish  pink,  red. 
Fifth  ring,  or  fifth  order, — -pale  bluish  green,  white,  pink. 
Sixth  ring,  or  sixth  order, — pale  blue-green,  pale  pink. 

Seventh  ring,  or  seventh  order, — very  pale  bluish  green,  very  pale  pink.     After  these,  the  colours  become  so 
faint  that  they  can  scarcely  be  distinguished  from  white. 

On  these  we  may  remark,  that  the  green  of  the  third  order  is  the  only  one  which  is  a  pure  and  full  colour,  that  of  63f>. 
the  second  being  hardly  perceptible,  and  of  the  fourth  comparatively  dull  and  verging  to  apple  green  ;  the  yellow 
of  the  second  and  third  order  are  both  good  colours,  but  that  of  the  second  is  especially  rich  and  splendid  ;  that  of 
the  first  being  a  fiery  tint  passing  into  orange.  The  blue  of  the  first  order  is  so  faint  as  to  be  scarce  sensible, 
that  of  the  second  is  rich  and  full,  but  that  of  the  third  much  inferior ;  the  red  of  the  first  order  hardly  deserves 
the  name,  it  is  a  dull  brick  colour;  that  of  the  second  is  rich  and  full,  as  is  also  that  of  the  third;  but  they  all 
verge  to  crimson,  nor  does  any  pure  scarlet,  or  prismatic  red,  occur  in  the  whole,  series. 

Phenomenon  2.  The  breadths  of  the  rings  are  unequal.    They  decrease,  and  the  colours  become  more  crowded,      ™*' 
as  they  recede  from  the  centre.     Newton  (to  whom  we  owe  the  accurate  description  and  investigation  of  their  Breadths' 
phenomena)  found  by  measurement  the  diameters  of  the  darkest  (or  purple)  rings,  just  when  the  central  black  the  rings 
spot  began  to  appear  by  pressure,  and  reckoning  it  as  one  of  them  to  be  as  the  square  roots  of  the  even  numbers  and  thick- 
0,  2,  4,  6,  &c. ;  and  those  of  the  brightest  parts,  of  the  several  orders  of  colours,  to  be  as  the  square  roots  of  the  nesses  at 
odd  numbers   1,  3,  5,   7,  &c.     Now  the  surfaces  in  contact  being  spherical,  and  their  radii  of  curvature  very  wh'ch  they 
great  in  proportion  to  the  diameters  of  the  rings,  it  follows  from  this  that  the  intervals  between  the  surfaces  at   P 
the  alternate  points  of  greatest  obscurity  and  illumination  are  as  the  natural  numbers  themselves  0,  1,  2,  3,  4, 
&c.     The  satne  measurements,  when  the  radii  of  curvature  of  the  contact  surfaces  are  known,  give  the  absolute 
magnitudes  of  the  intervals  in  question.     In  fact,  if  r  and  /  be  the  curvatures  of  two  spherical  surfaces,  a  convex 
and  concave,  in  contact,  and  D  the  diameter  of  any  annulus  surrounding  their  point  of  contact,  the  interval  of 
the  surfaces  there  will  be  the  difference  of  the  versed  sines  of  the  two  circular  arcs  having  a  common  chord  D. 
Now  (fig.  130)  if  A  E  be  the  diameter  of  the  convex  spherical  surface  A  D,  we  have  EA  :  A  D  1 1  A  D  :  D  B 

AD-         Da  D'2  1 

•  =  — —  r,  and  in  like  manner  B  C  =   -^—  /,   so  that D 4  (r  —  r1)  =  D  C,  the  interval  of  the 


A  E  8  b  8 

surfaces  at  the  point  D.  Thus  Newton  found,  for  the  interval  of  the  surfaces  at  the  brightest  part  of  the  first 
ring,  one  178000dth  part  of  an  inch  ;  and  this  distance,  multiplied  by  the  even  natural  numbers  0,  2,  4,  6,  8,  &c. 
gives  their  distance  at  the  black  centre  and  the  darkest  parts  of  the  purple  rings,  and  by  the  odd  ones  1,  3,  5,  &c. 
their  intervals  at  the  brightest  parts. 

Phenomenon  3.  If  the  rings  be  formed  between  spherical  glasses  of  various  curvatures,  they  will  be  found  to      638. 
be  larger  as  the  curvatures  are  smaller,  and  vice  versa ;  and  if  their  diameters  be  measured  and  compared  with  Invariable 
the  radii  of  the  glasses,  it  will  be  found,  that,  provided  the  eye  be  similarly  placed,  the  same  colour  is  invariably  relatl°"  he- 
produced  at  that  point,  or  that  distance  from  the  centre  where  the  interval  between  the  surfaces  is  the  same.  Coio™s  ^ 
Thus,  the  white  of  the  first  order  is  invariably  produced  at  a  thickness  of  one  178000th  of  an  inch  ;  the  purple,  thicknesses 
which  forms  the  limit  of  the  first  and  second  orders,  at  twice  that  thickness.     So  that  there  is  a  constant  rela-  of  plates, 
tion  between  the  tint  seen  and  the  interval   of  the  surfaces  where  it  appears.     Moreover,  if  the   glasses   be 
distorted  by  violent   and   unequal  pressure,   (as  is  easily  done   if  thin  lenses  be  used,)  the  rings   lose  their 
circular  figure,  and  extend  themselves  towards  the  part  where  the  irregular  pressure  is  applied,  so  as  to  form  a 
species  of  level  lines   each  marking  out  a  series  of  points  where  the  surfaces  are  equidistant.     Thus,  too,  if  a 


4G4  L  I  G  H  T. 

Light,      cylinder  be  laid  on  a  plane,  the  rings  pass  into  straight  lines  arranged  parallel  to  its  line  of  contact,  but  following    Part  II. 
— ~\— -  the  same  law  of  distance  from  that  line  as  the  rings  from  their  dark  centre,  and  if  the  glasses  be  of  irregular  v_^,  ,™. 
curvature,  as  bits  of  window  glass,  the  bands  of  colour  will   follow   all  their  inequalities ;    yet   more,  if  the 
pressure  be  very  cautiously  relieved,  so  as  to  lift  one  glass  from  the   other,  the  central  spot  will  shrink  and 
disappear,  and  so  on  ;    each  ring  in  succession  contracting  to  a  point,  and  then  vanishing,  so  as  to  bring  all  the 
more  distant  colours   successively  to  the  centre,  as  the  glasses  recede  from   absolute  contact.     From  all  these 
phenomena  it  is  evident,  that  it  is  the  distance  between  the  surfaces  only  at  any  point  which  determines  the 
colour  seen  there. 

639.  Phenomenon  4.     This  supposes,  however,  that  we  observe  them  with  the  eye  similarly  placed,  or  at  the  same 

Effect  of      angle  of  obliquity.     For  if  the  obliquity  be  changed  by  elevating  or  depressing  the  eye,  or  the  glasses,  the 

obliquity  of  diameters  (but  not  the  colours)  of  the   rings  will  change.     As  the  eye  is  depressed,  the  rings  enlarge ;  and  the 

incidence.     same  tmt  which  before  corresponded  to  an  interval  of  the   178000th  of  an  inch,  now  corresponds  to  a  greater 

interval.     This  distance  (-r-nrSv-ir)  is  determined  by  measures  taken  nearly  at,  and  reduced  by  calculation  exactly 

to,  a  perpendicular  incidence.     At  extreme  obliquities,  however,  the  diameters  of  the  several  rings  suffer  only  a 

certain  finite  dilatation,  and  Newton's  measures  led  him  to  the  following  rule  :  viz.   "  That  the  interval  between 

the  surfaces  at  which  any  proposed  tint  is  produced,  is  proportional  to  the  secant  of  an  angle  whose  sine  is  the 

Jirst  of  106  arithmetical  mean  proportionals  between  the  sines  of  incidence  and  refraction,  into  the  glass  from  the 

air,  or  other  medium  included  between  the  surfaces,  beginning  with  the  greater  ;"  or,  in  algebraic  language,  the 

relative  index  of  refraction  being  p.,  and  0  the  angle  of  incidence,  and  p  that  of  refraction  of  the  ray  as  it  passes 

out  of  the  rarer  medium  into  the  denser ;  then,  if  t  be  the  interval  corresponding  to  a  given  tint  at  the  oblique 

incidence  0,  and  T  at  a  perpendicular  incidence,  we  shall  have 

t  =  T  .  sec  u  where  sin  u  =  sin  6 (sin  6  —  sin  p) 

but  sin  p  =  —  .  sin  0,  consequently  we  have 

106  -) 

t  =  T  .  sec  u ;  sin  u  = —  .  sin  0  .  =  — — -         •  .  sin  0. 

107  107  p 

640.  To  see  the  rings  conveniently  at  extreme  obliquities,  a  prism  maybe  used,  laid  on  a  convex  lens,  as  in  fig.  132. 
Fig.  132.  If  the  eye  be  placed  at  K,  the  set  of  rings  formed  about  the  point  of  contact  E  will  be  seen  in   the  direction 
Rings  seen  j£  H,  and  as  the  eye  is  depressed  towards  the  situation  I,  where  the  ray  I G  intromitted  from  I  would  just  begin 
through  a  to  sujfer  totai  reflexion,  the  rings  are  seen  to  dilate  to  a  certain  considerable  extent.     When  the  eye  reaches  I, 

the  upper  half  of  the  rings  disappears,  being  apparently  cut  off  by  the  prismatic  iris  of  Art.  555,  which  is  seen 
in  that  situation,  but  the  black  central  spot  and  the  lower  half  of  the  rings  remains  ;  but  when  the  eye  is  still 
further  depressed  the  rings  disappear,  and  leave  the  central  spot,  like  an  aperture  seen  in  the  silvery  whiteness 
of  the  total  reflexion  on  the  base  of  the  prism,  and  dilated  very  sensibly  beyond  the  size  of  the  same  spot  seen 
in  the  position  K  H  :  thus  proving,  that  the  want  of  reflexion  on  that  part  of  the  base  extends  beyond  the  limits 
of  absolute  contact  of  the  glasses,  and  that,  therefore,  the  lower  surface  interferes  with  the  action  of  the  upper, 
,  and  prevents  its  reflexion  while  yet  a  finite  interval  (though  an  excessively  minute  one)  intervenes  between 
them.  Euler  has  made  this  an  objection  to  the  undulatory  theory,  but  the  objection  rests  on  no  solid  grounds, 
as  it  is  very  reasonable  to  conclude,  that  the  change  of  density  or  elasticity  in  the  ether  within  and  without  a 
medium  is  not  absolutely  per  saltum,  but  gradual.  If  so,  and  if  the  change  take  place  without  the  media,  the 
approach  of  two  media  within  that  limit,  within  which  the  condensation  of  the  ether  takes  place,  will  alter  the 
law  of  refraction  from  either  into  the  interval  separating  them. 

641.  In  order,  however,  to  see  to  the  greatest  advantage  the  colours  refi""t«d  by  a  plate  of  air  at  great  obliquities, 
Fringes        the  following  method,  first  pointed  out  by  Sir  William   Herschel,  may  be  employed.     On  a  perfectly  plane  glass 
scenwhenaor  metallic  mirror,  before  an  open  window,  lay   an  equilateral   prism,  having  its  base  next   the  glass  or  mirror 
prism  is       very  tn]jy  p]anei  and  looking  in  at  the  side  AC,  fig.  133,  the  reflected  prismatic  iris,  a,  b,  c,  will  be  seen  as  usual 
plane  "hss    in  the  direction  E  F>  where  a  ray  from  E  would  just  be  totally  reflected.     Within  this  iris,  and  arranged  parallel 
Fig.  133.      to  it,  are  seen  a  number  of  beautiful  coloured  fringes,  whose  number  and  distances  from  each  other  vary  with 

every  change  of  the  pressure  ;  their  breadths  dilating  as  the  pressure  is  increased,  and  vice  versa.  They  do  not 
require  for  their  formation,  that  the  surfaces  should  be  exceedingly  near,  being  seen  very  well  when  the  prism  is 
separated  from  the  lower  surfaces  by  the  thickness  of  thin  tissue  paper,  or  a  fine  fibre  of  cotton  wool  interposed, 
but  in  this  case  they  are  exceedingly  close  and  numerous.  If  the  pressure  be  moderate,  they  are  ne  irly  equi- 
distant, and  are  lost,  as  it  were,  in  the  blue  iris,  without  growing  sensibly  broader  as  they  approach  it.  As  the 
intervals  of  the  surfaces  is  diminished,  they  dilate  and  descend  towards  the  eye,  appearing,  as  it  were,  to  come 
down  out  of  the  iris.  They  do  not  require  for  their  formation  a  perfect  polish  in  the  lower  surface.  An  emerierf 
glass,  so  rough  as  to  reflect  no  regular  image  at  any  moderate  incidence,  shows  them  very  well.  The  experi- 
ment is  a  very  easy  one,  and  the  phenomena  so  extremely  obvious  and  beautiful,  that  it  is  surprising  it  should 
not  have  been  noticed  and  described  by  Newton,  especially  as  it  affords  an  excellent  illustration  of  his  law 
above  stated  To  understand  this,  let  EH,  E  K,  E  L  be  any  rays  from  E  incident  at  angles  somewhat  less 
than  that  of  total  reflexion  on  the  base ;  they  will  therefore  be  refracted,  and,  emerging  at  the  base  B  C,  will  be 
reflected  at  M  N,  (the  obliquity  of  the  reflexion  being  so  great,  that  even  rough  surfaces  reflect  copiously  and 
regularly  enough  for  the  purpose,  Art.  558,)  and  will  pursue  the  courses  HDPp,  K  F  Q  9,  LGRr,  &c.  entering 
the  prism  again  at  P,  Q,  R.  Reciprocally,  then,  rays  p  P,  q  Q,  &c.  incident  at  P,  Q,  &c.  in  these  directions, 


LIGHT.  465 

will  entel  the  eye  at  E  after  traversing  the  interval  B  C  N  M,  and  being  reflected  at  M  N,  and  will  affect  the  eye     Tart  III. 
'  with  the  colour  corresponding  to  that  obliquity  and  that  interval  between  the  surfaces  which  is  proper  to  each.  >—  -•>,—• 
If  then  we  put,  as  above,  0  for  the  exterior  angle  of  incidence  of  the  ray  D  H  on  the  base  of  the  prism,  and 
take 


106  /»+  1  10 

sin  u  =  --  -  -  .  sin  0  =  -  ~  -  .  sin  p  =  k  .  sin 
107  p.  107 

the  tint  seen  in  the  direction  E  H  will  (abstraction  made  of  the  dispersion  at  the  surface  A  C)  be  the  same  with 
that  reflected  at  a  perpendicular  incidence,  by  a  plate  of  air  of  the  thickness  T  =  t  .  cos  «  =  <  -/  1  —  If  .  sin/)4, 
where  t  =  the  distance  between  the  surfaces  B  C,  M  N.  There  will,  therefore,  appear  a  succession  of  colours 
in  the  several  consecutive  situations  of  the  line  E  II,  analogous  to  those  of  the  coloured  rings,  (except  in  so  far 
as  the  dispersion  of  the  side  A  C  alters  the  tints  by  separating  their  component  rays.) 

But  the  whole  series  of  colours  will  not  be  seen,  because  those  which  require  greater  obliquities  than  that  at       642. 
which  total  reflexion  takes  place,  cannot  be  formed.     In  fact,  the   angle,  reckoned   from  the  vertical  at  which  a 
tint  corresponding  to  a  thickness  T  in  the  rings  would  be  formed,  is  given  by  the  equation 


sm  p  =  -r 


taking  /t  =  —  for  glass,  which  it  is  very  nearly.     Now,  according  to  this,  the  central  tint,  or  black  of  the  first 
order,  which  is  formed  when  T  =  0,  requires  that 


sin'  =  T  = 


107 


which  being  greater  than  — -  shows  that  this  tint  lies  above  the  situation  of  the  iris,  and  cannot  therefore  be 

1 
seen.    Tlie  first  visible  tint  will  be  that  close  to  the  iris,  where  sin  p  =  —  which  gives 


~: — '  =  0-079  ' 

nearly,  or        9r.     Hence  it  appears,  that  these  fringes  would  be  seen,  by  an  eye  immersed  in  the  prism,  when 

the  interval  between    its  base  and  the  glass  it  rests  on  is  more  than  12  times  that  at  which  clours  are  formed 

13  1 

at  a  perpendicular  incidence,  t.  e.  at  12'25  X  ,  or  about — th  of  an  inch,  which  is  about  the  thickness 

of  fine  tissue  paper.  Moreover,  from  this  value  of  T,  we  see  that  the  first  tint  immediately  visible  below  the 
iris  ascends  in  the  scale  of  the  rings  (i.  e.  belongs  to  a  point  nearer  their  centre)  as  the  value  of  t  diminishes, 
or  as  the  prism  is  pressed  closer  to  the  glass  ;  and  this  explains  why  the  fringes  become  more  numerous,  and 
appear  to  come  out  of  the  iris  by  pressure.  With  regard  to  their  angular  breadth,  (still  to  an  eye  immersed  in  the 

1   inch 

prism.)     If  we  put  e  =  ,  we  have,  putting  pa,  plt  &c.   for  the  values  of  p,  corresponding   to  the  several 

orders  of  visible  tints, 


very    nearly,   sin  p,  =   —  (  1  —  0-079  .  — £  )  and  so  on.     The  sines  then  of  the  incidences  at  which  the  several 

/»   \  t   / 

orders  of  colours  are  developed,  beginning  at  the  iris,  increase  in  arithmetical  progression,  so  that  the  fringes  must 
be  disposed  in  circular  arcs  parallel  to  the  iris,  and  their  breadths  must  be  nearly  equal,  and  greater  the  greater 
the  pressure  or  the  less  t  is,  all  which  is  conformable  to  observation.  The  refraction  of  the  side  of  the  prism 
between  the  eye  and  the  base,  however,  disturbs  altogether  the  succession  of  colours  in  the  fringes,  and 
in  particular  multiplies  the  number  of  visible  alternations  to  a  gce.at  extent,  in  a  manner  which  will  be  evi- 
dent on  consideration.  We  have  been  rather  more  particular  in  explaining  the  origin  of  these  fringes,  and 
referring  them  to  the  general  phenomena  observed  by  Newton,  because  up  to  the  present  time  we  believe  no 
strict  analysis  of  them  has  been  given,  as  well  as  on  account  of  the  great  beauty  of  the  phenomenon  itseF.  If 
we  hold  the  combination  up  to  the  light,  and  look  through  the  base  of  the  prism  and  the  glass  plate,  so  as  to 
see  the  transmitted  iris  of  Art.  556,  its  concavity  will,  in  like  manner,  be  seen  fringed  with  bands  of  colours  of 
precisely  similar  origin.  To  return  now  to  the  rings  seen  between  convex  glasses. 

Phenomenon  5.     If  homogeneous  light  be  used  to  illuminate   the  glasses,  the  rings  are  seen  in  much  greater 
VOL.  rv.  3  p 


466  LIGHT. 

Light,      number,  and  the  more  according  to  the  degree  of  homogeneity  of  the  light.     When  this  is  as  perfect  as  possible, 
_.— „  -^-  as,  ibr  instance,  when  we  use  the  flame  of  a  spirit  lamp  with  a  salted  wick,  as  proposed  by  Mr.  Talbot,  they  are  ' 
Phenomena  literally  innumerable,  extending  to  so  great  a  distance  that  they  become  too  close  to  each  other  to  be  counted,  or 
even  distinguished  by  the  naked  eye,  yet  still  distinct  on  using  a  magnifier,  but  requiring  a  higher  and   higher 
Tioht"  power  as  they  become  closer,  till  we  can  pursue  them  no  farther,  and  disappearing  from  their  closeness,  and  not 
from  any  confusion  or  running  of  one   into  the  other.     Moreover,  they  are  now  no  longer  composed  of  various 
colours,  but  are  wholly  of  the  colour  of  the  light  used  as  an   illumination,  being  mere  alternations  of  light  and 
obscurity,  and  the  intervals  between  them  being  absolutely  black. 

644.  Phenomenon  6.     When  the  illuminating  light  is  changed  from  one  homogeneous  ray  to  another,  as  when,  for 
Contraction  instance,  the  colours  of  the  prismatic  spectrum  are  thrown  in  succession  on  the  glasses  at  their  point  of  contact, 
of  the  rings  at  sucn  an  angle  as  to  be  reflected  to  the  eye,  then,  the  eye  remaining  at  rest,  the  rings  are  seen  to  dilate  and 
rVranlble    contract  m  magnitude  as  the  illumination  shifts.     In  red  light  they  are  largest,  in  violet  least,  and  in  the  inter- 
—.j3"8     '    mediate  colours  of  intermediate  size.     Newton,  by  measuring  their  diameters,   ascertained  that  the   interval   of 

the  surfaces  or  thickness  of  the  plate  of  air,  where  the  violet  ring  of  any  order  was  seen,  is  to  its  thickness, 
where  the  corresponding  red  ring  of  the  same  order  is  formed,  nearly  as  9  :  14  ;  and,  determining  by  this  method, 
the  thickness  of  the  plate  of  air  where  the  brightest  part  of  the  first  ring  was  formed,  when  illuminated  in  suc- 
cession  by  the  several  rays  proceeding  from  the  extreme  red  to  the  extreme  violet,  he  ascertained  those  thick- 

thecoloured  nesses  jo  De  the  halves  of  the  numbers  already  set  down  in  the  second  column  of  the  Table,  p.  453,  expressed  in 
ring*.  x 

parts  of  an  inch,  and  which  answer  to  the  values  of  -^— ,  or  the  lengths  of  a  semiundulation  for  each  ray. 

645.  This  phenomenon  may  be  regarded  as  an  analysis  of  what  takes  place  when  the  rings  are  seen  in  white  light ; 
Synthesis  of  for  in  that  case  they  may  be  regarded  as  formed  by  the  superposition  one  on  the  other  of  sets  of  rings  of  all  the 
the  coloured  simple  colours,  each  set  having  its  own  peculiar  series  of  diameters.     The  manner  in  which  this  superposition 
p.ngs'm       takes  place,  or  the  synthesis  of  the  several  orders  of  colours,  may  be  understood  by  reference  to  fig.  134,  where 

the  abscissae  or  horizontal  lines  represent  the  thicknesses  of  a  plate  of  air  between  two  glasses,  supposed  to 
increase  uniformly,  and  where  R  R',  RR",  &c.  represent  the  several  thicknesses  at  which  the  red,  in  the  system 
of  rings  illuminated  by  red  rings  only,  vanishes,  or  at  which  the  darkness  between  two  consecutive  red  rings  is 
observed  to  happen,  while  R  r,  Rr',  Rr",  &c.  represent  those  •  which  the  brightness  is  a  maximum.  In  like 
manner,  let  0  0',  0  0'',  &c.  be  taken  equal  to  the  several  thicknesses  at  which  the  orange  vanishes,  or  at  which 
the  black  intervals  in  the  system  of  orange  rings  are  seen,  and  so  on  for  the  yellow,  green,  blue,  indigo,  and 
violet  rings.  So  that  R  R',  0  0',  Y  Y',  &c.  are  to  each  other  in  the  ratio  of  the  numbers  in  column  2  of 
the  above  Table,  (Art.  575.)  Then  if  we  describe  a  set  of  undulating  curves  as  in  the  figure,  and  at 
any  point,  as  C  in  A  E,  draw  a  line  parallel  to  AV,  cutting  all  these  curves;  their  several  ordinates,  or  the 
portions  of  this  line  intercepted  between  the  curves  and  their  abscissae,  will  represent  the  intensity  of  the 
light  of  each  colour,  sent  to  the  eye  by  that  thickness  of  the  plate  of  air.  Hence,  the  colour  seen  at  that 
thickness  will  be  that  resulting  from  the  union  of  the  several  simple  rays  in  the  proportions  represented  by  their 
ordinates. 

g4fi  The  figure  being  laid  down  by  a  scale,  we  may  refer  to  it  to  identify  the  colours  of  particular  points.     Thus, 

Synthesis  of  ^Tsi  at  lne  thickness  0,  or  at  A  the  origin  of  the  tints,  all  the  ordinates  vanish,  and  this  point,  therefore,  is  black. 
the  several  As  the  thickness  of  the  plate  of  air  increases  from  0  while  yet  very  small,  it  is  evident,  on  inspection,  that  the 
orders  of  ordinates  of  the  several  curves  increase  with  unequal  rapidities,  those  for  the  more  refrangible  rays  more  rapidly 
colouti.  tnan  tnose  for  tne  ]ess>  go  that  the  first  jeeDie  ]jffht  wnjch  appears  at  a  very  small  thickness  A  1,  will  have  an 
excess  of  blue  rays,  constituting  the  pure  but  faint  blue  of  the  first  order,  (Art.  635.)  At  a  greater  thickness, 
however,  as  A  2,  the  common  ordinate  passes  nearly  through  the  maxima  of  all  the  curves,  being  a  little  short  of 
that  of  the  red,  and  a  little  beyond  that  of  the  violet.  The  difference,  however,  is  so  small,  that  the  several 
colours  will  all  be  present  nearly  in  the  proportions  to  constitute  whiteness,  and  being  all  nearly  at  their  maxi- 
mum, the  resulting  tint  will  be  a  brilliant  white.  This  agrees  with  observation  ;  the  white  of  the  first  order 
being,  in  fact,  the  most  luminous  of  all ;  beyond  this  the  violet  falls  off  rapidly,  the  red  increases,  and  the  yellow 
is  nearly  at  its  maximum,  so  that  at  the  thickness  A3  the  white  passes  into  yellow,  and  at  a  still  greater 
thickness,  A  4,  where  the  violet,  indigo,  blue,  and  green,  are  all  nearly  evanescent,  the  yellow  falling  otf,  and 
the  orange  and  red,  especially  the  latter,  in  considerable  abundance,  the  tint  resulting  will  be  a  fiery  orange, 
growing  more  and  more  ruddy.  At  B  is  the  minimum  of  the  yellow,  i.  e.  of  the  most  luminous  rays.  Here 
then  will  be  the  most  sombre  tint.  It  will  consist  of  very  little  either  of  orange,  green,  blue,  or  even  indigo ; 
but  a  moderate  portion  of  violet  and  a  little  red  will  produce  a  sombre  violet  purple,  which,  since  the  more  re- 
frangible rays  are  here  all  on  the  increase,  while  the  less  are  diminishing,  will  pass  rapidly  to  a  vivid  blue,  as  at 
the  thickness  denoted  by  A  5.  At  6,  where  the  ordinate  passes  through  the  maximum  of  the  yellow,  there  is 
almost  no  red,  very  little  orange,  a  good  deal  of  green,  very  little  blue,  and  hardly  any  indigo  or  violet  Here 
then  the  tint  will  be  yellow  verging  to  green,  but  the  green  is  diminishing  and  the  orange  increasing,  so  that  the 
yellow  rapidly  loses  its  green  tinge,  and  becomes  pure  and  lively.  At  7  the  predominant  rays  are  orange  and 
yellow,  being  so  copious  that  the  little  red  and  violet  with  which  they  are  mixed  does  not  prevent  the  tint  from 
being  a  rich,  high-coloured  yellow.  At  8  a  full  orange  and  copious  red  are  mixed  with  a  good  deal  of 
indigo  and  a  maximum  of  violet,  thus  producing  a  superb  crimson.  At  C  we  have  again  a  minimum  of 
yellow ;  but  there  being  at  the  same  time  a  maximum  of  red  and  indigo,  this  point,  though  dark  in  com- 
parison of  that  on  either  side,  will  still  be  characterised  by  a  fine  ruddy  purple.  This  completes,  and  as  we 
see  faithfully  represents,  the  second  order  of  colours.  At  9,  10  we  see  the  origin  of  the  vivid  green  of  the  third 
order,  in  the  comparative  copiousness  of  green,  yellow,  and  blue  rays  at  the  former  point,  and  of  yellow,  green, 


LIGHT.  467 

and  violet  at  the  latter,  while  the  red  and  orange  are  almost  entirely  absent,  and  thus  we  may  pursue  all  the    Pan  III. 
>  tints  in  the  scale  enumerated  in  Art.  635  with  perfect  fidelity.  ^ — v— — "' 

As  the  thickness  increases,  however,  it  is  clear  that  rays  differing  but  little  in  refrangibility  will  differ  much  in       647. 
intensity,  as  the  smallest  difference  in  the  lengths  of  the  bases  of  their  curves  being  multiplied  by  the  number  of  Degradation 
times  they  are  repeated,  will  at  length  bring  about  a  complete  opposition,  so  that  the   maximum  of  one  ray  will  of  tlie  l"Ui- 
fall  at  length  on  a  minimum  of  another  differing  little  in  refrangibility,  and  not  at  all  in  colour.     Thus,  at  con- 
siderable thicknesses,  such  as  the  10th  or  20th  order,  there  will  coexist  both  maxima  and  minima  of  every  colour; 
since  each  colour,  in  fact,  consists  not  of  rays  of  one  definite  refrangibility,  but  of  all  gradations  of  refrangibility 
between  certain  limits.     In  consequence,  the  tints,  as  the  thickness  increases,  will  grow  less  and  less  pure,  and 
will  at  length  merge  into  undistinguishable  whiteness,  which,  however,  for  this  very  reason,  will  be  only  half  as 
brilliant  as  the  white  of  the  first  order,  which  contains  all  the  rays  at  their  maximum  of  intensity. 

Phenomenon  1 '.     Such  are  the  phenomena  when  a  plate  of  air  is  included  between  two  surfaces  of  glass.     It  is      648 
not  however  as  air,  but  as  distance,  that  it  acts ;  for  in  the  vacuum  of  an   air-pump  the  rings  are  seen  without  Colours 
any  sensible  alteration.     If,  however,  a  much   more  refracting  medium,  as  water  or  oil,  be  interposed,  the  dia-  r<jfle< 
meters  of  the  rings  are  observed  to  contract,  preserving,  however,  the  same  colours  and  the  same  laws  of  their  S;iferent 
breadths  ;  and  Newton  found  by  exact  measurements,  that  the  thicknesses  of  different  media  interposed,  at  which  media. 
a  g-iven  tint  is  seen,  are  in  the  inverse  ratio  of  their  refractive  indices.     Thus,  the  white  of  the  first  order  being 

produced  in  vacuo  or  air  at  the  178000th  of  an  inch,  will  be  produced  in  water  at  part  of  that  thickness. 

He  found,  moreover,  that  the  law  stated  in  Art.  639  for  the  dilatation  of  the  rings  by  oblique  incidence,  holds 
equally  good,  whatever  be  the  nature  of  the  interposed  medium.  Hence  it  follows,  that  in  dense  media  the 
dilatation  at  great  obliquities  is  much  less  than  in  rare  ones,  and  that  in  consequence  a  given  thickness  will  re- 
flect a  colour  much  less  variable  by  change  of  obliquity  when  the  medium  has  a  high  refractive  power  than  when 
low.  Thus,  the  colours  of  a  soap  bubble  vary  much  less  by  change  of  incidence  than  those  of  a  film  of  air,  and 
much  more,  on  the  other  hand,  than  the  iridescent  colours  on  polished  steel,  which  arise  from  a  film  of  oxide 
formed  on  the  heated  surface. 

Phenomenon  8.     Surfaces  of  glass,  or  other  denser  medium  enclosing  the  thin  plate  of  a  rarer,  are  not  how-      649. 
ever  necessary  to  the  production  of  the  colours ;  they  are  equally,  and  indeed  more  brilliantly,  visible  when  any  Colours  -e- 
very  thin  laminae  of  a  denser  medium  is  enclosed  in  a  rarer,  as  in  air,  or  in  vacuo.     Thus,  soap  bubbles,  exceed-  fleeted  by 
ingly  thin  films  of  mica,  &c.  exhibit  the  same  succession  of  colours,  arranged  in  fringes  according  to  the  variable  f?aP  '>"b" 
thickness  of  the  plates.     The  following  very  beautiful  and  satisfactory  mode  of  exhibiting  the  fringes  formed  by    es> 
plates  of  glass  of  a  tangible  thickness  has  been  imagined  by  Mr.  Talbot.     If  a  bubble  of  glass  be  blown  so  thin 
as  to  burst,  and  the  glass  films  which  result  be  viewed  in  a  dark  room  by  the  light  of  a  spirit  lamp  with  a  salted 
wick,  they  will  be  seen  to  be  completely  covered  with  striae,  alternately  bright  and  black,  in  undulating  curves 
parallel  to  each  other  according  to  the  varying  thickness  of  the  film.     Where  the  thickness  is  tolerably  uniform, 
the  striae  are   broad  ;  where  it  varies  rapidly,  they  become  so  crowded  as  to  elude  the  unassisted   sight,  and 
require  a  microscope  to  be  discerned.     If  the  film  of  glass  producing  these  fringes  be  supposed  equal  to   the 
thousandth  of  an  inch  in  thickness,  they  must  correspond  to  about  the  89th  order  of  the  rings,  and  thus  serve  to 
demonstrate  the  high  degree  of  homogeneity  of  the  light ;  for  if  the  slightest  difference  of  refrangibility  existed, 
its  effect  multiplied  eighty-nine  times  would  become  perceptible  in  a  confusion  and  partial  obliteration  of  the 
black  intervals.     In  fact,  the  thickness  of  a  plate  at  which  alternations  of  light  and  darkness  or  of  colour  can 
no  longer  be  discerned,  is  the  best  criterion  of  the  degree  of  homogeneity  of  any  proposed  light,  and  is,  in  fact, 
a  numerical  measure  of  it.     This  experiment  is  otherwise  instructive,  as  it  shows  that  the  property  of  light  on 
which  the  fringes  depend  is  not  restricted  to  extremely  minute  thicknesses,  but  subsists  while  the  light  traverses 
what  may  be  comparatively  termed  considerable  intervals. 

Phenomenon  9.     When  the  glasses  between  which  the  reflected  rings  are  formed  are  held  up  against  the  light,       650. 
a  set  of  transmitted  coloured  rings  is  seen,  much  fainter,  however,  than  the  reflected  ones,  but  consisting  of  tints  Transmitted 
complementary  to  those  of  the  latter,  i.  e.  such  as  united  with  them  would  produce  white.     Thus  the  centre  is  colours' 
white,  which  is  succeeded  by  a  yellowish  tinge,  passing  into  obscurity,  or  black,  which  is  followed  by  violet  and 
blue.     This  completes  the  series  of  the  first  order.     Those  of  the   second  are  white,  yellow,  red,  violet,  blue  : 
of  the  third,  green,  yellow,  red,  bluish  green,  after  which  succeed  faint  alternations  of  red  and  bluish  green,  the 
degradation  of  tints  being  much  more  rapid  than  in  the  reflected  rings. 

It  was  to  explain  these  phenomena  that  Newton  devised  his  doctrine  of  the  fits  of  easy  reflexion  and  trans-       651. 
mission,  mentioned  in  the  9th  postulate  of  Art.  526.    This  doctrine  we  shall  now  proceed  to  develope  further,  and  Newton's 
apply,  as  he  has  done,  to  the  case  in  question.     In  addition  then  to  the  general  hypothesis  there  assumed,  it  will  exp'anatio* 
be  necessary  to  assume  as  follows  :  uf  tlle 

The  intervals  at  which  the  fits  recur,  differ  in  different  rays  according  to  their  refrangibilities,  being  greatest  for  thm'plates. 
the  red  and  least  for  violet  rays,  and  for  these,  and  the  intermediate  rays,  in  vacuo,  and  at  a  perpendicular  inci-       652 
dence,  are  represented  in  fractions  of  an  inch  by  the  halves  of  the  numbers  in  column  2  of  the  Table,  Art.  575.    Laws  of 

In  other  media,  the  lengths  of  the  intervals  in  the  course  of  a  molecule  at  which   its  fits  recur  are  shorter,  in  the  fits- 
the  ratio  of  the  index  of  refraction  of  the  medium  to  unity.  653. 

At  oblique  incidences,  or  when  a  ray  traverses  a  medium  after  being  intromitted  obliquely,  (at  an  angle  =r  0      C^A 
with  the  internal  perpendicular,)  the  lengths  of  the  fits  are  greater  than  at  a  perpendicular  incidence,  in  the 
ratio  of  radius  to  the  rectangle  between  the  cosine  of  6  and  the  cosine  of  an  arc  ?/,  given    by   the    equation 
_  106  p-\-  1 

3r2 


468  LIGHT. 

Light.          Let  us  now  consider  what  will  happen  to  a  luminous  molecule,  the  length  of  whose  fits  in  any  medium  is  J  X,    Part  III 
— •— - v-"—''  which,  having  been  intromitted  perpendicularly  at  the  first  surface,  and  traversing  its  thickness  (=  f),  reaches  the  v^-^^—. 

655.  second.     First,  then,  if  we  suppose  t  an  exact  multiple  of  J  X,  it  is  evident  that  the  molecule  will   arrive  at  the 
Explanation  second  surface  in  precisely  the  same  phase  of  its  fit  of  transmission   as   at  the  first.     Of  course  it  is  placed  in 

•  lhb  "h8*  l^e  ver^  same  circumstances  in  every  respect,  and  having  been  transmitted  before  must  necessarily  be  so  again, 
momentous"  Thus  every  ray  which  enters  perpendicularly  into  such  a  lamina  must  pass  through  it,  and  cannot  be  reflected  at 
light.  its  second  surface.  On  the  other  hand,  if  the  thickness  of  the  lamina  be  supposed  an  exact  odd  multiple  of 

£  X,  &c.  every  molecule  intromitted  at  its  first  surface  will  on  its  arrival  at  the  second  be  in  exactly  the  contrary 
phase  of  its  fits,  and,  having  been  before  in  some  phase  of  a  fit  of  transmission,  will  now  be  in  a  similar  phase  of 
a  fit  of  reflexion.  It  will,  therefore,  not  necessarily  be  transmitted ;  but  a  reflexion,  more  or  less  copious,  will 
take  place  at  the  second  surface  in  this  case,  according  to  the  nature  of  the  medium  and  its  general  action  on 
light.  For  it  will  be  remembered,  that  every  molecule  in  a  fit  of  reflexion  is  not  necessarily  reflected.  It  is 
disposed  to  be  so ;  but  whether  it  will  or  no,  will  depend  on  the  medium  it  moves  in  and  that  on  which  it 
impinges,  and  on  the  phase  of  its  fit.  Now  conceive  an  eye  placed  at  a  distance  from  a  lamina  of  unequal 
thickness,  so  as  to  receive  rays  reflected  at  a  very  nearly  perpendicular  incidence  from  it.  It  is  evident,  that  in 
virtue  of  the  reflexion  from  the  first  surface,  which  is  uniform,  it  will  receive  equal  quantities  of  light  from  every 
point.  But  with  regard  to  the  light  reflected  from  the  second  the  case  is  different ;  for  in  all  those  parts  where 
the  thickness  of  the  lamina  is  an  exact  even  multiple  of  £  X,  none  will  be  reflected,  while  in  all  those  where  it  is 

an  exact  odd  multiple  of  -— - ,  a  reflexion  will  take  place  ;  and  since  each  molecule  so  reflected  retraces  the  path 
by  which  it  arrived,  and  therefore  describes  again  the  same  multiple  of  — ;  its  total  path  described  within  the 

lamina,  when  it  has  reached  the  first  surface  again,  will  be  an  exact  multiple  of  — - ,  and  therefore  it  will  pene- 
trate that  surface  and  reach  the  eye.  In  consequence,  in  virtue  of  the  reflexion  at  the  second  surface  alone,  the. 
lamina  would  appear  black  in  every  part  where  its  thickness  =:  0,  or ,  or ,  &c.,  and  bright  in  those  parts 

where  its  thickness  =   ,  or  — ,  — ,  &c.  ad  infinitum.     In  the  intermediate  thicknesses  it  would  have  a 

4  44 

brightness  intermediate  between  these  and  absolute  obscurity  ;  so  that  on  the  whole,  the  lamina  would  appear 
marked  all  over  with  dark  and  bright  alternating  fringes,  just  as  we  see  it  actually  does  in  the  experiment 
described,  (Art.  649.)  The  uniform  reflexion  from  the  first  surface  superposed  on  these,  will  not  prevent  their 
inequality  of  illumination  from  being  distinctly  seen. 

656.  Hence  it  is  evident,  that  if  we  take  the  abscissa  of  a  curve  equal  to  thickness  of  the  lamina  at  any  point,  and 
Of  the         the  ordinate  proportional  to  the  intensity  of  the  light  reflected  from  the  second  surface,  and  returned  through  the 
rings  seen     first)  this  curve  will  be  an  undulating  line,  such  as  we  have  represented  in  fig.  134,  touching  the  abscissa  at  equal 
by  white      distances  equal  to  the  length  of  a  whole  fit  of  a  ray  of  the  colour  in  question.     Now  these  distances  for  rays  of 

different  colours  being  supposed  such  as  we  have  assumed  in  Art.  652,  the  construction  of  Art.  645  holds 
good,  and  when  white  light  falls  on  the  lamina,  its  second  surface  will  reflect  a  series  of  colours  of  the  composi- 
tion there  demonstrated,  and  such  as  we  actually  observe,  but  diluted  with  the  light  uniformly  reflected  from 
every  point  of  the  first  surface. 

If  the  lamina  instead  of  a  vacuum  be  composed  of  any  refracting  medium,  the  tints  will  manifestly  succeed 
each  other  in  a  similar  series,  but  the  thickness  at  which  they  are  produced  will  be  to  that  in  a  lamina  of  vacuum, 
in  the  ratio  of  the  lengths  of  the  fits  in  the  two  cases,  that  is,  in  the  proportion  of  i  :  the  index  of  refraction  of 
the  medium.  Thus  the  rings  seen  between  two  object  glasses  including  air,  ought  to  contract  when  water,  oil, 
&c.  is  admitted  between  them,  as  they  are  found  to  do,  and,  by  measure,  in  that  precise  ratio. 

657.  At  oblique  incidences,  0  being  the  angle  of  intromission  into  the  lamina,  t  .  sec  0  is  the  whole  path  of  the  ray 
Of  the  rtila-  between  the  first  and  second  surfaces,  and  since  J  X  .  sec  6  .  sec  u  is  the  length  of  the  fits  of  the  given  ray  at 
tation  of  the  this  obliquity,  in   order  that  the  luminous   molecule  may  arrive  at  the  second  surface  in  the   same  phase,  and 
""?* '          therefore  be  reflected  with  equal  intensity,  it  must  in  this  space  have  passed  over  the  same  number  of  these  fits  ; 

2  t     sec  0 

incidences.   nence  we  must  have =  constant,  or  t  proportional  to  sec  v,  which  agrees  with  observation. 

X  .  sec  0  .  sec  « 

g5g  All  the  light  which  is  not  reflected  at  the  second  surface  passes  through  it,  and  forms  the  transmitted  series  of 

Of  the         colours.     These,  therefore,  consist  of  the  whole  incident  light  (=  1)   minus  that  reflected  at  the  first  surface, 

transmitted  (which  will  be  a  small  fraction,  and  which  we  will  call  a,)  minus  that  reflected  at  the  second  surface.     Now  this 

"ngs-  last  will  be  a  periodical  function  whose  minimum  is  0,  and  its   maximum   can   never  exceed   a,  because  the 

reflexion  at  the  second  surface  of  a  medium  cannot  be  stronger  than  at  the  first  at  a  perpendicular  incidence. 

(2  t\*  C          /       2  C^\~\ 

sin  — -  j  ,  and  thus  we  have  1  -  a  -j  1  -f  (  sin  —  Vj-  for  the  intensity  of  this 

particular  coloured  ray  in  the  transmitted  series,  and  a[  sin  —  I    in   the    reflected.      Hence  it   is  evident,    that 

owing  to  the  smallness  of  a,  the  difference  between  the  brightest  and  darkest  part  of  the  transmitted  series  will 
be  small  in  comparison  with  the  whole  light,  and  thus  the  alternations  in  homogeneous  light  ought  to  be  (as 
they  are)  much  less  sensible  than  iu  the  reflected  rings,  and  the  tints  in  white  light  much  more  pallid  and  dilute. 


LIGHT.  469 

tignt.          Thus  we  see  that  the  Newtonian  hypothesis  of  the  fits  affords  a  satisfactory-enough  explanation,  or  rather    Part  III. 

— V"""'  represents  with  exactness  all  the  phenomena  above  described.     It  has  been  even  asserted,  that  this  doctrine  is  ^~-^^-~ 

really  not  an  hypothesis,  but  nothing  more  than  a  pure  statement  of  facts ;   for  that,  first,  in  point  of  mere  fact,       659. 

the  second  surface  of  the  lamina  does  send  light  to  the  eye,  in  the  bright  parts  of  the  fringes,  and  does  not  send 

it  in  the  dark  parts;  and,  secondly,  that  this  is  the  same  thing  with  saying  that  the  light  which  has  traversed  a 

thickness  =  (2  n  -j-  1)  —  is,  and  that  which  has  traversed  2  n  ——  is  not  susceptible  of  being  reflected.    And, 

in  truth,  if  only  one  ray  could  be  regarded  as  being  concerned,  and  were  the  light  reflected  at  the  first  surface 
of  the  lamina  altogether  out  of  the  question,  this  way  of  stating  it  would  be  strictly  correct.  But,  if  it  can  be 
shown,  that,  on  any  other  hypothesis  of  the  nature  of  light,  (as  the  undulatory,)  the  second  link  of  this  argument 
is  invalid ;  and  that  though  the  second  surface,  like  the  first,  may  reflect  in  every  part,  without  regard  to  its 
thickness,  its  full  average  portion  of  the  light  that  is  incident  on  it  ;  yet  that  afterwards,  by  reason  of  the 
interference  of  rays  reflected  from  the  first  surface,  such  light  does  not  reach  the  eye  (being  destroyed  in  every 

point  of  its  course)  from  those  parts  where  the  thickness  is  an  even  multiple  of  — -,  then  it  is  evident,  that  the 

Newtonian  doctrine  is  something  more  than  a  mere  aliter  statement  of  facts,  and  is  open  to  examination  as  a 
theory. 

Let  us  now  see,  therefore,  what  account  the  undulatory  theory  gives  of  these  phenomena.     We  will  begin,      6fiO. 
for  a  reason  which  will  presently  appear,  with  the  transmitted  rings.     Conceive,  then,  a  ray,  the  length   of  Explanation 
whose  undulations  in  any  medium  is  X,  to  be  incident  perpendicularly  on  the  first  surface  of  a  lamina  of  that  of  the 
medium  whose  thickness  is  =  t;   and  (for  simplicity)   let  its  surfaces  be  supposed  parallel,  then  it  will   be  translmtte^ 
divided  into  two  portions,  the  first  (  =  a)  reflected,  and  the  second  (=  1  —a)  intromitted.     Let  0  be  the  phase  ""f^"",^6 
of  this  portion  at  reaching  the  second  surface.     Here  it  will  be  again  divided  into  two  portions,  the  one  hypothesis. 
reflected  back  into  the  medium  and  equal  to(l  —  a)  .a,  or  (a  being  small)  very  nearly  to  <z,  and  the  remainder 
(1  —  a)  —  a  (1  —  a),  or  nearly  1  —  2  a,  transmitted.     These  portions,  supposing  no  undulation,  or  part  of  an 
undulation,  gained  or  lost  in  the  act  of  transmission  or  reflexion,  will  both  be  in  the  phase  0.     The  reflected 

portion  will  again  encounter  the  first  surface  in  the  phase  0  -f-  2  it .  — ,  will  there  be  again  partially  reflected, 

n 

with  an  intensity  equal  to  a  X  a  r=  a',  and  the  portion  so  reflected  will  reach  the  second  surface  in  the  phase 

2  t 
6  -f-  2  ir  .  -"—,  and  will  there  be  transmitted  with  an  intensity  =  (1  —  a)  .  a8,  or  nearly  =  a*.      Now,  the 

\i 

reflexions  being   all  perpendicular,  this   portion  will    be  confounded  with   the   portion   1   —  2  a  transmitted 

without  any  reflexion;  and  putting  «  =  •v'l  —  2  a  =  1  —  a,  nearly,  and  a  :=  */a?  =  a,  a  and  a' will  represent 
the  amplitudes  of  vibration  of  the  ethereal  molecule  at  the  posterior  surface,  which  each  of  these  rajs  tend  to 
impress  on  it.  Hence,  its  total  excursion  from  rest  will  be  represented  by 


that  is 

(1  -  a)  cos  O  -\- 


/  2<\ 

cos  0  -}-  a1 .  cos  I  0  -\-  2  TT  .  — -  1, 

(Q  t  \ 
0  -f  2ir  .  -^—  I. 

(2  <\ 
9  -f  2  JT  . J  -  a  .  cos  0. 


The  first  term  of  this  is  independent  of  t,  and  represents,  in  fact,  the  incident  ray  in  the  state  in  which  it  would 
arrive  at  the  second  surface,  had  no  reflexions  taken  place.  The  other  two  .terms  represent  rays  the  former  of 

which  evidently  is  in  complete  discordance  with  the  latter,  and  destroys  it  when  t  is  any  odd  multiple  of  — ,  (or  of 

the  half  length  of  one  of  Newton's  fits,  a  fit  being,  as  we  have  seen  above,  equal  to  half  an  undulation,)  thus 
leaving  the  ray  at  its  emergence  of  the  same  intensity  as  it  would  have  had  were  the  lamina  away  ;  but  when  t 

is  any  odd  multiple  of  half  a  fit,  then  the  value  of  cos  l0-\-Zir.--\=—  cos  0 ;  and  the  emergent  raj- 
is  in  this  case  represented  by  (1  —  2  a)  .  cos  0,  being  less  than  the  incident  ray  by  twice  the  light  reflected  at 
the  first  surface. 

Thus  if  the   thickness  of  the  plate  be  different  in   different   parts,   the    light   transmitted   through    it    to      gfij 
the  eye  will  not  be  uniform,  but  will  have  alternate  maxima  and  minima  corresponding  to  the  thicknesses  0  Origin  of 
X  2  X          3  X  the'bright 

—7- ,      — —  ,       — — ,    &C.  and  dark 

444  •          *     i_ 

rings  in  ho- 
lt we  apply  to  the  expression  above  given,  the  general  formula  Art.  (613)  for  the  composition  of  rays  in  one  ""H6"60" 
plane,  we  shall  find  for  the  intensity  A4  of  the  ray  finally  emergent, 


470 


LIGHT. 


A«  =  (I  -  «r)s  +  2  a  (I  -  a)  .  cos  2  *•  .  — - 

\ 


, 

«« 


Part  III. 


=  1  -  4  a  (1  -  a)  .  sin  (- 
=  1  -  4  a  .  sin  fz  v  —  j 


-2  v 


663. 

Transmit- 
ted tints  in 
white  light 
expressed 
alge- 
braically. 


which  shows  that  the  several  maxima  are  equal  to  the  incident  ray,  and  the  minima  to  that  ray  diminished  by 
four  times  the  light  reflected  at  the  first  surface.  The  difference  of  phase  between  the  simple  and  composite 
emergent  ray,  or  the  value  of  B  in  the  formula  cited,  is  given  by  the  equation, 

sin  B  =  —  .  sin  f  2  ar  .  — -  j  =  a  .  sin  (  2  •*  .  — —  j ,  neglecting  A«, 

so  that  for  such  media  as  have  not  a  very  high  refractive  power,  this  difference  is  always  small.  It  is,  however, 
periodical,  and  differs  for  different  thicknesses. 

Suppose  now,  instead  of  homogeneous  light,  white  light  to  lall  on  the  lamina,  and  let  us  represent  a  ray  of 
such  light,  as  in  Art.  488,  by  C  +  C'-f-  C"-f-  &c.,  or  by  S  (C),  C,  C',  &c.  being  the  intensity  of  the  several 
elementary  rays  of  all  degrees  of  refrangibility,  then  will  the  transmitted  compound  twain  be  represented  in  tint 
and  intensity  by 


C    l  -  4  a  .  sin 


J    7T 

t  V 
*/ 

}  +  c' 

<  1  —  4 

a  .  sin 

(-4)*}+- 

S. 

C  jl 

-  4  a. 

sin  (2  *  -f  y  }  • 

-  4 

«)  + 

C(4a 

-  4a. 

sin  (  2  ir 

X   /    j 

>). 

S(C) 

+  4a 

.  s  [c 

.  cos  1  2 

LLY) 

x  ;  r 

or  by 


Now  this  is  the  same  with 


=  (1  -  4  a) 

The  first  term  of  this  expression  represents  a  beam  of  white  light  of  the  intensity  1  —  4  a.  The  second,  a 
compound  tint  of  the  intensity  4  a,  which,  diluted  with  the  above-mentioned  white  light,  forms  the  pallid  tints 
of  the  transmitted  series.  If  we  disregard  this  dilution,  and  consider  only  the  tint  in  its  purity  as  it  would 
appear  were  the  white  light  suppressed,  its  expression 


40.  S 


=  4  a  I  S  (C) 


-  S  (  C  .  sin  (2  TT 


indicates  that  it  is  complementary  to  the  tint  represented  by 

S  JC  .  sii 

But  if  we  conceive  a  curve  whose  abscissa  =  t,  and  whose  ordinate  is  C  .  sin  f  2  TT  .  — —  J ,  it  is  evident  that 

this  will  be  precisely  the  undulating  curve  represented  for  each  prismatic  ray  in  fig.  134  ;  and  taking  the  sum  of 
all  the  ordinates  so  drawn  for  each  colour  in  the  spectrum,  we  have  the  identical  construction  from  which  we 
derived  the  colours  of  the  reflected  rings  in  Art.  645.  If,  then,  we  take  the  series  of  tints  so  composed,  and 
thence  deduce  their  complements  to  white  light,  and  dilute  these  complementary  colours  with  white,  in  the 
proportion  of  4  a  rays  of  the  complementary  colour  to  1  —  4  a  of  white,  we  shall  have  the  series  of  transmitted 
tints  which  ought  to  result  from  the  doctrine  of  interferences,  and  which,  in  fact,  is  observed. 

664.  In  the  case  of    oblique  transmission,  let  AC,   B  D,  fig.  135,  be  the  surfaces  of  the  lamim,    and  A  a  its 

Case  of        thickness  ;  and  let  A  E  be  the  surface  of  a  wave  of  which  the  point  A  has  just  reached  the  first  surface  of  the 

oblique         lamina  ;  and  let  S  A,  S  C,  perpendicular  to  it,  represent  rays  emanating  from  one  origin  S,  then  will  a  partial 

»ra"slnission  reflexion  take  place,  and  its  intensity  will  be  diminished  in  some  certain  ratio  1  :  1  —  a  depending  on  the  angle 

of  incidence.     The  transmitted  wave  will  be  bent  aside,  taking  the  position  A  ft,  and  advancing  along  A  B  the 

refracted  ray  ;  so  that  when  it  reaches  the  position  B  F,  the  wave  without  the  lamina  will  have  the  corresponding 

position  FG.     Here  another  partial  reflexion  will  take  place  depending  on  the  interior  incidence,  and  we  may 

denote  by  (1  —  o)  (1  —  a)  the  transmitted  portion,  and  by  (1  —  a)  .  a  the  reflected  portion.     These  portions  set 

off  together,  from  B,  the  former,  with  the  velocity  V  due  to  the  exterior  medium,  along  the  line  B  H  parallel  to 

S  A,  forming  a  wave  which  (provided  S  be  sufficiently  distant)  may  be  regarded  as  a  plane  of  indefinite  extent 

moving  uniformly  with  that  velocity  along  B  H.     The  latter  portion  proceeds  along  B  C,  according  to  the  law 

of  reflexion,  with  the  velocity  V  due  to  the  medium  of  which  the  lamina  is  composed  till  it  reaches  C,  where  it 

undergoes  another  partial  reflexion,  and  proceeds  back  along  the  line  C  D  with  a  diminished  intensity  =  (1  —  a) 


LIGHT.  471 

l.;ght.     .  o?,  but  with  the  same  velocity  V  till  it  reaches  D,  having  described  a  space  =  BC+CD  =  2AB  with  that    Part  III 
»-v~»/'  velocity.     At  D  it  undergoes  another  partial  reflexion,  and  only  a  portion  =  (1—  a)  (1  —  a)  .  as  is  transmitted,  ^—  y—  - 
which  sets  off  from  D  along  the  line  D  I  (parallel  to  B  H)  with  the  velocity  V,  that  is,  with  the  same  velocity 
as  the  wave  along  B  H.     This  wave  may  also  be  regarded  as  a  plane  of  indefinite  extent  perpendicular  to  D  1, 
and  therefore  parallel  to  the  former.     But  they  are  not  coincident  ;   for  the  former,  having  the  start  of  the  latter, 
will  have  come  into  a  position  I  H  K  in  advance  of  the  position  D  L  M  taken  by  the  latter,  and  both  the  waves 
moving  forwards  now  with  the  same  velocity  V  will  preserve  this  distance  for  ever  unaltered.     The  interval  L  H 
we  may  term  the  interval  of  retardation.     To  determine  it,  we  have  to  consider  that  the  space  B  H  is  described 
by  the  former  wave  with  a  velocity  V,  while  the  latter  describes  B  C  -j-  0  D  with  the  velocity  V,  and  therefore 


=  2<.  sec 


putting  ft.  for  the  relative  index  of  refraction  of  the  lamina,  p  for  the  angle  of  refraction  a  A  B,  and  t  for  the 
thickness  A  a,  because  V  :  V  ; '.  /t  :  1. 

Again,  B  L  =  B  D  .  cos  D  B  L  ==  D  B  .  sin  <j)  (<j>  being  the  angle  of  incidence  corresponding  to  p  the  angle  of 
refraction,)  =  2  a  B  .  sin  0  =  2  t  .  tan  p  .  sin  0.     Therefore  the  whole  interval  of  retardation  is  equal  to 

2  t    n 
2  t  {  n  .  sec  p  —  tan  o  .  sin  <f>  }  =   : — •  (I  —  sin  />*)  =  2  /t  t .  cos  p 

because  sin  0  =  ft .  sin  p. 

Thus,  in  virtue  of  the  two  internal  reflexions,  each  wave  which  before  entering  the  medium  was   single,  will      555 
after  quitting  it  be  double,  being  followed  at  the  constant  interval  2  /» t .  cos  p  by  a  feebler  wave  of  the  intensity 
above  determined.     The  same  being  true  of  every  wave  of  the  system  of  which  the  ray  consists,  these  two 
systems  (considered  as  of  indefinite  duration)  will  be  superposed  on,  and  interfere  with  each  other,  according  to 
the  general  principles  before  laid  down. 

Let  X,  be  the  length  of  an  undulation  in  the  lamina,  then  will  ft,  X  represent  that  of  an  undulation  in  the  sur-       666. 
rounding  medium.     This  is  obvious,  because  the  velocity  in  the  latter  being  to  that  in  the  former  as  ft :  1 ;    and  Undulationi 
the  same  number  of  undulations  being  propagated  in  the  same  time  through  a  given  point  in  both  cases,  they  snorter  in 
must  be  more    crowded,    and  therefore  occupy  less  space  in  the  one  than  the  other  in  the  ratio   of  the  ^eftl 
velocities. 

Hence  the  differences  of  phases  between  the  interfering  systems  at  any  point  will  equal  ,,,.* 

interval  of  retardation  2  t  .  cos  p  2  t< 

2ir  .-  —  =  2  if  .  - =  2  w  .  — - —  ,  putting  tf  =  t  .  COS  p, 

14  n,  ™  Ai  tor  tne 

transmitted 
and  therefore  the  final  resulting  wave  will  be  expressed  by  the  equation  ray 

(  /  2  if  \  ) 

X  =  •/  (1  -  a)  (1  -  o)  j  cos  0  +  a  .  cos  (  0  -f-  2  ir  .  — —  I  L 

which  being  resolved  in'o  the  fundamental  form  A  .  cos  (0  -f-  B),  as  before,  gives 

A'  =  (1  -  a)  (1  -  «)  .  {  1  +  2  a  .  cos  (2  TT  .  1*)  -f-  «t  j  , 

and  .    J_/0_       2*" 

sin  B  = 


/y/  1  +  2  o  .  cos  ( 2  IT  .  ^—\ 


Such  are  the  general  expressions  for  the  intensity  and  change  of  origin  of  the  compound  transmitted  ray.  668 
It  is  evident,  however,  that  when  a  and  a  are  small,  which  they  always  necessarily  are  in  any  but  extreme  cases,  Case  of 
this  value  of  A"  reduces  itself  by  neglecting  their  powers  and  products  to  moderate 

ublianiliet. 

(1  —  a  -\-  a)  —  4  a  .  sin  (  2  ir  .  —— 

which  is  exactly  analogous  to  the  expression  in  Art.  662,  for  the  case  of  perpendicular  incidence ;  and  shows, 
that  with  the  exception  of  a  very  trifling  difference  in  the  degree  of  dilution,  the  same  laws  of  alternation 
in  brightness,  in  homogeneous  light,  and  of  tint  in  white  light,  must  hold  good  in  both  cases. 

But  there  is  one  essential  difference.      The  same  tints  will  arise  in  the  case  of  oblique  incidence   at  the      ....„ 
thickness  t,  which  in  that  of  perpendicular  incidence  is  produced  at  the  thickness  t .  cos  p,  because  if  =  t .  cos  p.  Dilatation 
Now  this  is  always  less  than  t,  and  therefore  the  tint  produced   at  oblique  incidences  at  the  given  thickness  Of  theVingi 
will  be  higher  in  the  scale  (or  correspond  to  a  less  thickness)  than  in  perpendicular  ;    and,  consequently,  the  explained, 
rings,  or  fringes,  so  seen  by  transmission  should  dilate  by  inclining  the  lamina  to  the  eye.     The  law  of  dilata- 
tion  evidently,  at  moderate  incidences,  coincides  nearly  with  Newton  s   rule;    for  this  gives,  on  reduction, 
neglecting  sin  p*, 


472 


LIGHT. 


Lijht. 


670. 
Deviation 
from  New- 
ton's rule  at 
great  obli- 
quities pro 
bably  ac- 
counted for. 

671. 

Origin  of 
the  reflected 
rings. 


672. 

Lass  of  half 
in  undula- 
tion. 


sec  u  —  sec  p 


—  1)  .  tan  p* 


673. 

Not  con- 
trary to 
dynamical 
principles 

674. 

Nor  to  the 
undulatnrjr 
doctrine. 


which  does  not  deviate  very  greatly  f  om  sec  p  at  moderate  incidences. 

At  great  incidences  the  case  is  different,  and  the  noncoincidence  of  the  results  of  the  undulatory  doctrine 
with  experiment  might  be  drawn  into  an  argument  against  it,  were  we  sure  that  the  law  of  refraction  at  extreme 
incidences,  and  with  very  thin  lamina,  does  not  vary  sensibly  from  that  of  the  proportional  sines.  This  is, 
indeed,  highly  probable,  as  M.  Fresnel  has  remarked,  (Mem.  stir  la  Diffraction,  $c.)  and  as  we  have  before 
had  occasion  to  observe.  The  inquiry  into  which  this  would  lead,  is,  however,  one  of  the  most  delicate  and  difficult 
in  physical  optics,  and  the  reader  must  be  content  with  this  general  notice  of  a  possible  explanation  of  one  of  the 
many  difficulties  which  still  beset  the  undulatory  doctrine. 

The  origin  of  the  reflected  rings  may  be  accounted  for  in  a  similar  way  from  the  partial  transmission  of  the 
waves  reflected  from  the  second  surface  back  through  the  first,  and  their  interference  with  the  waves  reflected 
immediately  from  the  first.  The  relative  intensities  of  these  waves,  (in  general,)  are  a  and  (I  —  a)  (1  —  a)  .  a; 
or,  in  the  case  where  a  and  a  are  both  small,  nearly  in  the  ratio  of  a  :  a,  and  at  a  perpendicular  incidence,  very 
nearly  in  the  ratio  of  equality.  Hence  their  mutual  destruction  in  the  case  of  complete  discordance  will  be  much 
more  complete  than  in  the  transmitted  rings,  and  the  colours  arising,  much  less  dilute  than  those  of  the  latter, 
agreeably  to  observation. 

There  is,  however,  one  consideration  of  importance  to  be  attended  to  in  the  application  of  the  undulatory  doc- 
trine to  the  reflected  rings,  which  at  first  sight  appears  in  the  light  of  a  powerful  argument  against  its  admis- 
sibility,  viz.  that  if  we  apply  the  same  reasoning  to  the  reflected,  as  we  have  already  done  to  the  transmitted, 
rings,  we  should  arrive  at  the  conclusion,  that  their  tints  should  be  precisely  the  same  and  in  the  same  order, 
beginning  with  a  bright  white  in  the  centre  ;  because  here,  the  path  traversed  by  the  ray  within  the  lamina 
vanishing,  the  waves  reflected  from  the  two  surfaces  ought  to  be  in  exact  accordance,  whereas  it  appears,  by 
observation,  that  the  reverse  is  the  case,  the  central  spot  being  black  instead  of  white.  It  becomes  necessary, 
then,  to  suppose,  that  in  this  case,  half  an  undulation  is  lost  or  gained  either  by  the  wave  reflected  from  the  first 
or  second  surface.  If  this  hypothesis  be  made,  the  phenomena  of  the  reflected  rings  are  completely  represented 
on  the  undulatory  system,  for  the  compound  wave  reflected  by  the  joint  action  of  the  two  surfaces  should  be 
represented  by  the  equation, 


x  =  ^~ 


~a.  cos  &  4- 


—  a) 


-  a)  .  cos  -f 


*     -~ 


and  if  this  be  put  equal  to  A  .  cos  (0  -f-  B)  we  get 


=  a  4-  a.  (1  -  a)  (1  -  a)  -  2  «/  aa  (I  -  a)  (1  -  a)  .  cos    z  w      - 


675. 


and  in  the  case  of  a  and  a  both  very  small 

_  f  J/   V  t 

A5  =  (^  a-  */a)°-  4-4.  v/ao.sinf  2*-  —  -j 

and  at  a  perpendicular  incidence,  where  t'  =  t  ,  and  where  a  and  a  may  be  supposed  equal 

/        t  V 

A2  =  4  a  .  sin  f  2  ir  —  I 

Thus  we  see,  that  in  this  case  the  total  intensity  of  the  compound  reflected  wave  4-  tna'  of  the  transmitted 
(Art.  662)  make  up  1,  the  intensity  of  the  incident  wave;  and  thus,  this  supposition  of  the  loss  or  gain  of  half 
an  undulation  is  in  no  contradiction  with  the  law  of  the  conservation  of  the  vis  viva. 

In  fact,  however,  if  we  consider  the  mode  in  which  the  undulations  are  propagated,  at  the  limit  between  two 
media,  we  shall  see  nothing  contrary  to  dynamical  principles  in  the  loss  of  half  or  any  part  of  an  undulation  in 
the  transfer  —  for  it  cannot  be  supposed,  that  the  density  or  elasticity  of  the  ether  changes  abruptly  at  the  sur- 
faces of  media,  but  that  there  intervenes  some  very  minute  stratum  in  which  it  is  variable.  In  this  stratum, 
therefore,  the  length  of  an  undulation  is  neither  exactly  that  corresponding  to  the  denser,  nor  to  the  rarer 
medium,  but  intermediate,  and  of  a  magnitude  perpetually  varying.  Therefore  the  number  of  undulations  to  be 
reckoned  as  added  to  the  phase  of  the  ray  in  traversing  this  stratum,  will  differ  from  what  it  would  be  if  one 
medium  terminated,  and  the  other  commenced  abruptly.  Without  knowing  the  law  of  density,  the  limits 
between  which  it  undergoes  its  change,  or  the  exact  mode  in  which  the  partial  reflexion  of  a  wave  traversing  it 
is  performed,  it  is  impossible  to  subject  the  point  to  strict  calculation,  we  must  rather  submit  to  be  taught  by 
experiment,  and  content  ourselves  with  such  conclusions  as  we  can  deduce  from  observation.  In  the  case 
before  us,  all  that  observation  teaches  us  is,  that  there  is  half  an  undulation  more  of  difference  in  the  phases  of 
two  rays  that  have  been  reflected  in  the  manner  last  considered,  than  in  those  of  the  two  whose  interference 
forms  the  transmitted  rays.  From  some  curious  experiments  of  Dr.  Young,  too,  we  may  gather  that  it  is  not  in 
all  cases  strictly  half  an  undulation  of  difference  to  be  reckoned,  but  rather  a  variable  fraction  depending  on  the 
nature  of  the  contiguous  media. 

The  formulae  of  Art.  672  show  that  it  is  only  in  the  case  of  perpendicular  incidence  that  the  tints  are  pure, 
and  that  in  all  others,  and  especially  at  great  obliquities,  where  a  and  a  differ  considerably,  there  will  be  a- 


I.  I  G  H  T.  4?<? 

Light.      dilution  of  white  light,  and  this  is  also  agreeable  to  experience.     At  a  perpendicular  incidence,  however,  the     Part  III. 
"^j"^  minima  of  each  homogeneous  colour  ought  to  be  absolutely  evanescent ;  so  that  if  we  were  to  remove  the  reflec-  v««pv"^»' 
lion  of  the  upper  surface  of  an  object  glass  laid  down  on  a  plate,  (or  use  a  prism,  so  as  to  prevent  its  reaching  E*p«rime»- 
the  eye,)  the  intervals  between  the  rings  in  homogeneous  light  ought  to  appear  absolutely  black.     In  the  New-  (""'v"e' 
Ionian  doctrine  this  should  not  be  the  case,  because  the  light  reflected  from  the  upper  surface  of  the  lamina  of  two  theories 
included  air  should  still  remain  even  in  the  minima  of  the  rings.     This  then  affords  a  positive  means  of  deciding 
between  the  two  theories.     M.  Fresnel  describes, an  experiment  made  for  this  purpose,  and  states  the  result  to 
be  unequivocally  in  favour  of  that  of  undulations.      (Diffraction  dela  Lumicre,  p.  11.) 

§  V.     Of  the  Colours  of  Thick  Plates. 

Under  certain  circumstances  rings  of  colours  are  formed  by  plates  of  transparent  media  of  considerable  thick-      676 
ness.     The  circumstances  under  which  they  appear,  in  one  principal  case,  are   thus  described  by  Newton,  who 
first  observed  them,  and  who  has  applied   his  doctrine  of  the  fits  of  easy  reflexion  and  transmission  to  explain 
them,  with  singular  ingenuity. 

"  Admitting  a  bright  sunbeam  through  a  small  hole  of  one-third  of  an  inch  in  diameter  into  a  dark  room,  it  Newton's 
was  received  perpendicularly  on  a  concavo-convex  glass  mirror  one  quarter  of  an  inch  thick,  having  each  surface  "Per" 
ground  to  a  sphere  of  six  feet  in  radius,  and  the  back  silvered.     Then  holding   a  piece  of  white   paper  in  the  Inirror  ° 
centre  of  its  concavity,  having  a  small  hole  in  the   middle  of  it  to  let  the  sunbeam  pass,  and  after  reflexion  at 
the   speculum  to  repass  through  it,  the  hole  was  observed  to  be  surrounded  with  four  or  five  coloured   concen- 
tric rings  or  irises,  just  as  the  rings  seen  between  object-glasses  surround  their  central  spot — but  larger  and 
more  diluted  in  their  colours".  ..."  If  the  paper  was  much  more  distant  from  the  mirror,  or  much  less  than 
six  feet,  the  rings  became  more  dilute  and  gradually  vanished.''.  ..."  The  colours  of  these  rings  succeeded  each 
other  in  the  order  of  those  which  are  seen  between  two  object  glasses,  not  by  reflected  but  by  transmitted  light, 

viz.  white,  tawny  white,  black,  violet,  blue,   greenish   yellow,  yellow,  red,  purple,"  &c "  The  diameters  of 

these  rings  preserved  the  same  proportion  as  those  between  the  object-glasses,  the  squares  of  the  diameters  of 
the  alternate  bright  and  dark  rings,  reckoning  the  central  white  as  a  ring  of  the  diameter  0,  forming  an  arith- 
metical progression,  beginning  at  0.  And  in  the  case  described,  the  diameter  of  the  bright  ring  measured 
respectively  0,  1-i-l,  2$,  2{i,  3^.".  ..."  Lastly,  in  the  rings  so  formed  by  reflectors  of  different  thicknesses,  their 
diameters  were  observed  to  be  reciprocally  as  the  square  roots  of  the  thicknesses.  If  the  back  of  the  mirror  was 
silvered,  the  rings  were  only  so  much  the  more  vivid." 

These  various  phenomena,  and  a  variety  of  similar  ones,  some  of  more,  some  of  less  complexity,  according  to  677 
the  variation  of  the  distance,  and  obliquity  of  the  mirror,  and  the  curvature  of  its  surfaces,  Newton  has 
explained  very  happily,  by  considering  the  fits  of  easy  reflexion  and  transmission  of  that  faint  portion  of 
the  light  which  is  irregularly  scattered  in  all  directions  at  the  first  surface  of  the  glass,  and  which  serves  to 
render  it  visible.  But  for  this  explanation  we  must  refer  to  his  Optics,  as  our  object  here  is  more  particularly 
and  distinctly  to  show  what  account  the  undulatory  doctrine  gives  of  this  phenomenon,  which  has  hitherto  been 
passed  over  rather  cursorily,  not  without  some  degree  of  obscurity. 

There  is  no  surface,  however  perfectly  polished,   so  free  from   small   scratches  and  inequalities   as  not  to       673. 
reflect  and  transmit,  besides   those  principal  rays  which  obey  the  regular  laws  of  reflexion  and  refraction,  as  Principle  of 
dependent   on  the   general   surface,  other,  very  much   feebler,  portions   scattered  in   all  directions,  by  which  the  explanation 
surface  is  rendered  visible  to  an  eye  anywhere   placed,  but  most  copiously  in  and   about  the  direction  of  the  j",       u"~ 
regularly  reflected  and  transmitted  rays.     It  is  the  interference  of  these  portions,  scattered  at  the  first  surface  by  SyStac,nry 
the  ray  in  passing  and   repassing  through  it,  nearly  in  its  own  direction,  that  the  rings  in  question  are  attributed 
in  the  undulatory  doctrine. 

Let  F  A  D,  E  B  G  be  the  parallel  surfaces  of  any  medium  exposed  perpendicularly  to  a  homogeneous  ray      679. 
emanating  from  a  luminous  point  C,  and  incident  at  A.     The  chief  portion  will  pass  straight  through  A,  and  be  Its  applica- 
reflected  back  from  B  to  A  again.     But  at  A  a  scattering  takes  place,  and  the  transmitted  ray  A  B  is  accom-  tion 
panied  by  a  diverging  cone  of  faint  rays  A  a,  A  b,  Ac,  &c.,  all  which  set  out  from  A  in  the  same  phase  of  their     g' 
undulations   with  the  principal  one  from   which  they  originate,  so  that  A  may  be  regarded  as  their  common 
origin.     Take  Q,  the  focus  of  rays  reflected  at  the  second   surf-.ce  conjugate  to  A  (if  the  surfaces  be   plane, 
Q  and  A  are  equidistant  from  B)  and  the  cone  of  scattered  rays,  with  the  regularly  reflected  ray  in  its  axis,  will 
after  reflexion  diverge  as  from  Q.     Again,  when  they  pass  into  the  air  again,  if  we  take  q  the  focus  conjugate 
to  Q  of  rays  refracted  at  the  surface  F  D,  they  will  after  refraction  diverge  from  q,  and  by  the  nature  of  foci  on 
the   unrlulatory  hypothesis,  the  undulations  will  be  propagated  in  the  air  as  if  they  had  a   common   origin  q 
placed  in  air ;   because,  after  refraction,  the  waves  have  the  form  of  spheres  diverging  from  q,  and  therefore 
every  portion  of  their  surfaces  are  equidistant  from  q  ;  had  they,  therefore,  really  emanated  from  q,  as  separate 
rays,  they  must  at  the  moment  of  such   emanation  have  been  all  in  one  phase.     Now,  when  the  reflected  beam 
reaches  A  a  portion  of  it  will  again  l>e  scattered  in  a  cone,  having  the  regularly  transmitted  ray  A  G  in  its  axis  ; 
and  the  rays  A  O,  A  N,  A  M,  &c.  of  this  cone  will  all  have  A  for  their  origin,  and  will  be  in  the  same  phase  at  their 
departure  from  A  with  the  ray  AG;   but  this  is  in  the  phase  it  would  have  had  as  emanated  from  q;  hence,  if 
we  consider  any  point  M  out  of  the  directly  transmitted  ray  A  G,  it  will  be  reached  at  once  by  a  wave  belonging 
to  each  diverging  cone,  the  one  along  q  M  from  q  and  the  other  along  AM  from  A,  and  the  difference  of  routes 
is  equal  to  q  A  -(-  A  M  —  qM.     Therefore,  when  M  is  very  nearly  coincident  with  G,  this   is  very  small  and  at 
G  vanishes,  or  the  waves  are  in  exact  accordance.     As  M  recedes  from  G  it  increases,  and  when  it  becomes, 

VOL.  <v.  3  q 


474 


LIGHT. 


Light. 


half  an  undulation,  the  waves  are  in  complete  discordance  and  annihilate  each  other,  and  so  on  alternately.     There      I'*rt  " 
fore,  as  this  is  true  of  all  rays  in  conical  surfaces  round  A  G  as  an  axis,  equally  inclined  with  A  M,  9  M,  if  we  place  Vs-"^/~-~ 
a  white  screen  at  G,  it  will  appear  marked  with  alternate  dark  and  bright  rings  round  a  bright  centre.  To  deter- 
mine tneir  diameters  we  need  only  put  </A-j-AM  —  qrMrrre.  — ,  or,  if  we  take  9  A  =  a,  A  G  =  r,  GM  =  y, 


a  -f  V  ^  -f  ^  - 
If  we  resolve  this  equation  neglecting  y2,  we  find 


(«+•! 


)!  -f  y*  =  n  — - 


680. 

Law  of  the 
diameters  of 
the  rings. 


681. 

Of  their 
colours. 

682. 

Concentra- 
tion of  the 
rings  from 
all  points  of 
the  surface. 
Fig.  137. 

683. 

Newton's 
experiment 
particularly 
considered. 


684. 


(a+r) 

which,  on  substituting  0,  1,  -2,  3,  &c.  in  succession  for  n,  shows  that  the  successive  diameters  of  the  alternate  dark 
and  bright  rings  are  in  the  progression  of  the  square  roots  of  those  numbers. 

If  the  thickness  of  the  plate  be  small  compared  to  the  distance  of  the  screen,  a  will  also  be  small,  and  the 
value  of  y  becomes 

y  =  r.^n. 

m 

which  shows  that  for  rays  of  a  given  refrangibility  the  diameters  of  the  rings  are  as  the  distance  of  the  screen 
directly,  and  the  square  root  of  the  thickness  of  the  plate  inversely. 

I^astly,  the  diameter  of  a  ring  of  the  same  order  in  different  homogeneous  lights,  are  as  the  square  roots  of  the 
lengths  of  their  undulations.  Now,  this  is  the  very  same  law  that  governs  the  diameters  of  the  rings  formed 
between  object-glasses.  Consequently,  if  instead  of  homogeneous  we  consider  white  light,  we  ought  to  have  a 
succession  of  coloured  rings  whose  tints  agree  precisely  with  the  transmitted  series  in  that  experiment. 

But  the  rays  so  formed,  by  rays  scattered  from  a  single  point  A,  would  be  too  feeble  to  be  visible.  If,  how- 
ever, we  suppose  the  surfaces  to  be  concentric  spheres  having  G  in  their  common  centre,  as  in  fig.  137,  then 
any  rays  G  A,  G  A'  falling  on  any  points  whatever  of  their  surfaces  will  depict,  on  screens  G  M,  G  M'  respect- 
ively perpendicular  to  them  as  G,  equal  systems  of  rings  having  G  in  their  common  centre  ;  and,  when  the  arc 
A  A'  is  not  very  great,  the  screens  may  be  regarded  as  coincident  (for  in  that  case  B  M  —  M  A  =r  B  M'—  MA1) 
and  the  rings  from  every  point  of  the  surface,  exactly  superposed  on  each  other,  and  being  thus  increased  in 
intensity  in  proportion  to  the  area  of  the  exposed  surface,  become  visible. 

Now  this  is  exactly  Newton's  case,  for  the  sun  being  a  luminary  of  a  considerable  diameter,  the  hole  in  the 
centre  of  the  spheres  may  be  regarded  as  a  portion  of  the  sun  of  that  size,  actually  placed  there.  Of  this, 
every  indivisible  point  may  be  regarded  as  the  origin  of  a  system  of  waves,  and  as  depicting  on  the  screen  its 
own  set  of  rings.  These,  were  the  hole  infinitely  small,  would  be  infinitely  more  clear  and  pure  in  their  tints 
than  the  transmitted  rings  between  object-glasses,  because  they  are  not  (as  in  those  rings)  diluted  with  the 
great  quantity  of  white  light  which  escapes  interference.  But  owing  to  the  size  of  the  hole,  their  centres  are 
not  exactly  coincident,  and  therefore  their  tints  mix  and  dilute  each  other,  and  that  the  more  the  larger  the 
hole  is. 

If  c  be  the  thickness  of  the  glass,  since  Q  is  the  conjugate  focus  of  A,  on  the  surface  B  whose  radius  we  will 

T  -+-  C  2  T  C 

call  r-fc  putting  G  A  =  r,  we  have,  by  Art.   249,   BQ=  -       -  .  c,  A  Q  =    -      — ;   and,  by  Art.  24S 


r  —  c 


A  9  =  a  =  - 


2cr 


2  c  -  /.  (r  -j-  c) 


,  taking  yu  for  the  refractive  index ;  and  when  c  is  small  compared  with  r,  we  get 


2c 


showing  that  the  diameters  of  the  rings  are  in   this  case  in  the  subduplicate  ratio  of  the  refractive  index  of  the 
glass  directly,  and  of  its  thickness  inversely. 

3                                                                               2 
685  ^  we  re(^uce  tn's  va'ue   to  numbers,  taking  /»  =  — ,  n  =  4,  r  =  6  feet  =  7fc  inches,  and  X  = =  the 

2 
length  of  an  undulation  for  yellovr  rays 


2"  r  89000 

nearly,  we   find,  for  the   diameter  of  the  second   bright  ring  in 


90000 
yellow  light,  (which  corresponds  to  the  brightest  part  of  the  same  ring  in  white,) 


.  4  =  2-35, 


686 

Case  of 
oblique 
incidence. 

687. 

Phenomena 
ooserved  liy 
the  Duke  of 
Chaulnes 
And 


'   4      90000 
which  agrees  almost  precisely  with  Newton's  measure  2f,  or  2'375. 

When  the  mirror  is  inclined  to  the  incident  beam  the  phenomena  become  more  complicated,  and  have  been 
elegantly  described  by  Newton,  (Optics,  book  ii.  part  iv.  obs.  10.)  In  this  case,  the  axes  of  the  two  interfering 
cones  of  scattered  rays,  which  are  always  the  incident  and  reflected  rays,  are  no  longer  coincident.  But  the 
same  principles  apply  equally  to  this  case  in  all  other  respects,  and  the  reader  may  exercise  himself  in  tracing 
their  consequences. 

The  Duke  de  Chaulnes  found  similar  rings  to  be  exhibited  when  the  surface  of  the  mirror  was  covered  with 
a  thin  film  of  milk  dried  on  it,  so  as  to  make  a  delicate  semitransparent  coating,  or  even  when  a  fine  gauze  or 
muslin  was  stretched  before  it;  see  the  account  of  his  experiments  in  the  Mem.  Acad.  Sci.  Paris,  1705  ;  and 


LIGHT.  475 

Light.      Sir  William   Herschel  (Phil.  Trans.  1807)  describes  a  pleasing  experiment,  in  which  rings  were  produced  by    Part 
~~^^~~>  strewing  liair  powder  in  the  air  before  a  metallic  mirror  on  which  a  beam  of  light  is  incident,  and  intercepting1  * 
the  reflected  ray  by  a  screen.     The  explanation  of  these  phenomena  seems,  however,  to  depend  on  other  appli-  j 
cations  of  the  general  principle,  and  will  be  better  conceived  when  we  come  to  speak  of  the  colours  produced 
by  diffraction. 

Dr.  Brewster,  in  the  Transactions  of  the  Royal  Society  of  Edinburgh,  has  described  a  series  of  coloured  fringes      "^ 
produced  by  thick  plates  of  parallel  glass,  which   afford  an  excellent  illustration  of  the  laws  of  periodicity  sjj.,s  re 
observed  by  the  rays  of  light  in  their  progress,  whether,  as  in  the  Newtonian  doctrine,  we  consider  them  as  sub-  fringes  seen 
jected  to  alternate  fits  of  easy  reflexion  and  transmission,  or,  as  in  the  undulatory  hypothesis,  as  passing  through  in  thick 
a  series  of  phases  of  alternately  direct  and  retrograde  motions  in  the  particles  of  ether,  in  whose  vibrations  they  plates- 
consist.     We   may  here  remark,  once  for  all,  that  the  explanations  which  the   undulatory  doctrine    affords   of 
phenomena  of  this  description,  may,  for  the  most  part,  be  translated  into  the  language  of  the  rival  hypothesis  ; 
so  as  to  afford,  with  more  or  less  plausibility  and  occasional  modifications,  a  result  corresponding  with  observa- 
tion.    It  is  not,  therefore,  among  phenomena  of  this  class  that  we  must  look  for  the  means  of  deciding  between 
them.     We  shall  adopt,  therefore,  in  the  remainder  of  this  essay,  the   undulatory  system,  not  as   being  at  all 
satisfied  of  its  reality  as  a  physical  fact,  but  regarding  it  as  by  far  the   simplest  means  yet  devised  of  grouping 
together,  and  representing  not  only  all   the  phenomena  explicable  by  Newton's   doctrine,  but  a  vast  variety  of 
other  classes  of  facts  to  which  that  doctrine  can  hardly  be  applied  without  great  violence,  and  much  additional 
hypothesis  of  a  very  gratuitous  kind. 

The  fringes  in  question  are  seen  when  two  parallel  plates  of  glass  of  exactly  equal  thickness  (portions  of  the  gag 
same  plate)  are  slightly  inclined  to  each  other,  (at  any  distance,)  and  through  them  both,  at  nearly  a  perpen-  Described 
dicular  incidence,  a  circular  luminary  of  1°  or  2°  in  diameter  (a  portion  of  the  sky,  for  instance)  is  viewed. 
There  will  in  this  case  be  seen,  besides  the  direct  image,  a  series  of  lateral  images  reflected  between  the  glasses, 
and  growing  fainter  and  fainter  in  succession  as  they  are  formed  by  2,  4,  6,  or  more  internal  reflexion^  ; 
and  of  which  all  but  the  first  is  so  faint  as  scarcely  to  be  visible,  except  in  very  strong  lights.  The  direct  image 
is  colourless  ;  but  the  reflected  one  is  observed  to  be  crossed  with  fifteen  or  sixteen  beautiful  bands  of  colour, 
parallel  to  the  common  section  of  the  surfaces  of  the  plates.  Their  breadth  diminishes  rapidly  as  the  inclination 
of  the  plates  increases.  When  the  plates  employed  were  0.12J  inch  in  thickness,  and  inclined  at  an  angle  of 
1°  11'  to  each  other,  the  breadth  of  each  fringe  measured  26'  50",  and  at  all  other  inclinations  their  breadth  was 
inversely  as  the  inclination.  At  oblique  incidences  its  fringes  are  seen  when  the  plane  of  incidence  is  at  right 
angles  to  the  principal  section  of  the  plates,  but  are  at  their  maximum  of  distinctness  when  parallel  to  it. 

To  understand  their  production,  let  us  call  the  surfaces  of  the  plates  in  order,  reckoning  from  that  on  which      690 
the  incident  light  first  falls,  A,  a,  B,  b;  and  let  us  consider  a  ray,  or  system  of  waves  emanating  from  a  common  Explained. 
origin  at  an  infinite  distance.     Then,  when  this  ray  falls  on  the  plates  it  will  at  every  surface  undergo  a  partial 
reflexion,  and  the  remainder  will  be  transmitted  ;  each  of  the  several  portions  will  be  again  subdivided  when- 
e\er  it  meets  either  surface.     So  that  either  image  will,  in  fact,  consist  of  several   emergent  rays,    parallel  in 
their  final  directions,  but  which  have  traversed  the  glasses  by  very  different  routes.     Thus  the  direct  or  principal 
image  will  consist  of 

1.  The  chief  portion  of  the  whole  incident  light,  refracted  at  A,  at  a,  at  B,  and  at  b,  and  emergent  parallel 
to  the  incident  ray,  which  we  will  represent  by  A  a  B  6. 

2.  A  portion   refracted  at  A,  reflected  at  a,  reflected  again  at  A,  refracted  again  at  a,  at  B  and  at  b,  and 
emergent  parallel  to  the  incident  beam.     This  we  will   denote  thus,  A  a'  A'  a  Bb  ;   the  letters  denoting  the 
surfaces,  the  accent  reflexion,  and  its  absence  refraction. 

3.  A  portion  which  has  undergone   two  similar  reflexions  in  the  interior  of  the  second  plate,  and  which  in 
the  same  manner  may  be  represented  by  A  a  B  6'  B'  6. 

4.  Other  portions  which  have  undergone  respectively  four,  six,  &c.  reflexions  to  infinity  within  either  of  the 
plates,  and  which  may  be  represented   by  such  combinations  as  A.a'  A.'  a'  A'a  B  b,  A  a  B  6'B'  b'  B'i,   or,  for 
brevity,  by  A  («'  A')Q  a  B  b,  A  a  B  (6'  B')li  b,  &c.  ;    but  these  latter  portions  are  too  faint  to  have  any  sensible 
influence  on  the  light  of  the  direct  image  with  which  they  are  confounded. 

The  first  lateral  reflected  image  will  consist  of  four  principal  portions  which  have  undergone  two  reflexions      ggi 
each,  viz. 

AaB'a'Bi;         A  a  B'a  A'a  B  b;         A.aBb'Ba'Bb; 


all  which  will  emerge  parallel.  Besides  these  there  are  infinite  others,  formed  by  a  greater  number  of  reflexions 
and  by  the  portions  A  a'  A'  a  of  the  incident  beam  reflected  within  the  first  glass;  but  these  are  all  too  faint 
materially  to  affect  the  image  in  question,  which  therefore  we  may  regard  as  composed  solely  of  the  four  rays 
just  enumerated.  Now  if  we  cast  our  eye  on  the  figure,  (138,)  we  see  the  course  pursued  by  each  of  these  Fie  l"s 
portions  1,  2,  3,  4  ;  and  it  is  evident  that  the  first  portion  has  traversed  the  thickness  twice,  and  the  interval 
between  the  glasses  three  times,  or  nearly;  neglecting  at  present  all  consideration  of  the  inclination  of  the 
plates  2  t  -(-  3  i.  In  like  manner,  the  portion  2  will  have  traversed  4  t  -f-  3  i  ;  the  portion  3,  4  t  -f  •  3  i;  and  the 
portion  4,  6  t  -(-  3  i.  Hence  it  appears  that  the  portions  1  and  4  differ  in  their  routes  by  nearly  four  times  the 
thickness  of  the  glass,  and  can  therefore  produce  no  colours  ;  but  the  other  portions,  at  a  perpendicular  inci- 
dence, would  not  differ  at  all,  and  at  very  small  inclinations  of  the  plates,  and  of  the  incident  rav,  will  only  differ 
by  reason  of  the  small  differences  of  the  inclinations  at  which  they  traverse  their  respective  thicknesses  and 
intervals.  They  will,  therefore,  interfere  so  as  to  produce  colour  ;  and  this  will  be  dependent  on  the  interval 
or  retardation  of  one  ray  behind  the  other,  arising  from  the  varying  obliquity  of  the  ray  which  enters  the  eye 

Now  when  we  look  at  a  luminous  image  of  sensible  magnitude,  the  rays  by  which  we  see  its  several  points      692 

30.2 


476  LIGHT. 

Light.      are  incident  in  all  planes,  and  at  all  inclinations.     Hence,  the  image  seen  will  appear  of  different  colours  in  its    Pftrl  "'• 
v_^v^«^'  different  points,  and  the  disposition  of  these  colours  will  follow   the  law,  whatever  it  be,  which  regulates  the  ^^  "v-~~- 
Isochro        interval  of  retardation.     The  colours,  therefore,  will  be  arranged  in  bands,  circles,  or  other  forms,  according  to 
matic  l.nes   the  form  of  the  curves  arising  geometrically  from  the  consideration  of  equal  intervals  of  retardation  prevailing 
defined.       m  every  pOjnt  of  their  course.     Such  curves,  now  and  hereafter,  we    shall  term   isochromatic  lines,  or  lines  of 

equal  tint,  measuring  in  all   cases  the  tint  numerically  by  the  number  of  undulations,  or  parts  of  an  undulation 

of  mean  yellow  light  to  which  the  interval  of  retardation  is  equal. 

693.  Let  us,  then,  first  consider  the  case  when  the  ray  is  incident  in  a  plane  perpendicular  to  the  common  section. 

Fig.  139.      In  this  case,  fig.  139,  let  KLMN  be  a  ray  formed  by  the  union  of  two  rays  SAaBfclKL  and  SC  KFGIIKL, 

whose  courses  through  the  system  are  similar  to  2  and  3,  fig.  138.     Draw  AD  perpendicular  to  S  C,  then  will 

the  interval  of  retardation  be  equal  to 


=  D  C  -f  (E  F  -  a  B)  +  (F  G  -  I  K)  +  2  (K  H  -  B  b), 

the  first  three  terms  being  performed  in  air,  the  last  in  glass.  Now,  without  entering  into  a  trigonometrical 
calculation,  it  is  evident  that  this  will  be  very  small  at  a  perpendicular  incidence,  and  will  increase  rapidly  as 
the  angle  of  incidence  varies  ;  and  that  (the  inclination  of  the  plates  remaining  constant)  it  will  increase  by 
nearly  equal  increments,  as  the  incidence  varies  by  equal  changes  from  0  on  either  side  of  the  perpendicular. 
Therefore,  in  a  direction  at  right  angles  to  the  common  section  of  the  surfaces  the  tints  will  vary  rapidly, 
increasing  on  either  side  of  the  perpendicular  incidence  ;  and  at  very  moderate  obliquities  on  either  side, 
the  interval  of  retardation  will  become  too  great  for  the  production  of  colour.  On  the  other  hand,  if  we 
conceive  the  rays  S  A,  S  C,  to  be  incident  in  a  plane  very  nearly  parallel  to  the  principal  section,  then  will  the 
points  K  and  G  be  situated,  not,  as  in  the  figure,  at  different  distances  from  P,  but  at  very  nearly  the  same  ;  so 
that  (whatever  be  the  incidence)  K  I  will  very  nearly  equal  GF,  and  for  the  same  reason  F  E  will  very  nearly- 
equal  aS.  Moreover,  in  this  case  GK  will  be  very  nearly  equal  to  FT,  and  the  angles  of  internal  incidence 
will  be  also  very  nearly  equal,  so  that  H  G  -f-  G  K  will  differ  very  little  from  B  b  +  b  I,  and  1  B  will  be  very 
nearly  equal  to  G  K,  and  therefore  to  I  F,  so  that  the  point  F  will  almost  exactly  coincide  with  B,  and  the  rays 
SAaB,  SCEF  will  coincide  almost  precisely,  making  D  C  =  0  ;  and  these  approximate  equalities  and  coin- 
cidences will  continue  for  great  variations  in  the  angle  of  incidence,  provided  the  plane  of  incidence  be  unaltered. 
The  interval  of  retardation,  then,  will  in  this  case  depend  very  little  on  the  angle  of  incidence  ;  so  that  in  a 
direction  parallel  to  the  common  section  of  the  surfaces,  the  tints  will  vary  but  little.  Hence  it  appears  that 
they  will  be  arranged  in  the  manner  described  by  Dr.  Brewster,  •••'-.  in  fringes  parallel  to  that  line.  Their 
general  analytical  expression  is,  however,  rather  too  complex  to  be  »....  -  .ct  down,  though  very  easily  investigated 
from  what  has  been  said. 

694.  By  intercepting   the  principal  transmitted  beam  in   the  direct  image,  and  receiving  on  the  eye  only  those 

Fig.  140.  portions  of  the  rays  going  to  form  it  whose  curves  are  as  in  fig.  140,  or  the  portions  Aa'A'oBA,  and 
AaB6'B'6,  Dr.  Brewster  succeeded  in  rendering  visible  a  set  of  coloured  fringes,  which  in  general  are  diluted 
and  concealed  in  the  overpowering  light  of  the  direct  beam.  They  originate  evidently  in  the  interference  of 
these  two  rays,  whose  courses  are  each  represented  by  4  t  +  i,  and  would  therefore  be  strictly  equal  were  the 
plates  exactly  parallel.  Their  theory,  after  what  has  been  said,  will  be  obvious  on  inspection  of  the  figure,  as 
well  as  those  of  all  the  rest  of  the  systems  of  fringes  which  he  has  described  in  that  highly  curious  and  inte 
resting  memoir. 

695  Mr.  Talbot  has  observed,  when  viewing  films  of  blown  glass  in  homogeneous  yellow  light,  and  even   in 

common  daylight,  that  when  two  films  are  superposed  on  each  other,  bright  and  dark  stripes,  or  coloured  bands 
ant'  fring6*  °f  irregular  forms,  are  produced  between  them,  though  presented  by  neither  separately.  These  are 
obviously  referable  to  the  same  principle,  the  interference  taking  place  here  between  rays  respectively  twice 
reflected  within  the  upper  lamina,  and  once  reflected  at  the  upper  surface  of  the  lower  lamina,  or  else  between 
rays  one  of  which  is  thrice  reflected  in  the  mode  represented  by  AaB'a'B'rc  A,  and  the  other  in  that  repre- 
sented by  AaB'a  A'o'A,  the  interval  between  the  glasses  being  supposed  to  be  exactly  equal  to  the  thickness 
of  the  upper  one  in  both  cases,  a  condition  which  is  sure  to  obtain  somewhere  when  the  laminse  are  curved.  A 
still  more  curious  and  delicate  case  of  the  production  of  similar  fringes  has  been  noticed  by  Professor  Amici,  to 
take  place  when  two  of  the  blue  feathers  of  the  wing  of  the  Papilio  Idas  (a  species  of  butterfly)  are  laid  on 
each  other  in  the  field  of  his  powerful  and  exquisite  microscopes.  These  feathers  he  describes  as  small  plates 
of  perfect  transparency,  and  uniformly  and  delicately  striated  over  their  whole  surface.  The  fringes  in  question 
are  formed  between  them,  and  vary  in  breadth,  form,  and  situation,  according  to  the  manner  in  which  the 
feathers  are  superposed.  Their  origin  seems  to  be  independent  of  the  striae  however,  and  is  easily  understood 
on  the  principles  above  explained.  The  same  may  be  said  of  the  colours  observed  by  Mr.  Nicholson  in  combi- 
nations of  parallel  glasses  of  unequal  thickness.  Suppose,  for  instance,  that  instead  of  the  plates  having 
exactly  equal  thicknesses,  their  thicknesses  t,  t'  differ  by  a  very  minute  quantity,  then  the  course  of  the  rays 
A  a1  A'  a  B  b  and  A  a  B  b'  B'  6  will  (at  a  perpendicular  incidence)  be  respectively  3  t  -f-  i  '•  -\-  t  '  ;md  t  +  i  -j-  3  t', 
(supposing  the  plates  strictly  parallel,)  and  the  difference  of  their  routes  is  2  I  —  2  f  ;  so  that  if  this  be  exceed- 
ingly minute,  colours  will  arise,  or,  if  not,  may  be  produced  by  a  slight  inclination  of  the  plates  to  each  other, 
and  so  of  an  infinite  variety  of  cases  which  may  arise. 


LIGHT.  477 

Light.  I'"'   "I- 

§  VI.      Of  the  Colours  of  Mired  Plates. 

The  colours  hitherto  described  have  been  referred  to  the  interference  of  rays  rigorously  coincident  with  each       ggg^ 
other  throughout  their  whole  course,  after  the  point  where  they  begin  to  be  superimposed.     Such    interfering  interference 
rays,  or  systems  of  waves,  being  united  into  a  point  on  the  retina,  that  point  is  agitated  by  the  sum  or  difference  of  rays  not 
of  their  actions,  and  the  sensation  produced  is  according.     But  if  this  coincidence  be  only  approximate,  as,  if  strictly 
two  systems  of  waves  be  propagated  from  origins  so  nearly  coincident  in  angular  situation  from  the  eye,  that  co 
their  images  formed  on  the  retina  shall  be  too  close  to  be  distinguishe%d  by  the  mind  from  the  image  of  a  single 
point,  the  impressions  produced  will  still  be  confounded  together;    or  rather,  we  ought  to  say,  the  mechanical 
action  on  one  point  will   be  propagated  through  tSe  substance  of  the  retina  to  the  other,  and  a  sensation  for 
responding  to  their  mean  or  average  effect  will  bb  produced.     If,  then,  the  rays   concentered    on   contiguous 
points  of  the  retina  be  in  exact  discordance,  and  of  equal  intensity,  a  mutual  destruction  will  take   place,  as  if 
they  fell  on  one  mathematical  point ;  if  in  exact  accordance,  they  will  increase  each  others  effects,  and  so  for  the 
intermediate  states. 

To  apprehend  this  more  fully,  we  must  consider  that  the  impression  of  light  appears  to  spread  on  the  retina      697. 
to  a  certain  extremely  minute  distance  all   around  the  mathematical  focus  of  the  rays  concentered  by  the  lenses  Irradiation. 
of  the  eye.     Thus  the  image  of  a  star  is  never  seen  as  a  point,  but  as  a  disc  of  sensible  size,  and  that  the  larger 
as  the  light  is  stronger.     Thus,  too,  the  bright  part  of  the  new  moon  is  seen,  as  it  were,  larger  than  the  faintly 
illuminated  portion  of  its  disc  projecting  beyond  it  as  an  acorn  cup  beyond  the  fruit,  &c.     This  effect  is  termed 
irradiation,  and  is  manifestly  the  consequence  of  an  organic  action  such  as  we  have  described. 

It  follows  from  this,  that  when  waves  emanate  from  origins  undistingiiishably  near,  they  may  be  regarded  in  698. 
their  effects  on  the  eye  as  emanating  from  origins  strictly  in  one  and  the  same  right  lines,  the  direction  of  the 
joint  ray;  and  the  laws  of  their  interferences  will  be  precisely  the  same,  considered  in  their  effect  on  vision,  as  if 
the  lenses  of  the  eye  were  away,  and  the  retina  were  a  mere  screen  of  white  paper,  on  a  single  physical  point  of 
which  (viz.  the  point  where  the  images  concentered  by  the  lenses  wot/Id  have  fallen)  the  interfering  undulations 
propagated  simultaneously  from  the  two  origins  fell,  and  agitated  it  with  a  vibration  equal  to  their  resultant. 

This  premised,  we  are  in  a  condition  to  appreciate  the  explanation  afforded  by  the  undulatory  doctrine  of  the  699. 
phenomena  of  mixed  plates.  They  were  first  noticed  (says  Dr.  Young)  by  him  "  in  looking  at  a  candle  through  two  Phenomena 
pieces  of  plate  glass  with  a  little  moisture  between  them.  He  thus  observed  an  appearance  of  fringes  resembling  °*  "'"" 
the  common  colours  of  thin  plates ;  pnd  upon  looking  for  the  fringes  by  reflexion,  found  that  the  new  fringes  " 
were  always  in  the  same  direction  as  the  others,  but  many  times  larger.  By  examining  the  glasses  with  a 
magnifier,  he  perceived,  that  wherever  the  fringes  were  visible,  the  moisture  was  intermixed  with  portions  of  air 
producing  an  appearance  similar  to  dew.''  "  It  was  easy  to  find  two  portions  of  light  sufficient  for  the  produc- 
tion of  these  fringes;  for  the  light  transmitted  through  the  water  moving  in  it  with  a  velocity  different  from 
that  of  light  passing  through  the  interstices  filled  only  with  air,  the  two  portions  would  interfere  with  each  other 
and  produce  effects  of  colour  according  to  the  general  law.  The  ratio  of  the  velocities  in  water  and  air  is  that 
of  three  to  four;  the  fringes  ought  therefore  to  appear  where  the  thickness  is  six  times  as  great  as  that  which 
corresponds  to  the  same  colour  in  the  common  case  of  thin  plates ;  and  upon  making  the  experiment  with  a 
plane  glass  and  a  lens  slightly  convex,  he  found  the  sixth  dark  circle  actually  of  the  same  diameter  as  the  first 
in  the  new  fringes.  The  colours  are  also  easily  produced  when  butter  or  tallow  is  substituted  for  water,  and 
the  rings  then  become  smaller  in  consequence  of  the  greater  refractive  density  of  the  oils;  but  when  water  is 
added  so  as  to  fill  up  the  interstices  of  the  oil,  the  rings  are  very  much  enlarged  ;  for  here  the  difference  ot 
velocities  in  water  and  in  oil  is  to  he  considered,  and  this  is  much  smaller  than  the  difference  between  air  and 
water.  All  these  circumstances  are  sufficient  to  satisfy  us  of  the  truth  of  the  explanation,  and  is  still  more 
confirmed  by  the  effect  of  inclining  the  plates  to  the  direction  of  the  light;  for  then,  instead  of  dilating  like  the 
colours  of  thin  plates,  these  rings  contract,  and  this  is  the  obvious  consequence  of  an  increase  of  the  lengths 
of  the  paths  of  the  light  which  now  traverses  both  media  obliquely,  and  the  effect  is  everywhere  the  same  as 
that  c/  a  thicker  plate.  It  must,  however,  be  observed,  that  the  colours  are  not  produced  in  the  whole  light 
that  is  transmitted  through  the  media  ;  a  small  portion  only  of  each  pencil  passing  through  the  water  contiguous 
to  the  edges  of  the  particle  is  sufficiently  coincident  with  the  light  transmitted  through  the  neighbouring  portions 
of  air  to  produce  the  necessary  interference  ;  and  it  is  easy  to  show  that  a  considerable  portion  of  the  light  that 
is  beginning  to  pass  through  the  water  will  be  dissipated  laterally  by  reflexion  at  its  entrance,  on  account  of 
the  natural  concavity  of  the  surface  of  each  portion  of  the  fluid  adhering  to  the  two  surfaces  of  the  glass,  and 
that  much  of  the  light  passing  through  the  air  will  be  scattered  by  refraction  at  the  second  surface.  For  these 
reasons  the  fringes  are  seen  when  the  plates  are  not  directly  interposed  between  the  eye  and  the  luminous 
object."  (Young,  Phil.  Trans.  1802;  Account  of  some  Cases  of  the  Production  of  Colours.)  To  see  the 
phenomena  to  advantage,  we  may  add,  it  is  only  necessary  to  rub  up  a  little  froth  of  soap  ami  water  almost  dry 
between  two  plane  glasses,  and  hold  them  at  a  distance  from  the  eye  between  it  and  a  candle,  or  the  reflexion 
of  the  sun  on  any  polished  convex  object.  If  two  slightly  convex  glasses,  or  a  plane  and  a  convex  one  be  used, 
the  colours  are  seen  arranged  in  rings. 


L  I  G  H  T. 

Pan  III. 

*•— ^v^— 

§  VII.   Of  the  Colours  of  Fine  Fibres  and  Striated  Surfaces. 

If  two  points  supposed  capable  of  reflecting  light  in  all  directions  (as  two  infinitely  small  spheres,  &c.)  be  so 

;nce  near  eac.i  other  as  to  appear  to  the  eye  as  one,  and  if  rays  from  a  common  origin  reflected  from  them  reach  the 

fleeted  from  eve'  tncv  w'"  interfere;    and  if  the  light  be  homogeneous,  its  intensity  will  vary  periodically,  with  an  interval  of 

point;  or      retardation  corresponding  to  the  difference  of  their  paths  ;    if  white,  the  colour  of  the  mixed  reflected  ray  will  be 

lines  very     the  same  as  if  it  had  been  transmitted  through  a  plate  of  air  of  a  thickness  equal  to  that  difference,  but  deprived 

jr each     of  jts  diluting  white.     Suppose  two  exceedingly  fine  cylindrical   polished  fibres  to  be  placed  at  right  angles  to 

f\g.'\4]       t'le  'me  °^  s'8'hl>  and  parallel  to  each  other,  as  in  fig.  141,  as  A  B  C,  a  b  c  ;  and  let  S  be  a  luminous  point  very 

distant  with  respect  to  the  interval  of  the  fibres,  and  E  the  eye,  placed  so  as  to  receive  the  reflected  rays  B  K, 

6  E,  which,  by  supposition,  are  near  enough  to  interfere.     Then  the  differences  of  phases  of  the  rays  on   the 

(S6-f  6E)  -  (SB  +  BE)  bx  +  by 

retina  is  evidently  equal  to  2  TT  x   — —  =  2  w  .  —  — ,  supposing  B  x  and  B  y 

\  A, 

perpendicular  to  S  6  and  b  E.  If,  then,  we  suppose  I  and  i  to  be  the  angles  of  incidence  of  the  rays  S  B,  E  B 
on  the  plane  in  which  the  axes  of  the  two  cylinders  AC,  ac  lie,  and  put  B  6  their  distance  equal  to  a,  we  have 
for  the  difference  of  phases 

2  TT  .    —  .  (sin  I  +  sin  i), 

\ 

Hence,  if  a  remain  the  same,  this  will  vary  with  the  obliquity  both  of  the  incident  and  reflected  ray  to  the  plane 
of  the  axes  of  the  fibres  ;  and,  therefore,  if  that  plane  be  turned  about  an  axis  parallel  to  the  fibres,  a  succession 
of  colours  analogous  to  the  transmitted  series  of  those  of  their  plates,  but  much  more  vivid,  will  be  seen,  as  if 
reflected  on  them. 

701.  Any  extremely  fine  scratch  on  a  well  polished  surface  may  be  regarded  as  havin .       concave,  cylindrical,  or, 
Colours  of     at  least,  a  curved  surface  capable  of  reflecting  the  light  equally  in  all  directions;  this  is  evident,  for  it  is  visible 

' on  in  all  directions.  Two  such  scratches,  then,  drawn  parallel  to  each  other,  and  then  turned  round  an  axis  parallel 
to  both  in  the  sunshine,  ought  to  affect  the  eye  in  succession  with  a  series  of  colours  analogous  to  those  of  thin 
plates.  This  is  really  the  ease.  Dr.  Young  found,  on  examining  the  lines  drawn  on  glass  in  Mr.  Coventry's 
micrometric  scales,  each  of  them  to  consist  of  two  or  more  finer  lines  exactly  parallel,  and  at  a  distance  of  about 
one  10,000th  of  an  inch.  Placing  the  scale  so  as  to  reflect  the  sun's  light  at  a  constant  angle,  and  varying  the 
inclination  of  the  eye,  he  found  the  brightest  red  to  be  produced  at  angles  whose  sines  were  in  the  arithmetical 
progression  1,  2,  3,  4. 

702.  In   the  beautiful  specimens  of  graduation  on   glass  and  steel   produced  by  Dr.  Wollaston,   Mr.  Barton,  and 
O'  systems    ]yj   Fraunhofer,  single  lines  exactly  parallel  to  each  other,  and  distant  in  some  cases  not  more  than  one  10,000th 

of  an  inch,  and  at  precisely  equal  intervals,  are  drawn  with  a  diamond  point.  If  the  eye  be  applied  close  to  a 
paralk-l  reflecting  or  refracting  surface  so  striated,  so  as  to  view  a  distant,  small,  bright  light  reflected  in  it,  it  will  be  seen 
lines.  accompanied  with  splendid  lateral  spectra,  which  evidently  originate  in  this  manner.  They  are  arranged  in  a 

straight  line  passing  through  the  reflected,  colourless  image,  and  at  right  angles  to  the  direction  of  the  striae. 
Their  angular  distances  from  each  other,  the  succession  of  their  colours,  and  all  their  other  phenomena,  are  in 
perfect  agreement  with  the  above  explanation.  Their  vividness  depends  on  the  exact  equality  of  distance 
between  the  parallel  lines,  which  causes  the  lateral  images  produced  by  each  pair  to  coincHe  precisely  in 
distance  from  the  principal  image,  and  thus  to  pmduce  a  multiplied  effect.  If  the  distance  of  the  lines  be 
unequal,  the  images  from  different  pairs,  not  coinciding,  blend  their  colours,  and  produce  a  streak,  or  ray  of 
white  light.  This  is  the  origin  of  the  rays  seen  darting,  as  it  were,  from  luminous  objects  reflected  on  irregularly 
polished  surfaces.  These  colouis  may  be  transferred,  by  impression  from  the  surface  originally  graduated,  to 
sealing  wax,  or  other  soft  body  ;  or  from  steel,  by  violent  pressure,  to  softer  metals.  It  is  in  this  way  that  those 
beautiful  striated  buttons  and  other  ornaments  are  produced,  which  imitate  the  splendour  and  play  of  colours 
of  the  diamond. 

703.  Dr.  Young  has  assimilated  the  colour  thus  produced   when  a  beam  of  white  light  strikes  on  a  succession  of 
parallel  equidistant  lines,  to  the  musical  tone  heard  when  any  sudden  sound  is  echoed  in  succession  by  a  series 
of  equidistant  bars  having  flat  surfaces  situated  in  a  direction  perpendicular  to  the  line  in  which  they  are  arranged, 

colours  of     f°r  instance,  an  iron  railing.     It  is  evident  that  such  echoes  will  reach  the  ear  in  succession,  at  precisely  equal 
striated         intervals  of  time,  each  being  equal  to  the  time  taken  by  sound  to  traverse  twice  the  space  separating  the  bars  ; 
surfaces        and  thus  producing  on  the  ear,  if  the  bars  be  sufficiently  numerous,  the  effect  of  a  musical  sound.   (Phil.  Trans. 
musi^a?*1"    18<^  '    ""  f^e  The°ry  of  Light  and  Colours.)     This   explanation,  however,    appears    to    us,  we   confess,  more 
tones  con-     ingenious  than  satisfactory.     The  pitch  of  the  musical  tone  produced  by  the  echoes  is  independent  of  the  sound 
sidertd.        echoed,  which  may  be  a  single  blow,  or  a  noise,  (i.  e.  a  sound  consisting  of  non-periodic  vibrations,)  and  requires 
for  its  production  a  number  of  echoing  bars  sufficient  to  prolong  the  echoes  a  sensible  time.     On  the  other  hand, 
the  light  reflected  from  parallel  striae  depends  for  its  colour  wholly  on  the  incident  ray,   being  red  in  red  light, 
yellow  in  yellow,  &c. ;  and  is  produced  equally  well  from  two  or  from  twenty,  as  from  a  million  of  such  reflecting 
lines.     The  intensity,  not  the  colour, — the  magnitude,  not  the  frequency  of  the  impression  made  on  the  retina  by 
the  reflected  ravs,  is  modified  by  their  interference.     We  think  it  necessary  to  point  out  this  defect  in  the  illus- 
tration in  question,  inasmuch  as  it  has  become  popular  for  its  ingenuity,  and  primd  facie  plausibility  ;  while,  in 
reality,  it  is  calculated  to  give  very  erroneous  impressions  of  the  analogy  between  sound  and  light. 


LIGHT.  479 

1  icht.  A  single  scratch  or  furrow  in  a  surface  may,  as  that  eminent  philosopher  has  himself  remarked,  produce  colours    Pan  HI. 

•— y— ^  by  the  interference  of  the  rays  reflected  from  its  opposite  edges.     A  spider's  thread  is  often  seen   to  gleam   in  v— >  — _• 
the  sunshine  with  the  most  vivid  colours.     These  may  arise  either  from  a  similar  cause,  or  from  the  thread  itself      704. 
as  spun  by  the  animal,  consisting1  of  several,  agglutinated  together,  and  thus  presenting  not  a  cylindrical,  but  a  Colours  of 
furrowed  surface.  •fljfcto'i 

The  phenomena  exhibited  by  light  reflected  from  and  refracted  through  the  polished  surface  of  mother  ofwe^Q^' 
pearl,  are,  no  doubt,  referable  in  great  measure  to  the  same  principle,  so  far  as  they  depend  on  the  structure  of  nlo,(jer 
of  the  surface.  Dr.  Brewster  has  described  them  in  a  most  curious  and  interesting  Paper,  (published  in  the  Of  pearl 
Phil.  Trans.  1814,  p.  397;)  and  a  writer  in  the  Edinburgh  Philosophical  Journal,  vol.  ii.  p.  117,  has  added 
some  further  particulars  illustrative  of  the  curious  and  artificial  structure  of  this  singular  body.  Every  one 
knows  thut  mother  of  pearl  is  the  internal  lining  of  the  shell  of  a  species  of  oyster.  It  is  composed  of  extremely 
thin  lamina?  of  a  tough  and  elastic,  yet  at  the  same  time  hard  and  shelly  substance,  disposed  parallel  to  the 
irregular  concavity  of  the  interior  of  the  shell.  When,  therefore,  any  portion  of  it  is  ground  and  polished  on  a 
plane  tool,  the  artificial  surface  so  produced  intersects  the  natural  surfaces  of  the  lamina;  in  a  series  of  undulating 
curves,  or  level-lines,  which  are  nearer  or  farther  asunder,  according  to  the  varying  obliquity  of  the  artificial  to 
the  natural  surfaces.  As  these  laminae  adhere  imperfectly  to  each  other,  their  feather-edges  become  broken  up 
by  the  action  of  the  powders,  &c.  used  in  grinding  and  polishing  them,  so  as  to  present  a  series  of  ridges  or 
escarpments  arranged  (when  any  very  small  portion  of  the  surface  only  is  considered)  nearly  parallel  to,  and 
equidistant  from  each  other,  which  are  distinctly  seen  with  a  microscope,  and  which  no  polishing  in  the  least 
degree  obliterates  or  impairs.  The  light  reflected,  therefore,  or  dispersed  on  their  edges,  will  interfere  and 
produce  coloured  appearances  in  a  direction  perpendicular  to  that  of  the  striae.  This  is,  in  fact,  their  situation  ; 
but  the  phenomena  are  modified  in  a  very  singular  manner  by  the  peculiar  form  of  the  edges  and  hollows, 
which  results,  no  doubt,  from  the  crystalline  structure  of  the  pearl.  That  it  is  the  configuration  only  of  the 
surface  on  which  they  depend,  is  evident  from  the  remarkable  fact,  that,  like  the  colours  described  in  Art.  702, 
they  may  be  transferred,  by  impression,  to  sealing  wax,  gum,  resin,  or  even  metals,  with  little  or  no  diminution 
of  their  brilliancy  ;  and  the  impression  so  transferred,  if  examined  by  the  microscope,  is  found  to  exhibit  a 
faithfiil  copy  of  the  original  striae,  though  sometimes  so  minute  as  hardly  to  exceed  one  3000th  of  an  inch  in 
their  distance  from  each  other.  For  a  particular  description  of  this  very  curious  and  beautiful  class  of  pheno- 
mena, however,  our  limits  oblige  us  to  refer  to  the  original  memoirs  already  cited,  especially  as  their  theory  is 
still  accompanied  with  some  obscurity. 

§  VIII.     Of  the  Diffraction  of  Light. 

When  an  object  is  placed  in  a  very  small  beam  of  light,  or  in  the  cone  of  rays  diverging  from  an  extremely      706. 
small  point,  such  as  a  sunbeam  admitted  through  a  small  pin-hole  into  a  dark  chamber,  or,  still  better,  through  Fringes 
an  opening  of  greater  size,  behind  which  a  lens  of  short  focus  is  placed,  so  as  to  form  an  extremely  minute  and  f"™e(l  e» 
brilliant  image  of  the  sun  from  which  the  rays  diverge  in  all  directions,  its  shadow  is   observed  to  be  bordered  '?"°,r  '"°  '; ' 
externally  by  a  series  of  coloured  fringes  which  are  more  distinct  the    smaller   the  angular  diameter  of  the  bodies  in  ;i 
luminous  point,  as  seen  from   the  object.     If  this  be  much  increased,   the   shadow  and  fringes  formed  by  its  small  beam 
several  points,  regarded  each   as  an  independent  luminary,  overlap  and   confuse  each   other,    obliterating  the  of  light. 
colours,  and  producing  what  is  called  the  penumbra  of  the  object ;  but  when  the  luminous  point  is  extremely 
minute,  the  shadow  is  comparatively  sharp,  and  the  fringes  extremely  well  defined. 

These  fringes  (which  were  first  described  by  Father  Grimaldi  in  a  work  entitled  Physico-Mathesis  de  Liimine,       707. 
Bologna,  16o5,  and  afterwards  more  minutely  by  Newton  in  the  third  book  of  his  Optics')  surround  the  shadows  of Tlieir 
objects  of  all  figures,  preserving  the  same  distance  from^every  part,  like  the  lines  along  the  sea-coast  in  a  map;  ^.""^'•t?'' 
only,  where  the  object  forms  an  acute,  salient  angle,  the  fringes  curve  round  it ;   and  where  it  makes  a  sharp, 
reentering  one  they  cross,  and  are  carried  up  to  the  shadow  at  each  side,  without  interfering  or  obliterating  each 
other.     In   white  light  three  only  are  to  be  seen,  whose  colours,  reckoning  from  the  shadow,  are  black,  violet, 
deep  blue,  light  blue,  green,  yellow,  red ;  blue,  yellow,  red  ;  pale  blue,  pale  yellow,  pale  rtd.     In  homogeneous 
light  they  are,  however,  more  numerous,  and  of  different  breadths,  according  to  the  colours  of  the  light,  being 
narrowest  in  violet  and  broadest  in  red  light,  as  in  the  coloured  rings  between  glasses ;  and  it  is  by  the  mutual 
superposition  of  the  different  sets  of  fring'es  for  all  the  coloured  rays  that   their  tints  are  produced,  and  their 
obliteration  after  a  few  of  the  first  orders  caused. 

The  fringes  in  question  are  absolutely  independent  of  the  nature  of  the  body  whose  shadow  they  surround,      708. 
and  the  form  of  its  edge.     Neither  the  density  or  rarity  of  the  one,  nor  the  sharpness  or  curvature  of  the  other,  Are  inde 
having  the  least  influence  on  their  breadth,  their  colours,  or  their  distance  from  the  shadow;   thus  it  is  indifferent  pendent  of 
whether  they  are  formed  by  the  edge  or  back  of  a  razor,  by  a  mass  of  platina  or  by  a  bubble  in  a  plate  of  glass,  ^slin   \he 
(which,  though  transparent,  yet  throws  a  shadow  by  dispersing  away  the   light  incident  on  it,)   circumstances  shadow, 
which   make  it  clear  that  their  origin  has  no  connection  with  the  ordinary  retractive  powers  of  bodies,  or  with 
any  elective  attraction  or  repulsions  exerted  by  them  on  light ;  for  such  forces  cannot  be  conceived  as  independent 
of  the  demity  of  the  body  exerting  them,  however  minute  we  might  regard  the  sphere  of  their  action. 

To  see  the  fringes  in  question,  they  may  be  received  on  a  smooth,  white  surface,  and  examined  and  measured       709. 
thereon  by  contrivances  which  readily  occur;  this  was  the  mode  pursued  by  Newton.     M.  Fresnel,  however,  M.  Fresnel  s 
having  (to  avoid  the  inconvenience  of  intercepting  the  light  by  the  interposition  of  the  observer)  received  them  on  an  metllod  °f 
crneried  glass  plate,  was  enabled,  by  placing  himself  behind  it,  to  approach  near  enough  to  examine  and  measure  '^fJJ 


480  LIGHT 

Light,      them  with  a  magnifier.     In  so  doing,  however,  he  observed,  that  when  thus  once  brought  under  inspection,  they    Part  III. 
— « "%— ••••  continued  visible,  and  were  indeed  much  brig-liter  and  more  distinct  in  the  focus  of  the  lens  (as  if  depicted  in  the  s^-^— ~ 
air)  even  when  the  emeried  glass  was  altogether  withdrawn  ;  and  this  fortunate  observation,  by  enabling'  him  to 
avoid  the  use  of  a  screen  altogether,  and  to  perform  all  his  measurements  of  their  dimensions  by  the  aid  of  a 
micrometer,  put  it  in  his  power  to  examine  them  with  a  degree  of  minuteness  and  precision  no  other  way  attain- 
able, and  fully  adequate  to  the  delicacy  of  the  inquiry :  for  it   is  manifest  that  the  fringes,  being  seen  as  they 
would  be  formed  if  received  on  a  screen  in  the  focus,  may  be  regarded  as  any  other  optical  image  formed  in  the 
focus  of  a  telescope,  viewed  with  any  magnifier,  and  treated  in  all  respects  as  such  images. 

710.  Whatever  mode  of  examining  them  we  adopt,  however,  we  shall  observe  the  following  facts: 

Tlieir  phe-  Phenomenon  1.  That,  ctsteris  paribm,  the  distances   from   each   other   and  from  the  border  of  the  shadow 

•t^Th  '  f"m'n'snes  as  the  screen  on  which  they  are   received,  or  the   plane  in  the  focus  of  the  lens  in  which   they  are 

'li-tances  formed,  approaches  the  border  of  the  opaque  body,  and  ultimately  coincides  with  it,  so  that  they  seem  to  have 

inter  te. '  their  origin  close  to  the  edge  of  the  body. 

711.  Phenomeno7i  2.  That  they  are  not,  however,  propagated  in  straight  lines  from  the  edge  of  that  body  to   a 
They  are  distance,  but  in  hyperbolic  curves,  having  their  vertices  at  that  edge  ;  and  therefore  that  it  is  not  one  and  the 
propagated  same  light  which  forms  one  and  the  same  fringe  at  all  distances  from  the  opaque   body.      To  explain  this, 
lines  conceive  the  distances  of  the  fringes  from  each  other  and  from  the  shadow  measured  accurately  at  a  great  variety 

of  distances  from  the  edge  of  the  body  ;  then,  were  they  propagated  in  straight  lines,  and  were  each  fringe  really 
the  axis  of  a  pencil  of  rays  emanating  from  a  point  at  that  edge,  their  intervals  and  distances  from  the  shadow 
ought  to  be  proportional  to  the  distances  from  the  edge  of  the  body;  but  it  is  not  so,  in  fact, — the  former 
distances  increasing  as  we  recede  from  the  opaque  body  much  more  rapidly  at  first,  and  less  so  as  we  recede, 
than  according  to  the  law  of  proportionality  ;  and  if  the  locus  of  each  fringe  be  laid  down  from  such  measures, 
Fig.  142.  '*•  w'"  he  found  to  be  an  hyperbolic  curve  having  its  convexity  outwards  or  from  the  shadow.  Thus  in  fig.  142 
O  is  the  luminous  point,  A  the  edge  of  the  body,  and  G  H  a  screen  perpendicular  to  the  straight  line  O  A,  C 
the  border  of  the  visible  shadow,  and  D,  E,  F  the  places  of  the  successive  minima  of  the  fringes  in  a  line  at 
right  angles  to  the  edge  of  the  shadow.  If  the  screen  be  brought  nearer  to  the  body  A  as  at  gh,  and  if  c,  d,  e,  f 
be  the  points  corresponding  to  C  D  E  F,  their  loci  will  be  the  hyperbolas  AeC,  A  d  D,  &c. 

712.  It  will  be  noticed  also  that  the  border  C  of  the  visible  shadow  is  not  coincident  with  B,  that  of  the  geometrical 
The  visible   one,  which  lies  in  the  straight  line  O  A,  grazing  the  edge  of  the  object.     The  deviation  is  difficult  to  perceive  in 
dUTersVom  '^e  shadow  of  a  large  body,  having  nothing  to  measure  from  ;  but  if  we  examine  those  of  very  narrow  bodies, 
the  geome-   as  "^  a  hair,  for  instance,  in  such  a  beam  of  light  as  described,  we  shall  find  on  measuring  the  total  breadth  of 
trical  one     the   shadow  a  full  proof  of  this.     This  fact  was  observed  by  Grimaldi.     The  limit    of  the  visible  shadow  also 
anuislarger.  follows  the  same  law  of  curvilinear  propagation   as   the  fringes.     Thus,  jNewtou  found  the  shadow  of  a  hair 

one  280th  of  an  inch  in  diameter  placed  at  12  feet  distance  from  the  luminous  point,  to  measure  at  4  inches 
from  the  hair  ^'T  inch,  or  upwards  of  4  diameters  of  the  hair,  at  two  feet,  -Jg  inch,  or  10  diameters;  while  at  10 
feet  it  measured  only  ^  inch,  or  35  diameters,  instead  of  120,  which  it  should  have  done  if  the  rays  terminating 
the  shadow  had  proceeded  in  straight  lines  ;  or  rather,  to  speak  more  correctly,  if  the  shadow  were  bounded  by 
straight  lines. 

To  account  for  these  remarkable  facts,  Newton  supposes  that  the  rays  passing  at  different  distances  from  the 

Newton  s     edges  of  bodies  are  turned  aside  outwards,  as  if  by  a  repulsive  force ;  and  that  those  nearest  are   turned  more 

thTdeflex-    asitle  than  lhose  more  remote.  as  in  fig.  143,  where  X  is  a  section   of  the   hair,  and  AD,   BE,   CF,   &c.  rays 

ion  of  light,  "hich  pass  at  diflerent  distances  beside  it,  and  which  are  turned  otf  at  angles  rapidly  diminishing  as  the  distance 

Ki».  Us!"      increases  in  directions  D  G,  E  H,  FI,  &c.     It  is  manifest  that  the  curve  W  Y  Z,  to  which   all  these  deflected 

rays  are  tangents,  and  within  which  none  can  enter,  will  be  convex  outwards  ;  and  its  curvature  will  be  greatest 

at  the  vertex  VV,   and  will   diminish   continually  as   it  recedes  from   X,    being,  in  fact,  the  caustic   of  all    the 

deflected  rays. 

714.  This  will  be  the  boundary  of  the  visible  shadow.     Tp  account  for  the  fringes,  he  supposes  (Optics,  book  iii. 
His  account  question  3)  that  each  ray  in  its  passage  by  the  body  undergoes  several  flexures  to  and  fro,  as  in  fig.  144  at  a, 

b,  c ;  and  that  the  luminous  molecules,  of  which  that  ray  consists,  are  thrown  oflT  at  one  or  other  of  the  points 
Fig°144.  °f  contrary  flexure,  or  other  determinate  points  of  the  serpentine  curve  described  by  them  according  to  the 
state  of  their  fits  in  which  they  there  happen  to  be,  or  other  circumstances;  some  outwards,  as  in  the  directions 
a  A,  b  B,  cC,  rfD,  and  others  we  may  suppose  inwards,  as  a  a,  b  /3,  c  7,  &c.  With  the  latter  we  have  here 
no  concern.  The  former,  it  is  evident,  will  give  rise  to  as  many  such  caustics  as  above  described,  as  there 
are  deflected  rays;  and  each  caustic,  when  intercepted  on  a  screen  at  a  distance,  will  depict  on  it  the  maximum 
of  a  fringe.  The  intervals,  however,  between  these  caustics,  or  minima  of  the  fringes,  .vill  not  be  totally  black ; 
because  the  rays  from  the  other  caustics,  after  crossing  on  the  confines  of  the  shadow,  or  interior  fringes,  wi  1 
pursue  their  course,  and  partially  illuminate  all  the  space  beyond.  Tims  the  fringes  should  be  less  numerous 
and  the  degradation  of  colour  more  rapid  than  in  the  coloured  rings. 

715.  This  theory  accounts  then  perfectly  for  the  curvilinear  propagation  of  the  fringes,  for  their  rapid  degradation, 
Newton's      for  their  apparently  originating  in  the  very  edge  of  the  body,  (since  each  caustic  will  actually  come   up  to  that 
anctrine        edge,  as  at  A,  fig.  142,)  and  for  the  remarkable   brightness  of  the   fringes,  especially  the    first,  which   really 
Kresnel'3       co"tains  in  itself  all  the  light  which  would  have  passed  into  the  region   B  C   between   the  visible  and  gcomc- 
objeciions     tric<il  shadows.     It  should  appear,  therefore,  that  M.  Fresncl,  in  the  objections  he  has  taken  against  these  points 
to  it  con-      of  the  Newtonian  doctrine  of  inflexion  in  his  excellent  work  .S'i/r  la  Diffraction  tit'  la  Ln/nr're,  (§  1,  p.  15,  17,  19,) 

must  have  formed  a  very  inadequate  conception  of  the  doctrine  he  opposes,  which,  if  viewed  in  the  light  he  has 
there  placed  it  in,  would  indeed  deserve  no  other  epithet  than  puerile,  and  must  be  looked  upon  as  quite  unworthy 
of  its  illustrious  author ;  and  were  these  the  only  didicuhii's  to  be  explained,  we  should  certainly  not  be  justified 


LIGHT.  481 

in  passing  a  hasty  sentence  on  it.     Other  objections  advanced  by  the  same  eminent  philosopher,  however,  are    Part  III. 
'  more  serious,  and  refer  to  a  phenomenon  of  which  the  doctrine  of  deflective  forces  seems  incapable  of  giving  v«— v-»« 
any  account ;    but  of  which,  in  justice  to  Newton  we  ought  to  add,  it  does  not  appear  that  he  was  aware,  or 
its  importance  could  not  fail  to  have  struck  him. 

Phenomenon  3.  All  other  things  remaining  the  same,  let  the  opaque  body  A  be  brought  nearer  the  luminous      716. 
point  O,  (fig.  142.)   The  fringes  then,  formed  at  the  same  distance  as  before  behind  A,  are  observed  to  dilate  con-  Dilatation 
siderablyin  breadth, — preserving,  however,  the  same  relative  distances  from  each  other,  and  from  the  border  of  the  *'  tne 
shadow.     This  fact  is  evidently  incompatible  with  the  idea  of  their  being  caused  by  any  deflecting  force  emanating  ™°Ses  °y 
from  the  opaque  body,  since  it  is  inconceivable  that  such  a  force  should  depend  on  the  distance  the  light  has  pr0ich  of 
travelled  from  another  point  no  way  related  to  the  body.  the  radiant 

To  explain  the  diffracted  fringes  on  the  undulatory  doctrine,  Dr.  Young  conceived  the  rays  passing  near  the  point. 
edge  of  the  opaque  body  to  interfere  with  those  reflected  very  obliquely  on  its  edge,  and  which  in  the  act  of      1¥l. 
reflexion  had  lost  half  an  undulation,  as  in  the  case  of  the  reflected  rings.     This  supposition  would,  in  fact,  D^  f  oung'J 
lead  us  to  conclude  the  existence  of  a  series  of  fringes  propagated  hyperbolically,   and  perfectly  resembling  " the"1" 
those  really  existing.     M.  Fresnel,  however,  has  shown  that  a  minute  though  decided  difference  exists  between  fringes  on 
their  places,  as  given  by  this  theory  and  by  direct  measurement ;  and  has,  moreover,  remarked,  that  were  this  the  undula- 
the  true  explanation,  they  could  hardly  be  supposed  absolutely  independent  of  the  figure  of  the  edge  of  the  tory  . 
opaque  body,  which  experience  shows  they  are  ;   and  that  in  cases  where  this  edge  is  extremely  sharp,  the  small  Q^""^,, 
quantity  of  light  which  could  be  reflected  from  it  would  be  insufficient  to  interfere  with  that  passing  by  it,  so  as  against"!". 
to  form  fringes  so  bright  as  we  see  them.     These  objections  appear  conclusive,  especially  as  the  supposition  of 
a  reflexion  on  the  edge  of  the  body  is  unnecessary,  since  a  more  strict  application  of  the  undulatory  doctrine, 
assisted  by  the  principle  of  interferences,  will  be  found  to  afford  a  full  and  precise  explanation  of  all  the  facts, 
regarding  the  opaque  body  as  merely  an  obstacle  bounding  the  waves  propagated  from  the  luminous  point  on 
one  side. 

To  show  this,  let  us  consider  a  wave  AMP   propagated  from  O,  and  of  which  all  that  part  to  the  right  of  A      718. 
(fig.  145)  is  intercepted  by  the  opaque  body  A  G;  and  let  us  consider  a  point  P  in  a  screen  at  the  distance  AB  Fresnel's 
behind  A,  as  illuminated  by  the  undulations  emanating  simultaneously  from  every  point  of  the  portion  AMF,  explanation 
according  to  the  theory  laid  down  in  Art.  628,  et  seq.     For  simplicity,  let  us  consider  only  the  propagation  of  Flg' 145> 
undulations  in  one  plane.     Put  A  O  =  a,  A  B  =  b,  and  suppose  X  =  the  length  of  an  undulation ;  and  drawing 
P  N  any  line  from  P  to  a  point  near  M,  put  PF=/iNM  =  s,  PB  =  x;  then,  supposing  P  very  near  to  B, 

and  with  centre  P  radius  P  M  describing  the  circle  Q  M,  we  shall  have  /=  PQ-f-  Q  N  =  */  (a-\-  !>)*-{-  x*  —a 

-f-  Q  N  =  b  -) (-  Q  N.     Now,  Q  N  is  the  sum  of  the  versed  sines  of  the  arc  s  to  radii  O  M  and  P  M, 

£  (ft-|-o) 

and  is  therefore   equal  to  ^-  +  ^  =    ^(^  +  j)  =  ^7  '  s* '   s°  that-  finally> 


-6    I  | 

~ 


2  (a  +  6)  2ab 

Now,  if  we  recur  to  the  general  expression  demonstrated  in  Art.  632,  for  the  motion  propagated  to  P  from 
any  limited  portion  of  a  wave,  we  shall  have  in  this  case  a  .  0  (0)  =  1,  because  we  may  regard  the  obliquity  of 
all  the  undulations  from  the  whole  of  the  efficacious  part  of  the  surface  A  M  N  as  very  trifling,  when  P  is  very 
distant  from  A  in  comparison  with  the  length  of  an  undulation.  And  as  we  are  now  only  considering  undu- 

lations propagated  in  one  plane,  that  expression  becomes  merely  V  =  fds  .  sin  2  TT  I  -—  —  J—  V  and  the  cor- 
responding expression  for  the  excursions  of  a  vibrating  molecule  at  P  will  be 


X  =/</..  cos  2*  _-. 
If  then  we  put  for  /"its  value,  and  take 

(t          b                cf         \  /H( 

'   T      "x  ~  2T(^+Tj/~  '  V  ~ 


and  consider  that  in  those  expressions  t  and  *  remain  constant,  while  s  only  varies,  the  latter  will  take  the 
form 


x  = 


•  {cos  0  -Sd  v-cos  (f 


which  shows  that  the  total  wave  on  arriving  at  P  may  be  regarded  as  the  resultant  of  two  waves  X'  .  cos  0 
and  X"  .  sin  0,  aiffering  in  their  origin  by  a  quarter-undulation,  and  whose  amplitudes  X'  and  X"  are  given  by 
the  expression 


the  integrals  being  taken  between  limits  of  v  corresponding  to  s  =  —  A  M,   and  s  =:  -|-  t».     Consequently, 
VOL.  iv.  8  R 


482 


LIGHT. 


Ligll!. 


sine* 

the  limits  of  v  must  b* 


Part  111. 


=~XV   (a+26) 


719. 

Rule  for 
determining 
the  illumi- 
nation of 
any  point  in 
the  screen. 

720. 
Maxima 
and  minima 
numerically 
estimated. 


Hence,  to  determine  the  intensity  of  the  light  at  any  point  P  on  the  screen,  we  must  first  of   all  calculate 
the  values  of  these  integrals;  and  having  thus  determined  X'  and  X",  the  square  root  of  the  sum  of  their  squares 


will  represent  the  amplitude  of  a  single  vibration,  the  resultant  of  both,  (Art.  615;)  and  the  sum  of 
their  squares  simply  (X'4  -f-  X//a),  the  intensity  ot'  the  light,  or  the  sensation  produced  in  the  eye. 

M.  Fresnel,  in  the  work  already  cited,  has  given  a  table  of  the  values  of  these  integrals  for  limits  succes- 
sively increasing  from  0  up  to  oo,  (at  which  latter  limit  each  is  equal  to  \,  as  may  readily  be  proved  ;)  and,  calcu- 
lating on  this,  he  finds  that  the  intensity  of  the  light,  without  the  limit  of  the  geometrical  shadow,  passes 
through  a  series  of  maxima  and  minima  according  to  the  following  table  : 

Table  of  the  Maxima  and  Minima  for  the  Exterior  Fringes,  and  of  the  Corresponding  Intensities  of  the  Light 

illuminating  them. 


Values  of». 

Intensities 
of  the  light. 

Values  of  >. 

Intensities 
of  the  light. 

First  maximum  .... 

1.2172 

2.7413 

Fourth  minimum    .  . 

3.9372 

1.7783 

First  minimum  .... 

1.8726 

1.5570 

Fifth  maximum  .... 

4.1832 

2.2206 

Second  maximum  .  . 

2.3449 

2.3990 

Fifth  minimum  .... 

4.4160 

1.8014 

Second  minimum   .  . 

2.7392 

1.6867 

Sixth  maximum  .... 

4.6069 

2.1985 

Third  maximum 

3.0820 

2.3022 

Sixth  minimum  .... 

4.8479 

1.8185 

Third  minimum  .... 

3.3913 

1.7440 

Seventh  maximum.  . 

5.0500 

2.1818 

Fourth  maximum  .  . 

3.6742 

2.2523 

Seventh  minimum  .  . 

5.2442 

1.8317 

721. 

Illumina- 
tion of  the 
border  of 
the  geome- 
trical 
shadow. 
722. 

Illumina- 
tion within 
the  ahadow. 


In  this  it  is  to  be  remarked,  that  no  minimum  is  zero,  and  that  the  difference  between  the  successive  maxima 
and  minima  diminishes  very  rapidly  as  the  values  of  v  increase,  which  explains  the  rapid  degradation  of  their 
tints. 

If  the  point  P  be  situated  on  the  very  edge  of  the  geometrical  shadow,  its  illumination  should  on  this  theory 
be  (I)2  +  (i)2  =  J.  To  compare  this  with  the  illumination  of  the  same  point,  were  the  opaque  body  removed, 
we  have  only  to  consider,  that  at  a  great  distance  from  the  shadow  the  light  must  be  the  same,  whether  the 
body  be  there  or  not.  Now  the  limit  to  which  the  maxima  and  minima  approximate  is  2,  which  therefore 
represents  the  uniform  illumination  beyond  the  fringes  ;  so  that  the  light  on  the  border  of  the  geometrical 
shadow  is  equal  to  £  of  the  full  illumination  from  the  radiant  point. 

Within  the  shadow  we  have  only  to  make  s  orv  negative.     This  does  not  alter  the  values  of  the  integrals, 


but  it  does  their  limits,  which  must  in  that  case  be  taken  not  from  v  =  — 
from  v  =  -4- 


2a 


to  4* 


2  a 


a  +  b)b\ 
to  +00.    The  computations  have  been   executed  by  M.  Fresnel,  who  finds 


723. 

Visible 
shadow 
larger    than 
•he  geome- 
trical. 
724. 


a  +  6)  6  X 

that  no  periodical  increase  or  decrease  here  takes  place,  but  that  the  light  degrades  rapidly  and  constantly 
within  the  geometrical  shadow  to  total  darkness. 

The  actual  visible  shadow  then  is  marked  by  no  sudden  defalcation  of  light,  and  it  will  depend  on  the  judgment 
of  the  eye  where  to  establish  its  termination.  If  we  regard  all  that  part  as  shadow  which  is  less  illuminated  than 
the  general  light  of  the  screen  beyond  the  fringes,  then  the  visible  shadow  will  extend  considerably  beyond  the 
geometrical  one,  and  this  explains  why  the  shadows  of  small  bodies  are  so  much  dilated,  as  we  have  seen 
they  are. 

If  we  would  know  the  breadths  of  the  several  fringes,  we  have  only  to  find  the  values  of  x  in  the  equation 


/(a  +  b)b    X 
v  a  2 


where  v  has  in  succession  the  several  values  set  down  in  the  foregoing  table.  If  we  consider  the  variation  of  x 
for  successive  values  of  a  and  6,  we  shall  see  the  origin  both  of  the  curvilinear  propagation  of  the  fringes,  and  of 
their  dilatation  on  the  approach  of  the  luminous  point.  In  fact  if  we  regard,  first,  the  relation  between  b  and  j, 
or  the  locus  of  any  fringe  regarded  as  a  curve,  having  the  line  A  B  for  an  abscissa  and  B  P  as  an  ordinate,  we 
\  /  Afl  \ 

have  £*  =  v*  —— (  b  H V  which  is  the  equation  of  an  hyperbola  having  its  convexity  outwards  and  passing 

\  ^  / 

through  A.    Secondly,  on  the  other  hand,  if  we  regard  a  as  the  variable  quantity  and  b  as  constant,  we  see  that  for 
one  and  the  same  distance  fiom  the  screen,  the  breadths  of  the  fringes  increase  as  a  diminishes ;  the  increments  of 


LIGHT.  483 

Light,      their  squares,  as  the  incident  rays  from  being  parallel  become  more  divergent,  being  directly  as  their  diver-   Part  III. 

•— v""""  gence.     Thirdly,  for   equal   values  of  X,  a,  and  6,  x  is  proportional  to  v ;  so  that  the  breadths  of  the   o°Tpralv_i   v    _' 

fringes  are  always  in  the  same  ratio  to  each  other,  and  form  a  progression  the  same  with  those  of  the  values  of 

v  in  the  foregoing  table.    Lastly,  the  breadths  of  the  fringes  for  different  coloured  rays  are  as  the  square  roots  of 

the  lengths  of  their  undulations. 

The  accordance  of  this  theory  with  experiment,  so  far  as  it  regards  the  distances  of  the  fringes  from  the  725. 
shadow  and  from  each  other,  has  been  put  to  a  severe  test  by  M.  Fresnel,  and  found  perfect.  It  were  to  be 
wished,  however,  that  he  had  stated  somewhat  more  precisely  the  instrumental  means  by  which  he  determined 
the  place  of  the  border  of  the  geometrical  shadow,  from  which  his  measures  are  all  stated  to  be  taken ;  and 
which,  being  marked  by  no  phenomenon  of  maximum  or  minimum,  might  be  liable  to  uncertainty  if  judged  of 
by  the  eye  alone.  This,  however,  in  no  way  invalidates  the  accuracy  of  the  final  conclusions,  as  the  intervals 
between  the  fringes  are  distinctly  marked,  and  susceptible  of  exact  measurement.  The  dilatation  of  the  fringes 
on  the  approach  of  the  luminous  point  is,  perhaps,  the  strongest  fact  in  favour  of  the  undulatory  doctrine,  and 
in  opposition  to  that  of  inflection,  which  has  yet  been  adduced.  It  seems  hardly  reconcilable  to  any  received 
ideas  of  the  action  of  corpuscular  forces,  to  suppose  the  force  of  deflection  exerted  by  the  edge  of  a  body 
on  a  passing  ray,  to  depend  on  the  distance  which  the  ray  has  passed  over  before  arriving  at  that  edge  from 
an  arbitrarily  assumed  origin.  M.  Fresnel  has  placed  this  argument  in  a  strong  light,  in  his  work  already  cited. 

Besides  the  exterior  fringes  above  described,  there  are  others  formed  in  certain   circumstances   within  the      726. 
shadows  of  bodies  which  afford   peculiarly  apt  illustrations  of  the  principle  of  interferences.     The  first  class  of  Fringes 
phenomena  of  this  kind  was  noticed  by  Grimaldi,  who  found  that  when  a  long,  narrow  body  is  held  in  a  small  observed  by 
diverging  beam  of  light,  the  shadow  received  on  a  screen  at  a  distance  will   be  marked  in  the  direction  of  its  "ri™ald' 
length  with   alternate  streaks  or  fringes  brighter  and  darker  than  the  rest.     These  are  more  or  less  numerous,  narrow 
according  as  the  distance  of  the  shadow  from  the  body  is  smaller  or  greater  in  proportion  to  the  breadth  of  the  shadows. 
latter.     To   study  the  phenomena  more  minutely,  Dr.  Young  passed  a  sunbeam  through  a  hole  made  with  a 
fine  needle  in  thick  paper,  and  brought  into  the  diverging  beam  a  slip  of  curd  one-thirtieth  of  an  inch  in  breadth, 
and  observed  its  shadow  on  a  white  screen  at  different  distances.     The  shadow  was  divided  by  parallel  bands, 
as  above  described,  but  the  central  line  was  always  white.     That  these  bands  originated  in  the  interference  of  Dr. Young's 
the  light  passing  on  both  sides  of  the  card,  Dr.  Young  demonstrated  beyond    all  controversy,   by  simply  fun(|an'en- 
intercepting  the  light  on  one  side  by  a  screen  interposed  between  the  card  and  the  shadow,  leaving  the  rays  f*rej,j'e r' 
on  the  other  side  to  pass  freely,  in  the  manner  represented  in  fig.   146,  where   O  is    the    hole,  A   B   the   card,  Fig.  146. 
E  F  its  shadow,  and  C  D  the  intercepting  body  receiving  on  its  margin  the  margin  of  the  shadow  of  the  edge 
B  of  the  body.      In  this  arrangement  all  the  fringes  which  had  before  uxisted  in  the  shadow  E  F  immediately 
disappeared,  although  the  light  inflected   on  the   edge  A   was  allowed  to   retain  its  course,  and  must  have 
necessarily   undergone  any  modification  it  was  capable  of  receiving  from  the  proximity  of  the  other  edge  B. 
The  same  result  took  place  when  the  intercepting  screen  was  placed  as  at  c  d  before  the  edge  B  of  the  body, 
so  as  to  throw  its  own  shadow  on  the  margin  B  of  the  card. 

Without  entering  minutely  into  the  rationale  of  this  phenomenon,  which,  however,  the  formula?  of  the  pre-      727. 
ceding  articles  enable  us  fully  to  do,  by  considering  the  illumination    of  any  point  X   between  E  and  F  as  Expla- 
arising  from  the  whole  wave  a  A  B  6,  minus  the  portion  A  B,  and  which  M.  Fresnel  has  done  at  full   length,  nation. 
and  with  great  success,  in  his  Memoir  already  so  often  cited ;    we  shall  content  ourselves  with  showing  how 
fringes  or  alternations  of  colour  must  originate  in  such   circumstances;    in   fact,   if  we  join  AX,  B  X,  the 
difference  of  routes  of  the  waves  arriving  at  X   by  the  paths  O  AX,  OB  X  is  equal  to  B  X  —  A  X.     It  is 
therefore  nothing  in  the  middle  of  the  shadow  E  F,  which  ought  therefore  to  be  illuminated  by  double  the  light 
deflected  into  the  shadow  at  that  distance  by  either  edge,  Art.  722,  which  will  be  less  as  the  object  is  larger, 
and  the  shadow  broader.     But  on  either  side  of  the  middle  B  X  —  A  X  increases ;    and  when  it  attains  a  value 
equal  to  half  an  undulation,  the  waves  are  in  complete  discordance,  and  therefore  the  middle  bright  portion  will 
be  succeeded  by  a  dark  band  on  either  side,  and  these  again  by  bright  ones,  and  so  on. 

An  elegant  variation  of  this  experiment  of  Dr.  Young  is  afforded  by  a  phenomenon  described  by  Grimaldi.       728. 
When  a  shadow  is  formed  by  an  object  having  a  rectangular  termination ;    besides  the  usual  external  fringes  Grima'.di's 
there  are  two  or  three  alternations  of  colours,  beginning  from  the  line  which  bisects  the  angle,  disposed,  within  crested 
the  shadow  on  each  side  of  it,  in  curves  which  are  convex  towards  the  bisecting  line,  and  which  converge  towards  frmSes' 
it  as  they  become  remote  from  the  angular  point.     These  fringes  are  the  joint  effect  of  the  light  spreading  into 
the  shadow  from  each  outline  of  the  object,  and  interfering  as  above ;  and  that  they  are  so,  is  proved  by  placing 
a  screen  within  a  few  inches  of  the  object,  so  as  to  receive  only  one  edge  of  the  shadow,  when  the  whole  of  the 
fringes  disappear.     If,  on  the  other  hand,  the  rectangular  point  of  the  screen  be  opposed  to  the  point  of  the 
shadow,  so  as  barely  to  receive  the  angle  of  the  shadow  on  its  extremity,  the  fringes  will  remain  undisturbed. 
(Young,  Experiments  and  Calculations  relating  to  Physical  Optics,  Phil.  Trans.,  1803.) 

Such  are  some  of  the  more  remarkable  appearances  produced  within  and  beyond  the  shadows  of  narrow       729. 
bodies.     Let  us  next  consider  the  effect  of  transmitting  a  beam  through  a  very  narrow  aperture.     And  the  first  J^e  °( 
case  is  when  the  aperture  is  circular.     Suppose,  for  instance,  we  place  a  sheet  of  lead,  having  a  small  pin-hole  through"* 
pierced  through  it,  in  the  diverging  cone  of  rays  from  the  image  of  the  sun,  formed  by  a  lens  of  short  focus,  and  small 
in  the  line  joining  the  centres  of  the  hole  and  focus  prolonged  place  a  convex  lens  or  eye-glass,  behind  which  circular 
the  eye  is  applied.     The  image  of  the  hole  will  be  seen  through  the  lens  as  a  brilliant  spot,  encircled   by  rings  aperture, 
of  colours  of  great  vividness,  which  contract  and  dilate,  and  undergo  a  singular  and  beautiful  alternation  of  tints, 
as  the  distance  of  the  hole   from  the  luminous  point  on  the  one  hand,  or  on  the  eye-glass  on  the  other,  is 
changed.     When  the  latter  distance  is  considerable,  the  central   spot  is  white,  and  the  rings  follow  nearly  the 
order  of  the  colours  of  thin  plates.     Thus,  when  the  diameter  of  the  hole  was  about  T'ffth  of  an  inch,  its  distance 

3  a2 


484 


LIGHT. 


Light. 


(a)  from  the  luminous  point  about  6  feet  6  inches,  and  its  distance  (6)  from  the  eye-lens  24  inches,  the  series   P»rt  HI. 
of  colours  was  observed  to  be,  '"— V»" 

1st  order.     White;  pale  yellow  ;  yellow;  orange;  dull  red. 


2d  order, 
brilliant. 
3rd  order. 
4th  order. 
5th  order. 
6th  order. 
7th  order. 


Violet ;  blue  (broad  and  pure ;)  whitish  ;  greenish  yellow  ;  fine  yellow  ;  orange  red,  very  full  and 


Purple  ;  indigo  blue ;  greenish  blue ;  pure,  brilliant  green  ;  yellow  green  ;  red. 
Good  green,  but  rather  sombre  and  bluish  ;  bluish  white  ;  red. 
Dull  green  ;   faint  bluish  white  ;  faint  red. 
Very  faint  green  ;  very  faint  red. 
A  trace  of  green  and  red. 

When  the  eye-lens  and  hole  are  brought  nearer  together,  the  central  white  spot  contracts  into  a  point  and 
vanishes,  and  the  rings  gradually  close  in  upon  it  in  succession,  so  that  the  centre  assumes  in  succession  the 
t  most  surprisingly  vivid  and  intense  hues.  Meanwhile  the  rings  surrounding  it  undergo  great  and  abrupt  changes 
and  sur-P°  'n  4ne'r  tints.  The  following  were  the  tints  observed  in  an  experiment  made  some  years  ago,  (.July  12,  1819,) 
rounding  the  distance  between  the  eye-glass  and  luminous  point  («  +  6)  remaining  constant,  and  the  hole  being  gradually 
rings.  brought  nearer  to  the  former. 


730. 

Table  of 
colours  of 


24.00 
18.00 

13.50 

10.00 
9.25 
9.10 
8.75 
8.36 
8.00 
7.75 
7.00 
6.63 

6.00 

5.85 
5.50 
5.00 
4.75 
4.50 
4.00 
3.85 
3.50 


Central  Spot. 


White 
White 

Yellow 

Very  intense  orange 
Deep  orange  red 
Brilliant  blood  red 
Deep  crimson  red 
Deep  purple 
Very  sombre  violet 
Intense  indigo  blue 
Pure  deep  blue 
Sky  blue 

Bluish  white 

Very  pale  blue 

Greenish  white 

Yellow 

Orange  yellow 

Scarlet 

Red 

Blue 

Dark  blue 


Surrounded  by 


Rings  as  in  the  foregoing  Article. 

The  two  first  rings  confused,  the   red  of  the  3rd  and  green  of  the  4th 
orders  splendid. 

Interior  rings  much  diluted,  the  4th  and  5th  greens  and  3rd,  4th  and  5th 
reds  the  purest  colours. 

All  the  rings  are  now  much  diluted. 

The  rings  all  very  dilute. 

The  rings  all  very  dilute. 

The  rings  all  very  dilute. 

The  rings  all  very  dilute. 

A  broad  yellow  ring. 

A  pale  yellow  ring. 

A  rich  yellow. 

A  ring  of  orange,  from  which  it  is  separated  by  a  narrow,  sombre  space. 
r  Orange  red,  then  a  broad  space  of  pale  yellow,  after  which  the  other  rings 
I          are  scarcely  visible. 

A  crimson  red  ring. 

Purple,  beyond  which  yellow  verging  to  orange. 

Blue,  orange. 

Bright  blue,  orange  red,  pale  yellow,  white. 

Pale  yellow,  violet,  pale  yellow,  white. 

White,  indigo,  dull  orange,  white. 

White,  yellow,  blue,  dull  red. 

Orange,  light  blue,  violet,  dull  orange. 


this  case. 


731 .  The  series  of  tints  exhibited  by  the  central  spot  is,  evidently,  so  far  as  it  goes,  that  of  the  reflected  rings  in  the 
Frcsnel's  colours  of  thin  plates.  The  surrounding  colours  are  very  capricious,  and  appear  subject  to  no  law.  They  depend, 
analysis  of  indeed,  on  very  complicated  and  unmanageable  analytical  expressions,  with  which  we  shall  not  trouble  the  reader, 
but  content  ourselves  with  presenting  the  explanation  given  by  M.  Fresnel  of  the  changes  of  tint  of  the  central  spot 
in  white  light,  and  its  alternations  of  light  and  total  darkness  observed  by  him  in  an  homogeneous  illumination. 
Let  then  a  and  6  be  the  distances  of  a  small  hole  whose  radius  is  r  from  the  luminous  point,  and  a  screen 
placed  behind  the  hole  perpendicularly  to  the  ray  passing  directly  through  its  centre.  Then  if  we  consider  any 
infinitely  narrow  annulus  of  the  hole  whose  radius  is  z,  and  breadth  d  z,  this  annulus  will  send  to  the  central 
point  of  the  screen  a  system  of  waves  whose  intensity  is  proportional  to  the  area  of  the  annulus,  or  2  <n  z  dz, 
but  whose  phase  of  undulation  differs  from  that  of  the  central  ray,  by  reason  of  the  difference  of  the  paths 
described  by  them.  Now,  calling/  the  distance  of  each  point  in  the  annulus  from  the  centre  of  the  screen,  we 
have/2  =  6*-f-  zs,  and,  in  like  manner,  if  f  be  the  distance  of  the  luminous  point  from  the  same  annulus, 
f*  —  a*  -f  x*,  so  that  (/  +  f)  —  (a  -f-  6)  the  difference  of  paths,  or  interval  of  retardation,  is  equal  to 

— -{ —  -| )  =  — —^-—?.     Hence,  the  general  expression  in  Art.  632  for  the  amplitude  of  the  total  wave, 

.  2  \  a          6  /  -2  ab 

incident  on  the  centre  of  the  screen  in  this  particular  case,  is  equivalent  to 

\t        z'(s  +  &)). 
"If" 


LIGHT. 

or.  integiuting,  which  from  the  peculiar  form  of"  the  differential  is  in  this  case  easy, 

a  b 


__ 
X  = 


a-\-b 

which,  extended  from  z  =  0  to  2  =  r,  gives 

a  6  \ 


(  (t         z*  (a  +  6)\  1 

-<  const  +  cos  2  w  .  I  —  --  -  —  —  —  -  I   > 
(.  \T  2ab\   /  J 


o  6  \    C  .    TT  (a  +  V)  r*  t        /       Tr(a+b)r>        \  t 

-  _     -  -|  sin     v        /      .  sin  2  TT-  +  (  cos  --  '—  —  -  1  )  .  cos  2  ir  — 
a+  b   (_  a  6  X  T-        \  •  »  X  / 

This  expresses,  as  we  have  before  remarked  in  a  similar  case,  (Art.  718,)  two  partial  waves  differing  by  a  quarter- 
undulation,  and  expressing  it,  as  in  that  case,  by  X  =  X'  .  cos  0  -f-  x"  •  sin  9>  where  0  =.  —  ,  we  find  for  the 
intensity  A*  of  their  resultant 


\a  +  b 

To  make  use  of  this,  however,  we  must  compare  it  with  what  would  be  the  direct  illumination  of  the  centre       732. 
of  the  screen,  if  the  aperture  were  infinite,  t.  e.  if  the  direct  light  from  the  luminous  point  shone  full  upon  it.  |'^"'fn^e 
To  this  case,  however,  neither  our  formula  nor  our  reasoning  are  applicable  ;   for  if  we  make  r  infinite  in  this  centrai  spot 
expression,  it  becomes  illusory,  and  presents   no  satisfactory  sense,  and  in  our  reasoning  we  have  neglected  to  compared 
consider  the  law  of  diminution  of  the  intensity  of  the  oblique  waves,  or  regarded  0  (0)  in  Art.  631  as  invariable,  with  the 
which  in  this  extreme  case  is  far  from  the  truth.     We  must,  therefore,  have  recourse  to  another  method.     Now,  J^^J'1"111" 
M.  Fresnel  has  demonstrated   (and  our  limits  oblige  us  to  take  his  demonstration  for  granted)  that   this  total  i?reslle'i>s 
illumination  is  equal  to  one-fourth  of  that  which  the  centre  of  the  screen  would   receive  from  an   opening   of  theorem. 
such  a  radius,  that  the  difference  of  routes  of  a  ray  passing  through  the  centre,  and  one  diffracted  at  the  circum- 

ference, shall  be  an  exact  semi-undulation,   i.  e.    in  which  -  ~  —  =  —  ,  or  r  =  \/  -     :  —  ;•     If  then  we 

2  a  b  2  a  -f-  o 

substitute  this  for  r  in  the  above  formula,  and  put  C  for  the  whole  illumination,  we  get,  071  the  same  scale, 

.  /«6XV 

c=+»     sm^ 

and,  consequently, 


,  /3V 

In  this  expression  r,  a,  b  are  independent  of  \,  and  therefore  the  value  of  A8  is  of  the  form  4  C  (  sin  2  TT  .  —  I        733. 

\  \/   The  colours 

la  -L.  M  r«  those  of  the 

where  B  =  -  —  —•  -£  —  .     Hence,  if  we  suppose  light  of  all  colours  to  emanate  from  the  luminous  point,  the  reflected 
4  a  b  rings. 

compound  tint  produced  in  the  central  point  of  the  screen  will  be  represented  by  S  •<  4  C  .  I  sin  2  it  —  \  r  and 

will  therefore,  by  Art.  673,  be  the  same  with  that  reflected  by  a  plate  of  air  whose  thickness  is  B,  or  —  -  —  —  — 

which  increases  as  b  diminishes  when  a  -f-  b  remains  constant.  Thus  we  see  the  origin  of  the  succession  of 
colours  of  the  central  spot  in  the  Table  above  recorded,  which  is  the  more  satisfactory,  as  that  experiment  was 
made  without  reference  to,  and  indeed  in  ignorance  of,  this  elegant  application  of  M.  Fresnel's  general  principles, 
the  merit  of  which  is  due  (as  he  himself  states)  to  M.  Poisson.* 

Another  very  curious  result  of  M.  Poisson's  researches  is  this,  that  the  centre  of  the  shadow  of  a  very  small       734. 
circular  opaque  disc,  exposed  to  light  diverging  from  a  single  point,  is  precisely  as  much   illuminated   by  the  Poisson's 
diffracted  waves  as  it  would  be  by  the  direct  light,  if  the  disc  were  altogether  removed.     We  cannot  spare  room  ^eo.r,?ln  !°r 
for  the  demonstration  of  this  singular  theorem.     It  has  been  put  to  the  test  of  experiment  by  M.  Arago,  with  natjonUj™t5,e 
a  small  metallic  disc  cemented  on  a  very  clear  and  homogeneous  plate  of  glass,  and  with  full  success.  centre  of  a 

When  the  light  is  transmitted  through  two  equal  apertures,  placed  very  near  each  other,  the  rings  are  formed  small  circii- 
about  each  as  in  the  case  of  one  ;  but  besides  these  arise  a  set  of  narrower,  straight,  parallel  fringes  bisecting  lar  shadow. 
the  interval  between  their  centres,  and  at  right  angles  to  the  line  joining  them.     If  the  apertures  be  unequal,      "35. 
these  fringes  assume  the  form  of  hyperbolas,    having    the  aperture  in  their  common  focus.      Besides  these  JH^j 
also  two  other  sets  of    parallel  rectilinear  fringes  (in  the  ease  of    equal  apertures)  go    off  in    the  form    of  iiiroug(,  two 
a  St.  Andrew's  cross  from  the  centre  at  equal  angles  with  the  first  set.     See  figures  147,  148.     When  the  apertures 
apertures  are  more  numerous  or  varied  in  shape,  the  variety  and  beauty  of  the  phenomena  are  extraordinary  ;  v"y  near 
but  of  this  more  presently.  Fi     M?"' 

M.  Fresnel  has  shown,  that  when  the  light  from  a  single  luminous  point  is  received  on  two  plane  mirrors  ^143 

•  The  coincidence  in  the  higher  orders  of  colours  was,  however,  ic  our  experiments  less  complete,  and  especially  the  green  of  the  third 
order,  which  was  wanting  altogether  in  some  cases. 


486 


LIGHT. 


very  slightly  inclined  to  each  other,  so  as  to  form  two  almost  contiguous  images,  if  these  be  viewed  with  a 
lens,  there  will  be  seen  between  them  a  set  of  fringes  perpendicular  to  the  line   joining  them.     These    are 
evidently  analogous  to  those  produced  by  the  two  holes  in  the  experiments  last  described.     The  experiment  is 
^e<icate  ?  f°r  >f  tne  surfaces  of  the  reflectors  at  the  point  where  they  meet  be  ever  so   little,  the  one  raised 
above  or  depressed  below  the  other,  so  as  to  render  the  difference  of  routes  of  the  rays  greater  than  a  very  few 
undulations,  no  fringes  will   be  seen.     But  it  is  valuable,  as  demonstrating  distinctly  that  the  borders  of  the 
inclined  to   apertures  in  the  preceding  experiment  have  nothing  to  do  with  the  production  of  the  fringes,  the  rays  being  in 
each  other,   this  case  abandoned  entirely  to  their  mutual  action  after  quitting  the  luminous  point.     An  exactly  similar  set  of 


Part  III. 


Fig.  149. 


•ct  of 

a  denser"1* 
medium  i 


Fig.  150. 
Displace- 


fringes  is  formed  if,  instead  of  two  reflectors,  we  use  a  glass,  plane  on  one  side,  and  on  the  other  composed  of 
two  planes,  forming  a  very  obtuse  angle,  as  in  fig.  149.  This  being  interposed  between  the  eye-lens  E  and  the 
luminous  point  S,  forms  two  images  S  and  S'  of  it  ;  and  the  interference  of  the  rays  S  E  and  S'  E  from  these 
images,  forms  the  fringes  in  question. 

Since  the  production  of  the  fringes  and  their  places  with  respect  to  the  images  of  the  luminous  point,  depends 
on  the  difference  of  routes  of  the  interfering  rays,  it  is  evident,  that  if,  without  altering  their  paths,  we  alter 
'^e  re'oc''y  °f  one  °f  them  with  respect  to  the  other,  during  the  whole  or  a  part  of  its  course,  we  shall  produce 
*'le  same  effect.  Now,  the  velocity  of  a  ray  may  be  changed  by  changing  the  medium  in  which  it  moves.  In  the 
one  of  two  undulatory  system,  the  velocity  of  a  ray  in  a  rarer  medium  is  greater  than  in  a  denser.  Hence,  if  in  the  path  of  one 
interferin  of  two  interfering  rays  we  interpose  a  parallel  plate  of  a  transparent  medium  denser  than  air.  (at  right  angles 
to  the  ray's  course,)  we  shall  increase  its  interval  of  retardation,  or  produce  the  same  effect  as  if  its  course  had 
been  prolonged.  If  then  a  thick  plate  of  a  dense  medium,  such  as  glass,  be  interposed  in  one  of  the  rays 
which  form  visible  fringes,  they  will  disappear  ;  because  the  interval  of  retardation  will  be  thus  rendered  suddenly 
equal  to  a  great  number  of  undulations,  whereas  the  production  of  the  fringes  requires  that  the  difference  of 
routes  shall  be  very  small.  If,  however,  only  a  very  thin  lamina  be  interposed,  they  will  remain  visible,  but 
shift  their  places.  Thus,  in  fig.  150,  let  S  A,  S  B  be  rays  transmitted  through  the  small  apertures  A,  B  from  the 
luminous  point  S,  and  received  on  the  screen  D  C  E,  these  forming  a  set  of  fringes  of  which  C,  the  middle  one, 
will  be  white.  Let  D,  E  be  the  dark  fringes  immediately  adjacent  on  either  side  ;  and  things  being  thus  disposed, 
j.".ennt  of  th*  let  a  thin  film  of  glass  or  mica  G  be  interposed  in  one  of  the  rays  S  A,  its  thickness  being  such  that  the  ray  in 
exnUuneif  '  *raversin&  •*  shall  just  be  retarded  half  an  undulation.  Then  will  the  rays  A  E,  B  E,  which  before  were  in  com- 
plete discordance,  be  now  in  exact  accordance,  and  there  will  be  formed  at  E  a  bright  fringe  instead  of  a  dark 
one.  On  the  other  hand,  the  ray  AC  will  now  be  half  an  undulation  behind  BC,  instead  of  in  complete 
accordance  with  it,  so  that  at  C  there  will  be  formed  a  dark  fringe,  and  so  on.  In  other  words,  the  whole 
system  of  fringes  will  be  formed  as  before,  but  will  have  shifted  its  place,  so  as  to  have  its  middle  in  E  instead 
of  in  C,  z.  e.  will  have  moved  from  the  side  on  which  the  plate  of  the  dense  medium  is  interposed.  It  is  evident, 
that  if  the  plate  G  be  thicker,  the  same  effect  will  take  place  in  a  greater  degree. 

To  make  the  experiment,  however,  it  must  be  considered  that  the  refractive  power  of  glass,  or  indeed  of  any 
Mo^eof        but  gaseous  media,  is  so  great,  that  any  plate  of  manageable  thickness  would  suffice  to  displace  the  fringes  so 
he'testof  ^ar  as  '°  '"row  them  wholly  out  of  sight.     But  we  shall  succeed,  if,  instead  of  a  single  plate  G  placed  over  one 
experiment  aperture  A,  we  place  two  plates  G,  g  of  very  nearly  equal  thicknesses,  (such  as  will  arise  from  two  nearly  con- 
tiguous fragments  of  one  and  the  same  polished  plate,)  one  over  each  aperture  ;  or  we  may  vary  the   thickness 
of  the  plate  traversed  by  either  ray  by  inclining  it,  so  as  to  bring  it  within  the  requisite  limits.     This  done,  the 
effect  observed  is  precisely  that  described  ;  the  fringes  shift  their  places  from  the  thicker  plate,  without  sustaining 
any  alteration  in  other  respects.     This  elegant  experiment  affords  a  strong   indirect  argument  in  favour  of  the 
undulatory  system,  and  in  opposition  to  that  of  emission,  since  it  proves  that  the  rays  of  light  are  retarded  in 
t'le'r  Passage  through  denser  media,  agreeably  to  what  the  undulatory  system  requires,   and  contrary  to   the 
conclusions  of  the  corpuscular  doctrine. 

MM.  Arago  and  Fresnel  have  taken  advantage  of  this  property,  to  measure  the  relative  refractive  powers  of 
different  gases,  or  of  the  same  in  different  states  of  temperature,  pressure,  hu:i.idity,  &c.     It  is  manifest,  that  if 
me   o  o      an^  considerable  portion  of  the  path  of  one  of  the  interfering  rays  be  made  to  pass  through  a  tube  closed  at 
determining  '3Otn  ends  with  glass  plates,  and  the  other  through  equal  glass  plates  only,  the  fringes  will  be  formed  as  usual. 
refractions    But  if  the  tube  be  exhausted,  or  warmed,  or  cooled,  or  filled  with  a  gas  of  different  refractive  density,  a  displace- 
of  ga»e*.      ment  of  the  fringes  will  take  place,  which  (if  they  be  received  in  the  focus  of  a  micrometer)  may  be   measured 
with  the  greatest  delicacy.     Knowing  the  amount  of  their  displacement,  as  compared  with  the  breadth  of  the 
fringes,  we  know  the  number  of  undulations  gained  or  lost  by  one  ray  on  the  other  ;   and  hence,  knowing  the 
internal  length  of  the  tube,  we  have  the  ratio  of  the  refracting  power  of  the  medium  it  contains  to  that  of  air. 
What  renders  this  method  remarkable  is,  that  there  is  actually  no  conceivable  limit  to  the  precision  of  which  it  it 
susceptible,  since  tubes  of  any  length  may  be  employed,  and  micrometers  of  any  delicacy. 

740.  The  phenomena  of  diffraction,  and  those  arising  from  the  mutual  interference  of  several  very  minute  pencils 

Praunhofer's  of  rays  emanating  from  a  common  origin,  have  been  investigated  by  M.  Fraunhofer  with  great  care  and  extra- 
experiments  ordinary  precision,  by  the  aid  of  a  very  delicate  apparatus  devised  and  executed  by  himself. 

This  apparatus  consisted  of  a  repeating,  12-inch  theodolite,  reading  to  every  4",  carrying,  attached  to  its 
terference.  horizontal  circle,  a  plane  circular  disc  of  six  inches  in  diameter,  having  its  axis  precisely  coincident  with  that  of 
the  theodolite,  and  having  its  own  particular  divisions  independent  of  those  of  the  theodolite.  In  the  centre  of 
this  disc  was  placed  vertically  a  metallic  screen,  having  in  it  one  or  more  narrow,  vertical,  rectangular  slits,  or 
other  apertures,  and  so  fixed  as  to  have  the  middle  of  its  aperture,  or  system  of  apertures,  exactly  coincident  with 
the  axis  of  the  instrument.  Attached  to  the  great  circle  of  the  theodolite,  horizontally,  was  a  telescope,  having  its 
object-glass  three  inches  and  a  half  from  the  centre,  and  its  axis  directed  exactly  to  it,  and  precisely  parallel  to 
the  plane  of  the  limb,  and  provided  with  a  delicate  micrometer,  whose  parallel  threads  were  exactly  vertical. 


Argument 

system 

739. 

Arago  and 

met  hod  of 


His  appa- 
ratus. 


LIGHT.  487 

Light.      The  instrument  being  insulated   on  a  support  of  stone,  a  beam  of  solar  light  was  directed    by  a    heliostat,     Part  III. 
»-v— ••  through  a  very  narrow  slit,  also  exactly  vertical,  having  a  breadth  of  one  hundredth  of  an  inch,  and  distant  463J  ^- v-^-' 
inches  from  the  centre  of  the  theodolite,  so  as  to  fall  on  the  screen,  and,  being  transmitted  through  its  apertures, 
to  be  received  into  the  telescope.     It  is  manifest  that  the  eye-glass  of  the  telescope  will  here  view  the  fringes,  &c. 
as  they  are  formed  in  its  focus.     The  magnifying  power  of  the  telescope  used   by  Fraunhofer  varied  from  30  to 
50  times. 

M.  Fraunhofer  first  examined  the  effect  produced  by  the  diffraction  of  the  light  through  a  single  slit, — the       741. 
breadth  of  which  he  first  determined  with  the  greatest  precision  by  means  of  a  micrometer-microscope,  with  Fringes 
which  he  assures  us  that  he  found  it  practicable  to  appreciate  so  minute  a  quantity  as  1 -50,000th  of  an  inch.  The  £' 
slit  being  then  placed  on  the  apparatus,  and  accurately  adjusted   before  the  object-glass  of  the  telescope,  which  narrow 
was  directed  exactly  to  the  aperture  in  the  heliostat,  the  image  of  the  latter  was  formed  in  its  focus,  accompanied  aperture, 
by  lateral  fringes,  which  by  the  effect  of  the  magnifying  power  were  dilated  into  broad  and  brilliant  prismatic 
spectra.     The  distances  of  the  red  ends  of  these  spectra  from  the   middle  point,  or  white  central  image,  were 
then   measured  accurately  by   means  of  the  micrometer.     The   result  of  a  great  number  of  experiments  with 
apertures  of  all  breadths  from  one-tenth  to  one-thousandth  of  an  inch,  agreed  to  astonishing  precision  with  each 
other,  and  with  the  following  laws,  viz.  that  (under  the  circumstances  of  the  experiment,) 

1.  The,  angles  of  deviation  of  the  diffracted  rays,  forming  similar  points  of  the  systems  of  fringes  produced  Their  laws 
by  different  apertures,  are  inversely  as  the  breadths  of  the  apertures. 

2.  That  the  distances  of  similar  rays  (the  extreme  red,  for  instance,)  from  the  middle  in  the  several  spectra,  s" 
constituting  the  successive  fringes,  form  in  each  case  an  arithmetical  progression  whose  difference  is  equal  to  its 

first  term. 

3.  That  calling  7  the  breadth  of  the  aperture,  in  fractions  of  a  Paris  inch,  the  angular  distances  L',  L",  L'", 
&c.  in  parts  of  a  circular  arc  to  radius  unity,  of  the  extreme  red  rays  in  each  fringe  from  the  middle  line,  are 

respectively  represented  by  L'  =  — ,  L"  =  2  .  — ,  L'"=3  .  — ,  &c.  where  L  =  0.0000211,  and  a  similar  law 

7  7  Tf 

holds  for  all  the  other  coloured  rays,  different  values  being  assigned  to  L  for  each. 

This  conclusion  agrees  perfectly  with  the  result  of  an  experiment  related  by  Newton  in  the  Hid  Book  of  his       742. 
Optics.   He  ground  two  knife  edges  truly  straight,  and  placed  them  opposite  to  each  other,  so  as  to  be  in  contact  Newton's 
at  one  end,  and  at  the  other  to  be  at  a  small  distance,  such  that  the  angle  included  between  them  was  about  e^P'nrr- 
1°  54',  thus  forming  a  slit  whose  breadth  at  their  intersection  was  evanescent,  and  at  4  inches  from  that  point  kn;fe  edges 
•|th  of  an  inch,  and  in  the  intermediate  points,  of  course,  of  every  intermediate  magnitude.     Exposing  this  in  a 
sunbeam  emanating  from  a  very  small   hole  at  15  feet  distance,  he  received  their  shadows   on  a  white  screen 
behind  them,  and  observed  that  when  they  were  received  very  near  to  the  knife  edges,  (as  at  half  an  inch,)  the 
fringes  exterior  to  (he  shadow  of  each  edge  ran  parallel  to  its  border  without  sensible  dilatation,  till  they  met  and 
joined  without  crossing,  at  angles  equal  to  that  contained  between  the  knife  edges.     But  when  the  shadows  were 
received  at  a  great  distance  from  the  knives,  the  fringes  had  the  form  of  hyperbolas,  having  for  one  asymptote  the 
shadow  of  the  knife  to  which  they  respectively  belonged,  and  for  the  other  a  line  perpendicular  to  that  bisecting  the 
angle  of  the  two  shadows,  each  fringe  becoming  broader  and  more  distinct  from  the  shadow  which  it  bordered,  as  it 
approached  the  angle.     These  hyperbolas  crossed  without  interfering,  as   represented  in  fig.  151.     Their  points  Fig- 151 
of  crossing,  Newton  found,  however,  not  to  be  at  a  constant  distance  from  the  angle  included  between  the  pro- 
jections of  the  knife  edges,  but  to  vary  in   position  with  the  distance  from  the  knives,  at  which  the  shadow  is 
received  on  the  screen  ;   and  hence,  he  says,  "  I  gather  that  the  light  which  makes  the  fringes  upon  the  paper,  is 
not  the  same  light  at  all  distances  of  the  paper  from  the  knives  ;  but  when  the  paper  is  held  very  near  the  knives, 
the  fringes  are  made  by  light  which  passes  by  their  edges  at  a  less  distance,  and   is  more  bent  than  when  the 
paper  is  held  at  a  greater  distance  from  the   knives."     Newton,  however,  left  these  curious   researches,  which 
could  hardly  have  failed  to  have  led  in  his  hands  to  a  complete  knowledgeof  the  principles  of  diffraction — unfinished ; 
being,  as  he  says,  interrupted  in,  and  unwilling  to  resume  them  :  doubtless,  owing  to  the  chagrin  and  opposition 
his  optical  discoveries  produced  to  him.     An  unmeet  reward,  it  must  be  allowed,  for  so  noble  a  work,  but  one  of 
which,  unhappily,  the  history  of  Science  affords  but  too  many  parallels. 

The  above  were  the  results  obtained  by  M.  Fraunhofer  when  the  two   edges  of  the  aperture  were  both  in  a       743 
plane  perpendicular  to  the  incident  rays  ;  but  when  the  same  effective  breadth  was  procured,  by  inclining  a  larger  Case  wlleB 
aperture  obliquely,  so  as  to  reduce  its  actual  breadth  in  the  ratio  of  the  cosine  of  its  incidence  to  radius,  or  by  Jjj 
limiting  the  incident  ray  by  two  opaque  edges  at  different  distances  from  the  object-glass  of  the  telescope,  the  we,^/  "' 
phenomena  were  very  different.     To  accomplish  this,  two  metallic  plates  were  fixed  upright  on  the  horizontal  different 
plate  of  the  theodolite,  having  their  edges  exactly  vertical,  and  precisely  at  opposite  extremities  of  a  diameter,  distances 
Then,  by  turning  the  plate  round  on  its  axis,  the  passage  allowed  to  the  light  between  them  could  be  increased  or  fro™  th* 
diminished  at  pleasure.     The  phenomena,  then,  were  as  follows.     When  the  opening  allowed  to  the  light  was  t°h^ej"  £, 
considerable,  as  0.02  or  0.04  inch  (Paris,)  the  fringes  were  exactly  similar  to  those  observed  when  the  edges  were 
equidistant  from  the  object-glass  ;  but  as  the  opening  diminished,  they  ceased  to  be  symmetrical  on  both  sides  of 
the  middle  line,  those  on  the  side  of  that  edge  of  the  aperture  nearest  to  the  telescope  becoming  broader  than 
those  on   the  other,  which,   on  their  part,  undergo    no  sensible  alteration.       As   the  aperture  contracts,  this 
inequality  increases,  till   at  length  the  dilated  fringes  begin  to  disappear  in  succession,   the   outermost  first, 
which  they  do  by  suddenly  acquiring  an  extraordinary  magnitude,  so  as  to  fill  the  whole  field  of  the  telescope, 
and  thus,  as  it  were,  losing  themselves.     While  these  are  thus  vanishing,  those  on   the  other  side  remain  quite 
unaltered  till  the  last  is  gone,  when  they  all  disappear  at  once,  which  happens  at  the  moment  that  the  opening 
is  reduced  to  nothing  by  the  two  edges  covering  each  other. 


LIGHT. 

Light.          When  the  aperture  placed  before  the  object-glass,  instead  of  being  a  straight  line,  was  a  small,  circular  hole,     P»rt  I'1- 
v™^-^-  and  the  aperture  of  the  heliostat,  in  like  manner,  a  minute  circle,  the  phenomena  of  live  rings  were  observed,  and  ^••v^' 

744.  their  diameters  could  be  accurately  measured  by   the  micrometer.     The    results  of  these  measurements  led 
Case  of  a     M.  Fraunhofur  to  the  following  laws  :   1st,  that  for  apertures  of  different  diameters,  the  diameters  of  the  rings 
small,circu-  are  inversely  as  those  of  the  apertures  forming  them  ;    2dly,  that  the  distances  from  the  centre  of  the  maxima 
ar  aperture.  Of  extremered  rays  (or  of  rays  of  any  given  refrangibility)  in  the  several  rings  of  one  and  the  same  system,  form 

an  arithmetical  progression,  whose  difference  is  somewhat  less  than  its  first  term.     Thus,  calling  7  the  diameter 

of  the  aperture,  and  putting  L  =  °-0000214  and  t  _  0-0000237)  he  found  L'  =  I,  L"  =  I  +  L,  11"  =  I  +  2  L, 

7  7 

&c.,  where  L'  L",  &c.  represent  the  angular  semidiametcrs  of  the  several  rings  expressed  in  arc  of  a  circle  to 
radius  unity.  The  near  coincidence  of  the  value  of  L  in  this  case,  with  that  in  the  case  of  a  linear  aperture,  and 
the  small,  but  decided  difference  of  the  values  of  the  first  term  of  the  progression  in  the  two  cases,  are  very 
remarkable. 

745.  When  the  aperture  was  a  very  narrow,  circular  annulus,  such  as  might  be  traced  with  a  steel  point  on  a  gilt 
Case  of  a     disc  of  glass,  of  whatever  diameter,  the  image  was  a  circular  spot,  surrounded  in  like  manner  by  coloured  rings, 
very  narrow,  the  diameters  of  which  depended  nowise  on  the  diameter,  but  only  on  the  breadth  of  the  annulus,  being  in  fact 

(as  might  be  expected)  the  very  same  as  the  intervals  between  similar  opposite  fringes,  on  both  sides  of  the 
central  line  in  the  image  produced  by  a  linear  aperture  of  equal  breadth. 

746.  But  the  most  curious  parts  of  M.  Fraunhofer's  investigations  are  those  which  relate  to  the  interference  of  rays 
Interference  transmitted   through  a  great  many  narrow  apertures  at  once.     When  these  apertures  are  exactly  equal,  and 
ofmanyrays  placed  at  exactly  equal  distances  from  one  another,  phenomena  of  a  totally  different  kind  from  those  originating 
through        'n  a  s'nff'e  aperture  are  seen.     In  his  first  experiments  of  this  kind  he  formed  a  grating  of  wire,  by  stretching 
gratings        a  verv  fine  w're  across  a  frame,  in  the  form  of  a  narrow,  rectangular  parallelogram,  whose  shorter  sides  were 

screws  tapped  in  the  same  die,  and  therefore  precisely  similar ;  across  these  screws  in  the  consecutive  intervals 
between  their  threads  the  wires  were  stretched,  and  of  course  could  not  be  otherwise  than  parallel  and  equidistant. 
The  diameter  of  the  wire  was  0.002021  Paris  inch,   the  intervals  between  them  each  0.003862,  and  the  grating 
consisted  of  260  such  wires.     When  this  apparatus  was   placed   precisely  vertical   before  the  object  glass  of  his 
telescope,  and  illuminated  by  a  narrow  line  of  light  0.01  inch  in  breadth,  also  exactly  vertical,  forming  the  aper- 
ture of  the  heliostat,  the  image  of  this  was  seen  in  the  telescope,  colourless,  well  defined,  and  in  all  respects  pre- 
cisely as  it  would  have  been  seen  without  the  interposition  of  any  grate  or  aperture  at  all,  occupying  the  centre  uf 
Spectra  of    the  field,  only  less  bright.     On  either  side  of  this  was  a  space  perfectly  dark,  after  which  succeeded  a  series  of 
the  second    prismatic  spectra,  which  he  calls  spectra  of  the  second  class,  not  consisting  of  tints  melting  into  each   other, 
class.  according  to  the  law  of  the  coloured  rings,  or  any  similar  succession  of  hues  depending  on  a  regular  degra- 

dation of  light,  but  of  perfectly  homogeneous  colours  ;  so  much  so,  as  to  exhibit  the  same  dark  lines  crossing  them 
as  exist  in  the  purest  and  best  defined  prismatic  spectrum.  In  the  disposition  of  things  already  described,  the 
first,  or  nearer  spectrum  is  completely  insulated,  the  space  between  it  and  the  central  image,  as  well  as 
between  it  and  the  second  spectrum,  being  quite  dark.  The  violet  ends  of  the  spectra  are  inwards,  and  the  red 
outwards ;  but  the  violet  end  of  the  third  spectrum  is  superposed  on  the  red  end  of  the  second,  so  as  in  place 
of  a  dark  interval  to  produce  a  purple  space ;  and  as  we  proceed  farther  from  the  middle,  the  spectra  become 
more  and  more  confounded,  but  not  less  than  thirteen  may  easily  be  counted  on  each  side  by  the  aid  of  a  prism 
refracting  them  transversely,  so  as  to  separate  their  overlapping  portions. 

747.  The  measurement  of  the  distances  of  similar  points  in  the  several  spectra  are  rendered   susceptible  of  the 
Ratio  of  the  utmost  precision  by  means,  of  the  dark  lines  which  cross  them.     A  very  remarkable  peculiarity  of  these  spectra 

must,  however,  be  here  noticed,  viz.  that  although  the  dark  lines  hold  exactly  the  same  places  in  the  order  of 
them"  colours,  or,  in  other  words,  correspond  to  precisely  the  same  degrees  of  refra/igibilily,  as  in  the  prismatic  spectra 
formed  by  refraction,  yet  the  ratio  of  the  intervals  between  them,  or  the  breadths  of  the  several  coloured  spaces, 
differ  entirely  in  the  two  cases.  Thus,  in  the  diffracted  spectra,  the  interval  between  the  lines  C  and  D  (fig.  94) 
is  very  nearly  dou'j'e  of  that  between  G  and  H,  while  in  a  spectrum  formed  by  a  flint-glass  prism  of  an  angle 
of  270,  the  proportion  is  reversed,  and  in  a  water  prism  of  the  same  angle  C  D  :  G  H  :  :  2  :  3. 

748.  In  the  diffracted  fringes  formed  by  a  single  aperture,  their  distances  (as  we  have  seen)  from  the  axis  depends 
Their  laws.  on]v  on  the  breadth  of  the  aperture,  being  inversely  as  that  breadth.     In  the  spectra  formed  by  a  great  •number, 

their  distances  from  the  central  image  depends  neither  on  the  breadths  of  the  apertures  nor  on  the  intervals 
between  them,  but  on  the  sum  of  these  quantities,  that  is,  on  the  distances  between  the  middle  points  of  the 
consecutive  apertures,  (or,  in  the  case  before  us,  on  the  distances  between  the  axes  of  the  wires.)  By  a  series  of 
measures  performed  with  the  utmost  care  and  precision  on  wire  gratings  of  a  great  variety  of  dimensions, 
M.  Fraunhofer  ascertained  the  following  laws  and  numerical  values. 

749.  1.  For  different  gratings,  if  we  call  7  the  breadth  of  each  of  the  interstices  through  which  the  light  passes,  and 
i  that  of  each  of  the  opaque  intervals  between  them,  the  magnitudes  of  spectra  of  the  same  order,  and  the  dis- 
tances of  similar  points  in  them  from  the  axis,  is  inversely  as  the  sum  -y-f-  &. 

750.  2.  The  distances  of  similar  points,  (t.  e.  of  similar  colours  or  similar  fixed  lines,)  in  the   several  consecutive 
spectra  formed  by  one  and  the  same  grating  from  the  axis,  constitute  an  arithmetical  progression  whose  difference 
is  equal  to  its  first  term. 

751.  3.  For  the  several  refrangibilities  corresponding  to  the  fixed  lines  B,  C,  D,  E,  &c.  the  first  term  of  this  pro- 
gression is  numerically  represented  by  the  respective  fractions  which   follow,  being  the   lengths  of  the   arcs,  or 
their  sines  to  radius  unity. 


LIGHT.  489 

Ugnt.  _  0.00002541  _  0.00001945_  __  0.00001464  ttut  IN. 

-~>  ~~  —  •  - 


0.0000'2422  0.00001794 

C  =  -  —  —  ;         F  =  -  ——  ;  &c. 

7  -f-  a 


0.00002175  _  0.00001587 

;        «  —  -  ~~~  ; 


These  results  were  all,  however,  deduced  from  gratings  so  coarse  as  to  allow  of  our  regarding  the  angles  of      752. 
diffraction  as  proportional  to  their  sines  ;    but  when  extremely  fine  gratings    are   employed,    the    spectra  are  Case  of 
formed  at  great  distances  from  the  axis,  and  the  analogy  of  other  similar  cases,  as  well  as  theory,  would  lead  us  extremely 
to  substitute  sin  B,  sin  C,  sin  D,  &c.  in  the  place  of  B,  C,  D,  &c.     This,  M.  Fraunhofer  found  by  experiment  gr°Syngs 
to  be  really  the  case.     The  construction  of  gratings  proper  for  these  delicate  purposes,  however,  was  no  easy 
matter.     Those  employed  by  him  were  nothing  more  than  a  system  of  parallel  and  equidistant  lines  ruled  on  Methods  of 
plates  of  glass  covered  with  gold-leaf,  or  with  the  thinnest  possible  film  of  grease  ;  by  the  former  of  these  constructing 
methods  he  found,  that  the  proximity  of  the  lines  might  be  carried  to  the  extent  of  placing  about  a  thousand  in  th 
the  inch,  but  when  he  would  draw  them  still  closer,  the  whole  of  the  gold-leaf  was  scraped  off.     When  the  sur- 
face was  covered  with  a  film  of  grease  so  ihin  as  to  be  almost  imperceptible  to  the  sight,  (although  the  intervals 
were  in  this  case  transparent,)  no  change  was  produced  in  the  optical  phenomena,  so  far  as  the  spectra  were 
concerned,  only  the  brightness  of  the  central  image  being  increased.     By  this  means  he  was  enabled  to  obtain 
a  system  of  parallel   lines  at  not  more  than  half  the  distance  from  each  other  that  could  be  produced  on  gold- 
leaf:  but  beyond  this  degree  of  proximity,  he  found  it  impossible  to  carry  the  ruling  of  equidistant  lines  on  any 
film  of  grease  or  varnish.     But  this  being  still  far  short  of  his  wishes,  he  had  recourse  to  actual  engraving  with 
a  diamond  point  on  the  surface  of  the  glass  itself,  and  by  this  means  was  enabled  to  rule  lines  so  fine  as  to  be 
absolutely  invisible  under  the  most  powerful  compound  microscope,  and  so  close  that  30,000  of  them  lie  in  a 
single  Paris  inch.     When  so  excessively  near,  however,  no  accuracy  of  machinery  will  ensure  that  perfect  equi- 
distance  which  is  essential  to  the  production  of  the  spectra  now  under  consideration,  and  he  found  it  impossible 
to  succeed  in  placing  them  nearer  than  0.0001223,  (or  about  8200  to  the  inch,)  with  such  a  degree  of  precision 
as  to  enable  him  to  distinguish  the  fixed  lines  in  the  spectra  ;   and,  if  it  be  considered,  that  a  deviation  to  the 
extent  of  the  hundredth  part  of  the  just  interval  frequently  occurring,  is  sufficient  to  obliterate  these,  and  that  to 
produce  the  spectra  in  sufficient  brightness  to  affect  the  eye,  some  hundreds  or  even   thousands   must  be  ruled, 
we  shall  be  enabled  to  form  some  conception  of  the  difficulties  to  be  encountered  in  researches  of  this   kind. 
For  a  detail  of  some  of  these,  and  of  the  methods  employed  by  him  to  count  their  number  and  measure  their  dis- 
tances, we  must  refer  to  his  original  Memoir,  (read  to  the  Royal  Bavarian  Academy  of  Sciences,  June  14,  1823.) 

In  the  course  of  these  researches,  M.  Fraunhofer  met  with  a  very  singular  and  instructive  peculiarity  in  one      753. 
of  the  engraved  glass-gratings  used  by  him  ;  which,  although  it  produced  spectra  equidistant  on  either  side  of  The  spectra 
the  axis,  jet   gave  always   those  on  one  side  a  much   greater  degree  of  brightness  than   those    on   the   other,  modified  by 
Attributing  this  to  iheform  of  the  furrows  being  sharper  terminated  on  one  side  than  on  the  other,  owing  either  Jh*  s°^™  '^ 
to  the  figure  of  the  diamond  point  or  the  manner  of  its  application,  he  endeavoured  to  produce  a  similar  struc-  the  gratings. 
ture  of  the  stria?  in  a  film  of  grease  spread  on  glass,  by  purposely  applying  the  engraving  tool  obliquely,  and  the 
attempt  proved  successful. 

When  the  incident  rays  from  the  opening  in  the  heliostat  fell  obliquely  on  the  grating,  it  might  be  supposed      754. 
that  the  phenomena  would  be  the  same  as  those  exhibited  by  a  closer  grating,  having  intervals  less  in  proportion  Case  of 
of  the  cosine  of  the  angle  of  incidence  to  radius.     But  the  analogy  of  the  unsymmetrical  fringes  produced  by  a  inched 
single  aperture,  whose  sides  lie  in  a  plane  oblique  to  the  incident  ray,  may  lead  us  to  expect  a  different  result,  u*""^e. 
and  experiment  confirms  the  surmise  ;   thus,  M.  Fraunhofer  found,  that  on  inclining  a  grating,  whose   intervals  lr;ca'i 
(•y-J-  £)  were  each  equal  to  0.  0001223  inch,  so  as  to  make  the  angle  of  incidence  55°  with  the  perpendicular,  spectra  of 
the  distance  of  the  first  fixed  line  D  from  the  axis  on  the  one  side  of  the  axis  was  1  5°  6',  and  on  the  other  no  less  the  second 
than  30°  33',  or  more  than  double. 

The  facts  deduced  by  M.  Fraunhofer  in  the  above  detailed  researches  are  certainly  extremely  curious.     The      755, 
most  interesting  and  remarkable  point  about  them  is  the  perfect  homogeneity  of  colour  in  the  spectra,  indicating  Theoretical 
a  saltus,  or  breach  of  continuity,  in  the  law  of  intensity  of  each  particular  coloured  ray  in  the  diffracted  beam,  considera- 
For  it  is  obvious,  that  taking  any  one  refrangibility  (that  corresponding  to  the  fixed  line  C,  for  example,)  the  tlons- 
expression  of  its  intensity  in  functions  of  its  distance  from  the  axis  must  be  (analytically  speaking)  of  such  a 
nature  as  to  vanish  completely  for  every  value  of  that  distance,  excepting  for  a  certain  series  in  arithmetical  pro- 
gression, or,  as  it  is  called,  a  discontinuous  function  ;   so  that  the  curve  representing  such  value,  having  the 
distance  from  the  axis  for  its  abscissa,  must  be  a  series  of  points  arranged  above  the  axis  at  equal  intervals  ;  or, 
at  least,  a  curve  of  the  figure   represented  in  fig.  151,  in  which  certain  extremely  narrow  portions,  equidistantly 
arranged,  start  up  to  considerable  distances  from  the  axis,  while  all  the  intermediate  portions  lie  so  close  to  that 
line  as  to  be  confounded  with  it.     The  manner  in  which  such  a  function  can  be  supposed  to  originate  from  the 

summation  of  a  series  of  the  values  offd  v  .  sin  -^-  v9  and/  d  v  .  cos  -^-  »',  (Art.  718.)  taken  successively  be- 

tween limits  corresponding  to  the  boundaries  of  the  several  interstices,  involves  too  many  complicated  consiJe 
rations  to  enter  into  in  this  place.     M.  Fraunhofer,  meanwhile,  states  the  following  general  expression,  as  the 
result  of  his  own   investigations  founded  on  the  principle  of  interferences.     Let  n  indicate  the  order  of  any 
VOL.  iv.  3  s 


490 


LIGHT. 


Ligln. 


Fraun- 
hofer's 
formula. 


spectrum,  reckoned  from  the  axis ;  e  the  distance  from  the  middle  of  one  interstice  to  that  of  the  adjacent  one 
=  ,y  _j_  £  ;  \  the  length  of  an  undulation  of  an  homogeneous  ray  ;  a  the  angle  of  incidence  of  the  ray  from  the 
luminous  point  on  the  grating ;  and  y  the  length  of  a  perpendicular  let  fall  from  the  micrometer  thread  of  the 
telescope,  (or  from  the  point  in  the  focus  of  its  object-glass,  where  that  particular  homogeneous  ray  in  that 
spectrum  is  found,}  on  the  plane  of  the  grating.  Then,  if  the  angular  elongation  of  that  ray  from  the  axis  be 
called  0("\  we  shall  have,  in  general, 


Part   III. 


cotan  tf*i  = 


-  (e  .  sin  <r  +  n  X)*  }    .    {  4  y*  -f  e3  -  (e  .  sin  a  -f  n  X)«  } 


2  y  (t  .  sin  <r  -j-  w  X) 

In  this  equation,  n  is  to  be  regarded  as  +  for  the  spectra  which  lie  on  the  side  of  the  axis  on  which  the  incident 
ray  makes  an  obtuse  angle  with  the  plane  of  the  grating,  and  negative  for  the  spectra  on  the  other  side.  This 
formula  he  states  to  be  rigorous,  and  independent  of  any  approximation.  When  y  is  very  great  (as  it,  in  fact, 
always  is,)  compared  with  e  and  X,  this  reduces  itself  simply  to 


cotan  flf"'  = 


_  ^6«  —  (e  .  sin  a  -f-  n  X)g 


or  sin     "  = 


e  .  sin  a  -{-  n  X 


756. 

Lengths  of 
undulations 
of  the  rays 
B,C,D,&c. 

assigned  by 
Fraunhofer. 


757. 
Diffracted 
spectra  pro- 
duced by 
reflexion. 

758. 
Alleged 
limit  to  the 
powers 
of  micro- 
scopes. 


759. 

Spectra 
produced 
by  compo- 
»ite  gra- 
ting*. 


Singular 
phenome- 
non noticed 
by  Fraun- 
hofer 

respecting 
the  inten- 
sity of  the 
spectra. 


760. 
Various 
stages  of 
the  pheno- 
mena. 
Spectra  of 
the  first  class 

701 


e  .  sin  a  -f-  n  X 

This  formula,  applied  to  M.  Fraunhofer's  measures  of  the  distances  of  the  same  fixed  lines  in  successive  spectra 
on  either  side  of  the  axis,  in  the  case  of  inclined  gratings,  represents  them  with  perfect  exactness.  When  the 

gratings  are  perpendicular  to  the  ray  a  =  0,  and  the  equation  becomes  sin  <(lt)  = ,  which  is  the  law  before 

noticed  for  symmetrical  spectra.  And  hence,  too,  it  appears  that  the  values  of  X,  or  the  lengths  of  the  undulations 
for  the  several  rays  designated  by  C,  D,  E,  &c.,  are  no  other  than  the  numerators  of  the  fractions  in  Art.  751, 
expressed  in  parts  of  a  Paris  inch,  which  thus  become  data  of  the  utmost  value  in  the  theory  of  light,  from 
the  great  care  and  precision  with  which  they  have  been  fixed*  and  for  the  possibility  of  identifying  them  at 
all  times. 

If  the  unruled  surface  of  the  glass  grating  be  covered  with  black  varnish,  and  the  light  reflected  from  the 
ruled  surface  be  received  in  the  telescope,  the  very  same  phenomena  are  seen  as  if  the  light  had  been  transmitted 
through  the  glass,  and  the  same  analytical  expression,  according  to  M.  Fraunhofer,  applies  to  both  cases. 

A  curious  consequence  of  this  expression  is,  that  if  f,  the  distance  between  the  lines,  be  less  than  X,  and  the 
light  fall  perpendicularly  on  the  grating,  so  that  sin  a  =  0,  we  shall  have  sin0n)  >  1,  and  therefore  (X"'  imagi- 
nary. It  appears,  therefore,  that  lines  drawn  on  a  surface  distant  from  each  other  by  a  less  quantity  than  one 
undulation  of  a  ray  of  light,  produce  no  coloured  spectra.  Hence,  such  scratches,  or  inequalities,  on  polished 
surfaces,  have  no  effect  in  disturbing  the  regularity  of  reflexion  or  refraction,  and  produce  no  dimness  or 
mistiness  in  the  image  ;  if  less  distant  from  each  other  than  this  limit.  M.  Fraunhofer  seems  inclined  to 
conclude  further,  that  an  object  of  less  linear  magnitude  than  X  can  in  consequence  never  be  discerned  by 
microscopes,  as  consisting  of  parts  :  a  conclusion  which  would  put  a  natural  limit  to  the  magnifying  power  of 
microscopes,  but  which  we  cannot  regard  as  following  from  the  premises. 

When  the  intervals  of  the  parallel  interstices  are  unequal,  and  disposed  with  no  regularity,  the  light  of  the 
diffracted  spectra  of  different  combinations  is  confounded  together,  and  a  white  misty  streak  at  right  angles  to 
the  direction  of  the  lines  arises  ;  but  when  they  are  regularly  unequal,  so  that  the  same  intervals  recur  in 
regular  periods,  if  we  call  E  (=  e'-f-  e"  -f-  e"'  -j-  &c.)  the  interval  between  any  two  distant  by  a  whole  period, 

we  shall  have,  for  the  law  of  the  lateral  spectra,  the  equation  sin  flf*'  =  -pp.     And  the  spectra  so   formed,  are 

Ei 

still  observed  to  consist  of  homogeneous  light,  exhibiting  the  fixed  lines  with  great  distinctness.  A  very  curious, 
and,  as  far  as  .concerns  the  practical  measurement  of  the  phenomena,  useful  observation  has  been  made  by 
M.  Fraunhofer  on  the  spectra  so  formed  by  these  composite  gratings,  viz.  that  although  they  follow  the  same  law 
in  respect  o.f  their  distances  from  the  axis,  yet  the  successive  spectra  differ  greatly  in  intensity,  some  being  so 
faint  as  to  be  scarce  perceptible,  while  the  immediately  adjacent  ones  will  often  be  very  intense.  Owing  to 
this  cause,  spectra  of  the  higher  orders,  which  in  a  simple  grating  the  interval  of  whose  interstices  is  represented 
by  E,  are  confused  and  obliterated  by  the  encroachment  of  those  adjacent,  are  often  very  distinct  when  formed 
by  a  composite  grating,  the  period  of  recurrence  of  whose  similar  interstices  is  E  =  e'  -j-  e"  -f-  e'"  -f-  &c.  Thus, 
M.  Fraunhofer  was  never  able,  through  a  simple  grating  to  see  the  fixed  lines  C  and  F  in  the  spectrum  of  the 
12th  order,  reckoning  from  the  axis,  while  in  a  composite  grating,  consisting  of  three  systems  of  lines  continually 
repeated,  whose  intervals  t',  e",  e'"  were  to  each  other  as  25  :  33  :  42,  these  fixed  lines  as  well  as  the  lines  D  and 
E,  were  distinctly  seen  in  the  12th  spectrum,  owing  to  the  almost  total  disappearance  of  the  10th  and  1 1th.  Nay, 
even  the  fixed  line  E  in  the  24th  spectrum  could  be  seen,  and  its  distance  from  the  axis  measured  with  this 
grating. 

Such  are  the  extreme  cases  of  the  phenomena  as  produced  by  a  single  aperture,  and  by  an  infinite,  or,  at 
least,  very  great  number ;  but  the  intermediate  steps  and  gradations  by  which  one  set  of  phenomena  pass  into  the 
other,  remain  to  be  traced.  When  a  single  interstice  is  left  open  in  a  grating,  the  spectra  are  formed  as  described 
in  Art.  741.  These,  M.  Fraunhofer  calls  spectra  of  the  first  class,  and  their  colours  are  not  homogeneous,  but 
graduate  into  one  another. 

When  two  contiguous  interstices  are  left  open,  the  spectra  of  the  first  class  appear  as  before ;  but  between  the 
axis  and  the  first  spectrum  on  either  side  appear  other  spectra,  which  M.  Fraunhofer  terms  imperfect  spectra  of 
the  second  class,  their  colours  being  similar  to  those  of  the  first  class,  and  no  fixed  lines  being  visible  in  them. 


LIGHT.  491 

Light.      When  three  adjacent  interstices  are   left  open,  a  third  set  of  spectra,  or  spectra  of  the  third  class,  are  formed    Part  III. 
"•v-^-'  between  the  axis  and  the  nearest  of  the  imperfect  spectra  of  the  second  class.     Besides  these,  no  new  classes  of  ~—  •  v—  ' 
spectra  arise  by  a  further  increase  of  the  number  of  interstices  ;   but  these  undergo  a  series  of  modifications  as  Spectra  of 
the  interstices  grow  more  numerous.     These  are  chiefly  as  follows  :  das' 

The  spectra  of  the  third  class  grow  narrower,  and  approach  the  axis,  till  at  last  they  run  together  and  form      ^.g,9 
by  their  union  the  colourless,  well-defined  image  of  the  opening  of  the  heliostat  in  the  axis  of  the  whole  pheno-  Moditica- 
menon.     By  a  series  of  exact  measurements,  M.  Fraunhofer  found  their  breadths  to  be  inversely  as  the  number  tionsof 
of  interstices  by  which  they  are  produced  in  the  same  grating,  and  inversely  as  the  intervals  of  the  interstices  for  these  spec- 
different  ones  ;  and  in  general,  that  7  -f-  6"  =  e  representing  this  interval,  m  the  number  of  interstices  used,  and  n  tra       '"" 
the  order  of  the  spectrum,  6  >J  the  distance  of  extremity  of  the  red  rays  in  that  spectrum  is  given  by  the  equation 


n  0.0000208  interfering 

&•">  =  —    X    -  .  rays. 

Formula  for 

As  the  spectra  of  the  third  class  contract  into  the  axis,  they  leave   a  dark   space  between  it  and  the  first  sPectra  of 
spectrum  of  the  second  class.    This  and  the  other  spectra  of  that  class  meanwhile  grow  continually  more  vivid  and  thlrf  c'ass- 
homogeneous  in  respect  of  colour  ;   till  at  length,  when  the  number  of  interfering  rays  is  very  much  increased,  Trans 
the  fixed  lines  begin  to  appear  in  them,  and  they  acquire  the  character  of  perfect  spectra  of  the  second  class,      from  jmper_ 
M.  Fraunhofer  next  examined  the  phenomena  produced  by  immersing  in  media  of  different  refractive  powers  feet  to  per- 
the  gratings  used,  when  he  found  all  the  phenomena  precisely  similar  ;   but  the  distances  at  which  the  several  fee'  spectra 
spectra  were  formed  from  the  axis,  to  be  less  than  when  in  air,  in  the  inverse  ratio  of  the  refractive  indices. 

A  very  beautiful  and  splendid  class  of  optical  phenomena  has  been  investigated  and  described  by  M.  Fraun-      IRA. 
hofer,  which  arise  by  substituting  for  the  gratings  used  in  the  above  experiments  very  small  apertures  of  regular  phenomena 
figures,  such   as  circles  and  squares,  either  singly  or  arranged  in   regular  forms,  in  great  numbers  ;    as,  for  of  gratings 
instance,  when  two  equal  wire  gratings  are  crossed  at  right  angles.     Fig.  151  is  a  representation  of  the  pheno-  immersed 
menon  produced  when  the  light  is  received  on  the  object-glass  of  the  telescope  through  two  circular  holes  of  the  '"  fluids- 
diameter  0.02227  inch,  placed   at  a  distance  of  0.03831   inch  centre  from  centre.     Each   compartment  is  a 
separate  spectrum.     In  the  bands  a  a,  bb  we  see  here  plainly  the  origin  and  minute  structure  of  the  vertical  and  0f"Vg,|tutlc 
crossed  fringes  described  in  Art.  735.     The  appearances  vary  as  the  number  of  apertures  is   increased,  the  minute 
spectra  growing  purer  and  more  vivid.     That  which  arises  when  two  equal  wire  gratings  are  crossed,  is  figured  apertures 
in  M.  Fraunhofer's  work,  and  is  one  of  the  most  magnificent  phenomena  in  Optics.  for  gratings. 

When  we  look  at  a  bright  star  through  a  very  good  telescope  with  a  low  magnifying  power,  its  appearance  is      766. 
that  of  a  condensed,  brilliant  mass  of  light,  of  which  it  is  impossible  to  discern  the  shape  for  the  brightness;  Rings  seen 
and  which,  let  the  goodness  of  the  telescope  be  what  it  will,  is  seldom  free  from  some  small  ragged  appendages  al)out.  tne 
or  rays.     But  when  we  apply  a  magnifying  power  from  200  to  300  or  400,  the  star  is  then  seen  (in  favourable  telescopes 
circumstances  of  tranquil  atmosphere,  uniform  temperature,  &c.)  as  a  perfectly  round,  well-defined  planetary 
disc,  surrounded  by  two,  three,  or  more  alternately  dark  and  bright  rings,  which,  if  examined  attentively,  are 
seen  to  be  slightly  coloured  at  their  borders.     They  succeed  each  other   nearly  at  equal  intervals  round  the 
central  disc,  and  are  usually  much  better  seen  and  more  regularly  and  perfectly  formed  in  refracting  than  in 
reflecting  telescopes.     The   central   disc,  too,  is  much   larger  in  the  former  than   in  the  latter  description   of 
telescope. 

These  discs  were  first  noticed  by  Sir  William  Herschel,  who  first  applied  sufficiently  high  magnifying  powers      767. 
to  telescopes  to  render  them  visible.     They  ;ire  not  the  real  bodies  of  the  stars,  which  are  infinitely  too  remote  Spurious 
to  be  ever  visible  with  any  magnifiers  we  can  apply  ;    but   spurious,  or  unreal  images,  resulting   from  optical  d'scs  of 
causes,  which  are  still  to  a  certain  degree  obscure.     It   is   evident,  indeed,   to  any  one  who  has  entered  into  ' 
what  we  have  said  of  the  law  of  interferences,  and  from   the  explanation  given  in  Art.  590  and   591   of  the 
formation  of  foci  on  the  undulatory  system,  that  (supposing  the  mirror  or  object-glass  rigorously  aplanatic)  the 
focal  point  in  the  axis  will  be  agitated  with  the  united  undulations,  in  complete  accordance,  from  every  part  of 
the  surface,  and  must,  of  course,  appear  intensely  luminous  ;    but  that  as  we  recede  from   the  focus  in   any 
direction  in  a  plane  at  right  angles  to  the  axis,  this  accordance  will  no  longer  take  place,  but  the  rays  from  one 
side  of  the  object-glass  will  begin  to  interfere  with   and    destroy  those  from   the  other,  so  that  at  a  certain 
distance  the  opposition  will  be  total,  and  a  dark  ring  will  arise,  which,  for  the  same  reason,  will  be  succeeded 
by  a  bright  one,  and  so  on.     Thus  the  origin   both  of  the  central  disc  and  the  rings  is  obvious,  though  to  Explanation 
calculate  their  magnitude  from  the  data  may  be  difficult.     But  this  gives  no  account  of  one  of  the  most  remark-  of  the 
able  peculiarities  in  this  phenomenon,  viz.  that  the  apparent  size  of  the  disc  is  different  for  different  stars,  being  rings  on  the 
uniformly  larger  the  brighter  the  star.     This  cannot  be  a  mere  illusion  of  judgment  ;  because  when  two  unequally  Princ'Ple  of 
bright  stars  are  seen  at  once,  as  in  the  case  of  a  close  double  star,  so  as  to  be  directly  compared,  the  inequality  f"ren"ces 
of  their  spurious  diameters  is  striking  ;  nor  can  it  be  owing  to  any  real  difference  in  the  stars,  as  the  intervention 
of  a  cloud,  which  reduces  their  brightness,  reduces  also  their  apparent  discs  till  they  become  mere  points.     Nor 
can  it  be  attributed  to  irradiation,  or  propagation  of  the  impression  from  the  point  on  the  retina  to  a  distance,  as 
in  that  case  the  light  of  the  central  disc  would  encroach  on  the  rings,  and  obliterate  them  ;    unless,   indeed,  we 
suppose  the  vibrations  of  the  retina  to  be  performed  according  to  the  same  laws  as  those  of  the  ether,  and  to 
De  capable  of  interfering  with  them  ;    in  which  case,  the  disc  and  rings  seen  on  the  retina  will  be  a  resultant 
system,  originating  from  the  interference  of  both  species  of  undulations. 

Not  to  enter  further,  however,  on  this  very  delicate  question,  we  shall  content  ourselves  with  stating  some  of     ^8. 
the  phenonena  we  have  observed,  as  produced  by  diaphragms,  or  apertures  of  various  shapes  variously  applied  P' 
to  mirrors  and  object-glasses,  and  which  form  no  inapt  supplement  to  the  curious  observations  of  Fraunhofer  on  j." 
the  effect  of  very  minute  apertures,  of  which  they  are  in  some  sort  the  converse.  vario«> 

3  S  2  figure*. 


492  LIGHT. 

Light.          When  the  whole  aperture  of  a  telescope  is  limited  by  a  circular  diaphragm,  whether  applied  near  to,  or  at  a 
^_v*w'  distance  from,  the  mirror  or  object-glass,  the  disc  and  rings  enlarge  in  the  inverse  proportion  of  the  diameter  of 

769.  the  aperture.     When  the  aperture  was  much  reduced  (as  to  one  inch,  for  a  telescope  of  7  feet  focal  length)  the 
Circular       spurious  disc  was  enlarged  to  a  planetary  appearance,  being  well  denned,  and  surrounded  by  one   ring  only, 
apertures,     strong  enough  to  be  clearly  perceived,  and  faintly  tinged  with  colour  in  the  following  order,  reckoning  from  the 

centre  of  the  disc.     White,  very  faint  red,  black,  very  faint  blue,  white,  extremely  faint  red,  black.     When  the 
aperture  was  reduced  still  farther  (as  to  half  an  inch)  the  rings  were  too  •faint  to  be  seen,  and  the  disc  was  enlarged 
to  a  great,  size,  the  graduation  of  light  from  its  centre  to  the  circumference  being  now  very  visible,  giving  it  a 
Fig.  152.     hazy  and  cometic  appearance,  as  in  fig  152. 

770.  When  annular  apertures  were  used  the  phenomena  were  extremely  striking,  and  of  great  regularity.     The 
Annular       exterior  diameter   of  the  annulus   being   three    inches,  and   the  interior    l£,    the  appearance  of  Capella  was 

res-  as  in  fig.  153,  and  of  the  double  star  Castor,  as  in  154.  As  the  breadth  of  the  annulus  is  diminished,  the  size 
of  the  disc  and  breadth  of  the  rings  diminish  also,  (contrary  to  what  took  place  in  Fraunhofer's  experiments 
with  extremely  narrow  annuli,  and  obviously  referring  the  present  phenomena  to  different  principles,)  at  the  same 
^j  °  time  the  number  of  visible  rings  increases.  Fig.  155,  156,  and  157  exhibit  the  appearance  of  Capella  with 
annular  apertures  of  5.5  inch  —  5  inch  (i.  e.  whose  exterior  diameter  =  5.5  and  interior  =:  5)  of  0.7  —  0.5,  of 
2.2  —  2.0.  In  the  last  case  the  disc  was  reduced  to  a  hardly  perceptible  round  point,  and  the  rings  were  so  close 
and  numerous  as  scarcely  to  admit  being  counted,  giving,  on  an  inattentive  view,  the  impression  of  a  mere 
circular  blot  of  light.  When  the  breadth  of  the  annulus  was  reduced  to  half  this  quantity,  the  intervals  between 
the  rings  could  no  longer  be  discerned.  The  dimensions  of  the  rings  and  disc,  generally,  seem  to  be  proportional 

r'-r 

to  . 

r 

771.  Besides  the  rings  immediately  close  to  the  central  disc,  however,  others  of  much  greater  diameter  and  fainter 
Another  set  light,  like  halos,  are  seen  with  annular  apertures,  which  belong  (in  Fraunhofer's  language)   to  spectra  of  a 
of  rings,      different  class.     With  a  single  annulus  they  are  too  faint  to  be   distinctly  examined,  but  with  an  aperture 
K'd  159       comPosed   of  two  annuli,  as  in  fig.  158,  they  are  very  distinct  and  striking,  presenting  the  phenomenon  in 

fig.  159,  (in  which  it  is  to  be  understood  that  light  is  represented  in  the  engraving  by  darkness  and  darkness 

by  light.) 

When  the  aperture  was  in  the  form  of  an  equilateral  triangle,  the  phenomenon  was  extremely  beautiful ;  it 
Image  pro-  consisted  of  a  perfectly  regular,  brilliant,  six-rayed  star,  surrounding  a  well-defined  circular  disc  of  great 
triangular*  ^r'Sntness-  The  ravs  do  not  unite  to  the  disc,  but  are  separated  from  it  by  a  black  rinsf.  They  are  very  narrow, 
aperttire.  an(^  perfectly  straight;  and  appear  particularly  distinct  in  consequence  of  the  total  destruction  of  all  t  fit  diffused 

light  which  fills  the  field  when  no  diaphragm  is  used ;    a  remarkable  effect,  and  much  more  than  in  the  mere 
Fig   160      proportion  of  the  light  stopped.     Fig.  160  is  a  representation  of  this  elegant  appearance.     The  same  arises 

when,  in  place  of  an  equilateral  triangle,  the  aperture  is  the  difference  of  two  concentric,  equilateral  triangles 

similarly  situated. 

773.  As  a  triangle  has  but  three  side'  and  three  angles,  it  seems  singular  that  a  six-rayed  star  should  be  produced. 
When  out    Supposing  three  to   arise  from  the  angles,  and  three  from  the  sides,  it  might  be  expected  that  some  sensible 
of  focus.      difference  should  exist  in  the  alternate  rays,  marking  their  different  origin.     When  the   telescope  is  in  perfect 

focus,  however,  all  the  rays  are  precisely  alike ;  but  if  thrown  out  of  focus,  their  difference  of  origin  becomes 
Fi»  161  apparent.  Fig.  161  represents  the  phenomenon  then  seen,  in  which  the  alternate  branches  are  seen  to  consist 
of  a  series  of  fringes  parallel  to  their  length,  and  the  others  of  small  arcs  of  similar  fringes  immediately  adjacent 
to  the  vertices  of  the  hyperbolas  to  which  they  belong,  and  which  consequently  cross  the  rays  in  a  direction 
perpendicular  to  their  length.  As  the  telescope  is  brought  better  in  focus,  the  hyperbolas  approach  their  asymp- 
totes, and  are  confounded  together  in  undistinguishable  proximity  ;  and  thus  three  rays  arise  composed  of  conti- 
nuous lines  of  light,  and  three  intermediate  ones  composed  of  an  infinite  number  of  discontinuous  points  placed 
infinitely  near  each  other.  To  represent  analytically  the  intensity  of  the  light  in  one  of  these  discontinuous  rays 
would  call  for  the  use  of  functions  of  a  very  singular  nature  and  delicate  management. 

774.  The  phenomenon  just  described  affords  in  certain  cases  a  very  perfect  position-micrometer  for  astronomical 
Application  uses.     If  the  diaphragm  be  turned  round,  the  rays  turn  with  it ;  and  if  a  brilliant  star  (as  a  Aquilse)  have  near 
to  the  con-  jt  a  very  smaii  one>  tne  diaphragm  may  be  so  placed  as  to  make  one  of  the  rays  pass  through  the  small  star, 
a^posi'tion"    which  thus  remains  like  a  bead  threaded  on  a  string,  and  may  be  examined  at  leisure.     If  then  the  position  of 
micrometer,  the  diaphragm  be  read  off  on  a  graduation  properly  contrived,  the  relative  situations  of  the  two  stars  become 

known.     We    have  satisfied  ourselves  by  trial   of  the  practicability  of  this  ;    and  by  proper  contrivances  the 

principle  may  be  made  available  in  cases  which  at  first  sight  appear  to  present  considerable  difficulties. 
775  When  three  circular  apertures,  having  their  centres  at  the  angles  of  an  equilateral  triangle,  were  used,  the 

Three  image  consisted  of  a  bright  central  disc.  Six  fainter  ones  in  contact  with  it,  and  a  system  of  very  faint  halo- 
circular  like  rings  surrounding  the  whole  as  in  fig.  162.  When,  however,  three  equal  and  similar  annular  apertures 
apertures,  were  thus  disposed,  the  appearance  when  in  focus  was  as  in  fig.  153,  being  exactly  the  same  as  if  two  of  them 

were  closed.     But  when  thrown  a  little  out  of  focus,  the  difference  was  perceived.     Fig.  163  represents  the 
Fig.  163.     appearance  in  this  case,  each  of  the  apertures  then  produces  its  own  central  disc  and  system  of  rings,  whose 

intersections  give  rise  to  the  system  of  intersectional  fringes  there  depicted.     As  the  telescope  is  brought  better 
Fig.  164.     in  focus  these  disappear,  and  the  phenomenon  is  as  in  fig.  164  ;    the  centres  gradually  approaching,  and   the 

rings  blending  till  the  point  of  complete  coincidence  is  attained. 
7»g  An  aperture  in  the  form  of  the  difference  between  two  concentric  squares  produced  not  an  eight,  but  a  four 

rayed  star.     The  rays,  however,  were  not,  as  in  the  case  of  the  triangular  aperture,  uninterrupted  fine  lines, 

gradually  tapering  away  fro;n  the  centre  to  their  extremities,  but  composed  of  distinct  alternating  obscure  and 


LIGHT.  493 

lJgn«-      bright  portions,  as  represented  in   %.  165.     The  portions  nearest  the  central  disc  (which  is  circular)  were  Part  IIL 
•~~\-~m*'  composed  of  bands  transverse  to  the  direction  of  the  rays,  and  tinged  with  prismatic  colour.     Similar  bands,  >— -v^*' 
no  doubt,  existed  in  the  more  distant  portions,  which  extended  to  a  great  length. 

An  aperture  consisting  of  fifty  squares,  each  of  about  half  an  inch  in  the  side,  regularly  disposed  at  intervals  pP*_r  "'^' 
so  as  to  leave  spaces  between  them  in  both  directions  equal  in  breadth  to  the  side  of  each,  produced  an  image      rm. 
totally  different  from  that  described   by  Fraunhofer  as   resulting  from  the  crossing  of  two  equal  very  close  Effect  of 
gratings,  though  the  distribution  and  shape  of  the  apertures  were  the  same  in  both  cases.     It  was  as  repre-  very  nume- 
sented  in  fig.  166,  consisting  of  a  white,  round,  central  disc,  surrounded  by  eight  vivid  spectra,  disposed  in  the  rous  S1uar« 
circumference  of  a  square,  beyond  which  were  arranged  in  the  shape  of  a  cross,  triple  lines  of  very  faint  spectra  S'r'iggS 
extending  to  a  great  distance. 

When  the  aperture  consisted  of  numerous  equilateral  triangles  regularly  disposed,  as  in  fig.  167,  the  image      778. 
presented  the  very  beautiful  phenomenon  represented  in  fig.  168,  consisting  of  a  series  of  circular  discs  arranged  F'g.  167. 
in  six  diverging  rays  from  the  central  one,  and  each  surrounded  with  a  ring.     The  central  disc  was  colourless  and 
bright ;   the  rest  more  and  more  strongly  coloured  and  elongated  into  spectra,  according  to  their  degree  of 
remoteness  from  the  centre.     These  are  only  a  few  of  the  curious  and  beautiful  phenomena  depending  on  the 
figures  of  the  apertures  of  telescopes,  which  afford  a  wide  field  of  further  inquiry,  and  one  at  least  as  interesting 
to  tlie  artist  as  to  the  philosopher. 


494 


LIGHT. 


Light. 


PART  IV. 


OF  THE  AFFECTIONS  OF  POLARIZED  LIGHT. 


§  I.  Of  Double  Refraction. 


779. 

Exceptions 
to  the  law 
of  ordinary j 
refraction 


Classes  of 
bodies  in 
which  it 
holds. 


780. 
Double 
refraction. 


WHEN  a  ray  of  light  is  incident  on  the  surface  of  a  transparent  medium,  a  portion  of  it  is  reflected,  at  an 
angle  equal  to  that  of  incidence,  another  small  portion  (-so  small,  however,  that  we  shall  neglect  its  consi- 
deration) is  dispersed  in  all  directions,  serving  to  render  the  surface  visible,  and  the  rest  enters  the  medium  and 
is  refracted.  The  law  of  refraction,  or  the  rule  which  regulates  the  path  of  this  portion  within  the  medium, 
has  been  explained  in  the  preceding  parts  ;  and  no  exceptions  to  it,  as  a  general  law,  have  hitherto  been  noticed. 
It  is,  however,  very  far  from  general ;  and,  in  fact,  obtains  only  where  the  refracting  medium  belongs  to  one  or 
other  of  the  following  classes,  viz. 

Class  1.  Gases  and  vapours. 

2.  Fluids. 

3.  Bodies  solidified  from  the  fluid  state  too  suddenly  to  allow  of  the  regular  crystalline  arrangement  of 

their  particles,  such  as  glass,  jellies,  &c.,  gums,  resins,  &c.,  being  chiefly  such  as  in  the  act  of 
cooling  pass  through  the  viscous  state. 

4.  Crystallized  bodies,  having  the  cube,  the  regular  octohedron,  or  the  rhomboidal  dodecahedron  for 

their  primitive  form,  or  which  belong  to  the  tessular  system  of  Mohs.  A  very  few  exceptions 
(probably  only  apparent  ones,  arising  from  our  imperfect  knowledge  of  crystallography)  exist  to 
the  generality  of  this  class. 

The  solid  bodies  belonging  to  these  classes,  moreover,  cease  to  belong  to  them  when  forcibly  compressed  or 
dilated,  either  by  mechanical  violence,  or  by  the  unequal  action  of  heat  or  cold,  which  brings  their  particles 
into  a  state  of  strain,  such  as  in  extreme  cases  to  produce  their  disruption,  as  is  familiarly  seen  in  the  cracking 
of  a  piece  of  glass  by  heat  too  suddenly  and  partially  applied.  The  class  of  fluids  too  admits  some  exceptions, 
at  least  when  very  minutely  considered ;  but  the  deviation  from  the  ordinary  law  of  refraction  in  these  cases  is 
of  so  microscopic  a  kind,  that  we  shall  at  present  neglect  to  regard  it. 

All  other  bodies,  comprehending  all  crystallized  media,  such  as  salts,  gems,  and  crystallized  minerals,  not 
belonging  to  the  system  above  mentioned  ;  all  animal  and  vegetable  bodies  in  which  there  is  any  disposition  to 
a  regular  arrangement  of  molecules,  such  as  horn,  mother  of  pearl,  quill,  &c. ;  and,  in  general,  all  solids  when 
in  a  state  of  unequal  compression  or  dilatation,  act  on  the  intromitted  light  according  to  very  different  laws, 
dividing  the  refracted  portion  into  two  distinct  pencils,  each  of  which  pursues  a  rectilinear  course  so  long  as  it 
continues  within  the  medium,  according  to  its  own  peculiar  laws,  but  without  further  subdivision.  This  pheno- 
menon is  termed  double  refraction.  It  is  best  and  most  familiarly  seen  in  the  mineral  termed  Iceland  spar, 
which  is,  in  fact,  carbonate  of  lime  in  a  regular  crystalline  form.  This  is  generally  obtained  in  oblique  parallel- 
epipeds, easily  reduced  by  cleavage  to  regular,  obtuse  rhomboids,  and  is  not  uncommonly  met  with  in  a  state  of 
limpid  transparency,  on  which  account,  as  well  as  by  reason  of  its  remarkable  optical  properties,  it  easily 
attracted  attention.  Bartholinus,  in  1669,  appears  to  have  been  the  first  to  give  any  account  of  its  double 
refraction,  which  was  afterwards  more  minutely  examined  by  Huygens,  the  first  proposer  of  the  undulatory 
theory  of  light,  whose  researches  on  this  phenomenon  form  an  epoch  in  the  history  of  Physical  Optics  little  if 
at  all  less  important  than  the  great  discovery  of  the  different  refrangibility  of  the  coloured  rays  by  Newton.  To 
Huygens  we  owe  the  discovery  of  the  law  of  double  refraction  in  this  species  of  medium.  Newton,  misled  by 
some  inaccurate  measurements,  (a  thing  most  unusual  with  him,)  proposed  a  different  one  ;  but  the  conclusions 
of  Huygens,  long  and  unaccountably  lost  sight  of,  were  at  length  established  by  unequivocal  experiments  by 
Dr.  Wollaston,  since  which  time  a  new  impulse  has  been  given  to  this  department  of  Optics;  and  the  successive 
labours  of  Laplace.  Malus,  Brewster,  Biot,  Arago,  and  Fresnel  present  a  picture  of  emulous  and  successful 
research,  than  which  nothing  prouder  has  adorned  the  annals  of  physical  science  since  the  developement  of  the 
true  system  of  the  universe.  To  enter,  however,  into  the  history  of  these  discoveries,  or  to  assign  the  share  of 
honour  which  each  illustrious  labourer  has  reaped  in  this  ample  field  forms  no  part  of  our  plan.  Of  the  splendid 
constellation  of  great  names  just  enumerated,  we  admire  the  living  and  revere  the  dead  far  too  warmly  and  too 
deeply  to  suffer  us  to  sit  in  judgment  on  their  respective  claims  to  priority  in  this  or  that  particular  discovery ; 
to  balance  the  mathematical  skill  of  one  against  the  experimental  dexterity  of  another,  or  the  philosophical 
acumen  of  a  third.  So  long  as  "  one  star  differs  from  another  in  glory, ' — so  long  as  there  shall  exist 
varieties,  or  even  incompatibilities  of  excellence, — so  long  will  the  admiration  of  mankind  be  found  sufficient 
for  all  who  truly  merit  it.  Waving,  then,  all  reference  to  the  history  of  the  subject,  except  in  the  way  of  inci- 
dental remark,  or  where  the  necessity  of  the  case  renders  it  unavoidable,  we  shall  present  the  reader  with  as 


LIGHT.  495 

Light,     systematic  an  account  as  we  are  able,  of  the  present  state  of  knowledge  with  respect  to  the  laws  and  theory  of    Part  IV. 
•*~v— •"'  Double  Refraction.     The  Huygenian  law  having  been  demonstrated  to  apply  rigorously  to  the  case  for  which  v-~-v~~-/ 
he  himself  devised  it,  as  well  as  to  a  very  large  class  of  other  bodies,  we  shall  begin  with  that  class,  and  proceed 
afterwards  to  consider  more  complicated  cases. 

In  all  crystallized  bodies,  then,  which  possess  double  refraction.it  is  found  that  that  portion  of  a  ray  of      78J. 
ordinary  light  incident  on  any  natural  or  artificially  polished  surface  which  enters  the  body  is  separated  into  two  Axes  of 
equal  pencils  which  pursue  rectilinear  paths,  making  with  each  other  an  angle  not  of  constant  magnitude,  but  ™* 
varying  according  to  the  position  which  the  incident  ray  holds  with  respect  to  the  surface,  and  to  certain  fixed 
lines,  or  axes  within  the  crystal,  and  which  lines  are  related  in  an  invariable  manner  to  the  planes  of  cleavage, 
or  other  fixed  planes  or  lines  in  the  primitive  form  of  the  crystal.     Now,  it  is  found  that  in  every  crystal  there 
is  at  least  one  such  fixed  line,  along  which  if  one  of  these  two  pencils  be  transmitted  the  other  is  so  also,  so 
that  in  this  case  the  two  pencils  coincide,  the  angle  between  them  vanishing.     Moreover,  no  crystal  has  yet  been 
discovered  in  which  more  than  two  such  lines  exist.     These  lines  are  called  the  optic  axes.     All  double  refracting 
crystals,  then,  at  present,  may  be  divided  into  such  as  have  one,  and  such  as  have  two,  optic  axes. 

When  a  ray  penetrates  the  surface  of  a  crystal  so  as  to  be  transmitted  undivided  along   the    optic  axis;      782. 
or  when,  moving  within  the  crystal  along   that   line,  it  meets    the  surface  and  passes   out,   whatever  be  the  R>y? 
inclination   of  the   surface,  its  refraction  is  always  performed  according   to  the  ordinary  law  of  the  propor-  "] 
tional  sines.     Thus,  in  this  particular  case,  the  crystal  acts  precisely  as  an  uncrystallized  medium,  (some  rare  axes  suffer 
instances  excepted,  of  which  more  hereafter.)  ordinary 

But  in  all  other  cases  the  law  is  essentially  different,  and  (for  one  portion  of  the  divided  pencil,  at  least)  refraction 
of  a   very  singular  and  complicated  nature.     This  we   shall  first  proceed    to  explain   in  the  simpler  case  of  onl-L'Rq 
crystals  with   one  optic   axis.      But,  first,   we  must   explain  somewhat  more  distinctly,  what  we  mean    by  w, 
axes  and  fixed  lines  within  a  crystal.     Suppose  a  mass  of  brickwork,  or  masonry,  of  great  magnitude,  built  of  meant  by 
bricks,  all  laid  parallel  to  each  other.     Its  exterior  form  may  be  what  we  please ;  a  cube,  a  pyramid,  or  any  other  axes  and 
figure.     We  may  cut  it  (when  hardened  into  a  compact  mass)  into  any  shape,  a  sphere,  a  cone,  or  cylinder,  &c. ;  n«cl  lines 
but  the  edges  of  the  bricks  within  it  lie  still  parallel  to  each  other ;  and  their  directions,  as  well  as  those  of  the  wlttlin  a 
diagonals  of  their  surfaces,  or  of  their  solid  figures,  may  all  be  regarded  as  so  many  axes,  i.  e.  lines  having  (so  cr 
long  as  the  mass  remains  at  rest)  a  determinate  position,  or  rather  direction  in  space,  no  way  related  to  the 
exterior  surfaces,  or  linear  boundaries  of  the  mass,  which  may  cut  across  the  edges  of  the  bricks  in  any  angles 
we  please.     Whenever,  then,  we  speak  of  fixed  lines,  or  axes  of,  or  within,  a  crystal,  we  always  mean  directions 
in  space  parallel  to  each  of  a  system  of  lines  drawn  in  the  several  elementary  molecules  of  the  crystal,  according 
to  given  geometrical  laws,  and  related  in  a  given  manner  to  the  sides  and  angles  of  the  molecules  themselves. 
We  must  conceive  the  axis,  then,  of  a  crystallized  mass  not  as  a  single  line  having  a  given  place,  but  as  any  line 
whatever  having  a  given  direction  in  space,  i.  e.  parallel  to  the  axis  of  each  molecule,  which  is  a  line  having  a 
determinate  place  and  position  within  it. 

In  the  remainder  of  this  section,  when  we  speak  of  the  axis  or  axes  of  a  crystallized  mass  or  surface  generally,      784. 
we  mean  the  direction  of  the  optic  axis  or  axes  of  its  molecules,  or  of  a  crystal  similar  and  similarly  situated 
to  any  one  of  them. 

Of  the  Law  of  Double  Refraction  in  Crystals  with  One  Optic  Axis. 

This  class  of  crystals  comprises  all  such  as  belong  to  Mohs's  rhombohedral  system,  or  which  have  the  acute  or      785. 
obtuse  rhomboid,  or  regular  six-sided  prism,  for  their  primitive  form,  as  well  as  all  which  belong   to  his  Enumera- 
pyramidal  system,  or  whose  primitive  form  is  either  the  octohedron  with  a  square  base,  the  right  prism  with  a  tionofcrys- 
square  base,  or  the  bi-pyramidal  dodecahedron.     All  such  crystals  Dr.  Brewster  has  shown  to  have  but  one  tal*  "*V'"S 
axis,  which  is  that  to  which  the  primitive  form  is  symmetrical,  viz.  in  the  rhomboid,  the  axis  of  the  figure,  or  axjs  fn 
line  joining  the  two  angles  formed  by  three  equal  plane  angles ;  in  the  hexagonal  prism,  the  geometrical  axis  classes, 
of  the  prism  ;  in  the  octohedron,  or  square  based  prism,  a  line  drawn  through  the  centre  of  the  base  at  right 
angles  to  it.     The  cases  in  accordance  with  the  rule  are  so  numerous,  and  the  exceptions,  once  believed  to  be 
so,  have  so  often  disappeared  on  the  attainment  of  a  more  perfect  knowledge  of  the  crystalline  forms  of  the 
excepted  minerals,  that  when  any  case  of  disagreement  seems  to  occur,  we  are  justified  in  attributing  it  rather 
to  our  own  incorrect  determination  of  this  datum,  than  to  want  of  generality  in  the  rule  itself. 

In  all  crystals  of  this  class,  one  of  the  two  equal  pencils  into  which  the  refracted  ray  is  divided  follows  the      786. 
ordinary  law  of  Snellius  and  Descartes,  having  a  constant  index  of  refraction  (/*),  or  invariable  ratio  of  the  sine  Refraction 
of  incidence  to  that  of  refraction,  whatever  be  the  inclination  of  the  surface  by  which  it  enters ;   so  that  its  °^  t'le  ot^}~ 
velocity  within  the  medium,  when  once  entered,  is  the  same  in  whatever  direction  it  traverses  the  molecules ;  "^  "jay  "J 
and  with  respect  to  this  ray  the  crystal  comports  itself  as  an  uncrystallized  medium.     This,  then,  is  called  the  crystals. 
ordinary  pencil. 

To  understand  the  law  obeyed  by  the  other,  or  extraordinary  portion  of  the  divided  pencil,  let  us  consider      737 
it  as  fairly  immersed  in  the  medium,  and  pursuing  its  course  among  the  molecules.     Then  its  velocity  will  not,  Huygens's 
as  in  the  case  of  the  ordinary  ray,  be  the  same  in  whatever  direction  it  traverses  them,  but  will  depend  on  the  law  for  the 
angle  it  makes  with  the  axis ;   being  a  minimum  when  its  path  within  the  crystal  is  parallel  to  the  axis,  and  a  velocity  of 
maximum  when  at  right  angles  to  it,  or  vice  versa;   and  in  all  intermediate  inclinations  of  an  intermediate  th<-.extra- 
magnitude  according  to  the  following  law.     Let  an  ellipsoid  of  revolution,  either  oblate  or  prolate,  as  the  case  ™'n 


496 


LIGHT. 


Light. 


788. 

Its  con- 
nection with 
the  law  of 
extraordi- 
nary refrac- 
tion. 


789. 
Investiga- 
tion of  the 
latter  from 
the  former 
law 


Expression 
for  the 
radius  of  the 
spheroid  of 
refraction. 


may  be,  be  conceived,  having  its  axis  of  revolution  coincident  in  direction  with  the  axis  of  the  crystal,  and  its  polar 
to  its  equatorial  radius  in  the  ratio  of  the  minimum  and  maximum  velocities  above  mentioned,  i.  e.  as  the  velocity 
of  a  ray  moving  parallel  to  that  of  one  perpendicular  to  the  axis.  Then  in  all  intermediate  positions,  the  radius 
of  this  spheroid  parallel  to  the  ray  will  represent  its  velocity  on  the  same  scale  that  its  polar  and  equatorial 
radii  represent  the  velocities  in  their  respective  directions. 

This  is  the  Huygenian  law  of  velocities,  in  its  most  simple  and  general  form.  It  does  not  at  first  sight  appear 
what  this  has  to  do  with  the  law  of  extraordinary  refraction  ;  but  the  reader  who  has  considered  with  the  requisite 
attention  what  has  been  said  in  Art.  539,  540,  with  prospective  reference  to  this  very  case,  will  easily  perceive 
that,  the  law  of  velocity  of  the  ray  within  the  medium  once  established,  it  becomes  a  mere  matter  of  pure 
Geometry  to  deduce  from  it  the  law  of  extraordinary  refraction,  whether  we  adopt  the  Corpuscular  theory,  and 
employ  Laplace's  principle  of  least  action,  as  in  that  Article  ;  or  whether,  preferring  the  Undulatory  hypothesis, 
we  substitute  for  this  principle  the  equivalent  one  of  swiftest  propagation,  as  explained  in  Art.  587,  588.  We 
should  observe,  however,  that  the  Huygenian  law,  as  just  stated,  is  worded  in  conformity  with  the  undulatory 
doctrine,  in  which  the  velocity  in  a  denser  medium  is  supposed  slower  than  in  a  rarer.  But  when  we  use  the 
principle  of  least  action,  we  must  invert  the  use  of  the  word,  or,  which  comes  to  the  same  thing,  suppose  the 
the  velocity  in  the  medium  to  be  inversely  proportional  to  the  radius  of  the  ellipsoid.  The  results  being 
necessarily  the  same  in  both  cases,  we  shall  use  at  present  the  language  of  the  Corpuscular  system. 

Retaining,  then,  the  notation  of  Art.  540,  the  law  of  refraction  will  be  derived  from  the  equation  V  .  S  -f  V  .  S 
=  a  minimum,  where  V  is  the  velocity  without,  and  V  that  within  the  medium,  and  where  S  and  S'  are  the  spaces 
described  without  and  within  it,  in  the  passage  of  a  ray  from  point  to  point.  Let  a  and  6  be  the  polar  and 
equatorial  semiaxes  of  the  ellipsoid  above  spoken  of,  (which  we  shall  call  the  ellipsoid  of  double  refraction,)  and 
let  n,  ft,  7  be  the  coordinates  of  the  point  (A)  without  the  crystal,  and  a',  ft',  7'  those  of  one  (B)  within  it, 
through  which  the  ray  is  supposed  to  pass,  and  x,  y,  z  the  coordinates  of  a  point  in  the  surface  of  the  crystal,  on 
which  it  must  be  incident,  so  as  to  be  capable  of  passing  from  A  to  B  in  the  manner  required  by  the  law  of 
extraordinary  refraction  ;  and  let  0  be  the  angle  which  the  interior  portion  S'  makes  with  the  axis  of  the  crystal. 
Then  will  the  radius  of  the  spheroid  parallel  to  this  portion  (by  conic  sections)  be  expressed  by 


Part  IV. 


ab 


ab 


' 

•f  6s  .  sin  0*  +  <z«  cos  04 

where  a  is  the  equatorial,  and  6  the  polar  radius  of  the  spheroid.     Now,  if  we  take  p  to  represent  the  index  of 


ordinary  refraction,  since  we  have,  generally,  V  =  —  —  ,  and  since,  when  r  =  b  the  extraordinary  and  ordinary 


const 


rays  coincide,  and  therefore  V  =p  V,  consequently  we  must  have  /•  V  = ,  and  const  =  6  ft  V,  so  that 

D 

we  shall  get 


790. 

Introduc- 
tion of  the 
principle  of 
least  action 
cr  swiftest 
propagation 
Fig.  169. 


791. 


In  general,  as  we  have  already  seen,  the  condition  of  least  action  affords  the  equation 

d{VS  +  V'S'}  =  0,orV.dS  +  V'.«/S'  +  S'.  dV'  =  0;     (2) 

But  to  make  use  of  this,  we  must  express  V,  S,  and  S',  in  terms  of  variable  quantities  relating  to  a  point  any 
how  taken  in  the  surface  of  the  crystal.  Whether  this  point  be  expressed  by  rectangular  or  polar  coordinates 
is  no  matter:  it  will  be  more  convenient,  however,  to  use  polar.  Let,  then,  C  (fig.  169)  be  the  point  of  inci- 
dence of  the  ray  A  C  on  the  surface  H  a  O  b,  and  about  C  as  a  centre  describe  a  sphere.  Let  Z  C  2  be  the  per- 
pendicular to  the  surface  at  C,  and  let  P  C^  be  the  position  of  the  axis  of  the  crystal.  The  plane  Z  P  H  zp  O  Z 
perpendicular  to  the  surface,  and  passing  through  the  axis,  is  called  the  principal  section  of  the  surface.  Let 
Z  A  a,  2  B  b  be  vertical  planes,  containing  the  incident  and  refracted  rays,  and  join  B  ;;  by  the  arc  of  a  great 
circle.  Then  it  is  evident,  that  this  arc  will  be  equal  to  <£. 

Suppose,  now,  the  axis  of  the  x  to  be  parallel  to  H  C  the  projection  of  the  axis  of  the  crystal,  and  since  we  may 
choose  the  plane  of  the  x,  y,  as  we  please,  let  it  coincide  with  the  refracting  surface,  so  that  2=0.  Then 
dropping  the  perpendiculars  A  M,  M  m,  B  N,  N  n,  and  putting  X  =  Z  P  =  z  p  =  angle  between  the  axis  and 
perpendiculars. 

•as  =  O  a  =  inclination  of  the  plane  of  incidence  to  the  principal  section. 

w7  =  O  6  =  inclination  of  plane  of  refraction  to  ditto. 

6  =  angle  Z  C  A  =  Z  A  =  angle  of  incidence 

ff  —  z  C  B  =  z  B  =  angle  of  refraction. 
We  shall  have  as  follows: 


consequently. 


AC  =S;  AM  =  7;  C  ro»  =  (o  -  *)'  ;  M  m*  =  (ft  - 


and,  si 


o  —  x  := 


7 
7  .  tan  8  .  cos  TS  ;   ft  —  y  =  7  .  tan  <?  .  sin  w  ;  S  = — 

similarly, 

of  -  x  =  7' .  tan  ff .  cos  ra';  /?'  -  y  =  7'  .  tan  ff .  sin  rss' ;  S'  =  -  — ; 


LIGHT.  197 

Uebt.      Now,  differentiating  these  equations,  and  considering  that  d  (a  —  x)  =  d  (a1  —  x)  and  d  (/3  —  y)  =  d  (ft1  —  y)     Part  IV. 
•^V'""*'  we  get  v«^^«», 

d  (tan  0  .  cos  TO)  =:  —  .  d  (tan  ff .  cos  TO')  ; 
Tf 

7' 
d  (tan  0  .  sin  TO)  =  —  .  d  (tan  5'  .  sin  TO')  ; 

which  equations,  developed  and  reduced,  afford  the  following, 

d  0         7'    /cos  0V  d  0         7' 

•y— .  =  — - .  I  -    —  I .  cos  (TO  —  TO')  ;    - — -  =  —  .  cos  0*  .  tan  d1  .  sin  (TO  —  TO')  ;  \ 

do         &     \cos  0  /  d  TO         7 

>          (4) 

d  TO         7'      sin  (TO'  —  TO)  d  TO        7'    tan  ^ 

=  —  . — r  :  =  —  , cos  (TO'  —  TO)  ; 

d  ff        7      tan  0  .  cos  0'2  d  TO'      '  f     tan  0 

which  are  necessary  conditions,  in  order  that  the  point  C  may  remain  on  the  surface. 

But  since  S,  S',  V  may  be  regarded  as  functions  of  ff  and  TO',  which  are  the  polar  coordinates  we  propose  to        792. 
use  as  independent  variables,  we  shall  have 


and,  moreover, 

7  .  sin  0 


COS02 

so  that,  substituting  their  values  in  the  equation  (2,)  we  get 

0  =  -fv  .  7  '  S'"  .  .  — 
C          cos  0*       dff 


.          - 

cos  e'°-     T  cos  6>'  '   d& 


:n  which  the  coefficients  of  each  of  the  two  independent  differentials  being  separately  made  to  vanish,  we  get 

=  —  V  .  —  .  — — -C°8      .  — .  _  V' .  tan  ff   \ 

d  0  7'  cos  0*  d  & 

\.  (o) 

d  V  7       sin  0  .  cos  0'      d  0 

d  is1  <y'  cos  0*  d  TO* 

In  these,  substituting  the  values  of  -—-.  and  - — found  in  equation  (4,)  we  obtain  the  following 

d  ff          dw 


dv  h      <7) 

r— ;  =  —  V  .  sin  0  .  sin  ff  .  sin  (TO  —  TO')  I 

d'BT 

These  are  the  very  same  equations  with  those  deduced  by  Laplace  and  Malus,  by  a  more  abstruse  and  compli- 
cated calculus,  from  the  primary  dynamical  relations  of  the  problem,  and  from  them  it  is  easy  to  express,  in 
general,  the  law  of  refraction  corresponding  to  any  given  law  of  velocities,  for  we  have  only  to  put  them  under 
the  form 

d  V/ 
V  .  sin  6  .  cos  -a    cos  zi7  -f-  V  .  sin  0  .  sin  TO  .  sin  TO'  =  —  V  .  sin  ff  —  cos  ff .  -r-r, 

1      dV 

V  .  sin  0  .  cos  TO  .  sin  TO*  —  V  .  sin  0  .  sin  TO  .  cos  TO*  =  — — -r  - — -. ; 

sin  &  dw 

and  multiplying  the  first  by  cos  TO',  and  the  second  by  sin  TO',  and  adding,  we  get 

V  .  sin  0  .  cos  ro  =  — — -  .  - — —  cos  ff  .  cos  TO*  .  -r— ;  —  sin  ff  .  cos  •a' .  V  ;  (b) 

sin  01     d  TO*  d  0 

Mid,  again,  multiplying  the  first  by  sin  TO',  and  the  second  by  —  cos  TO',  and  adding,  we  find 

V  .  sin  0  .  sin  TO  = : — -.    -r—.  -  cos  &  .  sin  TO'  .  -— ;  -  sin  0f  .  sin  TO*  .  V;  (9) 

sin  v    dvr  d  0' 


VOL.  IV 


3T 


498  LIGHT. 

Light.      Now,  the  second  members  of  these  equations,  (when  V  the  velocity  of  the  extraordinary  ray  is  any  function  of  0     Fart  IV. 
— V—— '  the  angle  it  makes  with  the  axis,  or  of  its  position  within  the  crystal,)  is  always  explicitly  given  in  terms  of  ff  v— v— ' 
and  BT%  so  that,  calling  P  and  Q  their  values  so  expressed,  we  have  at  once 


P 


tan  TO  =  -- ;  cos  -a  =  -  — ;  sin  0  ~  ^  P1  +  Q2  ; 

V  p«  -f-  Q» 

so  that  -a  and  0  are  directly  expressed  in  terms  of  TO' and  0' ;   and,  therefore,  the  direction  in  which  a  ray,  moving 
anyhow  within  the  crystal  will  emerge,  is  known,  and  vice  vend. 
793.  It  only  remains  to  execute  these  processes  in  the  case  before  us.     To  this  end  (for  simplicity)  we  shall  put 

V  =  1,  and  suppose  (since  a  and  b,  the  semiaxes  of  the  spheroid,  are  arbitrary)  b  =  — ,  or  p,  =  — ,  and  put  W 


for  the  radical  vo»Tcos  0s  +  68  .  sin  0S  when  we  shall  have 

W  .,  _  aa  -  62      cos  0 

Now  in  the  spherical  triangle  Z  B  p  we  have,  the  side  Zp  =  \  ;  Z  B  =  ff,  angle  p  Z  B  =  TO',  and  sidep  B  =  0, 
therefore,  by  spherical  trigonometry, 

cos  0  =  cos  X .  cos  6'  -f-  sin  X  .  sin  ff  .  cos  TO',  (1 0) 

and  differentiating  separately  with  respect  to  ff  and  TO', 

d  .  cos  0 

d .  cos  0  .  •/./•_/ 

—  =  —  sin  X  .  sin  0'  ,  sin  TO*. 

d  TO 

lii  then,  we  write  these  values  in  the  partial  differences  of  V  in  the  equations  (8)  and  (S,)  they  will  become 

sin  0  .cos  -ss  = 1  W1.  sin  0'.  cos  TO'  4-  (a?  —  b-)  cos  0  [sin  X  ( 1  -  cos  w'2.  sin  0'2)  —  cos  X.  sin  0'.  cos  0' .  cos  TO7]  {• 

ah  Vf(  ) 

sin#.sinTO= \  W4.sin0'.  sin  TO7  —  (a9  —  6s)  cos  0  [sin  X.  sin  TO',  cos  TO'.  sinO1*  +  cos  X.  sin  ff .  costf'.sinTO']  >. 

a  b  W( 

In  these,  let  b*  +  (tf  —  62)  cos  0*  be  put  for  W«,  and,  bearing  in  mind  that  the  value  of  cos  0  is  as  given  in 
the  equation  (10,)  we  shall  see  that  they  will  reduce  themselves  respectively  to 

sin  0 .  cos  TO  = r-=;  ")  &5  •  sin  0' .  cos  TO'  +  (as  -  b*)  .  sin  X  .  cos  0  f 

a  b  W    L 

that  is,  by  reason  of  (10,) 

(a*  -  61)  .  cos  X  .  sin  X  .  cos  0'  -f  (a7 .  sin  X-  -f  W-  .  cos  X2)  .  cos  TO'  .  sin  & 

—  sinO  .  cos  TO  = TTy — 

and  )..       (11) 

62  .  sin  0'  .  sin  TO' 

—  sin  0  .  sin  TO  = j-== — 

794.          These  equations,  conjointly  with  the  equations  expressing  the  value  of  W  in  terms  of  cos  0,  and  of  cos  0  in 
terms  of  &  and  TO',  afford  a  complete  solution  of  the  problem  in  the  case  when  a  ray  passes  out  of  a  crystal  into 
air,  and  suffice  to  determine  both  the  inclination  of  the  refracted  ray  to  the  surface,  and  the  inclination  oi"  the 
plane  in  which  it  lies  to  the  principal  section. 
For  brevity,  let  us  put 

a«  .  sin  X« -j- ft8  .  cos  X«  =  A;     a2.  cosX3  +  6s.  sinX»  =  B  ;    (a8  -  65)  .  sinX.  cos  X  =  C;         (12) 

and,  dividing  the  second  of  the  equations  (11)  by  the  first,  we  find 

b*  .  tan  0' .  sin  TO' 


which  gives  immediately  the  inclination  of  the  plane  of  emergence  to  the  principal  section,  or,  as  it  is  sometimes 
termed,  the  azimuth  of  the  emergent  ray. 


LIGHT. 


499 


Light.          Reciprocally,  if  having  given  the  angle  of  incidence  and  azimuth  of  a  ray  incident  externally  on  the  crystal,     Part  IV. 


we  would  find  the  angle  of  refraction  and  azimuth  of  the  intromitted  ray,  we  must  find  &  and  TO'  from  the  above 
equations  in  terms  of  0  and  TO.     This  may  be  thus  accomplished  : 

Take  x  =  tan  & .  cos  TO',  and  y  =  tan  ff  .  sin  TO*, 

1 


then 


and,  moreover, 


a*  +  y*  =.  tan  6",  and  cos  ff*  = 


795. 

Given  the 
path  of  the 
ray  without, 
required 
that  within 
the  crystal. 


tan  •a  = 


A  I 


now,  since  W*  =  6«  +  (a1  —  b')  .  cos  0«, 

=  cos  0'"  -)  - — -r  +  (a*  -  6s)  (cos  \  +  sin  X.  .  tan  0* .  cos  TO')')- 
v.  cos  6*  •  J 

the  second  of  the  equations  (11)  becomes,  by  squaring, 

{68  ~)         6« 

— -IT-  +  (a1  -  b1)  .  (cos  X  +  sin  \  .  tan  6' .  cos  TO ')«>•=:  —  (tan  0' .  sin  TO')', 
cos  W1  J        a* 

that  is, 

o«  (sin  0  .  sin  w)«  {  6«  (1  +  «•  +  y»)  +  (a«  -  ft2)  (cos  X  +  x  .  sin  X)2  }  =  62  y2, 
that  is,  developing 

a2  .  (sin  0  .  sin  TO)2  {  A*2  +  2  C  x  +  B  +  62  y2  }  =  62  y2. 

62  y  Ify  C 

Now  we  have  A  x  +  C  =  -,  and  ,r  — — -. 

tan  TO  A .  tan  TO        A 

And,  on  substitution,  this  equation  will  be  found  to  take  the  form  p  y2  +  q  =  0,  and  being  resolved  to  give 

a2 .  sin  0  .  sin  TO 


y  —  tan  ff  .  sin  TO'  =r 


-/A  —  a2  .  sin  02  (A  .  sin  TO2  -{-  62 .  cos  TOS 
and  substituting  this  in  the  value  of  x,  we  find 

a?  If  sin  9  .  cos  TO 


(14) 


*  =  tan  &  .  cos  TO'  = 


*/  A  -  a2  .  sin  02  {  A  .  sin  TO2  +  b2  .  cos  TO2  } 


_£ 
A 


(15) 


These  equations  are  identical  with  those  demonstrated  by  Malus  in  his  Theorie  de  la  Double  Refraction,  with  some 
slight  differences  of  notation  only,  arising  from  our  having  reckoned  TO  and  TO'  from  the  opposite  point  of  the  circle. 

The  values  of  A,  B,  and  C  depend  only  on  a,  b,  and  X,  that  is,  on  the  peculiar  nature  of  the  crystal,  which       796. 
determines  the  ratio  of  the  axes  of  the  spheroid  of  double  refraction,  and  on  the  inclination  of  the  axis  to  the  Particular 
surface  on  which  the  ray  is  incident.     The  former  are  constant  for  one  and  the  same  crystal,  however  the  surface  aPP''cat'on> 
be  placed  ;  the  latter  is  constant  for  any  given  surface.     Hence  it  appears,  that  the  general  law  of  extraordinary 
refraction,  when  we  confine  ourselves  to  the  consideration  of  a  surface  given  in   position  with  respect  to   the 
axis,  resolves  itself  into  an  infinite  variety  of  particular  laws,  some  of  which  we  shall  now  consider. 

Case  1.     X  =  0,  the  surface  perpendicular  to  the  axis  ;  A  =  62  ;  B  =  a4  ;  C  =  0,  and  the  equations  (14)  and       797. 
(15)  become  1st.  When 

sin  0  .  sin 


tan  0'  .  sm  TO  =  -y-  . 
t> 


;  tan  0f  .  cos  TO'  = 


lar  to  the 
surface. 


these  equations  (as  well  as  Equation  13)  give  TO'  =  TO,  so  that  in  this  case  the  plane  of  refraction  is  the  same  with 
that  of  incidence,  and  the  extraordinary  ray  is  not  deviated  out  of  the  vertical  plane.     Hence,  we  get  simply 


a*  sin  0 

tan  0'  -  —   . 

o        A/  1  -  a*  .  sin 


(16) 


which  expresses  the  law  of  extraordinary  refraction  in  this  case.     If  0  =  0,  0'  =  0,  or  the  ray  incident  perpen- 

G?  1 

dicularly  passes  unrefracted  along  the  axis.     If  0  =.  90°,  tan  &  =  -  —  f  Now  if  we   put  b  —  —  and 


a  =  —  T,  this  becomes 

X 


tan  0'  =±  s~ 


(17) 


which,  fi  and  p.'  being  each  greater  than  unity,  is  always  real,  so  that  the  ray  can  enter  the  crystal  however  oblique 
its  incidence. 


500  LIGHT. 

Light.          Case  2.  When  the  axis  lies  in  the  surface,  or  \  =  90°  ;  A  =  a1 ;  B  =  6" ;  C  =  0,  and  the  equations  become       *""*  'v- 

a  .  sin  0  .  sin  ra 

tan  ff .  sin  -a  =  (18) 

2d-  when  V  1  -  sin  6»  {  a?  .  sin  TO'  +  6» .  cos  TO'  } 

the  axis  lies 

in  the  »ur-  62  sin  e  .  cos  TO 

tan «- .  cos  *  =  —  .     ,.         ,  ,,,f    0    _,   _.  ,   ..    _===;  (19) 

(20) 


o2  /  /.  V 

tan  is  —  -jj- .  tan  -a  =  I  — p  1   .  tan  OT  . 


The  latter  of  these  equations  shows  that  the  extraordinary  ray  deviates  from  the  plane  of  incidence.  The  amount 
of  this  deviation  is  nothing  when  the  plane  of  incidence  coincides  with  the  principal  section,  but  increases  on 
either  side  of  it  till  it  attains  a  certain  magnitude,  the  deviation  being  from  the  axis,  or  the  plane  of  refraction 
making  a  greater  angle  with  the  axis  than  that  of  incidence.  The  two  planes  then  approach  each  other,  and 
when  OT  =  90°,  tan  -EJ  =  cc,  tan  ro'  =  cc,  and,  consequently,  is'  =  90°,  or  the  plane  of  refraction  coincides  with 
that  of  incidence. 

799.  The  equations  (18)  and  (19)  show  that  in  the  present  case,  the  refracted  ray  does  not  describe  a  conical  surface 
Case  of  about  the  perpendicular  when  the  incident  one  does  so,  and  therefore  that  the  law  of  refraction  varies  in  every 
•efraction  m  different  azimuth.  Two  cases  deserve  express  notice,  viz.  those  in  which  the  plane  of  incidence*is  coincident 
pal  section  w''^  *ne  P"ncipa'  section,  and  when  perpendicular  to  it.  In  the  former,  w  =  0  and  w'  =  0,  so  that  we  have 

62  sin  0 

tan0'  =  -  .     .          --  .  (21) 

a  l-^ 


A  remarkable  relation  holds  good  in  this  case  between  the  angles  of  refraction  of  the  ordinary  and  extraordinary 
ray,  their  tangents  being  to  each  other  in  a  given  ratio.  In  fact,  if  we  find  (#')  =  the  angle  of  refraction  for  the 

ordinary  ray,  we  have  sin  (01)  =  —  .  sin  0  =:  b  .  sin  0,  and,  consequently, 

tan  9  =  L  .  _SJ"_^L-  =  A  .  tan  (fl).  (M) 

a       V  1  -  sm  (<O2        a 
In  the  latter  case,  when  the  plane  of  refraction  is  at  right  angles  to  the  axis,  CT  =  TO-'  =:  90°,  and  we  get 

tan  tf  =          a-sm°         .  sin  0  -  a  .  sin  0.  (23) 

V  1  -  a5  .  sin  6* 

800.  In  this  case,  therefore,  the  sine  of  incidence  is  in  a  given  ratio  to  that  of  refraction,  and  the  extraordinary 
Case  of  re-  , 
fraction  at    refractiOn  is  performed  according  to  the  same  law  as  the  ordinary,  only  with  a  different  index,  viz.  ft',  or  —  .instead 

to  the  prin-  j 

cipalsec-     of  ft,  or  —  .     Hence,  if  we  consider  only  this  particular  case,  the  medium  will  appear  to  have  two  indices  of 

tion.  n 

refraction,  an  ordinary  and  an  extraordinary  one. 

801.  It  was  by  a  careful  examination  of  these  cases,  that  Dr.  Wollaston  was  enabled  to  verify  the  Huygenian  law. 
Experimen-  fne  circumstance  last  mentioned  puts  it  in  our  power  to  determine  in  the  case  of  any  particular  crystal  the  axes  of 

its  spheroid  of  double  refraction.     We  have  only  to  cut  a  prism  of  it,  having  its  refracting  angle  parallel  to  the 
mining  the    ax's-  an^  ascertain  its  indices  of  refraction  according  to  the  principles  laid  down  in  the  former  part  of  this  Essay, 

r'  eroid  of  \  \ 

ble  re-    and  calling  them  ft.  and  /,  the  semiaxes  of  the  spheroid  will  be  respectively  —  and  —r.     Thus,  in  the  instance 
fraction.  f-  f- 

of  carbonate  of  lime,  which  Malus  examined  with  the  utmost  care,  he  found  the  two  values  of  a  and  b  to  be 
respectively  equal  to  the  numbers  0.67417  and  0.60449,  having  determined  ft'  =  1.4833,  and  ft  =  1.6543. 
(Thkorie.  de  la  Double  Refraction,  p.  199.) 

802.  In  this  arrangement,  however,  it  is  not  possible  to  decide  simply  from  the  phenomena  of  refraction,  which  is 
the  ordinary,  and  which  the  extraordinary   ray.     There  are,  however,  infallible  and  easy  criteria,  as  we  shall 
speedily  show.     Meanwhile,  we  may  for  the  present  content  ourselves  with  observing,  that  as  a  moderate  devia- 
tion from  the  exact  azimuth  OT  =  90°  imparts  to  the   extraordinary  ray  a  deviation  from  the  plane  of  incidence 
which  does  not  happen  to  the  ordinary  one,  this  may  serve  for  a  criterion  to  distinguish  them  in  certain  cases. 

803.  The  square  of  the  velocity  of  the  ordinary  ray  within  the  medium  is  /tsVJ,  or  /t4,  that  is.  —  ,  and  is   constant. 
Law  of  the  ^  ,.         .       ,  2 

f  the  extraordinary  is  V,s>  or         _  that  ig  to  say>  £2£^  +  !!L^( 


or,  V"  =  —  -  (—  -  —     .  sin  0!. 

6*  6"         a 


LIGHT.  501 

Light.      The  square  of  the  velocity  of  the  extraordinary  ray  is  therefore  (in  the  corpuscular  doctrine)  diminished  by  a  quantity     Pan  IV. 
«— y^^  proportional  to  the  square  of  the  sine  of  the  inclination  of  the  ray  within  the  crystal,  to  the  axis.    We  say  dimin-  ^— v— •—' 
ished,  in  the  algebraical  sense  of  the  word,  supposing  a  >  6,  this  agrees  with  common  parlance ;  but  if  a  <  6,  Division  of 
then  it  will  be  increased.     This  gives  rise  to  the  subdivision  of  the  crystallized  bodies  now   treated  of  into  two  fT813'8 
classes,  which  have  by  some  been  termed  attractive  and  repulsive :  by  others,  positive  and  negative,  which  seems  j°v'°  P°^'~ 
preferable,  as  the  former  phrases  involve  theoretical  considerations.     Positive  crystals  are,  then,  such  as  have  a  negative. 

less  than  b,  or  in  which  the  spheroid  of  double  refraction  is  prolate.     In  these  the  coefficient  —    (  —  —    — -  \ 

which  we  call  k  is  positive,  and  the  square  of  the  velocity,  or  c*  -j-  k  .  sin  0',  (where  »  =  —    =r  velocity  of  the 

ordinary  ray  within  the  medium,)  is  increased  by  the  action  of  the  medium,  and  is  a  minimum  in  the  axis.  In  the 
negative  class  the  coefficient  k  is  negative,  a  >  6,  or  the  spheroid  of  double  refraction  is  oblate.,  and  the  velocity 
of  the  extraordinary  ray  is  a  maximum  along  the  axis.  In  positive  crystals,  therefore,  the  index  of  ordinary 
refraction  (/i)  is  less  than  that  of  extraordinary  ;  in  negative,  greater.  To  the  former  class  belong  quartz,  ice, 
zircon,  apophyllite,  (when  uniaxal  ;)  and  to  the  latter,  Iceland  spar,  tourmaline,  beryl,  emerald,  apatite,  &c. 
The  negative  class,  as  far  as  our  present  knowledge  extends,  far  out-numbers  the  positive  among  natural  and 
artificial  crystals.  They  were  first  distinguished  by  M.  Biot. 

In  the  undulatory  doctrine  the  velocity  is  the  reciprocal  of  what  it  is  in  the  corpuscular  doctrine,  and  is       §04. 
therefore  directly  as  the  radius  of  the  spheroid  of  double  refraction.     Hence  a  wave  propagated   within    the  Undulationj 
crystal  from  any  point  will  run  over  in  the  same  time  in  different  directions,  distances  proportional  to  the  radii  propagated 
of  the  spheroid  parallel  to  those  directions ;  and  therefore  at  any  instant  the  surface  of  the  whole  wave  will  be  '"."P1-6" 
itself  a  spheroid  similar  to  the  spheroid  of  double  refraction.     This  is  Huygens's  conception  of  the  subject.     It 
requires  us  to  regard  the  crystal,  or  the  ether  within  the  crystal  through  which  the  undulation  is  propagated,  as 
having  different  elasticities  in  different  directions.     As  far  as  regards  the  molecules  of  a  solid  body  there  is  no 
apparent  impossibility  or  improbability  in  such  an  idea,  but  the  contrary ;  but  if  we  regard  the  propagation  of 
the  light  within  the  medium  to  take  place  by  the  elasticity  of  the  ether  only,  we  must  then  suppose  its  molecules 
in  crystallized  bodies  to  be  in  a  very  different  physical  state  from  what  they  are  in  free  space,  and  either  to  be 
in  some  manner  connected  with  the  solid  particles,  (forming  atmospheres,  for  instance,  about  them,)  or  as 
subjected  to  laws  of  mutual  action  which  approximate  to  those  governing  the  molecules  of  solid  bodies  ;  and 
partaking,  themselves,  of  a  regular  crystalline  arrangement  and  mutual  dependency. 

To  pursue  the  particular  applications  of  the  general  formulae  (13,)  14,)  and  (15)  farther,  would  be  far  beyond      §05. 
our  limits.     The  reader  who  is  curious  on  this  very  interesting  part  of  Physical  Optics,  and  who  wishes  to  be  Malus's' 
delighted  and  instructed  by  a  combination  of  consummate  mathematical  skill  with  sound  experimental  research,  further 
which  may  deservedly  be  cited  as  a  model  of  the  kind,  will  find  every  thing  which  relates  to  the  subject  in  its  researclles- 
best  form  in  the  work,  already  so  often  cited,  of  Malus,  Theorie  de  la  Double  Refraction,  which  gained  the  mathe- 
matical prize  of  the  French  Institute  in  1810.     To  the  theory  of  the  internal  reflexion  of  the  extraordinary  ray 
which  offers  many  remarkable  particularities,   as  there  delivered,  we  must  especially  refer  him,  as  well  as  to 
his  investigation  of  the  foci  of  lenses  formed  of  doubly  refracting  crystals,  of  which  we  shall  here  only  extract  Foci  of  a 
the  results,  in  the  single  case  of  a  double  convex  lens  having  the  axis  of  double  refraction  in  the  direction  doubly 
of  the  axis  of  the  lens.  refracting 

Let  r,  r1  be  the  radii  of  the  anterior  and  posterior  surfaces  of  the  leiis,  both  supposed  convex. 

d  =  distance  of  the  radiant  point  in  the  axis. 
a,  b  =  the  equatorial  and  polar  radii  of  the  spheroid  of  double  refraction,  as  above. 

D  =  distance  of  the  conjugate  focus  behind  the  lens  for  extraordinary  rays. 

A  =  extraordinary  focal  length  for  parallel  rays. 

F  =  ordinary  focal  length  for  parallel  rays. 
Then  shall  we  have  for  the  general  expression  of  D, 

ePbdrr1  _          -  6  r  / 

~~~" 


d  (r  +  /)  (2  6«  -  a?  -  a«  6)  -  a»6  r  r1  '  (r  +  /)  (1  -6)  ' 

If  the  lens  be  equi-convex,  or  r  =  r1, 

a2  b  r  d  a?br 


2  (2  6«  -  a2  -  a4  6)  d  -  a*  6  r  '  2  (2  6s  -  a'  -  a"  6)  ' 


F=-  A-F=-2F. 


2  (1  -  6)  '  '  2  6s  -  a«  -  a*  6  ' 

In  the  case  of  Iceland  spar,  these  last  equations  become 

D  =  - r  .  88,2286 ;         F  =-  r  .  0,7642  ;         D  -  F  =- F  .  114,4546; 
and  in  the  case  of  rock  crystal  (quartz) 

D  =  -  r  .  0.9628  ;        F  =  -  r  .  0,8958 ;         D  -  F  =  -  F.0,0748. 

To  represent,  in  general,  the  course  of  any  extraordinarily  refracted  ray,  Huygens  has  giving  the  following 
construction,  (fig.  170.)  Let  H  E  D  be  the  elliptic  section  of  the  spheroid  of  double  refraction  by  the  surface, 
and  R  C  the  incident  ray  falling  on  C  its  centre,  and  B  C  K  the  orthographic  projection  of  the  ray  R  C  on  the 


502 


LIGHT. 


811. 

Form  and 


Light,  surface.  Let  H  M  E  be  the  portion  of  the  spheroid  within  the  crystal,  whose  axis  passes  through  C,  and  may  be 
v— ~\— •-'  anyhow  inclined  to  the  surface.  Then  will  the  surface  of  this  spheroid  be  the  boundary  of  the  wave  propagated 
Huygens's  from  c  as  a  centre,  after  the  lapse  of  a  given  time.  Draw  C  O  in  the  plane  R  0  K  at  right  angles  to  R  C^  and 
£°°sf™cex.  make  OK  (perpendicular  to  C  K,  or  parallel  to  RC)  equal  *.o  the  space  described  by  light  in  the  medium 
traordinary  exterior  to  the  crystal  in  the  same  given  time.  This  will  determine  the  point  K  in  the  line  B  C  K.  Through  K 
refraction,  draw  K  T  perpendicular  to  B  K,  and  about  K  T  as  an  axis  let  a  plane  revolve  passing  through  K  T,  till  it  touches 
Fig.  170.  the  surface  of  the  spheroid  in  I.  Join  C  I,  and  C  I  is  the  extraordinary  refracted  ray. 

807.  The  demonstration  of  this  construction  (granting  the  principle  of  spheroidal  undulations)  is  evident,  if  we 
Demonstra-  consider  the  manner  in  which  the  general  wave,  a  perpendicular  to  whose  surface  forms  what  we  term  a  ray  of 
the"  rind     Iight'  ^at  least  *n  sin»'y  refractinjr  media,)  arises  from  the  reunion  of  all  the  elementary  waves  propagated  from 
pie  of^ne-  every  Part  of  tne  surface,  (Art.  586.)     In  this  construction,  if  we  conceive  a  plane  wave  from  an  infinitely  distant 
roidal  no-    luminary  perpendicular  to  RC  to  move  along  RC,  every  point  in  the  line  CK  will  become  in  succession,  and 
dulations.     every  point  in  the  line  C  D  perpendicular  to  C  K,  or  parallel  to  KT  simultaneously,  a  centre  of  vibration.     The 

general  wave,  therefore,  will  be  a  surface  touching  all  ellipsoids  described  about  each  point  of  the  surface,  having 
their  axes  parallel,  their  generating  ellipses  similar,  and  their  linear  dimensions  proportional  to  the  distance  of 
their  centre  from  the  line  KT.  Of  course  it  can  be  no  other  than  the  tangent  plane  I  KT  drawn  as  above. 

808.  This  then  will  be  the  form  and  position  of  the  general  wave  within  the  crystal.     Now  if  we  consider  only  that 
very  minute  portion  of  it  which  emanates  from  C,  it  is  evident  that  I  is  the  corresponding  point  in  it;    and 
therefore  C  I  is  necessarily  the  direction  of  the  ray,  because  I  is  the  point  on  which  that  portion  of  the  general 
wave  transmitted  through  a  very  small  aperture  at  C  would  fall. 

809.  Thus  we  see,  that  in  the  case  of  the  extraordinary  ray,  we  are  no  longer  to  regard  the  ray  as  a  perpendicular 
Oblique       to  the  surface  of  the  wave.     It  is  propagated  obliquely  to  that  surface.     So  soon,  however,  as  the  wave  emerges 

'"  into  the  ambient  medium,  the  usual  law  of  perpendicular  propagation  is  restored, 
ordinary  ™~      To  show  the  identity  of  the  law  of  extraordinary  refraction  resulting  from  this  construction  with  that  expressed 

810.  ty  the  general  equations  (13,)  (14,)  and  (15,)  we  have  only  to  translate  it  into  analytical  language.     This  has 
been  done  by  Malus,  in  his  work  above  referred  to ;  and  the  reader  may  also  consult  Biot's  Traite  General  de 
Physique,  for  a  more  elementary  exposition  of  the  process,  which  is  one  of  considerable  complexity,  for  which 
reason  we  shall  not  embarrass  ourselves  with  it  here. 

Some  very  remarkable  and  important  consequences  follow  from  this  mode  of  viewing  the  subject.     It  appears 
that  when  a  plane  wave  is  incident  on  a  doubly  refracting  surface,  the  transmitted  extraordinary  wave  is  also 
position  of  plane,  and  advances  with  a  uniform  velocity  in  a  direction  oblique  to  itself.     Consequently  the  velocity  is  also 
cHnar"'™0''"  uniform  in  a  direction  perpendicular  to  itself.    Moreover,  its  common  section  with  the  surface  is  always  parallel  to 
ray-  j£  Tt  or  to  tne  cornmon  section  of  the  incident  wave  with  the  same  surface.     Hence,  it  is  evident,  that  it  moves 
in  the  same  way  as  an  ordinarily  transmitted  wave  would  do,  and  at  any  instant  has  the  same  position  that  such 
a  wave  would  have,  provided  the  index  of  refraction  in   the  latter  case  were  properly   assumed.     The   only 
difference  is,  that  the  motions  of  the  vibrating  molecules,  of  which   they  respectively  consist,  are  executed   in 
different  planes.     Now,  when  this  wave  emerges  from  the  medium,  it  obeys  the  same  laws  as  on  its  entry,  only 
reversed ;  so  that  it  still  continues  a  plane  wave,  and  its  common  section  with  the  surface  of  emergence  remains 
unaltered. 

Hence  it  follows,  that  if  we  cut  a  prism  of  any  doubly  refracting  crystal  with  one  axis,  and  transmit  through 
it  a  ray  incident  in  a  plane  at  right  angles  to  the  edge  of  the  prism,  the  ordinary  and  extraordinary  ray  will  both 
emerge  in  that  plane,  and  their  separation  will  take  place  in  a  plane  containing  the  incident  and  ordinarily- 
refracted  ray,  and  will  therefore  be,  apparently,  such  as  would  arise  from  attributing  two  ordinary  refractive 
powers  to  the  medium.  It  is  only  when  the  edge  of  the  prism  is  oblique  to  the  plane  of  incidence,  that  the 
extraordinary  ray  can  deviate  from  the  plane  containing  the  incident  and  ordinarily  refracted  rays. 

We  see,  then,  that  in  the  theory  of  extraordinary  refraction,  it  is  necessary  to  consider,  as  distinct,  two  things, 
Velocity  of  which,  in  that  of  ordinary,  are  one  and  the  same,  viz.  the  velocity  of  the  luminous  waves,  and  the  velocity  of  the 
twu>"»OKand  Ta^s  °f  '*£*'•  This  distinction  will  require  to  be  very  carefully  kept  in  view  hereafter,  when  we  come  to  treat 
of  rays  of  of  the  law  of  refraction  in  crystals  with  two  axes  of  double  refraction.  For  this,  however,  we  are  not  yet 
light  dis-  prepared,  as  the  knowledge  of  this  law  presupposes  an  acquaintance  with  a  multitude  of  facts  relative  to  the 
tinguished.  polarization  of  light,  of  which  we  have  yet  said  nothing.  It  will  suffice  here  to  mention,  that  the  whole  doctrine 
Theory  of  of  double  refraction  has  recently  undergone  a  great  revolution  ;  one,  indeed,  which  may  be  said  to  have  changed 
the  face  of  Physical  Optics,  in  consequence  of  the  researches  of  M.  Fresnel.  It  had  all  along  been  taken  for 
granted,  that  in  crystals  possessed  of  double  refraction,  one  of  the  pencils  followed  the  ordinary  law  of  propor- 
tional sines.  It  had,  moreover,  been  ascertained,  hy  experiments  hereafter  to  be  related,  that  the  difference  of 
the  squares  of  the  velocities  between  the  two  pencils  is  in  all  cases  proportional  to  the  product  of  the  sines  of 
the  angles  contained  between  the  extraordinary  ray  (as  it  was  termed)  and  the  two  axes,  or  directions  in  which 
the  refraction  is  single.  It  was  hence  concluded,  that  the  velocity  of  the  extraordinary  pencil  was  in  all  cases 

represented  by  J  v*  -f-  k  .  sin  0  .  sin  0',  v  being  that  of  the  ordinary  one,  and  k  a  constant  depending  on  the 
nature  of  (he  crystal,  and  <p,  0'  the  angles  in  question.  This  granted,  there  would  be  no  difficulty  in  deter- 
mining the  form  of  the  surface  of  double  curvature,  which  should  be  substituted  for  the  Huygenian  spheroid; 
so  as  to  render  the  same  construction  with  that  described  in  Art.  806,  or  the  general  formula:  in  Art.  792,  appli- 
cable to  this  case.  In  fact,  if  we  call  a  the  semi-angle  between  the  two  axes,  and  conceive  three  coordinates  x, 
y,  z,  of  which  JT  bisects  that  angle,  the  plane  of  the  x,  y  containing  both  axes,  it  is  easy  to  see,  by  spherical 
trigonometry,  that  we  must  have 


Part  IV. 


812. 

Conse- 
quences in 
double 
refraction 
through 
prisms. 
813. 


double  re- 
fraction i 
bi-axal 
crystals 
deferred, 

and  why, 


L  [  G  H  T.  503 

Light.  a  .  cos  a  -f-  y  .  sin  a  ,  _  x  .  cos  a  —  y  .  sin  «  Part  IV. 

•"—  '  • 


Hence,  since  r  (  ^  I*  -)-  y8  -j-  z*)  the  radius  of  the  surface  of  the  wave,  is  always  equal  to 

1  1 


V  , 

v  c*  +  k*  .  sin  0  .  sin  0' 

a  simple  substitution  would  give  at  once  the  equation  of  its  surface  as  referred  to  the  three  coordinates  x,  y,  z  ; 
namely, 

0  =  (If  -  o4)  (jr8  +  y*  +  z2)8  +  2  (jr8  +  y'  +  z8)  (us  -  /ts  *8  .  cos  a8  -  it2  y'  .  sin  a«) 

+  &8  (x8  .  cos  a«  -f  y*  .  sin  a2)2  -  1, 

which  it  would  be  easy  then  to  transform  into  functions  of  r,  •&,  and  0,  as  required  for  the  application  of  the 
general  analytical  formulae  by  the  usual  substitutions 

z  =  r  .  sin  0  ;         y  =:  r  .  sin  0  .  sin  OT  ;         x  =  r  .  sin  0  .  cos  w. 

The  researches  of  M.  Fresnel,  however,  as  before  remarked,  have  destroyed  the  basis  on  which  this  theory 
rested,  by  demonstrating  the  non-existence  of  an  ordinarily  refracted  ray  in  the  case  of  crystals  with  two  axes. 
The  theory  which  he  has  substituted  in  its  place,  however,  and  which  it  is  impossible  to  regard  otherwise  than 
as  one  of  the  finest  generalizations  of  modern  science,  we  must  reserve  for  a  more  advanced  place  in  this  essay. 
We  shall  now  proceed  to  treat 

Of  the  Polarization  of  Light. 

The  phenomena  which  belong  to  this  division  of  our  subject  are  so  singular  and  various,  that  to  one  who  has  §14. 
only  studied  the  subject  of  Physical  Optics  under  the  relations  presented  in  the  foregoing  pages,  it  is  like  enter- 
ing into  a  new  world,  —  so  splendid  as  to  render  it  one  of  the  most  delightful  branches  of  experimental  inquiry  ; 
and  so  fertile  in  the  views  it  lays  open  of  the  constitution  of  natural  bodies,  and  the  minuter  mechanism  of  the 
universe,  as  to  place  it  in  the  very  first  rank  of  the  physico-mathematical  sciences,  which  it  maintains,  by  the 
rigorous  application  of  geometrical  reasoning  its  nature  admits  and  requires.  The  intricacy  as  well  as  variety 
of  its  phenomena,  and  the  unexampled  rapidity  with  which  discoveries  have  succeeded  each  other  in  it,  have 
hitherto  prevented  the  possibility  of  embodying  it  satisfactorily  in  a  systematic  form  ;  but,  after  the  rejection  of 
numberless  imperfect  generalizations,  it  seems  at  length  to  have  acquired  that  degree  of  consistency  as  to  enable 
us  —  not,  indeed,  to  deduce  every  phenomenon,  by  distinct  steps,  from  one  general  cause  —  but  to  present  them, 
at  least,  in  something  like  a  regular  succession  ;  to  show  a  mutual  dependence  between  their  several  classes, 
which  is  a  main  step  to  a  complete  generalization  ;  and  to  dispense  with  the  bewildering  detail  of  an  immense 
multitude  of  individual  facts,  which,  having  served  their  purpose  in  the  inductive  process,  must  in  future  be 
considered  as  having  their  interest  merged  in  that  of  the  laws  from  which  they  flow. 


§  II.  General  Ideas  of  the  Distinction  between  Polarized  and  Unpolarized  Light. 

In  all  the  properties  and  affections  of  light  which  we  have  hitherto  considered,  we  have  regarded  it  as  315 
presenting  the  same  phenomena  of  reflexion  and  transmission,  both  as  respects  the  direction  and  intensity  of 
the  reflected  or  transmitted  beam,  however  it  may  be  presented  to  the  reflecting  or  refracting  surface,  provided 
the  angle  of  incidence,  and  the  plane  in  which  it  lies,  be  not  varied.  And  this  is  true  of  light  in  the  state  in 
which  it  is  emitted  immediately  from  the  sun,  or  from  other  self-luminous  sources.  A  ray  of  such  light,  incident 
at  a  given  angle  on  a  given  surface,  may  be  conceived  to  revolve  round  an  axis  coincident  with  its  own  direction  ; 
or,  which  comes  to  the  same  thing,  the  reflecting  or  refracting  surface  may  be  actually  made  to  revolve  round  the 
ray  as  an  axis,  preserving  the  same  relative  situation  to  it  in  all  other  respects,  and  no  change  in  the  phenomena 
will  be  perceived.  For  instance,  if  in  a  long  cylindrical  tube  we  fix  a  plate  of  glass,  or  any  other  medium  at 
any  angle  of  inclination  to  the  axis ;  and  then,  directing  the  tube  to  the  sun,  turn  the  whole  apparatus  round  on 
its  axis,  the  intensity  of  the  reflected  or  refracted  ray  will  suffer  no  variation,  and  its  direction  (if  deviated)  will 
revolve  uniformly  round  with  the  apparatus,  so  that  if  received  on  a  screen  connected  invariably  with  the  tube,  it 
will  continue  to  fall  on  the  very  same  point  in  all  parts  of  its  rotation.  Or  we  may  receive  the  light  from  a 
piece  of  white  hot  iron  at  any  angle  on  any  medium,  and  its  phenomena  will  be  precisely  the  same,  whether  the 
iron  be  at  rest,  or  be  made  to  revolve  round  an  axis  coincident  with  the  direction  of  the  ray. 

But,  if  instead  of  employing  a  ray  immediately  emitted  from  a  self-luminous  source,  we  subject  to  the  same      816 
examination  a  ray  that  has  undergone  some  reflexions,  refractions,  or  been   in   any  one  of  a  great  variety  of  Polarized 
ways  subjected  to  the  action  of  material  bodies,  we  find  this  perfect  uniformity  of  result  no  longer  to  hold  good.  rays  have 
It  is  no  longer  indifferent  in  what  plane,  with  respect  to  the  ray  itself,  the  reflecting  or  refracting  surface   is  actlu'red 
presented  to  it.     It  seems  to  have  acquired  sides  ;  a  right  and  left,  a  front  and  back  ;  and  the  intensity,  though  f^  ™la" 
not  the  direction  of  the  reflected  or  transmitted  portion,  depends  materially  on  the  position  with  respect  to  these  external 

space. 


504  LIGHT. 

Light,  sides,  in  which  the  plane  of  incidence  lies,  though  every  thing  else  remains  precisely  the  same.  In  this  state  it  is 
— -V— —•  said  to  be  polarized.  The  difference  between  a  polarized  and  an  ordinary  ray  of  light  can  hardly  be  more  readily 
Illustration,  conceived  than  by  assimilating  the  latter  to  a  cylindrical,  and  the  former  to  a  four-sided  prismatic  rod,  such  as  a 
lath  or  a  ruler,  or  other  long,  flat,  straight  stick.  It  is  evident  that  the  cylinder,  if  inclined  to  any  surface  at  a 
given  angle  in  a  given  plane,  may  be  turned  round  its  own  axis  without  altering  its  relations  to  the  plane,  while 
those  of  the  prism  will  vary  essentially  according  to  the  position  of  its  sides.  Let  us  suppose,  for  instance,  (it 
is  but  a  simile,  which  we  do  not  wish  the  reader  to  dwell  on  for  a  moment,  or  to  imagine  that  any  analogy  is 
hereafter  intended  to  be  established,)  that  we  had  occasion  to  thrust  such  a  rod  into  a  surface  composed  of 
detached  fibres,  all  lying  in  one  direction,  or  of  scales  or  laminae  arranged  parallel  to  one  another,  we  should 
find  a  much  greater  facility  of  penetration  on  presenting  its  broad  side  in  the  direction  of  the  laminae  or  fibres, 
than  transverse  to  them.  A  thin  sheet  may  be  slipped  between  the  bars  of  a  grating,  which  would  present  an 
insuperable  obstacle  to  it  if  presented  cross-wise. 

817.  But,  to  be  more  particular,  and  to  give  a  more  clear  conception  of  the  marked  distinction  which  exists  between 
Property  of  a  polarized  and  an  unpolarized.  ray.     There  are  many  crystallized  minerals,  which  when  cut  into  parallel  plates 

B  tourma-  are  sufficiently  transparent,  and  let  pass  abundance  of  light  with  perfect  regularity,  but  which,  nevertheless,  at 
other  crys-  'ts  emergence  's  found  to  have  acquired  that  peculiar  modification  here  in  question.  One  of  the  most  remark- 
tais.  able  of  these  is  the  tourmaline.  This  mineral  crystallizes  in  long  prisms,  whose  primitive  form  is  the  obtuse 

rhomboid,  having  its  axis  parallel  to  the  axis  of  the  prism.  The  lateral  faces  of  these  prisms  are  frequently  so 
numerous  as  to  give  them  an  approach  to  a  cylindrical  or  cylindroidal  form.  Now  if  we  take  one  of  these 
crystals,  and  slit  it  (by  the  aid  of  a  lapidary's  wheel)  into  plates  parallel  to  the  axis  of  the  prism  of  moderate 
and  uniform  thickness,  (about  -fo  of  an  inch,)  which  must  be  well-polished,  luminous  objects  may  be  seen 
through  them,  as  through  plates  of  coloured  glass.  Let  one  of  these  plates  be  interposed  perpendicularly 
between  the  eye  and  a  candle,  the  latter  will  be  seen  with  equal  distinctness  in  every  position  of  the  axis  of  the 
plate  with  respect  to  the  horizon,  (by  the  axis  of  the  plate  is  meant  any  line  in  it  parallel  to  the  axes  of  its 
molecules,  or  to  the  axis  of  the  prism  from  which  it  was  cut.)  And  if  the  plate  be  turned  round  on  its  own 
plane,  no  change  will  be  perceived  in  the  image  of  the  candle.  Now,  holding  this  first  plate  in  a  fixed  position, 
(with  its  axis  vertical,  for  instance,)  let  a  second  be  interposed  between  it  and  the  eye,  and  turned  round  slowly  in 
its  own  plane,  and  a  very  remarkable  phenomenon  will  be  seen.  The  candle  will  appear  and  disappear  alternately 
at  every  quarter  revolution  of  the  plate,  passing  through  all  gradations  of  brightness,  from  a  maximum  down 
to  a  total,  or  almost  total,  evanescence,  and  then  increasing  again  by  the  same  degrees  as  it  diminished  before. 
If  now  we  attend  to  the  position  of  the  second  plate  with  respect  to  the  first,  we  shall  find  that  the  maxima  of 
illumination  take  place  when  the  axis  of  the  second  plate  is  parallel  to  that  of  the  first,  so  that  the  two  plates 
have  either  the  same  positions  with  respect  to  each  other  that  they  had  in  the  original  crystal,  or  positions  differing 
by  180°,  while  the  minima,  or  evanescences  of  the  image,  take  place  exactly  90°  from  this  parallelism,  or  when 
the  axes  of  the  two  plates  are  exactly  crossed.  In  tourmalines  of  a  good  colour,  the  stoppage  of  the  light  in 
this  situation  is  total,  and  the  combined  plate  (though  composed  of  elements  separately  very  transparent  and  of 
the  same  colour)  is  perfectly  opake.  In  others  it  is  only  partial ;  but  however  the  specimens  be  chosen,  a  very 
marked  defalcation  of  light  in  the  crossed  position  takes  place.  We  shall  at  present  suppose  that  the  specimens 
employed  possess  the  property  in  question  in  its  greatest  perfection.  Now  it  is  evident  that  the  light  which  has 
passed  through  the  first  plate  has  acquired  in  so  doing  a  property  totally  distinct  from  those  of  the  original  light 
of  the  candle.  The  latter  would  have  penetrated  the  second  plate  equally  well  in  all  its  positions  ;  the  former  is 
incapable  altogether  of  penetrating  it  in  some  positions,  while  in  others  it  passes  through  readily,  and  these 
positions  correspond  to  certain  sides  which  the  ray  has  acquired,  and  which  are  parallel  and  perpendicular 
respectively  to  the  axis  of  the  first  plate.  Moreover,  these  sides  once  acquired,  are  retained  by  the  ray  in  all  its 
future  course,  (provided  it  be  not  again  otherwise  modified  by  contact  with  other  bodies,)  for  it  matters  not  how 
great  the  distance  between  the  two  plates,  whether  they  be  in  contact  or  many  inches,  yards,  or  miles  asunder, 
not  the  least  variation  is  perceived  in  the  phenomenon  in  question.  If  the  position  of  the  first  plate  be  shifted, 
the  sides  of  the  transmitted  ray  shift  with  it,  through  an  equal  angle,  and  the  second  will  no  longer  extinguish 
it  in  the  position  it  at  first  did,  but  must  be  brought  into  a  position  removed  therefrom,  by  an  angle  equal  to 
that  through  which  the  first  plate  has  been  made  to  revolve. 

818.  A  great  many  other  crystallized  bodies   besides  the  tourmaline  possess  this  curious  property,  and  several  in 
Selection  of  great  perfection.     The  tourmaline,  however,  is  one   easily   procured,  and  being  exceedingly  useful  in  optical 
proper         experiments,  we  would  recommend  the  reader  who  has  any  desire  to  familiarize   himself  with  the  practical 

es'  manipulations  of  this  branch  of  optical  science,  to  provide  himself  with  a  good  pair  of  corresponding  plates  of 
this  mineral,  cut  and  polished  as  above  directed.  The  colour  is  a  point  of  great  .moment.  Those  of  a  blue 
or  green  colour  possess  the  property  in  question  very  imperfectly  ;  the  yellow  varieties,  unless  when  verging  to 
greenish  brown,  are  equally  improper,  the  best  colour  is  a  hair-brown,  or  purplish  brown,  and  they  may  be  slit 
and  polished  by  any  lapidary. 

819.  But  it  is  not  only  by  such  means  that  the  polarization  of  a  pencil  of  light  may  be  operated,  nor  is  this  the  only 
Various        character  which  distinguishes  polarized  from  ordinary  light.     We  shall,  therefore,  describe  in  order,  the  principal 
modes  of      means  by  which  the  polarization  of  light  may  be  performed,  and  the  assemblage  of  characters  which  are  inva- 
pntanzmg     j-jably  found  to  coexist  in  a  ray  when  polarized. 

The  chief  modes  by  which  the  polarization  of  light  may  be  eff,  cted,  are 
1st.   By  reflexion  at  a  proper  angle  from  the  surfaces  of  transparent  media. 

2d.   By    transmission    through    a  regularly    crystallized    medium   possessed   of  the    property    of  double  re- 
fraction. 

3d.  By  transmission  through  transparent,  uncrystallized  plates  in  sufficient  number,  and  at  proper  angles. 


LIGHT.  505 

Light.          4th.  By  transmission  through  a  variety  of  bodies,  such  as  agate,  mother-of-pearl,  &c.  which  have  an  approach     Part  IV- 
— "v*™"'  to  a  laminated  structure,  and  an  imperfect  state  of  crystallization.  x— "y-""' 

The  characters  which  are  invariably  found  to  coexist  in  a  polarized  ray,  being  the  chief  of  those  by  which  it       820. 
may  be  most  easily  recognised  as  polarized,  are —  Characters 

1.  Incapability  of  being  transmitted  by  a  plate  of  tourmaline,  as  above  described,  when  incident  perpendicu-  "[zeadpr° 
larly  on  it,  in  certain  positions  of  the  plate  ;  and  ready  transmission  in  others,  at  right  angles  to  the  former.        flight?' 

2.  Incapability  of  being  reflected  by  polished  transparent  media  at  certain  angles  of  incidence,  and  in  certain 
positions  of  the  plane  of  incidence. 

3.  Incapabiltiy  of  undergoing  division  into  two  equal  pencils  by  double  refraction,  in  positions  of  the  doubly 
refracting  bodies,  in  which  a  ray  of  ordinary  light  would  be  so  divided. 

Besides  which,  there  might  be  enumerated  a  vast  variety  of  other  characters,  which,  however,  it  will  be  better 
to  regard  as  properties  at  once  of  polarized  light,  and  of  the  various  media  which  affect  it.     It  cannot  fail  to  be 
remarked,  that  all    these  characters  are  of  the  negative  kind,  and  consist  in  denying  to  polarized  light  properties 
which  ordinary  light  possesses,  and  that  they  are  such  as  affect  the  intensity  of  the  ray,  not  its  direction.     Thus,  Affect  the 
the  direction  which  a  polarized  ray  will  take  under  any  circumstances  of  the  action  of  media,  is  never  different  intensity 
from  what  an  unpolarized  ray  might  take,  and  from  what  a  portion  of  it  at  least  actually  does.     For  instance,  *?d  not  the 
when  an  unpolarized  ray  is  separated  by  double  refraction  into  two  equal  pencils,  a  polarized  ray  will  be  divided  (h'| 
into  two  unequal  ones,  one  of  which  may  even  be  altogether  evanescent,  but  their  directions  are  precisely  the 
same  as  those  of  the  pencils  into  which   the  unpolarized  ray  is  divided.     Hence  we  may  lay  it  down  as  a 
general  principle,  that  the  direction  taken  by  a  polarized  ray,  or  by  the  parts  into  which  it  may  be  divided  by 
any  reflexions,  refractions,  or  other  modifying  causes,  may  always  be  determined  by  the  same  rules  as  apply  to 
unpolarized  light ;   but  that  the  relative  intensities  of  these  portions  differ  from  those  of  similar  portions  of 
unpolarized  light,  according  to  certain  laws  which  it  is  the  business  of  the  optical  inquirer  to  ascertain. 


§  III.     Of  the  Polarization  of  Light  by  Reflerion. 

When  a  ray  of  direct  solar  light  is  received  on  a  plate  of  polished  glass  or  other  medium,  a  portion  more  or       821. 
less  considerable  is  always  reflected.     The  intensity  of  this  portion  depends  only  on  the  nature  of  the  medium  L'gnt 
and  on  the  angle  of  incidence,  being  greater  as  the  refractive  power  of  the  former  is  greater,  and  as  the  ray  falls  P°';lrl«d  t>) 
more  obliquely  on  the  surface.  But  it  is,  moreover,  found,  that  at  a  certain  angle  of  incidence,  (which  is  therefore 
called  the  polarizing  anyle,)  the  reflected  ray  possesses  all  the  characters  above  enumerated,  and  is  therefore 
polarized. 

This  remarkable  fact  was  discovered  by  Malus  in  1808,  when  accidently  viewing,  through  a  doubly  refracting      822. 
prism,  the  light  of  the  setting  sun  reflected  from  the  glass  windows  of  the  Luxembourg  Palace  in  Paris.     On  Discovery 
turning  round  the  prism,  he  was  surprised  to  observe  a  remarkable  difference  in  the  intensity  of  the  two  images;    y 
the  -nost  refracted  alternately  surpassing  and  falling  short  of  the  least  in  brightness,  at  each  quadrant  of  the 
revolution.     This  phenomenon  connecting  itself  in  his  mind  with  similar  phenomena  produced  by  rays  which  had 
undergone  double  refraction,  and  with  which,  from  the  researches  he  was  then  engaged  in,  he  was  familiar,  led 
him  to  investigate  the  circumstances  of  the  case  with  all  possible  attention,  and  the  result  was  the  creation  of  a 
new  department  of  Physical  Optics.     So  true  it  is,  that  a  thousand  indications  pass  daily  before  our  eyes  which 
might  lead  to  the  most  important  conclusions.     The  seeds   of  great   discoveries  are  everywhere  present  and 
floating  around  us,  but  they  fall  in  vain  on  the  \mprepared  mind,  and  germinate  only  where  previous  inquiry  has 
elaborated  the  soil  for  their  reception,  and  awakened  the  attention  to  a  perception  of  their  value. 

To  make  this  new  property  acquired  by  the  reflected  ray  evident  by  experiment,  let  any  one  lay  down  a  large  823. 
plate  of  glass  on  a  black  cloth,  on  a  table  before  an  open  window,  and  placing  himself  conveniently  so  as  to  look  Experiment 
obliquely  at  it,  let  him  view  the  reflected  light  of  the  sky,  (or,  which  is  better,  of  the  clouds  if  not  too  dark,) 
from  the  whole  surface,  which  will  thus  appear  pretty  uniformly  bright.  Then  let  him  close  one  eye,  and  apply 
before  the  other  a  plate  of  tourmaline,  cut  as  above  directed,  so  as  to  have  its  axis  in  a  vertical  plane.  He  will 
then  observe  the  surface  of  the  glass,  instead  of  being  as  before  equally  illuminated,  to  have  on  it,  as  it  were, 
an  obscure  cloud,  or  a  large  blot,  the  middle  of  which  is  totally  dark.  If  this  be  not  seen  at  first,  it  will  come 
into  view  on  elevating  or  depressing  the  eye.  If  the  inclination  of  a  line  drawn  from  the  centre  of  the  dark 
spot  to  the  eye  be  measured,  it  will  be  found  to  make  an  angle  of  about  33°  with  the  surface  of  the  glass.  If 
now,  keeping  the  eye  fixed  on  the  spot,  the  tourmaline  plate  (which  it  is  convenient  to  have  set  in  a  small 
circular  frame  for  such  experiments)  be  turned  slowly  round  in  its  own  plane,  the  spot  will  grow  less  and  less 
obscure,  and  when  the  axis  of  the  tourmaline  it  parallel  to  the  reflecting  surface,  (or  horizontal,)  will  have  dis- 
appeared completely,  so  as  to  leave  the  surface  equally  illuminated,  and,  on  continuing  the  rotation  of  the  tourma- 
line, will  appear  and  vanish  alternately. 

It  appears  from  this  experiment,  that  the  ray  which  has  been  reflected  from  the  surface  of  the  glass  at  an       824 
inclination  of  33°,  or  an  incidence  of  57°,  has  thereby  been  deprived  of  its  power  to  penetrate  a  tourmaline 
plate  whose  axis  lies  in  the  plane  of  incidence.     It  has  therefore  acquired  the  same  character,  or  (so  far  as  this 
goes,  at  least)  undergone  the  same  modification  as  if,  instead  of  being  reflected  on  glass,  it  had  been  transmitted 
through  a  tourmaline  plate,  whose  axis  was  perpendicular  to  the  plane  of  reflexion. 

It  has,  moreover,  acquired  all  the  other  enumerated  characters  of  a  polarized  ray.     And,  first,  it  has  become 

VOL  iv.  3  u 


506  L  I  G  H  T. 

Light.       incapable  of  reflexion  at  the  surface  of  glass,  or  other  transparent  media  at  certain  definite  angles,   and  in     Par|  Iv- 
""""T1""""  certain  positions  of  the  plane  of  incidence.     To  show  this  experimentally,  let  a  piece  of  polished  glass  have  one  V"'"Y~~~ 

.  •       of  its   surfaces  roughenetl,  and  blackened  with   melted  pitch  or  black  varnish,   so  as  to  destroy  its  internal 

ThlT'oiT'"'  ren"ex'011'  anc*  let  tn's  be  ti*ed  on  a  stand,  so  as  to  be  capable  of  varying  at  will  the  inclination  o'f  its  polished 

rized  ray       surface  to  the  horizon,  and  of  turning  it  round  a  vertical  axis  in  any  azimuth.     A  very  convenient  stand  of  this 

incapable  of  kind  is  figured  in  fig.  171,  consisting  of  a  cylindrical  support  A  sliding  in  a  vertical  tube  B,  attached  to  a  round 

a  second  re-  base  F  like  a  candlestick,  and  carrying  an  arm  C,  which  can  be  set  to  any  angle  of  inclination  to  the  horizon  by 

flenon,  &c.  means  of  a  stiff  shoulder  joint  D.     To  this  arm  the  blackened  glass  E  is  fixed,  having  its  plane  parallel  to  the 

axis  of  the  joint  D.     Let  this  apparatus  be  set  on  a  table,  so  that  the  rays  reflected  from  a  pretty  large  plate  of 

glass  G,  at  an  angle  of  about  57°  (of  incidence)  shall   be  received  on  the  glass  E,  which  ought  to  be  inclined 

with  its  polished  surface  looking  downwards,  and  making  an  angle  of  about  73°  with  the  horizon,  see  Art.  842. 

Then  let  the  observer  apply  his  eye  near  the  glass  E,  so  as  to  see  the  glass  G  reflected  in  it,  and  slowly  turn 

the  stand  F  round  in  a  horizontal  plane,   keeping  always  the   reflected  image  of  G   in  view.     He   will   then 

perceive,  that  at  a  certain   point  of  the   rotation  of  the  stand,  the  illumination  of  this  image,  which  in  other 

situations  is  very  bright,  will  undergo  a  rapid  diminution,  and  at  last  wholly  disappear,  and  (if  the  glass  G  be 

large  enough)  the  same  appearance  of  a  cloud  or  large  dark  spot  will  then  be  visible  upon  it.     If  the  inclination 

of  the  arm  C  D  be  correct,  it  will  be  easy  to  find  such  a  position  by  turning  the  stand  a  little  backwards  and 

forwards,  as  shall  make  the  centre  of  this  spot  totally  black  ;  if  not,  bring  it  to  as  great  a  degree  of  obscurity  as 

possible  by  the  horizontal  motion,  then,  holding  fast  the  stand,  vary  a  little  one  way  or  another  the  inclination  of 

the  reflector  E,  and  a  very  complete  obscurity  will  readily  be  attained. 

826.  Another,  and,  for  some  experimental  purposes,  a  better  way  of  exhibiting  the  same  phenomenon,  is  to  take 
Another        two  metallic  or  pasteboard  tubes,  open  at  both  ends,  and  fitting  into  each  other  so  as  to  turn  stiffly.     Into  each 
mo^f  "*        of  these,  at  the  end  remote  from  their  junction,  fix  with  wax,  or  in  a  frame,  a  plate  of  glass,  blackened  at  the 
expe'rimen^   ^ack  as  a'3ove  described,  so  as  to  make  an  angle  of  33°  with  the   axis  of  the  tube,  as  represented  in  fig.  172. 
Fig.  172.       Then  having  placed  the  tube  containing  one  of  the  plates  (A)  so  that  the  light  from  any  luminary,  reflected  at 

the  plate  shall  traverse  the  axis  of  the  tube,  fix  it  there,  and  the  reflected  ray  will  be  again  reflected  at  B,  and 
on  its  emergence  may  be  received  on  a  screen  or  on  the  eye.  Now  make  the  tube  containing  the  reflector  B 
revolve  within  the  other,  so  that  that  reflector  shall  revolve  round  the  ray  A  B  as  an  axis,  preserving  the  same 
inclination.  Then  will  the  twice  reflected  ray  revolve  with  equal  angular  motion,  and  describe  a  conical 
surface.  But  in  so  doing,  it  will  be  observed  to  vary  in  intensity,  and  at  two  points  of  the  revolution  of  the  tube 
B  will  disappear  altogether.  Now  if  we  attend  to  the  position  of  the  reflectors  at  this  moment,  it  will  be  found 
that  the  planes  of  the  first  and  second  reflexion  make  a  right  angle. 

827.  By  repeating  these  experiments  with  all  sorts  of  reflecting  media,  and  determining  by  exact  measurement  the 
angles  at  which  the  original  ray  must  be  incident    that   polarization  shall  take  place,  and  those  at  which  a 
polarized   ray   ceases   to  be   reflected,  the  following  laws  have  been  ascertained   to  hold  good,  previous  to 
announcing  which  a  definition  will  be  necessary. 

82S.  Definition.     The   plane   of  polarization  of  a  polarized  ray  is  the   plane  in  which  it  must  have   undergone 

Plane  of       reflexion,  to  have  acquired  its  character  of  polarization  ;  or  that  plane  passing  through  the  course  of  the  ray 

polarization  perpendicular  to  which  it  cannot  be  reflected  at  the  polarizing  angle  from  a  transparent  medium  ;  or,  again,  that 

plane  in  which,  if  the  axis  of  a  tourmaline  plate  exposed  perpendicularly  to  the   ray  be  situated,  no  portion  of 

the  ray  will  be  transmitted.     Also,  a  polarized  ray  is  said  to  be  polarized  in  its  plane  of  polarization,  as  just 

defined. 

829.  The  plane  of  polarization  of  any  polarized  ray  is  to  be  considered  as  one  of  the  sides  of  the  ray  which  thus, 
Sides  of  a     in  all  its  future  progress,  carries  with  it  certain  relations  to  surrounding  fixed  space,  which   must  be  regarded, 
polarized      while  they  continue  unchanged,  as  inherent  in  the  ray  itself,  and  as  having  no  further  any  relation  to  the  parti- 
cular mode  in  which  they  originated. 

830.  The  laws  of  polarization  by  reflexion  are  these  : 

Laws  of  po-       Law  1.     All  reflecting  surfaces  are  capable  of  polarizing  light  if  incident  at  proper  angles;    only,  metallic 
larizationby  bodies,  or  bodies  of  very  high  refractive  powers,  appear  to  do  so  but  imperfectly,  the   reflected   ray  not  entirely 
•e^xion.      disappearing  in  circumstances  when  a  perfectly  polarized  ray  would  be  completely  extinguished.     Of  this  more 
hereafter. 

831.  Law  2.     Different  media  differ  in  the  angles  of  incidence  at  which  they  polarize  light ;  and  it  is  found,  that. 
Law  2.         these  angles  may  always  be  determined  from  the  following  simple  and  elegant  relation,  discovered  by  Dr.  Brewster 
Brewster's    after  a  laborious  examination  of  an  infinite  variety  of  substances. 

The  tangent  of  the  polarizing  angle  for  any  medium  is  the  index  of  refraction  belonging  to  that  medium. 
Thus,  the  indices  of  refraction  of  water,  crown-glass,  and  diamond,  being  respectively  1.336,  1.535,  and  2.487, 
their  respective  polarizing  angles  will  be  53°  11',  56°  55',  and  68°  6'.     For  diamond,  however,  or  bodies  of  very 
high  refractive  powers,  we  must  understand  by  the  polarizing  angle,  that  angle  of  incidence  at  which  the  reflected 
ray  approximates  most  nearly  to  the  character  of  a  ray  completely  polarized. 

832  It  follows  from  this  law,  that  one  and  the  same  medium  does  not  polarize  all  the  coloured  rays  at  the  same 

All  the         angle,  and  that  therefore  the  disappearance  of  the  reflected  pencil  can  never  be  total,  except  where  the  incident 
colours  not   ray  is  homogeneous.     This  will  account  in  some  degree  for  the  want  of  complete  polarization  of  a  white  ray, 
[lolanzod  at  reflected  at  any  angle  from  highly  refractive  media,  which  are  generally  also  highly  dispersive.     Of  the  reality 
dence"0'        °^ tne  'act'  ll  *s  easv  *°  sat'sfy  oneself  by  a  very  simple  experiment,  which  we  have  often  made.     Receive  a  sun- 
beam  on  a  plane  glass,  with   the  back  roughened  and  blackened,  at  an   incidence  (6)  nearly  equal  to  the 
polarizing  angle  (a,)  and  let  the  reflected  ray  pass  into  a  darkened  room,  and  fall  on  another  similar  glass, 
which  may  fce  held  in  the  hand,  so  as  to  reflect  the  ray  in  a  plane  at  right  angles  to  that  of  the  first  reflection,  and 


LIGHT.  507 

Light,      also  at  an  angle  (#")  nearly  equal  to  the  polarizing  angle  (a1)  of  the  second  plate.     It  will   be  easy  to  find  a     Part  IV. 
«~y— — «•  position  where  the  reflected  ray  (which  must  be  received  on  a  white  screen)  very  nearly  vanishes  ;  but  no  adjust-  v—v-""/ 
ment  of  the  angles  of  incidence  &  and  &  will  produce  a  total  disappearance.     When  the  disappearance  is  most  P«>ved  by 
nearly  total,  the  reflected  light  is  coloured  of  a  neutral  purple  ;   the  yellow,  or  most  luminous  rays,  being  now  e*Pennient- 
totally  extinguished.     In  this  position,  if  6  remain  constant,  and  &  the  incidence  on  the  second  plate  be  varied 
a  little  on  one  side  or  the  other  of  the  polarizing  angle  a',  the  reflected  ray  assumes  on  the  one  hand  a  pretty 
intense  blue-green,  and  on  the   other  a  ruddy  plum  colour  or  amethyst  red.     The  several  changes  of  tint, 
arisin    from  variatios  of 


C  0'  <  a' ;         Reflected  ray, 
t.     0  <  a  ;  •<    Intermediate, 

/        /)'    ^Nfc.     fi?    • 
\_  *f^     **    ) 

f    £tf    _^"*    «' 
I       P     ^_    (I    . 
^N    *»    3 

=  a;  1    ff  =  a?i 

[      \j    ^>-  fr   ! 

v_         **^         y 

(    ff<a>; 

a. ;  <    Intermediate, 

Ltf>a?i 


arising  from  variations  of  incidence  on  both  piates,  were  observed  to  be  as  follows : 

Strong  green. 

1st.     0<a;-{    Intermediate, White. 

Pale  red  or  amethyst. 

Strong  blue  green. 

2d.     6  =.  a ;  -|    0'  =  a' ;          Neutral  purple. 

Strong  plum  colour. 

Light  greenish  blue. 

3d.     0  >  a  ;  -{    Intermediate, White. 

Strong  red,  or  plum  colour. 

The  rationale  of  these  changes  of  colour  will  be  more  evident  when  we  have  announced  the  following  law, 
which  expresses  one  of  the  most  general  and  distinguishing  characters  of  polarized  light. 

Law  3.     When  a  polarized  ray  (no  matter  how  it  acquired  its  polarization)  is  incident  on  a  reflecting  surface       833. 
of  a  transparent,  or  other  medium  capable  of  completely  polarizing  light,  in  a  plane  perpendicular  to  that  of  the  Law  3. 
ray's  polarization,  and  at  an  angle  of  incidence  equal  to  the  polarizing  angle  of  the  medium,  no  portion  of  the  _Non-reflex- 
ray  will  be  reflected.     If  the  medium  be  of  such  a  nature  as  to  be  capable  only  of  incompletely  polarizing  light,  lblllty 
a  portion  will  be  reflected,  but  much  less  intense  than  if  the  incident  ray  were  unpolarized.  BehTiif 

It  is  evident  that  this  property  may  be  employed  to  distinguish  polarized  from  common  light,  as  well  as  that  of  certain,  and 
extinction  by  a  plate  of  tourmaline.  It  is,  however,  much  less  convenient  though  better  adapted  for  delicate  what  cases. 
inquiries. 

The  polarizing  angle  for  white  light  is,  in  fact,  the  angle  for  the  most  luminous  or  mean  yellow  rays ;  and       83-i. 
when  the  two  reflexions,  in  planes  at  right  angles  to  each  other,  are  made  at  this  angle,  the  yellow  rays  only  Explanation 
totally  escape  reflexion,  but  a  very  small  portion  both  of  the  red  and  blue  end  of  the  spectrum  are  reflected,  and  of  the 
form  a  feeble  purple  beam,  such  as  above  described.     The  polarizing  angle  for  red  rays  being  less  than  for  violet,  ^"'ast '" 
it  is  evident  that  when  either  9  or  0'  is  equal  to  the  polarizing  angle  for  red,  it  will  be  less  than  that  of  yellow,  experiment. 
and  still  less  than  that  of  blue  and  violet  rays ;  thus,  the  red  disappears  most  completely  from  the  reflected  beam 
in  those  cases  when  0  or  &  are  less  than  a  or  a',  leaving  an  excess  of  the  green  and  blue  rays,  and  vice  versa  in 
the  converse  cases.     Thus,  too,  if  0  be  <  a,  and  at  the  same  time  ff  <  a,  the  colour  produced  will  be  a  more 
intense  green  than  if  the  incidences  deviated  opposite  ways  from  the  polarizing  angles;  and  it  is  evident,  that 
a  compensation  may  arise  from  the  effect  of  such  opposite  deviations  giving  an  intermediate  white  ray,  exactly 
as  we  see  to  have  happened. 

Some  very  remarkable  consequences  follow  from  the  law  announced  by  Dr.  Brewster  for  finding  the  polarizing       835. 
angle,  which  may  be  presented  in  the  form  of  distinct  propositions.     Thus, 

Prop.  1.     When  a  ray  is  incident  on  a  transparent  surface,  so  that  the  reflected  portion  shall  be  completely       836. 
polarized,  the  reflected  and  refracted  portions  make  a  right  angle.     For  0  being  the  angle  of  incidence,  we  have  Consequen- 

tan  p  =  it  and  p,  being  the  angle  of  refraction,  sin  p  —  '- = =  cos  0.     Therefore  p  =  90°  -  0,  but  0  !?w  °l  P°la' 

p         tan  0  ruation. 

being  the  angle  of  incidence  is  also  that  of  reflexion,  and  p  -f-  0  is  therefore  equal  to  the  supplement  of  the 
angle  between  the  reflected  and  refracted  rays,  which  is  therefore  a  right  angle.  Q.  E.  D. 

Prop.  2.     When  a  beam  of  common  light  is  incident  at  the  polarizing  angle  on  a  parallel  plate  of  a  transparent      837. 
medium,  not  only  the  portion  reflected  at  the  first  surface,  but  also  that  reflected  internally  at  the  second,  and  Polarization 
the  compound  reflected  ray,  consisting  of  both  united,  are  polarized.  by  in|ernal 

Since  sin  p  =  cos  0,  and  since  p  is  also  the  angle  of  incidence  on  the  second  surface,  we  shall  have  tan  p  =  reflexi011- 

cotan  0  = =  —  =r  index  of  refraction  out  of  the  medium.     Hence,  p  is  the  angle  of  polarization  for  rays 

'nternally  incident,  and  therefore  that  portion  of  the  beam  which,  having  penetrated  the  first  surface,  falls  on  the 
second,  being  incident  at  its  polarizing  angle,  the  portion  reflected  here  will  also  be  polarized,  and  being  again 
incident  on  the  first  surface,  in  the  plane  of  its  polarization,  that  part  of  it  which  is  transmitted  will  (as  we  shall 
see  hereafter)  suffer  no  change  in  its  plane  of  polarization,  so  that  both  it  and  the  first  reflected  ray  will  come 
off  polarized  in  the  same  plane.  Q.  E.  D. 

Carol.  1.     Hence,  to  obtain  a  stronger  polarized  ray,  we  may  dispense  with  roughening  or  .blackening  the       838. 
posterior  surface,  provided  we  are  sure  that  the  surfaces  are  truly  parallel. 

If  a  series  of  parallel  plates  be  laid  one  on  the  other  so  as  to  form  a  pile,  the  portions  reflected  from  the       639. 
several  surfaces  all  come  off  polarized  in  the  same  plane,  and  by  this  means  a  very  intense  polarized  ray  may  be  v^- 
obtained.     It  can  never,  however,  for  a  reason  we  shall  presently  state,  contain  more  than  half  the  incident  °n  inl'^fw 
light,  whatever  be  the  number  of  plates  employed.  polarized 

3  U  2  beam. 


508  LIGHT 

Light.          For  a^reat  variety  of  optical  experiments,  a  pile  consisting  of  ten  or  a  dozen  panes  of  common  window-glass 
—V^''  set  in  a  frame,  is  of  great  use  and  very  convenient.     Such  a  pile  laid  down  before  an  open  window  affords  a 
840.      dispersed  beam,  each  ray  of  which  is  polarized  at  the  proper  angle,  and  of  great  intensity  and  very  proper  for 
the  exhibition  of  many  of  the  phenomena  hereafter  to  be  described. 

Prop.  3.  If  a  ray  be  completely  polarized  by  reflexion  at  the  surface  of  one  medium,  and  the  reflected  ray 
completely  transmitted  or  absorbed  at  that  of  a  second,  Required  the  inclination  of  the  two  surfaces  to  each 
other  ? 

Let  a  and  «'  be  the  polarizing  angles  of  the  respective  media;  then,  since  the  planes  of  reflexion  are  at  right 
angles  to  each  other,  and  a,  a'  are  the  angles  of  incidence,  if  we  call  I  the  inclination  required,  we  shall  have  by 
Art.  104,  cos  I  =  cos  a  .  cos  a'.  Now,  if  p,  p!  be  the  refractive  indices  of  the  media,  we  have  tan  o  =  p, 
tan  a'  =  «',  and  therefore 

tan  I  =  v^i  +  p*  _f-  pi  n'*. 
842.  Carol.  I.     If  the  media  be  both  alike, 


tan  I  =  p,  .  */2  __      ;  or  cos  I  = 


. 

Thus,  in  the  case  of  crown-glass,  /*  =  1.535  and  I  =  72°  40',  as  in  Art.  825. 

843.  By  the  help  of  this  law,  connecting  the  angle  of  polarization  with  the  refractive  index,  we  may  easily  deduce 
Method  of  the  one  from  the  other.  This  affords  a  valuable  and  ready  resource  in  cases  to  which  other  methods  can  hardly 
determining  jje  appijg^  for  ascertaining  the  refractive  powers  of  media,  which  are  either  opake,  or  in  such  small  or  irregularly 
indices'  by  snaPed  masses,  that  they  cannot  be  used  as  prisms.  For  ascertaining  the  angle  of  polarization,  only  one 
polarization,  polished  surface,  however  small,  is  necessary,  and  we  have  only  to  receive  a  ray  reflected  from  it  on  a  blackened 
glass,  or  other  similar  medium  of  known  refractive  index,  at  the  polarizing  angle,  and  in  a  plane  perpendicular 
to  that  at  which  it  is  reflected  by  the  surface  under  examination.  For  this  purpose  it  is  convenient  to  have  the 
glass  plate  (or,  which  is  better,  a  polished  plate  of  obsidian  or  dark  coloured  quartz)  set  in  a  tube  diagonally, 
so  as  to  reflect  laterally  the  ray  which  traverses  the  axis  of  the  tube.  At  the  other  end,  the  substance  to  be 
examined  must  be  fixed  on  a  revolving  axis  perpendicular  to  the  axis  of  the  tube,  and  having  its  plane  adjusted 
so  as  to  be  parallel  to  the  former,  which  must  then  be  turned  round  till  the  dispersed  light  of  the  clouds, 
reflected  by  it,  is  entirely  extinguished  by  the  obsidian  plate,  and  the  inclination  of  the  reflecting  surface  to  the 
axis  of  the  tube  in  this  situation  may  be  measured  by  a  divided  circle,  connecting  with  the  axis  of  rotation.  By 
this  means  we  may  ascertain  the  polarizing  angles,  and  therefore  the  refractive  indices  of  the  smallest  crystals, 
or  of  polished  stones,  gems,  &c.,  set  in  such  a  manner  as  not  to  admit  of  other  modes  of  examination.  To 
insure  a  fixed  zero  point  on  the  graduated  circle,  the  following  mode  (among  many  others)  may  be  resorted  to. 
Let  a  polished  metallic  reflector  or  small  piece  of  looking-glass  be  permanently  attached  to  the  revolving  axis,  so 
that  its  plane  shall  be  perpendicular  to  the  axis  of  the  tube,  when  the  index  of  the  divided  circle  marks  0°  0'.  This 
adjustment  being  made  once  for  all,  let  the  surface  to  be  examined  be  attached  by  wax  or  otherwise,  not  to  the 
axis  itself,  but  to  a  ring  turning  stiffly  on  it.  Then,  bringing  the  image  of  the  sun,  or  any  very  distant  object, 
sufficiently  bright  or  well  defined,  seen  in  the  reflector,  to  coincide  with  any  other  equally  well  defined,  and 
also  at  a  great  distance,  alter  the  attachment  of  the  substance  by  pressure  on  the  wax,  and  by  turning  round  the 
ring,  till  a  similar  coincidence  is  obtained  when  the  eye  is  transferred  to  it.  Then  we  are  assured  that  the  two 
surfaces  are  parallel,  and  that  therefore  the  reading  off  on  the  circle  measures  the  true  angle  between  the  axis 
of  the  tube  and  the  perpendicular,  or  the  angle  of  reflexion,  or  at  least  differs  from  it  only  by  a  constant 
quantity,  which  may  be  ascertained  at  leisure,  and  applied  as  index  error.  (This  mode  of  bringing  a  movable 
surface  to  a  fixed  position  with  respect  to  the  divisions  of  an  instrument,  is  applicable  to  a  great  variety  of 
cases,  and  is  at  once  convenient  and  delicate.) 

844.  Dr.  Brewster  has  remarked,  that  glass  surfaces  frequently  exhibit  remarkable,  and  apparently  unaccountable, 
Irregular      deviations  from  the  general  law  ;  but  on  minute  examination  he  found  that  this  substance  is  liable  to  a  superficial 
polarization  tarnish,  or  formation  of  infinitely  thin  films  of  a  different  refractive  power  from  the  mass  of  glass  beneath.     As 
lurfaras*        tne  P°'ar'zed  raY  never  penetrates  the  surface,  its  angle  of  polarization  is  determined  solely  by  this  film,  which 

is  too  thin  to  admit  of  any  direct  measure  of  its  refractive  index.  When  this  tarnish  has  gone  to  a  great  extent, 
scales  of  glass  detach  themselves,  as  is  seen  in  very  old  windows,  (especially  those  of  stables,)  and  even  in  green 
glass  bottles  which  have  long  lain  in  damp  situations,  and  which  acquire  a  coat  actually  capable  of  being  mistaken 
for  gilding. 

845.  In  metallic  or  adamantine  bodies,  which  polarize  light  but  imperfectly,  that  angle  at  which  the  reflected  beam 
Action  of      approaches  nearer  in  its  character  to  those  described  as  of  polarized  light,  is  to  be  taken  for  the  angle  of  pola- 

rization, and  from  it  the  refractive  power  may  still  be  found.  The  results  deduced  by  this  means  for  metallic 
bodies,  agree  with  those  obtained  from  the  quantity  of  light  reflected,  in  assigning  very  high  refractive  powers  to 
them.  Thus,  for  steel  the  polarizing  angle  is  found  to  be  above  71°,  and  for  mercury  76£°,  and  their  indices  of 
refraction  are,  therefore,  respectively  2.85  and  4.16.  This  latter  result,  indeed,  differs  greatly  from  that  of  Art 
594,  but  the  observations  are  so  uncertain,  and  the  angle  of  greatest  polarization  so  indefinitely  marked,  (not 
to  mention  the  errors  to  which  a  determination  of  the  reflective  power  itself  is  liable  to,)  that  we  cannot 
expect  coincidence  in  such  determinations.  Perhaps  5.0  may  be  taken  as  a  probable  index. 

846.  The  law  of  polarization  announced  by  Dr.  Brewster  is  general,  and  applies  as  well  to  the  polarization  of  light 
at  the  separating  surfaces  of  two  media  in  contact,  as  at  the  external   or  internal  surface  of  one  and  the  same 
medium.     He  has  attempted  to  deduce  from  it  several  theoretical  conclusions,  as  to  the  extent  and  mode  of 
action  of  the  reflecting  and  refracting  forces,  for  which  we  must  refer  the  reader  to  his  Paper  on  the  subject 
Philosophical  Transactions,  1916 


LIGHT.  509 

If  a  ray  be  reflected  at  an  angle  greater  or  less  than  the  polarizing  angle,  it  is  partially  polarized,  that  is  to     Part  IV. 
say,  when  received  at  the  polarizing  angle  on  another  reflecting  surface,  which  is  made  to  revolve  round  the  *— ~v— ~- ' 
reflected  ray  without  altering  its  inclination  to  it,  the  twice  reflected  ray  never  vanishes  entirely,  but  undergoes      847. 
alternations  of  brightness,  and  passes  through  states  of  maxima  and  minima  which  are  more  distinctly  marked  Partial  pola- 
according  as  the  angle  of  the  first  reflexion  approaches  more  nearly  to  that  of  complete  polarization.     The  same  rlzat¥ln- 
is  observed  when  a  ray  so  partially  polarized  is  received  on  a  tourmaline  plate,  revolving  (as  above  described) 
in  its  own  plane.     It  never  undergoes  complete  extinction,  but  the  transmitted  portion  passes  through  alternate 
maxima  and  minima  of  intensity,  and  the  approach  to  complete  extinction  is  the  nearer  the  nearer  the  angle  of 
reflexion  has  been  to  the  polarizing  angle.    We  may  conceive  a  partially  polarized  ray  to  consist  of  two  unequally  How 
intense  portions  ;  one  completely  polarized,  the  other  not  at  all.  It  is  evident  that  the  former,  periodically  passing  conceived, 
from  evanescence  to  its  total  brightness,  during  the  rotation  of  the  tourmaline  orreflector,  while  the  latter  remains 
constant  in  nil  positions,  will  give  rise  to  the  phenomenon  in  question.     And  all  the  other  characters  of  a  par- 
tially polarized  ray  agreeing  with  this  explanation,  we  may  receive  it  as  a  principle,  that  when  a  surface  does  not 
completely  polarize  a  ray,  its  action  is  such  as  to  leave  a  certain  portion  completely  unchanged,  and  to  impress 
on  the  remaining  portion  the  character  of  complete  polarization.     Thus  we   must  conceive  polarization  as  a 
property  or  character  not  susceptible  of  degree,  not  capable  of  existing  sometimes  in  a  more,  sometimes  in  a  less, 
intense  state.     A  single  elementary  ray  is  either  wholly  polarized  or  not  at  all.     A  beam  composed   of  many 
coincident  rays  may  be  partially  polarized,  inasmuch  as  some  of  its  component  rays  only  may  be  polarized,  and 
the  rest  not  so.     This  distinction  once  understood,  however,  we  shall  continue  to  speak  of  a  ray  as  wholly  or 
partially  polarized,  in  conformity  with  common  language.     We  shall  presently,  however,  obtain  clearer  notions 
on  the  subject  of  unpolarized  light,  and  see  reason  for  discarding  the  term  altogether. 

If  a  ray  be  partially  polarized  by  reflexion,  Dr.  Brewster  has  stated  that  a  second  reflexion  in  the  same  plane      848. 
renders  this  polarization  more  complete,  or  diminishes  the  ratio  of  the  unpolarized  to  the  polarized  light  in  the  Polarization 
reflected  beam  ;   and  that  by  repeating  the  reflexion,  the  ray  may  be  completely  polarized,  although  none  of  the  ty  several 
angles  of  reflexion  be  the  polarizing  angle.     Thus  he  found,  that  one  reflexion  from  glass  at  56°  45'  of  incidence,  re 
two  at  incidences  of  62°  30'  or  at  50°  20',  three  at  65°  33'  or  at  46°  30',  four  at  67°  33'  or  43°  51',  and  so  on,  ° 
alike  sufficed  to  operate  the  complete  polarization  of  the  ray  finally  reflected,  provided  all  the  reflexions  were 
made  in  one  plane.     At  angles  above  82°,  or  below  18°,  more  than  100  reflexions  were   required  to   produce 
complete  polarization. 

§  IV.     Of  the  Laws  of  Reflexion  of  Polarized  Light. 

When  polarized  light  is  reflected  at  any  surface,  transparent  or  otherwise,  the  direction  of  the  reflected  portion      849. 
is  precisely  the  same  as  in  the  case  of  natural  light,  the  angle  of  reflexion  being  equal  to  that  of  incidence ;  the 
laws  we  are  now  to  consider  are  those  of  the  intensity  of  the  reflected  light,  and  of  the  nature  of  its  polarization 
after  reflexion. 

One  essential  character  of  a  polarized  ray  is,  its  insusceptibility  of  reflexion  in  a  plane  at  right  angles  to  that      850. 
of  its  polarization  when  incident  at  a  particular  angle,  viz.  the  polarizing  angle  of  the  reflecting  surface.     In  Intensity  of 
this  case,  the  intensity  I  of  the  reflected  ray  is  0.     In  all  other  cases  it  has  a  certain  value,  which  we  are  now  to  ""^j^j 
inquire.     Let  us  suppose,  then,  to  begin  with  the  simplest  case,  that  the  polarized  ray  fails  on  the  reflecting  rayinc;jent 
surface  at  a  constant  angle  of  incidence,  equal  to  its  polarizing  angle,  and  that  the  reflecting  surface  is  turned  at  the  pola- 
round  the  incident  ray  as  an  axis,  so  that  the  plane  of  incidence  shall  make  an  angle  (=  a)  of  any  variable  mag-  rizing  angle 
nitude  with  the  plane  of  polarization.     It  is  then  observed,  as  we  have  seen,  that  when  a  =  90°  or  270°,  we  have  >n  any  PUne- 
1=0,  and  when  a  =  0°,  or  180°,  I  is  a  maximum.     Hence,  it  is  clear  that  I  is  a  periodic  function  of  a,  and  the 
simplest  form  which  can  be  assigned   to  it  (since  negative  values  are  inadmissible)  is  I  =  A  .  (cos  a)8.     This 
value,  which  was  adopted  by  Malus  on  no  other  grounds  than  those  we  have  stated,  is  however  found  to  represent 
the  variation  of  intensity  throughout  the  quadrant,  with  as  much  precision  as  the  nature  of  photometrical  experi- 
ments admits,  and  we  must  therefore  receive  it  as  an  empirical  law  at  present,  for  which  any  good  theory  of 
polarization  ought  to  be  capable  of  assigning  a  reason  a  priori. 

A  remarkable  consequence  follows  from  this  law.     It  is  that,  so  far  as  the  intensity  of  the  reflected  ray  is       « 
concerned,  an  ordinary  or  unpolarized  ray  may  be  regarded  as  composed  of  two  polarized  rays,   of  equal  ri^j^°  a" 
intensity,  having  their  planes  of  polarization  at  right  angles  to  each  other.     For  such  a  compound  ray  being  equjvaient 
incident  on  a  reflecting  surface,  as  above  supposed,  if  a  be  the  inclination  of  the  plane  of  polarization  of  one  to  two  pola- 
portion  to  that  of  incidence,  90  —  a  will  be  that  of  the  other,  and,  therefore,  since  rized  ones. 

A  .  (cos  a)'  +  A  .  (cos  .  90  -  <t)8  =  A,  (a) 

the  reflected  ray  will  be  independent  of  a,  and  therefore  no  variation  of  intensity  will  be  perceived  on  turning 
the  reflecting  surface  round  the  incident  ray  as  an  axis,  which  is  the  distinguishing  character  of  unpolarized  light. 
Any  such  pair  of  rays  as  here  described  are  said  to  be  oppositely  polarized. 

When  the  polarized  ray  is  not  incident  at  the  polarizing  angle,  but  at  any  angle  of  incidence,  the  law  of      852. 
intensity  of  the  reflected  ray  is  more  complicated.     M.  Fresnel  has  stated  the  following  as  the  general  expression  Fresnel's 
for  it.    Let  the  intensity  of  the  incident  ray  be  represented  by  unity,  and  calling,  as  before,  a  the  inclination  of  the  j?*"^*1  Iaw 
plane  of  incidence  to  that  of  primitive  polarization,  and  i  the  angle  of  incidence,  i'  the  corresponding  angle  of  intens;ty  Of 
refraction.     Then  will  the  intensity  of  the  reflected  ray  be  represented  by  a  reflected 

ray 


510  LIGHT. 

Light.  sinf(i-z*)  ,    tan'(i-ir)  Part  IV. 

*— v— ' '  *  —      .    i  /•  •    i     -/x  •  COS'  a  +  - —  .      .  Sill'  a.  (6)  ^-v-^ 

sin'  (i  +  i')  tan8  (z  +  l  ) 

This  formula  is  in  some  degree  empirical,  resulting  partly  from  theoretical  views,  of  which  more  hereafter,  and 
being  not  yet  verified,  or  indeed  compared  with  experiment,  except  in  particular  cases,  by  M.  Arago,  whose 
results,  so  far  as  they  go,  are  consonant  with  it. 

853.  it  will  be  well  to  examine  some  of  these.     And  first,  then,  when  a  =  90°,  and  i  =  the  polarizing  angle  of  the 
Particular     reflectjng.  surfacei  we  have  by  (835  and  836)  i  +  i'  =  90°,  and  therefore  tan  (j  +  i')  =  CD,  so  that  1  =  0.     In 
examined.    'nese  circumstances,  then,  the  reflected  ray  is  completely  extinguished,  which  agrees  with  fact. 

854.  2dly.     When  the  incidence  is  perpendicular,  we  have,  in  this  case,  both  i  and  i1  vanishing,  and  each  term  of  I 

lar'uic'i- 1CU"  tekes  tne  f°rm  ~TT-     Now  at  the  limit  we  have  (/t  being  the  refractive  index)  i  =  ft .  i',  and  very  small  arcs  being 

equal   to  their  sines  or  tangents,  we  have  sin  (i  —  i')  ~  i'  (/»  —  1) ;  sin  (i  +  £')  =*"(/*+  1),  and  so  for  the 
tangents.     Consequently, 


which  agrees  with  the  expression  deduced  by  Dr.  Young  and  M.  Poisson,  (Art.  592,)  for  the  intensity  of  the 
reflected  ray  in  the  case  of  unpolarized  light.  And  if  we  regard  the  unpolarized  ray  as  composed  of  two  rays, 
each  of  the  same  intensity,  (=  J)  polarized  in  opposite  planes,  the  reason  of  the  coincidence  will  be  evident. 

855.  3d.     When  a  —  0,  or  the  plane  of  polarization  coincides  with  the  plane  of  incidence,  we  have,  in  general, 

sin'O'-Q 

~  sin'  (i  +  I')' 

856.  4th.     When  a  =  90°,  or  when  the  plane  of  polarization  is  at  right  angles  to  the  plane  of  incidence, 

I  =  tan*  <*  "  '">  oo 

tan8  (i  +  i1) 

857.  5th.     When  a  =  45, 

Intensity  of  J  sin2  (i  —  i')         tan2  (t  -  Q~fr 

=  *  + 


This  last  is  the  same  result  with  that  which  would  result  from  the  supposition  of  two  equal  rays  polarized, 
the  one  in,  the  other  perpendicularly  to,  the  plane  of  incidence,  and  each  of  half  the  intensity  with  the  incident 
beam.  It  is  therefore  the  general  expression  for  the  intensity  of  a  ray  of  natural  or  unpolarized  light  reflected 
at  an  incidence  =  i  from  the  surface.  The  expressions  in  Art.  592  apply  only  to  perpendicular  incidences.  We 
are  thus  furnished  very  unexpectedly  with  a  solution  of  one  of  the  most  difficult  and  delicate  problems  of  experi- 
mental Optics.  Bouguer  is  the  only  one  who  has  made  any  extensive  series  of  photometrical  experiments 
on  the  intensity  of  light  reflected  from  polished  surfaces  at  various  angles,  but  his  results  are  declared  by 
M.  Arago  to  be  very  erroneous,  which  is  not  surprising,  as  the  polarization  of  light  was  unknown  to  him,  and  its 
lajws  might  affect  the  circumstances  of  his  experiments  in  a  variety  of  ways. 

858.  One  only  need  be  mentioned,  as  every  optical  experimentalist  ought  to  be  aware  of,  and  on  his  guard  against 
Polarization  it,  it  is  that  the  light  of  clear,  blue  sky,  is  always  partially  polarized  in  a  plane  passing  through  the  sun,  and  the 
of  the  light   part  from  which   the  light  is  received.     The  polarization  is  most  complete  in  a  small  circle,  having  the  sun  for 
of  the  sky.    jts  pOje>  an(j  jtg  ra<j;us  about  78°,   (according  to   an  experiment  not  very  carefully  made.)     Now  the  semi- 
supplement  of  this  (which  is  the  polarizing  angle)  is  51°,  which  coincides  nearly  with  the  polarizing  angle  of 
water,  (52°  45'.)     Thus  strongly  corroborating  Newton's  theory  of  the  blue  colour  of  the  sky,  which  he  conceives 
to  be  the  blue  of  the  first  order,  reflected  from  particles  of  water  suspended  in  the  air.     Dr.  Brewster  is  the  first, 
we  believe,  who  noticed  this  curious  fact.    But  to  return  to  our  subject. 

859.  When  the  incident  ray  is  only  partially   polarized,  we  may  regard  it  as  consisting  of  two  portions  :  the  one, 
Case  of  a    which  we  shall  represent  by  a,  completely  polarized  in  a  plane,  making  the  angle  n  with  that  of  incidence  ;  the 
ray  partially  /I    —   a\ 

polarized,     other  =  1  —  a  in  its  natural  state,  or,  if  we  please,   composed  of  two  portions  I   — - —  I,  one  polarized  in  the 

plane  of  incidence,  and  one  at  right  angles  to  it.     The  intensity  of  the  reflected  portion  of  the  former  is  equal  to 

sin*  (i  —   i'1  tan2  (i  —  i') 

cos*  a  +  a  .  — n-5 —  .  sin'  o, 

*         * ..  __ i  /  _•    i    ,/\ 


**  •        >     9   s  •  'l\    »«*•      ~       I      **  •  , 

sin'  (i  +  i )  tan 

and  that  of  the  latter  will  be  represented  by 

1  —  a  r  si 

2      \  sTna  (i  +  i')     '    tan8  (i  +  i') 
therefore,  their  sum,  or  the  total  reflected  light,  will  be 


i  -  i')        tan*  (i  -  i') 


sin*  (i  —  i1)       1  +  a  .  cos  2  a        tan'  (i  —  i')      1  —  a  .  cos  2 
~:~s  /'v    i    ;>\     '  "T"  . 


tan8 


The  above  formulae,  it  must  be  observed,  apply  only  to  the  case  of  reflexion  from  the  surfaces  of  uncrystallized 
media.  The  consideration  of  those  where  crystallized  surfaces  are  concerned,  cannot  be  introduced  in  this  part 
of  the  subject. 


LIGHT.  511 

jght.          When  the  plane  of  reflexion  coincides  with  that  of  the  primitive  polarization  of  the  ray,  the  polarization  is  not    part 
•v— -^  changed  by  reflexion.     Hence,  at  a  perpendicular  incidence  it  is  unchanged.     But  in  other  relative  situations  \^- 

of  the  two   planes  above-mentioned,  the  case  is  different,  and  it  becomes  necessary  to  inquire  what  change       860. 

reflexion  produces  in  the  state  and  plane  of  polarization  of  the  ray.     Now  it  is  found,  as  we  have  already  seen,  Position  of 

that  when  the  reflection  takes  place  in  the  plane  of  primitive  polarization,  if  the  incident  ray  be  only  partially  '^J™? 

polarized,  the  reflected  one  will  be  more  so,  in  that  plane.     But  if  the  incident  ray  be  completely  polarized,  it  Jjf  (he  re|°" 

retains  this  character  after  reflexion,  (except  in   one  remarkable  case,)  and  only  the  plane  of  polarization  is  flected  rav. 

changed.     Now,  according  to  M.  Fresnel,  the  new  plane  of  polarization  will  make  an  angle  with  the  plane  of 

reflexion,  represented  by  /3,  such  that 

cos  (i  +  i') 

tan  /3  =  -  ~^r  .  tan  o. 

cos  (i  -  i1) 

According  to  this  formula,  the  plane  of  polarization  coincides  with  the  plane  of  incidence  when  i-}-  i'  =  90°.  Now 
this  is  precisely  the  case  when  the  ray  falls  at  the  polarizing  angle  on  the  reflecting  surface.  If  a  =  90°,  or  the  ray 
before  incidence  be  polarized  in  a  plane  perpendicular  to  the  plane  of  incidence,  it  will  continue  to  be  so  after 
reflexion,  since  in  that  case  we  have  tan  /3  =  CD,  or  f)  =  90°. 

The  formula  has  been  compared  by  M.  Arago  with  experiment  only  in  one  intermediate  case,  viz.  when  861. 
a  —  45°,  and  the  coincidence  of  the  results  with  experiment  at  a  great  variety  of  incidences,  and  over  a  range  of 
values  of  /3  from  -f  38°  to  -  44°,  both  in  the  case  of  glass  and  water,  is  as  satisfactory  as  can  be  desired.  The 
particulars  of  this  interesting  comparison  will  be  found  in  Annales  de  Chimie,  xvii.  p.  314.  It  may  be 
observed  also,  that  these  results  of  M.  Fresnel  support  one  another,  the  latter  being  concluded  from  the  former 
by  considerations  purely  theoretical,  so  that  every  verification  of  the  one  is  also  a  verification  of  the  other. 

When  the  polarized  ray  is  reflected  from  a  crystallized  surface,  the  intensity  of  the  reflected  portion  is  no       862. 
longer  the  same,  but  depends  on  the  laws  of  double  refraction,  in  a  manner  of  which  more  hereafter.     Whether,  Reflexion 
or  how  far,  the  laws  above  stated  hold  good  for  metallic  surfaces,  remains  open  to  inquiry. 

faces. 

§  V.    Of  the  Polarization  of  Light  by  ordinary  Refraction,  and  of  the  Laws  of  the  Refraction  of  Polarized  Light. 

When  a  ray  of  natural  or  unpolarized  light  is  transmitted  through  a  plate  of  glass  at  a  perpendicular  incidence,       863. 
it  exhibits  at  its  emergence  no  signs  of  polarization  ;   but  if  the  plate  be  inclined  to  the  incident  ray,  the  trans-  Polarization 
milled  ray  is  found  to  be  partially  polarized  in  a  plane  at  right  angles  to  the  plane  of  refraction,  and  therefore  v  refrac- 
at  right  angles  to  the  plane  of  polarization  of  the  portion  of  the  reflected  ray  which  has  undergone  that  modifi- 
cation.    The  connection  between  the  polarized  portions  of  the  reflected  and  refracted  pencils  is,  nowever,  still 
more  intimate,  since  M.  Arago  has  shown  by  a  very  elegant  and  ingenious  experiment  that  these  portions  are  Arago'slaw. 
always  of  equal  intensity.     This  law  may  be  stated  thus  :  When  an  unpolarized  ray  is  partly  reflected  at,  and 
partly  transmitted  through,  a  transparent  surface,  the  reflected  and  transmitted  pencils  contain  equal  quantities  of 
polarized  light,  and  their  planes  of  polarization  are  at  right  angles  to  each  other. 

Hence  it  appears,  that  the  transmitted  ray  contains  a  maximum  of  polarized  light,  when  the  light  is  incident       864. 
at  the  polarizing  angle  of  the  medium,  and  this  maximum  is  equal  to  the  quantity  of  light  the  surface  is  capable 
of  completely  polarizing  by  reflexion.     Now  in  all  media  known,  this  is  much  less  than  half  the  incident  light, 
consequently  the  transmitted  portion  can  never  be  wholly  polarized  by  a  single  transmission. 

When  a  ray  is  totally  reflected  at  the  inner  surface  of  a  medium,  there  is  no  transmitted  portion,  an-1  it  is  a       865 
remarkable  coincidence  with  the  above  law,  that  in  this  case  the  reflected  beam  contains  no  polarized  portion 
whatever. 

With  regard  to  the  portion  of  light  which  has  passed  through  the  surface,  and  has  not  acquired  polarization,      866. 
M.  Arago   maintains  that  it  remains  in  the  state  of  natural  or  totally  unpolarized  light.     Dr.  Brewster,  on  the  Polarization 
other  hand,  concludes  from  his  experiments,  that,  although  not  polarized,  it  has  undergone  a  physical  change,  *>y  several 
rendering  it  more  largely  susceptible  of  polarization  by  subsequent  transmission  at  the  same  angle.     The  qnes-  JjJ^U," 
tion,  in  a  theoretical  point  of  view,  is  a  material  one,  and  apparently  very  easily  decided.'    The  facility,  however,  5jons 
is  only  apparent,  and  as  we  have  no  title  to  decide  it  on  the  grounds  of  our  own  experience,  we  shall  content 
ourselves  with  reasoning  on  the  conclusions  to  which  the  two  doctrines  lead.     Let  1  be  the  light  incident  on  the 
first  surface  of  a  glass  plate  at  the  polarizing  angle,  and,  after  transmission  through  both  surfaces,  let  a  -f  b  be  the 
intensity  of  the  transmitted  beam,  (and  of  course  1  —  a  —  6  that  of  the  reflected,)  and  let  a  be  the  polarized 
portion  and  6  the  unpolarized.     When  a  -f-  b  falls  on  another  plate  at  the  same  angle,  the  portion  a  being  pola- 
rized in  a  plane  perpendicular  to  that  of  incidence,  and  incident  at  the  polarizing  angle,  will  be  totally  trans- 
mitted, and  itsplane  of  polarization  (as  may  be  proved  by  direct  experiment)  in  this  case  iindergoet  no  change. 
Hence  the  portion  a  will  be  transmitted  (supposing  no  absorption)  undiminished  through  any  number  of  sub- 
sequent plates.     With  regard  to  the  portion  o,  if  this  be  to  all  intents  and  purposes  similar  to  natural  light,  it 
will  be  divided  by  reflexion  at  the  second  plate  into  two  portions,  the  first  of  which  =  6  .  (1  —  a  —  b)  being 
reflected  wholly  polarized,  and  the  other  =  6  (a  -f-  6)  will  be  transmitted.     Of  this,  the  portion  b  a  will  be  pola- 
rized in  a  plane  at  right  angles  to  that  of  refraction,  and  will  therefore  be  afterwards  transmitted  undiminished 
through  all  the  subsequent  plates.     But  the  portion  6s  will  be  unpolarized  light,  and  will  be  again  divided  by 
the  third  plate,  and  so  on.     Thus,  there  will  be  ultimately  transmitted  a  pencil,  consisting  of  a  polarized  portion 


512  LIGHT. 

v  ^  '^  _^ ,  =:  a  +  6  <z  -j-  is  a  -f-  ....  6*"'  a  sr  a  .  -   — -,  and  an  unpolarized  portion  =  6",  so  that  no  finite  number  of  v  _r  ^  -i_  • 

plates  could  ever  completely  polarize  the  whole  transmitted  beam. 

867.  On  the  other  hand,  if  the  unpolarized  portion  b  of  the  transmitted  beam  a  +  b  be  more  disposed  than  before, 
Dr.^Brew-    as  Dr.  Brewster  conceives,  to  subsequent  polarization,  the  progression  above  stated,   instead  of  converging 
sier's  theory  according  to  the  law  of  a  geometric  progression,  will   converge  more  rapidly,  or  may  even  suddenly  terminate 
polarization'  under  certain  physical  conditions.     Now,  Dr.  Brewster  states  it  as  a  general   law,  deduced  from  his  own  experi- 
Brewsier's    ments,  that  If  a  peneil  of  light  be  incident  on  a  number  of  uncrystallized  plates,  inclined  at  the  same  or  different 
general  law.  angles,  but  all  their  surfaces  being  perpendicular  to  the  plane  of  the  first  incidence,  the  total  polarization  of  the 

transmitted  pencil  will  commence  when  the  mm  of  the  tangents  of  the  angles  of  incidence  on  each  plate  is  equal 
to  a  certain  "  constant  quantity  due  to  the  refractive  power  of  the  plates,  and  the  intensity  of  the  incident  pencil  " 
This  last  phrase,  which  makes  the  number  and  position  of  the  plates  necessary  to  operate  total  polarization, 
depend  on  the  intensity  of  the  incident  light,  shows  evidently  that  the  total  polarization  here  understood,  is  not 
mathematically,  but  only  approximative^  total.  In  fact,  he  states,  this  constant  quantity  for  crown  glass  plates, 
and  for  the  flame  of  a  wax  candle  at  10  feet  distance,  to  be  equal  to  the  number  41.84.  In  other  words,  the 
remainder  of  unpolarized  light  for  this  intensity  of  illumination,  becomes  insensible.  Considered  in  this  light, 
we  regard  Dr.  Brewster's  experiments  as  by  no  means  incompatible  with  the  law  of  decrease  indicated  by  the 
geometric  progression  above-mentioned  and  the  contrary  sense  which  has  been  put  upon  this  expression  by 
M.  Arago,  or  his  commentator,  (Encyciop.  Brit.  Supp.,  vol.  vi.  part  2,  Polarization  of  Light,)  appears  to  us 
strained  beyond  what  strict  criticism  authorizes. 

Conceiving,  then,  as  we  do,  that  no  decided  incompatibility  in  matter  of  fact  exists  between  the  statements  of 
these  distinguished  philosophers,  we  cannot  but  regard  as  most  simple,  that  doctrine  which  recognises  no  change 
of  physical  character  in  the  unpolarized  portion  of  either  the  transmitted  or  reflected  beam.  (See  Art.  848.) 

868.  In  what  has  been  above  said  of  the  polarization  of  the  transmitted  ray,  we  have  not  taken  into  consideration 
Internal        that  part  of  the  light  reflected  at  each  surface  which  is  reflected  back  again,  and  traversing  (partially  at  least)  all 
reflexions     the  plates,  mixes  with  the  transmitted  beam,  and,  being  in  an  opposite  plane,  destroys  a  part  of  its  polarization. 

If  a  pile  of  parallel  glass  plates  be  exposed  to  a  polarized  ray,  so  that  the  angle  of  incidence  be  equal  to  the 
869        polarizing  angle,  and  then  turned  round  the  ray  as  an  axis  preserving  the  same  inclination,  the  following  pheno- 

Phenomena  me"a  take  Place  : 

of  piles  of  1.  When  the  plane  of  incidence  is  at  right  angles  to  that  of  the  raj's  polarization,  the  whole  of  the  incident 
platej  ex-  light  is  transmitted,  (except  what  is  destroyed  by  absorption  within  the  substance  of  the  glass,  or  lost  by  irregular 
^arii'li  reflexion  from  the  inequalities  in  the  surface  arising  from  defective  polish,)  and  this  holds  good  whatever  be  the 
lenT'"  number  of  the  plates.  The  polarization  of  the  transmitted  ray  is  unaltered. 

"-Z.  As  the  pile  revolves  round  the  incident  ray  as  an  axis,  a  portion  of  the  light  is  reflected,  and  this  increases 
till  the  plane  of  incidence  is  coincident,  with  the  plane  of  primitive  polarization,  when  the  reflected  light  is  a 
maximum.  Now,  M.  Arago  assures  us,  that  the  quantity  of  polarized  light  reflected  from  each  plate  is  greater  in 
proportion  to  the  intensity  of  the  incident  beam  than  if  natural  light  had  been  employed ;  and  the  same  pro- 
portion holding  good  at  each  plate,  the  transmitted  ray,  however  intense  it  may  have  been  at  first,  will  be 
weakened  in  geometrical  progression  with  the  number  of  plates,  and  at  length  will  become  insensible  ;  so  that 
in  this  situation  the  pile  will  present  the  phenomenon  of  an  opaque  body.  In  this  reasoning,  the  light  reflected 
backwards  and  forwards  between  the  plates  is  neglected  ;  but  as  it  is  all  polarized  in  the  same  plane,  and  as  in 
this  situation  the  reflexions,  however  frequent,  produce  no  change  in  its  plane  of  polarization,  all  the  reflected 
rays  are  in  the  same  predicament ;  and,  supposing  the  number  of  plates  very  great,  the  total  extinction  of  the 
transmitted  light  will  ultimately  (though  less  rapidly)  take  place. 

870.  Hence,  a  pile  of  a  great  number  of  glass  plates  inclined  at  an  angle  equal  to  the  complement  of  the  polarizing 
Phenomena  angle  (35°  x)  to  a  polarized  ray  ought  to  present  the  same  phenomenon  with  a  plate  of  tourmaline  cut  parallel 
of  piles  of    to  the  axis  of  its  primitive  rhomboid,  alternately  transmitting  and  extinguishing  the  whole  of  the  light  in  the 
plates,  and   successjve  quadrants  of  its  rotation,  and  being  thus  either  opaque  or  transparent,  according  to  its  position.     The 
line  pUtes    analogy,  however,  cannot  fairly  be  pushed  farther,  s;>  as  to  deduce  from  this  principle  an  explanation  of  the  phe- 
conipared.    nomena  of  the  tourmaline  ;  for,  although  it  be  true  that  a  plate  of  tourmaline  so  cut,  is  composed  of  lamime 

inclined  to  its  surface,  these  laminae  are  in  optical  contact;  and,  moreover,  their  position  with  respect  to  the 
surface  is  not  the  same  in  plates  cut  in  all  directions  around  the  axis,  because  although  an  infinite  number  of 
plates  may  be  cut  containing  the  axis  of  a  rhomboid  in  their  planes,  only  three  can  have  the  same  relation  to  its 
several  faces,  parallel  to  which  the  component  laminK  must  be  supposed  to  lie.  Moreover,  the  phenomena  are 
not  produced,  unless  the  tourmaline  be  coloured.  The  analogy  between  piles  of  glass  plates  and  lamina;  of  agate 
(of  which  more  presently)  is  also,  we  are  inclined  to  think,  more  apparent  than  real. 

871.  A  pile  of  plates  such  as  described  above  presents,  moreover,  the  same  difference  of  phenomena  when  exposed 
Further        to  polarized  and  unpolarized  light,  that  a  plate  of  tourmaline  does  ;  since  in  the  latter  case,  supposing  the  pile 
analogy.       sufficiently  numerous,  one  half  the  incident  light  is  transmitted,  completely  polarized  in  a  plane  perpendicular  to 

that  of  incidence. 

872.  The  laws  which  regulate  the  polarization  of  a  pencil  transmitted  by  a  transparent   surface,  inclined  at  any 
proposed  angle  to  the  incident  ray,  and  in  any  plane  to  that  of  its  primitive  polarization  (supposing  it  polarized) 
remain  open  to  experimental  investigation. 


LIGHT.  513 

l.iaht.  Part  IV. 

§  VI.   Of  the  Polarization  of  Light  by  Double  Refraction. 

When  a  ray  of  natural  light  is  divided  into  two  by  double  refraction,  in  such  a  manner  that  the  two  pencils  at      873. 
their  final  emergence  remain  distinct  and  susceptible  of  separate  examination,  they  are  both  found  completely  Light  poU- 
polarized,  in  different  planes,  exactly,  or  nearly,  at  right  angles   to  each  other.     To  show  this,  take  a  pretty  "zejj.  b'r 
thick  rhomboid   of  Iceland    spar,  and,    covering   one  side   of   it  with    a  blackened   card,  or   other   opaque  refjaction 
thin  substance,  having  a  small  pinhole  through  it,  hold  it  against  the  direct  light  of  a  window  or  a  candle,  with  oppositely 
the  covered  surface  from  the  eye.     Two  images  of  the  pinhole  will  then  be  seen  :  one,  undeviated  from  the  line  in  the  two 
joining  the-  eye  and  the  real  hole,  by  the  ordinarily  refracted  rays ;   and  the  other,  deviating  from  that  line,  in  a  Pfncils 
plane  parallel  to  the  principal  section  of  the  surface  of  incidence,  by  the  extraordinary.     These  images  will 
appear,  to  the  naked  eye,  of  equal  brightness ;  but,  if  we  interpose  a  plate  of  tourmaline,  (as  already  described,)  ™roofS '" 
and  turn  the  latter  about  in  its  own  plane,  they  will  be  rendered  unequal,  and  will  appear  and  vanish  alternately  thereof, 
at  every  quarter  revolution  of  the  tourmaline  ;  the  ordinary  image  being  always  at  its  maximum  of  brightness, 
and  the  extraordinary  one  extinct,  when  the  axis  of  the  tourmaline  plate  is  perpendicular  to  the  principal  section 
of  the  surface  of  incidence,  and  vice  vend  when  parallel  to  it. 

The  same  thing  happens,  when,  instead  of  examining  the  two  images  through  a  tourmaline  plate,  we  receive      874. 
their  light  on  a  glass  plate  inclined  at  the  polarizing  angle  to  it,  and  turn  this  plate  round  the  ordinary  ray  Experiment 
as  an   axis.     The  images  will  appear  and  disappear  alternately,  as  the  reflector  performs  successive  quadrants  va"e(1- 
of  its  revolution. 

Hence,  we  see  that  the  two  pencils  are  completely  and  oppositely  polarized ;  the  ordinary  pencil  in  a  plane      875. 
passing  through  the  axis  of  the  rhomboid  ;  the  extraordinary  one  in  a  plane  at  right  angles  to  it. 

The  same  phenomenon  is  much  better  seen  by  using  a.  prism  of  any  double  refracting  crystal,  having  such  a      876. 
refracting  angle  as  to  give  two  distinctly  separated  images  of  a  distant  object,  (as  a  candle.)     These  appear  and  Another 
disappear  alternately  at  quarter  revolutions  of  a  tourmaline  plate  or  glass  reflector,  and  are  of  equal  brightness  experiment" 
at  (he  intermediate  half-quarters. 

Double  refraction,  then,  polarizes  the  two  refracted  pencils  oppositely,  into  which  an  unpolarized  incident  ray      877. 


but  inclined  downwards,  against  the  reflected  light  from  the  glass.     Then,  generally  speaking,  two  images  of  t'lfrough 
the  pinhole  will  be  seen,  but  of  unequal  intensities  ;    and,  if  we  turn  round  the  rhomboid,  in  the  plane  of  the  doubly 
covered  side,  these  images  will  be  seen  to  vary  perpetually  in  their  relative  brightness,  the  one  increasing  to  a  max-  refracting 
imum,  while  the  other  vanishes  entirely,  and  so  on  reciprocally.     When  the  principal  section  of  the  rhomboid  is  in  metl'a- 
the  plane  of  reflexion  (i.  e.  of  polarization)  of  the  incident  ray,  the  ordinary  image  is  a  maximum  ;  the  extra- 
ordinary is  extinct,  and  Dice  versa  when  these  two  planes  make  a  right  angle.     The  experiment  may  be  advan- 
tageously varied  by  using  a  doubly  refracting  prism  ;  and,  while  looking  through  it  at  the  polarized  image  of  a 
candle,  turning  it  round  slowly  in  the  plane  bisecting  its  refracting  angle. 

This  experiment  leads  us  to  the  following  remarkable  law,  vit.  that  if  a  ray,  at  its  incidence  on   a  doubly      878. 
refracting  surface,  be  polarized  in  the  plane  parallel  to  the  principal  section,  it  will  not  suffer  bifurcation,  but  Unequal 
will  pass  wholly  into  the  ordinary  image  ;  if,  on  the  other  hand,  its  plane  of  primitive  polarization  be  perpen-  dl'vl!!0?  °^ 
dicular  to  the  principal  section,  it  will  pass  entirely  into  the  extraordinary  image.     In  intermediate  positions  of(,e^w'^en 
the  plane  of  primitive  polarization,  bifurcation  takes  place,  and  the  ray  is  unequally  divided  between  the  two  the  two 
refracted  pencils,  in  every  case  except  when  the  plane  of  primitive  polarization  makes  an  angie  of  45°  with  the  refracted 
principal  section.     In  general,  if  a  be  the  angle  last  mentioned,  and  A  the  incident  light,  (supposing  none  lost  Penc'ls- 
by  reflexion,)  A  .  cos8  a  will  be  the  intensity  of  the  ordinary,  and  A  .  sin*  a  of  the  extraordinary  pencil,  their 
sum  being  A. 

All  these  changes  and  combinations  are  exhibited  in  the  following  remarkable  experiment  of  Huygens,  which,      879. 
reasoned  on  by  himself  and  Newton,  first  gave  rise  to  the  conception  of  a  polarity,  or  distinction  of  sides,  in  the  Huygens's 
rays  of  light  when  modified  by  certain  processes.     Take  two  pretty  thick  rhomboids  of  Iceland  spar,  (which  exPerlmeo' 
should  be  very  transparent,  as  they  are  easily  procured,)  and  lay  them  down  one  upon  the  other,  so  as  to  have 
their  homologous  sides  parallel,  or  so  that  the  molecules  of  each  shall  have  the  same  relations  of  situation  as  if 
the  two  rhomboids  were  contiguous  parts  of  one  larger  crystal.     They  should  be  laid  on  a  sheet  of  white  paper 
having  a  small,  very  distinct,  and  well-defined  black  spot  on  it.     This  spot  then  will  be  seen  double  through  the 
combined  crystals,  as  if  they  were  one,  (a,  fig.  173,)  and  the  line  joining  the  images  will  be  parallel   to  the  Fig.  137. 
principal  section  of  either.     Now,  let  the  upper  crystal  be  turned  slowly  round  in  a  horizontal   plane  on  the 
lower,  and  two  new  images  will  make  their  appearance  between  the  two  first  seen,  which,  at  first,  are  very  faint, 
as  at  6,  fig.  173,  and  form  a  very  elongated  rhombus  with  the  two  former.     They  increase,  however,  in  intensity, 
while  the  other  pair  diminishes,  till  the  angle  of  rotation  of  the  upper  crystal  is  45°,  where  the  appearance  of  the 
images  is  as  at  c.     Continuing  the  rotation,  the  rhomb  approaches  to  a  square,  as  at  d,  and  the  two  original  images 
have  become  extremely  faint ;  and  when  the  rotation  is  just  90°,  they  will  have  disappeared  altogether,  leaving 
the  others  diagonally  placed,  as  at  e.     As  the  rotation  still  proceeds,  they  reappear  and  increase  in  brightness,  till 
the  angle  of  revolution  =  90°  -\-  45°  =  135°,  when  the  images  are  all  equal,  as  at  f;  after  which  the  original 
images  still  increasing,  and  the  others  diminishing,  the  appearance  g  is  produced,  which,  on  the  completion  of 
a  precise  half  revolution,  passes  into  h  by  the  union  of  both  the  original  images  into  one,  and  the  total  evanes- 

VOL.  IT.  3  X 


514 


LIGHT. 


Light. 


880. 


881. 
Use  of  an 
achromatic 
double 
refracting 
prism. 


Fig.  174. 
First  achro- 
matized by 
glass. 


882. 

Dr.  Wollas- 
ton's  mode 
of  doubling 
the  separa- 
tion of 
images. 
Fig.  175. 


883. 
Action  of 
crystals 
possessing 
on  double 
refraction. 


cence  of  the  other  pair.     In  this  oase.  only  single  refraction  (apparently)  happens  ;  or,  rather,  the  double  refrac-     Fart  IV 
tions  of  the  two  rhomboids  taking  place  in  opposite  directions,  and   being  equal   in  amount,  compensate  each  v— v"™*1 
other.     Unless,  however,  the  rhomboids  be  of  exactly  equal  thickness,  this  precise  compensation  will  not  take 
place,  ard  the  images  will  remain  distinct,  though  at  a  minimum  of  distance.     We  may  express  the  four  images 
thus : 

O  o,  the  image  ordinarily  refracted  by  both  rhomboids. 

O  e,  the  image  refracted  ordinarily  by  the  first,  and  extraordinarily  by  the  second. 

E  o,  the  image  refracted  extraordinarily  by  the  first,  and  ordinarily  by  the  second. 

E  e,  the  image  refracted  extraoidinarily  by  both. 

Then,  if  A  be  the  intensity  of  the  incident  light,  supposing  none  lost  by  reflexion  or  absorption, 

O  o  =  J  A .  cos4  a  =  ~Ee;     Oe=JA.  sin2  a  =  E  o, 
and  the  sum  of  all  the  four  images  =  A. 

The  same  phenomena  (with  some  unimportant  variations)  take  place  when  we  apply  two  doubly  refracting 
prisms  one  behind  the  other  close  to  the  eye,  and  view  a  distant  object  through  them,  turning  one  round  on 
the  other.  The  rationale  of  these  phenomena  follows  so  evidently  from  the  laws  stated  in  Art.  875  and  878, 
that  it  will  not  be  necessary  to  enlarge  on  it. 

The  property  of  a  double  refraction,  in  virtue  of  which  a  polarized  ray  is  unequally  divided  between  the  two 
images,  furnishes  us  with  a  most  convenient  and  useful  instrument  for  the  detection  of  polarization  in  a  beam 
of  light,  and  for  a  variety  of  optical  experiments.  It  is  nothing  more  than  a  prism  of  a  doubly  refracting 
medium  rendered  achromatic  by  one  of  glass,  or  still  better,  by  another  prism  of  the  same  medium  properly 
disposed,  so  as  to  increase  the  separation  of  the  two  pencils.  The  former  method  is  simple;  and,  when  large 
refracting  angles  are  not  wanted,  the  uncorrected  colour  in  one  of  the  images  is  so  small  as  not  to  be  trouble- 
some. It  is  most  convenient  to  make  the  refracting  angle  such  as  to  produce  an  angular  separation  of  about  2° 
between  the  images.  Thus,  in  fig.  174,  let  A  B  C  G  F  be  a  prism  of  Iceland  spar,  cut  in  such  a  manner  (we 
will  at  present  suppose)  that  the  refracting  edge  C  G  shall  contain  the  axis  of  the  crystal ;  and  let  it  be  achro- 
matized as  much  as  possible  by  a  prism  of  glass  C  D  E  P  G.  Then,  if  Q  be  a  small,  colourless,  luminous  circle 
of  about  a  degree  or  two  in  apparent  diameter,  as  seen  by  an  eye  at  O,  the  interposition  of  the  combined  prisms 
will  divide  it  into  two,  Q  and  9.  Now,  if  the  light  of  Q  be  completely  unpolarized,  these  two  will  remain 
exactly  of  equal  intensity  while  the  prism  AB  C  G  is  turned  round  in  a  plane  at  right  angles  to  the  line  of  vision. 
But  if  any  polarity  exist  in  the  original  light,  the  two  images  Q,  q  will,  in  turning  round  the  prism,  appear  alter- 
nately more  and  less  bright  one  than  the  other  ;  and  being  always  seen  immediately  side  by  side,  the  least 
inequality,  and  consequently  the  least  admixture  of  polarized  light  in  the  incident  beam,  will  be  detected. 

Iceland  spar,  from  its  very  great  double  refraction,  is  commonly  used  for  these  prisms ;  but  it  is  so  soft,  and 
its  structure  so  lamellar,  as  to  be  difficult  to  polish,  and  still  more  so  to  preserve  polished.  We  have  found  quartz 
and  limpid  topaz  to  answer  extremely  well.  The  following  ingenious  mode  of  rendering  available  the  low  double 
refraction  of  the  former,  due  to  Dr.  Wollaston,  is  here  eminently  useful.  Let  ABCD  abed  and  E  FGHefgh 
(fig.  175)  be  two  halves  of  a  hexagonal  prism  of  quartz  (the  form  it  affects)  produced  by  a  section  parallel  to  two  of 
the  sides.  In  the  vertical  face  A  D  da  draw  any  line  L  K  parallel  to  the  sides,  and  therefore  to  the  axis  of  the  prism, 
(which  is  also  that  of  double  refraction,)  and  join  C  L,  ck.  Then  a  plane  CL  Arc  will  cut  off  a  prism  CLKrfcD, 
having  L  k,  D  d,  or  C  c,  for  its  refracting  edges,  either  of  which  is  parallel  to  the  axis.  Again,  in  the  other  half 
of  the  prism  join  E/"and  H  g,  and  cut  the  prism  by  a  plane  passing  through  these  lines  ;  then,  regarding  either 
portion  as  a  double  refracting  prism,  having  for  refracting  edges  the  lines  E  H,  fg,  these  will  have  the  axis  of 
double  refraction  perpendicular  to  their  refracting  edges ;  and,  in  particular,  the  axis  will  lie  in  the  faces  HE  eh, 
or  FG  gf  at  right  angles  to  H  E  or  fg.  If,  then,  we  take  care  to  make  the  refracting  angle  C  L  D  of  the 
prism  C  L  K  d  c  D  equal  to  that  of  the  edge  II  E  of  the  prism  H  E  efg  h  ;  and  if  we  make  these  two  prisms 
act  in  opposition  to  each  other,  placing  the  edge  H  E  opposite  to  D  d,  and  the  edge  h  e  opposite  to  K  L ;  and 
having  thus  brought  the  two  surfaces  D  L  k  d  and  H  E  e  A  in  contact,  cement  them  together  with  mastic,  or 
Canada  balsam,  it  is  evident,  that  their  principal  sections  will  be  at  right  angles  to  each  other;  and  therefore 
only  two  images  will  be  formed,  the  whole  of  the  extraordinary  ray  of  the  one  prism  passing  into  the  ordinary 
image  of  the  other,  and  vice  vend.  Now,  to  see  how  this  acts  to  double  the  separation  of  the  images,  let  us 
conceive  m  n  to  be  a  luminous  line  viewed  through  one  of  the  prisms  with  its  edge  downwards  and  horizontal. 
It  will  be  separated  into  two  images,  e  and  o,  the  one  more  raised  than  the  other.  Suppose  the  ordinary  image 
to  be  most  refracted.  Then,  if  we  interpose  the  other  prism  with  its  edge  upwards,  both  these  images  will 
be  refracted  downwards  ;  but  the  ordinary  image  o,  which  was  before  moat  raised,  now  undergoing  extraordinary 
refraction,  is  least  depressed,  and  comes  into  the  position  o  e,  while  the  extraordinary  one  e,  which  was  before 
least  raised  is  now  most  depressed,  and  comes  into  the  situation  eo ;  and  it  is  evident  that  (the  refracting  angles 
being  equal,  and  the  double  refraction  of  the  two  prisms  the  same)  the  line  o  e  will  fall  as  far  short  of  the  ori- 
ginal line  m  n,  as  eo  surpasses  it,  viz.  by  a  quantity  equal  to  the  distance  between  the  two  first  images  o  and  e ; 
so  that  the  distance  between  the  twice  refracted  images  is  double  that  of  those  which  have  undergone  only  one 
refraction.  We  have  found  this  combination  extremely  advantageous,  as  quartz  takes  a  very  perfect  polish,  and 
from  its  hardness  is  not  liable  to  injury  from  scratches. 

Crystals  which  have  no  double  refraction  may  be  regarded  as  limits  of  those  which  have,  or  as  crystals  in 
which  the  two  rays  are  propagated  with  equal  velocity,  and  therefore  undergo  no  bifurcation  ;  or,  in  other  words, 
in  which  the  images  formed  coincide.  In  this  case  we  should  expect  to  find  no  polarization  of  the  emergent 
light,  because  the  two  pencils,  being  polarized  at  right  angles  to  each  other,  form  together  a  single  ray  having 
the  characters  of  unpolarized  light.  This  is  verified  by  experiment.  The  light  transmitted  by  fluor  spar,  for 


LIGHT.  515 

instance,  exhibits  no  signs  of  polarization,  unless  so  far  as  the  ordinary  action  of  the  surface  {roes.     We  are  awnre    Part  IV. 
of  no  experiments  indicating  how  far  the  action  of  the  surfaces  of  feebly  double  refracting  crystals  may  modify  v— — \-— ' 
their  polarizing  forces,  or  rather  their  effects  on  a  ray  which  has  penetrated  below  the  surface ;    or,  in  other 
words,  how  far  piles  of  crystallized  laminae  may  have  an  analogous  or  different  action  from  those  of  uncrystallized. 
Dr.  Brewster,  indeed,  found  piles  of  mica  films  to  polarize  light  by  transmission,  like  glass  piles,  but  the  subject 
is  open  to  further  inquiry. 

§  VII.   Of  tfte  Colours  exhibited  by  Crystallized  Plates  when  exposed  to  Polarized  Light,  and  of  the  Polarized 

Rings  which  surround  their  Optic  Axes. 

Qft4 

This  splendid  department  of  Optics  is  entirely  of  modern  and,  indeed,  of  recent  origin.  The  first  account  of  the 
colours  of  crystallized  plates  was  communicated  by  M.  Arago  to  the  French  Institute  in  181 1,  since  which  period, 
by  the  researches  of  himself,  Dr.  Brewster,  M.  Biot,  M.  Fresnel,  and,  latterly,  also  of  M.  Mitscherlich,  and  others, 
it  has  acquired  a  developement  placing  it  among  the  most  important  as  well  as  the  most  complete  and  systematic 
branches  of  optical  knowledge.  As  might  be  expected,  under  such  circumstances,  as  well  as  from  the  state  of 
political  relations,  and  the  consequent  limited  intercourse  between  Britain  and  the  Continent  at  the  period  men- 
tioned, an  immense  variety  of  results  could  not  but  be  obtained  independently,  and  simultaneously,  or  nearly 
simultaneously,  on  both  sides  of  the  channel.  To  the  lover  of  knowledge,  for  its  own  sake, — the  philosopher, 
in  the  strict  original  sense  of  the  word, — this  ought  to  be  matter  of  pure  congratulation  ;  but  to  such  as  are 
fond  of  discussing  rival  claims,  and  settling  points  of  scientific  precedence,  such  a  rapid  succession  of  interesting 
discoveries  must,  of  course,  afford  a  welcome  and  ample  supply  of  critical  points,  the  seeds  of  an  abundant 
harvest  of  dispute  and  recrimination.  Regarding,  as  we  do,  all  such  discussions,  when  carried  on  in  a  spirit  of 
rivalry  or  nationality,  as  utterly  derogatory  to  the  interests  and  dignity  of  science,  and  as  little  short,  indeed,  of 
sacrilegious  profanation  of  regions  which  we  have  always  been  accustomed  to  regard  only  as  a  delightful  and 
honourable  refuge  from  the  miserable  turmoils  and  contentions  of  interested  life,  we  shall  avoid  taking  any  part 
in  them  ;  and,  taking  up  the  subject  (to  the  best  of  our  abilities  and  knowledge)  as  it  is,  and  avoiding,  as  far  as 
possible,  all  reference  to  misconceived  facts  and  over-hasty  generalizations,  which  in  this  as  in  all  other  depart- 
ments of  science,  have  not  failed  (like  mists  at  daybreak)  to  spread  a  temporary  obscurity  over  a  subject 
imperfectly  understood,  shall  make  it  our  aim  to  state,  in  as  condensed  a  form  as  is  consistent  with  distinctness, 
such  general  facts  and  laws  as  seem  well  enough  established  to  run  no  hazard  of  being  overset  by  further 
inquiry,  however  they  may  merge  hereafter  in  others  yet  more  general ; — a  consummation  devoutly  to  be 
wished. 

The  general  phenomenon  of  the  coloured  appearances  to  which  this  section  is  devoted,  may  be  most  readily    .     °"*4 
and  familiarly  shown  as  follows.     Place  a  polished  surface  of  considerable  extent  (such  as  a  smooth  mahogany  m'^^  of 
table,  or,  what  is  much  better,  a  pile  of  ten  or  a  dozen  large  panes  of  glass  laid  horizontally)  close  to  a  exhibiting 
large  open  window,  from  which  a  full  and  uninterrupted  view  of  the  sky  is  obtained;  and  having  procured  a  the  colours 
plate  of  mica,  of  moderate  thickness,  (about  a  thirtieth  of  an  inch,  such  as  may  easily  be  obtained,  being  sold  of  crystal- 
in  considerable  quantity  for  the  manufacture  of  lanterns,)  hold  it  between  the  eye  and  the  table,  or  pile,  so  as  1" 
to  receive  and  transmit  the  light  reflected  from  the  latter  as  nearly  as  may  be  judged  at  the  polarizing  angle.  ;„  mjca_ 
In  this  situation  of  things,  nothing  remarkable  will  be  perceived,  however  the  plate  of  mica  be  inclined;  but  if 
instead  of  the  naked  eye  we  look  through  a  tourmaline  plate,  having  its  axis  vertical,  the  case  will  be  very  different. 
When  the  mica  plate  is  away,  the  tourmaline  will  destroy  the  reflected  beam,  and  the  surface  of  the  table,  or 
pile,  will  appear  dark  and  non-reflective  ;    at  least  in  one  point,  on  which  we  will  suppose  the  eye  to  be  kept 
steadfastly  fixed.     No  sooner  is  the  mica  interposed,  however,  than  the  reflective  power  of  the  surface  appears  to 
be  suddenly  restored  ;    and   on  inclining  the  mica   at  various  angles,  and  turning  it  about  in  its  own  plane, 
positions  will  readily  be  found  in  which  it  becomes  illuminated   with   the  most  vivid  and  magnificent  colours, 
which  shift  their  tints  at  the  least  change  of  position  of  the  mica,  passing  rapidly  from  the  most  gorgeous  reds 
to  the  richest  greens,  blues,  and  purples.     If  the  mica  plate  be  held  perpendicular  to  the  reflected  beam,  and 
turned  about  in  its  own  plane,  two  positions  will  be  found  in  which  all  colour  and  light  disappears  ;    and  the 
reflected   ray  is  extinguished,  as  if  no  mica  was  interposed.     Now,  if  we  draw  on  the  plate  with  a  steel  point  Two  re- 
two  lines  corresponding  to  the  intersection  of  the  mica  with  a  vertical  plane  passing  through  the  eye  in  either  markable 
of  these  two  positions,  we  shall  find  that  they  make  an  exact  right  angle.     For  the  moment,  let  us  call  these  lines  sections  of 
A  and  B  ;  and  let  a  plane  drawn  through  the  line  A,  perpendicular  to  the  plate,  be  called  the  section  A  ;  and  one  ,'|  e  "yst"' 
similarly  drawn  through  the  line  B,  the  section  B.     Then  we  shall  observe  further,  that  when  we  turn  the  plate 
Irom  either  of  these  positions,  45°  round,  in  its  own  plane,  so  that  the   sections  A  and   B  shall  make   angles 
of  45°  with  the  plane  of  reflexion,  (i.  e.  of  polarization  of  the  incident  ray,)  the  transmitted  light  will   be  a 
maximum. 

If  the  thickness  of  the  mica  do  not  exceed  ^ffth  of  an  inch,  it  will  be  coloured  in  this  position ;  if  materially      886. 
greater,  colourless  ;  and  if  less,  more  and  more  vividly  coloured,  and  with  tints  following  closely  the  succession  ^aw  of  .t'le 
of  the  reflected  series  of  the  colours  of  thin  plates,  and,  like  them,  rising  in  the  scale,  or  approaching  the  b°*d   t '" 
central  tint  (black)  as  the  thickness  is  less.     The  analogy  in  this  respect,  in  short,  is  complete,  with  the  excep-  perpendi- 
tion  of  the  enormous  difference  of  thickness  between  the  mica  plate  producing  the  tints  in  question,  and  those  cular 
required  to  produce  the  Newtonian  rings.     It  appears  by  measures  made  in  the  manner  hereafter  to  be  described,  incidence, 
that  the  tint  exhibited  by  a  plate  of  mica  exposed  perpendicularly  to  the  reflected  ray,  as  above  described,  is 
the  same  with  that  reflected  by  a  plate  of  air  of  T0gth  part  of  the  thickness  of  the  mica  employed. 

3x2 


516 


LIGHT. 


S87. 

i!it"(l  •e"hh" 
twose'ction 
»bove 
mentioned, 


Light.  If  the  mica  (still  exposed  perpendicularly  to  the  ray)  be  turned  round  in  its  own  plane,  the  tint  does  not 
'  change,  hut  only  diminishes  in  intensity  as  its  section  A  or  B  approaches  the  plane  of  polarization  of  the  inci-  ' 
dent  light.  When,  however,  the  plate  is  not  exposed  perpendicularly,  this  invariability  no  longer  obtains  ;  and 
^e  c*lan§'es  °f  tmt  appear  >n  the  last  degree  capricious  and  irreducible  to  regular  laws.  In  two  situations, 
s  nowever>  'he  phenomena  admit  a  simple  view.  These  are  when  the  sections  A  and  B  are  both  45°  from  the 
plane  of  polarization,  and  the  mica  plate  is  inclined  backwards  and  forwards  in  the  plane  of  one  or  the  other  of 
these  sections.  This  condition  is  easily  attained  by  first  holding  the  plate  perpendicularly  to  the  reflected  ray  ; 
then  turning  it  in  its  own  plane  till  the  lines  A,  B  are  each  45°  inclined  to  the  vertical  plane,  then  finally  causing 
it  to  revolve  about  either  of  these  lines  as  an  axis.  It  will  then  be  seen  that  when  made  to  revolve  round  one  of 
them  (as  A)  or  in  the  plane  of  the  section  B.  the  tint,  if  white,  will  continue  white  at  all  angles  of  inclination  ; 
but  if  coloured,  will  descend  in  the  scale  of  the  coloured  rings,  growing  continually  less  highly  coloured,  till  it  passes, 
after  more  or  fewer  alternations,  into  white;  after  which,  further  inclination  of  the  plate  will  produce  no  change. 
On  the  other  hand,  if  made  to  revolve  round  B,  or  in  the  plane  of  A,  the  tints  will  rise  in  the  scale  of  the  rings  ; 
and  when  the  mica  plate  is  inclined  either  way,  so  as  to  make  the  angle  of  incidence  about  35°  3',  will  have 
attained  its  maximum,  corresponding  to  the  black  spot  in  the  centre  of  Newton's  rings.  In  this  position  of  the 
plate,  the  reflected  beam  is  totally  extinguished  by  the  tourmaline,  as  if  the  sections  A  or  B  had  been  vertical. 
But  if  the  angle  of  incidence  be  still  further  increased  the  colours  reappear,  and  descend  again  in  the  scale  of 
the  rings,  passing  through  their  whole  series  to  final  whiteness.  We  take  no  notice  here  of  a  slight  deviation  from 
the  strict  succession  of  the  Newtonian  colours,  which  is  observed  in  the  higher  orders  of  the  tints,  as  we  shall 
have  more  to  say  respecting  it  hereafter. 

We  see,  then,  that  the  sections  A  and  B,  though  agreeing  in  their  characters  in  the  case  of  a  perpendicular 
exposure  of  the  mica,  yet  differ  entirely  in  the  phenomena  they  exhibit  at  oblique  incidences.  If  the  incidence 
take  place  in  the  plane  of  the  section  B,  the  tint  descends,  on  both  sides  of  the  perpendicular,  ad  infinitum. 
While,  if  the  incidence  be  in  the  section  A,  it  rises  to  the  central  black,  which  it  attains  at  equal  incidences  on 
either  side  of  the  perpendicular  (35°  3'),  and  then  descends  again  ad  infinitum,  or  to  the  composite  white  at  the 
other  extreme  of  the  scale. 

The  section  A,  then,  (which,  for  this  reason,  we  will  call  the  principal  section  of  the  mica  plate,)  is  characte- 
rised by  containing  two  remarkable  lines  inclined  at  equal  angles  to  the  surface  of  the  plate,  along  either  of 
which,  if  a  polarized  ray  be  incident,  its  polarization  will  not  be  disturbed  by  the  action  of  the  plate.  To  satisfy 
ourselves  of  this,  we  have  only  to  fix  the  mica  to  the  extremity  of  a  tube,  so  as  to  have  the  axis  of  the  tube 
inclined  at  an  angle  of  35°  3'  to  the  perpendicular  (or  54°  57'  to  the  plate)  in  the  plane  of  the  section  A  ;  then 
directing  the  axis  of  the  tube  to  the  centre  of  the  dark  spot,  or  the  reflecting  surface,  it  will  be  seen  to  continue 
(Jarii;j  an(j  remain  so  while  the  tube  makes  a  complete  revolution  on  its  axis.  Now,  this  could  not  be  if  the 
mica  exercised  any  disturbing  power  on  the  plane  of  polarization.  Hence,  we  conclude,  that  the  two  lines  in 
question  possess  this  remarkable  property,  viz.  that  whatever  be  the  plane  of  polarization  of  a  ray  incident  along 
either  of  them,  it  remains  unaltered  after  transmission.  For,  although  in  the  experiment  above  described,  the 
plane  of  polarization  remained  fixed,  and  that  of  incidence  was  made  to  revolve,  it  is  obvious  that  the  reverse 
process  would  come  to  the  very  same  thing. 

Now,  this  character  belongs  to  no  other  lines,  however  chosen,  with  respect  to  the  plate.  If  we  fix  the  plate 
on  the  end  of  the  tube  at  any  other  angle,  or  in  any  other  plane  with  respect  to  the  axis  of  the  latter,  although 
two  positions  in  the  rotation  of  the  tube  will  always  be  found  where  the  disappearance  of  the  transmitted  ray 
takes  place,  in  no  other  case  but  that  of  the  two  lines  in  question  will  this  disappearance  be  total,  or  nearly  so, 
in  all  points  of  its  revolution. 

The  refracting  index  of  mica  being  1.500,  an  angle  of  incidence  of  35°  3'  corresponds  to  one  of  refraction  = 
22°  31'.  Hence,  the  position  of  the  lines  within  the  mica  corresponding  to  these  external  lines  is  22J°  inclined 
to  the  perpendicular,  and  the  angle  included  between  them  45°.  These,  then,  are  axes  within  the  crystal, 
bearing  a  determinate  relation  to  its  molecules.  Dr.  Brewster  has  termed  them  axes  of  no  polarization,  a  long 
name.  M.  Fresnel,  and  others,  have  used  the  phrase  optic  axes,  to  which  we  shall  adhere.  As  this  term  has 
before  been  ;  pplied  to  the  "  axes  of  no  double  refraction,"  we  must  anticipate  so  far  as  to  advertise  the  reader 
that  these,  and  the  "  axes  of  no  polarization,"  are  in  all  cases  identical. 

Having,  by  the  criteria  above  described,  determined  the  principal  section,  and  ascertained  the  situation  of 
the  optic  axes  of  the  mica  plate  under  examination,  let  the  plate  be  inclined  to  the  polarized  beam,  so  that  the 
latter  shall  be  transmitted  along  the  optic  axes,  the  principal  section  A  making  an  angle  of  45°  with  the  plane  of 
polarization  ;  and  let  the  eye  (still  armed  with  the  tourmaline  plate,  with  its  axis  vertical)  be  applied  close  to 
the  mica.  A  splendid  phenomenon  will  then  be  seen.  The  black  point  corresponding  to  the  direction  of  the 
optic  axis  will  be  seen  to  be  surrounded  with  a  set  of  broad,  vivid,  coloured  rings,  of  an  elliptic,  or,  at  least,  oval 
form,  divided  into  two  unequal  portions  by  a  black  band  somewhat  curved,  as  represented  in  fig.  176.  Thi.« 
band  passes  through  the  pole,  or  angular  situation  of  the  optic  axis,  about  which  the  rings  are  formed  as  a 
centre.  Its  convexity  is  turned  towards  the  direction  of  the  other  axis,  and  on  that  side  the  rings  are  also 
broader.  If,  now,  the  other  axis  be  brought  into  a  similar  position,  a  phenomenon  exactly  similar  will  be. 
seen  surrounding  its  place,  as  a  pole.  If  the  mica  plate  be  very  thick,  these  two  systems  of  rings  appear  wholly 
detached  from,  and  independent  of,  each  other,  and  the  rings  themselves  are  narrow  and  close  ;  but  if  thin  (as  a 
30th  or  40th  of  an  inch)  the  individual  rings  are  much  broader,  and  especially  so  in  the  interval  between  the 
poles,  so  as  to  unite  and  run  together,  losing  altogether  their  elliptic  appearance,  and  dilating  towards  the  middle 
(or  in  the  direction  of  a  perpendicular  to  the  plate)  into  a  broad  coloured  space,  beyond  which  the  rings  are  no 
longer  formed  about  each  pole  separately,  but  assume  the  form  of  reentering  curves,  embracing  and  including 
both  poles.  Their  nature  will  presently  be  stated  more  at  large. 


888. 

Characters 
ui  the  two 
most 

remarkable 
sections. 

889. 
The 

principal 
sertiou 
defined. 
Contains 
the  two 
optic  axes. 
Characters 
ol  these 
axes. 


890. 


691. 

Position  of 
the  optic 

axes  in  mi 


892 

The  pola- 
rized rings 
about  the 
optic  axes. 
General  de- 
scription of 
their  pheno- 


L  I  G  H  T.  517 

I.i(fht.          If  preserving  the  same  inclination  of  the  mica  plate  to  the  visual  ray,  it  be  turned  about  it  as  an  axis,  the   Part  IV. 
-~Y-—'  b-ack  band  passing  through  the  pole  will  shift  its  place,  and  revolve  as  it  were  on  the  pole  as  a  centre  with  double  *•— -v-^«-' 
the  angular  velocity,  so  as  to  obliterate  in  succession  every  part  of  the  rinp-s.     When  the   plate  has  made  45°      893. 
of  its  revolution,  so  as  to  bring  its  principal  section  into  the  plane  of  polarization  of  the  incident  beam,  this  Further 
band  also  coincides  in  direction  with  that  plane,  and  is  then  visibly  prolonged,  so  as  to  meet  that  belonging  to  Par1 
the  set  of  rings  about  the  other  pole  ;    and   is  crossed  at  the  middle  point  between  the  poles  by  another  dark 
space  perpendicular  to  it,  or  in  the  plane  of  the  section  B,  presenting  the  appearance  in  fig.  177.  Fig.  177. 

These  phenomena,  if  a  tourmaline  be  not  at  hand,  may  be  viewed,  (somewhat  less  commodiously,  unless  the      894. 
mica  plate  be  of  considerable  size,)  by  using  in  its  place  the  reflector  figured  in  fig.  170,  or  by  a  pile  of  glass  Other 
plates  interposed  obliquely  between  the  eye  and  the  mica.     In  this  manner  of  observing  them,  the  colours  are  mi|  ?*  j^ 
surprisingly  vivid,  no  part  of  the  red  and  violet  rays  being  absorbed  more  than  the  rest ;  whereas  the  tourmalines  [hese  phe. 
generally  exert  a  considerable  absorbing  energy  on  these  rays  in  preference  to  the  rest,  and  thus  the  contrast  ofnomena, 
colours  is  materially  impaired.     On  the  other  hand,  however,  from  the  greater  homogeneity  of  the  transmitted 
light,  the  rings  are  more  numerous  and  better  defined ;  and  in  this  respect  the  phenomenon  is  greatly  improved 
by  the  use  of  homogeneous  light. 

We  have  taken  mica  as  being  a  crystallized  body  very  easily  obtained  of  large  size,  and  presenting  its  axes 
readily,  and  without  the  necessity  of  artificial  sections.  It  is  thus  admirably  adapted  for  obtaining  a  general 
rough  view  of  the  phenomena,  preparatory  to  a  nicer  examination.  From  the  wide  interval  between  its  axes, 
however,  and  the  considerable  breadth  of  its  rings,  it  is  less  adapted,  when  employed  as  above  stated,  to  give  a 
clear  conception  of  the  complicated  changes  which  the  rings  undergo,  on  a  variation  of  circumstances.  For 
this  reason  we  shall  now  describe  another  and  much  more  commodious  mode  of  examining  the  systems  of 
polarized  rings  presented  by  crystals  in  general,  which  has  the  advantage  of  bringing  the  laws  of  their  pheno- 
mena so  evidently  under  our  eyes  as  to  make  their  investigation  almost  a  matter  of  inspection. 

It  is  evident,  that  when  we  apply  the  eye  close  to,  or  very  near  a  plate  of  mica,  or  other  body,  and  view,       896. 
beyond  it,  a  considerable  extent  of  illuminated  surface,  each  point  of  that  surface  will  be  seen  by  means  of  a  ray  General 
which  has  penetrated  the  plate  in  a  different  direction  with  respect  to  the  axes  of  its  molecules;  so  that  we  may  principle  of 
consider  the  eye  as  in  the  centre  of  a  spherical  surface  from  all  points  of  which  rays  are  sent  to  it,  modified  ™*wi°  s  °he 
according  to  the  state  of  primitive  polarization,  and  the  influence  of  the  peculiar  energies  of  the  medium,  corre-  rings. 
spending  to  the  direction  in  which  they  traverse  it,  and  the  thickness  of  the  plate  in  that  direction. 

Any  means,  therefore,  by  which  we  can  admit  into  the  eye  through  the  plate  and  tourmaline  a  cone  of  rays  Periscopic 
nearly  or  completely  polarized  in  one  general  direction,  or  according  to  any  regular  law,  will  afford  a  sight  of  tourmaline 
the  rings ;  and  therefore  exhibit,  at  a  single  view,  a  synopsis,  as  it  were,  of  the  modifications  impressed  on  an  aPParatu* 
infinite  number  of  rays  so  polarized  traversing  the  plate  in  all  directions.     The  property  of  the  tourmaline  so 
often  referred  to  puts  it  in  our  power  to  perform  this  in  a  very  elegant  and  convenient  manner,  by  the  aid  of  the 
little  apparatus  of  which  fig.  178  is  a  section.     ABCD  is  a  short  cylinder  of  brass  tube,  the  end  of  which,  AC,  Fig.  178. 
is  terminated  by  a  brass  plate,  having  an  aperture  a  b,  into  which  is  set  a  tourmaline  plate  cut  parallel  to  the 
axis:  hgik  is  another  similar  brass  cylinder,  provided  with  a  similar  aperture  and  a  similar  tourmaline  plate  G, 
and  fitted  into  the  former  so  as  to  allow  of  the  one  being  freely  turned  round  within  the  other  by  the  milled  edges 
B  D,  hk.     A  lens  H  of  short  focus,  set  in  a  proper  cell,  is  screwed  on  in  front  of  the  tourmaline  G,  so  as  to 
have  its  focus  a  little  behind  its  posterior  surface,  (that  next  the  eye,  O.)     Between  the  two  surfaces  AC,  gi 
is  another  short  cylinder  of  thin  tube  cd,  carrying  a  brass  plate  with  an  aperture  somewhat  narrower  than  those 
in  which  the  tourmalines  are  set,  and  on  which  any  crystallized  plate  F  to  be  examined  may  be  cemented  with 
a  little  wax.     This,  with  the  cylinder  to  which  it  is  fixed,  is  capable  of  being  turned  smoothly  round  within  the 
cylinder  ABCD  by  means  of  a  small  pin  e  passing  through  a  slit  /made  in  the  side,  and  extended  round  so 
as  to  occupy  about  120°  of  the  circumference ;  by  which  a  rotation  to  that  extent  may  be  communicated  to  the 
crystallized  plate  F  in  its  own  plane  between  the  tourmaline  plates.     The  pin  e  should  screw  into  the  ring  cd, 
that  it  may  be  easily  detached,  and  admit  the  ring  and  plate  to  be  taken  out  for  the  convenience  of  fixing  on  it 
other  crystals  at  pleasure. 

The  use  of  the  lens  H  is  to  disperse  the  incident  light,  and  thus  equalize  the  field  of  view  when  illuminated       897. 
by  any  source  of  light,  whether  natural  or  artificial,  as  well  as  to  prevent  external  objects  being  distinctly  seen  Mode  of 
through  it,  which  would  distract  the  attention  and  otherwise  interfere  with  the  phenomena.     The  rays  converged  a*t'00  cf 
by  the  lens  to  a  focus  within  the  crystallized  plate  F,  afterwards  diverge  and  fall  on  the  eye  C\  after  traversing  rjusapra" 
the  plate  in  all  directions  within  the  limit  of  the  field  of  view.     As  by  this  contrivance  they  pass  through  a 
very  small  portion  of  the  crystal,  there  is  the  less  chance  of  accidental  irregularities  in  its  structure  disturbing 
the  regular  formation  of  the  rings,  since  we  have  it  in  our  power  to  select  the  most  uniform  portion  of  a  large 
crystal.     The  rays,  after  passing  through  the  lens,  are  all  polarized  by  the  tourmaline  G,  in  planes  parallel  to 
its  axis ;  and  passing  through  the  eye  in  this  state,  if  the  crystal  F  be  not  interposed,  the  rays  will,  or  will  not, 
penetrate  the  second  tourmaline,  according  as  its  axis  is  parallel  or  perpendicular  to  that  of  the  first.     In  con- 
sequence, when  the  cylinder  carrying  the  former  is  turned  round  within  that  carrying  the  latter,,  the  field  of  view 
is  seen  alternately  bright  and  dark. 

When  the  crystallized  substance  F  is  interposed,  provided  it  be  so  disposed  that  one  or  other  of  its  optic  axes      ygg 
is  situated  any  where  in   the  cone  of  rays  refracted  by  the  lens,  so  that  one  of  them  shall  reach  the  eye  by  Selection  of 
traversing  the  axis,  the  polarized  rings  are  seen.     If  both  the  axes  of  the  crystal  (supposing  it  to  have  more  crystals. 
than  one)  fall  within  the  field,  a  set  of  rings  will  be  seen  round  both,  and  may  be  studied  at  leisure.     In  order 
to  bring  the  whole  of  their  phenomena  distinctly  under  view,  it  is  requisite  to  select  such  crystals  as  have 
their  axes  not  much  inclined  to  each  other,  so  as  to  allow  the  rings  about  both  to  be  seen  without  the  necessity 
of  looking  very  obliquely  into  the  apparatus.     In  mica  the  axes  are  rather  too  far  removed  for  this.     The-  best 
crystal  we  can  select  for  the  purpose  is  nitre. 


518  LIGHT. 

Light.         Nitre  crystallizes  in  long,  six-sided  prisms,  whose  section,  perpendicular  to  their  sides,  is  the  regular  hexag-on.     Part  'V. 
-••>/—•••  They  are  generally  very  much  interrupted  in  their  structure  ;    but  by  turning  over  a  considerable  quantity  of  v<™ "****"' 

899.  the  ordinary  saltpetre  of  the  shops,  specimens  are  readily  found  which  have  perfectly  transparent  portions  of 
some  extent.     Selecting  one  of  these,  cut  it  with  a  knife  into  a  plate  above  a  quarter  of  an  inch  thick,  directly 

iarin°  across  the  axis  of  the  prism,  and  then  grind  it  down  on  a  broad,  wet  file,  till  it  is  reduced  to  about  ^th  or  Jth  inch 
and  polish-  'n  thickness;  smooth  the  surfaces  on  a  wet  piece  of  emeried  glass,  and  polish  them  on  a  piece  of  silk  strained 
ing  it.  very  tight  over  a  strip  of  plate  glass,  and  rubbed  with  a  mixture  of  tallow  and  colcothar  of  vitriol.  This  ope- 
ration requires  practice.  It  cannot  be  effected  unless  the  nitre  be  applied  wet,  and  rubbed  till  quite  dry, 
increasing  the  rapidity  of  the  friction  as  the  moisture  evaporates.  It  must  be  performed  in  gloves,  as  the  vapour 
from  the  fingers,  as  well  as  the  slightest  breatn,  dims  the  polished  surface  effectually.  With  these  precautions 
a  perfect  vitreous  polish  is  easily  obtained.  We  may  here  remark,  that  hardly  any  two  salts  can  be  polished 
by  the  same  process.  Thus,  Rochelle  salt  must  be  finished  wet  on  the  silk,  and  instantly  transferred  to  soft 
bibulous  hnen,  and  rapidly  rubbed  dry.  Experience  alone  can  teach  these  peculiarities,  and  the  contrivances 
(sometimes  very  strange  ones)  it  is  necessary  to  resort  to  for  the  purpose,  of  obtaining  good  polished  sections  of 
soft  crystals,  especially  of  those  easily  soluble  in  water. 

900.  The  nitre  thus  polished  on  both  its  surfaces  (which  should  be  brought  as  near  as  possible  to  exact  parallelism) 
Rings  ex-    ;s  to  be  placed  on  the  plate  at  F ;    and  the  tourmaline  plates  being  then  brought  to  have  their  axes  at  right 

angles  to  each  other  (which  position  should  be  marked  by  an  index  line  on  the  cylinders)  the  eye  applied  at  O, 
and  the  whole  held  up  to  a  clear  light,  a  double  system  of  interrupted  rings  of  the  utmost  neatness  and  beauty 

Fig.  179.     will  be  seen,  as  represented  in  fig.  179.     If  the  crystallized  plate  be  made  to  revolve  in  its  own  plane  between 

the  tourmalines  (which  both  remain  unmoved)  the  phenomena  pass  through  a  certain  series  of  changes  periodi- 

180.     caiiv>  returning,  at  every  90°  of  rotation,  to  their  original  state.     Fig.  180  represents  their  appearance  when  the 

Fi'f  182  rotation  is  just  commenced;  fig.  181,  when  the  angle  of  rotation  is  22J°,  or  67 £°;  and  fig.  182,  when  it  equals 
45°.  When  the  tourmalines  are  also  made  to  revolve  on  each  other,  other  more  complicated  appearances  are 
produced,  of  which  more  presently.  We  shall  now,  however,  suppose  them  retained  in  the  situation  above 
mentioned,  i.e.  with  their  axes  crossed  at  right  angles,  and  proceed  to  study  the  following  particulars  : 

1.  The  form  and  situation  of  the  rings. 

2.  Their  magnitudes  in  the  same  and  different  plates. 

3.  Their  colours. 

4.  The  intensity  of  the  illumination  in  different  parts  of  their  periphery. 

The  situation  of  the  rings  is  determined  by  the  position  of  the  principal  section  of  the  crystal,  or  by  that  of 
Situation  of  the  optic  axes  within  its  substance.  These  in  nitre  lie  in  a  plane  parallel  to  the  axis  of  the  prisms,  and  per- 
"  pendicular  to  one  or  other  of  its  sides.  It  is  no  unusual  thing  to  find  crystals  of  this  salt  whose  transverse 
SL-ction  consists  of  distinct  portions,  in  which  the  principal  sections  make  angles  of  60°  with  each  other ;  indi- 
cating a  composite  or  macled  structure  in  the  crystal  itself.  These  portions  are  divided  from  each  other  by 
thin  films,  which  exhibit  the  most  singular  phenomena  by  internal  reflexion,  on  which  this  is  not  the  place  to 
enlarge.  In  an  uninterrupted  portion,  however,  the  forms  of  the  rings  are  as  represented  in  the  figures  above 
referred  to,  their  poles  subtending  at  the  eye  an  angle  of  about  8°.  Now,  it  is  to  be  remarked,  that  as  the  plate 
is  turned  round  between  the  tourmalines,  although  the  black  hyperbolic  curves  passing  through  the  poles  shift 
their  places  upon  the  coloured  lines,  and  in  succession  obliterate  every  part  of  them  ;  forming,  first,  the  black 
cross  in  fig.  179,  by  their  union;  then  breaking  up  and  separating  laterally,  as  in  fig.  180,  and  so  on.  Yet  the 
rings  themselves  retain  the  same  form  and  disposition  about  their  poles,  and,  except  in  point  of  intensity,  remain 
perfectly  unaltered  ;  their  whole  system  turning  uniformly  round  as  the  crystallized  plate  revolves,  so  as  to 
preserve  the  same  relations  to  the  axes  of  its  molecules.  Hence  we  conclude,  that  the  coloured  rings  are  related 
to  the  optic  axes  of  the  crystal,  according  to  laws  dependent  only  on  the  nature  of  the  crystal,  and  not  at  all  on 
external  circumstances,  such  as  the  plane  of  polarization  of  the  incident  light,  &c. 

902.  The    general   form  of  the  rings,  abstraction   made  of  the  black  cross,  is  as  represented   in    fig.   183.     If 

Form  of  the  we  regard  them   all  as    varieties   of  one  and  the  same  geometrical  curve,   arising  from   the  variation  of  a 
rings,  parameter  in  its  equation,  it  will  be  evident  that  this  equation  must,  in  its  most  general  form,  represent  a  re- 

entering  symmetrical  oval,  which  at  first  is  uniformly  concave,  and  surrounds  both  poles,  as  A ;  then  flattens  at 
Fie  "lisa  CS  tne  s'c'es>  and  acquires  points  of  contrary  flexure,  as  B  ;  then  acquires  a  multiple  point,  as  C ;  after  which  it 
breaks  into  two  conjugate  ovals  D  D,  each  surrounding  one  pole.  This  variation  of  form,  as  well  as  the  general 
figure  of  the  curves,  bears  a  perfect  resemblance  to  what  obtains  in  the  curve  well  known  to  geometers  under 
the  name  of  the  lemniscate,  whose  general  equation  is 

(*4  +  y*  +  ^^  a*  (*a  +  4  x*), 

when  the  parameter  6  gradually  diminishes  from  infinity  to  zero ;  2  a  representing  the  constant  distance 
between  the  poles. 

903  The  apparatus  just  described  affords  a  ready  and  very  accurate  method  of  comparing  the  real  form  of  the 

Verified  by  rings  with  this  or  any  other  proposed  hypothesis.     If  fixed  against  an  opening  in  the  shutter  of  a  darkened 

enperiment.  room,  with  the  lens  H  outwards,  and  a  beam  of  solar  light  be  thrown  on  the  latter,  parallel  to  the  axis  of  the 

apparatus,  the  whole  system  of  rings  will  be  seen  finely  projected  against  a  screen  held  at  a  moderate  distance 

from  E.     Now,  if  this  screen  be  of  good  smooth  paper  tightly  stretched  on  a  frame,  the  outlines  of  the  several 

rings  may  easily  be  traced  with  a  pencil  on  it,  and  the  poles  being  in  like  manner  marked,  we  have  a  faithful 

representation  of  the  rings,  which  may  be  compared  at  leisure  with  a  system  of  lemniscates,  or  any  other  curve 

graphically  constructed,  so  as  to  pass  through  points  in  them  chosen  where  the  tint  is  most  decided.     This  has 


LIGHT.  519 

Light      accordingly  been  done,  and  it.  has  been  found  that  lemniscates  so  constructed  coincide  throughout  their  whole     Part  IV. 

™~v~~"^  extent,  to  minute  precision,  with  the  outlines  of  the  ring's  so  traced,  the  points  graphically  laid  down  falling  on  ' _        ' 

the  pencilled  outlines.  The  graphical  construction  of  these  curves  is  rendered  easy  by  (he  well-known  property 
of  the  lemniscate,  in  which  the  rectangle  under  two  lines  P  A  X  P' A  drawn  from  the  poles  to  any  point  A  in 
the  periphery  is  invariable  throughout  the  whole  curve.  This  is  easily  sliown  from  the  above  equation,  and  the 
value  of  this  constant  rectangle  in  any  one  curve  is  represented  by  a  X  b. 

When  we  shift  from  one  ring  to  another,  a  remains  the  same,  because  the  poles  are  the  same  for  all.     To      904. 
determine  the  variation  of  b,  let  the  rings  be  illuminated  with  homogeneous  light,  (or  viewed  through  a  red  Variation  of 
glass,)  and  outlined  by  projection,  as  above.     Then,  if  we  determine  the  actual  value  of  alt  by  measuring  the  llle  Para" 
lengths  of  two  lines  PA,  P' A  drawn  from  P,  P'  to  any  point  of  each  curve;  and,  calculating  their  product,  (to  arithmetic 
which  a  6  is  equal,)  it  will  be  found  that  this  product,  and  therefore  the  parameter  b,  increases  in  the  arithmetical  progression 
progression  0,  1,  2,  3,  4,  fyc.  for  the  several  dark  intervals  of  the  rings  beginning  at  the  pole,  and  in  the  progres-  from  ring  to 
sion  J,  f,  f,  &c.  for  the  brightest  intermediate  spaces.     To  ensure  accuracy,  the  mean  of  a  number  of  values  of  ""8- 
PA  X  P'A,  at  different  points  of  the  periphery,  may  be  taken  to  obviate  the  effect  of  any  imperfection  in  the 
crystal.  , 

This,  then,  is  the  law  of  the  magnitudes  of  the  successive  rings  formed  by  one  and  the  same  plate.     But  if  we      905 
determine  the  value  of  the  same  product  for  plates  of  nitre  similarly  cut,  but  of  different  thicknesses,    or  Effect  of 
of  the  same  reduced  in  thickness  by  grinding,  it  will  be  found  to  vary  inversely  as  the  thickness  of  the  plate,  varying  the 
ceeteris  paribus.  thickness  of 

The  colours  of  the  polarized  rings  bear  a  great  analogy  to  those  reflected  by  thin  plates  of  air,  and  in  most  "le X^' 
crystals  would  be  precisely  similar  to  them  but  for  a  cause  presently  to  be  noticed.     In  the   situation  of  the  i^e  coiour 
tourmaline  plates  here  supposed  (crossed  at  right  angles)  they  are  those  of  the  reflected  rings,  beginning  with  a  of  therin»s 
black  centre,  at  the  pole.      If  examined  in  the  situation  of  fig.  179,  and  traced  in  a  line  from  either  pole 
cutting  across  the  whole  system,  at  right  angles  to  the  line  joining  the  poles,  they  will  almost  precisely  follow 
the  Newtonian  scale  of  tints.     For  the  present  we  will  suppose  that  they  do  so  in  all  directions.     It  is  evident, 
then,  that  each  particular  tint  (as  the  bright  green  of  the  third  order,  for  instance)  will  be  disposed  in  the  form 
of  a  lemniscate,  and  will  have  its  own  particular  value  of  the  product  a  b.     The  tint,  then,  may  be  said  to  be 
corresponding  to, — dependent  on, — or,  if  we  will,  measured  by  a  b.     In  conformity  with  this  language  the  Numerical 
coloured  curves  have  been  termed,  and  not  inaptly,  iinchromatic.  lines.     Now,  in  the  colours  of  thin  plates,  we  mcasure  °f 
have  seen  that  these  tints  arise  from  a  law  of  periodicity  to  which  each  homogeneous  ray  is  subject ;  and  that 
(without  entering  at  this  moment  into  the  cause  of  such  periods)  the  successive  maxima  and  minima  of  each  par-  niatic"!^™" 
ticular  cojoured  ray  passed  through,  in  the  scale  of  tints,  correspond  to  successive  multiples  by  -£,  -|,  -f,  ^,  &c.  of 
the  period  peculiar  to  that  colour.     In  the  colours  of  thin  plates,  the  quantity  which  determines  the  number. of 
periods  is  the  thickness   of  the  plate  of  air,  or  other  medium  traversed  ;    and  the  number  of  times  a  certain 
standard  thickness  peculiar  to  each  ray  is  contained  therein,  determines  the  number  of  periods,  or  parts  of  a 
period,  passed  through.     In  the  colours,  and  in  the  case  now  under  consideration,  the  number  of  periods  is  Law  of  pe- 
proportional  to  the  product  (0  x  0')  of  the  distances  from  either  pole,  for  one  and  the  same  thickness  of  plate, —  riodicity. 
and  for  different  plates  to  t  the  thickness, — and,  therefore,  generally,  to  0  X  #'  X  t,  provided  we  neglect  the 
effect  of  the  inclination  of  the  ray  in  increasing  the  length  of  the  path  of  the  rays  within  the  crystal,  or  regard 
the  whole  system  of  rings  as  confined  within  very  narrow  limits  of  incidence. 

This  condition  obtains  in  the  case  here  considered,  because  of  the  proximity  of  the  axes  in  nitre  to  each  other      907. 
and  to  the  perpendicular  to  the  surfaces  of  the  plate.     But  in  crystals  such  as  mica,  or  others  where  they  are  Transition 
still  wider  asunder,  it  is  not  so  ;  and  the  projection  of  the  isochromatic  curves  on  a  plane  surface  will   deviate  from  mtre 
materially  from  their  true  form,  which  ought  to  be  regarded  as  delineated  on  a  sphere  having  the  eye,  or  rather  a  *°  °'t  " 
point  within  the  crystal,  for  a  centre.     In  such  a  case,  it  might  be  expected  that  the  usual  transition  from  the  whose  axes 
arc  to  its  sine  should  take  place  ;  and  that,  instead  of  supposing  the  tint,  or  value  of  a  b,  to  be  proportional  are  farther 
simply  to  9  x  0'  x  t,  (putting  0  —  A  P,  and  <?'  =  AP7,)  we  ought  to  have  it  proportional  to  sin  0  x  sin  ffx  asun(^er- 
length  of  the  path  of  the  ray  within  the  crystal.     Now  (putting  p  for  the  angle  of  refraction,  and  t  for  the 
thickness  of  the  plate)  we  have  t  .  sec  p  =  length  of  the  ray's  path  within  the  crystal.     If,  then,  we  put  n  for 

the  number  of  periods  corresponding  to  the  tint  a  b  for  the  ray  in  question,  and  suppose  h  =  ,  or  the 

71 

unit  whose  multiples  determine  the  order  of  the  rings,  we  shall  have 

a  J               t  General  ex- 

n  =  — -—  =   - ,-  .  sin  0  .  sin  &' .  sec  p,                (a)                                        pression  for 

h              n,  the  tint 

,  polarized 

and  h  = .  sin  6  .  sin  6'.  (b)  by  any 

n  .  COS  p  crystallized 

plate. 

If,  then,  the  suppositions  made  be  correct,  we  ought  to  have  the  function  on  the  right  hand  side  of  this  last 
equation  invariable,  in  whatever  direction  the  ray  penetrates  the  crystallized  plate,  and  whatever  be  the  order  of 
the  tint  denoted  by  n.  We  shall  here  relate  only  one  experiment,  to  show  how  very  precisely  the  agreement  of 
this  conclusion  with  fact  is  sustained. 

A  ray  of  light  was  polarized  by  reflexion  at  a  plate  of  perfectly  plane  glass,  and  transmitted  through  a  plate      908. 
of  mica,  having  its  principal  section  45°  inclined  to  the  plane  of  primitive  polarization,  and   the  mica  plate  Experiment 
made  to  revolve  in  the  plane  of  its  principal  section  about  an  axis  at  right  angles  thereto,  (or  about  the  axis  B,  verifying 
Art.  885.)     In  this  state  of  things,  if  viewed  through  a  tourmaline  as  above  described,  or  by  other  more  refined  '  ' 


520 


LIGHT 


means  presently  to  be  noticed,  the  succession  of  tints  exhibited  by  the  mica  was  that  of  a  section  of  the  rings     Plrl  'v- 
in  fi<r.  182,  made  by  a  line  drawn  through  both  the  poles.     To  render  the  observation  definite,  a  red  glass  was  S—"V~""1' 
interposed  so  as  to  reduce  the  rings  to  a  succession  of  red  and  black  bands,  and  the  angles  of  incidence  corre- 
sponding to  the  maxima  and  minima  of  the  several  rings  very  accurately  measured.     These  are  set  down  in 
Col.  2  of  the  following  table.     Col.  1  contains  the  values  of  n,  0  corresponding  to  the  pole,  £  to  th«  first 
maximum,  1  to  the  first  minimum,  1£  to  the  second   maximum,  and  so  on.     The  third  column  contains  the 
angles  of  refraction  computed  for  an  index  1.500  ;   the  fourth  and  fifth,  those  of  0  and  ff ;  the  sixth,  those  of  A 
deduced  from  the  above  equation,  and  which  ought  to  be  constant.     The  excesses  above  the  mean  are  stated 
in  the  last  column,  and  show  how  very  closely  that  equation  represents  the  fact.     The  thickness  of  the  mica  was 
0.023078  inches  =  t. 


Values  of  n. 

Angles  of  in- 
cidence. 

Angles  of 
refraction  =  f  . 

Values  of  1. 

Values  of  V. 

Values  of  A. 

Excesses  above 
the  mean. 

0.0 

35°  3'  30" 

22°31'   0" 

0°    0'     0" 

45°  V    0" 

0.5 

32  55  20 

21   14  40 

1  16  20 

43  45  40 

0,032952 

-  0.000195 

1.0 

30  34  40 

19  49  30 

2  41  30 

42  20  30 

0.033622 

-f  0.000475 

1.5 

28  15  40 

18  24     0 

470 

40  55     0 

0.033035 

-  0.000112 

2.0 

25  34  20 

16  43  30 

5  47  30 

39   14  30 

0.033327 

-f  0.000180 

2.5 

22  46  20 

14  57   15 

7  33  45 

37   28   15 

0.033143 

-f  0.000001 

3.0 

19  35  40 

12  55  10 

9  35  50 

35  26  10 

0.033058 

-  0.000089 

3.5 

15  48  40 

10  27  50 

12     3   10 

32  58  50 

0.033026 

-  0.000121 

4.0 

10  48  50 

7   11    10 

15   19  53 

29  42   10 

0.033010 

-  0000137 

909  Proceeding  thus,  and  measuring  across  the  system  of  rings  in  all  directions  for  plates  of  various  crystals  and 

General  of  a"  thicknesses,  it  has  been  ascertained,  as  a  general  fact,  that  in  all  substances  which  possess  the  property  of 
establish-  developing  periodical  colours  by  exposure  to  polarized  light  in  the  manner  described,  the  tint  (n),  or  rather 
ment  of  the  tj,e  number  of  periods  and  parts  of  a  period  corresponding,  in  the  case  of  a  ray  of  given  refrangibility,  to  a 

thickness  t,  an  angle  of  refraction  p,  and  a  position  within  the  crystal,  making  angles  0  and  Of  with  the  optic 

axes,  is  represented  by  the  equation 


law. 


t  .  sec  p 


X  sin  9  .  sin  0', 


Case  of  a 
crystal 
formed  into 
a  sphere. 


910. 

Methods  of 
viewing  the 
rings  at 
great  obli- 
quities. 


Fig.  184. 


911. 

Rings  in 
crystals 
with  on* 
axis. 
Fig.  185. 


h  being  a  constant  depending  only  on  the  nature  of  the  crystal  and  the  ray.  Were  the  crystal  of  a  spherical  form, 
instead  of  a  parallel  plate,  t.  sec  />,  which  represents  the  path  traversed  by  the  ray  within  it,  must  be  replaced 
by  a  constant  equal  to  the  diameter  of  the  sphere,  and  in  that  case  the  tint  would  be  simply  proportional  to 
the  product  of  the  sines  of  0  and  0'.  This  elegant  law  is  due  to  M.  Biot,  though  it  is  to  Dr.  Brewster's  inde- 
fatigable and  widely  extended  research  that  we  owe  the  general  developement  of  the  splendid  phenomena  of  the 
polarized  rings  in  biaxal  crystals.  It  appears,  then,  from  this,  that  if,  on  the  surface  of  a  sphere  formed  of  any 
crystal,  curves  analogous  to  the  lemniscate,  or  having  sin  6  X  sin  tf  constant  for  each  curve,  and  varying  in 
arithmetical  progression  from  curve  to  curve,  be  described, — then,  if  the  sphere  be  turned  about  its  centre  in  a 
polarized  beam,  as  above  described,  the  tint  polarized  at  every  point  of.  each  curve  will  be  the  same,  and  in 
passing  from  curve  to  curve  will  obey  the  law  of  periodicity  proper  to  the  crystal. 

There  is  hardly  any  character  in  which  crystals  differ  more  widely  than  in  the  angular  separation  of  their  optic 
axes,  as  the  table  annexed  to  the  end  of  this  article  will  show.  This,  while  it  affords  mo*  valuable  criteria  to 
the  chemist  and  mineralogist,  in  discriminating  substances  and  pointing  out  differences  of  structure  and  com- 
position which  would  otherwise  have  passed  unnoticed,  renders  the  investigation  of  their  phenomena  difficult, 
since  it  is  frequently  impossible,  by  any  contrivance,  to  bring  both  the  axes  under  view  at  once  ;  and  neces- 
sitates a  variety  of  artifices  to  obtain  a  sight  of  the  rings  about  both.  It  is  often  very  easy  to  cut  and  polish 
crystallized  bodies  in  some  directions,  and  very  difficult  in  others.  However,  by  immersing  plates  of  them  in 
oil,  and  turning  them  round  on  different  axes,  or  by  cementing  on  their  opposite  sides  prisms  of  equal  refracting 
angles  oppositely  placed,  as  in  fig.  184,  we  may  look  through  them  at  much  greater  obliquities  than  without  such 
aid ;  and  thus,  by  increasing  the  range  of  vision  to  nearly  a  hemisphere,  avoid  in  most  instances  the  necessity 
of  cutting  them  in  different  directions. 

When  the  two  axes  coalesce,  or  the  crystal  becomes  uniaxal,  the  lemniscates  become  circles  ;  and  the  black 
hyperbolic  lines,  passing  through  the  poles,  resolve  themselves  into  straight  lines  at  right  angles  to  each  other, 
forming  a  black  cross  passing  through  the  centre  of  the  rings,  as  in  fig.  185.  In  this  case  the  tint  is  repre- 
sented by  t  .  sin  0* ;  and  in  the  case  of  plates,  where  t,  the  thickness,  is  considerable,  or  where,  from  the  other- 
wise peculiar  nature  of  the  substance  the  rings  are  of  small  dimensions,  0  is  small,  and  therefore  proportional 
to  its  sine ;  so  that  in  passing  from  ring  to  ring  6*  increases  in  arithmetical  progression.  Hence  the  diameters 
of  the  rings  are  as  the  square  roots  of  the  numbers  0,  1,  2,  3,  Ac. ;  and  therefore  their  system  is  similar,  with 
the  exception  of  the  black  cross,  to  the  rings  seen  between  object-glasses.  Carbonate  of  lime  cut  into  a  plate 
at  right  angles  to  the  axis  of  its  primitive  rhomboid,  exhibits  this  phenomenon  with  the  utmost  beauty.  The 
most  familiar  instance,  however,  may  be  found  in  a  sheet  of  clear  ice  about  an  inch  thick  frozen  in  still  weather 
A  pane  of  window-glass,  or  a  polished  table  to  polarize  the  light,  a  sheet  of  ice  freshly  taken  up  in  winUw 


LIGHT.  521 

Light.      produce  the  rings,  and  a  broken  fragment  of  plate  glass  to  place  near  the  eye  as  a  reflector,  are  all  the  apparatus     Part  IV. 
•—•%—•  required  to  produce  one  of  the  most  splendid  of  optical  exhibitions.  ^*~-v~~s 

If  9  be  not  very  small,  the  measure  of  the  tint,  instead  of  t  .  sin  6*,  is  t  .  sec  p  .  sin  6*.  We  have  seen  that  in  912. 
uniaxal  crystals,  sin  0*  is  proportional  to  the  difference  of  the  squares  of  the  velocities  v  and  v1  of  the  ordinary  Analogy 
and  extraordinary  ray,  or  to  u'2  —  t>*.  Now,  if  we  denote  by  r  and  /  the  times  taken  by  these  £wo  rays  to  ^^""J1'* 

traverse  the  plate,  we  have  v  =  -      —  and  v'  =  -          —  ;    therefore  t  .  sec  .  p  sin  0*  is  proportional  to  rhecfrin ~>s 

and  those 
/I  1   \  (T  +  T)(T-TJ       ,,  produced  by 

(t  .  sec  />)3  x  (^  -   —  ),  that  .s,  to  -  /)a        '   .  (<  sec  /,)».  fhe  law  ofy 

interference 

or  (which  is  the  same  thing)  to  (v  +  v')  .  v  v1  (T  —  T').  But,  neglecting  the  squares  of  very  small  quantities,  of 
the  order  t/  —  v  and  T  —  T',  for  such  they  are  in  the  immediate  neighbourhood  of  the  axis,  the  factors  v  -f-  tf  and 
v  v'  are  constant ;  so  that  the  tint  is  simply  proportional  to  T  —  T',  the  difference  of  times  occupied  by  the  two 
rays  in  traversing  the  plate  ;  or  the  interval  of  retardation  of  the  slower  ray  on  the  quicker.  This  very  remark- 
able analogy  between  the  tints  in  question  and  those  arising  from  the  law  of  interferences,  was  first  perceived 
by  Dr.  Young ;  and,  assisted  by  a  property  of  polarized  light  soon  to  be  mentioned,  discovered  by  Messrs. 
Arago  and  Fresnel,  leads  to  a  simple  and  beautiful  explanation  of  all  the  phenomena  which  form  the  subject  of 
this  section,  and  of  which  more  in  its  proper  place. 

The  forms  of  the  rings  are  such  as  we  have  described,  only  in  regular  and  perfect  crystals  ;  every  thing  which       913 
disturbs  this  regularity,  distorts  their  form.     Some  crystals  are  very  liable  to  such  disturbances,  either  arising  Circum- 
from  an  imperfect  state  of  equilibrium,  or  a  state  of  strain  in  which  the  molecules  are  retained,  or  to  actual  sl 
interruptions  in  their  structure.     Thus,  specimens  of  quartz  and  beryl  are  occasionally  met  with,  in  which  the  j-jj^,  t|)e 
single  axis  usually  seen  is  very  distinctly  separated  into  two,  the  rings  instead  of  circles  have  oval  forms,  and  the  rjngs. 
black  cross  (which  in  cases  of  a  well  developed  single  axis  remains  quite  unchanged  during  the  rotation  of  the  cry- 
stallized plate  in  its  own  plane)  breaks  into  curves  convex  towards  each  other,  but  almost  in  contact  at  their  vertices, 
at  every  quarter  revolution.     Cases  of  interruption  occur  in  carbonate  of  lime  very  commonly,  and  in  muriacite 
perpetually  ;   and  the  effects  produced  by  them  on  the  configurations  of  the  rings  rank  among  the  most  curious 
and  beautiful  of  optical  phenomena.     They  have  not,  however,  been  anywhere  described,  and  our  limits  will 
not  allow  us  to  make  this  article  a  vehicle  for  their  description. 

The  form  of  the  rings  being,  then,  considered,  let  us  next  inquire  more  minutely  into  their  colours.     These       914. 
being  all  composite,  and  arising  from  the  superposition  on  each  other  of  systems  of  rings  formed  by  each  homo-  Colours  of 
geneous  ray,  we  can  obtain  a  knowledge  of  their  constitution  only  by  examining  the  rings  in  homogeneous  the  rai's- 
light.     This  is  easy,  for  we  have  only  to  illuminate  the  apparatus  described  above  by  homogeneous  light  of  all 
degrees  of  refrangibility  from  red  to  violet,  by  passing  a  prismatic  spectrum  from  one  end  to  the  other  over  the 
illuminating  lens  H,  the  eye  being  applied  as  usual  at  O,  and  observe  the  changes  which  take  place  in  the  rings, 
in  passing  from   one  coloured  illumination  to  another;    and,  if  necessary,  measure  their  dimensions.     This  is 
readily  done,  either  by  projecting  them  on  a  screen  in  a  darkened  room,  as  described  in  Art.  903,  or  by  detaching 
the  lens  H,  fig.  178,  and  simply  looking  through  the  apparatus  at  a  sheet  of  white  paper  strongly  illuminated 
by  the  rays  of  a  prismatic  spectrum,  where  the  rings  will  appear  as  if  depicted  on  the  paper,  and  their  outlines 
easily  marked,  or  their  diameters  measured.     The  following  are  the  general  facts  which  may  thus  be  readily 
verified. 

First,  in  the  case  of  crystals  with  a  single  axis,  the  rings  remain  circular,  and  their  centres  are  coincident  for       915. 
all  the  coloured  rays,  but  their  dimensions  vary.     In  the  generality  of  such  crystals,  their  diameters  for  different  I".  crystals 
refrangibilities  follow  nearly  the  law  of  the  Newtonian   rings,  when  viewed  in   similar  illuminations ;   their  "™? 
squares  (or  rather  the  squares  of  their  sines)  being  proportional,  or  nearly  so,  to  the  lengths  of  the  fits,  or  of  the  Deviations 
undulations  of  the  rays  forming  them.     This  law,  howerer,  is  very  far  from  universal  ;   and  in  certain  crystals  is  from  New- 
altogether  subverted.     Thus,  in  the  most  common  variety  of  apophyllite,  (from  Cipit,  in  the  Tyrol, — not  from  ton's  scale 
Fassa,  as  is  commonly  stated,)  the  diameters  of  the  rings  are  nearly  alike  for  all  colours,  those  of  the  green  rings  '"  'j1?  al'°" 
being  a  very  little  less;   those  formed  by  rays  at  the  confines  of  the  blue  and  indigo  exactly  equal,  and  those  ^ 
of  violet  rays  a  little  greater  than  the  red  rings.     It  is  obvious,  that  were  the  rings  of  all  colours  exactly  equal, 
the  system  resulting  from  their  superposition  would  be  simple  alternations  of  perfect  black  and  white,  continued 
ad  infinitum.     In  the  case  in  question,  so  near  an  approach  to  equality  subsists,  that  the  rings  in  a  tourmaline 
apparatus  appear  merely  black  and  white,  and  are  extremely  numerous,  no  less  than   thirty-five  having  been 
counted,  and  many  of  those  too  close  for  counting  being  visible  in  a  thick  specimen. 

When  examined  more  delicately,  colours  are,  however,  distinguished,  and  are   in  perfect  conformity  with  the       gig 
law  stated,  being  for  the  first  four  orders  as  follow  : 

First  order.     Black,  greenish  white,  bright  white,  purplish  white,  sombre  violet  blue. 

Second  order.  Violet  almost  black,  pale  yellow  green,  greenish  white,  white,  purplish  white,  obscure  indigo 

inclining  to  purple. 

Third  order.     Sombre  violet,  tolerable  yellow  green,  yellowisn  white,  white,  pale  purple,  sombre  indigo. 
Fourth  order.    Sombre  violet,  livid  grey,  yellow  green,  pale  yellowish  white,  white,  purple,  very  sombre 
indigo,  &c. 

Carbonate  of  lime,  beryl,  ice,  and  tourmaline  (when  limpid)  are  instances  of  uniaxal  crystals,  in  whose  rings       9' 7 
the  Newtonian  scale  of  tints  is  almost  exactly  imitated ;  and,  consequently,  the  intervals  of  retardation   of  the 
ordinary  and   extraordinary  rays  of  any  colour  on  one  another,  are  proportional  to  the  lengths  of  their  undu- 
lations.    On  the  other  hand,  in  the  hyposulphate  of  lime,  we  are  furnished  with  an  instance  of  more   rapid 

VOL.  iv.  3  Y 


522  LIGHT. 

Light,      degradation  of  tints,  and  therefore  of  a  more  rapid  variation  of  the  interval  just  mentioned.    The  following  was     Fart  IV 
v-™ **s"^/  the  scale  of  colour  of  the  rings  observed  in  this  remarkable  crystal :  v_v_— 

'hate  of8"'"          First  order'     Black>  verv  faint  sky  Dlue-  Pretty  strong  sky  blue,  very  light  bluish    white,  white,  yellowish 

white,  bright  straw  colour,  yellow,  orange  yellow,  fine  pink,  sombre  pink. 
Second  order.  Purple,  blue,  bright  greenish  blue,  splendid  green,  light  green,  greenish  white,  ruddy  white, 

pink,  tine  rose  red. 

Third  order.     Dull  purple,  pale  blue,  g-reen  blue,  white,  pink. 
Fourth  order.    Very  pale  purple,  very  light  blue,  white,  almost  imperceptible  pink. 
After  which  the  succession  of  colours  was  no  longer  distinguishable. 

918.  A  degradation  still  more  rapid  has  been  observed  in  certain  rare  varieties  of  uniaxal  apophyllite,  accompanied 
Other  re-     with  remarkable  and  instructive  phenomena.     In  these,  the  diameters  of  the  rings  (instead  of  diminishing  as  the 
"""s'of6     refrangibility  of  the  light  of  which  they  are  formed  increases)  increase  with  great  rapidity,  and  actually  become 
deviation      'nfin»te  f°r  rays  of  intermediate  refrangibility  ;   after  whic'h  they  again  become  finite,  and  continue  to  contract 

up  to  the  violet  end  of  the  spectrum,  where,  however,  they  are  still  considerably  larger  than  in  the  red  rays.  In 
consequence  of  this  singularity,  their  colours  when  illuminated  with  white  light  furnish  examples  of  a  complete 
inversion  of  Newton's  scale  of  tints.  The  following  were  the  tints  exhibited  by  two  varieties  of  the  mineral  in 
question,  in  one  of  which  the  critical  point  where  the  rings  become  infinite  took  place  in  the  indigo,  and  in  the 
other  in  the  yellow  rays.  In  the  former  they  were 

First  order.     Black,  sombre  red,  orange,  yellow,  green,  greenish  blue,  sombre  and  dirty  blue. 

Second  order.  Dull  purple,  pink,  ruddy  pink,  pink  yellow,  pale  yellow   (almost  white,)  bluish  green,  dull 
pale  blue. 

Third  order.     Very  dilute  purple,  pale  pink,  while,  very  pale  blue. 
In  the  latter  variety,  the  tints  were 

First  and  only  order.     Black,  sombre  indigo,  indigo  inclining  to  purple,  pale  lilac  purple,  very  pale  reddish 
purple,  pale  rose  red,  white,  white  with  a  hardly  perceptible  tinge  of  green. 

919.  The  doubly  refracting  energy  of  a  crystal  may  be  not  improperly  measured  by  the  difference  of  the  squares 
Relation       of   the  velocities  of   an  ordinary  and  extraordinary  ray  similarly  situated  with  respect  to  the   axes  ;    but  as 
between  the  this  difference,  for  rays  variously  situated  in  one  and  the  same  crystal,  is  proportional  to  sin  6*,  or  in  biaxal 
of  the  rings  crysta's  to  sm  "  •  sm  ^-  the  intrinsic  double  refractive  energy  of  any  crystal  may  be  represented  by 

and  the  1-4  —  7/2 

doubly  e  =    .         — : — — ;  (c) 

refractive  sln  f.ftnW 

regarding  this  henceforth  as  the  definition  of  this  energy,  we  have,  in  uniaxal  crystals,  e  =  — : —  — ,  and 
this  will  evidently  measure  the  actual  amount  of  separation  of  two  such  rays  when  emergent  from  the  crystal. 

t       SCO  O  /       SCC  O 

If  in  this  we  put  for  v  and  v'  their  equals  — '• —       -  and  -   '— -t — — ,  we  shall  have,  after  reduction, 

vi  _  „'.>  _  v  v>  (i,  i  v>) .  _^_n:i .        (C) 

t  .  sec  /> 

In  a  parallel  plate,   perpendicular  to  the  axis  and  in  the  immediate  vicinity  of  the  axis,  v'  and  sec  p  may 

be  regarded  as  constant,  and  v2  —  V1*  is  proportional   to  T'  —  T,  the  interval  of  retardation  of  one  ray  on  the 

other,  to  which  the  tint  in  white  light  and  the  number  of  periods  and  parts  of  a  period  in  homogeneous  light 

(to  which,  for  brevity,  we  will  continue  to  extend  the  term  tint)  are  proportional.     We  see,  then,  that  in  such 

cases  the  intrinsic  double  refracting  energy  is  directly  as  the  tint  polarized,  and  inversely  as  sin  0*,  and  therefore 

also  inversely  as  the  squares  of  the  diameters  of  the  rings.     As  the  rings  increase  in  magnitude,  then,  ceeteris 

paribus,  the  double  refractive  energy  diminishes  ;  and  hence  a  very  curious  consequence  follows,  viz.  that  in  the 

two  cases  last  mentioned  it  vanishes  altogether  for  those  colours  where  the  rings  are  infinite ;  in  other  words, 

that  although  the  crystal  be  doubly  refractive  for  all  the  other   coloured  rays,  there  is  one  particular  ray  in  the 

Case  of        spectrum  (viz.  the  indigo  in  the  former,  and  the  yellow  in  the  latter  case)  with  respect  to  which  its  refraction  is 

sials  at    s;ngie<     jn  jne  passage  through  infinity,  there  is  generally  a  change  of  sign.     In  the  instances  in  question  this 

tractive,       change  takes  place  in  the  value  of  e  or  v*  —  vn,  which  passes  from  negative  to  positive.     And  the  spheroid  of 

repulsive,     double  refraction  changes  its  character  accordingly  from  oblate  to  prolate,  passing  through  the  sphere  as  its 

and  neutral,  intermediate  state.     The  manner  in  which  this  may  be  recognised,  without  actually  measuring,  or  even  perceiving 

its  double  refraction,  will  be  explained  further  on. 

920.  For  crystals  with  two  axes  we  have  only,  at  present,  the  ground  of  analogy  to  go  upon  in  applying  the 
Application  above  formula  and  phraseology  to  their  phenomena.     The  general  fact  of  an  intimate  connection  of  the  double 

axal  refracting  energy  with  the  dimensions  of  the  rings,  is  indeed  easily  made  out ;  for  it  is  a  fact  easily  verified  by 
experiment,  that  all  crystals,  whether  with  one  or  two  axes,  in  which  the  rings  or  lemnincates  formed  are  of 
small  magnitude  in  respect  of  the  thickness  of  the  plate  producing  them,  are  powerfully  double  refractive, 
and  vice  versa  ;  and  that,  generally  speaking,  the  separation  of  the  ordinary  and  extraordinary  pencils  is,  ceetcrix 
paribus,  greater  in  proportion  as  the  rings  are  more  close  and  crowded  round  their  poles.  In  uniaxal  crystals, 
in  which  the  laws  of  double  refraction  are  comparatively  simple,  there  is  little  difficulty  in  submitting  the  point 
to  the  test  of  direct  experiment  and  exact  measurement,  and  it  is  found  to  be  completely  verified.  In  biaxal, 
however,  such  precise  and  direct  comparison  is  more  difficult,  and  calls  for  a  knowledge  of  the  general  laws  of 
double  refraction.  The  analogy,  however,  supported  by  the  general  coincidence  above  mentioned,  is  too  strong 
to  be  refused ;  and,  as  we  advance,  will  be  found  to  gain  strength  with  every  step. 


LIGHT. 


523 


Liglii.          Ill  biaxal  crystals,  similar  deviations  from  exact  proportionality  between  the   lengths  of  the   periods  of  the    Part  IV. 
K~S,~—  •'  several  coloured  rays  and  those  of  their  undulations,  or  fits,  exist  ;    but  their  effect  in  disturbing  the  colours  of  —  ~v-^/ 
the  rings  is  interfered  with,  and  frequently  masked  by,  another  cause,  which  has  no  existence  in  uniaxal  crystals,      921. 
viz.  that  the   optic  axes  differ  in  situation,  within  one  and  the  same  crystal  for  the   differently  refrangible  Separation 
homogeneous  rays;   and,  therefore,  that  the  elementary  lemniscates,  whose  superposition  forms  the  composite  of  tlle  °Pl.lc 
fringes  seen  in  a  white  illumination,  differ  not  only  in  magnitude  but  in  the  places  'of  'their  poles  and  the  interval  ferent|v 
between  them.     To  make  this  evident  to  ocular  inspection,  take  a  crystal  of  Rochelle  salt,  (tartrate  of  soda  and  refrangible 
potash,)  and  having  cut  it  into  a  plate  perpendicular  to  one  of  its  optic  axes,  or  nearly  so,  and  placed  it  in  a  rays  in 
tourmaline  apparatus,  let  the  lens  H  be  illuminated  with  the  rays  of  a  prismatic  spectrum,  in  succession,  begin-  h'axiil 
ning  with  the  red  and  passing  gradually  to  the  violet.     The  eye  being  all  the  time  fixed  on  the  rings,  they  will  crJ'3ta  " 
appear  for  each  colour  of  perfect  regularity  of  form,  remarkably  well  defined,  and  contracting  rapidly  in  size  as 
the  illumination  is  made  with  more  refrangible  light  ;  but  in  addition  to  this,  it  will  be  observed,  that  the 
whole  system  appears  to   shift  its  place  bodily,  and  advance  regularly  in  one  direction   as  the   illumination 
changes  ;  and  if  it  be  alternately  altered  from  red  to  violet,  and  back  again,  the  pole,  with  the  rings  about  it, 
will  also  move  backwards  and  forwards,  vibrating,  as  it  were,  over  a  considerable  space.     If  homogeneous  rays 
of  two  colours  be  thrown  at  once  on  the  lens,  two  sets  of  rings  will  be  seen,  having  their  centres  more  or  less 
distant,  and  their  magnitudes  more  or  less  different,  according  to  the  difference  of  refrangibility  of  the  two  species 
of  light  employed. 

Since  the  plate  in  this  experiment  is  supposed  to  have  its  surfaces  perpendicular  to  the  mean  position  of  the       922. 
optic  axis,  the  cause  of  these  appearances  cannot  be  found  in  a  mere  apparent  displacement  of  the  rings  by  All  i  he  axes 
refraction  at  the  surface,  existing  to  a  greater  extent  for  the  violet  than  the  red  rays,  add  to  which,  that  the  angle  ''e  '"  '''* 
which  their  poles  describe,  is  neither  the  same  in  magnitude  nor  direction  for  different  crystals.     In  some,  the 
optic  axes  approach  each  other  in  violet  light,  and  recede  in  red  ;  while  in  others  the  reverse  is  the  case.     In  all,  section. 
however,  so  far  as  we  are  aware,  the  optic  axes  for  all  the  coloured  rays  lie  in  one  plane,  viz.  the  principal 
section  of  the  crystal.     This  is  rendered  matter  of  inspection  by  cutting  any  crystal  so  that  both  axes  shall  be 
visible  in  the  same  plate,  and  placing  it  with  its  principal  section  in  the  plane  of  primitive  polarization.     In  this 
state  of  things  the  first  ring  about  each  pole,  as  in  fig.  179,  is  seen  divided  into  two  halves,  and  puts  on,  if 
the  plate  be  pretty  thick,  the  appearance  of  two  semi-elliptic  spots,  one  on  each  side  of  the  principal  section. 
These  spots  are  observed  to  be  differently  coloured  at  their  two  extremities  •  <n  some  crystals  the  ends  of  the 
spots,  as  well  as  the  segments  of  the  rings  adjacent  to  them,  which  are  iurned  towards  each  other,  being 
coloured  red,  and  the  other,  or  more  distant  ends,  with  blue  ;  and  in  others,  the  reverse.     In  some  crystals  this 
coloration  is  slight,  and  in  a  very  few,  imperceptible  ;  but  in  others  it  is  so  great,  that  the  spots  are  drawn 
out  into  long  spectra,  or  tails  of  red,  green,  and  violet  light  ;  and  the  ends  of  the  rings  are  in  like  manner 
distorted   and  highly  coloured,  presenting  the  appearance  in  fig.  186.     This  is  the  case  with  Rochelle  salt,  Fig.  186. 
above  mentioned.     If  these  spectra  be  examined  with  coloured  glasses,  or  with  homogeneous  light,  they  will  be 
seen  to  be  composed  as  in  fig.  187,  by  the  superposition  of  well  defined  spots  of  the  several  simple  colours 
arranged  in  lines  on  each  side  of  the  principal  section.     In  the  case  of  Rochelle  salt,  the  angular  extent  of 
these  spectra,  within  the  medium,  which  measures  the  interval  between  the  optic  axes  for  violet  and  red  rays, 
amounts  to  no  less  than  10°. 

Dr.  Brewster  has  given  the  following  list  of  crystals  presenting  these  phenomena,  which  he  has  divided  into 
two  classes,  according  to  his  peculiar  and  ingenious  views. 


.  187. 


^ 

ster  s  list  of 


Class  I. 

Nitre. 

Sulphate  of  baryta. 

Sulphate  of  strontia. 

Phosphate  of  soda. 

Tartrate  of  potash  and  soda. 

Supertartrate  of  potash  and  soda. 

Arragonite. 

Carbonate  of  lead.  (?) 

Sulphato-carbonate  of  lead. 

Class  II. 
Topaz. 

Mica. 
Anhydrite. 
Native  borax. 
Sulphate  of  magnesia. 


Undassed. 

Chromate  of  lead. 

Muriate  of  mercury. 

Muriate  of  copper. 

Oxynitrate  of  silver. 

Sugar. 

Crystallized  Cheltenham  salts. 

Nitrate  of  mercury. 

Nitrate  of  zinc. 

Nitrate  of  lime. 

Snperoxalate  of  potash. 

Oxalic  acid. 

Sulphate  of  iron. 

Carbonate  of  lead.  (?) 

Cymophane. 

Felspar 

Benzoic  acid. 

Chromic  acid. 

Nadelstein  (Faroe.) 


viations  of 
u  fronl 


To  which  list  a  great  many  more  might  be  added.     Bicarbonate  of  ammonia,  indeed,  is   the  only  biaxal  crystal 

we  have  examined  in  which  the  optic  axes  for  all  colours  appear  to  be  strictly  coincident.  934 

This  separation  of  the  axes  of  different  colours  explains  a  remarkable   appearance  presented  by  the  rings  of  Phenomena 
all  biaxal  crystals,  when  placed  with  their  principal  section  45°  inclined  to  the  plane  of  polarization  of  the  incident  of  the  vir- 
light.     It  is  universally  observed  that,  in  traversing  the  whole   system  of  rings   in  the  plane  of  the  principal  lual  Pnl<is 

3  y  2  explained. 


524  LIGHT. 

Light,  (section,  the  nearest  approximation  to  Newton's  scale  of  colours  is  obtained  by  assuming,  for  the  origin  of  the  P»rt  IV. 
V~""V"-'  scale,  not  the  poles  themselves,  but  other  points  (which  have  been  called  virtual  poles,  though  improperly)  lying  '— ' -v— 
either  between  or  beyond  them,  according  to  the  crystal  examined,  and  at  a  distance  from  them,  inva- 
riable for  each  species  of  crystal,  whatever  be  the  thickness  of  the  plate.  In  consequence,  the  poles 
themselves  are  not  absolutely  black,  but  tinged  with  colour ;  and  their  tint  descends  in  the  scale  as  the  thickness 
of  the  plate  increases,  and  as,  in  consequence,  one,  two,  or  more  orders  of  rings  intervene  between  them  and  the 
points  from  which  the  scale  originates.  These  points  are  observed  to  lie  between  the  poles  in  all  crystals  which 
have  the  blue  axes  nearer  than  the  red,  such  as  Rochelle  salt,  borax,  mica,  sulphate  of  magnesia,  topaz ;  and 
beyond  them  for  those  in  which  the  red  axes  include  a  less  angle  than  the  blue,  as  sulphate  of  baryta,  nitre, 
arragonite,  sugar,  hyposulphite  of  strontia ;  and  this  fact,  as  well  as  the  constancy  of  their  distance  from  the 
poles  when  the  thickness  of  the  plate  is  varied,  renders  their  origin  evident.  In  fact,  since  the  violet  rings  are 
smaller  than  the  red,  if  the  centre  about  which  the  former  are  described,  instead  of  being  coincident  with  that 
of  the  latter,  be  shifted  in  either  direction,  carrying  its  rings  with  it,  some  one  of  the  violet  rings  will  necessarily 
be  brought  up  to,  and  fall  upon  a  red  ring  of  the  same  order  ;  and  the  same  holding  good  with  the  intermediate 
rays,  provided  the  law  which  determines  the  separation  of  the  different  coloured  axes  be  not  very  different  from 
that  which  regulates  the  dimensions  of  the  rings  of  corresponding  colours,  the  point  of  coincidence  of  a  red 
and  violet  ring  of  the  same  order  will  be  nearly  that  of  a  red  and  green,  or  any  intermediate  colour.  The  tint, 
then,  at  this  point  will  be  either  absolutely  black,  (if  they  be  dark  rings  which  are  thus  brought  to  coincidence,) 
or  white,  if  bright ;  and  from  this  point  the  tints  will  reckon  either  way  with  more  or  less  exactness,  accord- 
ing to  the  same  scale  which  would  have  held  good  had  the  points  of  coincidence  been  the  poles  themselves. 
Should,  however,  the  two  laws  above  mentioned  differ  very  widely,  an  uncorrected  colour  will  be  left  at  the 
point  of  nearest  compensation,  just  as  happens  when  two  prisms  whose  scales  of  dispersion  are  dissimilar  are 
employed  to  achromatise  each  other.  To  what  an  extent  the  disturbance  of  the  Newtonian  scale  of  tints  may 
be  carried  by  this  and  the  other  causes  already  explained,  the  reader  may  see  by  turning  to  the  table  of  tints 
exhibited  by  Rochelle  salt  inPM.  Trans.  1820,  part  i. 

925.  We  come  next  to  consider  the  law  of  the  intensity  of  the  illumination  of  the  rings  in  different  parts  of  their 
T«o  suppo-  periphery  ;  but  this  part  of  their  theory  will  require  us  to  enter  more  fundamentally  into  the  mode  in  which  their 
sitionsasto  formation  is  effected,  and  to  examine  what  modifications  the  polarized  ray  incident  on  the  crystallized  plate 
0  undergoes  in  its  passage  through  it,  so  as  to  present  phenomena  so  totally  different  from  those  which  it  would 
crystals  in  have  offered  without  such  intervention.  It  is  evident  then,  first,  that  since  the  ray,  if  not  acted  on  by  the  plate, 
forming  the  would  have  been  entirely  stopped  by  the  second  tourmaline,  but,  when  so  acted  on,  is  partially  transmitted  so  as 
rings.  to  exhibit  coloured  appearances  of  certain  regular  forms  ;  that  the  crystallized  plate  must  have  either  destroyed 

altogether  the  polarization  of  that  part  of  the  light  which  is  thereby  enabled  to  penetrate  the  second  tourmaline, 
or,  if  not,  must  have  altered  its  plane  of  polarization,  so  as  to  allow  of  a  partial  transmission.     Between  these 
Doctrine  of  two  suppositions  it  is  not  difficult  to  decide.     Were  the  portion  of  light  which  passes  through  the  second  tourma- 
polarization  ]jne  an(j  forms  the  rings  wholly  depolarized,  that   is,  restored  to  its  original  state   of  natural  light,   since  the 
re  '  remainder,  its  complement  to  unity,  which  continues  to  be  stopped  by  the  tourmaline,  retains  its  state  of  polariza- 
tion unaltered,  it  is  evident,  that  each  ray  at  leaving  the  crystallized  plate  would  be  composed  of  two  portions, 
one  unpolarized  (=  A),  the  other  (=  1  —  A)  polarized.     Of  these,  the  half  only  of  the  first  (£  A)  would  be 
transmitted   by  the  second  tourmaline.     Now,  suppose  this  to  be  turned  round  in  its  own  plane  through  any 
angle  (=  a)  from  its  original  position,  then  the  unpolarized  portion  will  continue  to  be  half  transmitted;    and 
the   polarized,  being   now  partially  also    transmitted,  (in  the  ratio  of  sin8  a  :  1,)  will   mix  with  it,  so  that  the 
compound  beam  will  be  represented  by 

£  A  +  (1  —  A)  .  sin*  a  —  sin5  a  -)-  —  .  cos  2  a. 

Now,  if  we  suppose  a  to  pass  in  succession  through  the  values  0,  45°,  90°,  135°,  180°,  &c.,  this  will  become 
respectively  J  A,  J,  1  —  J  A,  \,  \  A,  &c.     Hence,  at  every  quarter  revolution  the  tints  ought  to  change  from 
those  of  the  reflected  rings  to  those  of  the  transmitted,  the   complements  of  the  former  to  white  light ;  and  at 
every  half  quarter  revolution  no  rings  at  all   should  be  seen,  but  merely  an  uniformly  bright  field  illuminated 
with  half  the  intensity  of  light  which  would  be  seen  were  the  second  tourmaline  altogether  removed. 
92fi.          But  the  phenomena  which  actually  take  place  are  very  different.     At  the  alternate  quadrants,  it  is  true,  the 
Phenomena  complementary  rings  are  produced,  and  the  appearance  is  as  represented  in  fig.  188.     The  black  cross  is  seen 
ot  the  com-  changed  into  a  white  one ;  the  dark  parts  of  the  rings  are  become  the  bright  ones  ;  the  green  is  changed  into 
lary  red,  and  the  red  into  green,  &c. ;  so  that  if  we  were  to  examine  no  farther,  the  fact  would  appear  to  agree  with 
Fig  188.      *ne  hypothesis.     But  in  the  intermediate  half  quadrants,  this  agreement  no  longer  subsists.     Instead  of  a  uni- 
formly illuminated  field,  a  compound  set  of  rings,  consisting  of  eight  compartments,  alternately  occupied  by 
the  primary  and  complementary  set,  is  seen,  presenting  the  appearance  of  fig.  191,  and  which  is  further  described 
in  Art.  935. 

927.  The  phenomena  then  are  incompatible  with  the  idea  of  depolarization.    It  remains  to  examine  what  account  can 

Hypothesis   be  given  of  them  on  the  supposition  of  a  change  of  polarization  operated  by  the  plate  ;  and  here  we  must 

of  a  change  remark  in  Umine,  that  this  cause  is  what  in  Newton's  language  would  be  termed  a  vera  causa,  a  cause  actually 

l"  in  existence;  for  we  have  already  seen  that  every  ray,  whether  polarized  or  not,  traversing  a  double  refracting 

medium  in  any  direction,  except  precisely  along  its  axis,  is  resolved  into  two,  polarized  in  opposite  planes.  When 

the  incident  ray  is  polarized,  these  portions   (generally  speaking)  differ  in  intensity,  and  though,  owing  to  the 

parallelism  of  the  plate  they  emerge  superposed,  their  polarization  is  not  the  less  real,  and  either  of  them  may  be 

suppressed,  and  the  other  suffered  to  pass,  by  receiving  them  on  a  tourmaline  properly  situated.     This  is  so  far 


LIGHT.  525 

Light.      agreeable  to  the  observed  fact,  when  the  tourmaline  plate  next  the  eye  is  removed,  the  rays  of  which  the  two  sets    Part  IV. 
-— y— —^  of  rings  consist,  coexist  in  the  transmitted  cone  of  rays  whose  apex  is  the  eye,  but,  being  complementary  to  each  v— ~v-— ^ 
other,  produce  whiteness.     This  may  be  made  matter  of  ocular  demonstration,  by  employing,  instead  of  a  Both  seis  of 
tourmaline,  which  absorbs  one  image,  a  doubly  refracting  achromatic  prism,  of  sufficiently  large  refracting  angle  r'"Ss  *ho-,\n 
to  separate  the  two  pencils  by  an  angle  greater  than  the  apparent  diameter  of  the  system  of  rings,  when  thea 
primary  set  will  appear  in  one  image,  and  its  complementary  set  in  the  other  ;  meanwhile,  to  return  to  our  tour- 
malines, since  the  two  sets  of  rings  seen  in  the  two  positions  of  the  posterior  tourmaline  are  complementary,  it 
follows,  that  all  the  rays  suppressed  in  one  position  are  transmitted  in  that  at  right  angles  to  it,  and  vice  versa  ; 
and,  as  a  necessary  consequence,  that  every  pair  of  corresponding  rays  in  the  primary  and  complementary  set  are 
polarized  in  opposite  planes. 

The  only  thing,  then,  which  appears  mysterious  in  the  phenomena  thus  conceived,  is  the  production  of  colour.      928. 
A  doubly  refracting  crystal,  which  receives  a  polarized  ray  of  whatever  colour,  divides  it  between  its  two  pencils,  M-  Blot's 
according  to  a  ratio  dependent  only  on  the  situation  of  the  planes  of  polarization  and  of  incidence,  and  of  the  do  tr!".e  of 
axes  of  the  crystal,  and  not  at  all  on  its  refrangibility.     How  then  happens  it,  that  at  certain  angles  of  incidence  ^la 
the  red  rays  pass  wholly  into  one  image,  and  the  green  or  violet  into  the  other,  while  at  other  incidences  the 
reverse  takes  place  :    whence,  in  short,  arises  the  law  of  periodicity  observed.     To  answer  this  question,  M.  Biot 
imagined  his  theory  of  alternate,  or  as  he  terms  it  movable  polarization,  according  to  which,  as  soon  as  a  pola- 
rized ray  enters  into  a  thin  crystallized  lamina,  its  plane  of  polarization  commences  a  series  of  oscillations,  or 
rather  alternate  assumptions  per  saltnm  of  two  different  positions,  one  in  its  original  plane,  the  other  in  a  plane 
making  with  that  plane  double  the  angle  which  the  principal  section  of  the  crystal  makes  with   it.     These 
alternations  he  supposes  to  be  more  frequent  for  the  more  refrangible  rays,  and  to  recur  periodically,  like  New- 
ton's fits  of  easy  reflexion  and  transmission,  at  equal  intervals  all  the  time  the  ray  is  traversing  the  crystal,  which 
intervals  are  shorter  the  more  inclined  its  path  is  to  the  axis  or  axes.     This  theory  is  remarkably  ingenious  in 
its  details ;  and  in  its  application  to  the  phenomena  of  the  rings,  though  open  (as  stated  by  its  author)  to  certain 
obvious  criticisms,  is  yet,  we  conceive,  capable  of  being  regarded  as  a  faithful  representation  of  most  of  their 
leading  features.     There  is,  however,  one  objection  against  it  of  too  formidable  a  nature  to  allow  of  its  being  Objection 
received  unless  explained  away,  if  any  other  can  be  devised  not  open  to  the  same  or  greater.     It  is,  that  it  requires  agamst  '<• 
us  to  consider  the  action  of  a  thin  crystal  on  light  as  totally  different,  not  merely  in  degree,  but  in  kind,  from 
that  of  a  thick  one,  while  yet  it  marks  no  limit  by  which  we  are  to  determine  where  its  action  as  a  thin  crystal 
ceases,  and  that  proper  to  a  thick  one  commences,  nor  establishes  any  gradations  by  which  one  mode  of  action 
passes  into  the  other.     A  thick  crystal,  as  we  know,  polarizes  the  rays  ultimately  emergent  from  it  in  two  planes, 
dependent  only  on  the  position  of  the  crystal  and  that  of  the  ray,  while  M.  Biot's  theory  makes  the  position  of 
the  plane  of  polarization  of  the  incident  ray  an  element  in  determining  their  ultimate  polarization  by  n  thin  one. 
Nor  are  we  in  this  theory  to  regard  as  thin  crystals  arAy  films  or  delicate  laminae.     A  plate  of  a  tenth  of -an  inch 
thick  or  more  may  be  a  thin  plate  in  some  cases  of  feebly  polarizing  bodies,  such  as  apophyllite,  &c. 

As  the  apparatus  employed  by  M.  Biot  for  studying  the  phenomena  of  the  colours  of  thin  crystallized  plates  929. 
offers  great  conveniences  for  the  measurement  of  the  angles  at  which  different  tints  are  produced,  and  for  their  M.  Blot's 
exhibition  in  their  state  of  greatest  purity  and  contrast,  we  shall  here  describe  it,  and  state  some  of  the  chief  ?eneral 
results  at  which  he  has  arrived.  A  (fig.  189)  is  a  plane  glass  blackened  at  the  posterior  surface,  or  a  plate  °f  described 
obsidian  inclined  at  the  polarizing  angle  to  the  axis  of  a  tube  A  B,  so  as  to  reflect  along  .it  a  polarized  ray  ;  (if  f\s,  189, 
greater  intensity  be  required,  we  may  use  a  pile  of  glass  plates,  taking  care  that  they  be  of  truly  parallel  surfaces,  190. 
and  placed  exactly  parallel  to  each  other.)  B  C  is  a  tube,  stiffly  movable  round  A  B  as  an  axis,  having  a 
graduated  ring  at  B,  read  off  by  a  vernier  attached  to  the  tube  A  B,  and  carrying  two  arms,  G  and  H,  through 
which  the  axis  of  a  swing  frame  E  passes,  which  can  thus  be  inclined  at  any  angle  to  the  common  axis  of  the 
tubes,  its  inclination,  or  the  angle  of  incidence  of  the  ray  reflected  along  the  axis  on  the  plane  of  the  frame 
being  read  off  by  an  index  on  the  divided  lateral  circle  D.  In  this  frame  is  an  aperture  F,  in  which  turns  a 
circular  plate  of  brass  having  a  hole  in  its  centre,  over  which  is  fastened  with  wax  the  crystallized  plate  to  be 
examined,  and  which  can  thus  be  turned  round  in  its  own  plane,  independently  of  any  motion  of  the  rest  of  the 
apparatus,  so  as  to  place  its  principal  section  in  any  azimuth  with  respect  to  the  plane  of  incidence.  We  have 
found  it  convenient  to  have  this  part  of  the  apparatus  constructed  as  in  fig.  190,  where  a  is  the  square  plate  of 
the  frame  ;  6  a  divided  circle  movable  in  it  and  read  off  by  an  index  ;  c,  d  is  a  circular  plate  movable  within 
the  divided  circle  to  admit  of  adjustment,  after  which  it  is  fastened  in  its  place  by  a  little  clamp,  so  as  to  turn 
with  the  circle  ;  this  carries  in  its  centre  another  swinging  circle  e,  moving  stiffly  on  its  axis,  and  having  in  the 
middle  an  aperture,  over  which  the  crystal  is  cemented,  thus  giving  room  for  an  adjustment  of  the  plane  of  the 
surface  of  incidence,  in  case  it  be  not  exactly  at  right  angles  to  the  principal  section  of  the  crystal,  an  adjustment 
very  useful  when  artificial  surfaces  are  under  examination,  which  it  is  hardly  possible  to  cut  and  polish  with 
perfect  precision.  It  is  also  convenient  for  some  experiments  to  have  a  second  frame  similar  to  the  first,  placed 
on  the  prolongation  of  the  arms  G,  H.  M  is  a  doubly  refracting  prism,  rendered  achromatic  either  by  a  prism 
of  flint  glass,  or,  still  better,  by  another  prism  of  the  same  doubly  refracting  medium.  Two  prisms  of  quartz, 
arranged  as  in  Art.  882,  are  very  convenient.  Their  angles  should  be  such,  that  when  placed  at  M  the  two 
images  of  a  small  aperture  P,  in  a  diaphragm  near  the  end  of  the  tube,  should  appear  almost  in  contact.  The 
prisms  so  adjusted  are  mounted  on  a  stand  N,  independent  of  the  other  apparatus,  and  capable  of  being  turned 
round  by  an  arm  K,  carrying  a  vernier,  by  whose  aid  the  angle  of  rotation,  or  position  of  the  plane  in  which  the 
double  refraction  takes  place,  can  be  read  off  on  a  divided  circle  L.  The  prism  should  be  so  adjusted  in  its  cell, 
that  when  the  vernier  reads  off  zero,  the  extraordinary  image  should  be  extinguished  ;  and  when  90°,  the  ordinary. 
Occasionally  a  tourmaline  plate  or  a  glass  reflector  may  be  substituted  for  the  prism. 

1o  use  this  apparatus,  the  crystallized  lamina  (which  we  will  at  present  suppose  to  be  a  parallel  plate  of  any      930. 


526  LIGHT. 

Light,     uniaxal   crystal,  having1  its  axis  perpendicular  to  the  plane  of  the  plate,)  is  to  be   placed  on  the  swing  frame     Part  IV- 
•— ••v^-*'  across  the  aperture,  and  being  adjusted  so  as  to  have  its  axis  directed  precisely  along  the  axis  of  the  tube  when  VS~P*V-""* 
Use  of  this  the  vernier  of  D  reads  off  zero,  which  is  readily  performed  by  the  various  adjustments  belonging1  to  the  frame, 
tus>    as  above  described,  the  instrument  is  ready  for  use.     The  attainment  of  this  condition  may  be  known  by  turning- 
the  tube  C  on  the  tube  A  B  as  an  axis,  when  the  extraordinary  image  of  the  aperture  P,  seen  through  a  doubly1 
refracting  prism,  ought  to  vanish  in  the  zero  position  of  the  vernier  K,  and  not  be  restored  in  any  part  of  the 
rotation  of  the  tube ;  for  it  is  manifest,  that  the  axis  is  the  only  line  to  which  this  property  belongs,  or  to  which 
all  the  rings  are  symmetrical.     It  is  then  evident,  that,  however  the  parts  of  the  apparatus  be  disposed,  1st,  the 
reading  off  of  the  vernier  D  will  give  the  angle  of  incidence  on  the  plate  ;  2d,  that  of  the  vernier  B,  the  angle 
made  by  the  plane  of  incidence  with  the  plane  of  primitive  polarization  ;  3d,  that  of  the  vernier  c  will  indicate 
the  angle  included  by  any  assumed  section  of  the  crystallized  plate  perpendicular  to  its  plane  with  the  plane  of 
incidence  ;  and,  lastly,  that  the  reading  of  the  vernier  K  will  give  the  angle  between  the  plane  of  primitive  polari- 
zation and  the  principal  section  of  the  doubly  refracting  prism. 

Suppose  now  we  adjust  the  vernier  B   to  zero,  it  will  then  be  found,  that  however  the  plate  E  be  situated,   or 
Us  applica-  whatever  be  the  incidence  of  the  ray,  only  the  ordinary  image  will  be  seen  (being  white,)  the  extraordinary  being 
phenomena  extinguished  (or  black.)     In  this  case  we  traverse  the  system  of  rings  in  the  direction  of  the  vertical  arm  of  the 
of  the  ring*  black  cross,  fig.  185,  of  the  primary,  and  the  white  one  of  the  complementary  set,  see  fig.  188.     The  phenomena 
of  one  axit.  are  the  same  if  we  set  the  vernier  B  to  90°,  and  then  turn  the  frame  E  on  its  axis,  thus  varying  the  incidence  in 
Fig.  188.     a  plane  at  right  angles  to  that  of  primitive  polarization,  or,  which  comes  to  the  same  thing,  traversing  the  rings 
along1  the  horizontal  arm  of  the  black  and  white  crosses.     In  intermediate  positions  of  the  vernier  B,  we  traverse 
the  ring's  along1  a  diameter,  making  an  angle  with  vertical  equal  to  the  reading  of  the  vernier.     In  this  case  the 
two  images  of  P  are  both  visible,  and  finely  coloured  ;  the  extraordinary  image  presenting  the  tint  of  the  primary 
rings  due  to  the  particular  angle  of  incidence  indicated  by  the  vernier  D  ;  the  ordinary,  that  of  the  comple- 
mentary system  corresponding  to  the  same  angle.     The  colours  of  the  two  images  are  thus  seen  in  circumstances 
the  most  favourable,  being  finely  contrasted  and  brought  side  by  side,  so  as  to  be  capable  of  the  nicest  comparison. 
It  is  when  the  vernier  D  reads  45°,  or  the  plane  of  incidence  is  45°,  inclined  to  that  of  primitive  polarization, 
that  the  contrast  of  the  two  images  is  at  its  maximum,  the  tints  in  the  extraordinary  image  being  then  most 
vivid,  and  those  in  the  ordinary  free  from  any  mixture  of  white   light.     In  general,  if  A  represent  the  light  of 
the  extraordinary  image  in  the  position  above  mentioned,  and  a  the  angle  read  off  on  the  vernier  B,  in  any  other 
position  of  the  plane  of  incidence,  the  two  images  in  this  new  position  (for  the  same  angle  of  incidence)  will  be 
represented  respectively  by 

A  .  (sin  2  a)2,  and  1  -  A  (sin  2  a)- 

that  is,  by  A  .  (sin  2  a)*,  and  (cos  2  a)2  +  (1  -  A)  .  (sin  2  a)J. 

The  former  of  these  expressions  indicates  a  ray  whose  tint  is  represented  by  A,  and  its  intensity  by  (sin  2  a)2 ;  the 
latter,  a  complementary  tint  1  —  A  of  the  same  intensity,  diluted  with  a  quantity  of  white  light,  whose  intensity 
is  represented  by  (cos  2  a)*. 

932.          These  expressions  represent  with  great  fidelity  the  tints  of  both  images,  the  intensity  of  the  extraordinary,  and 
Agreement    the  apparent  degree  of  dilution  of  the  ordinary  one  ;  and  since  a  ray  A  polarized  in  a  plane  making  an  angle  2  a 
the  *°r~    with  the  principal  section  of  the  doubly  refracting  prism,  would  be  divided  between  the  extraordinary  and  ordinary 
M*  Blot's      'maf?e  in  tne  ratio  of  (sin  2  o)s :  (cos  2  «)2,  it  follows,  that  if  we  regard  the  pencil  at  its  emergence  from  the  cry- 
hypothesis,   stallized  plate  as  composed  of  two  portions,  one  (=  A)  polarized  in  the  above  named  plane,  the  other  (=:  1  —  A) 
preserving1  its  primitive  polarization,  the  two  pencils  formed  by  the  doubly  refracting  prism  will  be  composed 
as  follows : 

Extraordinary  image.  Ordinary  image. 

1st.  From  the  pencil  A A  (sin  2  «)"  A  .  (cos  -2  a)1 

2d.   From  the  pencil  (1  -  A) 0  1  -  A 


Sum    A  (sin  2  a)4         1  —  A  +  A  .  cos  2  «' 

=  1  -  A  .  (sin  2  a)» 

Office  of  the  which  are  identical  with  those  above.     Thus  we  see,  that  the  facts  are  so  far  perfectly  conformable  to  M.  Blot's 
doubly         hypothesis  of  movable  polarization,  and   that  we  are  even  necessitated  to   admit   it,  provided   we  take  it  for 
"isnTor^     granted,  that  the  rings  exist  actually  formed  and  superposed  in  the  pencil  emergent  from  the  crystallized  lamina, 
tourmaline.  an^  that  the  office  of  the  doubly  refracting  prism  is  merely  to  analyze  the  emergent  pencil,  and  separate  the  two 
sets  from  each  other.     But  if  the  objection  mentioned  above  against  that  doctrine  be  really  well  founded,  this 
assumption  cannot  be  correct,  and  we  are  then  driven  to  conclude,  that  the   doubly  refracting  prism,  or  tourma- 
line, or  glass  reflector,  interposed  between  the  eye  and  the  crystallized  plate,  performs  a  more  important  office 
than  merely  to  separate  the  tints  already  formed;'  and  that,  in  fact,  they  are  actually  produced  by  its  action, — the 
crystallized  plate  only  preparing  the  rays  for  the  process  they  are  here  finally  to  undergo. 

933.  To  explain  how  this  may  be  conceived  to  happen  will  form  the  object  of  another  Section.  Meanwhile  we  will 
here  only  add,  that  the  transition  from  uniaxal  to  biaxal  crystals  is  readily  made.  We  have  only  to  consider,  that 
by  varying  the  angle  of  incidence,  (the  line  bisecting  the  angle  between  the  optic  axes  being  supposed  perpen- 
dicular to  the  plane  of  the  plate,)  we  cross  the  rings  in  a  line  passing  through  their  centre  of  symmetry  O,  fig.  183, 
and  makincr  an  ansrle  with  their  nrincioal  diameter  PP.  eaual  to  the  angle  read  off  on  the  vernier  B,  and  that 
ny  turning  the  plate  in  its  own  plane,  or  varying  the  angle  read  off  by  the  vernier  c,  we  in  effect  make  the  system 
traversed  pass  through  the  successive  states  represented  in  fig.  179,  180,  181,  182,  changing,  not  the  tint,  but 
the  intensity  of  the  extraordinary  image. 


LIGHT.  527 

Light.          When  the  doubly  refracting  prism  is  turned  in  its  cell,  the  tints  grow  more  dilute,  and  when  placed  in  an    part  jy 

— v— ^  azimuth  a,  that  is,  when  its  principal  section  is  placed  in  the  plane  of  incidence,  both  images  are  colourless,  but  i _  • 

of  unequal  brightness.     This  accords  with  M.  Biot's  doctrine  of  movable  polarization  ;  for  if  we  grant  that  the       934 
pencil  A  is  polarized  in  a  plane  making  an  angle  2  a  with  that  of  primitive  polarization,  it  will  make,  now,  an  Effect  of 
angle  =  a  with  that  of  the  principal  section  of  the  prism,  and  A  .  (sin  a)-  will  be  that  part  of  the  extraordinary  turning  the 
image  arising  from  the  pencil  A  ;  on  the  othei   hand,  the  pencil   1  —  A  retaining  its  original  polarization,  P"*"* '"  its 

0  —  A)  .  sin  a*  will  be  the  portion  of  the  extraordinary  image  produced  by  it  in  the  new  position  of  the  prism, 
and  the  sum,  or  the  whole  image,  will  be  simply  1   x  sin  a8,  which  being  independent  of  A,  or  of  the  tint, 
indicates  that  the  image  is  colourless.     In  the  same  manner  it  may  be  shown,  that  the  ordinary  image  will  equal 

1  X  cos  a',  and  their  intensities  will,  therefore,  be  to  each  other  as  sin  o*  to  cos  a",  and  will  be  equal  at  45°  of 
azimuth  ;  all  which  is  conformable  to  fact. 

The  motion  of  the  prism  in  its  cell  corresponds  to  a  rotation  of  the  posterior  tourmaline  in  its  own  plane  in       935. 
the  tourmaline  apparatus.     The  general  appearance  presented  by  the  rings  of  a  single  axis,  when  this  rotation  is  Effect  of 
not  a  precise  quadrant,  is  represented  in  fig.  191,  and  the  succession  of  changes  being  as  follows  :  At  the  first  turm"6 j.'1" 
commencement  of  the  rotation  the  arms  of  the  black  cross  appear  to  dilate  ;   they  grow  at  the  same  time  fainter,  a^"™^^ 
and  segments  of  the  complementary  rings  appear  in  them,  whose  bright  intervals  correspond  to  the  dark  ones  of  each  other. 
the  primary  set,  their  red  to  the  green  portions  of  that  set,  and  vice  vend.     The  junction  of  the  two  sets  is  marked  Fig.  191. 
by  a  faint  white  or  undecided  tint.  As  the  rotation  proceeds,  the  primary  segments  contract  in  extent,  and  become 
more  diluted  with  white,  while  the  secondary  extend,  and  grow  more  decided  ;  at  the  same  time  the  centre  of  the 
system  grows  gradually  bright,  and  when  the  rotation  has  attained  90°,  the  whole  has  assumed  the  appearance 
in  fig.  189.     The  phenomena  are  precisely  analogous  in  the  rings  of  biaxal  crystals.     The  least  deviation  from 
exact  rectangularity  in  the  tourmalines  gives  rise  to  complementary  segments  in  the  dark  hyperbolic  curves 
answering  to  the  arms  of  the   black  cross,  and  to  a  corresponding  dilution  and  contraction  of  the  primary 
segments,  which  at  last  disappear  altogether  in  the  undistinguishable  whiteness  of  a  pair  of  white  hyperbolas 
precisely  similar  to  the  black  ones  of  the  primary  rings  in  their  perfect  state. 

Hitherto  we  have  considered  the  rings  as  so  narrowed  by  the  thickness  of  the  plate,  as  to  be  all  contracted       936 
within  a  compass  round  the  poles  which  the  eye  can  take  in  at  once ;  but  if  the  thickness  be  greatly  diminished,  T'ms  Pr°- 
this  will  no  longer  be  the  case  ;  and,  instead  of  rings  of  a  distinguishable  form,  we  shall  see  only  broad  bands  y"™1}^ 
of  colour  extending  to  great  distances  from  the  poles,  and  even  visible  when  the  axes  themselves  are  so  much  .,]a{es  at 
inclined  to  the  surfaces  of  the  plate  as  to  be  quite  out  of  sight ;  or  even  when  the  axes  actually  lie  in  the  plane  great  dis- 
of  the  plate.     This  is  the  case  with  the  laminae  into  which  sulphate  of  lime  readily  splits ;  the  axes  lie  in  (heir  lances  from 
plane,  so  that  to  see  the  rings  in  them,  we  must  form  artificial  surfaces  perpendicular  to  the  lamina,  a  difficult  lhe  axes- 
and  troublesome  operation,  from  the  extreme  softness  and  fissile  nature  of  the  substance.     The  phenomena  of 
the  colours  of  this  crystal  were  early  studied,  and  almost  of  necessity  misconceived,   till   Dr.  Brewster,  by 
exhibiting  the  real  axes,  showed  that  they  form  only  a  particular  case  of  the  general  phenomenon  we  have  already 
dwelt  on. 

Adhering  to  the  denominations  employed  in  Art.  885 — 888,  let  us  call  the  plane  containing  the  two  axes,  the      937. 
section  A  ;    that  perpendicular  to  it,  and  passing  througli  the  line  which  bisects  the:r  lesser  included  angle,  the  Phenomena 
section  B  ;  and  that  which  similarly  passes  through  the  line  bisecting  their  greater  included  angle,  and  is  perpen-  °h;*  slnKle 
dicular  to  both  the  others,  the  section  C.     If  the  crystal  have  but  one  axis,  the  sections  A  and  B  pass  through  it, 
and  C  is  at  right  angles  to  it.     Then  if  the  lamina  contains  both  axes,  its  plane  will  be  that  of  the  section  A,  and 
the  other  two  sections  will  intersect  it  in  two  lines  (B  and  C)   at  right  angles  to  each  other.     Conceive,  now,  a 
polarized  ray  to  pass  through  such  a  lamina  at  a  perpendicular  incidence.     Then  if  the  plane  of  polarization 
coincide  with  either  of  the  sections  B  and  C,  its  polarization  will  be  undisturbed,  and  the  whole  of  the  trans- 
mitted light  will  pass  into  the  ordinary  image.     But  if  the  plate  be  turned  round  in  its  own  plane,  the  extra- 
ordinary image  will  reappear  and  become  a  maximum  at  every  45°  of  the  plate's  rotation ;    and  if  it  be  suffi- 
ciently thin,  will  exhibit  some  one  of  the  colours  of  the  rings,  and  the  tints  will  descend  regularly  in  the  scale  as 
the  thickness  is  increased,  the  thickness  being  a  measure  of  the  tint,  conformably  to  the  general  law  in  Art.  907, 
of  which  this  is  only  a  particular  case. 

When  two  such  plates  are  laid  together,  with  their  sections  B  and  C  corresponding,  it  is  evident  that  they  are       938. 
in  the  same  relation  as  if  they  formed  part  of  one  and  the  same  crystal ;   and  we  might  therefore  expect  to  find  Phenomena 
what  really  happens,  viz.  that  such  a  compound  plate  polarizes  the  same  tint  that  a  single  plate  equal  to  the  sum  °j 
of  the  thicknesses  would  do.     But  if  they  be  crossed,  i.  e.  laid  so  together  that  the  section  B  of  the  one  shall  pjrpendicu- 
coincide  with  the  section  C  of  the  other,  M.  Biot  has  shown  that  the  tint  polarized  is  that  due  to  the  difference  |ar  jncj. 
of  their  thicknesses.     If,  therefore,  this  difference  be  exactly  nothing,  the  crossed  plates  will  be  exactly  neutra-  dence. 
lized,  at  least  at  a  perpendicular  incidence,  and  that  whatever  be  their  thickness.     (To  procure  two  plates  of 
exactly  the  same  thickness,  we  have  only  to  choose  a  clear  and  truly  parallel  plate  terminated  by  fresh  surfaces  of 
fissure,  and  break  it  across.) 

When,  however,  the  incidence  is  not  perpendicular,  such  a  compound  plate  as  described  will  still  exhibit  colours       939. 
which  vary  in,  apparently,  a  very  irregular  manner  as  the  incidence  changes,  and  with  different  degrees  of  I"*??1161" 
rapidity  in  different  planes.     The  tourmaline  apparatus  here  renders  signal  service  in  rendering  the  law  of  these  " 
tints,  at  first  sight  extremely  puzzling,  a  matter  of  inspection.     When  such  a  crossed  plate  is  placed  between  the 
tourmalines,  crossed  at  right  angles,  it  exhibits  the  singularly  beautiful  and  striking  phenomenon  represented  in 
fig.  192,  in  which  the  tints  are  those  of  the  reflected  scale  of  Newton,  the  origin  being  in  the  black  cross.     If  the  pia  jgg 
tourmalines  be  parallel,  the  complementary  colours  are  produced  with  equal  regularity,  as  in  fig.  193.     If  the  f°   ]93 
compound  crystal  be  turned  round  in  its  own  plane,  the  figures  turn  with  it,  but  undergo  no  change  other  than 
an  alternation  of  intensity,  being  at  a  maximum  of  brightness  when  the  arms  of  the  cross  are  parallel  and 


528  LIGHT. 

v  1''g'lt       perpendicular  to  the  plane  of  orig-inal  polarization,  and  vanishing1  altogether  when  they  make  angles  of  45°  with 
v™*v~"-'  that  plane.     If  the  plates  be  not  crossed  exactly  at  right  angles,  or  be  not  precisely  of  equal  thickness,  other  ' 
phenomena  arise  which  it  is  easier  for  the  reader  to  produce  for  himself  than  to  read  a  detailed  account  of.     The 
same  may  be  said  of  the  very  splendid  but  complicated  phenomena  produced  by  crossing  two  equally  thick 
plates  of  biaxal  crystals,  such  as  mica,  topaz,  &c.  having  the  section  A  at  right  angles  to  their  surfaces. 
940.  Regarding,  however,  at  present  only  the  tint  produced  at  a  perpendicular  incidence,  it  is  found  that  when  any 

LAW  of        number  of  plates  of  one  and  the  same  medium,  of  any  thicknesses,  are  superposed  with  their  homologous  sections 
ed  corresponding,  the  tint  polarized  is  that  due  to  the  sum  of  their  thicknesses  ;  but  when  any  one  or  more  of  them 
perposition   nave  their  sections   B  and  C  at  right  angles  to  the  homologous  sections  of  the  others,  the  tint  is  that  due  to  the 
of  similar     sum  of   the    thicknesses  of  those  placed    one  way,  minus  the  sum  of  those  of  the  plates  placed  the   other 
plates.          way.     In  algebraical  language,  if  we  call  I,  I',  t",  &c.  the  thicknesses,  and  regard  as  negative  those  of  the  plates 
laid  crosswise,  the  tint  T  polarized  by  the  system  will  be  that  due  to  the  thickness  t  -f  ^  -f-  <"  -j-  &c. 

When  the  ray  is  made  to  traverse  a  plate  of  quartz,  zircon,  carbonate  of  lime,  or  any  other  uniaxal  crystal  cut  so 

Law  of  tmls  as  (o  contain  the  axis  of  double  refraction,  the  same  law  of  the  tints  holds  good,  the  tint  T  being  proportional  to 

f>y°dissf-       t'le  th'ckness  '  °f  the  plate,  and  for  any  given  plate  we  have  T  =  k  t,  k  being  a  constant  depending  on  the  nature 

milar  plates  °f  tne  p'ate.     Now,  if  several  plates  of  different  uniaxal  crystals  be  superposed,  of  which  t,  t',  &c.  are  the  thick- 

nesses, and  if  a  negative  value  of  t  be  supposed  to  denote  a  transverse  position  of  the  axis  of  the  plate,  the 

resultant  tint  will  be  represented  by 

&c. 


In  this  equation,  if  the  plates  be  all  of  one  substance,  k,  k',  &c.  are  all  alike  ;  but  if  they  be  different,  k  is 
Opposite  to  be  regarded  as  a  negative  quantity  for  all  such  crystals  as  belong  to  M.  Biot's  repulsive  class,  (Art.  803,)  such 
'totes  o  as  carb°nate  °f  l'me  ?  an(l  positive  for  all  such  (quartz,  for  instance)  which  belong  to  his  attractive  class.  Thus, 
positive  and  eacn  term  in  the  above  equation  may  change  its  sign  from  two  causes,  either  from  a  change  in  the  nature  of  the 
negative  crystal,  or  from  a  change  of  90°  in  its  azimuth. 
crystals.  The  above  is  only  a  particular  case  of  a  more  general  law  which  maybe  thus  announced,—  The  lint  ultimately 

943.  produced  is  proportional  to  the  interval  of  acceleration  or  retardation  of  the  ordinary  ray  on  the  extraordinary, 
General        g^fo  iraversing  the.  whole,  system  ;  the  partial  acceleration  or  retardation  in  each  plate  being  proportional  to  the 

length  of  the  path  described  within  the  plate,  multiplied  by  the  square-ofthe  sine  of  the  angle  which  the  transmitted 
ray  makes,  internally,  with  the  optic  axis  of  the  plate,  if  it  have  but  one  axis,  or  to  the  product  of  the  sines  of  its 
inclination  to  either,  if  it  have  two;  and  this  law  holds  good  for  all  positions  of  the  plates,  and  all  arrange- 
ments of  them  one  among  the  other.  Thus  (to  instance  its  application)  in  the  case  of  two  similar  and  equal 
plates  crossed  at  right  angles  ;  by  the  laws  of  polarization,  the  ray  which,  after  its  transmission  through  the  first 
plate  is  ordinary,  is  refracted  extraordinarily  by  the  second,  and  vice  versd  ;  thus  the  two  rays,  on  entering  the  second 
plate  exchange  velocities  ;  and,  therefore,  when  finally  emergent,  since  the  thickness  of  the  second  is  equal  to 
that  of  the  first,  the  one  ray  will  have  lost  ground  on  the  other  in  its  second  transmission  just  as  much  as  it 
gained  it  in  its  first  ;  and  thus  the  interval  of  retardation  and  the  tint  will  be  reduced  to  nothing. 

944.  From  this  it  appears,  that  if  two  uniaxal  plates  cut  at  right  angles  to  the  axis  be  superposed,  and  adjusted 
Supcrposi-    so  as  to  have  their  axes  precisely  coincident,  the  system  of  rings  will  have  their  diameters  diminished  if  the 

plates  be  both  attractive  or  both  repulsive  ;    but  enlarged,  if  their  characters  be  opposite.     The  experiment  is 
rightSa'm'les  rat'ler  delicate  ;    but  if  made  with  care,  placing  the  plates  on  one  another  with  soft  wax,  and  adjusting  their 
to  tneir  °      surfaces  by  pressure  to  the  exact  position,  it  succeeded  perfectly  in  the  hands  of  Dr.  Brewster. 
axes.  This  affords  a  means,  independent  of  any  measurement  of  the  separation  of  the  ordinary  and  extraordinary 

945.  pencils,  of  ascertaining  whether  an  uniaxal  crystal  be  attractive  or  repulsive  ;   for  if  its  rings  be  dilated  by 
Method  of    combining  it  with  a  thin  plate  of  carbonate  of  lime,  cut  at  right  angles  to  the  axis,  it  is  positive  ;   if  contracted, 
wheihe'r'a"g  neffat've-     A  simpler  and  readier  method  still  is  to  fasten  on  a  plate  of  the  substance  under  examination,  so  cut 
crystal  be     as  to  show  the  rings,  a  plate  of  sulphate  of  lime  of  moderate  thickness,  and  then,  interposing  it  between  the 
positive  or    tourmalines,  to  turn  it  about  in  its  own  plane.     A  position  will  be  found  where  the  rings  are  unaltered.     In  this 
negative.      situation  the  section   B  or  C  of  the  sulphate  of  lime  is  in  the  plane  of  primitive  polarization.     If  the  com- 

pound plate  be  turned  45°  from  this  situation,  it  will  now  be  observed  (if  the  thicknesses  of  the  two  plates  be 
properly  proportioned)  that  the  rings  in  two  opposite  quadrants  are  entirely  obliterated  ;  and  that  in  the  other 
two  they  are  removed  to  a  much  greater  distance  from  the  centre,  forming  segments  of  larger  circles,  much  closer 
together  ;  and  in  which  the  tints,  instead  of  commencing  from  the  centre,  commence  from  a  black  interval 
between  two  adjacent  white  rings  in  the  midst  of  the  system,  and  thence  descend  in  the  scale  both  inwards  and 
outwards.  In  this  state  of  things,  the  position  of  the  sulphate  of  lime,  with  respect  to  the  tourmalines, 
must  be  carefully  noted;  and  the  crystallized  plate  being  detached,  a  plate  of  carbonate  of  lime,  (perpendicular 
to  its  axis,)  or  of  any  other  known  uniaxal  crystal,  must  be  substituted  for  it  ;  and  the  sulphate  of  lime 
replaced  in  the  same  position.  If,  then,  it  be  found,  that  the  same  two  quadrants  of  the  rings  are  obliterated  in 
this,  as  in  the  former  case,  and  the  new  set  of  rings  in  the  other  quadrants  be  also  similarly  situated,  —  then 
the  crystal  examined  is  of  the  same  character  as  the  carbonate  of  lime,  or  other  crystal  used  as  a  standard  of 
comparison  ;  but  if,  on  the  other  hand,  the  quadrants  where  the  rings  were  obliterated  in  the  former  case  be 
those  where  the  new  rings  are  formed  in  the  latter,  then  the  characters  of  the  two  substances  are  opposite.  If 
the  crystallized  plate  be  too  thin,  or  of  too  feeble  polarizing  power  to  exhibit  these  phenomena  with  necessary 
distinctness,  we  must  place  it  in  azimuth  45°  on  the  divided  apparatus  described  in  a  former  article  (929  ;)  and, 
fixing  conveniently  in  the  polarized  beam  a  very  thin  plate  of  sulphate  of  lime  also  in  azimuth  45°,  ascertain, 
by  making  the  crystal  revolve,  whether  its  tints  have  been  raised  or  depressed  in  this  plane  by  the  action  of  the 
sulphate  ;  then,  removing  the  crystal,  replace  it  with  a  standard  one,  and  repeat  the  observation  without  touching 


LIGHT.  529 

tight.      the  sulphate.     If  both  crystals  have  their  tints  raised,  or  both  depressed,  their  characters  are  similar ;  it  they  be     Part  IV. 
•—V""*1  contrarily  affected,  dissimilar.     An  analogous  mode  of  observation  applies  to  biaxal  crystals.  v^ , 

§  VIII.   On  the  Interferences  of  Polarized  Rays. 

In  repeating  the  experiments  of  Dr.  Young  on  the  law  of  interference  it  occurred  to  M.  Arago,  that  it  946. 
would  be  worth  while  to  examine  whether  the  state  of  polarization  of  the  interfering  rays  would  cause  any  Origin  of 
modification  in  the  phenomena.  The  experiment  was  easy  in  the  case  where  both  rays  had  the  same  polarization, 
being,  in  fact,  the  ordinary  case ;  but  when  the  interfering  rays  were  required  to  have  a  different  state  of  pola- 
rization, it  will  easily  be  conceived  that  it  must  be  a  matter  of  great  delicacy  and  difficulty  to  superadd  this 
condition  to  the  others  called  for  by  the  nature  of  the  case,  which  requires  that  the  interfering  rays  should 
emanate  at  the  same  instant  from  a  common  origin,  and  should  have  executed  the  same  precise  number  of 
undulations  or  periods  (within  a  very  few  units)  between  their  origin  and  the  point  where  their  interference  is 
observed.  For  it  is  not  possible  to  change  the  state  of  polarization  of  a  ray  without  either  altering  its  course, 
or  transmitting  it  through  some  medium  in  which  more  or  fewer  undulations  are  executed  in  the  same  space. 
The  joint  ingenuity  of  himself  and  M.  Fresnel,  who  was  associated  with  him  in  this  interesting  inquiry,  how- 
ever, soon  found  means  of  obviating  the  difficulties  and  delicacies  of  the  subject,  and  the  results  of  their  expe- 
riments have  been  embodied  by  them  in  the  following  laws  : 

1.  That  two  rays  polarized  in  one  and  the.  same  plane  act  on  or  interfere  with  each  other  just  as  natural      947. 
rays,  so  that  the  phenomena  of  interference  in  the  two  species  of  light  are  absolutely  the  same.  Laws  °f '"' 

2.  That  two  rays  polarized  in  opposite  planes  (i.  e.   at  right   angles  to    each   other)  have  no   appreciable  ^,rferen^e 
action  on  each  other,  in  the  very   same  circumstances  where  rays  of  natural  light  would  interfere  so   as   to  ii»|,t. 
destroy  each  other.  °949 

3.  That  two  rays  primitively  polarized  in  opposite  planes  may  be  afterwards  reduced  to  the  same  plane  ofpola-       949 
Titation,  without  acquiring  thereby  the  power  of  interfering  with  each  other. 

4.  That  two  rays  polarized  in  opposite  planes,  and  then  reduced  to  similar  states  of  polarization,  interfere       950 
like  natural  rays,  provided  they  belong  to  a  pencil  the  whole  of  which  was  primitively  polarized  in  one  and  the 

tame  plane. 

5.  In  the  phenomena  of  interference  produced  by  rays  which  have  undergone  double  refraction,  the  place  of  the      951, 
coloured  fringes  is  not  alone  determined  by  the  difference  of  routes  or  velocities,  but  that  in  certain  circumstances 

a  difference  of  half  an  undulation  must  be  allowed  for. 

Such  are  the  laws  of  interference  of  polarized  pencils,  as  stated  by  Messrs.  Arago  and  Fresnel.     We  use  in       952. 
their  enunciation,  and  indeed  throughout  the  sequel  of  this  part  of  the  doctrine  of  Light,  the  language  of  the 
undulatory  system,  as  really  the  most  natural,  and  adapting  itself  with  the  least  violence  and  obscurity  to  the 
facts.     The  reader  may,  if  he  please,  substitute  that  of  the  corpuscular  hypothesis  and  the  Newtonian  fits,  super- 
adding  that  of  a  rotation  of  the  luminous  molecules  about  their  axes,  with  M.  Biot ;   or  simply  content  himself 
with  a  bare  enunciation  of  facts,  and  with  general  terms  expressive  of  the  existing  conditions  of  periodicity, 
without   much  trouble,  and  only  a  little  circumlocution,  but  with  a  great  sacrifice  of  clearness  of  conception. 
With  respect  to  the  laws  themselves,  the  first  is  easily  verified  ;    we  have  only  to  repeat  any  of  the  experiments  Experiment 
on  the  interference  of  rays  emanating  from  a  common  origin,  described  in  our  section  on  that  subject,  substi-  tal  verifica- 
tuting  polarized  instead  of  natural  light,  and  the  results  will  be  precisely  similar,  and  that  in  whatever  plane  tion  of  "i« 
the    light   be   polarized.      Rays,  then,  polarized  in  the  same  plane,   interfere  as  natural   rays  under  similar  first  '**'• 
circumstances. 

The  verification  of  the  second  law  is  more  difficult  and  delicate.     The  conditions  of  the  production  of  colours       953. 
by  interference  require  that  the  interfering  rays  should  emanate  simultaneously  from  a  common  origin,  or  form  Difficulties 
parts  of  one  and  the  same  wave  proceeding  therefrom  as  a  centre  ;  and  should  have  performed,  at  the  point  P*c"  lar-'° 
where  their  interference  is  examined,  the  same  number  of  undulations  in  their  respective  routes,  within  a  very 
few  units.     Now  at  their  leaving  their  origin  they  could  not  be  otherwise  than  in  the  same  state  of  polarization  ; 
and  as  they  are  required  to  arrive  at  the  point  of  interference  in  opposite  states,  a  change  of  polarization  must 
be  operated  on  one  or  both  rays,  either  by  reflexion,  transmission,  or  double  refraction,  after  leaving  their  origin, 
and   that  without  altering,  more  than  by  a  few  undulations,  the  difference  of   their  routes.     Now,  when  we 
consider  how   minute  a  quantity  an  undulation  is,  it  is  easy  to  conceive  the  delicacy  required  in  adjusting  the 
parts  of  any  apparatus  constructed  for  this  purpose,  or  the  peculiar  contrivances  which  must  be  resorted  to  to 
render  such  extreme  and  almost  impracticable  nicety  unnecessary. 

Several   ingenious  and  elegant  methods  of  making  the  experiment  have  been  devised  by  the  authors   last       954. 
named,  of  which  we  shall  content  ourselves  with  stating  one  or  two  of  the  easiest  and  most  satisfactory.     And,  Verifica- 
first,   the  origin  of  the  interfering  rays  being  the  image  of  the  sun  at  the  focus  of  a  small  lens,  as  we  shall  lion  of  ttle 
suppose  it  throughout  this  section,  (unless  the  contrary  be  expressly  said,)  it  is  evident  that  if  we  interpose  Mcond  law 
between  the  eye  and  this  image  a  rhomboid  of  Iceland  spar,  there  will  be  formed  two  images  separated  from 
each  other  by  a  space  which  will   he  greater  the  thicker  is   the  rhomboid  ;   but  which  will  always   (unless 
extremely  thick  rhomboids  be  used)  be  very  small ;  so  that  the  single  luminous  point  will  now  be  resolved  into 
two,  very  near  each  other,  and  which,  by  the  laws  of  polarization,  send  to  the  eye  rays  polarized  in  opposite 
planes.     But  in  this  disposition  of  things,  the  condition  of  near  equality  of  routes  is  subverted ;  for  the  ordinary 
and  extraordinary  pencils  pursue  different  paths  within   the  crystal,  and  with  very  different  velocities ;  so  that 
a  difference  will  thus  arise  in  the  total  number  of  undulations  executed  by  each,  sufficient  to  destroy  all  evidence 

VOL.  iv.  3  z 


530  LIGHT 

Light,     of  interference  by  the  production  of  coloured  fringes.     To  obvial  ?  this  diHculty,  M.  Fresnel   sawed   in  half  a     P"t  IV. 

^•••V" "^  rhomboid  of  Iceland  spar,  the  two  halves  of  which  must  of  necessity  have,  at  their  line  of  separation  and  its 

ex  ^riment*  'mme^'ate  confines,  precisely  equal  thicknesses.     These   halves  he  placed  one  on  the  other,  only  turning  one 

90°  round   in  azimuth,  so  as  to  have  their  principal  sections  at  right  angles.     In  this  state,  a  pencil  enterin"' 

bisected       them  nearly  at  the  intersection  of  the  planes  of  separation  would  at  its  final  emergence  be  divided,  not  into  four, 

rhomboid,    but  into  two  only,  (see  Art.  879,)  the  ray  ordinarily  refracted  in  the  rirst  half  having  undergone  extraordinary 

refraction  in  the  second,  and  vice  versd.     The  two  rays,  therefore,  have  exchanged  velocities  and  directions,  in 

the   second  transmission  ;    and,  therefore,  when  emergent,  will  have  described  exactly  equal  paths  with  equal 

velocities  in  each  respectively,  and  will  differ  only  in  their  states  of  polarization,  which  will  be  at  right  angles 

to  each  other.     We    have  here,  then,  a  case  in  which  pencils  diverge  from  two  points  side  by  side,  and  in  a 

state  in  all   other  respects  proper  for  interfering ;  nevertheless,  when  we  look  for  the  fringes  which  ought   to 

be  formed  under  such   circumstances,  (and  which  with   natural  light  would  be  seen,  see  Art.  735  and  736,) 

none  are  visible.     Their  absence,  then,  must  be  owing  to  the  opposite  state  of  polarization  of  the  inter- 

rering  rays. 

•>;>•  (        M.  Arago,  to  make  the  same  experiment,  employed  a  process  independent  of  double  refraction.     Two  fine 
erlnfe°nts s"ts  wcre  ma(^e  in  a  ^'m  plate  of  copper,  through  which  rays  from  the  common  origin  were  transmitted,  and 
with  mica    f°rmed  fringes  (in  their  natural  state)  when  viewed  by  an  eye  lens  in  the  manner  described,  (Art.  709.)     He 
piles.  now  prepared  two  piles  of  pieces  of  very  thin  mica,  or  films  of  blown  glass  laid  one  on  the  other,  fifteen    in 

number,  and  then  divided  this  compound  plate  in  half  by  a  sharp  instrument,  so  that  the  halves,  in  the  imme- 
diate neighbourhood  of  the  line  of  division,  could  not  be  otherwise  than  of  almost  exactly  equal  thickness. 
These  piles,  when  exposed  at  an  incidence  of  30°  to  a  ray,  were  found  to  polarize  the  portion  transmitted 
almost  completely.  They  were  then  placed  before  the  slits  so  as  to  receive  and  transmit  the  rays  from  the 
luminous  point  at  precisely  that  incidence,  and  through  spots  which  were  very  near  each  otter  in  the  undivided 
state  of  the  pile.  They  were,  moreover,  so  arranged,  (being  set  on  revolving  frames,)  that  the  plane  of 
incidence  could  be  varied  (and  therefore  that  of  polarization)  by  turning  either  round  in  azimuth  without  alter- 
ing its  inclination  to  the  ray,  or  varying  the  spot  through  which  the  ray  passed.  And  it  was  then  found,  that 
when  both  piles  were  placed  so  as  to  polarize  the  rays  in  parallel  planes,  as,  for  instance,  when  both  were 
inclined  directly  downwards,  or  one  directly  down  and  the  other  directly  up — the  fringes  were  formed  as  if  the 
piles  were  away;  but  where  one  of  the  piles  was  turned  round  the  incident  ray  as  an  axis  through  90°,  and  so 
placed  as  to  polarize  the  rays  transmitted  by  it  at  right  angles  to  the  other,  the  fringes  totally  disappeared,  nor 
could  they  be  restored  by  inclining  either  pile  a  little  more  or  less  to  the  incident  ray  in  the  plane  of  incidence, 
the  effect  of  which  would  be  to  alter  gradually  the  length  of  the  ray's  path  within  the  pile  without  changing 
its  polarization,  and  thus,  to  compensate  any  slight  inequality  which  might  still  subsist  in  their  thicknesses. 
In  intermediate  positions  the  fringes  appeared,  but  always  the  more  vividly  the  nearer  the  planes  of  polariza- 
tion approached  to  exact  parallelism,  thus  attaining  their  maximum,  and  undergoing  total  obliteration  at  each 
quadrant  of  the  rotation  of  either  pile,  (the  other  being  at  rest.) 

956.          A  plate  of  tourmaline  carefully  worked  to  exact  parallelism,  and  bisected,  would  answer  equally  well  with  the 

Tourmaline  transparent  piles  to  polarize  the  rays ;  but  the  tourmaline  should  be  selected  of  very  homogeneous  texture,  such 

plates  sub-  are  not  easy  to   meet   with,  though   they  maybe  found;    and  in  this  manner  the  experiment  is  perfectly  easy 

theUpMe8.0r  ant*  sat'sfactory-     O"e  half  the  tourmaline  is  fixed  over  one  aperture,  the  other  movable  in  a  cell  in  its  own 

plane  over  the  other.     The  same  phenomena  will  then  be  observed  by  turning  round  the  movable  tourmaline  as 

with  the  oblique  pile  in  the  last  experiment. 

An  experiment  still  more  simple,  and  equally  conclusive,  is  the  following,  of  M.  Fresnel.     He  placed  before 
.  Fresnel's  tne  sheet  of  copper  (having,  as  before,  two  narrow  slits  in  it  very  near  each  other)  a  single  thin  parallel  lamina 
ta^eitp^ri-    °^  su'phate  °f  Hme.     Now,  as  this  body  possesses  double  refraction,  each  pencil  would  be  divided  into  two — 
ment.          an  ordinary  and  an  extraordinary  one — which,  according  as  they  emanate  from  the  right  or  left  hand  slit,  we 
Analysis  of  will  term  R  o,  Re,  and  Lo,  Le.     If  natural  light  be  used  to  illuminate  the  slits,  these  pencils  will  be  of  equal 
the  pola-     intensity,  but  those  marked  e  will  be  polarized  oppositely  from  those  marked  o.     We  may  then  form  four  *om- 
-.zed  tints,    binations  :   1.  Ro  may  interfere  with   L  o  ;  2.  R  e  may  interfere  with  Le;    3.  R  o  with  Le;  4.  RewithLo. 
Now  of  these,  Ro  and  Lo  are  similarly  polarized,  and  they  have  described  equal  paths  with  equal  velocities; 
therefore,  supposing  them  capable  of  interference,  they  will  give  rise  to  a  set  of  fringes  corresponding  exactly 
to  the  middle  of  the  line  joining  the  two  slits,  or,  as  we  may  express  it,  in  the  axis  of  the  apparatus.     The 
same  may  be  said  of  R  e  and  L  e.     These  two  sets  of  fringes  will  therefore  be  superposed,  and  appear  as  one  of 
double  intensity.     Again,  Ro  may  be  combined  with  Le;  but  as  these  two  rays  have  traversed  the  sulphate  in 
different  directions  and  with  different  velocities,  those  rays  of  each  pencil  which  meet  in  the  axis  will  differ  by 
too  many  undulations  to  produce  colour ;    and  if  the  pencils  interfere,  the   place  of  the  fringes  will,  instead  of 
the  axis,  be  shifted  towards  the  side  where  the  pencil  has  the  greatest  velocity,  (Art.  737,)   and  that  the  more, 
the  thicker  the  lamina  of  sulphate,  so  that  if  taken  of  a  proper  thickness,  this  set  of  fringes  may  be  removed 
entirely  out  of  the  reach  of  the  middle  set,  and  should  be  seen  independent  of  it.     In  like  manner,  the  pencil 
Re  may  interfere  with  L  o,  and  give  rise  to  another  set  of  lateral  fringes;   but  as  the  ray  which  in  the  former 
combination  was  the  swifter,  in  this  is  the  slower,  this  set  will  lie  on  the  opposite  side  of  the  middle  set,  sup 
posing  it  produced  at  all ;  and  thus  there  should  be  seen  three  sets  of  fringes,  one  bright,   in  the  middle, 
and  two  fainter  on  either  side.     But,  in  fact,  only  one  set  is  seen,  viz.  the  middle  set.     Therefore  the  combina- 
tion of  the  rays  R  o  and  L  e,  L  o  and  R  e,  which  are  polarized  oppositely,  produce  no  fringes,  i.  e.  they  do  not 
interfere. 

But  if  we  cut  the  lamina  in  half,  and  turn  one  half  a  quadrant  round  in  its  own  plane,  these  rays  are  tnen 
reduced  to  the  same  polarization ;   and  the  rays  R  o  and  L  o,  Re  and  L  c,  which  in  the  former  case  gave  rise  to 


L  I  G  H  T.  531 

Light,      the  central  fringes,  are  now  placed  in  opposite  states  of  polarization  ;  and  it  is  accordingly  found  that  the  central    Part  IV 
w-v— '  fringes  have  disappeared  entirely,  and  that  two  lateral  sets  formed  respectively  by  Ro  and  Le,  Re  and  L  o,  \— — y-—'' 
have  started  into  existence.     If  we  turn  the  lamina  slowly  round,  these  will  gradually  fade  away,  and  the  central  Experiment 
reappear  and  become  brighter,  and  so  on  alternately ;    thus  affording  a  convincing  proof  of  the  truth  of  the  van 
second  of  the  laws  above  enunciated. 

The  experiment  related  by  Messrs.  Arago  and  Fresnel  in  support  of  their  third  law  is  as  follows  :  Resuming      959. 
the  arrangement  of  Art.  955  or  956,  and  placing  the  piles  or  tourmalines  so  as  to  polarize  the  two  pencils  Verification 
oppositely,  let  a  doubly  refracting  crystal  be  placed  between  the  eye  and  the  sheet  of  copper,  with  its  principal  j^J1'*  thlrd 
section  45°  inclined  to  either  of  the  planes  of  polarization  of  the  interfering  rays.     Each  pencil  will  then  divide 
itself  by  double  refraction  into  two  of  equal  intensity,  and  polarized  in  two  planes  at  right  angles,  one  of  which 
is  the  principal  section  itself.     We  ought,  therefore,  to  expect  to  see  two  systems  of  fringes,  one  produced  by 
the  interference  of  the  ordinary  ray  from  the  right  hand  aperture  (Ro)  with  that  of  the  left  (L  o,)  and  the  other 
by  that  of  Re  with  Le;  yet  no  fringes  are  seen.     The  experiment  may  be  varied  by  substituting  for  the  doubly 
refracting  prism  a  tourmaline,  or  pile,  with  its  principal  section  in  azimuth  45°.     This  must  reduce  to  a  common 
polarization  all  the  rays  which  traverse  it,  viz.  the  half  of  each  pencil,  yet  no  fringes  are  seen,  and  therefore  no 
interference  takes  place. 

The  following  experiment  is  adduced  in  the  Memoir  cited  in  support  of  the  fourth  and  fifth  of  the  above      9dO. 
laws.     A  lamina  of  sulphate  of  lime  is  perpendicularly  exposed  to  a  polarized  pencil  diverging  from  a  minute  ExPeri 
point,  and  immediately  behind  it  is  placed  a  plate  of  brass  pierced  with  two  very  small  holes  near  together.  ^"fVAh* 
The  principal  section  of  the  lamina  is  to  be  placed  at  an  angle  of  45°  with  the  plane  of  primitive  polarization,  fourth  and 
In  consequence,  from  each  of  the  holes  (right,  R, — and  left,  L)  will  emerge  a  ray  composed  of  two  equal  rays,  fifth  laws. 
Ro  and  Re,  and  Lo,  Le  oppositely  polarized,  viz.  at  angles  +  45°  and  —  45°  with  the  plane  of  primitive 
polarization,  which  we  will  suppose  vertical.     In  this  situation  of  things  a  rhomboid  of  Iceland  spar  is  placed 
between   the  two  holes,  and  the  focus  of  the  eye  lens  employed  to  view  the  fringes,  with  its  principal  section 
vertical,  i.  e.  making  again  with  that  of  the  lamina  angles  of  45°  either  way.     Each  of  the  four  rays  then  above 
mentioned  will  be  divided  into  two  equal  rays,  an  ordinary  and  an  extraordinary,  thus  giving  rise  in  all  to  the 
eight  rays 

Roo,  Reo ;  Loo,  Leo;  Roe,  Ree ;  Loe,  Lee. 

These  rays  are  received  on  the  eye  lens,  and  conveyed  into  the  eye.     Let  us  now  examine  their  respective  route 
and  states  of  polarization. 

First,  then,  the  rays  Ro  and  lie,  after  quitting  the  lamina,  are  parallel;  and  by  reason  of  the  very  small  961. 
thickness  of  it,  may  be  regarded  as  superposed,  being  undistinguishable  from  each  other ;  but  they  have 
described  within  the  lamina  different  paths  by  different  velocities,  so  that  on  emerging  they  will  differ  in  phase, 
by  an  interval  of  retardation  proportioned  to  the  thickness  of  the  lamina,  and  which  we  will  call  d,  so  that  a, 
being  the  phase  of  the  ray  R  o,  x  -f-  d  will  be  that  of  R  e.  The  very  same  may  be  said  of  L  o  and  L  e.  More- 
over^  the  two  rays  of  either  of  these  pairs  respectively  are  oppositely  polarized,  viz.  in  planes  +  45°  and  —  45° 
from  the  vertical.  This  we  may  represent  at  once  thus : 

Ray  Phase.  Plane  of  Polarization. 

R  o  x  +  45° 

Re  x  +  d  -45° 

L  o  x  +45° 

Le  x  +  d  -  45° 


Next,  the  portions  into  which  either  of  these  rays  is  subdivided,  in  traversing  the  rhomboid,  follow  in  their  962. 
passage  through  it  different  paths,  and  have  different  velocities;  but  all  which  are  refracted  ordinarily  have  one 
common  direction  and  velocity ;  and  so  of  those  refracted  extraordinarily ;  hence,  between  the  ordinary  and 
extraordinary  rays  here  produced,  will  arise  a  difference  of  phase  which  we  shall  call  £,  so  that  if  x  be  the  phase 
of  any  ordinary  ray,  x  -\-  o  will  be  that  of  the  corresponding  extraordinary  one  ;  and  their  planes  of  polarization 
will  be  opposed,  and  will  form  angles  respectively  =  0  and  90°  with  the  vertical.  Thus  the  circumstances  will 
stand  thus  : 

A.  B. 


|0 


Ray.  Phase.  Plane  of  Polarization. 

Roo  X  0° 

Reo  x  +  d  O1 

Loo  x  0° 

Leo  *  +  d  0° 


Ray.  Phase.  Plane  of  Polarization. 

Roe  x  +  S                          90° 

Ree  x  +  d  +  S                  90° 

Loe  x  +  S                          90° 

Lee  x  -4-  d  +  6                  90 


° 


These  eight  pencils  are  all  equal  in  intensity,  and  all  those  contained  in  the  first  set  (marked  A)  will  meet  in  963. 
one  part  of  the  field  of  view,  while  those  marked  B  (on  account  of  the  thickness  of  the  rhomboid,  which  we 
here  suppose  considerable,  so  as  to  produce  a  sensible,  and  even  a  large  separation  of  the  ordinary  and  extra- 
ordinary pencils)  will  meet  in  another,  distant  from  the  point  of  concourse  of  (A)  by  an  interval  proportional  to 
the  thickness  of  the  rhomboid,  and  which  we  will  here  suppose  so  large  as  to  throw  the  fringes  (if  any)  there 
produced,  entirely  out  of  the  way  of  mixing  with  those  produced  at  the  concourse  of  A.  Let  us  then  consider 
separately,  the  pencils  of  rays  of  the  parcel  A,  and  see  what  interferences  can  take  olace.  And  first,  Roo  may 


a  z  2 


532 


L  I  G  H  T. 


Light, 


964. 


965. 


966. 

Variation 

'  ' 

Allowance 
of  half  an 
jndulation. 


967. 

Application 
•o  the 
colours  of 


968. 

Why  co- 
lours  are 

ecre,n 

thiiMTystal 
lized  plate 
alone. 


969. 

Fig.  194. 
Explanation 

°f  'rhe  c°- 

p'olarized  * 
rings. 


combine  with  Loo,  and  since  their  difference  of  phase  is  zero,  they  will  interfere  in  the  axis  of  the  apparatus; 
^  and  their  planes  of  polarization  being  coincident,  there  is  no   reason  why  fringes  should  not  there  be  pro- 
duced  by  their  concourse.     The  same  holds  good  of  the  combination  R  i-  o  and   L  eo,  and,  consequently,  there 
will  be  superposed  on  each  other  in  the  axis  two  sets  of  fringes,  producing  cue  of  double  brilliancy. 

Next,  Roo  may  interfere  with  Leo;  but  there  being  a  constant  difference  of  phases  d  in  favour  of  the  latter, 
the  fringes  produced  by  their  concourse  will  lie  to  the  left  of  the  axis,  by  an  interval  proportional  to  the  thickness 
of  the  lamina  of  sulphate,  and  will  be  seen  separately.  Similarly,  the  concern  se  of  the  pencils  Reoand  Loo  will 
determine  the  production  of  another  set  of  lateral  fringes  ;  but  the  difference  of  phases  d  being  in  this  case  in  favour 
of  the  right  hand  pencil,  this  system  will  be  situated  as  much  to  the  right  of  the  axis  as  the  other  was  to  the  left. 

Thus  in  the  ordinary  image  three  sets  of  fringes  ought  to  be  seen,  and  in  the  extraordinary,  by  a  similar 
reasoning,  as  many.  Now,  in  fact,  this  is  the  case,  and  the  phenomena  are  seen  on  making  the  experiment  pre- 
cisely as  here  described.  But  it  is  evident  that  the  rays  which  form  the  lateral  fringes,  by  their  interferences,  are 
precisely  those  which,  at  their  leaving  the  sulphate,  had  opposite  polarizations,  but  have  been  afterwards  reduced 
to  similar  polarization  by  the  action  of  the  rhomboid. 

If  instead  of  a  rhomboid  of  sensible  double  refraction  we  substitute  a  plate  of  sulphate  of  lime,  or  of  rock 
crystal,  so  thin  as  to  produce  no  visible  separation  of  the  pencils,  the  fringes  produced  by  the  pencils  B  will  be 
superposed  on  those  arising  from  the  interference  of  the  pencils  A,  and  we  should  expect  therefore,  instead  of  six, 
to  see  three  sets  of  fringes,  the  middle  one  being  still  the  brightest.  But,  in  fact,  we  see  but  one  set,  and  the 
lateral  fringes  vanish  altogether.  This  remarkable  result  proves  that  the  colours  resulting  from  the  concourse  of 
the  rays  ordinarily  refracted  by  the  rhomboid,  are  complementary  to  those  resulting  from  that  of  the  extraordinary 
rays  .  antj  therefore  that  we  must  allow  half  an  undulation  to  be  gained  or  lost  when  we  would  pass  from  one  set 
to  the  other,  precisely  as  in  the  phenomena  of  the  reflected  and  transmitted  colours  of  thin  plates. 

One  of  the  most  important  consequences  of  these  laws,  is  that  they  supply  the  defective  link  in  the  chain  which 
connects  the  doctrine  of  undulations  with  the  colours  of  crystallized  laminae  as  described  in  the  last  section.  It 
had  been  already  remarked  (as  we  have  seen)  by  Dr.  Young,  that  the  passage  of  the  ordinary  and  extraordinary  rays 
witn  different  velocities  through  the  crystallized  plate,  would  give  rise  to  that  difference  of  physical  condition  of 
tne  ravs  at  t'le'r  er»ernence  which  would  lead  to  the  production  of  colours  ;  but  the  difficulty  remained  to  explain, 
not  why  colours  were  produced  in  certain  circumstances,  but  why  they  were  not  produced  in  all,  in  short,  what 
share  the  polarization  of  the  incident,  and  the  analysis  of  the  emergent  rays,  had  in  the  production  of  the  phe 
nomena. 

To  see  the  nature  of  this  difficulty  more  clearly,  imagine  a  wave  proceeding  from  a  distant  radiant  point 
to  be  incident  on  a  very  thin  crystallized  lamina.  It  will  be  subdivided  into  two,  each  traversing  the  plate 
jn  a  different  direction  and  with  its  own  proper  velocity,  and  each  of  them  emerging  parallel  to  its  original  direc- 
tion.  The  incident  wave  will,  therefore,  after  emergence  be  resolved  into  two  parallel  to  each  other,  but  sepa- 
rated  by  a  small  interval  equal  to  the  interval  of  retardation.  Now  the  hindmost  of  these  ought,  according 
to  the  law  of  interferences,  to  interfere  with  a  subsequent  wave  of  the  system  to  which  the  foremost  belongs, 
and  thus  periodical  colours  should  arise  on  merely  looking  against  the  sky  through  such  a  lamina  without  any 
other  apparatus.  Why  then  are  none  seen?  To  this  the  law  of  Messrs.  Arago  and  Fresnel  afford  a  satisfactory 
answer.  The  two  systems  of  waves  into  which  the  incident  system  is  resolved  are  oppositely  polarized,  and 
therefore,  though  all  other  conditions  be  satisfied,  incapable  of  interfering. 

To  understand  how  the  colours  of  the  polarized  rings  must  be  conceived  to  be  produced  by  interference,  let  us 
take  the  simplest  case  when  a  polarized  ray,  A  B,  fig.  194,  is  incident  on  any  thin  crystallized  plate  B,  whose 
principal  section  is  45°  inclined  to  the  plane  of  primitive  polarization.  Let  A  be  the  system  of  waves  which 
constitutes  the  incident  ray  ;  then  in  its  passage  through  the  crystallized  lamina  it  will  be  divided  into  systems 
®  am*  ^  °^  e(lual  intensities,  polarized  in  planes  +  45°  and  —  45°  inclined  to  that  of  primitive  polarization, 
and  the  one  lagging  a  few  undulations  behind  the  other,  so  as  to  interfere,  as  represented  in  the  figure,  and  con- 
stituting the  parallel  rays  C  F  and  D  G.  Let  these  now  be  received  on,  and  transmitted  through,  a  doubly 
refracting  prism  F  G  H  L  placed  with  its  principal  section  in  the  plane  of  primitive  polarization,  or  45°  inclined  to 
that  of  the  crystallized  lamina.  Then  will  each  of  the  incident  rays  be  again  subdivided,  C  F  into  II  M  and 
I  P,  and  D  G  into  KN  and  L  Q,  all  of  equal  intensity.  Of  these,  H  M  and  K  N  emerge  parallel,  as  also  K  N 
and  LQ  respectively.  Now  the  systems  of  waves  O  and  E  which  follow  each  other  at  a  certain  interval  d  will 
continue  to  do  so  in  both  the  refracted  rays,  as  if  they  formed  one  compound  system  ;  so  that  each  of  the  pencils 
H  M  K  N  and  I  P  L  Q  will  consist  of  a  double  system  of  waves  O  e  and  E  e,  O  o  and  E  o  respectively.  The  former 
pair  following  each  other  at  the  interval  d,  and  the  latter  at  the  interval  d  i  £  undulation,  (by  reason  of  the  demon- 
strated fact,  that  in  passing  from  the  ordinary  to  the  extraordinary  system  half  an  undulation  must  be  allowed. 
See  Art.  966.)  Now  as  each  ray  of  these  pairs  respectively  have  similar  polarizations,  viz.  those  of  the  pair 
ordinarily  refracted  (O  o  and  E  o)  in  the  plane  of  the  principal  section  of  the  prism,  and  those  of  the  extra- 
ordinary pair  O  e  and  E  e  in  a  plane  at  right  angles  to  it,  there  is  no  reason  why  interference  should  not  take 
place,  and  the  consequence  must  be,  the  production  of  complementary  colours  in  the  two  pencils  finally  emergent 

corresponding  to  the  intervals  of  retardation  d  and  d  -j-  -5-.  which  is  just  what  really  happens. 


Part  IV. 


Ex  lanation      Conceive  now  another  ray  incident  on  B  in  the  direction  A  B,  but  polarized  in  a  plane  at  right  angles  to  that 
of  tne"com"  °f  tne  ray  considered  in  the  last  paragraph.     Then  this  will  undergo  precisely  the  same  series  of  divisions  an 
plementary  subdivisions  as  the  former.     But  the  intervals  of  retardation  will  be  different  ;   for  its  plane  of  polarization  when 
tints.  incident  on  B  being  now  related  to  the  plane  of  ordinary  refraction,  as  that  of  the  other  ray  at  its  'ncidence  was 


LIGHT.  533 

Lignt.       to  the  extraordinary,  and  vice,  versa,  a  difference  of  half  an  undulation  must  (as  already  explained)  be  admitted     part  jy 
«— v-^-'  in  the  relative  position  of  the  two  systems  of  waves  O,  E,  at  their  emerg-ence,  from  this  cause,  independent  of  i 
the  interval  of  retardation  within  the  plate ;  so  that  if  d  were  the  interval  in  the  former  case,  d  —  ^  X  will  be  the 
difference  now,  and,  after  passing  through  the  prism,  we  shall   have  for  the  intervals  of  retardation  in  the  two 
binary   pencils,    instead  of  d  and  d  +  ^  X  which   they  were   before,  d  —  J  X  and  d.     Hence  the  two  pencils 
will  exchange  colours  when   the  polarization  of  the  incident  light  is  varied  by  a  quadrant,   and  this  is  also 
conformable  to  fact.     If  this  reasoning  be  not  thought  conclusive,  the  reader  is  referred  forwards  to  Art.  983 
and  984. 

Next,  let  the  incident  ray  be  unpolarized.     This  case,  as  we  have  seen  Art.  851,  is  the  same  with  that  of  a       971, 
ray  consisting  of  two  equal  rays  oppositely  polarized,  and  therefore  in  each  pencil  will  coexist,  superposed  on  Why  co- 
each  other,  the  primary  and  complementary  colour  arising  from   either  portion,  which  being  of  equal   intensity  lours  are  not 
will  neutralize  each  other's  colours  and  the  emergent  pencils  will  be  white,  and  each  of  half  the  intensity  of  the  Pro(lljced  b.Y 
incident  beams.     This  then  is  the  reason  (on   this  doctrine)  why  we  see  no  colours  when  the  light  originally  J|°h°  ar 
incident  on  the  crystallized  plate  is  unpolarized 

Thus,  the  theory  of  interferences,  modified  by  tne  principles  above  stated,  affords,  as  we  see,  an  explanation       972. 
of  the  colours  of  crystallized  plates  totally  distinct  from  that  of  movable  polarization.     The  only  delicacy  in   its  M.  Fresnel's 
application  to  all  cases,  lies  in  the  determination  which  of  the  emergent  pencils  must  be  regarded  as  having  its  general  ruh 
interval  of  retardation  increased  by  half  an  undulation.     M.  Fresnel  gives  the  following  rule  for  this  essential 
point.   (Note   on    M.  Arago's  Report  to   the   Institute  on   a  Memoir  of  M.  Fresnel  relative  to   the  colours  of  to'allow  for 
doubly  refracting  lamina?,  Annales  de  Cfiimie,  vol.  xvii.  p.  80.*)     The  image  whose  tint  corresponds  precisely  to  the  half  un- 
the  difference  of  routes,  is  that  in  which  the  planes  of  polarization  of  its  constituent  pencils  after  having  been  sepa-  dulation 
rated  from  each  other,  are  brought  together  by  a  contrary  motion,  while,  on  the  other  hand,  the.  pencils  whose  j?au>ej  or 
planes  of  polarization  are  brought  to  coincidence  by  a  continuance  of  the  same  motion  by  which  they  were  sepa- 
rated, produce  by  their  reunion  the  complementary  image.     To  understand  this  better,  let  P  C  be  the  plane  of  F'S-  ^S- 
primitive  polarization  projected  on  that  of  the  paper,  to  which  let  us  suppose  the  ray  perpendicular,  C  O  that  of 
the  principal  section  of  the  crystallized  lamina,  and  C  S  that  of  the  principal  section  of  the  doubly  refracting 
prism  ;  then  the  incident  pencil  polarized  in  the  plane  P  P'  will  after  penetrating  the  lamina  be  divided  into  two, 
one  O  polarized  in  the  plane  C  O,  the  other  E  in  the  plane  C  E  perpendicular  to  it.     Now,  C  O  may  always  be 
so  taken  as  to  make  an  angle  not  greater  than  a  right  angle  with   C  P,  and  C  E  so  as  to  have   C  P  between  C  E 
and  C  O  ;  so  that  the  plane  C  P  may  be  conceived  to  open  or  unfold  itself  like  the  covers  of  a  book,  into  C  O  and 
C  E,  one  on  either  side.     Again,  C  S  may  always  be  regarded  as  making  an  angle  not  greater  than  a  right  angle 
with  C  O,  and  when  the  ray  O  resolves  itself  into  two  (O  o  and  O  e)  by  refraction   at  the  prism,  its  plane  of 
polarization  C  O  may  be  conceived  to  open  out  into  the  two  C  S  and  C  T  at  right  angles  to  each  other,  including 
C  O  between  them  ;  and  in  like  manner  the  ray  E  will  resolve  itself  into  two  E  o  and  E  e,  and  its  plane  of  pola- 
rization C  E  will   open  out  into  the  two  C  S  and  C  T',  having  C  E  between  them   in  the  case  of  fig.  195  (a), 
and  into  C  S'  and  C  E  in  that  of  fig.  195  (6)  ;  in  the  former  case  C  T'  is  a  prolongation  of  C  T,  in  the  latter  C  S' 
is  a  prolongation  of  C  S.     The  rays  O  o  and  E  o  then  which   make  up  the  ordinary  pencil,  have,  in  the  case  of 
fig.  (<z),  been  each  brought  to  a  coincident  plane  of  polarization  C  S   by  two  motions  in  contrary  directions,  as 
represented  by  the  arrows,  and  the  extraordinary  ones  O  e  and  E  e  have  been  separated  and  brought  back  to  a 
coincident  plane  by  motions  continued  in  the  same  direction  for  each  respectively.     The  reverse  is  the  case  in 
fig.  6.     In  the  case  then  of  fig.  a  the  colours  of  the  ordinary  pencil   O  o  -f-  E  o  will  be  those  which  correspond 
precisely  to  the  difference  of  routes,  and  those  of  the  extraordinary  one  O  e  +  E  e  will  correspond  to  that  differ- 
ence plus  half  an  undulation,  while  in  that  of  fig.  6  the  reverse  happens.     This  rule  is  empirical,  i.  e.  is  merely  a 
result  of  observation.     It  is  clear  that  the  principle  of  the  conservation  of  the  DM  viva  in  this,  as  in  the  colours 
of  uncrystallized  plates,   requires  that  the  two  images  should  be  complementary  to  each   other,  and  therefore 
half  an  undulation  must  be  gained  or  lost  by  one  or  the  other  pencil,  but  which  of  the  two  is  to  be  so  modified 
we  have  no  me;ms  of  knowing  a  priori. 

This  once  determined,  however,  we  have  no  difficulty  in  deducing  the  formulae  of  intensity  and  other  circum-       973 
stances  of  the   phenomena  when  the  azimuth  of  the  crystallized  plate  is  arbitrary,  instead  of  being,  as  we  have 
hitherto  supposed,  limited  to  45°.     The  analytical  expressions  of  the  intensity  of  the  pencils  we  must  reserve  for 
our  next  section. 

§  IX.     Of  the  application  of  the  Undulatory  Doctrine  to  the  explanation  of  the  phenomena  of  Polarized  Light 

and  of  Double  Refraction. 

The  phenomena  of  double  refraction  and  polarization,  as  exhibited  in  the  experiments  of  Huygens  on  Iceland       0,74 
spar,  were  regarded  by  Newton  and  his  followers  as  insuperable  objections  to  the  undulatory  doctrine,  inasmuch  Newton's 
us  it   appeared  to  them  impossible,  by  reason  of  the  qiiaqudversum  pressure  of  an  elastic  fluid,  to  conceive  an  objections 
undulation  as  having  a  different  relation  to  different  regions  of  space,  or  as   possessing  sides.     "  Are  not,"  says  against  the 
Newton,  "  all  hypotheses  erroneous  in  which  light  is  supposed  to  consist  in  pressure  or  motion  propagated 


*  This  Memoir  was  read  to  tne  Institute,  Oct.  7,  1816.  A  Supplement  was  received  Jan.  19, 
June  4,  18'21.  And  while  every  optical  philosopher  in  Europe  has  been  impatiently  expecting  ii 
unpublished,  and  is  known  to  us  only  by  meagre  notices  in  a  periodical  Journal. 


,  1818.     M.  Arago's  report  on  it  was  read 
its  appearance  for  seven  years,  it  lie*  a>  yet 


534  LIGHT. 

Light,      through  a  fluid  medium?"  ........  "for  pressures  or  motions  propagated  from  a  shining  body  through  an  uni- 

v—  "v*™''  form   medium,  must  be  on  all  sides  alike,  whereas  it  appears  that  the  rays  of  light  have  different  properties  in  v 
their  different  sides."  ........  "  To  me,  this  seems  inexplicable,  if  light  be  nothing  else  than  pressure  or  motion 

propagated  through  ether."     Opticks,  book  iii.  quest.  28.     And,  again,  quest.  29  ;  "  Are  not  rays  of  light  very 
small  bodies  emitted  from  shining  substances?1'  ........  "  The  unusual  refraction  of  Iceland  crystal  looks  very 

much  as  if  it  were  performed  by  some  kind  of  attractive  virtue  lodged  in  certain  sides  both  of  the  rays  and  of  the 
particles  of  the  crystal.''  ......  "  I  do  not  say  this  virtue  is  magnetical.  —  It  seems  to  be  of  another  kind.     I  only 

say,  that,  whatever  it  be,  it  is  difficult  to  conceive  how  the  rays  of  light,  unless  they  be  bodies,  can  have  a  per- 
manent virtue  in  two  of  their  sides  which  is  not  in  their  other  sides,  and  this,  without  any  regard  to  their  position 
as  to  the  space  or  medium  through  which  they  pass." 

975.  Although  we  have  no  knowledge  of  the  intimate  constitution  of  elastic  media,  or  the   manner  in  which  their 
Examined,   contiguous  particles  are  related  to  each  other  and  affect  each  other's  motion,  yet  it  is  certain  that  the  mode  and 

laws  of  the  propagation  of  motion  through  them  by  undulation  cannot  but  depend  very  materially  on  this  con- 
nection. The  only  analogies  we  have  to  guide  us  into  any  inquiry  into  these  laws,  are  those  of  the  propagation 
of  sound  in  air  or  water,  and  of  tremors  through  elastic  solids,  and  along  tended  chords  and  surfaces  ;  and  such  is 
the  extreme  difficulty  of  the  subject  when  taken  up  in  a  purely  mathematical  point  of  view,  that  we  are  forced  to 
have  recourse  to  these  analogies,  and,  dismissing  in  the  present  state  of  science  the  vain  hope  of  embracing  the 
whole  subje.ct  in  analytical  formulae,  suffer  ourselves  to  be  instructed  by  experience,  as  to  what  modifications  the 
peculiar  constitution  of  vibrating  media  may  produce  in  the  propagation  of  motion  through  them.  Now,  when 
sound  is  propagated  through  air  or  water,  in  which  the  molecules  are  at  least  supposed  to  have  no  mutual  con- 
nection but  to  be  capable  of  moving  with  equal  facility,  and  to  be  restored  to  their  places  with  equal  elastic- 
forces,  in  whatever  direction  they  are  displaced,  and  in  which,  moreover,  it  is  (at  least  theoretically)  taken  for 
granted,  that  the  motion  of  any  molecule  has  an  equal  tendency  to  set  in  motion  those  adjacent  to  it,  in  what- 
ever direction  these  may  be  situated  with  respect  to  it;  it  is  difficult  to  conceive  that  the  motion  of  a  molecule  in 
the  surface  of  a  wave,  at  some  distance  from  the  centre  whence  the  sound  emanates,  can  be  performed  otherwise 
than  in  the  direction  of  the  radius,  or  at  right  angles  to  the  surface  of  the  wave;  so  that  in  this  case  the  motion 
of  the  vibrating  molecules  must  coincide  with  the  direction  of  the  rays  of  sound,  and  there  appears,  therefore,  no 
reason  why  such  rays  should  bear  different  relations  to  the  different  regions  of  space  surrounding  them,  whether 
right  or  left,  above  or  below;  for  the  ray  being  regarded  as  an  axis,  all  parts  of  the  sphere  round  it  are  similarly 
related  to  it. 

976.  But  if  we  conceive  a  connection  of  any  kind,  such  as  may  possibly  be  established  by  repulsive  and   attractive 
forces,  or  magnetic  or  other  polarities  subsisting  between  the  molecules  of  the  vibrating  medium,  the  case  is 
altered.     It  will  no  longer  then  follow  of  necessity,  that  the  individual  motion  of  each  molecule  is  performed  in 
the  direction  in  which  the  general  wave  advances,  but  it  may  be  conceived  to  form  any  angle  with  that  direction, 
even  a  right  angle.    A  familiar  instance  of  such  a  mode  of  propagation  may  be  seen  in  the  wave  which  runs  along 
a  long  stretched  cord,  struck,  shaken,.or  otherwise  disturbed  at  one  end.    The  direction  of  the  wave  is  the  length 
of  the  cord,  and  that  of  the  motion  of  each  molecule  lies  in  a  plane  perpendicular  to  it.    Now  this  is  precisely  the 

Fresnel's      kind  of  propagation  which  M.  Fresnel   conceives  to  obtain   in   the  case  of  light.     He   supposes   the  eye   to  be 

theory  of      affected  only  by  such  vibrating  motions  of  the  ethereal  molecules  as  are  performed  in  planes  perpendicular  to  the 

vibrations     directions  of  the  rays.     According  to  this  doctrine,  a  polarized   ray  is  one  in  which  the  vibration  is  constantly 

performed  in  one  plane,  owing  either  to  a  regular  motion  originally  impressed  on  the  luminous  molecule,  or  to 

some  subsequent  cause  acting  on   the  waves  themselves,  which  disposes  the  planes  of  vibration  of  their  mole- 

cules all  one  way.     An  unpolarized  ray  may  be  regarded  as  one  in  which  the  plane  of  vibration    is  per- 

petually varying,  or   in  which  the  vibrating  molecules  of  the  luminary  are  perpetually  shifting  their  planes  of 

motion,  and  in   which  no  cause  has  subsequently  acted  to  bring  the   vibrations  thus  excited  in  the   ether  to 

coincident  planes. 

977.  The  analogy  of  the  tended  cord  (which  appears  to  have  suggested  itself  to  Dr.  Young  on   considering  the 
Propagation  optical  properties  of  biaxal  crystals  in  1818)  will  help  our  conception  greatly.     Suppose  such  a  cord  of  indefinite 
of  light        length,  stretched  horizontally,  and  one  end  of  it  being  held  in  the  hand,  let  it  be  agitated  to  and  fro  with  a 


.a^     motion  perpendicular  to  the  length  of  the  cord.     Then  will  a  wave  or  succession  of  waves  be  propagated  along  it, 

waves  alone  an(^  every  molecule  of  the  cord  will,  after  the  lapse  of  a  time  proportional  to  its  distance  from  the  hand,  begin 

a  stretched    to  describe  a  line  or  curve  similar  and  similarly  situated  to  that  described  by  the  extremity  at  which  the  agitation 

cord.  originates.     If  the  original  agitation  be  regularly  repeated  and  constantly  confined  to  one  plane,  the  same  will 

be  true  of  the  motion  of  each  molecule,  and  the  whole  extent  of  the  cord  will  be  thrown  into  the  form  of  an  undu- 

-.ting  curve  lying  in  one  plane,  so  far  as  the  motion  has  reached.     In  this  case  it  will  represent  a  polarized  ray 

or  system  of  waves.     If,  after  a  few  vibrations  in  one  plane,  the  extremity  be  made  to  execute  a  few  in  another, 

and  then  again  in  another,  and  so   on,  so  that  the   plane  of  vibration  shall  assume  in  rapid  succession  all  pos- 

sible situations,  since  each  molecule   obeys  exactly  the   same  law  of  motion  with   the  extremity,  the  curve  will 

consist  of  portions  lying  in  all  possible  planes,  and  since  by  reason  of  the  propagation  of  the  undulation  along  it, 

every  point  of  it  is  in   succession   agitated  by  the   motion  of  every  other,  all   these  varied  vibrations  will  run 

through  any  given  point  of  it,  and  were  a  sentient  organ  like  the  human   retina  stationed  there,  the  impression 

it  would  receive  would  be  analogous  to  that  excited  in  the  eye  by  an  unpolarized  ray  of  light. 

<;*S.  It  may  be  objected  to  this  mode  of  conceiving  the  luminiferous  undulations,  that  the  molecules  of  the  ether,  if 

Ohjectsins    it  be  a  fluid,  such  as  we  have  hitherto  all  along  regarded  it,  cannot  be  supposed  connected  in  strings,  or  chains 

eons'dertd.  [j^g  those  of  a  tended  cord,  but  must  exist  separate  and  independent  of  each  other.    But  it  is  sufficient  for  our  pur- 

pose to  admit  such  a  degree  of  lateral  adbesion  (we  hesitate  to  term  it  viscosity)  as  may  enable  each  molecule  in 

its  motion  not  merely  to  push  before  it  those  whi"H  lie  directly  in  the  line  of  its  motion,  but  to  drag  along 


LIGHT.  535 

Light,  with  it  those  which  lie  on  either  side,  in  the  same  direction  with  itself.  Or,  acknowledging  at  once  tne  1'art  IV. 
— v^—'  difficulty,  since  light  is  a  real  phenomenon,  we  are  not  to  expect  it  to  be  produced  without  a  mechanism  VN—V™I/ 
adequate  to  so  wonderful  an  effect.  We  do  not  hesitate  to  attribute  to  the  fluids  which  are  imagined  to  account 
for  the  phenomena  of  heat,  electricity,  magnetism,  &c.  properties  altogether  repugnant  to  our  ordinary  notions 
of  fluids,  and  why  should  we  deny  ourselves  the  same  latitude  when  light  is  to  be  accounted  for.  It  is  true 
the  properties  we  must  attribute  to  the  ether  appear  characteristic  of  a  solid  than  of  a  fluid,  and  may  be 
regarded  as  reviving  the  antiquated  doctrine  of  a  plenum.  But  if  the  phenomena  can  be  thereby  accounted 
for,  i.  e.  reduced  to  uniform  and  general  principles,  we  see  no  reason  why  that,  or  any  still  wilder  doctrine, 
should  not  be  admitted,  not  indeed  to  all  the  privileges  of  a  demonstrated  fact,  but  to  those  of  its  represen- 
tative, or  locum  tenens,  till  the  real  truth  shall  be  discovered.  Assuming  it,  then,  with  M.  Fresnel,  as  a  pos- 
tulatum,  that  the  vibrations  of  the  ethereal  molecules  which  constitute  light  are  performed  in  planes  at  right 
angles  to  the  direction  of  the  ray's  progress,  let  us  see  what  account  can  be  given  of  the  phenomena  of 
polarized  light. 

And  first,  then,  of  the  interference  of  two  polarized  rays,  whether  polarized  in  the  same,  or  different  planes.       979. 
The  plane  of  polarization  in  this  doctrine  may  be  assumed  to  be  either  that  in  which  the  vibrations  are  executed,  Explanatioa 
(i.  e.  a  plane  passing  through  the  direction  of  the  ray  and  the  line  described  by  each  of  the  vibrating  molecules  ° 
in  its  excursion,)  or  one  perpendicular  to  it,  which  we  please.     Reasons,  presently  to  be  stated,  render  the  latter  interference 
preferable,  but  at  present  it  is  a  matter  of  indifference  which  we  assume.     Now,  in  §  3,  Part  III.  we  have  on  this 
investigated  at  length,  with  a  view  to  the  present  inquiry,  the  modes  of  vibration  which  result  from  the  combi-  doctrins. 
nation  of  any  assigned  vibrations,  whether  executed  in  the  same  or  different  planes ;   and  it  follows  from  the 
purely  mechanical  principles  there  laid  down,  1st,  That  the  combination  of  two  vibrations  executed  in  the  same 
plane,  produces  a  resultant  vibration  in  the  same  plane,  which  may  be  of  any  degree  of  intensity  from  the  sum 
to  the  difference  of  the  intensities  of  its  component  vibrations,  according  to  the  difference  of  their  phases.    Now, 
each  of  these  systems  of  vibration  represents  a  polarized  ray;   so  that  rays  polarized  in  the  same  plane  ought, 
on  these  principles,  to  be   capable  of  destroying  or  reinforcing  each  other  by  interference,  as  we  see  they  do. 
But  the  case  is  otherwise  when  the  component  vibrations  are  executed  in  different  planes,  for  in  that  case  it  i« 
obvious  that  they  never  can  destroy  each  other  completely  so   as  to  produce  rest.     The  general  case  of  non- 
coincident  planes  of  vibration  is  analyzed  in  Art.  618  ;  and  in  Art.  621  we  see,  that  even  when  each  of  the 
component  vibrations  is   rectilinear,   the  resultant  is  elliptic  ;    so  that  each  molecule  of  the   ether  performs 
continual  gyrations  in  one  direction,  and  never  can  be  totally  quiescent. 

Thus  we  see  that  the  interference  of  rays  similarly  polarized,  and  the  non-interference  of  those  dissimilarly,       9SO. 
is  a  necessary  consequence  of  the  hypothesis  we  are  considering;  and  indeed  was  the  phenomenon  which  first  Analogy 
suggested  it.     It  may  be  familiarly  explained  by  the  analogy  of  our  tended  cord.     Conceive  such  a  cord  to  of  lh« 
have  its  extremity  agitated  at  equal  regular  intervals  with  a  vibratory  motion  performed  in  one  plane,  then  it  ^c 
will  be  thrown,  as  we  have  seen,  into  an  undulatory  curve,  all  lying  in  the  same  plane.     Now,  if  we  superadd 
to  this  motion  another,  similar  and  equal,  but  commencing  exactly  half  an  undulation  later,  it  is  evident  that  the 
direct  motion  every  molecule  would  assume,  in  consequence  of  the  first  system,  will  at  every  instant  be  exactly 
neutralized  by  the  retrograde  motion  it  would  take  in  virtue  of  the  other ;    and,  therefore,  each  molecule  will 
remain  at  rest,  and  the  cord  itself  be  quiescent.     But  if  the  second  system  of  motions  be  performed  in  a  plane 
at  right  angles  to  the  first,  the  effect  will  evidently  only  be  to  distort  the  figure  of  the  cord  into  a  curve  of  double 
curvature,  which,  in  the  general  case,  will  be  an  elliptic  helix,  and  will  pass  into  the  ordinary  circular  one  when 
the  two  component  vibrations  differ  in  phase  by  a  quarter  of  an  undulation,  or  90°.   (See  Art.  627.  Carol.) 

In   this  case  the  extremity  of  the  cord  describes  a  circle  with  a  continuous  motion,  and  this  motion  is  imi-        981. 
tated  by  each  molecule  along  its  whole  length.     It  is  easy  to  make  this  a  matter  of  experiment ;  we  have  only  Case  of  a 
to  hold  in  our  hands  the  end  of  a  long  stretched  cord,  or  grasp  it  firmly  in  any  part  of  its  extent,  and  work  the  ro|atory  o: 
part  held  round  and  round,  with  a  regular  circular  motion,  and  we  shall  see  the  cord  thrown  into  a  helicoidal  ^™on- 
curve,  each  portion  of  which  circulates  in  imitation  of  the  original  source  of  the  motion 

But  experience  shows,  not  merely  that  two  equal  rays  polarized  at  right  angles  do  not  destroy  each  other  for       98-2 
any  assignable  difference  of  origins,  but,  that  whatever  be  this  difference,  the  intensity  of  the  resultant  ray  remains  Resultant  of 
absolutely  the  same.     Now  this  is  also  a  necessary  consequence  of  the  theory  of  transverse  vibrations.     To  show  two  ra.v« 
this,  we  need  only  refer  to  the  expressions  for  A,  B,  C  in  equation  (7,)  Art.  619,  resuming  at  the  same  time  the  °[Ja0r]"eejy 
notation  and  reasoning  of  that  article.     The  intensity  of  the  impression  made  on  the  eye  by  any  ray  being  Investigated 
proportional   to  the  vis  viva,  is  represented  by  the  sum  of  the  several  vires  vivee  in   the  three  rectangular 
directions,  or  by  A2  +  B°  -f-  C«,  that  is,  by 

a*_f-  53  _^_  C2  _|_  att  _|_  j'i  _j_  C"L  _^_  2  a  a',  cos  (p  -  p')  +  2  b  b' .  cos  (q  -  q')  -\- -2  c  c  .  cos  (r  -  /). 

Now  if  we  assume  the  directions  of  the  coordinates  x  and  y  to  be  those  transverse  to  that  of  the  ray,  and  the 
one  in  the  plane  of  polarization  of  one  ray,  the  other  in  that  of  the  other,  at  right  angles  to  it,  and  that  of  z 
in  the  direction  of  the  ray  itself,  we  have 

a'  =  0,     6=0      e  =  0,     d  =  0  ; 
and  therefore  the  above  expression  for  tbe  intensity  becomes 

A°-       B-'-f  O=  a*+  6'2, 
whicn  is  independent  of  p  —  p',  q  —  q\  r  —  r,  the  difference  of  phases,  and  is  equal  to  tile  sum  01  tne  mteu- 


536  L  I  G  H  T. 

I jght.     sities  of  the  separate  rays.     And  we  may  remark,  by  the  way,  that  no  other  supposable  mode  of  vibration  but  ^Part  ^v 
— • "V"""*'  that  in  question,  in  which  c  and  (/,  the  amplitudes  of  vibration  in  the  direction  of  the  ray  vanish,  could  produce     "" "V~" 

the  same  result.     (Fresnel's   Considerations  Theoriques  sitr  la  Polarization  de  la  Liimiere.      Bulletin  de  Ut 

Sociile  Philamatique,  October,  1824.) 
983.  Let  us  now  consider  what  will  happen  when  a  ray  polarized  in  any  plane  is  resolved  into  two  polarized 

Rationale  of  in  any  other  two  planes  at  right  angles   to  each  other,  and  these  again   reduced  to  two  others  also  at  right 
the  rule  for  ...  .  ..  _.-       ... 


Fig.  195.  hypothesis  assumed)  at  right  angles  to  the  plane  of  primitive  polarization.  When  this  ray  is  divided  into  two 
others  oppositely  polarized,  the  vibrations  are  of  course  resolved  into  two  others  performed  in  planes  at  right 
angles  to  each  other.  Let  C  O  and  C  E  be  the  projections  of  these  planes,  which  are  therefore  perpendicular 
to  the  planes  of  polarization  of  the  two  new  rays  respectively.  Suppose  that  at  any  instant  the  molecule  C  of 
the  primitive  ray  is  moving  from  C  in  the  direction  C  P;  then  this  motion,  if  resolved  into  two,  will  give  rise 
to  two  motions,  one  in  the  direction  from  C  towards  O,  the  other /rom  C  towards  E.  If  each  of  these  motions) 
be  again  resolved  into  two,  in  planes  whose  projections  are  S  C  S'  and  TCT',  at  right  angles  to  each  other,  that 
in  the  direction  C  O  will  produce  two  motions,  one  in  the  direction  C  S,  and  the  other  in  the  direction  C  T ; 
and  on  the  other  hand  the  motion  in  the  direction  C  E  will  produce  one  in  the  direction  C  S,  and  the  other  (in 
the  case  of  fig.  195,  a)  in  the  direction  CT'  opposite  to  C  T.  Thus  the  two  resolved  motions  in  the  plane  S  S' 
will  conspire,  but  those  in  the  plane  TT'  will  oppose,  each  other.  In  the  case  of  fig.  195,  b,  the  reverse  will 
happen  ;  the  motions  in  the  plane  T,  T'  conspiring,  and  those  in  the  plane  S  S'  opposing,  each  other.  For  sim- 
plicity of  conception,  however,  we  will  confine  ourselves  to  the  former  case.  If,  now,  we  pass  from  the  consi- 
deration of  the  vibrations  to  that  of  the  rays,  it  will  appear  that  we  have,  in  fact,  resolved  the  original  ray 
polarized  in  the  plane  P  P'  into  two,  polarized  in  planes  perpendicular  respectively  to  C  O  and  C  E  ;  and  these 
again,  finally,  each  into  two,  viz.  one  polarized  in  the  perpendicular  to  S  S',  and  one  perpendicular  to  TT*. 
The  two  portions  polarized  perpendicular  to  S  S'  form  one  ray,  and  those  perpendicular  to  TT'  another;  but  in 
the  former,  the  component  portions  tend  to  strengthen, — in  the  latter,  to  destroy  each  other.  Hence,  if  we 
consider  the  two  former  portions  as  having  a  common  origin,  we  must  regard  the  latter  as  differing  by  hah 
an  undulation. 

984.  Hitherto  we  have  supposed  the  second  resolution  of  the  rays  to  take  place  at  the  same  point  C  in  the  course 
of  the  ray  as  the  first,  but  this  may  not  be  the  case,  and  several  cases  may  be  imagined ;  first,  we  may  suppose 
the  two  portions  into  which  the  ray  is  first  resolved  to  run  on  in  the  same  line  with  equal  velocities  ;  and  after 
describing  any  given  space,  to  be  then  resolved,  at  another  point  C'  (whose  projection  in  the  figure  will  coincide 
with  C)  into  the  final    rays  S  S'  and  TT'.     It  is  evident  that  this  will  make  no  difference  in  ihe  result,  for  the 
phases  in  which  each  ray  arrives  at  C'  will  be  alike  ;  and  after  the  second  resolution  the  conspiring  vibrations 
in    the  direction  S  S'  will  still  he   in  the  same  phase,  and  the  opposing  ones  in  the  plane  TT1  must  still  be 
regarded  as  in  opposite  phases,  i.  e.  as  differing  by  half  an  undulation.     Or,  secondly,  we  may  suppose,  that, 
owing  to  any  cause,  the  two  resolved  rays  do  not  travel  with  equal  velocity,  (as  in  the  case  where  the  reso- 
lution is  performed  by  double  refraction.)     In  this  case,  if  i  be  the  interval  of  retardation  of  the  one  ray  on 
the  other  when  they  arrive  at  C',  i  will  represent  the  difference  of  phases  of  the  two  rays  at  the  instant  of  their 
second  resolution.     Consequently,  when  resolved,  the  final  ray,  whose  vibrations  are  performed  in  S  S',  will  be 
the  mm;  and  that  whose  vibrations  are  performed  in  TT',  the  difference  of  two  rays,  one  in  a  certain  phase  (0), 
the  other  in  the  phase  0  -J-  i  ;   or,  which  is  the    same  thing,  the  former  will   be  the  sum  of  two  components 
in  the  phases  0  and  0  -j-  i ;  the  latter,  the  sum  of  two  in  the  phases   0  and  0  -j-  i  -)-  180°,   so  that  still  the 
difference  of  half  an  undulation  is  to  be  applied.     In  the  case  of  fig.  195,  6,  if  we  pursue  the  same  reasoning, 
it  will  appear  that  this  difference  still  subsists,  but  must  be  applied  conversely,  viz.  to  the  compound  ray  whose 
vibrations  are  performed  in  C  S. 

985.  We  have  here,  then,  the  theoretical  origin  of  the  allowance  of  half  an  undulation,  in  those  cases  where  it  is 
required  to  account  for  the  polarized  tints,  Art.  966,  and  of  the  rule  laid  down  in  Art.  972  for  its  correct  appk- 
cation.     However  arbitrary  the  assumption  may  have  appeared  as  there  presented,  and  however  singular  it 
may  have  seemed  to  make  the  affections  of  a  ray  at  one  point  of  its  course  dependent  on  those  which  it   had 
at  a  former  instant,  we  now  see  that  the  whole  is  a  direct  and  very  simple  consequence  of  the  ordinary  elemen- 
tary rules  for  the  composition  and  resolution  of  motions.     It  is  worthy  of  notice,  that  the  fact  was  ascertained 
before  the  theory  of  transverse  vibrations  was  devised,  so  that  this  theory  has  the  merit  of  affording  an  a  priori 
explanation  of  what  had  previously  all  the  appearance  of  a  mere  gratuitous  hypothesis. 

986.  In  conceiving  the  resolution  of  a  ray  into  two  others  polarized  in  different  planes,  we  may  be  aided  by  the 
Application  analogy  of  the  tended  cord,  which  we  have  before  had  occasion  to  refer  to.     In  fig.  196  let  A  B  be  a  stretched 
ot  the  ana-  cor(j)  branching  at  B  into  the  two  B  C  and  B  D,  making  a  small  angle  with  each  other  at  B,  and  having  either 
stretched  *   e1ua'  or  unequal  tensions.     Suppose  the  plane  in  which  the  two  branches  lie  to  he  (for  illustration's  sake)  hori- 
cord.            zontal,  and  let  the  extremity  A  of  the  single  cord  be  made  to  vibrate  regularly  in  a  vertical  plane ;   or,  at  least 
Fig.  196.     let  the  vibrations  of  the  cord,  before  arriving  at  B,  be  reduced  to  a  vertical  plane  by  means  of  a  small  polished 

vertical  guide  I  K,  against  which  the  cord  shall  press  lightly,  and  on  which  it  may  slide  freely  without  friction. 
Beyond  the  point  of  bifurcation  B,  and  at  such  a  distance  that  the  excursions  of  the  molecule  B  shall  subtend 
no  sensible  angle  from  them,  let  two  other  such  polished  guiding  planes  be  placed,  inclined  at  different  angles 
to  the  horizon,  and  making  a  right  angle  with  each  other.  Suppose  now  B  to  make  any  excursion  from  its 
point  of  rest,  then  were  the  plane  E  F  parallel  to  I  K,  the  molecule  of  the  branch  B  C  contiguous  to  E  F  would 
ilide  on  E  F  through  a  space  equal  to  the  whole  excursion  of  B  ;  but  since  it  is  inclined  to  I  K  at  an  angle 


LIGHT.  537 

Ujht.  (:=  0)  a  part  only  ot  the  motion  of  B  will  be  employed  in  causing  this  molecule  to  glide  on  E  F,  and  the  iVtIV. 
— \— —s  remainder  will  cause  the  cord  to  bend  over  and  press  on  the  obstacle ;  but  by  reason  of  the  minuteness  of  the  v— -^— • 
excursions  of  B,  this  bending  and  the  resistance  of  the  obstacle  and  consequent  loss  of  force  will  be  very  minute 
and  may  be  neglected.  Now,  since  the  pressure  of  the  obstacle  removes  the  cord  from  the  position  it  would 
have  taken  had  no  obstacle  existed,  in  a  direction  perpendicular  to  its  surface,  it  is  easy  to  see  that  the 
amplitude  of  excursion  of  the  contiguous  molecule  on  the  plane  E  V  must  be  to  that  of  B  as  cos  6  to  radius  ; 
and,  therefore,  calling  a  the  amplitude  of  B's  excursions,  a  .  cos  0  will  be  that  of  the  molecule  contiguous 
to  E  F,  and  of  course  that  of  every  subsequent  molecule  of  the  branch  B  C.  Here  the  part  of  B's  motion, 
which  is  perpendicular  to  E  F,  is  not  expended  or  destroyed  in  bending  the  cord  B  C  over  the  obstacle,  but 
remains  in  activity,  and  exerts  itself  on  the  branch  B  D,  causing  it  to  glide  on  the  plane  G  H  ;  and  the  ampli- 
tude of  the  excursions  of  the  molecule  in  contact  with  this  plane  will  in  like  manner  be  represented  by  a  .  cos 
(inclination  of  G  H  to  I  K,)  that  is,  by  a  .  cos  (90  —  0),  or  by  a  .  sin  0.  The  vis  viva,  then,  in  each  of  these 
respective  planes  is  represented  by  a* .  cos  0*  and  a1 .  sin  G*,  whose  sum  is  equal  to  a2,  the  initial  vis  viva. 

If  we  decompose,  in  like  manner,   the  maximum  velocity  a  of  the  ethereal  molecule  C  (fig.  195)   in  the       987. 
direction  C  P  into  two  in  the  respective  directions  C  O  and  C  E,  we  get  a  .  cos  0  and  a  .  sin  0  for  the  elementary  Rationale  of 
velocities;    and  since  the  amplitudes,  Cfeteris  paribu*,  are  as  the  velocities,   (Art.  610,)  the  amplitudes  of  the  Maluf's 
component  rays  will  be  respectively  a  .  cos  0  and  a  .  sin  0  •  and  their  intensities,  which  are  as  the  squares  of  the  ™tens°.r    J 
amplitudes,  (Art.  605,)  will  be  n2 .  cos  (P  and  as .  sin  0".     Now  this  is  the  very  law  propounded  by  Malus  for  the  the  comple- 
intensities  of  the  two  portions  into  which  a  polarized  ray  is  divided  by  double  refraction,  and  of  which  the  mentary 
theory  of  transverse  vibrations  gives,  as  we  see,  a  simple  and  rational  a  priori  account,  thus  raising  it  from  rajs. 
a  mere  empirical  law  to  the  rank  of  a  legitimate  theoretical  deduction. 

We  have  not  done  with  the  analogy  of  the  tended  cord.     What  we  have  shown  in  Art.  986  is  independent  of      988. 
the  tensions  of  the  branches  into  which  the  cord  is  divided,  and  relates  only  to  the  amplitudes  of  their  excur-  Case  of  the 
sions  from  rest  when   thrown  into  vibration.     But  the  velocity  with  which  the  waves,  once  produced,  will  be  two  ™~ 
propagated  along  either  branch  depends  solely  on  its  tension.     Nothing,  however,  prevents  the  tensions  of  the  durations"" 
two  branches  from  being  very  different ;  for,  whatever  be  the  ratio  of  two  forces  applied  in  the  directions   B  C  propagated 
and  B  D,  they  may  be  balanced  at  B  by  a  proper  force  applied  along  any  other  line  as  B  A.     Hence  the  waves  with 
will  run  along  B  C  and  B  D  with  different  velocities.     Similarly,  if  we  conceive,  that  owing  to  the  peculiar  different 
constitution  of  crystallized  bodies,  and  the  relation  of  their  particles  to  the  ether  which  pervades  them,  its  mole-  ve 
cules  are  more  easily  displaced,  or  5'ield  to  a  less  force  in  certain  planes  than  in  others  ;  or,  in  other  words, 
that  it  possesses  different  elasticities  in  different  directions ;   then  will  the  planes  of  polarization  assumed  by 
the  resolved  portions  of  the   rays  determine  the  elasticities  brought  into  action,  and,  by   consequence,    the 
velocities  of  their  propagation.     Now  we  have,  in  a  former  section,  shown  that  the   bending  of  a  ray  at  the 
confines  of  a  medium  depends  essentially  on  its  velocity  within  as  compared  with  that  without,  by  the  analytical 
relations  deduced  from  the  "  principle  of  swiftest  propagation."    A  difference  of  velocity,  therefore,  draws  with 
it,  as  a   necessary  consequence,  a  diversity  t,f  path  ;    and  thus  the  bifurcation,  or  double  refraction  of  a  ray 
incident    on  a   crystallized   surface,    presents  no    longer    any  difficulty   in  theory,   provided    we    can    find    an 
adequate  reason  for  the  resolution  of  its  vibrations  into  two  determinate  planes  at  the  moment  of  its  entering 
Ihe  crystal. 

Let  us  take  (with  M.  Fresnel,  Annales  de  Chimie,  xvii.   p.  179  et  seq.)  the  case  of  a  crystal  with  one  axis. 
We  may  regard  this,  or  rather  the  ether  within  it,  modified  in  its  action  by  the  molecular  forces  of  the  crystal,  Expla"a'j<"i 
as  an  elastic  medium  in  which  the  elasticity  in  a  direction  perpendicular  to  the  axis  is  different  from  that  in  a  nomen«  Of 
direction  parallel  to  it,  that  is,  in  which  the  molecules  are  more  easily  compressible  in  the  one  than  in  the  other  double 
direction  ;  but,  equally  so  in  all  directions  perpendicularly  to  the  axis,  on  whatever  side  the  pressure  be  applied,  refraction 
To  aid  our  conceptions  in  imagining  such  a  property,  we  may  assimilate  an  uniformly  elastic  medium  to  an  in.  cr>'stals 
assemblage  of  thin,  elastic,  hollow,  spherical  shells  in  contact;  and  such  a  medium  as  we  are  considering,  to  a  wl. 
similar  assemblage  of  oblate  or  prolate  hollow  ellipsoids,  arranged  with  all  their  axes  parallel  to  one  common  direc- 
tion, which  is  that  of  the  axis  of  the  crystal.*    It  is  evident  that  the  resistance  of  the  spherical  assemblage  to  pressure 
must  be  the  same  in  all  directions,  but  that  of  the  spheroidal  must  differ  according  as  the  pressure  is  applied 
perpendicularly  or  parallel  to  the  axis.     Thus,  it  is  easy  to  crush  an  egg  by  a  force  applied  in  the  direction 
of  its  shorter  diameter,  which  will  yet  sustain  a  violent  pressure  applied  at  the  extremities  of  its  longer.     It  is, 
moreover,  evident,  if  any  molecule  of  such  an  assemblage  were  disturbed,  so  as  to  throw  it  into  vibration,  that, 
provided  always  the  amplitude  of  its  excursions  were  extremely  small  compared  to  the  diameter  of  each  ellipsoid, 
the  immediate  tendency  of  the  vibration  will  be  to  communicate  motion  to  two  strata  only  of  molecules,  viz.  that 
in  which  the  axis  and  equator  of  the  disturbed  molecule  lie  respectively,  since  it  is  only  at  the  poles  and  equator 
that  they  touch,  and  therefore  only  through  these  points  that  motion   can  be  communicated  from  one  to  the 
other.     Consequently,  any  motion  communicated   to  a  molecule  of  such  a  mass  could  only  be  propagated  by 
vibrations  performed  in   planes  parallel  and  perpendicular  to  the  axis.     Hence,  if  a  vibratory  motion  in  any 
plane  be  propagated  into  such  an  assemblage  of  particles  from  without,  it  will  immediately,  on  its  reaching  it, 

•  The  idea  of  spheroidal  molfculei  in  Iceland  spar  suggested  itself  to  Huygens  (raiher  fancifully,  perhaps)  as  a  means  by  winch  spne- 
roidal  undulations  might  be  propagated  through  it,  (0/>.  Reliq.  torn.  i.  Tructiitus  de  Lumiue,  p.  70,  cited  by  Wollaston,  Phil.  Tram.  cm. 
p.  58  ;)  and  the  last-named  eminent  1'hilosopher,  in  the  Bakerian  Lecti  ,e  for  1813,  has  most  ingeniously  shown  how  such  molecules  may  be 
combined  to  build  up  crystals,  having  the  primitive  forms  and  cleavage-  of  acuie  and  obtuse  rhomboids.  It  is  true,  that  in  all  this  there  is 
much  hypothesis;  and  it  should  be  observed,  too,  that  the  crystallogriphic  stiucture  would  require  oblate  spheroids,  where  in  the  text  we 
have  employed  prelate,  and  vice  versa*.  But  we  intend  there  only  &n  analogy,  not  a  theory.  It  would  be  easy  to  devise  hypothetical 
modes  of  action  where  these  forms  might  be  reversed  if  needful. 

VOi:,        IV  4      A 


538  LIGHT. 

l.iijht.      he  resolved  into  two,  in  the  planes   above  named  ;    and  these,  hy  reason  of  the   different  elasticities,  will   be 
^-~v-— '  propagated  with  different  velocities. 

990.  The  reader  must  not  suppose  that  this  is  intended  for  an  account  of  the  real  mechanism  of  crystallized  bodies. 
Bifurcation    jt  js  merely  intended  to  show  that  it  is  not  absurd,  or  contradictory  to  sound  mechanical  principles,  to  assume 
of  me  re-      ^^  ^^  ^y  ^e  tnejr  constitution,  that  vibrations  can  only  be  propagated  through  them  by  molecular  excur- 

s'ons  executed  in   planes   parallel  and  perpendicular  to   their  axes.     Assuming,  then,  that  such  is  the  case,  the 
vibrations  of  a  ray  incident  on  such  a  crystal  will  be  resolved  into  two,  performed  in  these  respective  planes,  and 
their  velocities  of  propagation  being  different,  the  rays  so  arising  will  follow  different  courses  when  bent  by 
refraction.     Let  us  first  consider  that  whose  vibrations  are  executed  in  planes  perpendicular  to  the  axis.     Since 
the  crystal  is  symmetrical  with  respect  to  its  axis,  and  equally  elastic  in  all  directions  perpendicular  to  it,  the 
Properties     velocity  of  propagation  of  this  portion  will  be  the  same  in  all  directions.    Its  index  of  refraction,  therefore,  will  be 
oftheordi-   constant,  and  the   refraction  of  this  portion  will  follow  the  ordinary  law.     Moreover,  its  plane  of  polarization 
nary  ray.       being  that  perpendicular  to  which  the  vibrations  are  performed,  will  necessarily  pass  through  the  axis,  in  which 
respect  it  also  agrees  with  the  ordinary  ray,  as  actually  observed. 

991.  The  extraordinary  ray  arises  from  the  other  resolved  portion  of  the  original  vibration,  which  is  performed  in  a 
Properties     plane  parallel  to  the  axis.       By    the  principle  of  transverse  vibrations,  it  is  also  performed  in  a  plane   per- 
oltheextra-  pendicnlar  to  the  ray.     If,  then,  we  suppose  a  plane  to  pass  through  the  extraordinary  ray  and  the  axis,  it  will 
'^'lained'15'  cut  a  P'ane  perpendicular  to  the  ray  in  a  straight  line,  which  will  be  the  direction  of  the  vibratory  motion.     This 

direction,  then,  is  inclined  to  the  axis  in  an  angle  equal  to  the  complement  of  that  made  by  the  extraordinary 
ray  with  the  latter  line,  and  therefore,  when  the  extraordinary  ray  is  parallel  to  the  axis,  the  line  of  vibration  is 
perpendicular  to  it,  and  vice  versa.  In  the  former  case,  the  elastic  force  resisting  the  displacement  of  the  mole- 
cules is  the  same  as  in  the  case  of  the  ordinary  ray,  and  therefore  the  velocities  of  both  rays  are  equal,  and 
their  directions  coincide,  and  thus  along  the  axis  there  is  no  separation  of  the  rays.  In  the  latter,  the  elasticity 
is  that  parallel  to  the  axis,  and  therefore  differing  from  the  former  by  the  greatest  possible  quantity.  Here,  then, 
the  difference  of  velocities,  and  therefore  of  directions  is  at  its  maximum.  In  intermediate  situations  of  the 
extraordinary  ray,  the  elasticity  developed  is  intermediate,  and  therefore  also  the  velocity  and  double  refraction. 
Thus  we  see,  that  according  to  this  doctrine  the  difference  of  velocities,  and  consequent  separation  of  the  pencils 
should  be  nothing  in  the  axis,  and  go  on  increasing  till  the  extraordinary  ray  is  at  right  angles  to  it,  which  is 
conformable  to  fact.  Lastly,  the  plane  of  polarization  of  the  extraordinary  ray  being  at  right  angles  to  the 
plane  of  vibration,  must  also  be  at  right  angles  to  a  plane  passing  through  the  ray  and  the  axis,  which  is  also 
conformable  to  fact. 

:>92  The  theory  of  M.  Fresnel  gives  then,  as  we  see,  at  least  a  plausible  account  of  the  phenomena  of  double 

refraction  in  the  case  of  uniaxal  crystals  ;  and  when  we  consider  the  profound  mystery  which,  on  every  other 
hypothesis,  was  admitted  to  hang  over  this  part  of  tha  subject,  we  must  allow  that  this  is  a  great  and  impor- 
tant step.  But  the  same  principles  are  equally  applicable  to  biaxal  crystals  with  proper  modifications,  and 
(which  is  a  strong  argument  for  their  reality)  lead,  when  so  applied,  to  conclusions  which,  though  totally  at 
variance  with  all  that  had  been  taken  for  granted  before,  on  the  grounds  of  imperfect  analogy  and  insufficient 
experiment,  have  been  since  verified  by  accurate  and  careful  experiments,  and  have  thus  opened  a  new 
and  curious  field  of  optical  inquiry  Nothing  stronger  can  be  said  in  favour  of  an  hypothesis,  than  that  it 
enables  us  to  anticipate  the  results  of  experiment,  and  to  predict  facts  opposed  to  received  notions,  and  mis- 
taken or  imperfect  experience. 

993  But  before  we  enter  on  this,  it  may  be  right  to  show  how  the  phenomenon  on  which  the  theory  of  movable 
Sxplana-      polarization  is  founded,  is  accounted  for  by  the  doctrine  of  transverse  vibrations.     According  to  this  theory,  as 
tion  of  the     soon  as  a  polarized  ray  enters  a  crystal,  it  commences  a  series  of  alternate  assumptions  of  one  or  other  of 
phenomena   two  planes  of  polarization,  in  the  azimuths  0°  and  2  i,  i  being  the  inclination  of  the  principal  section  to  the 

plane  of  primitive  polarization  :  the  plane  assumed  being  in  azimuth  0°,  when  the  thickness  traversed  is  such 
'  as  to  render  the  interval  of  retardation  of  the  ordinary  on  the  extraordinary  ray  0,  or  any  whole  number  ot 
undulations,   and   in   azimuth  2  i  when  it    is  any  whole  odd  number  of   semi-undulations.      Suppose  a  ray 
polarized  in  the  azimuth  0  to  be  incident  perpendicularly  on  a  crystallized  lamina,  having  its  principal  sec- 
tion in  the  azimuth  i,  then  it  will  be  resolved  into  two,  the  vibrations  of  which  are  respectively  performed  in  the 
principal  section,  and  perpendicular  to  it.     Consequently,  if  we  represent  by  unity  the  amplitude  of  the  original 
Case  of        vibrations,  those  of  the  two  resolved  vibrations  will  be  equal  respectively  to  sin  i  and  cos  i.     Now,  the  thick- 
compleie      ness  of  the  plate  being  first  supposed  such  as  to  render  the  interval  of  retardation  an  exact  number  of  undula- 
accordance.  tions,  these  rays  will  emerge  from  the  lamina  in  exact  accordance,  and  being  parallel,  the  systems  of  waves  of 
which  they  consist  will  run  on  together.     Being  polarized,  however,  in  opposite  planes  they  will  neither  destroy 
each  other,  nor  produce  a  compound   ray  equal   to   their   sum,  but  their   resultant  must  be  determined  as  in 
Art.  623.     For  we  have  here  the  case  of  rectilinear  vibrations,  in   complete  accordance,  of  given   amplitudes, 
and  making  a  given  angle  (90°,)  so  that  the  result  there  obtained  is  immediately  applicable  to  this  case,  and 
the  resultant  vibration  will  be,  first,  rectilinear,  so  that  the  compound  ray  will  appear  wholly  polarized  in  one 
plane:  and,  secondly,  its   amplitude  will  be,  both  in  quantity  and   direction,   the  diagonal  of  a  parallelogram 
whose  sides  are  the  amplitudes  of  the  component  vibrations.     Consequently,   it  will  be  identical  with  that  by 
whose  resolution  these  were  produced,  and  therefore  the  resultant,  or  emergent  compound  ray  will  be,  in  respect 

994  both  of  its  polarization  and  intensity,  precisely  similar  to  the  original  incident  one. 

Fig.'  197.  When  the  difference  of  paths  within  the  crystal  is  an  exact  odd  multiple  of  half  an  undulation,  the  waves  at 

Case  of  their  egress  from  the  posterior  surface  will  be  in  complete  discordance.  But  their  resultant  may  still  be 
o.impletc  determined  by  the  same  rule,  regarding  either  of  the  rays  as  negative,  i.  e.  as  having  its  vibrations  executed 
;„  the  Oj,pOSjte  direction.  F'  T  suppose  the  molecule  C  moving  in  the  direction  C  P,  with  the  velocity  C  P 


LIGHT.  539 

Light,     (fig.    197)  at  the  entry  of  the  ray,   then  the  resolved  velocities  in  the  planes    C  O  and  C  E  will  be  repre-      Part  TV. 
— v — :'   sented  in  quantity  and  direction  by  C  O  and  C  E.     But  at  their  egress,  the  vibrations  in  the  direction  C  E  v— v— ' 
having  gained  or  lost  a  half  undulation  on  those  in  C  O,  if  C  O  represent  the  quantity  and  direction  of  motion 
of  the  molecule  C  in  that  plane,  C  E'  equal  and  opposite  to  C  E  will  represent  its  motion  in  the  other  plane, 
and  this,  combined  with  C  O  will  compose,  not  the  original  motion  C  P,  as  in  the  former  case,  but  C  Q,  making 
an  equal  angle  with  C  O  on  the  other  side.     The  resultant  ray,  then,  instead  of  being  polarized  in  the  plane 
of  the  incident  one,   (i.  e.  perpendicular  to  CP)  will  be  polarized  in  a  plane  perpendicular  to  C  Q,  makinf 
an  angle  equal  to  P  C  Q  (=  2  P  C  O  =  2  i)  with  CO. 

When  the  difference  of  routes  is  neither  an  exact  number  of  whole,  or  half  undulations,  the  vibrations  of       995 
the  resultant  ray  (by  Art.  621)  will  no  longer  be  rectilinear,  but  elliptic  ;   and  in  the  particular  case  when  the 
interval  of  retardation  is  a  quarter  or  an  odd  number  of  quarter  undulations,  it  will  be  circular.      In  this  case, 
the  emergent  ray,  varying  its  plane  of  vibration  every  instant,   will  appear   wholly  depolarized,  so  as  to  give 
two  equal  images  by  double  refraction  in  all  positions  of  the  analysing  prism. 

These  several  consequences  may  be  rendered  strikingly  evident  by  a  delicate  and  curious  experiment  related       996. 
by  M.  Arago.     Let  a  polarized  pencil,  emanating  from  a  single  radiant  point,  be  incident  on  a  double  rhomboid  Experiment 
of  Iceland  spar,  composed  of  two  halves  of  one  and  the  same  rhomboid,  superposed  so  as  to  have  their  principal  exnlbmn8  1 
sections  at  right  angles  to  each  other.     Then  the  emergent  rays  will  emanate  as  if  from  two  points  (see  Art.  879)  e^^f'" 
near  each  other,  and  polarized  in  opposite  planes.     Let  these  two  cones  of  rays  be  received  on  an  emeried  glass,  interference 
or  in  the  focus  of  an  eye-lens,  so  that  the  glass  or  field  of  view  shall  be  illuminated  at  once  by  the  light  of  both, 
which  being  oppositely  polarized  will  exhibit  no  fringes  or  coloured  phenomena,  but  merely  a  uniform  illumina- 
tion ;   and  let  all  the  light  but  that  which  falls  on  a  single  very  small  point  of  the  field  of  view  be  stopped  by  a 
plate  of  metal,  with  a  small  hole  in  it,  so  as  to  allow  of  examining  the  state  of  polarization  of  the  compound  ray 
illuminating  this  point,  separately  from  all  the  rest.     Then  it  will  be  seen,  on  analysing  its  light  by  a  tourmaline 
or  double  refracting  prism,  that,  when  the  spot  examined  is  distant  from  both  radiants  by  the  sa'me  number  of 
undulations,  although  in  fact  composed  of  two  rays  oppositely  polarized,  (as  may  be  proved  by  stopping  one  of 
them,  and  examining  the  other  singly,)  yet  it  presents  the  phenomenon  of  a  ray  completely  polarized  in  one 
plane,  which  is  neither  that  of  the  one  or  the  other  of  its  component  rays,  but  the  original  plane  of  polarization  of 
the  incident  light.    Suppose  now,  by  a  fine  screw  we  shift  gradually  the  place  of  the.  metal  plate  so  as  to  bring  the 
hole  a  little  to  one  or  the  other  side  of  its  former  place.     The  ray  which  illuminates  it  will  appear  to  lose  its  pola- 
rized character  as  the  motion  of  the  plate  proceeds,  and  at  length  will  offer  no  trace  of  polarization;  continuing  the 
motion,  and  bringing  in  succession  other  points  of  the  field  of  view  under  examination,  the  light  which  passes 
through  the  hole  will  again  appear  polarized,  at  first  partially,  and  at  length  totally ;  not,  however,  as  before,  in  the 
plane  of  primitive  polarization,  but  in  a  plane  making  with  it  twice  the  angle  included  between  it  and  the  principal 
section  of  the  first  rhomboid,  and  so  on  alternately.     Thus  we  are  presented  with  the  singular  phenomenon  of  two 
rays  polarized  in  planes  at  right  angles,  which  produce  by  their  concourse  a  ray  either  wholly  polarized  in  one  or 
the  other  of  two  planes,  or  not  polarized  at  all,  according  to  the  difference  of  routes  of  the  rays  before  their  union. 

In  1821,  M.  Fresnel  presented  to  the  Academy  of  .Sciences  of  Paris  a  Memoir,  containing  the  general  appli-       997. 
cation  of  the  principle  of  transverse  vibrations  to   the   phenomena  of  double  refrae'.ion  and  polarization  as  Fresnel's 
exhibited  in  biaxal  crystals,  which  was  read  in  November  of  that  year.     A  brief  report  on  the  experimental  ?enera' 
parts  of  this  Memoir  by  the  Committee  of  the  Academy  appointed  to  examine  it,  about  half  a  dozen  pages,  was  jhe°i|?r  °^ 
published  in  the  Annales  tie  Chitnie,  vol.  xx.  p.  337,  recommending  it  to  be  printed  as  speedily  as  possible  in  refraction 
the  collection    of  the  Mfrnoires  des  Savans   Strangers.     We  are  sorry  to  observe,  that  this  recommendation 
has  not  yet  been  acted  upon,  and  that  this  important  Memoir,  to  the  regret  and  disappointment  of  men  of  science 
throughout  Europe,   remains  yet  unpublished  ;  though  we  trust  (from   the  activity  recently  displayed  by  the 
Academy  in   the   publication  of  their  Memoirs  in  arrear)   this  will  not  long  continue  to  be  the  case.  *     An 
abstract  by  the  author  himself,  which  appeared    in  the  Bulletin  de  la  Societe  Philomatique  of  1822,  and  was 
subsequently  reprinted  in  the  Annales  de  Cfiimie,  1825,  enables  us,   however,  to  present  a   sketch,  though  an 
imperfect  one,  of  its  contents,  supplying  to  the  best  of  our  ability  the  demonstration  of  the  fundamental  pro- 
positions, and  reaping  a  melancholy  gratification  from  the  inadequate  tribute,  which,  in  thus  introducing  for 
the  first  time  to  the  English  reader  a  knowledge  of  these  profound  and  interesting  researches,  we  are  enabled 
to  pay  to   departed  merit.     His  saltern  accumiilem  donis — et  fungar  inani  munere.     For  even  at  the  moment 
when  we  are  recording  his  discoveries,  their  author  has  been  snatched  from  science  in  the  midst  of  his  brilliant 
career  by  a  premature   death,   like  his  hardly  less  illustrious  contemporary,  Fraunhofer,  the  early  victim  of  a 
weakly  constitution  and  emaciated  frame,  unfit  receptacles  for  minds  so  powerful  and  active. 

M.  Fresnel   assumes,  as  a  postulatum,  that  the  displacement  of  a  molecule  of  the   vibrating  medium  in  a       998. 
crystallized  body  (whether  that  medium  be  the  ether,  or  the  crystal  itself,   or  both   together,  in  virtue  of  some  General  ex- 
mutual  action  exercised  by  them  on  each  other,)  is  resisted  by  different  elastic  forces,  according  to  the  different  Pressron .*** 
directions  in  which  the  displacement  takes  place.     Now  it  is  easy  to  conceive,  that  in  general  the  resultant  of  *rC^S0'[ca 

medium  in 

*  This  delay  has  been  productive  of  a  singular  consequence,  which  will  suffice  to  show  the  small  degree  of  publicity  which  labours,  even  vestigated. 
the  most  important,  can  acquire  by  the  circulation  of  such  notices  as  those  mentioned  in  the  text.  So  lately  as  December  1826,  the 
Imperial  Academy  of  Sciences  of  Petersburg  proposed  as  one  of  their  prize  questions  for  the  two  years  1827  and  1828,  the  following,  "  To 
deliver  the  optical  system  of  waves  from  all  the  objections  which  have  (as  it  appears)  with  justice,  been  urged  against  it,  and  to  apply  it  to 
the  polarization  and  double  refraction  of  light"  In  the  programma  announcing  this  prize,  M.  Fresnel's  researches  on  the  subject  are  not 
alluded  to  (though  his  Memoir  on  Diffraction  is  noticed,)  and  it  is  fair  to  conclude,  were  not  then  known  to  the  Academy.  Precisely  one 
month  before  the  publication  of  this  programma,  the  Royal  Society  of  London  awarded  their  Rumford  Medal  to  M.  Fresnel,  "  for  his  appli- 
cation of  the  undulatory  theory  to  the  phenomena  of  polarized  light,  and  for  his  important  experimental  researches  and  discoveries  in  physical 
optics."  Our  readers  will  be  gratified  to  know,  that  the  valuable  mark  of  this  high  distinction  reacped  him  a  few  days  before  his  death 

4  A  2 


540  LIGHT. 

L.g»t.  all  the  molecular  forces  which  act  on  a  displaced  molecule,  is  not  necessarily  parallel  to  the  direction  of  its  dis-  Part  IV. 
>—  x—  -»-  placements  when  the  partial  forces  are  unsymmetrically  related  to  this  direction,  but  the  proposition  may  be  v—  •  -v— 
demonstrated  a  priori,  as  follows.  Suppose  three  coordinates  x,  y,  and  z,  to  represent  the  partial  displacements 
of  any  molecule  M  in  their  respective  directions,  and  r  (=  v  x"1  -j-  y*  -f-  z9)  the  total  displacement,  making  angles 
«.  /3,  7,  respectively  with  the  axes  of  the  x,  y,  z,  so  that  x  =  r  .  cos  a,  y  =  r  .  cos  /3,  z  —  r  .  cos  7.  Now,  since  in 
this  theory  we  assume  that  the  displacements  of  the  molecules  are  infinitely,  or  at  least  extremely  small  com- 
pared with  the  distances  of  the  molecules  inter  se,  it  is  evident  that  whatever  be  the  law  of  molecular  action,  the 
force  resulting  from  any  displacement  must  (cceteris  paribus)  be  proportional  to  the  linear  magnitude  of  that  dis- 
placement, and  can,  therefore,  be  only  of  the  form  r  .  0,  where  0  is  some  unknown  function  of  the  angles  a,  ft,  7, 
Principle  of  or  their  cosines.  And,  moreover,  since  such  infinitely  small  displacements,  in  whatever  direction  made,  neither 
partial  dis-  alter  the  angular  position,  nor  distance  of  the  displaced  molecule  among  the  rest,  by  any  sensible  quantity,  all 
placements.  tne;r  forces  will  act  on  it  in  its  displaced  position  in  the  same  manner  as  before.  Hence  the  total  force  deve- 
loped by  the  simultaneous  displacements  x,  y,  z,  or  by  the  single  displacement  r  must  be  equivalent  to  (or  the 
statical  resultant  of)  the  three  which  would  be  developed  independently  by  the  several  partial  displacements 
x,  y,  z.  Now  the  force  originating  in  the  partial  displacement  x  alone  will  result  from  r  0  by  making  r  =  x  and 
0  equa'  *o  a,  where  a  is  the  same  function  of  1,  0,  0,  that  0  is  of  cos  c,  cos  f),  cos  7.  a  therefore  is  a  con- 
stant depending  only  on  the  position  of  the  axes  of  the  x,  y,  z  with  respect  to  the  molecules  of  the  crystal. 
And  when  this  partial  force  =  a  x  is  resolved  into  the  directions  of  these  several  axes,  since  its  direction  (what- 
ever it  be)  is  determinate,  the  resolved  portions  can  only  be  of  the  form  Ax,  A'  x,  A"  x,  where  A,  A',  A"  are  in 
like  manner  dependent  only  on  the  position  of  the  coordinates  x,  y,  z  with  respect  to  the  molecules,  and  not  at 
all  on  a,  /3,  7,  which  are  arbitrary,  and  where  Ae  -(-  A'2  -f-  A"'2  =  a2.  The  same  being  true  of  the  partial  forces 
brought  into  play  by  the  displacements  y  and  z,  it  follows  that  the  total  force  arising  from  the  displacement  r 
must  be  the  resultant  of  the  three  forces 

f=Ax  +  By  +  Cz,    f'  =  A'x  +  B'y  +  C'z,    f"  =  A"  x  -j-  B"  y  +  C"  z, 

respectively  parallel  to  the  axes  of  the  x,  y,  z,  where  the  coefficients  are  independent  of  a,  /3,  7,  and  where,  in  like 
manner,  B*  +  B*  +  B''2  =  6;,  C8  -j-  C'2  -f-  C"2  =  c2.  But  we  have  x  =  r  .  cos  a,  y  =  r  .  cos  ft,  z  =  r  .  cos  7,  so 
that  if  we  put 

f  =.  r    {  A  .  cos  a  -f-  B  .  cos  ft  +  C  .  cos  7  }  , 

f  =r    {  A'  .  cos  a  -f  B'  .  cos  ft  +  C'  .  cos  7  }  , 
/"  =  r    {  A",  cos  a  +  B".  cos  /3  -f-  C".  cos  7  }  , 

the  resultant  of  f,f',f"  will  be  the  force  urging  the  displaced  molecule. 

099.  New  these  forces  acting  in  the  directions  of  the  coordinates  may  each  be  decomposed  into  two,  one  in  the 

Expression    direction  of  the  displacement  r,  and  the  other  at  right  angles  to  it  in  the  planes  respectively  of  r  and  x,  r  and  y, 
of  the  elas-  r  and  z,  the  sum  of  the  former  will  be 

ticity  in  any  „          ,,  ,,,  ,,,, 

assigned     '  F  =  /.  COS  a  +  /'  .  COS  ^  +  /"  .  COS  7, 

"i"1'      which  is  the  whole  force  tending  to  urge  the  displaced  molecule  directly  to  its  position  of  equilibrium.     The  latter 

will  be  respectively  equal  to/,  sin  «,  /'  .  sin  ft,  and/"  .  sin  7  ;   but  as  they  act,  although  in  one  plane,  yet  not  in 

the'direc-'0  *'le  same  direction,  they  will  not  destroy  each  other,  unless  they  be  to  each  other  in   the  ratio  of  the  sines  of  the 

tion  of  dis-    angles  they  make  with  each  other's  direction.     But  it  is  evident,  that  since  a,  /3,  7  are  arbitrary,  this  condition 

placement,    cannot  hold  good  in  general,  because  it  furnishes  two  equations,  which,  taken  in  conjunction  with  the  relation 

cos  a*  _j-  Cos  /?  +  cos  7*  =  1,  suffice   to  determine   a,  ft,  7.     Hence  it  follows,  that  the,  displaced  molecule  is, 

except  in  certain  cases,  urged  by  the.  elastic  forces  of  the  medium  obliquely  to  the  direction  of  its  displacement. 

1000  Mr.  Fresnel  next  goes  on  to  observe,  that  in  general  every  elastic  medium  has  three  rectangular  axes,  in  the 

Axes  of  '      direction  of  which,   if  a  molecule  be  displaced,   the  resultant  of  the  molecular  forces  urging  it  will  act  in  the 

elasticity      direction  of  its  displacement.     These  are  the  excepted  cases  just  alluded  to,  and  to  the  axes  possessing  this  pro- 

defined  and  pertv     (which  he  regards   as   the    true   fundamental   axes   uf  the   crystal,)    he  gives    the  name   of  Axes  of 

investigated 


To  demonstrate  this  proposition  we  must  observe,  that,  by  mechanics,  in  order  that  the  resultant  of  three 
rectangular  forces//',/"  shall  make  angles  «,/3,  7  with  their  three  directions,  and  therefore  be  coincident  in  direc- 
tion with  r,  they  must  be  to  each  other  in  the  ratio  of  the  cosines  of  these  angles,  and  therefore  we  must  have  the 
following  equations  expressive  of  this  condition, 

/         cos  a      /     _  cos  a      /'  _  cos  /3 
/'  "~  cos/3  '   f"  ~  cos  7  '   f'  ~  cos  7' 

These  three  equations  are  in  general  equivalent  to  two  only,  but  when  combined  with  the  equation 
cos  a*  _)_  cos  fp  -j-  cos  7s  =  1  resulting  from  the  geometrical  conditions  of  the  case,  they  suffice  to  determine 
a,  ft,  and  7  ;  and  if  we  put  u,  v,  w  for  the  cosines  of  these  angles,  furnish  the  following  system  of  equations 
which  every  axis  of  elasticity  must  satisfy. 

(Au+Bv  +  Cw)v-  (A'u  +  B'v  +  C'w)u, 
(Au  +  Bv  +  Cw)w=(A"u  +  B"v  +  C"w)u 
(A'u  -f  B'v+'.C'w)  w  =  (A"u  -f  B"»  +  C"  w)  v 

M2  +  V*  -f  W2  =   1. 


L  I  G  H  T.  541 

Light.          Suppose  by  elimination  we  have  derived  from  these  equations  the  position  of  one  axis  of  elasticity,  then  it  will     Part  IV. 
••— v——  follow  of  necessity,  that  two  others  must  exist,  at  right  angles  to  it  and  to  each  other.     To  prove  this,  we  'v-^-v—^ 
must  consider  the  connection  between  the  partial  forces  developed  by  any  displacement  of  the  molecule  M,  and      1001. 
the  molecular  attractions  and  repulsions  of  the  medium.    Let  0  be  the  action  of  any  molecule  d  m  on  M,  which  we  Three  exist 
suppose  to  be  exerted  in  the  direction  of  their  line  of  junction,  and  to  be  a  function  of  their  mutual  distance  p.  m  any  cr 
Then,  if  we  suppose  M  displaced  by  any  arbitrary  quantities  S  x,  Sy,  &  z  (infinitely  small  in  comparison  with  p)  \D^"IO  ' 
in  the  direction  of  the  three  coordinates,  we  have  each  other. 


drj)  x  y  z 

and  putting  0  =  —  ,     and  —  =  cos  X,     —  —  cos  /»,     —  =  cos  v, 

dp  p  p  p 

we  have  60  =  0'.       £  j  .  cos  X  +  J  y  •  cos  ft  +  5  p  .  cos  v  }  . 

Consequently,  since  the  force  of  the  molecule  d  m,  resolved  into  the  directions  of  the  coordinates,  is  respectively 
equal  to 

(0  +  50)  dm.—  ,    (0  +  S0)dm.-^.,     and(0+X0)dwi.—  , 
P  P  P 

the  sum  of  all  these  throughout  the  medium  will  be  the  total  action  on  M  ;  but  since  in  the  original  position  of 
the  molecule  M  it  is  in  equilibrio,  we  have 

/0dm.—  =0,    /0dm.  2-  =0,  and/0  dm.  —  =  0, 

so  that  the  whole  action  of  the  medium  on  M  in  its  displaced  situation  will  be,  in  the  three  directions  General 

<T  r  y  ^9  relation 

/  —  d  m  .  6  0,          /   —  dm.  50,  /   —  dm.£0;  between  tne 

f  P  f  partial  elas- 

that  is,  in  the  direction  of  the  x,  ticities. 


'  d  m  .  {  cos  \»  X  *  +  cos  ft  .  5  y  +  cos  v*  .  S  g  }  ; 

$x,  &y,  $  x,  are  the  partial  displacements  of  M  in  the  directions  of  the  coordinates,  and  are,  therefore,  the  same 
we  denoted  in  Art.  998  by  x,  y,  z.  Restoring  these  denominations,  we  see  that,  on  this  hypothesis,  (the  most 
natural  which  can  be  formed  respecting  the  mode  of  molecular  action)  the  coefficients  A,  B,  C,  can  be  no  other 
than  the  following, 

A=/0'  dm  .  cosX8,     B  —f  0'  dm.  cos  A,  .  cos  ,u,     C  =  /0'  dm  .  cos  X.  cos  v; 
and  by  similar  reasoning  we  find 

A'  =/0'dm.  cos  \  .cos/<,     B'  =/  0'  d  m  .  cos  /**,  C1  =  /0'dm  .  cos/t  .  cos  v; 

A"  =  f  0'  d  m  .  cos  X  .  cos  v,     B'  =  f  0'  a  m  .  cos  p,  .  cos  v,     C''  =  f  0'  d  m  .  cos  v8  ; 
and,  consequently,  the  following  relations  must  necessarily  subsist  between  these  coefficients 

B  =  A',     C  =  A",     C'  =  B". 

^  This  premised,  suppose  we  have  determined  one  axis  of  elasticity  of  the  medium  by  the  foregoing  equations.      1002 
Since  the  positions  of  the  axes  of  the  coordinates  are  arbitrary,  we  are  at  liberty  to  suppose  that  of  the  x  coin- 
cident with   the   axis  so  determined,  which   renders  A'  =  A"  —  0,  and  consequently  B  =  0  and   C  =  0,   and 
B"  =  C',  because  the  relations  above  demonstrated  are  general  and  independent  of  any  particular  situation  of 
the  axes.     The  equations  of  Art.  1000  then  become  One 

A  «  v  =  (B'  v  +  C'  w)  u,     \uw=  (B"  v  +  C"  w)  v, 


(B'u  +  C'w)w  =  (C'u  +  C"w)c,     WS  +  KS  +  w*=  1.  of  the  other 

determined, 

Now  if  we  put  u  =  0,  or  a  =  90°,  the  two  former  of  these  are  satisfied  without  any  i  elation  supposed  between 
v  and  w,  so  that  if  we  determine  these  from  the  two  latter  only,  the  whole  system  will  be  satisfied.  These 
(making  u  =  0)  give  at  once  by  elimination 


1 

where  m  =  —  -  —  ;..     Now  since  m*  is  necessarily  positive,  4  wia  +  1  is  so,  and  is  >  1  ;  therefore  —  = 

^  4  m>  +  1 

is  real  and  <  1,  consequently  w1  and  *>!  are  both  positive,  and  therefore  v  and  w  both  real,  and  less  than  unity 
Hence  it  follows,  that  there  are  necessarily  two  axes  at  right  angles  to  the  x  which  satisfy  the  conditions  of  axes' 
of  elasticity,  and  the  opposite  signs  of  v  and  w  show  that  they  are  at  right  angles  to  each  other. 

For  simplicity,  therefore,  we  will  in  future  suppose  the  directions  of  the  coordinates  to  bs  coincident  with  those 
of  the  axes  of  elasticity,  so  as  to  make 


542  LIGHT. 

Ligh".  A  *=  a,  A'  =  A"  =  0  ;     B'  =  A,  B  =  B"  =  0  ;     C"  =  c,  C  =  C'  =  f) ;  Part  IV. 

~~^<s~~~'  then  we  have  by  Art.  998  for  the  partial  forces,  ^— -,,— . 

f  —  a  x  =  a  r  .  cos  a,      /'  =  b  y  =  >>  r  .  cos  ft,      f"  —  c  z  =  c  r  .  cos  -/, 
and  by  999, 

F  ~  r  {  a  .  cos  n?  +  6  .  cos  ft1  -)-  c .  cos  7*  } 

for  the  whole  force  urging  the  molecule  M  in  the  direction  of  the  r,  generally  assumed,  in  which  it  will  be 
observed  that 

a  =  f<t>'  •  cosX9rfm,       b  =f  <j>' .  cos  ^  d  m,       c  =  f  (f>' .  cos  v1  d  m. 

1004.  M.  Fresnel  next  conceives  a  surface,  which  he  terms  the  "  Surface  of  Elasticity,"  constructed  according  to  the 
The  surface  following  law : — on  each  of  the  axes  of  elasticity,  and  on  every  radius  r  drawn  in  all  directions,  take  a  length 
dcfaadhuK?  Pr°Port'ona'  to  tne  st|uare  root  of  the  elasticity  exerted  on  the  displaced  molecule  by  the  medium  in  the  direc- 
investigated  tion  of  the  radius,  or  to  V  F.     Then  if  we  call  R  this  length,  or  the  radius  vector  of  the  surface  of  elasticity,  we 

shall  have 

R*  =  {  a  r  .  cos  a8  -)-  b  r .  cos  /38  +  n  r ,  cos  7*  }  X  const. 

Its  radius     The  values  of  R  parallel  to  the  axes  are  then  had  by  the  equation 
vector  ex- 
pressed. R8  =  const  ar,     R!  =  const  x  6  r,     R«  =  const  x  c  r 

which  (for  brevity,  as  we  shall  have  no  further  occasion  to  recur  to  our  former  denominations)  we  shall  express 
simply  by  a1,  b*,  c8,  so  that  the  equation  of  the  surface  of  elasticity  will  be  of  the  form 

R*  =  a2 .  cos  Xs  +  6« .  cos  Y'  -f  <?  .  cos  Z«, 

where  X,  Y,  Z,  now  stand  for  a,  ft,  7,  the  angles  made  by  R  with  the  axes  of  the  coordinates. 

1005.  Let  us  now  imagine  a  molecule  displaced   and  allowed  to  vibrate   in  the  direction   of  the  radius    R,  and 
Velocity       retained  in  that  line,  or  at  least  let  us  neglect  all  that  part  of  its  motion  which  takes  place  at  right  angles  to 
ofpolariza-  tue  ra(J'us  vector.     Then  the  force  of  elasticity  by  which  its  vibrations  are  governed  will  be  proportional  to  R*, 
tion  of  an     and  the  velocity  of  the  luminous  wave  propagated  by  means  of  them,  in  a  direction  transverse  to  them  (or  at 
interior        right  angles  to  R)  will  be  proportional  to  R,  so  that  the  surface  of  elasticity  being  known,  the  velocity  of  a  wave 
wave  deter-  transmitted  through  the  medium  in  a  given  direction,  and  with  a  given  plane  of  polarization  will  be  had  at  once 

as  follows.  Parallel  to  the  surface  of  the  wave,  and  at  right  angles  to  its  plane  of  polarization  draw  a  straight 
line.  This  will  be  the  direction  of  the  vibrations  by  which  the  wave  is  propagated.  Parallel  to  this  line  draw  a 
radius  vector  to  the  surface  of  elasticity,  and  it  will  represent  the  wave's  velocity. 

1006.  The  equation  of  the  surface  of  elasticity,  if  we  put  for  R,  cos  X,  cos  Y,  cos  Z,  their  values  in  terms  of  three 
Equation  of  coordinates  will  become 

the  surface  f^t    i    „!  _|_  -,»)*  =  at  xi  ±  J*  yt  _L  <*  z*. 

of  elasticity.  .  v 

It  is,  therefore,  in  general  a  surface  of  the  fourth  order.  If  we  suppose  it  cut  by  a  plane  passing  through  its 
centre,  whose  equation  must  therefore  in  general  be  of  the  form  mx-\-  ny  +  p  z  =  0,  the  curve  of  intersection 
will  be  a  species  of  oval  whose  diameters  are  not  necessarily  all  equal. 

1007.  Suppose  now  any  molecule  set  in  vibration  in  this  plane,  then  at  any  period  of  its  motion  it  will  not  be  uro-ed 
Resolution    directly  to  its  point  of  rest  but  obliquely,  so  that  it  will  not  describe  a  straight  line,  but  will  circulate  in  a  curve 
of  an  inci-    more  or  less  complicated ;  its  motion  in  this,  however,  will  always  be  resolvable  into  two  vibratory  rectilinear  ones  at 
dent  wave    right  angles  to  each  other,  one  parallel  to  the  greatest,  and  the  other  to  the  least  diameter  of  the  section.    Each  of 

these  vibratory  motions  will,  by  the  laws  of  motion,  be  performed  independently  of  the  other,  and  therefore  the  motion 

propagated  through  the  crystal  will  affect  every  molecule  of  it  in  the  same  way  as  if  two  separate  and  independent 

Polarized  in  rectilinear  vibrations  (at  right  angles  as  above)  were  propagated  through  it,  with  different  velocities.  Consequently 

opposite       every  system  of  waves  propagated  from  without  into  the  crystal,  will  necessarily  on  entering  it  be  resolved  into  two 

planes.         propagated  with  different  velocities,  and  polarized  in  planes  at  right  angles  to  each  other,  viz.  those  parallel 

respectively  to  the  greatest  and  least  diameter  of  a  section  of  the  surface  of  elasticity  parallel   to   the  plane  of 

either  wave.     And  as  every  difference  in  the  velocities  of  two  waves  propagated  parallel  to  each  other  through 

a  medium,  gives  rise  to  a  corresponding  difference  in  their  planes  at  their  emergence  from  it  into  another,  where 

they  assume  a  common  velocity,  these  waves  will  at  their  egress  no  longer  be  parallel,  and  the  rays  which  are 

perpendicular  to  them  will  be  inclined  to  each  other,  thus  producing  the  phenomena  of  double  refraction ;  and  it 

is  evident  that  the  waves  at  their   egress  must   retain   the  planes  of  polarization  they  received   in  the  crystal, 

because  any  molecule  of  the  exterior  medium  at  the  junction  of  the  media  will  begin  to  move  only  in  the  plane 

in  which  it  was  displaced  by  the  contiguous  molecule  in  the  medium. 

1008.  This  theory  then  accounts  perfectly  both  for  ilte  bifurcation  of  the  emergent  ray,  and  the   opposite  polariza- 
tions of  the  two  portions  into  which  it  is  divided.     These  portions  will  coincide  in  direction,  and  there  will  be 
no  double  refraction  when  the  section  of  the  surface  of  elasticity  above  mentioned  is  (if  such  can  ever  be  the 
case)  a  circle,  because  all  its  radii  being  then  equal,  the  elasticity  is  the  same  in  all  directions,  and  all  vibrations 
performed  in   it  will  have  equal  periods,  so  that   in  this  case  the  resolution  of  the  incident  wave  into  two  no 
longer  takes  place,  nor  is  its  plane  of  polarization  changed.     Now  the  section  in  question  becomes  a  circ'e, 
when  x*  -f-  y"  +  2*  =  const  =r  r*,  or  when  a8  3?  -f-  6s  y8  +  c8  z«  =  r4.  Combining  these  with  mx  +  ny  +pz  =  0, 
we  get 


LIGHT.  543  . 

Light.  r<  =  rz  (j,;  +  yi  _f-  z"-),  Part  IV. 

p'  r4  =  r1  (/  a;8  +  ^8  y8  +  (wi  *  +  n  y)1), 
and  ;>2r4  =  p*  a*  z*  +  p*  b*  y*  +ti>  (mx 


and  equating  these,  and  considering  that  the  equation  thence  resulting  ought  to  be  verified  independently  of  any 
particular  values  of  i,  y,  we  get 

r'  (m-  +  p>)  =  a*  1?  +  m*  c\  Investiga- 

tion  of  the 

m  n  rs  =  m  n  c8,  optic  axes. 

r*  (p8  +  n-)  =  b*  p*  +  n-  c8. 
These  equations  cannot  be  satisfied  except  by  supposing  either  m,  n,  or  p  to  vanish,  or  the  section  in  question  to 

/  „  V      a1  -  If 

pass  through  one  or  other  of  the  axes.    If  we  suppose  m  =  0,  we  have  r  =  a,  I  —  I  =  -  ,  which  shows  that 

\P  /       c2  —  a2 

—  )  cannot  be  positive,  and  of  course  —  not  real,  unless  a,  the   semiaxis  of  the  surface  through   which  the 
P/  P 

section  passes,  be  that  intermediate  in  length  between  6  and  c,  the  other  two  semiaxes. 

It  appears  then,  that  the  surface  of  elasticity  admits  of  two  circular  sections  and  no  more,  formed  by  diametral      1009 

71 

planes  passing  through  the  mean  axis  of  the  surface,  and  (since  —  has  two  values  equal  but  of  opposite  signs) 

that  these  sections  are  both  equally  inclined  to  each  of  the  other  two  axes.  The  normals  to  these  sections  are 
the  directions  of  no  double  refraction,  or  the  optic  axes  of  the  crystal.  Of  these,  then,  there  will  be  two  and  two 
only,  in  all  crystals  which  possess  three  unequal  axes  of  elasticity,  and  rays  propagated  along  them  will  suffer 
neither  double  refraction,  nor  change  of  polarization. 

The  position  of  these  axes  depends  wholly  on  the  values  of  a,  b,  c,  the  semiaxes  of  the   surface  of  elasticity.      1010. 
We  have,  however,  no  other  measure  of  the  elasticity  of  the  medium  than  the  velocity  with  which  the  rays  are  Dispersion 
propagated  through  it  ;   and  if,  as  the  phenomena  of  ordinary  dispersion  indicate,  the  rays  of  different  colours  be  of  '')e  axes 
propagated  in  one  and  the  same  medium  with  velocities  somewhat  different,  (an  effect  which  might  result  from  "J]^''61" 
certain  suppositions  relative  to  the  extent  of  the  sphere  of  action  of  its  molecules  compared  with  the  lengths  of  e 
an  undulation,)  the  semiaxes  a,  b,  c,  which  must  be  taken  proportional  to  the  velocities  of  propagation,  must  be 
supposed  to  vary  a  little  for  waves  of  different  lengths.     Now  this  variation  may  not  be  in  the  same  ratio  for  all 

the  three  semiaxes,  and  thus  a  variation  in  the  values  of—  will  arise.     But  —  is  the  tangent  of  the  inclination 

P  P 

of  the  plane  of  section  to  the  plane  of  the  x  y,  or  of  half  the  angle  the  two  circular  sections  make  with 
each  other,  i.  e.  the  cotangent  of  half  the  angle  between  the  optic  axes,  which  will  thus  vary,  and  give 
rise  to  that  separation  of  axes  of  different  colours,  and  their  distribution  over  a  certain  angle,  in  the  plane 
containing  any  two  of  the  same  colour,  which  observation  shows  to  exist,  (Art.  921  and  922.) 

The  general   laws  of  double  refraction  flow  with  great  facility  from  these  principles.      We  have  only  to 
resume  the  construction  and  reasoning  of  Art.  806  and  807,  et  seq.,  substituting  for  the  ellipsoid  of  revolution,  Application 
which  the  Huygenian  theory  assumes  as  the  figure  of  a  wave  originating  in  any  molecule  of  the  crystal,  the  of  the  Huy- 
surface,  whatever  it  be,  which,  in  the  general  case,  terminates  a  wave  so  propagated,  and  investigating  the  point  ge"ian  con- 
of  contact  I  (fig.  170)  of  this  surface  with  a  plane  IKT  passing  through  the  line  KT  drawn  as  there  described.  Suru( 
There  is  this  difference,  however,  in  the  two  cases,  or,  at  least,  in  the  method  of  treating  them,  that  in  the  case 
theory  there  stated  the  form  of  the  wave  is  made  a  matter  of  arbitrary  assumption,  in  the  present  case  it  is 
to  be  determined  it  priori.     This  will  render  it  necessary  to  depart  in  some  respects  from  the  course  before 
adopted.     If  we  know,  a  priori,  the  form  of  the  wave,  the  position  of  the  tangent  plane  is  given  ;  vice  versd, 
if  we  can  determine  the  position  of  this  plane  in  all  cases,  a:  priori,  the  figure  of  the  wave,  which  must  be 
such  as  to  touch  all  such  planes,  under  the  conditions  of  the  case,  becomes  known. 

Now,  in  Art.  807,  it  is  shown  that  the  tangent  plane  is  in  all  cases  coincident  with  the  position  assumed     1012 
within  the  crystal,  by  the  surface  of  a  plane  indefinite  wave  propagated  from  an  infinitely  distant  luminary,  per-  Direction 
pendicular  to  the  line  of  incidence  R  C.     It  follows,  moreover,  from  Art.  81  1,  that  if  we  know  the  velocity  with  "id  velocity 
which  such  a  plane  wave  advances  within  the  crystal  in  a    direction    perpendicular  to  its    surface,  we  may  of  a  P'anl> 
calculate  its  inclination  to  the  surface  of  incidence  by  the  law  of  ordinary  refraction,  assuming  an  index  of  w' 
refraction  which  is  to  that  of  the  ambient  medium  as  the  velocity  of  the  wave  before  incidence  is  to  its  velocity 
within  the  medium  perpendicular  to  its  own  surface.     The  reader  will  here  keep  in  view  the  distinction  noticed 
in  Art.  813  between  the  velocity  of  the  wave  and  that  of  the  ray  conveyed  by  it,  whose  direction,  generally 
speaking,  is  oblique  to  its  surface.     Now  the  velocity  of  a  wave  within  the  medium  in  any  direction  is  given 
by  the  equation  of  the  surface  of  elasticity,  whose   radius  vector   expresses    it  in  all  cases.     But  it  has  been 
shown,  that  every  vibration  impressed  on  the  molecules  of  the  crystal  is  resolved  into  two  rectilinear  ones  propa- 
gated with  velocities  proportional  to  the  greatest  and  least  diameters  of  that  section  of  the  surface  of  elasticity 
which  is  parallel  to  the  plane  in  which  they  are  performed.     Now  it  is  the  same  thing,  (as  far  as  the  law  of  double 
refraction  is  concerned,)  whether  we  regard  the  bifurcation  to  take  place  by  the  separation  of  a  single  exterior 
ray  into  two  interior  ones,  or  a  single  interior  into  two  exterior.     We  will  take  the  latter  case,  and  suppose  the 


544  LIGHT 

Ught.      ordinary  and  extraordinary  plane  waves  to  be    parallel  within  the    medium.      Their  velocities  may  then  he     IVt  rv. 
•  -_—  °^-^'  investigated  as  follows  :  the  equation  of  the  surface  of  elasticity  being  "—  v—  • 

of'lTordi-  R4  =  a2  x*  +  b'yi  +  <*  z8, 

nary  and 

extraordi-     if  we  take,  for  the  equation  of  the  second  plane, 

nary  plane 

wave  inves-  z  =  m  *  +  n  y, 

and  put  V  for  the  maximum  or  minimum  radius  vector  of  the  surface  in  the  section  in  question,  V  will  be  the 
value  of  R,  which  makes  d  R  =  0,  and  therefore  will  be  given  by  elimination  from  the  following  system  of 
equations 

Vs  =  a»  -f-  y*  -f-  s\ 

V«  =  a8  *8  +  64  y*  +  c2  z!, 

2    =  m  x  -f-  »  V. 

and  their  differentials,  regarding  V  as  constant.  This  elimination,  which  is  complicated  enough,  must  be  con- 
ducted as  follows  :  first,  if  among  the  differential  equations  we  eliminate  d  x,  dy,  dz;  and  for  z  in  the  whole 
system  substitute  its  value,  we  shall  get,  putting  p  =  a8  -  68  ;  q  =  a1  —  c*  ;  r  =  ft8  -  c2  ; 

V»  =  (a8  +  m*  c8)  x"  +  (b*  +  n*ct)yt+2mnc'x  y, 
V8  =  (1  +  m8)  x>  +  (1  +  ?is)  y'  +  2  m  n  x  y, 
0  =  mnq.v'  -  mnryt  +  kxy, 
where  k  =  p  +  n*  q  -    m?  r  =  (1  -f-  7i*)  </  —  (1  -f-  wi*)  r. 


- 

These,  by  elimination,  give  the  following,  in  which 

M=  fc8 


M  it8  =  Vs  rv  -  c8)  {  (I  +  ?is)  k  +  2  m'  n1  r  }  -  r  k  Vs, 
M  y1  =  -  Vs  (V  -  c8)  {  (1  +  m*)  *  -  2  m*  n2  9  }  -f-  r  q  V, 
M  ,r  y  =  -  mra  {  (1  +  »s)  9  +  (1  +  m4)  r  }  V*  (V*  -  <*)  +  2  OT  M  g  r  Vs  ; 

and  by  equating  the  square  of  the  last  of  these  to  the  product  of  the  two  first,  we  find,  after  all  reductions,  the 
following  equation  for  determining  V  : 

(V*  -  a*)  (V«  -  68)  +  m'  (V*  -  6»)  (V«  -  c8)  +  «2  (V»  -  ««)  (V  -  e8)  =  0. 

1013.          The  roots  of  this  equation  determine  the  maximum  and  minimum  values  of  the  radius  vector  in  the  plane  of 
General       section,  and  therefore  the  velocities  of  ordinary  and  extraordinary  plane  waves  moving  parallel  to  each  other 
equation  of  wjthin   the  crystal,  and  these  found,  the  figure  of  the  wave  becomes  known,  from  the  condition  that  its  surface 
ite  I**™"  must  always  be    a  tangent  to   a  plane  distant  by  the  quantity  V  from  the  secant  plane  whose  equation  is 
•  from  a         z  ~  m  x  +  n  y  ;    and  that,  whatever  be  the  values  of  m  and  n.     Its  investigation  is  therefore  reduced  to  a 
point  in  the  purely  geometrical  problem.     Required  the  equation  of  a  curve  surface,  which  shall  touch  every  plane  parallel 
medium.      to  a  plane  whose  equation  is  z  =mx  -(-  ny  ;    and   distant  from  it  by  a  quantity  V,   a  function  of  m  and  n 
given  by  the  above  equation,  which,  being  resolved,  will  be  found  to  lead  to  the  following  equation 


6"  y«  -(-  ^z8)  (**  +  y'+z*)  -  a'  (b'  +  c8)  x8  -  i8  («'  +c'-)y>\  _ 
-  c«  (a8  +  hi)  2"  +  a8  b"  c8  )  ~ 

1014.          The   surface  represented  by  this  equation  is,   generally  speaking,  of  the  fourth  order,  and  consists  of  two 

Nonexist-     distinct  surfaces,  or  sheets,  (nappes.)     One  of  these,  by  its  contact  with  the  plane  in  question,  determines  the 

enceofthe  direction  of  the  ordinary,  and  the  other  of  the  extraordinary  ray.     Now,  it  is   important  to  remark,   that  this 

rtesian     eqUatjOn,  so  long  as  particular  values  are  not  assigned  to  a,  b,  c,  is  not  decomposable  into  quadratic  factors,  so 

fraction  in    that  neither  of  the  sheets  of  which  it  consists  is  spherical,  or  ellipsoidal  ;  and,  consequently,  neither  the  ordinary 

biaxal          nor  the  extraordinary  ray  follows  either  the  Cartesian  or  Huygenian  law  of  refraction.     This  is  a  consequence 

crystals.       too  remarkable  not  to  have  been  put  to  the  test  of  experiment.     Two  methods  have  been  put  in  practice  by 

M.  Fresnel  for  this  purpose.     The  first  consisted  in  measuring  directly  the  velocities  of  the  two  rays  in  plates  of 

topaz  cut  in  different  directions  with  respect  to  their  axes  by  the  method  explained  under  the  head  of  inter- 

ferences, (Art.  738  and  739.)     Since  a  difference  of  velocity  of  the  interfering  rays  displaces  the  diffracted  fringes 

as  a  difference  of  thickness  would  do,  it  is  manifest  that  if,  in  two  plates  differently  cut,  but  of  precisely  the 

same  thickness,  the  fringes  formed  by  the  ordinary  rays  are  differently  displaced  when  the  plates  are  combined 

successively  with  one  and  the  same  equivalent  plate  of  glass,  or  any  other  standard  medium,  their  velocity  cannot 

be  the  same  in  both  plates  ;  and  if  such  difference  be  observed  to  take  place,  both  in  the  fringes  formed  by  the 

interference  of  the  ordinary  and  of  the  extraordinary  rays  severally,  with  a  compensated  pencil,  it  is  clear  that 

neither  can  have  a  constant  velocity.     Now  the  condition  of  equal  thickness  is   secured  by  cementing  the 

two  plates  edge  to  edge,  and  grinding  and  polishing  them  together,  and  carefully  examining  the  surfaces  alter 

the  operation,  to  be  satisfied  of  their  precise  continuity,  which  may  be  done  by  the  reflected  image  of  a  distant 

object,  and  yet  more  delicately  by  pressing  slightly  on  them  a  convex  lens  of  long  focus,  over  their   line  of 

junction.     If  the  coloured  rings  formed  between  the  surfaces  be  uninterrupted,  we  are  sure  that  this  condition 


L  I  G  H  T  545 

is   rigorously  satisfied.     The  experiment  so  made,   M.  Fresnel  found  to  confirm  the  conclusion  to  which  the     l'"rl 
above  theory  leads.     But  in  corroboration  of  this  important  result,  the  following  method  was  also  used. 

In  topaz  the    extraordinary  refraction  is  stronger  than  the  ordinary;    so  that  the  ordinary  ray,  when  the      1015. 
two  are  separated    by  a  prism   of    that  medium,   may  be  at  once    recognised,    by  being    the  least  deviated.  -^not''er  tx 
M.  Fresnel  procured   two   prisms  to   be  cut  from  one   topaz,  in  both  of  which  the  base   was  parallel  to   the  ''roviMhe  '° 
cleavage  planes,  and  therefore  perpendicular  to  a  line  bisecting  the  angle  between  the  optic  axes  and  to  the  same. 
principal  section  of  the  crystal,   i.  e.  to  the  mean  axis  of  elasticity  ;   but  in  one  the   plane  of  the   refracting 
angle  was  coincident  with,  and  in  the  other  perpendicular  to,  that  section,   these  being  the  planes  in  which  the 
difference  between  the  velocities  of  the  ordinary  ray  is  the  greatest,  as  is  easily  seen  from  what  has  above  been 
said.     These  prisms  were  cemented  side  by  side,  so  as  to  have  their  bases  in  one  plane  and  their  refracting  edg-es 
in    one    straight    line  ;    and  were    then    very  carefully  ground    and    polished    to    plane    surfaces,  so  that   the 
refracting  angles  in  both  could  not  be  otherwise  than  precisely  equal.     In  this  situation  the  compound  prism 
ABC,  fig.  199,   1,  (which   is  seen  in  perspective  in  fig.  199,  2,)  whose  refracting  angle  ABC  was  about  92°, 
was  achromatised  by  two  prisms  C  B  A  and  D  C  A  of  crown  glass,  in  which  circumstances  a   slight,  uncompen- 
sated  refraction  remained  in  favour  of  the  topaz  prism.     Looking  now  through  the  side  E  B,  the  whole  combi 
nation  was  turned  round  the  refracting  edge  as  an  axis,  till  the  image  of  a  distant  object,  a  black   line  on  a 
white  ground,  appeared  stationary  ;   so  that,  the  refracted  rays,  both  ordinary  and  extraordinary,  must  have  tra- 
versed the  prisms  very  nearly  parallel  to  the  b:ise,  or  at  right  angles  to  the  mean  axis,  but  in  the  different  planes 
above  mentioned  in  each.     Now  it  was  observed,  that  the  least  refracted  image  of  the  black  line  so  seen,  that 
is  the  ordinary  one,  was   broken   at    the  junction   of  the  two   prisms,  being  more  deviated  by  one  than  by  the 
other,  while  the  most  refracted  or  extraordinary  image  formed   a  continuous  line  in   both.     This  latter  fact 
(whteh,  at  first  sight,  would  lead  us  to  suspect  that  the  extraordinary  image  had  been  mistaken  for  the  ordinary 
one)  is  a  consequence  of  the  theory  ^bove  explained,  and  is  an  additional  confirmation  of  it. 

When  two  of  the  axes  of  eh.iiicity  (as  6  and  c,  for  instance)  are  equal,  the  general  equation  of  the  surface  of      1016. 
the  wave  becomes  decomposable  into  two  factors,  and  may  be  put  under  the  form  Case  of 

z"  -  V)  \a*x*  +  b*.  (f  +  2'-)  -  a8  b*  }  =  0,  [ 


•••Hch  is  the  product  of  the  equation  of  a  sphere  with  that  of  an  ellipsoid  of  revolution.  In  this  case  the  two 
circular  sections  coincide  with  the  plane  of  the  y  z,  and  the  two  optic  axes  with  the  axis  of  the  x.  We  have 
here  then  the  case  of  uniaxal  crystals,  and  are  thus  furnished  with  an  a  priori  demonstration,  both  of  the  Huy- 
genian  law  of  elliptic  undulations,  in  the  case  of  the  extraordinary  wave  in  such  crystals,  and  of  the  constancy 
of  the  index  of  refraction  in  that  of  the  ordinary.  The  manner  in  which  this  results  as  a  corollary  from  the 
general  case  is  at  once  elegant  and  satisfactory. 

M.  Fresnel  gives  the  following  simple  construction  for  the  curve  surface  bounding  the  wave  in  the  case  of     1017. 
unequal  axes,  which  establishes  an  immediate  relation   between  the  length  and  direction  of  its  radii.     Conceive  Genuine- 
an  ellipsoid  having  the  same  semiaxes  a,  6,  c  ;  and  having  cut  it  by  any  diametral  plane,  draw  perpendicular  tlon    ,tllet 
to  this  plane  from   the  centre  two  lines,  one  equal  to  the  greatest,  and  the  other  to  the  least,  radius  vector  of  the  ^ij'jp^oid'  "' 
section.     The  loci  of  the  extremities  of  these  perpendiculars  will  be  the  surfaces  of  the  ordinary  and  extraordi- 
nary waves  ,  or,  in  other  words,  their  lengths  will  be  the  lengths  of  the  radii  of  the  waves  in  those  directions, 
and  will  therefore  measure  the  velocity  of  the  two  rays  propagated  in  those  directions,  in  the  same  way  as  the 
radii  of  the  Hnygenian  ellipsoid  are  proportional  to  the  velocities  of  the  extraordinary  ray  in  their  direction. 

Finally,  if  we  divide  unity  by  the  squares  of  the  two  semiaxes  of  a  diametral  section  of  the  ellipsoid,  the      1018. 
difference  of  these  quotients  will   be  found  to  be  proportional  to  the  product  of  the  sines  of  the  angles  which  Origin  of 
the  perpendicular  to  this  section  makes  with  the  two  normals  to  the  planes  of  the  circular  sections  of  the  '  le  ' 
ellipsoid.     Now,  in  all  the  crystals  hitherto  known,  these  sections  differ  very  little  from  the  circular  sections  of  of^heTJo" 
the  surface  of  elasticity,  and  may,  without  sensible  error,  be  supposed  to  coincide  with  them  ;  consequently,  the  sines. 
two  normals  in  question  may  be  taken  for  this  purpose  as  the  optic  axes  of  the  crystal.     We  have  thus  the 
origin  of  that  law,  deduced  from  the  phenomena  of  the  coloured  lemniscates,   which  makes  the   difference   of 
the  squares  of  the  reciprocal  velocities  proportional  :o  the  product  of  the  sines  made  by  the  ray  with  the  optic 
axes  ;  and  thus  the  phenomena  of  the  polarized  rings  are  all  made  to  depend  on  the  same  general  principles. 

Such  is  the  beautiful  theory  of  Fresnel  and  Young,  (for  we  must  not  in  our  regard  for  one  great  name  forget  1019 
the  justice  due  to  the  other,  and  to  separate  them  and  assign  to  each  his  share  would  be  as  impracticable  as  invi- 
dious, so  intimately  are  they  blended  throughout  every  part  of  the  system  ;  early,  acute,  and  pregnant  suggestion 
characterising  the  one,  —  and  maturity  of  thought,  fulness  of  systematic  developement,  and  decisive  experimental 
illustration,  equally  distinguishing  the  other.  If  the  deduction  in  succession  of  phenomena  of  the  greatest  variety 
and  complication  from  a  distinctly  stated  hypothesis,  by  strict  geometrical  reasoning,  thr  ugh  a  series  of  inter- 
mediate steps,  in  which  the  powers  of  analysis  alone  are  relied  on,  and  whose  length  and  complexity  is  such 
as  to  prevent  all  possibility  of  foreseeing  the  conclusions  from  the  premises,  be  a  characteristic  of  the  truth 
of  the  hypothesis,  —  it  cannot  be  denied  that  it  possesses  that  character  in  no  ordinary  degree  ;  but,  however 
that  may  be,  as  a  generalization  the  reader  will  now  be  enabled  to  judge  whether  the  encomium  we  passed  on 
it  in  a  former  Article  be  merited.  We  can  only  regret  that  the  necessary  limits  of  this  Essay,  which  is  already 
extended  greatly  beyond  our  original  design,  forbid  our  entering  farther  into  its  details. 

The  axes  of  elasticity  are  those  which  M.  Fresnel   regards   as  the  fundamental   axes  of  a  doubly  refractive      1020. 
medium.     The  optic  axes  can  in  no  view  of  the  subject  be  regarded  as  such,  for  several  obvious  reasons.     First,  Dr-  Brew. 
they  are  seldom  symmetrically  situated  relative  to  fundamental  lines  in  the  crystalline  form  ;  secondly,  because  s'er  sl 
they  vary  in  position  according  to  the  colour  of  the  incident  light  ;  thirdly,  because  it  is  found  that  for  one  and  zjJ 
the  same  coloured  illumination,  and  in  the  same  crystal,  their  situation  varies  by  a  variation  of  temperature. 

VOL.  iv  4  B 


546  LIGHT. 

Light.  This  important  fact  has  been  lately  ascertained  by  M.  Mitscherlich,  and  we  shall  presently  have  occasion  to  speak 
further  of  it.  From  all  these  reasons  it  follows,  that  we  can  regard  them  only  as  resultant  lints,  to  which  no 
a  priori  properties  can  be  supposed  to  belong,  but  which  simply  satisfy  the  condition  v  —  tf  =  0,  according  to 
the  laws  which  regulate  the  constitutions  of  the  functions  v,  t/,  the  velocities  of  the  two  rays,  in  terms  of  those 
quantities  which  we  may  regard  as  fundamental  data,  and  tile  situation  of  the  ray  within  the  medium.  The  axes 
of  elasticity  themselves  may,  perhaps,  be  regarded  as  mere  resultants  from  the  equations  of  Art.  1000,  and 
determined  from  other  remoter  data  dependent  on  the  fundamental  lines  in  the  crystalline  form,  and  the  intensity 
and  distribution  of  the  molecular  forces  within  it.  Accordingly,  Dr.  Brewster  considers  the  optic  axes  as  the 
resultants  of  others  which  he  terms  polarizing  axes,  and  from  which  he  conceives  to  emanate  polarizing  forces 
producing  the  phenomena  of  the  rings  and  of  the  double  refraction  and  polarization  observed.  We  shall  not 
here  stop  to  examine  into  the  propriety  of  these  terms.  The  reader  who  may  have  doubts  on  the  subject  will, 
in  what  follows,  mentally  substitute  other  and  more  general  phrases  in  their  place  expressive  of  relation  and 
causality,  while  we  proceed  to  state  the  assumptions  with  which  he  sets  out,  and  the  conclusions  he  very  inge- 
niously deduces  from  them. 

1021.          Postulate  1.    A  polarizing  axis,  when   single,  has  the  characters  of  an  axis  of  no  double  refraction,  and  is 
A  single        coincident   with  the   axis  of  the   Huygenian   spheroid  in  such  crystals  as  have  but  one.     A  positive  axis  acts 
polar-ing     as  the  axis  in  quartz,  &c.  may  be  supposed  to  do,  and  a  negative,  as  that  of  carbonate  of  lime,  &c. 
aX'i'     o  Post.  2.  The  polarizing  force  of  a  single  axis  in  any  medium  is  proportional   to,  and  measured  by,  the  tint 

.,  .  ~.~  developed  in  the  ordinary  and  extraordinary  pencils  into  which  a  doubly  refracting  prism  analyzes  a  polarized 
force  raJ''  which  has  traversed  a  given  thickness  of  the  medium. 

1023.  Carol.  1.  The  polarizing  force  of  a  single  axis  in  the  same  medium   is  as  the  square  of  the  sine  of  the  angle 
made  by  the  ray  traversing  it  internally,  with  the  axis. 

1024.  Carol.  2.  The  same  force  is  also  inversely  as  the  thickness  necessary  to  be  traversed  at  a  given  angle  to 
develope  the  same  or  equal  tints.     This  may  be  regarded  as  the  intrinsic  polarizing  force  or  intensity  of  the  axis. 

1025.  Post.  3.   When  two  axes  exist  in  one  medium  and  operate  together,  they  polarize  a  tint  whose  measure  (see 
Composi-      Art.  906)  is  the  diagonal  of  a  parallelogram  whose  sides  measure,  on  the  same   scale,  the  tints  which  would  be 
tion  of  tints  polarized  by  either,  separately,  and  include  between  them  an  angle  double  of  the  mutual  inclination  of  two  pianos 

passing  through  the  ray  and  either  axis  respectively. 

1026  Carol.  I,  If  t  and  t'  be  the  numerical  measures  of  the  tints  polarized  by  either  of  two  axes  separately,  T  that 
Formula  for  P°larized  by  their  joint  action,  andC  the  angle  between  the  planes  just  described,  the  tint  T  will  be  given  by  the 
the  com-      equation  T"  =  t1  -f  2  it' .  cos  2  C  -f-  t\ 

pound  tint. 

1027  Carol.  2.  If  a  and  b  represent  the  intensities  of  the  axes,  and  a  and  ft  the  angles  which  the  ray  makes  with 
each  respectively,  we  have  t  =  a  .  sin  a2 ;  t'  =  6  .  sin  fP,  and 

T  =  (a  .  sin  a8)'  +  (b  .  sin  /3»)»  +  2  a  b  .  sin  a*  .  sin  ft'- .  (1  -  2  .  sin  C«), 

=  {  a  .  sin  a'  +  b  .  sin  ft'  }*  —  4  a  b  .  sin  a* .  sin  /3s .  sin  C*, 
or  else  T4  =  {  a  .  sin  a*  -  b  .  sin  /3s  }2  -j-  4  a  b  .  sin  a2  .  sin  ft' .  cos  C2. 

1028.  If  7  be  the  angle  contained  between  the  polarizing  axes,  since  n,  ft,  7  are  the  sides  of  a  spherical  triangle, 
and  C  the  angle  included  between  the  sides  a  and  ft,  or  opposite  to  7,  we  have 

cos  a  .  cos  ft  —  cos  7 

cos  C  = —  -  , 

sin  a  .  sin  ft 

and  if  this  be  written  for  cos  C  in  the  latter  of  the  expressions  above  given  for  T-,  we  find  on  reduction 
T2  =  {  a  .  sin  a2  -f  6  .  sin  /32  }s  —  4  a  b  {  1  — -  cos  a2  —  cos  ft*  —  cos  7*  +  2  .  cos  a  .  cos  ft  .  cos  7  }  . 

1029.  Carol.  If  the  polarizing  axes  be  at  right  angles  to  each  other,  7  =  90°  and  cos  7  =  0,  and  the  expression  for 
the  compound  tint  becomes  T*  =  {  a  .  sin  o*  +  6  .  sin  fi°-  J*  —  4  a  b  (sin  a*  —  cos  /32). 

1030  Proposition.   Two  rectangular  polarizing  axes,  cither  both  positive  or  both  negative,  being  given,  two  ot/iT  nien, 

or  fixed  lines,  may  be  found,  such  that  calling  0  and  Q1  the  angles  made  with  them  respectively  by  a  ray  traversing 
a  spherical  portion  of  the  medium,  the  lint  polarized  shall  be  proportional  to  sin  0  .  sin  (/.* 

Resultant  Let  A  C  and  B  C  (fig.  199)  be  the  two  polarizing  axes  including  a  right  angle,  of  which  let  B  C  be  the  more 
axes  arising  powerful.  Let  O  C  be  a  ray  penetrating  the  crystal  in  that  direction;  and  in  a  plane  P  C  Q  perpendicular  to 
from  the  A  C  B,  draw  any  two  lines  PC,  Q  C,  making  equal  angles  with  B  C,  either  of  which  we  will  represent  by  i. 
joint  action  Tnen  jf  ft  sphere  about  C  as  a  centre  be  conceived,  it  will  intersect  the  planes  A  C  B,  P  C  Q,  OCA,  O  C  B, 
tang'u'lar''0  O  C  P,  O  C  Q  in  lines  of  great  circles  B  A,  P  B  Q,  O  A,  O  B,  O  P,  O  Q,  and  we  shall  have  P  B  =  Q  B  =  x, 
polarizing  OA  =  «,  OB  =  /3,  O  P  =  #,  O  Q  =  0';  and  by  Spherical  Trigonometry,  from  the  triangle  O  B  P,  we  have 


F'g-  199. 


cos  O  B  P  (=  sin  O  B  A  =  sin  A  O  B  .  **"  °A  =  sin  a  .  sin  C,  since  A  B  =  90°  ) 
\  sin  A  B  / 


cos  ft  .  cos  x  —  cos  0 
sin  ft  .  sin  x 


*  M.  Biot  appears  lo  have  first  noticed  the  fact  announced  in  this  proposilion,  viz.  that  Dr.  Brewster's  hypothesis  of  polarizing  axes  lends 
to  a  result  mat/iemalicalty  identical  with  his  own  elegant  law  of  ttie  product  of  tlie  sines.  He  has,  however,  suppressed  his  demonstration. 
Dr.  Brewster's  verification  of  this  coincidence  of  results  seems  to  have  been  founded  on  a  numerical  comparison  of  Biot's  experiments  on 
sulphate  of  lime  with  his  own  theory. 


LIGHT.  547 

Light,      and  therefore  —  cos  0  =:  sin  a  .  sin  ft  .  sin  *  .  sin  C  —  cos  ft  .  cos  x.  Part  IV. 

~"s^~*    and  similarly  from  the  triangle  O  B  Q,  since  O  B  Q  =  90°  -f-  O  B  A,  we  obtain  a  second  relation  ^^  "V^"*1 

-f-  cos  ff  =  sin  o  .  sin  ft  .  sin  x  .  sin  C  -f-  cos  /3  .  cos  x  ; 
and,  adding  and  subtracting,  (putting,  for  brevity's  sake,  cos  Of  =  p,  cos  °  =  <jr,) 

p  +  q  =  2  .  cos  j3  .  cos  x  ;        p  —  q  =  2  .  sin  a  .  sin  /3  .  sin  x  .  sin  C. 

These  equations  express  the  geometrical  relations  subsisting  between  the  lines  PC,  Q  C,  and  the  axes  AC,  BO: 
and,  if  combined  with  the  equations  of  Art.  1028  and  1029,  suffice  to  eliminate  a,  /3,  and  C,  and  to  express 
T  in  terms  of  x,  0,  and  ff  alone.  To  execute  this,  we  have  by  the  equations  just  demonstrateu 

f  JL-L3L  \  —  cos  /3»  ;         (  P  ~.q     )  =  sin  a*  .  sin  /3s  .  sin  C2  ; 
\2  .  cos  x)  \2  .  sin*/ 

and  in  the  latter,  putting  1  —  cos  C3  for  sin  C4,  and  for  cos  C2  its  value  given  by  Art.  1028,  which,  since  7  = 
90°,  becomes  simply 

sin  a8  .  sin  /3s  .  cos  C*  =  cos  a2  .  cos  /3J, 

we  have  (  -  --  —  )  r=  sin  a?  .  sin  /3s  —  cos  a'  .  cos  |3*, 

\2  .  sin  x/ 

=  sin  a2  —  cos  /39. 
Hence  we  get,  for  the  values  of  sin  a8  and  sin  /3a, 


and,  substituting  these  in  the  equation  of  Art.  1029, 

T8  =  \b  H  --  —  -  (p  +  qY  +   -  -  -  (p  —  a)*  \    -    —  -  (p  -  o)f. 
I  4  .  cos  xi  ^      ^          4  .  sin  x*   w  )  sin  ^   V^ 

Such  is  the  general  form  of  the  expression  for  the  tint,  when  referred  to  arbitrary  axes  in  the  manner  here  sup- 

posed, and  it  is  complicated  enough  ;    but  if  we  fix  the  position  of  the  new  axes  so  as  to   make  sin  j;a  =  —  j- 

0 

the  complication  disappears  ;    we  have  then   ---  —  =   —  ,  and  --  =    —  ,  so   that   the  value  ot 

4  .  stn  x*  4  4  .  cos  *2  4 

T*   reduces  itself  to 


=  6"  (1  _  pt)  (1  _  f)  —  b'  .  sin  0'  .  sin  0", 

restoring  the  values  of  p  and  q,  or  cos  0'  and  cos  0,  consequently 

T  =  -  b  .  sin  0  .  sin  tf. 

The  negative  sign  is  prefixed  for  the  reason  stated  further  on  in  Art.  1034. 

Thus  we  see  that  the  combined  action  of  the  two  axes  in  the  manner  here  supposed,  on  Dr.  Brewster's  prin-      103  . 
ciples,  will  give  rise  to  a  series  of  isochromatic  lines  arranged  in  the  form  of  sphero-lemniscates  about  two  poles 
P,  Q,  determined  by  the  condition 

.    /   intensity  of  the  feebler  axis 

sin  B  P  =  sin  B  Q  =   \/  —  :  —  ; 

intensity  ot  the  stronger 

and  *he  lines  C  P,  C  Q  so  determined  have  therefore  the  character  of  the  optic  axes  in  biaxal  crystals,  and  may 
be  designated  with  Dr.  Brewster  by  the  name  of  resultant  axes.  We  must  be  careful,  however,  not  to  confound 
a  resultant  with  a  polarizing  axis  in  this  theory. 

If  the  polarizing  axes  be  not  of  the  same  denomination,  as  if  one  be  positive  and  the  other  negative,  the  1032 
value  of  sin  B  P  becomes  imaginary,  and  the  tints  cannot  be  so  arranged.  But  if  we  suppose  the  new  axes  to  Combina 
tie  in  this  case  in  the  same  plane  with  the  polarizing  ones,  as  in  fig.  200,  all  other  things  remaining,  we  ti(">  of  » 
have  here  positive 

cos  O  B  A  =  +  cos  O  B  Q,         and  cos  O  B  A  =  -  cos  O  B  P,  with  a 

negative 

cos  a  cos  (3  .  cos  x  —  cos  &'  axii. 

but  cos  O  B  A  =  --  :  —  —  ,        and  cos  O  B  Q  =  - 

sin  p  sin  /3  .  sin  x 

so  that  we  find  cos  0'  =  p  ==  cos  ;3  .  cos  x  -f-  cos  «  .  sin  *.; 

4  B  2 


548  L  I  G  H  T. 

and  similarly  cos  0  —  q  =  cos  /3  .  cos  x  —  cos  a  .  sin  i,  Part  i 

whence,  by  adding  and  subtracting',  we  get  at  once  x>—  v 

p  -  q  j>  +  n 

cos  a  =  ~—~-  ;  cos  /3  =    4 

2  .   Sill  X 

which,  substituted  in  the  value  of  T2,  give 

•P  =     «  +  6) 


.r4          cos  x*  /        4  \sinx8        cos  JT/     2    j 

a  6  2  a  6  (sin  x8  —  cos  x8) 

-4a6  +      .          -        -  (7,2  +  0)+  —i-  pa. 

sin  x2  .  cos  of  sin  x*  .  cos  x* 

Now,  if  in  this  we  suppose  -     —  -j  --  =  0,  or  tan  x1  =  --  —  ,  it  will,  on  substitution  and  reduction,  take 

sin  x1        cos  x*  b 

the  form 

(i-P'H1  -?')  >>'  •  sin  0'  .  sin  0* 


cos  X*  cos  x* 

and  T  =  -  --  -  .  sin  0  .  sin  ff  • 

cos  r 

that  is,  restoring  the  value  of  x,  (  since  tan  x4  =  —     ,-,  and  therefore  cos  x1  =  -  -  —  1,  finally, 

\  b  b  —  a/ 

T  =  -  (6  —  a)  .  sin  0  .  sin  6'. 

1033.          Thus,  in  this  case  also,  the  isochromatic  lines  are  sphero-lemniscates,  and  the  only  difference  is   that  their 

Portion  of   poles  lie  now  in  the  plane  of  the   polarizing  axes,  instead  of  at  right  angles  to  it  ;  and  that  whereas  in   the 

lesuitant  •  _ 

axes  in  /    fl 

tWs  case,     former  case  the  semi-angle  between   them  (T)  was  given  by  the  equation   sin  x  =    \/   -j-,  that  is,  cos  x  = 


\/  ,  in  this  it  is  given  by  the  equation  cos  x  =  \/ . 

v     o  —  a 


1034.  Carol.  1.  In  the  case  when  a  =  6,  or  when  the  two  polarizing  axes  are  of  the  same  denomination  and  of  equal 
Cases  of  the  intensity,  we  have  sin  x  =1,  or  x  =  90°,  so  that  the  angle  between  the  resultant  axes  being  180°,  they  form  one 

resolution     strajg.nt  line,  the  lemniscates  become  circles,  and  the  single  resultant  axis  has  now  the  characters  of  a  polaritins 

of  a  smglo         -II  i     •   •  •      •  j.  ,   . 

mis  into       axis.     Hence,  vice  versa,  a  single  polarizing  axis,  in  any  direction,  may  be  resolved  into  two  others  equal  in 

two.  intensity,  at  right  angles  to  it  and  to  each  other,  and  of  an  opposite  denomination  to  the  resolved  axis.     This 

follows  from  the  negative  sign  of  T,  which  is  prefixed  in  extracting  the  square  root  in  Art.  1030  and  1032  ; 
because  in  the  case  supposed,  when  the  arc  A  B  is  90°  the  angle  C  or  AO  B  is  necessarily  greater  than  90°,  and 
2  C  the  angle  of  the  parallelogram  of  tints  >  ISO0;  so  that  the  diagonal  will  be  to  be  measured  backwards 
through  the  angle,  or  must  be  a  negative  quantity. 

1035.  Carol.  2.  Since  a  single  axis  is  equivalent  to  two  equally  intense  axes  of  an  opposite  character  at  right  angles 
Composi-     to  it  and  to  each  other,  if  we  superadd  to  both  another  equal  axis  also  of  the  opposite  kind,  and  in  the  direction 
tion  of  three  Qj-  ^ne  nrst;  t|,is  WJH  destroy  the  effect  of  the  first,  and  therefore  the  combination  of  three  equal  and  similar  axes 
angular         arising  on  the  other  side  at  right  angles  to  each   other,  will  be  equivalent  to  none   at   all.     Thus,  three  equal 
»xes.            rectangular  axes  of  the  same  character  destroy  each  other's  effects.     This  is  Dr.  Brewster's  account  of  the  want 

of  polarization  and  double  refraction  in  crystals  whose  primitive  form  is  the  cube,  regular  octohedron,  &c., 
and  whose  secondary  forms  indicate  a  perfect  symmetry  in  their  molecules  with  respect  to  three  rectangular 
axes. 

1036.  There  is  no  necessity  to  pursue  further  the  general  subjects  of  this  species  of  composition  of  axes  and  of  tints. 
Indeed,  it  appears  to  us  that  the  rule  for  the  parallelogram  of  tints,  as  laid  down  by  Dr.  Brewster,  becomes 
inapplicable  when  a  third  axis  is  introduced  ;   for  this  obvious  reason,  that  when  we  would  combine  the  com- 
pound tint  arising  from  two  of  the  axes  (A,  B)  with  that  arising  from  the  action  of  the  third  (C,)  although  the 
sides  of  the  new  parallelogram  which  must  be  constructed  are  given,  (viz.  the  compound  tint  T,  and  the  simple 
tint  t",)  yet  the  wording  of  the  rule  leaves  us  completely  at  a  loss  what  to  consider  as  its  angle,  inasmuch  as  it 
assigns  no  single  line  which  can  be  combined  with  the  axis  C  in  the  manner  there  required,  or  which  quoad  koc 
is  to  be  taken  as  a  resultant  of  the  axes  A,  B.     For  further  information  therefore  on  this  subject  we  shall  content 
ourselves  with  referring  the  reader  to  his  original  Paper  in  the  Transactions  of  the  Royal  Society,  1818. 

§  X.     Of  Circular  Polarization. 

1037.  The  first  phenomena  referable  to  the  class  of  facts  to  whose  consideration  this  section  will  be  devoted,  were 
noticed  by  M.  Arago  in  his  Memoir  published  among  those  of  the  Institute  for  1811  on  the  colours  of  crystal- 
lized plates.     He  observed  that  when  a  polarized  ray  was  made  to  traverse  at  right  angles  a  plate  of  rock  crysta' 


LIGHT. 


519 


(quartz)  cut  perpendicularly  to  the  axis  of  double  refraction,  on  analyzing  the  emergent  ray  by  a  doubly  refracting    Part  IV. 
'  prism,  the  two  images  had  complementary  colours,  and  that  these  colours  changed  when  the  doubly  refracting  v— >..— •' 
prism  was  made  to  revolve ;    so  that  in  the  course  of  a  half  revolution,  the  extraordinary  image  (for  example)  Phenomena 
which  at  first  was  red,  became  in  succession  orange,  yellow,  yellow-green,  and  violet,  after  which  the  same  series  of  °0|j',!-™t"n 
tints  would  of  course  recur.    It  is  evident  that  this  is  just  what  would  take  place,  supposing  the  several  coloured 
rays  at  their  emergence  from  the  rock  crystal  to  be  polarized  in  different  planes;  and  to  this  conclusion  M.  Arago 
came  in  a  second  Paper,  subsequently  read  to  the  Institute.     The  subject  was  resumed  by  M.  Biot,  in  a  Paper 
published  in  the  Mem  de  I'Inst.,  1812 ;  and  his  labours  were  completed  in  a  second  extremely  interesting  Paper 
read  to  that  body  in  September,  1818. 

When  a  polarized  ray  is  made  to  traverse  the  axis  of  Iceland  spar,  beril,  and  other  uniaxal  crystals,  we  have      1038. 
seen  that  it  undergoes  no  change  or  modification  ;  and  that  when  analyzed  at  its  egress  by  a  doubly  refracting  Rotatory 
prism,  having  its  principal  section  in  the  plane  of  primitive  polarization,  the  ordinary  image  will  contain   the  phenomena 
whole  ray,  or  the  complementary  tints  will  be  white  and  black.     Quartz,  however,  is  an  exception  to  this  rule.  °  <luar 
A  polarized  ray  transmitted,  however  precisely,  along  its  axis,  is  still  coloured  and  subdivided,  and  that  the  more 
evidently,  the  thicker  is  the  plate.     If  we  place  on  a  proper  apparatus,  such  as  that  described  in  Art.  929  and 
figured  in  fig.  189,  a  very  thin  plate  of  this  body,  and  turn  round  the  analyzing  prism  M  in  its  cell,  till  the  extra- 
ordinary image  is  at  its  minimum  of  brightness,  it  will  in  this  position  have  a  sombre  violet,  or  purple  tinge, 
because  the  yellow  or  most  luminous  rays,  which  are  complementary  to  purple,  are  now  completely  extinguished. 
Let  the  angle  of  rotation  of  the  prism  in  its  cell,  measured  on  the  divided  circle  R,  and  which  in  this  case  will 
be  small,  be  noted  ;  and  then  let  the  rock  crystal  plate  be  detached,  and  another  cut  from  the  same  crystal,  but 
of  twice  the  thickness,  be  substituted.     The  tint  of  the  extraordinary  image  will  no  longer  be  violet ;   but  if  the 
prism  be  made  to  revolve  through  an  additional  equal  arc  in  the  same  direction,  the  violet  or  purple  tint  will  be 
restored,  and  the  minimum  of  brightness  attained ;    and,  in  general,  if  the  thickness  of  the  plate  (always  sup- 
posed cut  from  the  same  crystal)  be  greater  or  less  in  any  ratio,  the  angle  of  rotation  through  which  the  prism 
must  be  moved  in  the  same  direction,  to  produce  a  minimum  of  intensity  and   a  purple  tint  in  the  extraordinary 
image,  is  increased  or  diminished  in  the  same  ratio.     In  consequence,  if  the  plate  be  sufficiently  thick,  one  or 
more  circumferences  will  be  required  to  be  traversed;  and  as  only  the  excesses  over  whole  circumferences  can  be 
read  off,  this  may  produce  some  confusion  or  doubt,  unless  we  take  care  to  use  a  succession  of  thicknesses  so 
gradually  increasing  as  not  to  allow  of  a  saltiis  of  a  whole,  or  a  half  circumference. 

From  this  experiment  we  collect,  that  the  plane  of  polarization  of  a  mean  yellow  ray  which  has  traversed  the      1039. 
axis  of  a  quartz  plate,  has  been  turned  aside  from   its  original  position,  through  an   angle  proportional  to  the  Rotation 
thickness  of  the  plate  ;  and,  therefore,  assumes  at  its  egress  a  position  the  same  as  it  would  have,  had  it  revolved  °[ the.  P'ane 
uniformly  in  one  direction,  during  every  instant  of  the  ray's  progress  through  the  plate.     The  same  holds  good  "j^0 
for  all  the  other  homogeneous  rays ;  but  to  prove  it,  we  must  abandon  the  use  of  white  light,  and  operate  with 
pure  rays  of  the  particular  colour  we  would  examine.    If  we  use  pure  red  light,  for  instance,  or  defend  the  eye  with 
a  pure  red  glass,  the  same  will  be  observed,  only  that  instead  of  a  violet  tint  and  a  minimum  of  light,  we  shall 
have  a  total  obliteration  of  the  extraordinary  pencil  when  the  prism  attains  ks  proper  position,  thus  proving, 
what  in  the  former  mode  of  observation   might   have  been  doubtful,  that  the  polarization  of  the  emergent  ray 
is  complete. 

In  examining  in  this  way  the  quantity  by  which  one  and  the  same  plate  of  quartz  turns  aside  the  planes  of      1040. 
polarization  of  the  different  homogeneous  rays,   M.  Biot  ascertained  that  the  more  refrangible  rays  are  more  Law  of  ro- 
energetically  acted  on  than  the,   less,  and  have  their   planes   of   polarization  deviated  through    a    greater  arc.  t?l'°V  of ''" 
According  to  this  eminent  philosopher,  the  constant  coefficient,  or  index,  which  represents  the  velocity  with  coloured 
which  the  plane  of  polarization  may  be  conceived  to  revolve,  is  proportional  to  the  square  of  the  length  of  an  rays, 
undulation  of  the  homogeneous  ray  under  consideration ;  so  that  if  we  call  X  the  length  of  an  undulation,  and 
t  the  thickness  of  the  plate,  the  deviation  produced  will  be  equal  to  k  .  X*  t,  k  being  a  certain  constant.     The 

18°.414 
value  of  this  constant  he  assigns  at  -  -  lsfi),,a>  when  t  is  reckoned  in  millimetres  ;  and  the  following  is  stated 

by  him  as  the  numerical  amount  of  the  deviations  in  degrees  (sexagesimal)  produced   by  one  millimetre  of 
thickness  of  rock  crystal  on  the  several  rays  : 


Designation  of  the  homogeneous  ray. 


Arc  of  rotation  cor- 
responding to  one 
millimetre. 


Extreme  red     

Limit  of  red  and  orange.  .  .  . 
Limit  of  orange  and  yellow 
Limit  of  yellow  and  green.  . 
Limit  of  green  and  blue     .  .  , 
Limit  of  blue  and  indigo    .  . . 
Limit  of  indigo  and  violet. .  . 
Extreme  violet 


17°.4964 
20°.4798 
22°.313S 


30°.04(JO 
34°.5717 
37°6829 
44°.0827 


LIGHT. 


Light. 
1041. 

Kight  and 
felt  handed 
quartz. 


1042. 

Phenomena 
of  plagie- 
dral  crystals 


1043. 

Superposi- 
tion of 
plates  »f 
-ock 
crystal. 

1044. 
Amethyst. 


1045. 

Rotatory 
phenomena 

.11  liquids. 


In  the  course  of  these  researches  M.  Biot  was  led  to  the  very  singular  discovery  of  a  constant  different  sub-  Pan  IV. 
sistinp:  in  different  specimens  of  rock  crystal,  in  the  direction  in  which  this  rotation  or  angular  shifting  of  the  *«— v~ 
plane  of  polarization  of  a  ray  traversing  them  takes  place.  In  some  specimens  it  is  observed  to  be  from  right 
to  left,  in  others  from  left  to  right.  To  conceive  this  distinction,  let  the  reader  take  a  common  cork-screw,  and, 
holding  it  with  the  head  towards  him,  let  him  turn  it  in  the  usual  manner,  as  if  to  penetrate  a  cork.  The  head 
will  then  turn  the  same  way  with  the  plane  of  polarization  of  a  ray  in  its  progress  from  the  spectator  through 
a  right-handed  crystal  may  be  conceived  to  do.  If  the  thread  of  the  cork-screw  were  reversed,  or  what  is  termed 
a  left-handed  thread,  then  the  motion  of  the  head  as  the  instrument  advanced  would  represent  that  of  the  plane 
of  polarization  in  a  left-handed  specimen  of  rock  crystal.  It  will  be  observed,  that  we  do  not  here  mean  to  say 
that  the  plane  of  polarization  does  so  revolve  in  the  interior  of  a  crystal,  but  that  the  ray  at  its  egress  presents 
the  same  phenomena  as  to  polarization  as  if\t  had  done  so.  This  is  necessary,  for  we  shall  see  presently  that 
a  very  different  view  of  the  subject  may  be  taken. 

In  crystals  which  present  this  remarkable  difference,  when  cut  and  polished,  and  when  the  external  indications 
of  cryst  illinefonn  are  obliterated,  no  other  difference  can  be  detected.  Their  hardness,  transparency,  refractive 
and  double  refractive  powers  are  the  same  ;  and,  with  the  exception  of  the  direction  in  which  it  takes  place, 
their  effects  in  deviating  the  planes  of  polarization  of  the  rays  which  traverse  them  are  alike.  Experiments 
subsequent  to  M.  Biot's  researches  have,  however,  established,  as  a  result  of  extensive  induction,  a  very  curious 
connection  between  this  direction  and  the  crystalline  forms  affected  by  individual  specimens.  In  the  variety  of 
crystallized  quartz,  termed  by  Hauy,  Plagiedral,  there  occur  faces  which  (unlike  those  in  all  the  more  common 
varieties)  are  unsymmetrically  related  to  the  axes  and  apices  of  the  primitive  form,  whether  regarded  as  the  rhomboid 
or  bipyramidal  dodecahedron.  Fig.  201  represents  such  a  crystal,  in  which  when  the  apex  A  is  set  upwards, 
the  faces  C,  C,  C,  are  observed  to  lean  all  in  one  direction,  viz.  to  the  right,  with  respect  to  the  axis,  as  if  dis- 
torted from  a  symmetrical  position  by  some  cause  acting  from  left  to  right  all  round  the  crystal.  When  the 
vertex  B  is  set  upwards,  the  same  distortion,  and  in  the  same  direction,  is  observed  in  the  plagiedral  faces 
D,  D,  D,  and  crystals  of  quartz  are  excessively  rare,  if  they  exist  at  all,  in  which  two  plagiedral  faces  leaning 
opposite  ways  occur.  Now  it  has  been  ascertained,  that  in  crystals  where  one  or  more  of  these  faces,  however 
minute  and  even  of  microscopic  dimensions,  can  be  seen,  we  may  thence  predict  with  certainty  the  direction  of 
rotation  in  a  plate  cut  from  it,  which  is  always  that  in  which  the  plagiedral  face  appears  to  lean  with  respect 
to  an  observer  regarding  it  as  the  reader  does  the  figure,  which  represents  a  right-handed  crystal.  Hence  we  are 
entitled  to  conclude,  that  whatever  be  the  cause  which  determines  the  direction  of  rotation,  the  same  has  acted  in 
determining  the  direction  of  the  plagiedral  faces.  Other  crystallized  minerals,  as  apatite,  &c.  also  present  pla- 
giedral and  unsymmetrical  faces  ;  but,  independent  of  their  extreme  rarity,  they  are  not  possessed  of  the  property 
of  rotation  ;  so  that  at  present  we  are  unable  to  say  whether  this  curious  law  be  general,  or  to  conjecture  to  what 
principles  it  will  hereafter  prove  to  be  referable. 

When  two  plates  of  rock  crystal  are  superposed,  if  they  be  both  right-handed  or  both  left,  their  joint  rotatory 
effect  will  be  the  sum  of  their  respective  ones,  i.  e.  each  ray's  plane  of  polarization  will  be  shifted  through  an 
angle  equal  to  the  sum  of  those  through  which  it  would  have  been  shifted  by  their  separate  actions.  If  their 
characters  be  opposite,  it  will  be  their  difference,  i.  e.  the  index  of  rotation  in  a  right-handed  crystal  being 
regarded  as  positive,  it  will  be  negative  in  a  left-handed  one. 

The  amethyst  (and,  possibly,  also  the  agate  in  some  cases)  presents  the  very  remarkable  and  curious  pheno- 
menon of  these  two  species  of  quartz  crystallized  together  in  alternate  layers  of  very  minute  thickness.  Accord- 
ingly, when  a  crystal  of  amethyst  is  cut  at  right  angles  to  the  axis,  and  examined  by  polarized  light  transmitted 
exactly  along  the  axis,  and  analyzed  as  usual,  it  offers  a  striped  or  fringed  appearance,  as  represented  in 
fig.  202,  variegated  with  different  colours,  according  to  the  different  planes  of  polarization  assumed  by  the  rays 
emergent  at  its  several  points,  and  presenting,  according  to  the  distribution  of  its  elements,  the  most  beautiful 
combinations  and  contrasts  of  coloured  fasciae  and  spaces.  For  a  particular  account  of  these  phenomena,  the 
reader  is  referred  to  a  Paper  by  Dr.  Brewster,  (Edinburgh  Transactions,  vol.  xi.)  who  first  observed  and  publicly 
described  them,  though  we  have  reason  to  believe  them  to  have  been  known  to  others  by  independent  observa- 
tion previous  to  the  publication  of  his  very  curious  and  interesting  Memoir.  The  layers  may  be  distinctly  seen 
cropping  out  to  the  surface  in  a  fresh  fracture  of  the  mineral,  and  imparting  that  peculiar  undulated  fracture 
which  is  the  chief  mmeralogical  character  of  this  substance  by  which  it  is  known  from  ordinary  quartz. 

But  the  phenomena  of  rotation  as  above  described  are  not  confined  to  quortz.  Many  liquids,  and  even 
vapours  exhibit  it,  a  circumstance  which  would  seem  very  unexpected,  when  we  consider  that  in  liquids  and 
gases  the  molecules  must  be  supposed  unrelated  to  each  other  by  any  crystalline  arrangement,  and  independent 
of  each  other ;  so  that  to  produce  any  such  phenomena,  each  individual  molecule  must  be  conceived  as  unsym- 
metrically constituted,  i.  e.  as  having  a  right  and  a  left  side.  M.  Biot  and  Dr.  Seebeck  appear  about  the  same 
time  to  have  made  this  singular  and  interesting  discovery ;  hut  the  former  has  analyzed  the  phenomena  with 
particular  care,  and  it  is  from  his  Memoir  above  cited  that  we  extract  the  following  statements.  The  liquids  in 
which  he  observed  aright-handed  rotatory  property,  according  to  our  sense  of  the  word  above  explained,  in  which 
the  observer  is  supposed  to  look  in  the  direction  of  the  ray's  motion,  are  oil  of  turpentine,  oil  of  laurel,  vapour 
of  turpentine  oil,  and  an  alcoholic  solution  of  artificial  camphor  produced  by  the  action  of  muriatic  acid  on  oil 
of  turpentine.  The  left-handed  rotation  was  observed  by  him  in  oil  of  lemons,  syrup  of  cane  sugar,  and  alco- 
holic solution  of  natural  camphor.  In  all  these,  the  intensity  of  the  action,  or  the  velocity  of  rotation,  was 
much  inferior  to  quartz.  The  following  are  their  indices  of  rotation,  or  the  arcs  of  rotation  produced  by  one 
millimetre  of  thickness  in  the  plane  of  oolurization  of  a  certain  homogeneous  red  ray  chosen  by  M.  Biot  for  a 
standard,  as  calculated  from  his  data. 


LIGHT.  551 

BnRifl't"h"»'dr''  rmlex  of  rntanon.  Left-handed.  Index  of  rotation.  Farl  iy 

'  lock  crystal       +  18°.414  Rock  crystal -  18°.414  v,X-L 

Oil  of  turpentine     +    0°.271  Oil  of  lemon    _     0°436 

Ditto,  another  specimen +    0°251  Concentrated  syrup  of  sugar     -     0°.554 

Ditto,  purified  by  repeated  distillations   -f-     0°.2S6 
Oil  of  laurel 

Solution   of  1753  parts  of  artificial") 
camphor  in  17359  of  alcohol    . .  j" 

It  follows  further  from  M.  Riot's  researches,  that  when  any  two  or  more  liquids  are  mixed  together,  or  com-      1046 
bined  with  plates  of  rock  crystal,  the  rotation  produced  by  the  compound  medium  will  be  always  the  sum  of  the  Law  of 
rotations  produced  by  the  several  simple  ones,  in  thicknesses  equal  to  their  actual  thicknesses  present  in  the  mution  ,n 
combination    the  thicknesses  in  mixed  liquids  being  assumed  in  the  ratio  of  the  volumes  of  each  respectivelv   mixlure' 
that  calling  F  the  compound  thickness,  and  R  the  resulting  index  of  rotation,  we  shall  always  have' 

R.  T=  r.  t +,->.{  +  r".t"  +  &c. 

where  r,  /,  &c.  are  the  indices  (with  their  signs)  of  the  elementary  i  igredients,  and  t,  f,  &c.  their  thicknesses 
Thus,  when  66  parts  by  measure  of  oil  of  turpentine,  having  the  index  +  0.253  are  made   to  act  against  38  of 
oil  of  lemon,  we  have 

+  66  X  0.251  -  38  x  0.436  =  0.002, 

so  that  (these  thicknesses  ought  almost  exactly  to  compensate  each  other  ;  and  such  was,  in  fact  the  result  ot 
M.  Biots  experiment,  the  whole  pencil  transmitted  being  found  to  retain  its  primitive  polarization  without  the 
least  trace  of  an  extraordinary  image.  Again,  when  into  two  tubes  of  the  same  bore,  but  of  verv  unequal 
lengths,  equal  quantities  of  oil  of  turpentine  were  poured,  and  the  rest  of  their  leng-ths  filled  with*  sulphuric 
ether,  which  has  no  rotatory  property,  or  in  which  r  =  0,  the  two  compound  thicknesses  thus  differently  con- 
stituted gave  identically  the  same  tints  in  all  positions  of  the  analyzing  prism.  Thus  we  see  that  dilution  or 
mixture  which  only  separate,  without  decomposing  the  molecules,  do  not  alter  their  rotatory  power  Nay  even 
when  reduced  to  vapour,  M  Biot  found,  that  oil  of  turpentine  still  preserved  its  property  and  peculiar  character: 
and,  had  not  the  explosion  of  his  apparatus  prevented  accurate  measures,  would  probably  enough  have  been 
found  to  retain  the  same  index  of  rotation  allowing  for  the  change  of  density.  From  these  circumstances  he 
concludes  that  the  rotatory  power  is  essentially  inherent  in  the  molecules  of  bodies,  and  carried  with  them  into 
all  their  combinations.  But  this  is  too  rapid  a  generalization  ;  for  neither  sugar  nor  camphor  in  the  solid  state 
possess  this  property,  though  examined  for  it  in  the  same  circumstances  as  quartz  is,  by  transmittiim-  the  pola 
nzed  ray  along  their  optic  axes;  and,  on  the  other  hand,  quartz  held  in  solution  by  potash,  or  (as  Dr"  Brewster 
has  found)  melted  by  heat,  and  thus  deprived  of  its  crystalline  arrangement,  manifests  no  such  propert.  Thi« 
obscure  part  of  chemical  optics  well  deserves  additional  attention. 

M.  Fresnel's  researches  have  been  directed  to  the  rotatory  phenomena  with  the  same  brilliant  success  which      1047 
has  distinguished  his  other  inquiries  into  the  nature  of  light  ;  and  he  has  shown  that  they  may  be  explained  by  Fresnel's 
conceiving  the  molecules  of  the  ether,  which  propagate  rays  along  the  axis  of  quartz,  or  rotatory  fluids,  instead  theor.V  of 
of  vibrating  in  straight  lines,  to  revolve  uniformly  in  circles,  in  the  manner  explained  in  Art.  627,  (where  we  circular  P°- 
have  shown  (Corol.)  that  such  a  mode  of  vibration  may  subsist,  and  must  arise  from  the  interference  of  two  larization 
rectangular  vibrations  of  equal  amplitude,  but  differing  in  phase  by  a  quarter  undulation,)  and   bv  admitting 
that,  in  virtue  of  some  peculiar  mechanism  in  the  molecules  of  the  media  in  question,  such  circular  vibrations, 
when  performed  from  right  to  left,  bring  into  play  an  elasticity  slightly  different  from  that  which  propag-ates 
them  forward  when  performed   in  the  contrary  direction.     The  colours  produced   by  such  media  he  conceives 
to  originate  in  the  interference  of  two  pencils  thus  circularly  polarized,  and  lagging  the  one  behind  the  other 
by  an  interval  of  retardation  proportioned  to  their  difference  of  velocities. 

But  to  make  this  last  hypothesis  admissible,  it  is  incumbent  on  us  to  show  that  the  phenomenon  which  nece«-      1048 
sarily  accompanies  a  difference  of  velocities,  viz.  a  bifurcation  of  the  pencil  in  the  act  of  refraction  at  oblique  Peculiar 
surfaces,  really  takes  place.     This  has  accordingly  been  shown  by  M.  Fresnel,  by  an  experiment  which    though  (1«ub'e  «- 
of  great  delicacy,  is  decisive  and  satisfactory.     From  a  crystal  of  quartz  he  procured  to  be  cut  a  prism  havinn-  fractio" 
its  refracting  angle   150°,  and  its  faces  equally  inclined  to  the  axis ;  so  that  a  ray  traversing  it  internally  parallel  Produced 
to  its  axis  should  be  incident  at  equal  angles,  viz.  of  75°  on  either  face.     As  this  is  too  great  to  allow  of  the  laHv'ILi 
ray's  egress,  he  cemented  on  the   surfaces  the  two  halves  of  another  precisely  similar  prism  cut  from  another  W  medit 
rock  crystal  of  an  opposite  rotatory  character.     Thus  in  fig.  203,  A  C  B  is  the  first  prism,  and  the  side  C  B  of  the 
second   prism  CB  E   being  cemented  on  to  C  B,  this   prism  is  bisected  by  the  plane  BD    and  the  half  of  it 
D  B  E  transferred  to  the  other  side,  and  cemented  with  its  side  B  C  in  contact  with  A  C,  thus  producing  the 
achromatic   parallelepiped   F  A  BD  ;    so  that  if  a  ray  be  incident  on  Q  in  the  direction  PQ  parallel  to  the  base 
A  B,  i.  e.  to  the  axis  of  the  two  crystals,  it  will  traverse  all  three  in  the  direction  of  the  axes  of  their  spheroids 
of  double  refraction ;  and,  therefore,  so  far  as  the  Huygenian  law  of  double  refraction  is  concerned   ou»-ht  to 
undergo  no  division.     Now  it  is  evident,  that  if  the  ray  P  Q  be  at  its  entry  into  AFC  divided  into  two 'circular!  v 
polarized  in  opposite  directions,  the  one  (R)   moving  quicker  than   the  other  (I,,)  then,  at  quitting-  the  surf-ice 
A  C,  a  bifurcation  must  take  place,  the  ray  R  beiri"-  fca.,/,  and  L  most  refracted.     In  this  state  they  are  incident 
on  the  medium  A  C  B,  and  now  the  portions  R  and  L,  by  reason  of  the  opposite  nature  of  the  media  exch-ino-P 
velocities  ;  so  that  R,  which  at  its  emergence  from  the  &c-r  A  C  r-f  F  A  C  was  leant  refracted  upwards  will  now 


552 


L  I  G  II  T. 


L'L'tlt. 


1049. 

Chirncters 
ol  cirvuUr 
polarization 


1050. 

Other  cha- 
racter? of 
circularly 
polarized 
rays. 


1051. 


1052. 

Anot'itr 
mode  of 
producing 
circular  pti 
larization 


1053. 


1054. 


1055. 

Tints  pro- 
duced by 
circularly 
polarized 
raj  I. 


be  most  refracted  downwards ;  and  thus  the  separation  of  the  images  will  be  doubled,  and  the  sarm  will  take 
place  at  the  common  face  C  B.  Thus  this  combination,  both  from  the  doubling-  of  the  separation,  and  the 
greatness  of  the  angles  of  incidence,  is  peculiarly  well  adapted  to  render  sensible  any  bifurcation,  or  difference 
of  velocities,  however  small,  which  may  exist  along  the  axis.  Accordingly,  with  the  compound  prism,  so  con- 
structed, a  double  refraction  is  produced  ;  and  the  two  rays  are  really  observed  to  emerge,  making  a  sensible 
•m<rle  with  each  other. 

But  it  is,  moreover,  observed,  that  though  thus  separated  by  a  real  double  refraction,  the  two  pencils  have  not 
acquired  the  characters  which  double  refraction  usually  impresses  on  the  ordinary  and  extraordin-ry  ra\s.  at 
their  emergence,  but  very  different  ones.  In  common  cases  of  double  refraction  the  two  emerg  .-.'iicils  are 
each  wholly  polarized  in  opposite  planes,  and  either  of  them  when  examined  with  a  doubly  r,.  ...cling  prisrr 
gives  two  unequal  images,  one  alternately  more  and  less  bright  than  the  other,  as  the  prism  revues  through 
successive  quadrants.  This  is  not  the  case  with  the  two  pencils  produced  in  the  case  before  ir  i 

First,  Either  of  them,  when  examined  with  a  doubly  refracting  prism,  gives  constantly  tw,  .  nagus  of  equal 
intensity,  in  whatever  plane  the  principal  section  of  the  latter  be  placed.  In  this  respect,  then,  thqy  present  the 
characters  of  unpolarized  light,  and  may  be  regarded  as  each  consisting  of  two  rays  polarized  at  right  angles  to 
each  other.  But 

Secondly,  They  differ  from  ordinary,  or  unpolarized  light,  in  a  very  remarkable  property,  which  was  first 
discovered  by  Fresnel,  and  is  a  chief  distinctive  character  of  this  kind  of  polarization.  Suppose  either  of  thorn 
to  be  incident  at  right  angles  on  the  surface  A  B  of  a  parallelepiped  of  crown  glass  of  the  refractive  index  1.51, 
having  its  angles  ABC  and  ADC  each  54J°,  it  will  then  be  totally  reflected  at  the  internal  surface  B  C  ;  and 
(if  the  parallelepiped  he  long  enough)  again  in  the  same  plane  at  the  opposite  surface  A  D,  and  will  emerge  at 
length  perpendicularly  through  the  surface  B  C.  But  the  emergent  ray,  instead  of  comporting  itself  as  ordinary 
light,  will  now  be  found  to  be  completely  polarized  in  a  plane  45°  inclined  to  that  in  which  the  reflections  were 
made,  whatever  may  have  been  the  position  of  that  plane.  If  both  the  pencils  be  treated  in  this  manner,  it  will 
be  found  that  the  one,  after  its  two  total  reflexions  will  assume  a  plane  of  polarization  45°  in  azimuth  to  the  right, 
and  the  other  45°  to  the  left  of  the  plane  of  the  reflexions. 

Thus  we  see  that  the  effect  of  double  refraction  along  the  axis  of  quartz  is  to  impress  on  either  of  the  emer- 
gent pencils  opposite  polarizations,  or  modifications,  of  a  nature  totally  distinct  from  that  given  to  a  ray  by 
ordinary  reflexion,  or  by  double  refraction  through  Iceland  spar,  &c. ;  and,  as  in  the  last  described  experiment, 
so  long  as  the  ray  enters  perpendicularly  into  the  first  surface  of  the  glass  parallelepiped,  it  is  indifferent  in  what 
plane  the  two  reflexions  are  operated,  and  since  when  presented  to  a  doubly  refracting  prism  in  any  plane  indif- 
ferently it  always  divides  itself  into  two  equal  pencils,  it  is  evident  that  the  ray  thus  modified  h:is  no  sides,  i.  e. 
no  particular  relations  to  certain  regions  of  space ;  and  therefore  that  the  epithet  circular  polarization,  apart 
from  all  theoretical  considerations,  may  be  naturally  applied  to  this  peculiar  modification.  But  the  characters 
above  described  are  not  the  only  ones  belonging  to  a  ray  thus  modified,  for 

Thirdly,  Such  a  ray  being  transmitted  through  a  thin  crystallized  lamina,  and  parallel  to  its  axis,  is  divided 
by  subsequent  double  refraction  into  two  rays  of  complementary  colours,  thus  marking  a  decided  difference 
between  it  and  a  ray  of  common  light ;  while,  on  the  other  hand,  these  colours  are  not  the  same  with  those 
which  would  arise  from  a  ray  of  light  polarized  in  the  usual  way  and  similarly  analyzed,  but  differ  from  them 
by  an  exact  quarter  of  a  tint,  either  in  excess  or  defect,  as  the  ease  may  be. 

Fourthly,  A  ray  so  modified  by  this  peculiar  double  refraction,  when  transmitted  again  along  the  axis  of 
rock  crystal,  or  through  columns  of  oil  of  turpentine,  of  lemons,  &c.,  and  then  analyzed  by  a  double 
refracting  prism,  gives  rise  to  no  phenomena  of  colour,  differing  in  this  from  polarized,  and  agreeing  with 
common  light. 

Another  independent  mode  of  impressing  on  a  ray  all  this  assemblage  of  characters  has  been  discovered  by 
M.  Fresnel.  .  It  consists  in  inverting  the  process  described  in  Art.  1049.  Thus,  into  the  side  CD  of  the  glass 
parallelepiped  there  mentioned,  let  a  common  polarized  ray  be  introduced  at  a  perpendicular  incidence,  the 
parallelepiped  being  so  placed  that  the  plane  of  internal  reflexion  at  the  side  AD  shall  be  45°  inclined  to  that 
of  its  primitive  polarization.  Then,  after  undergoing  two  total  internal  reflexions  at  G  and  F,  it  will  emerge  at 
E  deprived  of  its  characters  of  ordinary  polarization  and  endowed  with  those  of  circular,  and  being  no  way 
distinguishable  from  one  of  the  pencils  produced  by  double  refraction  along  the  axis  of  rock  crystal 

It  remains  to  show,  however,  that  the  characters  here  described,  as  impressed  on  a  ray  by  transmission  along 
the  axis  of  rock  crystal,  are  really  those  which  ought  to  belong  to  a  ray  propagated  by  circular  vibrations.  And, 
first,  it  follows  from  Art.  627,  that  this  latter  ray  is  the  resultant  of  two  rays  polarized  at  right  angles,  and  dif- 
fering in  their  phases  by  a  quarter  undulation.  It  must,  therefore,  of  necessity  possess  the  first  character,  vit. 
that  of  division  into  two  equal  pencils  by  double  refraction  in  any  plane,  for  the  same  reason  that  unpolarized 
light  is  so  divided,  the  difference  of  phases  having  nothing  to  do  with  this  character. 

In  the  next  place,  a  ray  propagated  by  circular  vibrations  when  incident  on  rock  crystal  in  the  direction  of  the 
axis,  will  (by  hypothesis)  be  propagated  along  it  by  that  elasticity  which  is  due  to  the  direction  of  its  rotation, 
the  wave  then  will  enter  the  crystal  without  further  subdivision,  and  there  will  be  no  difference  of  paths,  or  iute' 
tering  rays  at  its  emergence ;  and,  of  course,  no  colours  produced  on  analyzing  by  double  refraction,  which  is 
another  of  the  characters  in  question. 

When  a  ray  propagated  by  circular  vibrations  is  incident  on  a  crystallized  lamina  it  may  be  regarded  as 
composed  of  two,  one  polarized  in  the  plane  of  the  principal  section,  the  other  at  right  angles  to  it,  of  aqnal 
intensity,  and  differing  in  phase  by  a  quarter  undulation.  Each  of  these  will  be  transmitted  unaltered,  and 
therefore  at  their  emergence  and  subsequent  analysis  will  comport  themselves  in  respect  of  their  interferences, 
just  as  would  do  the  two  portions  of  a  ray  primitively  polarized  in  azimuth  45°,  and  divided  into  two  by  the 


LIGHT.  553 

Light.      double  refraction  of  the  lamina,  provided  that  a  quarter  undulation  be  added  to  the  phase  of  one  of  these  latter     Part  IV 
— -/•-•"*•  rays.     Now  such  rays  will,  as  we  have  shown  at  length  in  Art.  969,  produce  by  the  interference  of  their  doubly  v«— v^~— ' 
refracted  portions,  the  ordinary  and  extraordinary  tints  due  to  the  interval  of  retardation  within  the  crystallize. 1 
lamina.     Hence,  in  the  present  case,  the  tints  produced  will  be  those  due  to  that  interval,  plus  or  minus  the 
quarter  of  an  undulation  added  to,  or  subtracted  from,  the  phase  of  one  of  the  portions  ;    and,   consequently, 
"rill  differ  one-fourth  of  a  tint,  or  order,  from  that  which  would  arise  from  the  use  of  a  beam  of  ordinary  polarized 
light  incident  in  azimuth  45°  on  the  lamina. 

There  v-«uains  but  one  more  character  of  the  rays  transmitted  along  the  axis  of  quartz,  which  we  must  show      1056. 
to  belon5';         u  ray  propagated  by  circular  vibrations,  viz.  that  described  in  Art.  1049.       But  in  order  to  this  it  Modifica- 
will  be  npcei'.ary  to  state  the  result  of  M.  Fresnel's  researches  on  the  modifications  which  light  undergoes  by  "^ 
total  reflex'nn  in  the  interior  of  transparent  bodies.  [--ht ),., 

When  a  ra  >larized  in  any  azimuth  is  incident  on  a  reflecting  surface  which  reflects  the  whole  of  the  inci-  total 
dent  light,  if  »ve  decompose  it  into  two,  the  one  having  its  vibrations  performed  parallel,  and  the  other  perpen-  reflexion, 
dicular  to  the  surface,  and  regard  each  of  these  as  independent  of  the  other ;  it  is  evident  that  the  reflexion  of 
these  portions  will  be  performed  under  very  different  circumstances,  the  ethereal  molecules  having  in  the  former 
case  to  glide  as  it  were  on  the  surface,  and  therefore  parallel  to  the  strata  in  which  their  density  is  constant,  while 
in  the  latter  each  molecule  in  the  act  of  vibration  will  pass  into  strata  of  variable  density.  The  reflexions 
therefore  will  be  performed  at  different  depths  in  the  two  cases  ;  and  from  this  cause  will  arise  a  difference  of 
route,  and  a  consequent  difference  of  phase  in  the  reflected  portions,  so  that  the  total  reflected  ray  will  no  longer 
be  capable  of  being  regarded  as  one  having  a  single  origin,  but  as  two  of  unequal  intensities,  oppositely  pola- 
rized, and  differing  in  phase  by  a  quantity  depending  on  the  angle  of  incidence  and  the  refractive  power  of  the 
medium.  PYom  peculiar  considerations,  of  a  delicate  nature,  and  depending  on  a  discussion  of  the  imaginary 
forms  assumed  by  the  general  expressions  for  the  intensity  of  a  ray  reflected  at  any  angle  (Art.  852)  when  applied 
to  the  case  of  total  reflexion,  M.  Fresnel  has  been  led  to  the  following  expression  for  the  difference  of  phases  (S) 
of  the  two  portions  in  question. 

_    2  ff  .  (sin  Q4  -  Qug  -f  1)  .  (sin  i)«  +  1 

G.«+l)(8inO<-  1 

where  fi  is  the  index  of  refraction,  and  i  the  angle  of  internal  incidence.  This  formula,  it  is  to  be  observed,  is 
given  by  him,  not  as  strictly  demonstrated,  but  merely  as  highly  probable,  as  an  interpretation  of  the  analytical 
meaning  of  the  imaginary  formula  alluded  to.  The  mode  of  its  deduction  being,  however,  independent  of 
experiment,  and  entirely  a  priori,  it  is  clear  that  if  found  verified  by  careful  experiment  in  circumstances 
properly  varied,  it  may  be  received  as  a  physical  law,  like  any  other  result  of  the  same  kind.  Now  we  have 
already  seen,  that  in  the  case  of  crown  glass,  where  fi  =:  1.51  and  i  =  54^°,  a  polarized  ray,  having  its  azimuth 
45°,  reckoned  from  the  plane  of  total  reflexion,  has  its  polarization  destroyed,  and  becomes  resolved  into  a  ray 
having  the  other  characters  of  a  resultant  from  two  differing  45°  in  p'hase,  by  two  total  reflexions  at  this  angle, 
(Art.  1056.)  But  if  in  the  above  formula  we  make  fi  =  1.51,  and  i  =  54°  37',  we  shall  find  S  =  45°,  and 
2  f  =  90°,  so  that  the  above  equation  is  verified  in  this  case.  M.  Fresnel  also  found  that  the  same  effect  was 
produced  by  three  reflexions  when  the  angle  of  incidence  was  69°  12',  and  by  four  when  74°  4"2',  both  agreeing 
with  the  formula  which  gives  in  the  former  case  5  =  ^  90°,  and  in  the  latter  S  =  j  90°,  for  the  difference  of  phase 
gained  or  lost  by  one  portion  on  the  other  at  each  reflexion.  Similar  verifications  were  obtained  by  performing 
two  reflexions  at  the  internal  surface  of  glass,  and  two  at  the  confines  of  glass  and  water  at  angles  of  68°  27'. 

It  appears,  then,  that  when  a  ray  polarized  in  azimuth  45°  undergoes  two  total  reflexions  at  the  angles,  and  [057 
in  the  manner  described,  it  becomes  circularly  polarized  ;  and  if  vice  versd,  the  two  elements  of  a  ray  so  circu-  Explana- 
larly  polarized  be  made  to  retrace  their  course,  they  will  reunite  into  a  ray  polarized  completely  in  one  plane,  tion  of  the 
Thus  we  see,  that  all  the  characters  of  the  rays  transmitted  along  the  axis  of  rock  crystal  agree  with  those  of  a  rotatory 
ray  so  compounded,  and  possessing  circular  polarization.  In  order,  then,  to  explain  the  phenomena  presented  Phe 
by  a  polarized  ray  when  incident  on  a  plate  of  this  substance  cut  at  right  angles  to  its  axis,  we  must  first  regard 
the  ray  as  resolved  into  two  others  (which  we  will  call  A  and  B)  of  equal  intensity  ;  the  one  A  polarized  in  a 
plane  45°  inclined  to  the  right,  the  other  45°  inclined  to  the  left  of  the  vertical,  (which,  to  fix  our  ideas,  we  shall 
take  for  the  plane  of  primitive  polarization.)  Now,  since  by  Art.  615  a  ray  polarized  in  any  plane  may  be 
regarded  as  equivalent  to  two  rays  each  of  half  its  intensity,  differing  in  their  phases  by  a  quarter  undulation, 
let  us  conceive  the  ray  A  as  resolved  into  two,  A  a  polarized  in  the  plane  -f-  45°,  and  having  its  phase 
advanced  -(-  .1  undulation,  and  anotlier  A  b  also  polarized  at  -f  45°,  but  having  its  phase  retarded,  or  —  ^ 
undulation,  so  that  A.  a  and  A  6  differ  £  undulation  in  their  phases.  Similarly,  let  B  be  regarded  as 
decomposed  into  Ba  polarized  at  —  45°,  and  having  its  phase  +  ^undulation,  and  B6  polarized  also  at 
-  45°,  but  having  its  phase  —  -J  undulation  different  from  B.  Thus  will  the  original  ray  be  resolved 
into  the  four  A.  a,  A.  b,  B  a,  B  6.  Now,  let  us  combine  these  two  and  two  in  a  cross  order,  then  A  a 
combined  with  B  b  will  be  equal  rays,  polarized  in  opposite  planes,  and  differing  £  undulation  in  their  phases, 
and  will  therefore  compose  one  circularly  polarized  ray,  in  which  the  rotation  is  from  right  to  left.  Similarly, 
the  pair  A  b.  Bo  will  compound  another  equally  intense  circularly  polarized  ray  having  its  rotation  the  contrary 
way.  Now  these  will  (ex  hypothesi)  be  transmitted  through  the  quartz  with  unequal  velocities,  and  thus  an 
interval  of  retardation  will  arise,  and  if  the  surface  of  egress  or  ingress  be  oblique  to  the  axis,  a  double  refrac- 
tion will  take  place ;  and  two  circularly  polarized  rays  will  emerge  in  different  directions,  as  experiments  show 
they  do.  If  perpendicular  they  will  emerge  superposed,  and  will  compound  one  ray.  Let  us  now  examine  what 
will  be  the  character  and  slate  of  polarization  of  this  compound  ray.  To  this  end  conceive  a  molecule  of  ether  C 
to  be  at  once  agitated  by  two  circular  motions  in  opposite  directions;  one  in  a  circle  equal  and  similar  to  A  P  in 
VOL.  iv.  4  c 


554  L  f  G  II  T. 

Light,  the  direction  A  P,  the  other  in  a  circle  equal  and  similar  to  B  Q,  and  in  the  direction  B  Q,  fig.  205.  Let  A,  B  Part  IV. 
*-— ~v— —'  be  two  molecules  setting:  °l't  at  once  from  A,  B  in  these  circles  with  equal  velocities,  then  will  the  motion  of  — —-v—— 
Fig.  206.  C  at  any  instant  be  equal  to  that  compounded  of  the  motions  of  A  and  B  at  that  instant.  When  A  comes  to  P 
let  B  come  to  Q,  then  arc  A  P  =  B  Q,  and  the  motions  at  P  and  Q  will  be  each  resolved  into  two,  those  of  which 
parallel  to  C  D  (a  perpendicular  to  P  Q)  conspire,  while  those  in  the  directions  P  I)  and  Q  D  parallel  to  P  Q 
oppose,  and  being'  equal  destroy  each  other;  thus  C  will  move  only  in  virtue  of  the  sum  of  the  two  former,  and 
its  vibrations  will  therefore  be  rectilinear,  and  in  the  plane  C  D  perpendicular  to  P  D  Q.  If  the  thickness  of 
the  plate  of  quartz  were  nothing,  or  such  that  the  interval  of  retardation  were  an  exact  number  of  undulations, 
A,  B  would  lie  at  opposite  extremities  of  a  diameter,  and  C  D  the1  new  plane  of  polarization  would  be  per- 
pendicular to  AM  that  diameter,  or  coincident  with  the  plane  of  primitive  polarization.  But  if  not,  the  quicker 
motion  will  have  pained  on  the  other  a  part  of  a  circumference.  M  B,  which  is  to  a  whole  circumference  as  the 
thickness  of  the  plate  is  to  that  which  would  produce  a  difference  of  a  whole  undulation  ;  and  at  the  emergence 
of  the  two  waves  into  air,  after  which  they  circulate  with  equal  velocity,  if  we  suppose  the  one  molecule  to  be 
setting  out  from  A,  the  other  will  be  setting  out,  not  from  M  the  opposite  extremity  of  the  diameter,  but  from  B, 
and  therefore  C  D  the  new  plane  of  polarization  (which  from  what  has  just  been  shown  must  always  bisect  the 
angle  A  C  B)  will  no  longer  be  coincident  with  C  N  the  primitive  plane  of  polarization,  at  right  angles  to  A  M, 
but  will  make  an  angle  1)  C  N  with  it  equal  to  half  B  C  M,  and  therefore  proportional  to  M  B,  or  to  the  interval 
of  retardation,  i.  e.  to  the  thickness  of  the  plate.  Thus  the  system  of  rays  emerging  from  the  rock  crystal  plate 
will  compound  one  ray  polarized  in  one  plane,  and  in  the  position  the  original  plane  would  have  had,  had  it  revolved 
uniformly  round  the  ray  as  an  axis  during  its  passage  through  the  plate.  Thus  we  have  a  complete  and  satis- 
factory explanation  of  the  apparent  rotation  of  the  plane  of  polarization,  as  observed  by  Biot  in  the  case  of  a 
homogeneous  ray. 

105S.  It  is  observed,  that  the  spectra  formed  by  the  double  refraction  of  rock  crystal  along  its  axis  are  very  highly  and 
unequally  coloured.  The  violet  rays  are  most  separated,  and  therefore  the  difference  of  velocities  of  the  two  rotating 
pencils  is  much  greater  for  violet  th  n  for  red  rays.  Consequently,  the  apparent  velocity  of  rotation  of  the 
plane  of  polarization  will  also  be  greater  for  the  violet  rays  in  the  same  proportion,  and  thus  arise  all  the 
phenomena  of  coloration  observed  and  described  by  M.  Biot.  It  is  scarcely  possible  to  imagine  an  analysis 
of  a  natural  phenomenon  more  complete,  satisfactory,  and  elegant.  With  regard  to  the  physical  reason 
of  the  difference  of  velocity  in  the  two  circular  polarized  pencils  within  the  quartz,  it  is  true  we  remain  in  the 
dark ;  but  the  fact  of  such  difference  existing  is  now  shown  to  be  no  hypothesis,  but  a  fact  demonstrated  by 
their  observed  difference  of  refraction,  and  by  the  observed  characters  of  the  two  emergent  rays. 

§  XI.   Of  the  Absorption  of  Light  by  Crystallized  Media. 

1059.  Crystallized  media,  endowed  with  the  property  of  double  refraction,  are  found  to  absorb  the  differently 
Absorption  coloured  rays  differently,  according  to  their  planes  of  polarization,  and  the  manner  in  which  these  planes  are 
of  polarized  presented  to  the  axis  of  the  crystal,  and  also  to  exert  very  different  absolute  absorbing  energies  on  rays  of  one 
double^re-  co'our  polarized  in  different  planes.  A  remarkable  instance  of  this  has  been  already  often  referred  to  in  the 
fracting  case  of  the  brown  tourmaline,  a  plate  of  which,  cut  parallel  to  the  axis,  absorbs  almost  entirely  all  rays  polarized 
crystals.  in  the  plane  of  the  principal  section,  and  lets  pass  only  such  among  oppositely  polarized  rays  as  go  to  con- 
stitute a  brown  colour. 

'  1060.          When  such  a  plate,  then,  is  exposed  to  natural  light,  since  at  the  entrance  of  each  ray  into  its  substance  it  is 

Property  of  resolved  into  two,  one  polarized  in  the  plane  of  the  principal  section,  and  one  perpendicular  to  it,  the  former  is 

the  tour-      absorbed  in  its  progress  by  the  action  of  the  crystal,  while  the  brown  portion  of  the  latter  escaping  absorption, 

but  retaining  at  its  egress  the  polarization  impressed  on   it,  after  traversing  the  plate,  appears  with  its  proper 

colour,  and  wholly  polarized  in  a  plane  at  right  angles  to  the  aris.     Thus  the  curious  phenomenon  of  the  pola- 

Explained.   rization  of  light  by  transmission  through  a  plate  of  tourmaline,  or  other  coloured  crystal,  is   explained,  or  at 

least  resolved  into  the  more  general  fact  of  an  absorbing  energy  varying  with  the  internal  position  of  the  plane 

of  polarization.     The  crystal,  in  virtue  of  its  double  refractive  property,  divides  the  ray  into  two,  and  polarizes 

them  oppositely ;   and  the  unequal  absorption  of  these  two  portions  tubtequently  causes  the  total  suppression  of 

one,  and  the  partial  of  the  other  of  the  portions  so  separated.     Thus  we  see  that  the  polarized  beam  obtained 

by  transmission  through  a  tourmaline  must  always  be  of  much  less  than  half  the  intensity  of  the  incident  light. 

106 1.          The  destruction  of  the  pencil  polarized  in  the  principal  section  is  not,  however,  sudden  ;  for  if  the  plate  of 

Gr.,UuaI       tourmaline  be  very  thin,  the  emerging  pencil  will  only  be  partially  polarized,  indicating  the  existence   in    it  of 

destruction  rays  belonging  to  the  other  pencil.     This  is  best  shown  by  cutting  a  tourmaline  into  a  prism  having  its  refract- 

ordinary       ing  edge  parallel  to  the  axis,  and  its  angle  small,  so  as  to  produce  a  wedge  whose  thickness  increases  not  too 

tay.  rapid'y.     If  we  look  through   this  at  a  distant  candle,  we  shall    see  only  one  image,  viz.  the  extraordinary 

through  the  back  of  the  wedge,  (if  thick  enough  ;)  but  :\s  tin-  eye  approaches  the  edge,  the  ordinary  image  appears 

at  first  very  faint,  but  increasing  in  intensity  till,  at  the  very  edge,  it  becomes  equal  to  the  other.     At  the  same 

time  the  colour  of  the  latter,  which  at  first  was  intense,   becomes  diluted  ;   and  the  images  approximate  not  only 

to  equality  of  light,  but  to  similarity  of  tint.     We  see  by  this,  too,  that  in  strictness  the  ordinary  pencil  is  never 

completely  absorbed  by  any  thickness,  however  great ;  but  as  it  diminishes  in   geometrical   progression  as  the 

thickness  increases  in  arithmetical,  the  absorption  may  for  ;ill  practical  purposes  be  regarded  as  total  at  moderate 

thicknesses 


L  I  G  II  T.  555 

Light  The  indefatigable  scrutiny  of  Dr.  BrewsUr,  to  wlio:n  we  owe  nearly  all  our  knowledge  on  this  subject,  has     part  |v 

•—  v"-""-'  shown  that  the  same  property  is  possessed  in  greater  or  less  perfection  by  the  greater  number  of  coloured  doubly  >__       _ 
refracting-  media;  and  the  expression  of  the  property  may  be  rendered  general  by  considering  all  doubly  refrac-      1062. 
tive  media  as  possessing  two  distinct  absorbing  powers  or  two  separate  scales  of  absorption  for  the  two  pencils,  Media  pos- 
or  (adopting  the  language  of  §  III.  part  2)  as  having  two  distinct  types,  or  curves  expressing  the  law  of  absorp-  sess  t«° 
tion  throughout  the  spectrum.     If  these  types  be  both  straight  lines  parallel  to  the  abscissa,  the  crystal  will  be  d'st"!r! 
colourless.     Such  are  limpid  carbonate  of  lime,  quartz,  nitre,  &c.     If  they   be  similar  and  equal  curves,  the  pov^rs'  ' 
medium,  although  coloured,  will  present  the  same  colour,  and  the  same  intensity  of  tint,  in  common  as  in  pola- 
rized light.     If  dissimilar,  or  if,  although  similar,  their  ordinates  are  in  :i  ratio  of  inequality,  the  character,  in 
the  former  case,  and  the  intensity  in  the  latter,  will  vary  on  a  variation  of  the  plane  of  polarization  of  the  inci- 
dent beam  ,  so  that  if  a  plate  cut  from  such  a  crystal  be  exposed  to  a  beam  of  polarized  white  light,  and  turned 
round  in  its  own  plane,  or  otherwise  inclined  to   the  beam,  its  colour  will  change  either  in  hue  or  depth  or 
both.     Dr.  Brewster  has  remarked   such    change  of  colour  and  the  phenomena  connected  with  it  in  a  great 
variety  of  crystals  both  with  one  and  two  axes,  of  which  he  has  given  a  list  in  a  most  interesting  Paper  on  the 
the  subject  in  the  Philosophical  Ttmuaatimu,  1819,  p.  1,  which  we  stronglv  recommend  to  the  reader's  perusal. 
It  may  be  familiarly  seen  in  a  prism  of  smoked  quartz  of  a  pretty  deep  tinge,  which  held  with  its  axis  in  the 
plane  of  polarization  appears  of  a  purple  or  amethyst  colour,  while  if  held  in  a  direction  at  right  angles  to  this 
position,  its  colour  is  a  yellow  brown. 

But  in  order  to  analyze  the  phenomena  more  exactly,  we  must  examine  the  two  pencils  separately.     To  this       1063 
end  Dr.  Brewster  took  a  rhomboid  of  yellow  carbonate  of  lime  of  sufficient  thickness  to  give  two  distinct  images  Absorption 
of  a  small  circular  aperture  placed  close  before  it,  and  illuminated  with  white  light,  when  he  observed  that  the  of  the  rays 
image  seen  by  extraordinary  refraction  appeared  of  a  deeper  colour  and  ess  luminous  than  the  other,  being  yu  'n  tn.e  two 

orange  yellow,  while  the  ordinary  image  was  a  yellowish  v.-hite.     He  found,  moreover,  that  the  difference  of  Pen.cll'ex~ 
'     J  J  „  ,J  run  i  ned  in 

colour  was  greater  as  the  paths  of  the  refracted  rays  within  the  crystal  were  more  inclined  to  the  axis,  being  0  crystalswiih 

when  the  rays  passed  along  the  axis,  and  a  maximum  when  at  right  angles  to  it.     If  we  denote  by  ¥„  and  Y,  one  axis. 
the  ordinates  of  the  curves,  expressing  the  law  of  absorption  as  in  Art.  490,  for  the  ordinary  and  extraordinary 
pencil  respectively,  these  will  both  therefore  decrease  as  we  proceed  from  the  red  to  the  violet  end  of  the  spectrum, 
corresponding  to  types  of  the  character  of  that  represented  in  fig.  114;  but  Y,  being  smaller,  and  decreasing 
more  rapidly  than  Y0.     Moreover,  since  Y0  =  Y.  in  the  axis,  and   since  as  we  recede  from  the  axis  Y,,  increases  Formutafor 
(because  the  colour  of  the  ordinary  pencil  becomes  whiter  and  more  luminous)  while  Y,  diminishes  by  the  same  the  light 
degrees,  (the  extraordinary  becoming  deeper  and  less  bright,)  we   shall  represent   both  these  changes  satis-  transmi«eii 
factorily  by  putting 

Y0  -  Y  (1  -f  k  .  sin  0«)  ;         Y.  =  Y  (1  —  k  .  sin  0J). 

These  give  Y0  -f-  Y,  =  2  Y  =  constant,  or  independent  of  0,  which  agrees  with  an  observation  of  Dr.  Brewster, 
that  in  every  situation  the  combined  tints  of  the  two  images  are  exactly  the  same  with  the  natural  colour  of 
the  mineral,  (which,  in  this  instance,  appears  to  have  been  alike  in  all  directions.) 

In  this  case,  then,  the  colour  of  a  plate  of  the  crystal  of  given  thickness  exposed  to  natural  light  will  be  the       1064. 
same,  whether  the  plate  be  cut  parallel  or  perpendicular  to  the  axis.     But  Dr.  Brewster  has  observed,  thu*.  this  Cases  of 
is  not  always  the  case,  but  that  great  differences  occasionally  exist  in  this  respect.     Thus  he  found,  that  in  some  two  distinct 
specimens  of  sapphire  the  colour  when  viewed  along  the  axis  was  deep  blue,  and  when  across  it  yellowish  green.  c°lours' 
In  Idocrase  an  orange-yellow  tint  is  seen  along  the  axis,  and  a  yellowish  green  across  it.     Specimens  of  tour- 
maline also  are  not  uncommon  in  which  the  tint  across  the  axis  is  green,  while  along  the  axis  it  is  deep  red  ; 
and,  in  general,  this  mineral  is  always  much  more  opaque  in  the  direction  of  the  axis  than  in  any  other;  so  much 
so,  indeed,  that  plates  of  a  very  moderate  thickness  cut  across  the  axis  are  nearly  impermeable  to  light.     One  of 
the  most  remarkable  instances  of  this  kind  we  have  met  with  is  a  variety  of  sub-oxysulphate  of  iron,  whicli 
crystallizes  in  regular  hexagonal  prisms,  and  which  viewed  through  two  opposite  sides  of  the  prism  is  light 
green,  but  along  the  axis,  a  deep  blood  red,  so  intense  that  a  thickness  of  -£$  inch  allows  scarcely  any  light  to 
pass.     It  is  obvious,  that  to  such  cases  the   formulae  of  the  last  article  do  not  extend.     But  a  slight  modifi-  Imestiga- 
cation  will  enable  us  to  embrace  the  phenomena  in  an  analytical  expression.     For  if  we  take  formulas  for 


ya  =  X.  +  Y.  .  sin  0*  ;         y.  =  X.  +  Y.  .  sin  6*  ;  ^eas  <=»«•• 

where  X0  ,  Y«  ,  &c.  as  well  as  y,  ,  y,  represent  functions  of  X  (the  length  of  an  undulation)  being  the  ordinates 
of  so  many  curves,  or  types  of  tints,  whose  relations'  are  to  be  determined,  we  have 

y.  +  y.=  (X.  +  X.)  +  (Y,  +  Y,)  sin  0*. 

Now  tliis  is  the  tint  which  a  sphere  of  the  medium  of  a  diameter  =  1  will  exhibit  when  viewed  by  natural  light 
along  a  diameter  inclined  0°  to  the  axis.  If  we  represent  by  A  and  B  the  ordinates  of  the  types  of  the  tints 
it  is  observed  to  exhibit  in  the  directions  of  the  axis,  and  perpendicular  to  it,  we  have,  when  0=0, 

y«  +  y.  =  A  =  x»  +  x,  ; 

and  when  0  =  90°, 

y.  +  y,  =  B  =  (X.  +  X.)  +  (Y.  +  Y.),  Expression 

whence  we  have  Y,  +  Y,  =  B  -  A  ;  ^*J 

and  the  tint  exhibited  by  ordinary  light  at  the  inclination  0  to  the  axis,  will  be  represented  by  transmitted 

in  common 

y,  +  y.  —  A  -f-  (B  -  A)  .  sin  6*,  ii(?ht 

=  A  .  cos  0»  -f  B  .  sin  0*. 

4  c  2 


556  LIGHT. 

Light      Thus  in  the  case  of  our  sub-oxysulphate  of  iron,  A  is  the  ordinate  of  the  type  of  a  deep  blood-red  tint,  and  B     P3r<  IV 
v~— v— '  in  like  manner  represents  a  bright  pale  green,  so  that  we  shall  have  at  any  intermediate  inclination  9  '— -v  — 

tint  =  (deep  red)  X  cos  0*  -J--  (light  green)  x  sin  0*, 

which  represents  faithfully  enough  the  gradual  passage  of  one  hue  into  the  other  as  the  inclination  changes. 
1065.          Suppose  now  the  incident  beam  polarized  in  any  plane,   and  let  the  plane  in  which  the  ray  and  the  axis  of 
When  illu-  the  sphere  lie  make  an  angle  =  a  with  that  plane.     Then  would  cos  a-  and  sin  a8  represent  the  intensities  of 
"olarizV31  the  ordinary  anj  extraordinary  pencils  which  superposed  make  up  the  emergent  beam,  were  the  crystal  limpid ; 
but  in  virtue  of  its  absorbent  powers,  they  will  be  reduced  respectively  to 

y,  =  cos  «s  (X,  +  Y.  sin  02),      and  y,  =  sin  a»  (X,  -f-  Y.  .  sin  0'), 

so  that  at  their  emergence  they  will  no  longer  make  up  white  light,  but  a  variable  tint  whose  type  has  for  its 
ordinate 

(X0 .  cos  «s  -f  X. .  sin  a"-)  -(-  (Y0 .  cos  «*  -j-  Y.  .  sin  a2)  .  sin  0*, 
in  which  it  will  be  recollected  that          X.  -f  X,  =  A,  and  Y.  -f-  Y.  =  B  —  A. 

To  determine  the  individual  values  of  X0,  &c.  however,  we  must  have  two  more  conditions,  and  these  will  be 
found  by  considering,  first,  that  in  the  direction  of  the  axis  the  tint  must  be  independent  of  a,  which  gives 
X,  .  cos  a4  -)-  X, .  sin  «s  independent  of  a,  and  therefore  X,=  X,,  and  either  of  them  =  A.  To  get  another 
condition,  let  the  tints  be  noticed  which  the  sphere  or  crystal  exhibits  when  its  axis  is  perpendicular  to  the  visual 
ray ;  and,  first,  coincident  with,  next,  perpendicular  to,  the  plane  of  polarization,  i.  e.  when  a  =  0,  and  a  =  90". 
These  are  respectively  X.  +  Y.,  and  X.  -f-  Y, ; 

and  calling  these  a  and  6,  we  have 

Y.  =  a  -  X.  =  a  -  A,     Y.  =  6  -  A. 
Hence  the  final  expression  for  the  tint  seen  in  polarized  light  will  be 

A  +  {  (a  —  A)  .  cos  a*  -j-  (6  —  A)  sin  a-  }  .  sin  &', 
that'is,  A  .  cos  0*  +  {  a  .  cos  a4  +  b  .  sin  «'  }  .  sin  O1, 

in  which  it  will  be  observed  that  a  and  6  are  complements  of  each  other  to  the  tint  B,  because 

a  +  6  -  X.  -f  Y0  -f  X.  +  Y,  =  B,  by  Art.  1064. 

1066.  Such  is  the  expression  for  the  apparent  hue  of  crystals   with  one   axis,  which   exhibit  a  variable  colour  in 
Dichroism.  common  or  polarized  light,  according  to  their  position  with  respect  to  the  incident  light.     The  phenomenon  in 

question  may  be  generally  termed  dichroism,  though  the  word  has  usually  been  applied  only  to  that  particular 
case  where  a  marked  change  in  the  character  of  the  tint  takes  place,  as  from  red  to  green,  &c. 

1067.  The  dichroism  of  biaxal  crystals  differs  in  many  of  its  phenomena  from   those   having  only  one  optic  axis. 
Dichroism    If  we  look  through  a  plate,  or  into  a  crystal  of  any  biaxal  mineral,  having  the  property  in  question,  illuminated 
in  biaxal      jjy  natural  light  in  such  a  direction  that  the  visual  ray  within  the  crystal  shall  pass  along,   and  in  the  immediate 
F7St206       neighbourhood  of,  one  of  the  axes,  we  shall  perceive  a  phenomenon  like  that  represented  in  fig.  206,  consisting 

of  two  similar  and  equal  sombre  spaces  A  B  one  on  either  side  of  the  pole  P,  and  of  the  principal  section  P  1", 
Colours  of  and  if  we  look  along  the  other  axis  P'  a  similar  pair  of  spaces  will  be  seen  in  its  neighbourhood.  In  the 
iolite.  mineral  called  dichroite  by  Hauy,  (on  account  of  the  striking  difference  of  its  colours  in  different  positions,)  or 

iolite  (from  its  violet  hue)  by  others,*  of  which  the  phenomena  have  been  described  by  Dr.  Brewster  in  the  Paper 
already  cited,  these  spaces  are  of  a  full  blue  colour,  while  the  intermediate  region  towards  O,  along  the  line  O  P  C, 
Phenomena  and  the  space  beyond  P  towards  C  are  yellowish  white.     In  epidote  the  sombre  spaces  are  brown,  and  the  region 
of  epidote.    around  O  and  in  the  principal  section  green,  of  a  greater  or  less  degree  of  dilution.     In  this  latter  mineral  (at 
least  in  some  of  its  more  ordinary  varieties  of  crystalline  form,  viz.  in  long  striated  prisms  much  flattened,  and 
terminated  by  dihedral  summits  placed  obliquely,  so  as  to  truncate  two  of  the  angles  of  the  prism)  the  pheno- 
mena are  seen  without  any  artificial  section,  merely  by  looking  in  obliquely,  across  the  axis  of  the  prism  ;    and 
the  same  is  true   of  many  other  minerals,  as,  for  instance,  the  axinite,   in   which  the  transition  of   colour  is 
extremely  remarkable  and  beautiful 

1068.  The  phenomena  of  dichroism  in  biaxal,  as  well  as  in  uniaxal  crystals,  are  evidently  related  to  the  optic  axes. 
Connection  an(j  depend  on  the  planes  of  polarization  assumed  by  the  intromitted  light,  during  its  transit  through  the  crystal 

f  the  phe-   ^Q  wnose  absorptive  power  it  is  subjected.     Now,  if  we  consider  the  form  and  situation  of  the  sombre  spaces 
wUh'the       where  the  greatest  absorptive  energy  is  exerted,  we  are  at  once  struck  by  their  analogy  with  those  occupied  by 
polarized      the  more  vividly  coloured  parts  of  the  rays  about  the  axes  in  the  situation  of  fig.  179.     That  figure  represents 
rings  and      (Art.  900)  the  extraordinary  set  of  rings  as  seen  in  a  crystal  whose  principal  section  is  in  the  plane  of  primitive 
°Plic  axes     polarization.     Fig.  207  represents  the  ordinary  or  complementary  set  as  seen  around  either  of  the  axes,  the 
pole  P,  and  the  principal  section  being  here  occupied  with  white  light,  and  very  bright,  in  consequence  of  its 
containing  the  whole  incident  light,  while  the  lateral  or  coloured  portions  occupied  by  the  rings  are  less  illu- 
minated, the  colours  originating  in  an  abstraction  of  certain  rays. 

Conceive  now  a  number  of  such  sets  of  coloured  rings  not  all  of  exactly  the  same  dimensions,  nor  having 

*  Mohs,  with  his  usual  contemptuous  disregard  of,  or  rather  hostility  to,  all  ordinary  convenience  and  received  usage,  chooses  to  call  this 
mineral  •' pritmatic  quartz''  Such  a  nomenclature  mutt  ere  long  work  out  us  own  destruction,  but  while  it  subsists  the  nuisance  is  intolerable. 
We  cannol  but  lament,  that  such  a  cause  should  exist  to  raise  up  prejudice  against  a  system  in  many  respects  so  useful  and  valuable 


LIGHT.  557 

Light,      precisely  the  same  pole,  but  very  nearly  so,  to  be  superposed  on  one  another,  then  would  the  colour*  be  obliterated    Part  IV. 
-~v—  -  and  blended  into  white  light  by  their  overlapping,  but  still  the  general  intensity  of  the  light  in  the  lateral  regions  ^—  -  v^-~ 
would  remain  much  feebler  than  in  the  principal  section,  and  the  effect  would  be  precisely  that  of  fig.  206,  viz.  Analogy  in 
two  sombre,  cloudy,  fan-shaped  spaces  traversed  by  a  narrow  ray  of  vivid  light,  opening  out  from  P  towards  C  T"8?60.'  of 
and  O.     Such  would  be  the  case  with  a  limpid  crystal,  supposing  such  a  slight  degree  of  confusion  of  structure  i'' 
as  to  produce  the  non-coincidence  ofthe  rays  from  all  its  molecules.     In  this  case,  however,  neither  of  the  spaces 
in  question  would  appear  coloured,  nor  would  the  phenomena  be  seen  at  all  without  the  use  of  polarized  light 
and  its  subsequent  analysis.     But  if  we  conceive   the  crystal,  instead  of  limpidity,  to  possess  the  property  of 
double  absorption,  the  suppressed  and  transmitted  portions  will  be,  not  white  light,  but  light  of  the  colour  of 
one  or  other  of  the  pencils  into  which  it  is  resolved  by  double  refraction,  according  to  its  plane  of  polarization 
and   the  thickness  of  the   medium  it  has  traversed  ;   and  the  analysis  of  the  emergent  ray  may  be  regarded  as 
performed,  at  least  imperfectly  by  the  difference  of  absorptive  powers  acting  differently  on  the  two  pencils.     In 
support  of  this  it  may  be  noticed,  that  when  we  examine  the  system  of  rings  in  the  usual  way,  by  polarized 
light,  in  crystals  presenting  the  above  phenomenon,  they  are  usually  found  to  be  very  irregular,  several  sets 
evidently  overlapping  and  interfering  with  one  another,  and  rendering  the  non-coincidence  of  all  the  axes  a 
matter  of  ocular  demonstration. 

In  Art.  931  we  investigated  the  law  of  intensity  ofthe  illumination  of  the  polarized  rings  in  different  parts  of     1069. 
their  periphery  for  uniaxal  crystals.     As  what  is  there  said  does  not  apply,  however,  to  biaxal  ones,  and  as  the  DIGRESSION 
present  subject  has  led  us  to  the  consideration  of  the  more  general  case,  it  will  not  be  irrelevant,  if  we  digress  Theory  ol 
at  this  point,  in  order  to  show,  what  modifications  the  statement  there  made  must  receive  to  embrace  the  phe-  rut$°riag» 
nomena  of  biaxal  crystals.  resumed" 

M.  Biot  has  stated  the  general  law  of  polarization  in  biaxal  crystals,  from  his  elaborate  researches  on  that  1070. 
subject  (Mem.  sur  les  Lois  Generates  de  la  Double  Refraction  et  Polarisation,  8fc.  Mem.  Acad.  Sci.  1819)  to  Biot'sge 
be  as  follows  :  neral  law 

If  two  planes  be  drawn  through  the  course  of  a  ray  within  a  crystal  and  through  the  two  optic  axes,  and  a  planes  of 
third  plane  bisecting  the  angle  included  between  the  two  former,  this  will  be  the  plane  of  polarization  if  the  ray  polarization 
be  an  ordinary  one  —  but  one  perpendicular  to  it  if  extraordinary.     Thus  in   fig.  209,  C  P  and  C  P'  being  the  in  biaxal 
optic  axes,  and  AC   a  ray  penetrating  the  crystal,  if  PA,  P'  A   be  joined   by   arcs   of  circles  on  the   sphere  crystals 
H  O  K  A  having  C  for  its  centre,  and  the  angle  PA  P'  be  bisected  by  the  arc  AN,  the  plane  AC  N  bisecting 
the  dihedral  angle  between  the  planes  P  C  A  and  P'  C  A  is  the  plane  of  ordinary  polarization,  and  a  plane  per- 
pendicular to  it  that  of  extraordinary.     This  is  the  law  of  fixed  polarization,  and  expresses  generally  the  planes 
of  polarization  assumed  by  the  two  rays  at  their  emergence  from  doubly  refracting  crystals.     It  is  a  consequence 
of  Fresnel's  general  theory,  (though   deducible  from   it  by  a  train  of  analytical  reasoning  far  too  intricate  and 
refined  to  allow  of  its  insertion  in  a  treatise  like  the  present,)  and,  having  been  experimentally  established  long 
before  that  theory  was  devised,  must  be  looked  on  as  a  strong  additional  proof  of  its  conformity  to  nature. 

The  doctrine  of  movable  polarization,  however,  which,  so  far  as  respects  the  phenomena  of  the  colours  and      1071. 
intensity  of  the  rings,  has  been  shown   by  M.  Biot  in  the  same  excellent  paper,  to  represent  with  fidelity  their  Doctrine  of 
various  affections,  whether  in  uniaxal  or  biaxal  crystals,  requires  the  resulting  ray  to  assume  at  its  emergence  a  movable 
plane  of  polarization  alternately  coincident  with,  and  making  with  the  primitive  plane  of  polarization  twice  the  poli 
angle  which  the  plane  of  fixed    polarization  so  determined  would  make;    so   that  if  we   draw  AM  (fig.  208)  hfaMlcrvs 
bisecting  the  angle  PAP',  the  emergent  ray  will  be  affected  by  subsequent  analysis,  as  if  polarized  either  in  the  tals. 
plane  of  primitive  polarization,  or  making  with  it  an  angle  equal   to  twice  C  M  A,  and  from  this  it  is  easy  to  Fig.  208. 
derive  the  law  of  intensity  in  question,  for  the  ray  by  which  the  point  A  of  the  rings  is  formed  consists  of  two  Law  of 
portions,  of  which  (A)  is  affected  by  subsequent  analysis  by  a  prism  of  Iceland  spar,  as  if  it  were  polarized  in  a  intensity  of 
plane  making  an  angle  2  C  M  A  =  Y'  with  the  plane  of  primitive  polarization,  in  which  we  suppose  the  principal  '*)'  rin"s  '" 
section  ofthe  analyzing  prism  to  be  placed,  and  the  other,  complementary  to  this  (1  —  A)   retains  its  primitive  pojnts'of 
polarization.     The  portion  A  then  will   be  divided  between  the  ordinary  and  extraordinary  image   in  the  pro-  their  peri 
portion  (cos  2  ^)'  :  (sin  2  YO2,  an<l  (considering  only  the  latter,)  A  being  its  intensity  at  its  emergence  from  the  phery. 
crystal,  A  .  (sin  2  Y')2  will  be  its  intensity  in  the  extraordinary  image,  or  in  the  primary  set  of  rings,  while  the 
whole  of  the  portion  1  —  A  will  pass  into  the  ordinary  or  complementary  set,  as  in  Art.  932,  so  that  we  have 
only  to  express  this  in  terms  ofthe  azimuth  ofthe  crystallized  plate  itself,  and  the  direction  of  the  ray  within  the 
crystal.     For  this  purpose,  put  a  =  angle  C  O  P  =  azimuth  of  the  principal  section  ofthe  plate  reckoned  from 
the  plane  of  primitive  polarization,  9  =  A  P,  tf  =  A  P',  and  let  us  (for  simplicity)  consider  only  at  present  the  case 
when  P  and  P'  are  near,  as  in  nitre,  so  that  arcs  of  circles  may  be  regarded  as  straight  lines,  and  spherical  as  plane 
triangles,  (see  Art.  907.)     Now  if  in  fig.  208  we  put  0  for  the  angle  P  N  A,  or  the  angle  made  by  the  plane  of 
ordinary  polarization  with  the  principal  section,we  shall  have  V  =  C  M  A  =  COP  +  MNO  =  COP-f-PNA 

/PAN2 

=  a  +  0.     To  find  0  we  have  only  to  consider  that  sin  02  =  I  =^r=  IX  sin  (P  A  N  =  £  P  A  P1)*  ;  but  since  N  A  Analyti- 

\rJN/  callye*- 

P  A  2  a  0  pressed. 

bisects  the  angle  of  the  triangle  PAP'  and  cuts  the  base,  P  N  =  P  P'  X  —  2    ~3«  and 

.r  A.  —  (—  A.  1          0  -J-  (? 

4  aa  _  (0  _  0f\* 


{  4  a2  —  (0       <?')*  } 

.„  t  <  —  '  ___  «  _ 


5.">8  LIGHT. 

A  more  symmetrical  value  of  0  will,  however,  be  had  by  expressing  the  value  of  sin  2  0,  which  being  equal  to      !'•'"'  IV. 
'  4  .  sin  0s  (1  —  sin  0*)  is  immediately  given  by  substitution  of  the  foregoing.     If  we  execute  the  reductions  we  v ./— * 

shall  find  that,  putting  S  for  -  '    '       =  half  the  sum  of  the  sides  of  the  triangle  PAP 

2  (0 -f- 0')  (0  -  <0       -/S  (S-fl)(S-0')  (S  -  sTa) 


2  </  S  (S  -  0)  (S  -  ff)  (S  -  2  a)  .  ...  ,,  ,  .    „  .  D/  . 

-  is  the  well  known  expression  for  the  sine  ot  the  angle  I  A  P  included 

between  the  sides  0,  0',  and  therefore  calling  this  angle  P,  we  have 

(6>  +  0')  (0  -  0') 
S!n20=-    -$F~    -•"nP- 

The  nature  of  this  expression  renders  the  transition  from  plane  to  spherical  triangles  easy,  and  we  may  conclude 
consequently,  that,  in  crystals  where  the  axes  make  any  angle  2  a,  that  if  we  take 

sin  (0  +  0') .  sin  (0  -  0') 

sin  2  0  = s—  -  -  sin  P, 

(sin  2  fl)a 

an(j  ^  =  a  -)-  0,  we  shall  still  have  the  intensity  of  the  extraordinary  rings  represented  by  A  (sin  2  ^)2,  and  that 

of  the  ordinary  by  1  —  A  +  A  .  (cos  2  ^y,  that  is,  1  -  A  (sin  2  ^)X  tneir  sum  being,  as  it  ought,  unity. 
1 07-)  The  black  cross  which  divides  the  system  of  the  primary  rings,  is  too  remarkable  a  feature  not  to  require  express 
Form  of  the  notice.  Its  form,  it  is  evident,  must  be  determined  by  the  condition  that  the  line  M  A  shall  be  everywhere  perpendi- 
folack  cross  cular  to  C  O  D,  in  whicli  circumstances  the  locus  of  A  will  be  a  curve  marking  out  its  central  or  blackest  portion, 
in  biaxal  >j<ne  problem  then  is  reduced  to  a  purely  geometrical  one,  Required  a  curve  P  A  such  that  a  line  drawn  from  A 
crystals  in-  D;sectjng  tne  angle  between  lines  A  P,  A  P'  drawn  to  two  given  points  P,  P',  shall  always  be  perpendicular  to  a 

<nven  line  COD.    To  resolve  this,  retaining  the  former  notation,  and  putting  O  M  =  x,  M  A  =  y,   O  A  =  r, 

we  have 

x  .  cos  a  +  y  .  sin  «        N 
cosAOP=  cos(AOM-  n)  =  -         —^-^  =  — , 

y  .  cos  a  —  x ,  sin  "        M 

sin  A  O  P  = =  — , 

r  r 

outtinir  N  and  M  for  the  respective  functions  in  the  numerators  of  the  fractions. 

Now  since  PAM  is  half  the  angle  PAP',  it  is  easy  to  see  that  we  must  have  2  x  angle  O'AM  =  PAO  -  PA  O 


But,  Cos=-rg       ;  cosP'AO= 

PO       aM 

and 

consequently  we  have,  first, 


sin 


2  xy      «M    <S'*  +  r*    -a'       £±±r£l  -  _i*L    (ff*      fin 
20  AM,  o'-T^-  '~      ~  e&         I  "  2>*00> 


and,  secondly, 

ci 
Now  we  have  further, 

yt-  =  d>  +  r '  +  •>  a  N  3  ; 

which  substi  uted  in  the  values  of  sin  2  O  A  M  and  cos  2  O  A  M  above,  give  the  equations 

x  y .  6  ff  =  a- .  M  N, 

(y!  —  a?)  .  6  6'  =  r4  +  a1  (M*      N*), 

and,  eliminating  9  9'  from  these,  we  obtain 

if  (y*  _  Xf)  .  M  N  =  xy  {  r4  +  a1  (M8  -  N')  }. 

In   this  it  only  remains  to  substitute  for  M  and  N  their  values  y  .  cos .«  -  *  .  sin  «,  and  y.  sina  +  c.CMa, 
which  done,  the  whole  will  be  found  divisible  by  r\  and  will  reduce  itaetf  to  the  very  simple  equation 


L  I  G  H  T.  559 

Light.  a*  Pa,t  jv. 

_     _m_.  x  y  =  a  .  sin  u  .  cos  «  =  —  .  sin  2  a. 

The  black  cross  then  is  an  hyperbola,  passing1  through  the  poles  P,  P',  and  having  the  planes  of  primitive  pola-  ax<,s6a,.e  6 
rization,  and  one  perpendicular  to  it  (C  D  and  c  d)  for  its  asymptotes,  and  which  as  a  approaches  to  0,  or  90°,  near  it  is  ar 
approaches  nearer  and  nearer  to  its  asymptotes,  with  which  it  at  last  coincides  in   the  limiting  case,  all  which  hyperbola 
particulars   are   exactly    conformable  to  fact,  and  may   easily   be  verified  by  turning  a  plate    of  nitre  round 
between    crossed   tourmalines.     When    the  inclination    of  the   axes  is  so  considerable,  that  the   rings  about 
both  poles  cannot  be  seen  at  once,  there  will  arise  modifications  from  the  substitutions  of  the  sines,  &c.  of  arcs 
for  the  arcs  themselves,  which  it  is  not  worth  while  to  enter  into. 

To  return  now  to  the  phenomena  of  dichroism.  That  portion  of  the  light  transmitted  by  a  biaxal  coloured  1073. 
medium  which  has  relation  to  the  optic  axes,  and  which  forms  the  sombre  brushes  of  colour  (in  fig.  206,)  and  Empirical 
the  bright  spaces  which  divide  them,  have  evidently  for  their  analytical  expression  a  function  of  the  form  formula 

Y.(cos20)'+B.(«dn20)';    (a)  Zphel 

where  Y  and  B  are  functions  of  X,  and  represent  the  ordinates  of  the  types  of  two  fundamental  tints,  <j>  represent-  dichroism 
ing  as  before  the  angle  PNA,  fig.  208,  or  the  angle  made  by  the  plane  of  ordinary  polarization  with  the  principal 
section.  But  besides  this,  the  phenomena  described  by  Dr.  Brewster,  as  exhibited  by  the  iolite,  require  us 
to  admit  two  other  portions,  which  may  be  more  naturally  referred,  not  to  either  of  the  optic  axes  but  to 
the  line  C  O  (fig.  209)  bisecting  them,  and  having  for  its  expression  a  function  of  the  form  a  .  cos  O  A*  + 
6  .  sin  O  A!.  In  this  mineral,  when  exposed  to  common  light  (or  to  polarized,  provided  we  place  its  principal  section 
at  right  angles  to  that  of  polarization,)  the  lateral  brushes  A,  B,  fig.  206,  are  blue,  and  the  bright  rays  which 
divide  them,  passing  through  the  poles  P,  P'are  white,  or  yellowish  white,  and  so  far  the  phenomena  agree  with 
the  expression  (a)  if  we  suppose  Y  to  represent  a  bright  yellowish  white,  and  B  a  blue.  But  according  to  that 
expression  alone,  the  blue  spaces  should  be  continued  down  to  the  equator  C  a  b  D,  fig.  206,  and  there  ought 
to  be  two  directions  C  D  and  a  b  in  which  the  mineral  viewed  transversely  to  the  axis  of  the  prism  (which  is 
perpendicular  to  the  plane  C  a  6  D)  should  appear  yellow,  and  two  others,  m  n  and  p  q,  in  which  it  should 
transmit  a  blue  colour,  while  in  the  direction  of  the  axis  O  it  should  appear  yellow.  Now,  on  the  contrary,  the 
equatorial  colour  is  nearly  uniform  and  pale  yellow,  while  that  along  the  axis  O  is  blue  ;  and  in  proceeding 
from  the  equator  toward  the  axis  O  of  the  prisrn,  the  yellow  diminishes,  and  the  blue  gains  strength,  whether 
we  set  out  from  C  and  D,  or  from  a  and  b,  precisely  as  would  be  indicated  by  the  other  formula 

y  .  (sin  O  A)*  +  6  .  (cos  O  A)8, 

y  representing  a  yellow  white  and  6.  a  blue  tint.     If,  therefore,  we  put  O  A  =  v,  the  joint  expression 
T  =  (Y  .  cos  2  <j?  +  B  .  sin  2  0")  -f-  (y  .  sin  *»  -J-  6  .  cos  v«)  ;    (6) 

will  be  found  to  represent  pretty  correctly  the  variations  of  colour  as  far  as  they  can  be  judged  of  by  the  eye. 
Thus,  at  O  where  v  =  o,  and  0  =  90°,  we  have  T  =  Y  -j-  b,  which  may  indicate  either  a  yellow,  a  white,  or  a 
blue,  according  as  we  suppose  Y  or  b  to  be  predominant.  The  fact  being,  that  the  tint  at  O  is  blue,  we  must 
suppose  the  latter  to  express  the  more  decided  colour.  As  we  proceed  from  O  along  the  sections  O  C,  O  D, 
or  O  a,  O  b,  in  both  of  which  sin  2  0  =  o,  we  have 

T  =  (Y  +  y  .  sin  V)  -J-  b  .  cos  *•  =  (Y  +  6)  +  (y  -  b)  .  sin  S 

Now  y  expressing  a  yellow  white  and  6  a  strong  blue,  y  —  b  will  express  a  proportionally  vivid  yellow,  and 
therefore  the  blue  tint  Y  +  6  seen  along  the  axis  will  be  diluted  with  more  and  more  yellow  as  we  approach  the 
equator;  at  P  P',  then,  (by  a  proper  assumption  of  numerical  values)  it  will  be  rendered  nearly  neutral,  after 
which  the  yellow  will  predominate,  and,  at  the  equator,  will  remain  alone  sensible,  the  expression  for  T  then 
becoming  T  =  Y  -J-  y,  at  the  points  C,  a,  b,  D.  Let  us  next  consider  the  case  when  cos  20  =  o,  or  0  =:  45°, 
that  is  to  say,  along  the  axes  or  most  intense  lines  of  the  lateral  brushes.  In  this  case  we  have 

T  =  B  -(-  (y  .  sin  x8  +  6.  cos  i/2)  =  (B  +  6  .  cos  v*)  -f-  y  •  sin  v*. 

Now  if  we  suppose  B  and  l>  to  represent  blue  tints,  since  (in  the  case  of  iolite)  the  angle  between  the  axes  or 
PP'  =  62°50'  and  O  P  =  31°  25',  we  have  in  the  immediate  vicinity  of  the  pules,  (sin  i')-  =  1  nearly,  and 
cos  ve  =f,  so  that  in  th?  immediate  neighbourhood  of  P  the  tint  of  the  most  intense  part  of  the  brushes  will  be 
B  +  ;j  b  +  J  y,  which,  on  very  reasonable  suppositions  of  the  numerical  values  of  B,  6  and  y  will  denote  a  full 
and  rich  blue.  But  as  we  approach  the  equator  at  m,  n,  p,  q,  cos  va  diminishing  and  sin  v!  increasing,  the  sombre 
tint  B  is  continually  more  feebly  reinforced  by  the  tint  6  .  cos  i>*  and  more  strongly  counteracted  by  y  .  sin  »*,  till 
at  length  it  will  be  overpowered,  and  the  colour  in  these  points,  as  in  C,  a,  D,  A,  will  be  yellow  only  somewhat 
less  decided  than  in  the  latter,  its  tint  being  represented  by  T  =  y  +  B  instead  n(  y  +  D. 

In  general,  if  we  put  A  for  the  tint  transmitted  along  the  axis  O  of  the  prisrn,  P  for  that  seen  along  the  poles,       1074 
L  for  that  of  the  lateral  branches  at  their  origin  close  to  the  poles,  and  E  for  the  mean  equatorial  tint,  we  shiill  Determiiia- 
have  for  determining  Y,  y,  B,  b,  the  equations  tion  o(  the 

coefficients 

A  =  Y  +  6,    2E  =  2y-f-B  +  Y,  <•„,„,,„, 


p  —  Y  _|.  y  .  Sin  a*  +  b  .  cos  a*  ;     L  =  B  -f-  y  .  sin  a*  +  b  .  cos  a', 

en  elimination  from  these,  it  will  appear  that  there  is  an  equation  ~.  condition  to  be  satisfied,  vu 

•2  (A  -  P)  =  (2  A  -  2  E       P  -f  L)  .  sin  a'  ;     (o) 


5(>0 


L  I  G  H  T. 


Light.      and  that  supposing  it  satisfied,  one  of  the  tints,  as  y,  will  (so  far  as  these  conditions  are  concerned)  remain     Pa"  IV. 
arbitrary,  and  the  others  will  be  given  by  the  equation  v—  x/-™» 


26  = 

in  which  y  must,  however,  be  such  as  to  render  Y,  B.  b  real  tints,  i.  e.  expressed  by  positive  numbers. 
1075.  To  apply  this,  for  example's  sake,  to  the  case  of  the  iolite,  let  us  regard  every  white  ray  as  consisting  of  two 
Application  complementary  rays  of  bright  yellow  and  bright  blue  of  equal  efficacy  ;  and  suppose  that  by  observation  we 
have  ascertained  its  equatorial  tint  E  to  be  a  very  pale  but  strongly  luminous  yellow  white,  consisting  of  110 
suc!i  yellow  rays,  and  99  such  blue  ones,  producing  a  joint  intensity  =  209.  Moreover,  let  the  tint  seen  along 
the  axis  of  the  prism  (A)  be  a  blue,  of  a  good  colour,  but  considerably  less  intensity,  represented  by  10  such 
yellow  -f-  20  such  blue  rays  =  30.  That  seen  along  the  optic  axes  (P)  to  be  a  white  represented  by  36  yellow 
-f-  36  blue  =  72,  and  that  of  the  most  intensely  coloured  portions  of  the  lateral  brushes  =  L  to  be  a  stronger 
blue  than  that  seen  in  the  axis  of  the  prism,  such  as  may  be  represented  by  28  yellow  -f-  66  blue  =  94. 
These  numbers  are  chosen  so  as  to  satisfy  the  equation  of  condition,  taking  a  =  30°,  aud  if  we  substitute  them 
we  shall  find 

y  +  y  =  1  14  yellow  -f-  84  blue  ;       B  -f  y  =  106  yellow  +  1  14  blue  ;       y  -  b  =  104  yellow  +  64  blue, 

y  remaining  indeterminate  ;  if  we  suppose  its  composition  to  be  m  yellow  +  n  blue,  we  may  determine  m  and 
n  by  the  two  conditions  that  b  shall  (as  we  have  before  supposed)  represent,  a  pure  blue  without  any  mixture 
of  yellow,  and  Y  a  very  pale  yellow,  such  as  would  result  from  a  mixture  of  yellow  and  blue  in  the  ratio  of  10 
to  9.  These  conditions  are  satisfied  by  taking  m  =  104  and  n  =  75  ;  so  that  we  have,  finally, 


Y  =     10  yellow  +     9  blue  ; 
y  =  104  yellow  +  75  blue  ; 


B  =  2  yellow  -f-  39  blue  ; 
6=0  yellow  -4-  11  blue  ; 


1076. 

Phenomena 
exhibited 


1077. 

Unequal 

ts  °^ 

colours  of* 
ihe  two 
pencih-. 


and  these  being  taken  for  the  values  of  the  coefficients  in  the  expression  (6)  Art.  1073,  it  will  be  found  on  trial 
to  reproduce  the  tints  actually  observed.  In  fact,  the  extreme  equatorial  tints  being  y  +  Y  and  y  +  B,  will  be 
respectively  represented  by  114  yellow  -j-  84  blue,  and  106  yellow  +  114  blue;  the  former  is  a  very  pale 
yellow,  but  highly  luminous,  being  equivalent  to  30  rays  of  yellow  diluted  with  168  of  white;  while  the  latter  is 
a  blue  so  pale  as  to  be  umlistinguishable  fiom  white,  and  also  highly  luminous,  being  equivalent  to  8  rays  of 
blue  diluted  with  212  of  white. 

The  reader  will  perceive  that  the  formula  in  question  is  merely  empirical,  and  that  more  numerous  experi- 
ments  than  we  possess  will  be  required  to  establish  or  disprove  it.  It  is  unfortunately,  however,  difficult  to 
meet  with  biaxal  crystals  sufficiently  dichromatic  for  the  purposes  of  decisive  experiment,  and  at  the  same  time 
^arSe  and  transparent  enough  to  admit  of  being  cut  into  the  forms  and  examined  in  the  directions  required, 
through  a  thickness  sufficient  for  a  full  developement  of  their  colours.  Such  are  indeed  hardly  less  rare  than 
the  most  precious  gems  ;  and  this  circumstance  is  a  great  obstacle  to  the  advancement  of  our  knowledge  in  one 
of  the  most  interesting  branches  of  optical  inquiry,  which  that  of  dichroism  certainly  deservus  to  be  considered. 
Among  artificial  crystals,  however,  there  is  room  to  suppose  that  subjects  fit  for  such  experiments  may  be  met 
with.  One  remarkable  instance  of  dichroism  among  these  has  been  mentioned  in  the  sub-oxysulphate  of  iron. 
To  this  we  may  add  the  potash-muriate  of  palladium,  which  exhibits  along  the  axis  of  the  four-sided  prism  in 
which  it  crystallizes  a  deep  red,  and  in  a  transverse  direction  a  vivid  green.  (VVollaston,  Phil.  Trans.  1804.  On 
a  new  metal  in  Crude  Platina.)  The  curious  property  of  the  pnrpurates  of  ammonia,  potash,  &c.  described  by 
Dr.  Prout,  (Phil.  Trans.  1808,)  which  by  transmitted  light  exhibit  an  intense  red,  and  by  reflected,  on  one 
surface,  a  dull  reddish  brown,  and  on  another  a  splendid  green,  appears  referable,  not  so  much  to  the  principles 
of  dichroism  properly  so  called,  as  to  some  peculiar  conformation  of  the  green  surfaces,  producing  what  may  be 
best  termed  a  superficial  colour,  or  one  analogous  to  the  colour  of  thin  plates,  and  striated  or  dotted  surfaces. 
A  remarkable  example  of  such  superficial  colour,  differing  from  the  transmitted  tints,  is  met  with  in  the  green 
fluor  of  Alston-moor,  which  on  its  surfaces,  whether  natural  or  artificial,  exhibits,  in  certain  lights,  a  deep  blue 
tint,  not  to  be  removed  by  any  polishing. 

Dr.  Brewster  has  shown  that  the  action  of  heat  often  modifies  in  a  very  remarkable  manner  the  colour  of 
doubly  refracting  crystals,  producing  a  permanent  change  in  the  scale  of  absorption  of  the  crystals  as  affecting 
one  of  the  pencils  and  not  the  other.  Thus,  having  selected  several  crystals  of  Brazilian  topaz  which  displayed 
no  change,  of  colour  by  exposure  to  polarized  light,  (and  in  which,  of  course,  the  types  of  both  absorptions 
must  have  been  alike,)  and  bringing  them  to  a  red  heat,  or  even  boiling  them  in  olive  oil,  or  mercury,  they  expe- 
rienced  a  permanent  change,  and  had  acquired  the  property  of  absorbing  polarized  light  unequally.  He  then 
took  a  topaz  in  which  one  of  the  pencils  was  yellow  and  the  other  pink  ;  and  by  exposing  it  to  a  red  heat,  he 
found  the  extraordinary  pencils  more  powerfully  acted  on  than  the  ordinary,  the  yellow  colour  being  discharged 
entirely  from  the  one,  while  only  a  slight  change  was  produced  in  the  pink  tint  of  the  other.  This  change  of  colour 
in  the  topaz  by  heat  (though  not  its  intimate  nature)  is  well  known  to  jewellers,  who  are  in  the  habit  of  thus 
developing  in  this  gem  a  colour  more  highly  prized.  It  is  remarkable,  that  while  hot  the  topaz  is  perfectly  colour- 
less, and  acquires  the  pink  colour  gradually  in  cooling.  By  the  repeated  action  of  very  intense  heat  Dr. 
Brewster  was  never  able  to  modify  or  remove  this  permanent  pink  tint.  How  far  violent  compression,  slow 
application,  and  abstraction  of  the  heat,  or  other  mollifying  circumstances,  might  prevent  its  dercloperaent,  i: 


L  I  G  H  T.  5G1 

Light,      would  be  interesting  to  examine  ;    since  we  cannot  help  being  otherwise  struck  by  the  force  of  the  argument    Part  IV. 
— ->,-"•••'  geologists  may  draw,  from  the  existence  in  rocks  of  a  mineral  which  mere  elevation  of  temperature  unaccompanied  ^— - \—~-> 
with  change  of  composition,  thus  irrevocably  alters. 

One   general  character  of  all  dichroite  bodies  is,  that  when  natural  light  is  transmitted  through  a  plate  of     1078. 
sufficient  thickness,  in  any  direction  not  coincident  with  one  of  the  optic  axes,  the  emergent  beam  is  wholly  or  General 
partially  polarized  by  reason  of  the  unequal  action  of  the  medium  on  the  two  pencils,  and  the  consequent  sup-  *j.  °f 
pression  of  one  of  them.     And,  in  general,  whatever  cause  tends   to  interfere  unequally  with   their  free  trans-  crvsl;l|s_ 
mission  through  a  medium,  will  produce  a   similar  effect.     Thus,  for  example,  if  the  continuity  of  a  doubly  Effects  oi 
refracting  medium  be  interrupted   by  a  film  of  any  uncrystallized  substance,  since  the  two  pencils  by  reason  ot'ui>crysta!- 
their  angular  separation  are  incident  on  this  film  at  different  angles  ;  and  since,  moreover,  their  relative  refractive  'ize(^ lnter 
indices,  with  respect  to  the  medium  composing  the  film,  differ,  they  will  undergo  partial  reflexion  at  the  film  in  fi"^'"5 
different  proportions,  and  thus  an  inequality  will  arise  in  the  parts  transmitted.     If  the  refractive  index  of  the 
film  be  precisely  equal  to  the  ordinary  refractive  index  of  the  crystal  (supposed,  for  simplicity,  to  be  uniaxal) 
the  ordinary  ray,  it  is  evident,  will   undergo  no  disturbance   or  diminution,  while  the  extraordinary   will    be 
changed  in  direction  and  diminished  in  intensity  by  partial  reflexion  at  its  ingress  and  egress,  at  every  such  film 
which  may  exist  in  the  medium.     If  the  films  be  extremely  numerous,  and  if,  moreover,  they  be  not  disposed 
in  planes,  but  in  undulatory  or  irregular  surfaces  through  the  medium,  this  will  make  no  difference,  so  far  as  the 
ordinary  ray  is  concerned,  which  will  still  pass  undisturbed  through  the  system,  (except  so  far  as  any  opacity  in 
the  matter  of  the  films  may  extinguish  a  portion  of  it ;)  but  the  extraordinary  ray  will  be  rendered  confused, 
and  dispersed,  its  egress  from  the  films  not  being  performed  (by  reason  of  their  curvature)  at  the  same  angles 
as  its  ingress,  and  that  irregularly,  according  to  their  varying  inclination.     Hence  will  arise  a  phenomenon  pre-  Phenomena 
cisely  such  as  is  presented  by  the  agate,  and  other  irregularly  laminated  bodies,  through  plates  of  which,  if  a  of  agate- 
luminary  be   viewed,  it  is  seen  distinctly,  but   as  if  projected  on   a  curtain   of  nebulous   light  ;    and    if  ex- 
amined with  a  tourmaline,  or  doubly  refracting  prism,  the  distinct  image,  and  the  nebulous  light,  are  found  to 
be  oppositely  polarized.      If  we  examine  a  piece  of  agate  with  a  magnifier,  the  laminated  structure  and  unequal 
refraction  of  the  laminae  are  very  apparent ;    it  appears  wholly  composed  of  a  set  of  exceedingly  close  layers, 
not  arranged  in  planes,  but  in  undulating  or  crinkled  lines  like  a  number  of  figures   of  333333   placed  close 
together.     The  planes  of  polarization  of  the  nebulous  and  distinct  image  are  parallel  and  perpendicular  to 
the   general  direction  of  the  layers,  which  through  any  very  small  portion  of  the  substance  is  generally  pretty 
uniform. 

But  the  film  interposed  may,  itself,  be  crystallized,  and  inserted  between  adjacent  portions  of  a  regular  crystal,  1079. 
according  to  the  crystallographic  laws  which  regulate  the  juxtaposition  of  the  molecules  at  the  common  surfaces  Action  of  c 
of  macled  or  hemitrope  crystals.  Let  A  D  E  F  (fig.  210)  be  such  a  plate  interrupted  by  a  crystallized  lamina  crystalllz.ed 
B  C  E  F,  bounded  by  parallel  planes,  and  let  us  consider  what  will  happen  to  a  ray  S  a  incident  at  a.  It  is  ^'j 
evident,  that  were  the  crystallized  lamina  away,  or  were  its  molecules  homologously  situated  with  those  of  the  tv.  210. 
portions  on  either  side  of  it ;  in  the  latter  case,  we  should  have  an  uninterrupted  crystal ;  in  the  former,  two 
prisms  disposed  with  their  principal  sections  parallel,  and  acting  in  opposition  to  each  other;  in  either  case,  the 
emergent  ordinary  and  extraordinary  pencils  separated  by  double  refraction  at  the  first  surface  will  emerge 
parallel  to  the  incident  ray,  and  therefore  to  each  other.  But  the  principal  section  of  the  crystallized  film  being 
non-coincident  with  those  of  the  two  prisms  ABE,  C  FG,  it  will  alter  the  polarization  of  the  portions  ab,  ac; 
and  in  place  of  their  being,  as  in  the  former  case,  each  refracted  singly  by  the  second  prism  CFG,  they  will  now 
each  be  refracted  doubly,  so  that  in  place  of  two  emergent  rays  there  will  now  be  four.  The  subdivision  of  the 
ra\s  within  the  interposed  lamina  may  evidently  be  disregarded,  for  they  will  be  refracted  in  passing  from  the 
film  into  the  second  prism  in  the  same  direction,  where  contiguous,  as  they  would  were  an  infinitely  thin  plate  of 
air  interposed.  Now,  in  that  case,  they  would  emerge  from  the  film  in  pairs  respectively  parallel  to  the  incident 
rays  ab,  ac,  and  therefore  to  each  other.  Hence  the  refraction  at  the  second  prism  will  be  precisely  the 
same  as  if  the  lamina  were  suppressed,  and  in  its  place  the  rays  ab,  ac  had  received  at  a  the  polarizations 
they  acquire  by  its  action.  Now,  these  being  in  opposite  planes,  it  is  evident  that  each  of  the  rays  a  b,  ac 
would  undergo  both  an  ordinary  and  an  extraordinary  refraction.  Let  us  denote  these  four  emergent  pencils 
so  arising  by  O  O,  O  E,  E  O,  E  E,  and  suppose  a  b  to  be  the  direction  taken  by  the  ordinary  refracted  portion 
of  S  a,  and  a  c  that  of  the  extraordinary.  Then,  since  O  O  has  been  refracted  ordinarily  by  the  prism  CFG, 
and  was  incident  on  it  in  the  direction  of  the  ordinary  ray  a  b,  its  direction  on  emerging  will  be  parallel  to  S  a. 
Similarly,  E  E  is  refracted  extraordinarily,  and  being  incident  in  the  direction  6  c  of  the  extraordinary  portion 
of  S  a,  it  also  will  emerge  parallel  to  S  «,  and  thus  the  two  rays  O  O,  E  E  will  emerge  parallel,  and  their 
systems  of  waves  will  be  superposed.  But  the  portions  O  E  and  E  O,  the  one  being  incident  in  the  ordinary 
direction,  but  refracted  extraordinarily,  the  other  incident  in  the  extraordinary  direction  and  refracted  ordinarily, 
will  neither  emerge  parallel  to  the  original  ray  S  a,  nor  to  each  other;  and  this  will  give  rise  to  two  lateral 
images,  one  on  each  side  of  the  central  or  direct  image,  which  will  have,  moreover,  an  intensity  equal  (except 
in  extreme  cases)  to  the  sum  of  those  of  the  lateral  images. 

If  the  film  E  B  C  V  be  very  thin,  or  if  either  of  iis  optic  axes  be  nearly  coincident  with  the  direction  in  which      1080. 
the  light  traverses  it,  the  difference  of  paths  and  velocities  within  it  will  give  rise  to  an  interference  of  the  pairs  Phenomena 
of  rays  going  to  form  either  pencil  emergent  from  the  film,  and  thus  will  arise  the  colours  of  the  rings  in  each  of  >nter- 
image.     Those  on   either  side  the  central  one  will  be  consequently  tinged  with  the  respective  colours  of   the  ™P  e 
primary  and   complementary  set  of  rings  ;  while  the  central  image,  being  formed  by  the  precise  superposition  of  Spar 
two  similar  complementary  pencils  will  appear  white. 

All  these  phenomena  actually  occur,  and  have  been  described  by  Dr.  Brewster,  and  explained  by  him  on  the 
principles  here  laid  down,  in  certain  not  uncommon  specimens  of  Iceland  spar,  which  are  interrupted  by  such 

VOL   iv.  4  D 


562 


LIGHT. 


Light. 


1081. 

Phenomena 
Df  irlio- 
cyclopha- 
nnus 
crystals. 


1082. 
Fig.  211. 


hemitrope  films,  passing  through  the  longer  diagonals  of  opposite  faces  of  the  primitive  rhomb.     If  we  look  at 
a  candle  through  such  an  interrupted  rhomb,  it  will  be  seen  accompanied  by  a  pair  of  lateral  images  such  as  ' 
here  described,  and  exhibiting  frequently  the  complementary  tints  with  great  splendour. 

If  the  luminary  from  which  the  ray  S  a  issues  be  small,  the  lateral  images  will  be  separated  by  a  dark  interval 
from  each  other  and  from  the  central  one,  but  if  large  they  will  overlap.  If  infinite  (as  where  the  uniform  light 
of  the  sky  is  viewed)  all  the  images  will  be  superposed.  But  the  field  of  view  will  not  necessarily  be  uniform 
and  white.  The  c>  ntral  image  will  form  an  intense  white  screen,  or  ground,  on  which  will  be  projected  the  lateral 
ones.  Now,  if  the  film  be  so  constituted  as  to  have  within  the  visible  field  of  view  of  one  only  of  the  lateral 
images  the  pole  of  one  of  its  sets  of  rings,  (which  will  be  the  case  whenever  one  of  its  optic  axes  is  not  very- 
remote  from  perpendicularity  to  the  surface  of  the  plate  A  D,  so  as  to  admit  of  one  of  the  rays  O  E  or  E  O 
traversing  the  film  in  the  direction  of  its  axis,)  that  set  of  rings  will  not  be  seen  projected  centrally  on  the  cor- 
responding set  complementary  to  it  of  the  other  lateral  image,  by  reason  of  the  angular  separation  of  these  two 
images.  Of  course  its  colours  will  not  be  neutralized,  and  it  will  be  visible  per  se,  though  very  faint,  being 
diluted  by  the  whole  white  light  of  the  central  image  (O  O,  E  E)  and  by  the  whole  visible  and  nearly  uniform 
portion  of  the  other  lateral  one  (O  E.) 

This  is  not  the  only  way  in  which  a  crystal  perfectly  colourless  may  exhibit  its  sets  of  rings  by  exposure  to 
common  daylight  without  previous  polarization,  or  without  subsequent  analysis  of  the  transmitted  pencil.  The 
general  mass  of  the  crystallized  plate  may  have  one  of  its  optic  axes  in  the  direction  of  the  visual  ray,  as  in 
fig.  211,  and  the  portion  of  it  C  Ddc  included  between  two  films  B  Ccb  and  DdeE  will  then  form  precisely 
such  a  combination  as  that  above  described,  and  will  exhibit  a  set  of  rings  feeble  in  proportion  to  the  rarity  and 
minuteness  of  the  films,  and  the  consequently  small  area  of  their  outeropping  surfaces  B  C,  D  E.  These  are  not 
hypothetical  cases.  Dr.  Brewster  states  himself  to  have  met  with  specimens  of  nitre  exhibiting  their  rings  per 
se.  Such  are  rare.  But  in  the  bicarbonate  of  potash  it  is  an  accident  of  continual  occurrence ;  and,  indeed, 
almost  universal.  The  films  in  both  cases  are  easily  recognised,  and  their  position  and  that  of  the  system  of 
rings  seen  leave  no  doubt  of  the  correctness  of  the  explanation  here  given.  Such  crystals,  of  which  more  will 
no  doubt  be  hereafter  recognised,  may  be  termed  idiocydophanous  till  a  better  term  can  be  thought  of. 


§  XII.    On  the  effects  of  Heat  and  Mechanical  Violence  in  modifying  the  action  of  Media  on  Light,  and  on  the 
application  of  the  Undulatory  Theory  to  their  explanation. 


10S3. 
General 
account  of 
the  phe- 
nomena. 


1084. 
Accompa- 
nied  by 
double  re* 
fraction. 

1085. 
Effect  of 
heat  ana- 
logous to 
that  of 
pressure 


1086. 


It  was  ascertained  independently,  and  about  the  same  time  by  Dr.  Seebeck  and  Dr.  Brewster,  that  when  glass, 
which  in  its  ordinary  state  offers  none  of  the  phenomena  of  double  refracting  media,  is  heated  or  cooled 
unequally,  it  loses  this  character  of  indifference,  and  presents  phenomena  of  coloration,  &c.  analogous,  in  many 
respects,  to  those  exhibited  by  doubly  refracting  cnstals.  If  the  heat  communicated  be  below  the  temperature 
at  which  glass  softens,  the  effect  is  transient,  and  vanishes  when  the  glass  attains  a  uniform  temperature 
throughout  its  substance,  whether  by  the  equable  distribution  of  the  caloric  throughout  its  mass,  or  by  its 
abstraction  in  cooling..  But  if  the  temperature  communicated  be  so  high  as  to  allow  the  molecules  of  the  glass 
to  yield  to  the  mechanical  forces  of  dilatation  and  contraction  produced  in  the  act  of  cooling  and  take  a  new 
arrangement,  the  effect  is  permanent,  and  glass  plates  so  prepared  have  many  points  of  resemblance  with  crys- 
tallized bodies.  Dr.  Brewster  afterwards  ascertained,  that  mechanical  compression  or  dilatation  applied  to 
glass,  jellies,  gums,  and  singly  refractive  crystals  (such  as  fluor  spar,  &c.)  is  capable  of  imparting  to  them  the 
same  characters.  If  the  medium  to  which  the  pressure  is  applied  be  perfectly  elastic,  like  glass,  the  effect,  like 
that  of  heat,  is  transient.  But  if  during  the  continuance  of  the  compression  or  dilatation,  the  particles  of  the 
medium  are  allowed  to  take  their  own  arrangement  and  state  of  equilibrium,  then  when  the  external  force  is 
withdrawn  a  permanent  polarizing  character  will  be  found  to  exist. 

As  periodical  colours  are  not  produced  in  phenomena  of  this  class  without  a  resolution  of  the  incident  light 
into  two  pencils  moving  with  different  velocities,  and  as  a  difference  of  velocities  is  invariably  accompanied  with 
a  difference  of  refraction  at  inclined  surfaces,  it  might  be  expected  that  media  thus  under  the  influence  of  heat 
or  pressure  should  become  doubly  refractive.  This  has  been  verified  by  direct  experiment  by  M.  Fresnel,  who 
has  shown  that  a  peculiar  species  of  double  refraction  is  thus  produced. 

As  the  unusual  heating  or  cooling  of  glass  and  other  substances,  is  well  known  to  produce  in  the  parts 
heated  or  cooled  a  corresponding  inequality  of  bulk,  and  thus  to  bring  the  parts  adjacent  into  a  state  of  strain  in 
all  respects  analogous  to  that  arising  from  mechanical  violence,  and  as,  in  fact,  the  effects  of  heat  in  communi- 
cating double  refraction  to  glass,  whether  transient  or  permanent,  are  all,  as  we  shall  see,  (with  one  very 
obscure  and  doubtful  exception)  commensurate  with  the  amount  of  the  strain  thus  transiently  or  permanently 
induced,  we  have  little  hesitation  in  regarding  the  inequality  of  temperature  as  merely  the  remote,  and  the 
mechanical  tension  or  condensation  of  the  medium  as  the  proximate  cause  of  the  phenomena  in  question,  and 
are  very  little  disposed  to  call  in  the  agency  of  a  peculiar  crystallizing  fluid,  endowed  with  properties  analogous  to 
those  of  magnetism,  electricity,  &c.,  to  account  for  the  phenomena,  still  less  to  regard  media  under  the  influence  of 
heat  or  pressure  as  in  any  way  thereby  rendered  more  crystalline  than  in  their  natural  state  of  equilibrium. 

In  gasiform,  or  fluid  media,  no  such  phenomena  are  observed  to  be  developed  by  either  heat  or  pressure;  the 
reason  is  obvious,  the  pressure  is  equally  distributed  in  all  directions,  and  the  elasticity  of  the  ether  (on  the 
undulatory  hypothesis)  preserves  its  uniformity. 

But  in  solids  the  case  is  different.     The   molecules  cannot   shift  their  places  one  among  the  other,  and  the 


L  I  G  H  T.  563 

Light.       effect  of  a  compression  in  any  direction  is,  Jlrnt,  to  urge  contiguous  particles   nearer  together  in  that  direction,     Part  IV. 
i»~\— "•"  and  thereby  to  call  into  action  their  repulsive  forces,  more  than  in  the  natural  state,  to  maintain  the  equilibrium;  ^•""V""' 
secondly,  but  much  more  slightly  to  urge  contiguous  particles  in  a  direction  perpendicular  to  that  of  the  pressure  M 
laterally  asunder,  by  reason  of  the  increase  of  the  oblique  repulsive  force  developed  by  the  approach  of  the  mole-  pre'°Sur°  on 
cules  in  the   line   of  pressure  to  those  which  lie  obliquely  to  that  line.      But  this  action,  which  in  fluids  would  the  mole- 
eause  a  motion  of  the  lateral  particles  out  of  the  way,  in  solids  is  ultimately  equilibrated  by  an   increase  of  the  cules  of 
attractive  forces  of  the  adjacent  molecules  in  a  line  perpendicular  to  the  line  of  pressure  ;   and  thus  we  see  that solids- 
every  external  force  applied  to  a  solid  is  accompanied  with  a  condensation  of  its  particles  in  the  direction  of  the 
to  ice  and  a  dilatation  in  a  perpendicular  direction.     It  is  probable,  however,  that  this  latter  is  extremely  minute, 
on  account  of  the  rapid  diminution  of  the  molecular  forces  by  increase  of  distance,  rendering  the  diagonal  action 
insensible.     But  the  former  may  easily  be  conceived  to  produce  in  the  ether,  in  virtue  of  its  connection  (what- 
ever it  be)  vith  the  molecules  of  refracting  media,  a  difference  of  elasticity  in  the  two   directions  in   question, 
accompanied  with  all   the  necessary  concomitants  of  interfering  pencils,   periodical  colours,  and  double  refrac- 
tion     The  effect  of  dilatation  will  be  the  converse  of  that  of  compression,  the  direction  of  maximum  elasticity 
in  the  one  case  being  that  of  minimum  in  the  other. 

These  views  are  in  perfect  accordance  with  the  experiments  described  by  Brewster  and  Fresnel  on  compressed      1087 
and  dilated  glass.     According  to  the  former  (Phil.  Trans.    1816.  vol.  106)  the  effect  of  pressure  on  the  opposite  Effects  of 
edges  of  a  parallelepiped  of  glass  is  to  develope  in   it  "  neutrai    and  "  depolarizing  axes,"  the  former  parallel  compression 
and  perpendicular  to  the  direction  of  the  pressure,  the  latter  45°  inclined  to  them  ;  in  other  words,  a  parallelepiped  descri')ecj- 
of  glass  so  compressed,  will  when  exposed  to  a  ray  polarized  in  the  pi  ne  parallel  or  perpendicular  to  the  sides 
to  which  the  pressure  is  applied,  produce  no  change  in  its  polarization  and  develope  no  periodical  colours,  while 
if  polarized  in  45°  of  azimuth  with  respect  to  those  sides,  it  will  develope  a  tint,  descending  in  the  scale  of  the 
coloured  rinses  as  the  pressure  increases. 

In  this  case,  if  the  pressure  be  uniformly  applied  over  the  whole  length  of  each  opposite  side,  the  elasticity  of  the      10S8. 
ether  in  every  point  of  the  plate  will  be  uniform  in  either  direction  at  every  point  of  the  plate,  being  a  maximum  in  Exp'anation 
one,  and  a  minimum  in  that  at  right  angles  to  it.    The  incident  light  therefore  if  polarized  in  azimuth  «°  will  resolve  Jj",  .j    un~ 
itself  into  two  pencils  of  unequal  intensity  (viz.  cos  a*  and  sin  a2)  polarized  in  these  two  planes,  and  differing  at  doctrine 
their  egress  by  an  interval  of  retardation  proportional   to  t  x  (t/  —  u),  where  t  is  the  thickness  traversed,  and 
t>'  —  v  the  difference  of  velocities  of  the  pencils,  which  when  received  on  a  double  refracting  prism  will  (as  in  the 
case  of  a  crystallized  plate  (Art.  969)  give  rise  to  complementary  periodical  tints   in  the  two  images,  the  extra- 
ordinary image  vanishing  when  a  =  0,  or  90,  and  the   contrast  being  a  maximum   at  45°.     It  is,  of  course, 
extremely  difficult  to  give  such  a  perfect  equality  of  pressure,  so  that  we  must  not  be  surprised   if  a  perfect 
uniformity  of  tint  over  the  whole  surface  of  the  glass  should  not  take  place.     In  the   experiment,  however, 
described  by  Dr.  Brewster  (Prop.  I.  of  the  Memoir  cited)  this  seems  to  have  been  the  case. 

If  we  suppose  the  elasticity  of  the  ether  in  compressed  glass  less  in  the  direction  of  the  force  applied  (and      1089. 
where  consequently  the  medium  is  densest,  according  to  the  general  law)  than  in  the  perpendicular,  the  contrary 
will  be  the  case  in  dilated.     Hence,  supposing  the  forces  equal,  in  two  similar  plates,  the  extraordinary  waves,  or 
those  whose  vibrations  are  performed  in  the  direction  of  the  pressure,  and  which  are  therefore  polarized  at  right 
angles  to  that  direction,  will  advance  most  rapidly  in  the  former  case,  the  ordinary  in  the  latter.     Consequently,  if  Opposite 
we  regard  the  interval  of  retardation  or  the  tint,  t  (v1  —  v)  as  negative  in  the  former  case,  it  will  be  positive  in  the  effects  of 
latter;  and  the  tints  in  the  two  cases  will  present  the  opposite  characters  of  those  exhibited  by  doubly  refracting  compressor 
crystals  of  the  two  classes  described  in  Art.  940,  et  seq.  see  also  Art.  803,  as  negative  and  positive,  or  repulsive  *." 
and  attractive.     Two  such  plates,  therefore,  placed  homologously,  or  with  the  directions  of  the  forces  coincident, 
ought  to  neutralize  each  other,  and  if  crossed  at  right  angles  should  reinforce  each  other;  and  in  general,  if  t  be 
the  thickness  and  f  the  compressing  force  applied  to  any  plate  (supposing  the  difference  of  velocities  to  be  pro- 
portional to  the  force,  and  regarding  dilating  forces  as  negative)  we  shall  have  for  homologous!)  situated  plates 

T  =  tint  polarized  by  any  number  of  plates 

Law  of  su- 
=  (f.t+f.t'+  f"  .  t"  +  &C.)  perposition 

In  the  case  of  crossed  plates  the  thicknesses  of  those  placed  transversely  are  tn  be  regarded  as  negative,  just  »s  in 
the  case  of  the  superposition  of  crystallized  plates.  All  these  results  are  conformable  to  the  experiments 
of  Dr.  Brewster. 

The  phenomena  of  contracted  and  dilated   glass   may  most  easily  and  conveniently  be  produced  by  bending      1090. 
a  long  parallel  plate  of  glass  having  its   longer  edges  polished,  and  passing  the  light  through  them   across  its  Tints  pro- 
breadth.     In  this  case,  as  in  all  cases  of  flexure,  the  convex  surface  is  in  a  state  of  dilatation,  and  the  concave  of  <Juced  b.V 
compression,  while  there  exists  a  certain  intermediate  line  or  boundary  between  these  oppositely  affected  regions    f 
in  which  the  substance  is  in  its  natural  state  of  equilibrium,  and  on  both  sides  of  which  neutral  line  the  degree 
of  strain  increases  as   we  recede  from   it  towards   either  surface.     Fig.  212  is  a  section  of  such  a   bent  plate,  Fig.  212. 
much  exaggerated,  through   which  light,  polarized  in  a  plane  45°  inclined  to  its  length,  has  been   passed   and 
analyzed  as  usual.    The  neutral  line  is  marked  by  a  divided  black  stripe,  and  the  tints  on  either  side  of  it  descend 
in  Newton's  scale,  being  arranged   in  stripes  disposed  according  to   the  lines  11,  22,  33,  44,  &c.     The   tints, 
however,  on  opposite  sides  of  the  neutral  line  have  opposite  colours,  being  positive  on  the  side  of  the  dilatation, 
or  towards  the  convexity,  and  negative  on  the  compressed  or  concave  side.     In  a  plate  of  glass  1.5  inch  broad,  state  of 
0.28  thick  und  six  inches  long,  Dr.  Brewster  developed  seven  orders  of  colours  before  the  glass  brok«  with  the  strain  ascer- 
bonding  force  applied.     This  experiment  affords  an  exceedingly  beautiful  illustration  of  the  action  of  compressing  'ained  by 
and  bending  forces  on  solids,  and  furnishes  ocular  evidence  of  the  state  of  str.'in   into  which  their  several  parts  tnetints 

4o  2 


564  L  F  G  H  T. 

Light,     are  brought  by  external  violence.     The  ingenuity  of  Dr.  Brevvsterlias  not  overlooked  its  application  to  the  useful     Part  IV. 
*^~V~-s  and  important  object  of  ascertaining  the  state  of  strain  and  pressure  on  the  different  parts  of  architectural  struc-  — —  /-• 
tures,  as  stone  bridges,  timber  framings,  &c.,  by  the  use  of  glass  models  actually  put  together  as  the  buildings 
themselves.     We  must  recollect  always,  however,  that  the  information  thus  afforded  will  only  be   distinct  when 
the  load  intended  to  be  sustained  is  many  times  the  weight  of  the  materials. 

1091.  If  a  plate  of  glass  be  subjected  to  several  distinct  compressions  and  dilatations  in  different  directions,  Dr. 
Efl'ects  of  Brewster  finds,  that  its  action  will  be  the  same  as  the  combined  action  of  several  plates  each  subjected  to  one  of  the 
several  co-  forces  employed.  Thus  a  square  of  glass  compressed  equally  on  all  its  four  edges  exerts  no  polarizing  action. 

If  a  pressure  be  applied  at  a  single  point  of  a  mass  of  glass,  or  rather  at  two  opposite  points,  it  will  diverge 
1092       from  these  points  in  all  directions  into  the  mass,  and  the  lines  of  equal  pressure,  which  are  in  fact  the  isochro- 
Piessure'     niatic  lines,  must  have  their  form  determined  in  some  measure  by  the  figure  of  the  compressing  screw  or  tool  at 
applied  ata  its  point  of  contact  with  the  glass,  for  this  figure  regulates  the  form  and  curvature  of  the  indentation  immediately 
point.  under  it.     Dr.  Brewster  has  figured  several  of  the  curves  produced  by  the  application  of  such  pressure  to  dif- 

ferent parts  of  the  same  parallelepiped  of  glass,  for  which  the  reader  is  referred  to  his  Paper,  as  well  as  for  a 
variety  of  beautiful  figures  produced  by  crossing  plates  differently  strained. 

1093.  M.  Biot  has  observed,  that  in  some  instances  glass  maintained  in  a  state  of  vibration  by  the  action  of  a  bow 
Effects  of     or  otherwise,  depolarizes  light,  i.  e.  restores  the  vanished  pencil.     This  is  a  necessary  consequence  of  the  alter- 
vibration.     nate  compressions  and  dilatations  which  follow  each  other  in  rapid  succession  in  all  the   vibrating  molecules. 

Nodal  lines  (see  ACOUSTICS)  being  exempt  from  such  variations  of  density  ought  to  be  marked  by  black  bands, 
and  may  thus,  perhaps,  be  rendered  evident  to  the  eye. 

1094.  When  masses  of  jelly  (especially  of  isinglass)  are  pressed  betweti.  plates  they  acquire  a  polarizing  action.     If 
Polarization  dilated   by  proper   management,  and  in   that  state  allowed  to   dry  and  harden,  the   character  so    impressed, 
by  com-       according  to  Dr.  Brewster,  is  permanent  when  the  dilating  force  is  removed  ;  to  explain  which,  we  must  consider 
jelHes           *nat  ^e    exleri°r   coats  indurate   more  rapidly    than    the    interior,  and   when    they   have    acquired   the    con- 
sistency of  a  solid,  they  will  be  capable  of  resisting  the  subsequent  contraction  of  the  interior  portions  and  keep- 
ing them  in  a  dilated  state,  even  when  the  original  dilating  force  is  removed.     That  force  only  served  to  deter- 
mine the  figure  and  dimensions  of  the  exterior  crust,  and  when  once  that  crust  is  fully  formed  and   indurated,  it 
becomes  capable  of  maintaining  them  without  the  further  aid  of  the  cause  which  gave  them  rise.     The  polarizing 
power  of  isinglass  thus  developed  is  very  great,  and  even  exceeds  that  of  some  doubly  refractive  crystals,  such  as 
beryl ;    a  plate  of  isinglass  whose  thickness  is  624  polarizing  the  tint  which  would  be  reflected  by  a  plate  of  air 
whose  thickness  is  unity,  while  a  plate  of  beryl  parallel  to  the  axis,  to  polarize  the  same   tint,  will  require  a 
thickness    =  720.     Glass  compressed,  or   dilated,  by  an  equal   force,  would  require   a  thickness  (according  to 
Dr.  Brewster)  =  12580  to  produce  the  same  tint. 

1095.  We  come  now  to  consider  the  transient  effects   of  unequal   temperature  below  the   softening  point  of  glass. 
Transient      The  immediate  effect  of  an  increase  or  diminution  of  temperature  in  one  point  of  a  piece  of  glass,  is  to  produce 
effects  of      a  mechanical  strain  on  all  the  surrounding  part,  which  if  the  difference  of  temperature  is  considerable,  is  of  the 

hT'softe'tr    utmost  violence,  and  capable  of  breaking  asunder  the  thickest  pieces  of  glass ;    an  effect  with  which  every  one  is 

ing  point,     familiar.     Now,  as  we  know  that  strain  alone  developes  a  polarizing  action,  the  rule  of  philosophy,  "  non  plures 

causas  admitti  debere,"  fyc.  which  forbids  the  admission  of  a  second  cause  when   one  adequate  to  the  effect  is 

known  to  be  in  action,  will  hardly  justify  us  in  attributing  a  peculiar  action  to  the  caloric,  independent  of  its 

power  of  altering  the  dimensions  of  matter. 

1096.  When  a  heated  iron  bar  is  applied  along  the  edge  of  a  parallelepiped   of  glass  held  in   a  polarized  beam, 
Case  of  a     analyzed  as  usual,  the  vanished  image  is  restored  in  various  degrees  of  intensity  in  different  parts  of  the  glass, 
rectangular  The  neutral  axes  are  parallel  and  perpendicular  to  the  heated  edge,  and  the  axes  in  whose  azimuth  the  tint 
Plate  °f        polarized  is  the  strongest,  at  45°  of  inclination.     If  held  in  that  azimuth,  the  first  effect  of  the  heat  is  to  produce 
aVoneedge  a  'me>  or>  as  **  were>  a  wave  of  white  light  at  the  heated  edge,  which  advances  gradually  upon  the  glass,  driving 

before  it  a  dark  and  undefined  wave.  Nearly  at  the  same  instant,  and  long  before  the  slightest  increase  oj  tem- 
perature can  have  reached  the  further  extremity  of  the  glass  plate,  a  similar  but  fainter  white  wave  advances  from 
the  edge  opposite  to  the  heated  one,  driving  before  it  a  similar  undefined  dark  wave  ;  and  at  no  perceptible 
interval  of  time  another  white  fringe  appears  in  a  very  diluted  state  about  the  centre  of  the  plate,  advancing 
equally  towards  the  heated  edge  on  one  side  and  that  most  remote  on  the  other,  and  thus  condensing  the  two 
undefined  dark  waves  into  two  black  fringes.  The  white  tints  are  succeeded  by  tints  of  a  lower  order  in  the 
scale  of  colour,  yellow,  red,  purple,  blue,  &c.,  till  at  length  the  whole  scale  of  the  colours  of  thin  plates  is  seen 
arranged  in  four  sets  of  fringes  parallel  to  the  heated  edge,  and  having  for  their  origins  the  black  fringes  above 
mentioned.  At  the  same  time,  other  lateral  fringes  are  produced  along  the  edge  perpendicular  to  the  heated  one. 
Thus  in  all  six  sets  are  seen  ;  two  exterior,  viz.  those  parallel  to  the  heated  edge,  and  outside  of  the  black  fringes ; 
two  interior,  in  the  same  direction,  but  between  the  black  fringes ;  and  two  terminal,  along  the  lateral  edges. 
Fig.  213.  The  whole  phenomena  is  as  represented  in  fig.  213.  The  fringes  along  the  heated  edge  A  B  are  most  distinct 
and  numerous,  those  along  the  opposite,  C  D,  less  so,  and  the  interior  and  terminal  fringes  least  of  all. 

1097.  As  glass  is  an  extremely  bad  conductor  of  heat,  and  as  culinary  heat  is  propagated  through  glass  entirely  by 
Action  of     conduction,  it  follows,  that  the  sudden  application  of  an  elevated  temperature  to  the  edge  A  B  must  produce  a 
heat  in         dilatation  in  it,  not  participated  in  by  the  rest  of  the  glass.     If,  therefore,  the  stratum  of  molecules  A  B  were 
straining      detached  from  the  rest  of  the  glass,  it  would  elongate  itself  so  as  to  project  at  its  two  ends  beyond  the  edges 

AC,  DB.  When  the  heat  of  this  stratum  communicated  itself  to  the  next,  that  also  would  elongate  itself,  but 
in  a  less  degree ;  and  thus  after  a  very  long  time,  during  which  the  heat  had  penetrated  to  the  farther  extremity 
of  the  glass,  its  outline  would  assume  the  form  a  C  D  b,  the  lines  aC,  6  D  being  certain  curves  depending  on 
the  law  of  propagation  and  the  time  elapsed.  This  would  be  the  state  of  things  were  the  glass  plate  composed 


LIGHT.  565 

Light,      of  discrete  strata,  each  of  which  could  dilate  independently  of  all  the  rest.     And  since  in  each  of  these  (regard 'd     PartlV. 
— -v^-'  as  infinitely  thin)  the  temperature  and  strain  would  be  uniform,  there  would  arise  no  polarizing  action.     But,  in  v—"-v-"1"'' 
reality,  the  case  is  quite  different ;  every  stratum  is  indissolubly  connected  along  its  whole  extent  with  the  strata 
adjacent,  and  can  neither  expand  nor  contract  without  forcing  them  to  participate  in  its  change  of  dimension. 
In  so  far,  then,  as  two  adjacent  strata  participate  in  the  change  of  temperature  they  expand  together;  but  when 
one  is  hotter  than  the  other,  the  former  is  found  to  expand  few,  and  the  other  more  than  it  they  were  inde- 
pendent.    Now  the  strain  thus  induced  on  any  stratum  is  not,  like  the  caloric  which  causes  it,  confined  by  the 
conducting  power  of  the  medium,  but  propagates  itself  instantly  (with  diminished  energy)  to  the  strata  beyond, 
by  reason  of  the  mutual  action  of  the  molecules. 

The  general  problem,  then,  to  investigate  the  actual  state  of  strain  of  any  molecule  at  any  moment  is  one  of      1098. 
pome  complexity,  inasmuch  as  it  depends  at  once  on  the  laws  of  the  slow  propagation  of  heat,  and  the  iristan-  State  of  th« 
taneous  but  variable  participation  of  change  of  figure  necessary  to  establish  among  the  particles  a  momentary  ™r'°"s  rt 
equilibrium  under  the  circumstances  of  temperature  at  the  time  ;    but,  without  attempting  minutely  to  analyze  *]J 
the  effects,  if  we  content  ourselves  with  acquiring  a  general  idea  how  they  arise,  we  shall   find  little  difficulty.  Strain 
For  in  fig.  214,  if  we  conceive  the  stratum  A  B  b  a  adjacent  to  the  border  A  B  to  be  dilated  by  the  heat,  the  rest  determined. 
of  the  glass  retaining  its  original  temperature  ;  if  this  stratum  could  expand  separately,  its  edges  A  a,  B  6  would  F'f>-  2H. 
project  out  beyond  the  general  edges  Co,  D  $ ;    and  if  we  regard  two  terminal  strata  C  A  E  G,   D  B  F  H,  as 
detached  from  the  interior  portion  C  D  j3  a,  and  free  to  move  by  the  force  applied  at  their  extremities  A,  B,  they 
would  be  raised  by  the  dilatation  of  the  portion  A  B  b  a   into  the  situation  represented  in  the  figure,  turning 
round  C,  D  as  fulerums,  and  leaving  triangular  intervals  C  a  a,  D  ft/3  vacant,  and  in  these  circumstances  there 
would  be  no  strain  on  any  part  of  the  system.     But  the  cohesion  of  the  glass  prevents  the  formation  of  these 
vacancies,  and  the  bars  or  levers  C  AE  G,  D  B  F  H  cannot  move  into  this  situation  without  dragging  with  them, 
and  therefore  distending  the  strata  of  C  D  /3  «.     Let  PQ  be  any  such  stratum,  and  let  it  be  distended  to  p  q. 
Then  by  its  elasticity  it  will  tend  to  draw  the  bars  C  A  E  G  and  B  D  H  F  together ;  and  its  action  will  therefore 
tend,  first,  to  produce  a  pressure  on  the  fulerums  C,  D,  urging  the  points  CD  together,  and  therefore  bringing 
the  stratum  C  D  into  a  state  of  compression.     Secondly,  to  produce  also  a  pressure  on  A  a,  B  6,  or  a  resis'ance 
to  the  dilatation  of  A  B  ha,  which  its  increased  temperature  would  naturally  produce.     It  will  therefore  tend  to 
compress  back  the  strata  of  A  B  6  a  into  a  smaller  length  than  what  would  be  natural  to  them  in  their  heated 
state,  i.  e.  to  bring  them  also  into  a  relatively  compressed  state.     Thirdly,  the  tension  of  p  q  being  sustained  at 
C,  D  and  A,  B,  will  tend  to  bend  inwards  the  levers  A  C  G  E,  B  D  H  F,  rendering  them  concave  at  the  edges 
G  E,  H  F,  and  convex  at  C  A,   D  B,  and  thus  distending  the   lines  CA,  D  B,  and  compressing  the  strata 
adjacent  to  E  G,  H  F. 

From  this  reasoning  it  is  clear,  that  the  glass,  in  consequence  of  these  various  strains,  will  assume  a  figure      1099. 
concave  on  all  its  edges,  but  chiefly  so  at  the  lateral  ones  AC,  DB,  as  in  fig  215  ;  and  that  the  state  of  strain  Production 
of  its  various  parts  will  be  as  there  expressed,  all  the  edges  being  compressed,  but  principally  A  B  and  C  D,  and  of  fringes  .if 
the  interior  distended.     The  limit  between  the  distended  and  compressed  portions  parallel  to  A  B  must  neces-  "j 
sarily  be  marked  by  neutral  lines  a  b,  c  d  on  either  side  of  which  the  strain  will  increase,  being  a  maximum  in  $•;.•.  215 
the  middle  and  on  or  near  the  edges.     Consequently,  it  ought  to  polarize  four  sets  of  fringes,  having  a  6,  c  d 
for  their  origins,  and  of  which  the  two  external  (or  those   between  these  lines  to  the  edge)  ought  to  have  a 
character  opposite  to  those  of  the  internal,  the  portion  of  the  intromitted  pencil  polarized  parallel  to  A  B  being 
propagated  faster  than  that  parallel  to  A  C  in   the  one   case,   and  slower  in  the  other.     This  opposition  of 
characteis  is  conformable  to  Dr.  Brewster's  observations,  who   states  {Phil.  Trans,  1816)  that  the  parts  of  the 
glass  which  exhibit  the  two  exterior  sets  of  fringes  (adjacent  to  the  edges  A  B,  CD)  have  "  the  structure  of" 
attractive  crystals,  while  the  parts  which  exhibit  the  interior  and  terminal  sets  have  that  of  repulsive  ones; 
meaning,  of  course,  in  the  1  'nguage  of  the  undulatory  doctrine,  that  the  order  of  velocities  of  the  doubly 
refracted  pencils  is  reversed  in  passing  from  one  region  of  the  glass  to  the  other,  for  of  its  actual  structure  we 
can  know  nothing.     That  the  terminal  fringes  ought  (as  observed)  to  have  the  same  character  as  the  interior  is  Th<s  termi- 
a  necessary  consequence  of  the  above  reasoning,  for  the  terminal  regions  D  B,  A  C  are  compressed  in  directions  nal  fringes. 
parallel  to  their  edges,  and  therefore  perpendicular  to  the  direction  in  which  the  central  portion  is  distended  ; 
and  we  have  already  seen  that  compression  in  one  direction  is  equivalent  (so  far  as  the  character  of  the  tints 
produced  is  concerned)  to  distension  in  that  perpendicular  to  it. 

Lastly,  the  black  lines  separating  the  terminal  fringes  from  the  interior  ones,  arise  from  the  combined  action      1100. 
of  the  tension  of  the  interior  region  parallel  to  A  B  (fig.  214)  exerting  itself  on  any  point  as  q  on  the  inner  Ntutrat 
border  of  the  terminal  portion  D  B  F  H,  (which  we  have  regarded  as  an  elastic  bar,  or  lever,)  and  the  distension  lines  sepa- 
of  the  line  D  B  al?o  exerting  itself  at  q,  and  arising  from  the  convexity  given  to  this  line.     In  virtue  of  these  ratmj;  adJ* 
two  forces,  every  point  q  in  a  certain  line  at  a  proper  distance  from  the  extreme  edge  H  F,  will   be  equally  f^",^ 
distended  in  opposite  directions,  and  will  therefore   be  in  a   neutral   state,  as  to  polarization,  and,  of  course, 
appear  black.     The  terminal  fringes  are  less  developed  than  the  rest,  because  they  arise  simply  from  the  flexure 
of  the  edges  H  F,  GE,  which  is  an  indirect  effect  of  the  principal  force,  and  is  very  small,,  (owing  to  the  small 
dilatability  of  glass  by  heat,  and  consequent   minuteness  of  the  versed  sine  of  the  curve  into  which   they  are 
distorted,)  a    '  the  line  of  indifference  separating  them  from  the  others  lies  near  the  edges  ;  for  the  same  reason, 
the  tension  o!  the  convex   line  D  B  being  small,  and  therefore  putting  itself  in  equilibrium  with  that  of  the 
distended  column  p  q  at  a  point  9  near  its  extremity,  where  it  is  evident  that  the   strain  parallel  to  p  q  must  be 
much  diminished  ;  the  greater  portion  of  the  whole  tension  ofp  q  being  resisted  by  the  spring  oflaminae  situated 
still  further  from  the  edge  than  D  B. 

If  a  lamina  of  glass,  uniformly  heated,  be  suddenly  cooled  at  one  of  its  edges,  the  reverse  of  all  these  effects      1101 
will  arise;    the  outer  column  AB«6  (fig.  214)  will  suddenly  contract  and  comptess  violently   the  columns 


5G6  LIGHT. 

Light,      beyond  a  ft,  from   which  no  heat  has  yet  been  abstracted,  and  drag  inwards  the  ends  of  the   terminal  levers     Part  IV. 
'  — v— -'  E  A  G  C,  B  F  II  D,  which  will  thus  be  violently  pressed  on  the  parts  ft  Q  and  a  P  as  fulcra ;  and  their  action  v  j-y~-_- 
Phenomena  being  thus  transmitted  to  the  opposite  edge  C  D  will  tend  to  lengthen  it,  and  thus  bring  it,  as  well  as  the  edge 
"lass  rectan   ^  ^'  'nto  a  Distended  state-     The  terminal  edges  will  also  be  sprung  outwards.     The  strain  on   every  point 
u-le  cooled     w'"  be  exactly  the  reverse  of  what  is  expressed  in  fig.  215,  and  a  corresponding  inversion  of  the  characters 
atone  edge,  of  the  tints  will  take  place  ;   all  which  is  agreeable  to  Dr.  Brewster's  observation,  (Prop.  14  of  the  Memoir  cited.) 
110:2.          When  a  crack  takes  place  in  a  piece  of  unequally  heated  glass,  (he  directions  and  intensities  of  the  straining 
Effect  of  a    forces  in   every  part,  which  depend  wholly  on  the  cohesion  of  its  molecules,  and  the  continuity  of  the  levers, 
springs,  &c.  into  which  it  may  be  mentally  conceived  to  be  divided,  is   suddenly  altered  ;    and  the  fringes  are 
accordingly  observed  to  take  instantly  a  new  arrangement,  and  assume  forms  related' to  the  ligure  of  that  part 
of  the  glass  which  preserves  its  continuity.     To  analyze   the  modifications   arising  from  variations  of  external 
figure  and  different  applications  of  the  heat,  would  be  to  involve  ourselves  unnecessarily  in  a  wilderness  of  com- 
plexity.    One  simple  case  may,  however,  be  noticed,  in  which  the  centre  of  a  circular  piece  of  glass  is  heated. 
Each  exterior  anmilus  of  this  will  be  placed  in  a  state  of  distension  parallel  to  its  circumference,  and  will 
drcular  a      compress  all  within  it  by  a  force  parallel  to  the  radius.     The  central  point  will  be  neutral,  being  equally  confined 
p'.ate heated  ln  a'l   directions,  and  the  annul!  adjacent  to  the  centre  will  in  like  manner  be  compressed  both  radially  and 
in  the  circumferentially.     The  radial  strain  continues  as  we  recede  from  the  centre,  but  the  circumferential  diminishes, 

centre.  and  at  length,  as  already  said,  changes  to  a  state  of  distension,  and  of  course  passes  through  a  neutral  state, 
thus  giving  rise  to  a  black  circle  and  concentric  fringes  of  opposite  characters,  the  whole  of  which  will  be  inter- 
sected by  the  arms  of  a  black  cross  parallel  and  perpendicular  to  the  plane  of  primitive  polarization,  and  which 
of  course  remains  fixed  while  the  plate  is  turned  round  in  its  own  plane. 

1 103.          There  is   only  one  experiment  of  Dr.  Brewster  which  seems  hostile  to  the  theory  here  stated.     He  made  u 
Singular    >  partial  crack  with  a  red-hot  iron  in  a  verv  thick  piece  of  glass,  and  allowed  it  to  close  by  long  standing,  which 

aftti^t    rtl   a  ."_  _    .J*.          .    "^ _ 


it  did,  so  as  to  disappear  entirely.     In  this  state,  the  glass,  when  unequally  heated,  exhibited  the  same  fringes, 


srfect  of  a 

allowed  to     a<<  ^'  no  crack  had  existed  ;  but  the  moment  the  crack  was  opened  by  a  slight  heat  applied  near  it,  they  suddenly 
close.  changed  their  figure,  and  assumed  that  due  to  the  portion  having  the  crack  for  a  part  of  its  outline.     It  seems, 

however,  that  a  very  great  adhesive  force  takes  place  between  the  surfaces  of  glass  when  thus  in  optical  contact; 
and  to  those  who  are  aware  how  the  free  expansion  and  contraction  of  dissimilar  metallic  bars  may  be  com- 
manded, and  the  bars  in  consequence  made  to  ply  on  change  of  temperature  by  mere  forcible  juxtaposition, 
without  soldering,  till  the  difference  of  expansion  has  reached  a  certain  point,  when  they  give  way  with  a  snap 
and  regain  their  state  of  equilibrium,  the  anomaly  will  not  appear  in  the  light  of  a  radical  objection.  (We 
think  it  not  improbable,  that  the  musical  sounds  said  to  issue  at  sunrise  from  Certain  statues,  may  originate  in  some 
pyrometrical  action  of  the  kind  here  alluded  to.  We  have  often  been  amused  by  a  similar  effect  produced  in  the 
bars  of  the  grate  of  a  Jire  place.) 

1 104.  Such  are,  in  general,  the  transient  effects  of  a  heat  below  the  softening  point  of  glass,  unequally  distributed 
Phenomena  through  its  substance.     But  if  a  mass  of  glass  be  heated  up  to,  or  beyond  that  point,  so  as  to  allow  its  mole- 
ot  unan-       cules  to  glide  with  more  or  less  freedom  on  one  another,  and  adapt  themselves  to  any  form  impressed  on   the 
nealed  glass  mass>  all(j  ^en  suddenly  cooled,  either  by  plunging  into  water,  or  by  exposure  to  cold  air,  the  heat  is  abstracted 

from  its  external  strata  with  so  much  greater  rapidity  than  it  can  be  supplied  by  conduction  from  within,  that 
they  become  rigid,  while  the  inner  portions  are  still  soft  and  yielding.  At  this  instant,  there  is  therefore  no  strain 
in  any  part ;  but,  the  abstraction  of  the  heat  still  going  on,  the  internal  parts  at  length  become  solid,  and  tend, 
of  course,  to  contract  in  their  dimensions.  In  this,  however,  they  are  prevented  by  the  external  cru-t  already 
formed,  which  acts  as  an  arch  or  vault,  and  keeps  them  distended,  at  the  same  time  that  these  latter  portions 
themselves  are  to  a  certain  extent  forced  to  obey  the  inward  tension,  and  are  strained  inwards  from  their  figure 
of  equilibrium.  Glass  in  this  state  is  said  to  be  unannealed.  If  the  cooling  has  been  sudden,  and  the  mass 
considerable,  it  either  splits  in  the  act  of  cooling,  or  flies  to  pieces,  when  cold,  spontaneously,  or  on  the  slightest 
scratch  which  destroys  the  continuity  of  its  surface;  and  the  pieces  when  put  together  again  (which,  however, 
is  seldom  practicable,  as  it  usually  flies  into  innumerable  fragments,  or  even  to  powder,  as  is  familiarly  shown 
Rupert's  in  the  glass  tears  called  Rupert's  drops,  which  exhibit  a  very  high  polarizing  energy  from  their  intense  strain, 
drops.  an(j  which  hurst  with  a  violence  amounting  to  explosion,  on  the  rupture  of  their  long  slender  tails)  are  found  not 

to  fit,  but  to  leave  a  slight  vacancy  ;  thus  satisfactorily  provinsr  the  state  of  unnatural  and  violent  distension  in 
which  its  interval  parts  have  been  held.  The  case  is  precisely  analogous  to  that  of  a  gelatinous  substance 
allowed  to  indurate  under  the  influence  of  dilating  forces.  (See  Art.  1094.) 

1105.  If  the  cooling  be  less  sudden,  and  carefully  managed,  the  glass,  though  much  more  brittle   than  ordinary 
Patterns       annealed  glass,  is  yet  susceptible  (with  great  caution)  of  being  cut  and  polished  ;   and  in  this  state,  if  polarized 

light  be  passed  through  it,  it  exhibits  coloured  phenomena  of  astonishing  variety  and  splendour,  forming  fringes, 
square  and'   'r'ses>  anc'  patterns  of  exquisite  regularity  and  richness,  according  to  the  form  and  size  of  the  mass,  and  the 
rectangular   degree  of  strain  to  which  it  is  subjected.      In  all  these  cases  if  the  external  form  be  varied,  the  pattern  varies  cor- 
unannealed    respondingly,  as  it  is«easy  to  perceive  it  ought;  for  if  any  part  of  t'he  exterior  crust  be  removed,  that  part  of  the 
lle*  strain  which  it  sustained  will  fall  on  the  remainder,  and  on  the  new  surface  produced.     Figures  216,  217,  and 

'^2i  ~  218.  represent  the  patterns  exhibited  b\  a  circular,  a  square,  and  a  rectangular  plate  of  about  ^  inch  thick, 
the  two  latter  being  placed  so  as  to  ha\e  one  side  parallel  to  the  plane  of  primitive  polarization.  Figure  21&1 
and  220  represent  the  patterns  shown  by  the  t  vo  latter  in  azimuth  45°,  and  fig.  221  that  arising  from  the 
crossing  of  two  plates  equal  and  similar  to  fur.  2:?(),  each  being  in  azimuth  45°.  In  all  these  cases  the  laws  of 
superposition  of  Art.  1089  are  observed,  when  similar  points  of  similar  plates  are  laid  together.  If  symme- 
trically, the  tints  polarized  is  the  same  as  would  be  polarized  by  one  plate  whose  thickness  is  their  sum;  if 
crosswise,  their  difference. 


L  I  G  H  T.  5(T 

l.'ght.          If  a  square  or  rectangular  plate  be  turned  about  in  its  own  plane,  from  azimuth  0°,  the  arms  of  tbe  black    Part  IV 
••-v^""'  cross  dividing  it  into  four  quarters  become  curved,  as  in  fig.  222,  and  pass  in  succession  over  every  part  of  tho  •— — \— —• 
disc  ;  thus  showing  that  the  positions  of  the  axes  of  elasticity  of  the  molecules  vary  for  every  different  point  of     1106. 
the  plate,  and  in  different  parts  of  it  have  every  possible  situation.     We  shall  not  here  attempt  to  analyze  the  Effect  oi 
mechanical  state  of  the  molecules  in  any  case,  as  it  would  lead  us  too  far  ;    but  merely  mention  an  experiment  '"™n'jgln 
of  Dr.  Brewster,  which  is  sufficient  to  show  the  conformity  of  our  theory  of  these  figures  with  fact.     According  undented 
to  this  excellent  observer,  the  fringes  parallel  to  the  edge  A  B  of  the  rectangle  (fig.  220)  are  similar  in  their  plate  in  its 
character  to  those  produced  by  setting  the  corresponding  edge  of  a  similar  plate  of  annealed  gl;iss  on  a  hot  iron.  <™n  plane. 
Now,  in  the  latter  case,  the  exterior  fringes  adjacent  to  A  B,  C  D  arise  from  a  compressed  state  of  the  columns  "••  ***• 
parallel  to  AB;  and  the  interior,  from  a  distended.     And,  in  the  unannealed  plate  the  distribution  of  the  forces  Relation  uf 
is  almost  exactly  similar  to  that  described  in  Art.  1098  and  1099.      In  fact,  such  a  plate  may  be  likened,  in  some  these  phe- 
respects,  to  a  frame  of  wood  over  which  an  elastic  surface  is  stretched  like  a  drum.     The  four  sides  will  all  be  "omena  t« 
curved  inwards  by  its  tension,  and  they  will  all  be  compressed  in  the  direction  of  their  length  by  the  direct  ^^jntw 
tension,  independent  of  the  secondary  effect  produced  by  their  curvature.     The  terminal   fringes   in  the  articles  i,eated 
referred  to  arise  solely  from  the  secondary  forces  thus  developed  ;   but  the  analogy  between  the  cases  would  be  annealed 
complete,  if,  instead  of  supposing  the  annealed  plate  heated  at  one  edge  only,  the   heat  were  applied  at  all  the  Pla'es. 
four  simultaneously,  by  surrounding  it  with  a  frame  of  hot  iron.     For  a  farther  account  of  the  beautiful  and 
interesting  phenomena  produced  by  unanneuled  glass,  we  must  refer  the  reader  to  Dr.  Brewster's  curious  Paper 
already  cited. 

M.  Fresnel  has  succeeded  in  rendering  sensible  the  bifurcation  of  the  pencils  produced  by  glass  subjected  to      1107. 
pressure,  by  an  ingenious  combination  of  prisms  having  their  refracting  angles  turned  opposite  ways,  and  of  which 
the  alternate  ones  are  compressed  in  planes  at  right  an  <les  to  each  other,  thus  (as  in  the  case  of  the  double 
refraction  along  the  axis  of  quartz)  doubling  the  effect  produced. 

The  effects  produced  by  unequal  heat  and  pressure  on   crystallized  bodies,  in   altering  their  relations  to   light      1108. 
transmitted  through  them,  are  less  sensibly  marked  than  in   uncrystallized,  being  masked   by  t 'IK  more  powerful  Kffectsof 
effects  produced  by  the  usual  doubly  refractive  powers.     In  crystals,  however,  where  these  powers  are  feeble,  or  unequal 
in  which  they  do  not  exist  in   any  sensible  degree,  fas  in   fluor  spar,  muriate  of  soda,  and  other  crystals  which  lle 
belong  to  the  tessular  system,   Dr.  Brewster   has  shown  that  a  polarizing  and  doubly-refractive  action   is  deve-  crv»ullin  I 
loped  by  these  causes  just  as  in  uncrystallized  ones  ;  and  M.  Biot,  by  applying  violent  pressure  to  crystallized  bodies. 
substances  while  viewing  through  them  their  systems  of  rings  in  the  immediate  vicinity  of  their  axes  where  the 
polarizing  action  is  very  weak,  has  sncc  eded  in  producing  an  evident  distortion  of  the  rings  from  the  regularity 
of  their  form,  thus  rendering  it   manifest,  that  it  is  only  the  extreme  feebleness  of  the   polarizing  action   so 
induced  in  comparison  with  the  ordinary  action   of  the   crystal,  which  prevents   its  becoming  sensible  in    all 
directions. 

In  applying  what  is  here  said  to  heat,  however,  we  consider  only  its  indirect  action,  or  that  arising  from  its      1109. 
untqual  distribution,  inducing  a  strain,  and  thus  resolving  itself  into  pressure,  as  above  shown.     But  Professor  Mitschn-- 
Mitscherlich  in  a  most  interesting  series  of  researches  (which  we  hope,  ere  long  to  see  embodied  in  a  regular  I*™**  re~ 
form,  but  of  which  at  present  only  the  most  meagre  and  imperfect  details  have  reached  us)  has  shown  that  the  (^dj]^5,,,0" 
action  of  heat  on  crystallized  bodies,  even  when  uniformly  distributed,   so  that  the  whole  mass  shall  be  at  one  ti0n  of 
and  the  same  temperature,  is  totally  different  from  what  obtains  in  uncrystallized      In  the  latter  (as  well  as  in  crystals  l>y 
crystals  of  the  tessular  system)  an  elevation  of  temperature,  common  to  the  whole  mass,  produces  an  equal  dila-  ''eat. 
tation  in  all  directions,  the  mass  merely  increases  in  dimensions,  without  change  of  figure.     In  crystals,  however, 
not  belonging  to  the  tessular  system,  i.  e.  whose  forms  are  not  symmetrical  relative  to  three  rectangular  axes, 
the  dilatation  caused  by  increase  of  temperature  is   so  far  from  being   the   same   in   all  directions,  that   in  some 
cases  a  dilatation  in  one  direction  is  accompanied  with  an  actual  contraction  in  another. 

Of  this  important  fact,  (the  most  important,  doubtless,  that  has  yet  appeared  in  pyrometry.)  M.  Mitscherlich      1110. 
has  adduced  a  remarkable  and  striking  instance  in  the  ordinary  Iceland   spar,  (carbonate  of  lime.)      This  sub-  Pyrometri- 
stance  when  heated,  dilates  in  the  direction  of  the  axis  of  the  obtuse  rhomboid  which  is  the  primitive  form  of  its  "j^f0^ 
crystals,  and  contracts  in  every  direction  at  right  angles  to  that  axis,  so  that  there  must  exist  an   intermediate  lan(i  Spar 
direction,  in  which  this  substance  is  neither  lengthened  nor  contracted  by  change  of  temperature.     A  necessary 
consequence  of  such  inequality  of  pyrometric  action  is,  that  the   angles  of  the   primitive  form  will  undergo  a 
variation,  the  rhomboid  becoming  less  obtuse  as  the  temperature  increases,  and  this  has  been  ascertained  to  be 
the  case  by  direct  measurement;   M.  Mitscherlich   having  found,  that  an   elevation  of  temperature  from  the 
freezing  to  the  boiling  point  of  water  puoduced  a  diminution  of  8' 30"  in  the  dihedral  angle  at  the  extremities  of 
the  axis  of  the  rhomboid,  (Bulletin  des  Sciences  puhlie  par  la  Societe  Philomatique  de  Paris,  1824,  p.  40.) 

M.  Mitscherlich  assured   himself  of  the   fact  in   question  by  direct  measurement   of  a  plate  of  Iceland   spar      1111. 
parallel  to  the  axis,  at  different  temperatures,  by  the  aid  of  the  "  Spherometer,"  a  delicate  species  of  calibre  con-  Molle  of 
trived  by  M.  Biot  for  measuring  the  thickness  of  any  laminar  solid  by  the  revolution  of  a  screw  whose  point  is  ^^ 
just  brought  into  light  contact  with  the  surface,  and  by  which  the  10,000th  of  an  inch  is  readily  appreciated  and 
measured.     The  experiment  is  necessarily  one  of  great  delicacy,  but  our  readers  may  assure  themselves  at  least 
of  the  general  fact  of  unequal  change  of  dimension   by  change  of  temperature,  by  a   very  simple   experiment 
requiring  almost  no  apparatus.      Let  a  small  quantity  of  the  sulphate  of  potash  and  copper,  (an   anhydrous  salt  Pyrometri- 
easily  formed   by  crystallizing  together  the   sulphates  of  potash  and  of  copper,)  be  melted  in  a  spoon  over  a  cal  property 
spirit  lamp.     The  fusion  takes  place  at  a  heat  just  below  redness,  and  produces  a  liquid  of  a  dark  green  colour.  "(. 
The  heat  being  withdrawn,  it  fixes  into  a  solid  of  a  brilliant  emerald  green   colour,  and  remains  solid  and  anj  copper 
coherent  till  the  temperature  sinks  nearly  to  that  of  boiling  water,  when  all  at  once  its  cohesion  is  destroyed  ;  a 
commotion  takes  place  throughout  the  whole  mass,  beginning  from  the  surface,  each   molecule,  as  if  animated 


068  L  I  G  II  T. 

Light.      starting  up  and  separating  itself  from  the   rest,  till,  in  a  few  moments,  the   whole   is   resolved  into  a  heap  of     Part  IV. 
^_ — v<— — •   incoherent  powder,  a  result  which  could  evidently  not  take  place,  had  all  the  minute  and  interlaced  crystals  of  "~— v-~— 
which  the  congealed  salt  consisted  contracted  equally  in  all  directions  by  the  cooling  process,  as   in  that  case 
their  juxtaposition  would  not  be  disturbed.     Phenomena  somewhat  similar,  and  referable  to  the  same  principles, 
liave  (if  we  remember  right)  been  encountered  by  M.  Achard  in  the  fusion  of  various  frits  for  glasses,  &c. 

1112  The  relation  of  the  optical  and  crystallographical  characters  of  bodies  is  so  intimate,  that  no  change  can  be 
Double  re-   supposed  to  take  place  in  the  latter  without  a  corresponding  alteration  in  the  former.     As  the  rhomboid  of  Ice- 
fraction  of    land  spar  becomes  less  obtuse   by  heat,  and  therefore  approximates  nearer  to   the  cube,  in  which  the  double 
crystals  ya-  refraction  is  nothing,  it  might  be  expected  that  the  power  of  double  refraction  should  diminish,  and  this  result 

has  been   verified   by  M.  Mitscherlich  by  direct  measurement.     More  recently,  the  same  distinguished  chemist 

and  philosopher  has  ascertained  the  still   more  remarkable  and  striking  fact,  that  the  ordinary  sulphate  of  lime 

™"  ao(?  or  gypsum,  which,  at  common  temperatures,  has  two  optic  axes  in  the  plane  of  its   lamina?,  inclined  at  60°  to 

Milj'liate  of  each  other,  undergoes  a  much  greater  change  by  elevation  of  temperature  ;   the  axes  gradually  approaching  each 

lime.  other,  collapsing  into  one,  and  (when  yet  further  heated)  actually  opening  out  again  in  a  plane  at  right  angles 

to  the   laminae,    thus  affording   a   beautiful    exemplification  of  fresnel's  theory  of  the   optic  axes   as  above 

explained. 

1113  This  singular  result  we  cite  from  memory,  having  in  vain  searched  for  the  original  source  of  our  information  ; 
but  it  might  have  been  expected,  from  the  low  temperature  at  which  the  chemical  constitution  of  this  crystal  is 
subverted,  by  the  disengagement  of  its   water,  that  the  changes  in  its  optical   relations  by  heat  would  be  much 
more  striking  than  in  more  indestructible  bodies.     We  have  not,  at  this  moment,  an  opportunity  of  fully  verify- 
ing the  fact ;    but  we  observe,  that  the  tints  developed  by  a  plate  of  sulphate  of  lime  now  before  us,  exposed  as 
usual  to  polarized  light,  rise  rapidly  in  the  scale  when  the  plate  is  moderately  warmed  by  the   heat  of  a  candle 
held  at  some  distance  below  it,  and  sink  again  when  the  heat  is  withdrawn,  which,  so  far  as  it  goes,  is  in  con- 
formity with  the  result  above  stated.     Mica,  on  the  contrary,  similarly  treated,  undergoes  no   apparent  change 
in  the  position  of  its  axes   or  the  size  of  its  rings,  though   heated  nearly  to   ignition.     The  subject  is  in  the 
highest  degree  interesting  and  important,  and   lays  open  a  new  and  most  extensive  field  for  optical  investiga- 
tion.    It  is  in  excellent  hands,  and  we  doubt  not  will,  ere   long,  form  a  conspicuous  feature  in  the  splendid 
series  of  crystallographical  discovery  which  has  already  so  preeminently  distinguished  its  author 

§  XIII.     Of  the  Use  of  Properties  of  Light  in  affording  Characters  for  determining  and  identifying  Chemical 
and  Mineral  Species,  and  for  investigating  the  intimate  Constitution  and  Structure  of  Natural  liodies. 

1114.          Newton,  who  "  looked  all  nature  through,"  was  the  first  to  observe  a  connection  between  the  refractive  powers 
Relation       of  transparent  media  and  their  chemical  properties.     His  well  known  conjecture  of  the  inflammable  nature  of 
between  the  the  diamond,  from  its  high  refractive  power,  so  remarkably  verified  by  the  subsequent  discovery  of  its  one  and 
ctive  ,  only  chemical  constituent,  (carbon,)  was,  perhaps,  less  remarkable  lor  its  boldness,  at  a  period  when  Chemistry 
chemical"     consisted  in  a  mere  jargon,  in  which  salt,  sulphur,  earth,  oil,  and  mercury  might  be  almost  indifferently  substi- 
composition  tuted  for  one  another,  than  it  would  have  been  fifty  years  later.     His  divination  of  the  inflammable  nature   of 
ol  bodies,     one  of  the  constituents  of  water  is   at  least  equally  striking  as  an  instance  of  sagacity,  and  even  more  remark- 
able, for  the  important  influence  which  its  verification  has  exercised  over  the  whole  science  of  Chemistry.    These 
instances  suffice  to  show  the  value  of  the  refractive  index,  either  taken  in  conjunction  with  the  specific  gravity 
of  a  medium,  or  separately  as   a  physical   character.      The  refractive  indices  of  a  vast  variety  of  bodies  have 
been  ascertained  by  the  labours  of  Newton  and  later  experimenters,  among  whom  Dr.  Brewster  and  Dr.  VVol- 
laston  have  been  the  largest  contributors  to  our  knowledge.     They  may  be  grouped  together  in  a  general  way, 
in  order  of  magnitude,  as  follovvs : 

1115  Clans  1.  Gases  and  vapours.     Refractive  index  from  1.000  to  1.002,  under  ordinary  circumstances  of  pressure 

Classing-    and  temperature. 

tion  of  bo-        Class  2.     p  =  1.05  ....    ft  =  1.45.     Comprising  the  condensed  gases  ;    ethereal,  spirituous,  and  aqueous 
dies  accord-  liquids  ;  acid,  alkaline,  and  saline  solutions,  (not  metallic.) 

mg  to  their       Class  3.    Comprising,  first,  almost  all  unctuous,  fatty,  waxy,  gummy,  and  resinous  bodies  ;  camphors,  balsams, 
densities6      vegetable  and  animal  inflammables,  and  all  the  varieties  of  hydro-carbon.     Secondly,  stones  and  vitreous  com- 
pounds, in  which  the  alkalis  and  lighter  alkaline  earths  in  combination  with  silica,  alumina,  &c.  are  the  predo- 
minant ingredients.     Thirdly,  saline  bodies  not  having  the  heavy  metals,  or  the   metallic  acids  predominant 

ingredients.  /*  =  1.40 1.60. 

Class  4.  Pastes,  (glasses  with  much  lead,)  an-!,  in  general,  compounds  in  which  lead,  silver,  mercury,  and  the 
heavy  metals,  or  their  oxides  abound.  Precious  stones,  simple  combustibles  in  the  solid  state,  including  the 
metals  themselves. 

fi  =  1.60  and  upwards. 

These  classes,  however,  admit  of  so  many  exceptions  and  anomalies,  and  are  themselves  so  vague  and  indefinite, 
that  we  shall  not  attempt  to  distribute  the  observed  indices  under  any  of  them,  but  rather  prefer,  for  conve- 
nience of  reference,  presenting  the  whole  list  in  the  form  of  a  Table,  arranged  in  order  of  magnitude,  in  which 
all  these  classes  are  mingled  indiscriminately — a  form,  in  some  measure,  consecrated  by  usage. 


LIGHT. 


569 


Light. 


Table  of  Refractive  Indices,  or  Values  of  /*  for  Rays  of  Mean  Ref Tangibility,  (unless  expressed  to  the  contrary.) 
Dr.  Wollasioris  results,  however,  are  all  (according  to  Dr.  Young,  Philosophical  Transactions,  vol.  xcii. 
p.  370,)  to  be  regarded  as  belonging  to  the  Extreme  Red  Rays. 

N.  B.   In  this  Table  the  authorities  are  referred  to  as  follows : 

Br.  Brewster,  Encyclop.  Ed.,  and  Treatise  on  New  Philosophical  Instruments.  Bos.  Boscovich. 

B.  Y.  Dr.  Young's  Calculations  of  Dr.  Brewster's  Unreduced  Observations.    Quarterly  Journal,  vol.  xxii. 
Bi.  Biot.  F.  Faraday.  Du.  Dulong.  M.  Mains.  N.Newton.  Fr.  Fraunhofer." 
W.  \v~ollaston,  Phil.  Trans.                   He.  From  our  own  observation.                    Eul.  Euler  the  younger. 

C.  and  H.,  authorities  cited  by  Dr.  Young  in  his  Lectures. 


Part  IV. 


1116. 


..  1.000000 

1  3394    Br 

GASES, 
at  the  freezing  temperature  and  pressure  ~ 

29i".922  =  0".76 
1  000138     Du. 

Ditto      

1  340       B  Y 

1  345      B  Y 

1.353       B.Y. 
.      1.339       B.Y. 
1.339       B.Y. 
1.343       Br. 

JU44     \Rr 

1  1.349  /Br- 
1  .344       Eul. 
1.372       H. 
1.347       B.Y 
1.596      Br. 
.     1.345       Br. 

Saliva    

1  000272     Du. 

Salt  water  (  1  sea  water)  

.      1.000294     Bi. 

Cryolite    

Azote    

.      1.009300     Du. 

Vinegar  (distilled) 

Nitrous  gas  

1.000303     Du. 
1.000340     Du. 
1.000385     Du. 
1.000443     Du. 
1.000449     Du. 

Ditto      

Acetic  acid  (  ?  strength)  

Jelly  fish  (Medusa  vEquora)    

1.000449     Du. 
1.000451     Du. 
1.000503    Du. 
1.000644     Du. 
1  000665     Du 

Port  wine  

1.351       Eul. 
1.351       B.Y. 
1.354       B.Y. 
1.356       He. 
1.356      B.Y. 
/  1.358      W. 
\  1.374       B.Y. 
1.360       W. 
f!36l       Br. 
1  1.359      B.Y. 
1.360       B.Y. 
1.360       B.Y. 
f  1.368      Br. 
1  1.379       B.Y. 
1.37        W. 
1.370       N. 
1.371       C. 
1.372       He. 
1.374       Br. 
1.377       B.Y. 
1.375       C. 
1.376      Br. 
1.392       He. 
1.395      B.Y. 
1.401       Br. 
1.4098     Bi. 
1.379       B.Y. 
1.384       He. 
1.386       B.Y. 
1.395       Br. 
1.395       B.  Y. 
f  1.396       Br. 
11.404       B.Y. 
1.406       Br. 
1.3767     Br. 
1.3786     Br. 
1.3990    Br. 
1.386       B.Y. 
1.428      B.Y. 
1.436       B.Y. 
1.410      B.Y. 

1.439       B.Y. 
0.3801     w 
tl.447J     W' 
1.463       Eul. 
1.406       B.Y. 
1.403      B.Y. 
1.40563  Kr. 
f  1.410       B.Y. 
1  1.410       W. 
1.412      C. 
1.411       B.Y. 
1.423       B.Y. 
1.426       Bi 

Human  blood    

S  1  h       tt  d  h  d  "a  

Saturated  aqueous  soluticn  cf  alum    

.  u  p  urette      y  ro0en    

Olefiant  gas  

1.000678     Du. 
1.000772     Du. 

Ether                        

Albumen  

.     1.000834     Du. 

1.001095     Du. 
1.001159    Du. 

Brandy  

Rum  

Vapour  of  sulphuric   ether  (boiling  point   a 

t 
1.001530     Du. 

Ditto  (S.  G.  0.866)  .... 

LIQUIDS  AND  SOLIDS. 

P  — 
Ether  expanded  by  heat  to  three  times  its  volume     1.0570     Br. 
Tabasheer  from  Vellore,  a  yellowish  transparent 
variety                11111      Rr 

Ditto  

Ditto  (rectified  spirits)     

Ditto      

Ditto      

Saturated  solution  of  salt     

First   new  fluid   discovered  by   Dr.  Brewster 
cavities  in  topaz  

in 
..      1.1311     Br. 

Muriatic  acid  (  ?  S.  G.)  
Ditto  (S.G.  1.134)  
Ditto      

Tabasheer,  transparent,  from  Nagpore    1.1454     Br. 
Ditto                 ditto         ditto       another  specimen     1.1503     Br. 

Ditto  (strong)    

New    Huid  discovered  by  Dr.  Brewster  in   ame- 
thyst, at  83$°  Fahr  1.2106     Br. 

Sweet  spirit  of  nitre    

Second  new  fluid  discovered  by  Dr.  Brewster 
topaz,  at  83°  Fahr  

m 
1.2946     Br. 

Pus 

Nitrous  oxide  liquefied  by  pressure     

{much  lew     lp 
than  water     j". 

Muriatic  acid  gas    ditto  ditto  1 
Carbonic  acid  gas  ditto  ditto  /'  '  """^  equa1' 

Ice  

1    boUimuch     1 
.  1      le»  than       [p. 
I       w«Kr          J 

1     1.307       Br. 
.!     1.3085     Br. 

Crystalline  lens  of  the  eye  (human  ?)  outer  coat 
Ditto                               ditto                  middle  coat 
Ditto                               ditto                  centre  .... 
Ditto  of  the  lamb's  eye,  outer  coat  

[    1.3100     W. 

f    rather  less     lp 

Ditto              ditto            middle  coat  

Cyanogen  liquefied  by  pressure  

f    perhagnless   !„ 

Ditto                 ditto                      

..      1.316       Br 

Ditto              ditto                 middle  coat 

f       equal  to      •>„ 

Ditto              ditto                 centre   

Water      

.\        water        Jr. 

(N. 
.     1.336      iVf. 

Sulphuretted  hydrogen  liquefied  by  pressure   . 
Ammonia  liquefied  by  pressure  

(B, 

{rather  greater  1  p 
than  water     /  r  • 
(  greater  than   i 
1     water,  and     1 
,  /   greater  than   >F. 
j   all  the  other   1 
(.liquefied  gasetj 

..      1.3366     Br. 

Ditto  of  the  pigeon  

Juice  of  orange  peel     

Solution  of  potash,  S.G.  1.416,  (ray  El     . 

Nitric  aeid  (S.G.  1.48)  

Nitric  acid    

Ditto  of  the  haddock    

..      1.341       B.Y. 

Vitreous        ditto  

..      1.336       W. 

VOL.  IV. 

4.E 

570 


LIGHT. 


1.426 


Liel.t.         Gluten  of  wheat,  dried    

Fresh  yolk  of  an  egg   «•«»      £• 

Sulphuric  acid  (S.G.  1.7)    1-429       N. 

Ditto      ditto    (?  S.  G.)     

Ditto 

Ditto 

Fluor  spar    


B.Y. 
B.Y. 


1.430 

1.435 

1.440 

(1.433 

11.436 


Oil  of  rhue 
Phosphorous  acid 


(1.433 
\  1.449 


1.437 
1.441 


n. 

Hydrophosphoric  acid,  cold    (1.442 

[1.446 
Spermaceti  (melted)   <  1.454 

Oilof  wax   1-452 

Oil  of  wormwood 1.453 

Bees  wax,  melted 1-453 

Oil  of  chamomile 1-457 

Ditto     1-476 

Oil  of  lavender 1-457 

Ditto     1-467 

Ditto     1-475 

Alum 1-457 

Ditto  (S.G.  1.714) 1-458 

Ditto 1-488 

Tallow  (melted)   1-460 

White  wax  (melted)     1-462 

(1.467 

Oilof  poppy    |1483 

Sulphate  of  magnesia  (double  ?  least  refraction). .  1.465 

I"  1.467 

Borax,  (S.G.  =  1.714)  <  1-467 

Ll- 4  75 


Oil  of  peppermint 
Oil  of  rosemary  . . 


(1. 
Oil  of  spermaceti    |  j . 


Ditto 


(1.468 
1 1.473 
(1.469 
11.472 
1.470 
.473 
M  .469 ) 
Oil  of  almonds (1470 1 

(M81 
11.483 

Oil  of  turpentine,  rectified 1.470 

Spirit  of  turpentine,  (S.  G.  0.874) 1.471 

Oil  of  turpentine 1.475 

Ditto 1-476 

Ditto  (common)  1.476 

Ditto     1-482 

Ditto     . 1-485 

Ditto  (common)   1.486 

Ditto,  S.  G.  =  0.885,  (ray  E) 1.47835 

f  1.467* 
1.469 

Oil  of  olives <  1.470 

1.4705 
.1 1.476 
(1.471 
1.473 


Oil 


(1-4 
il  of  bergamot  •{  j  4 


Oil  of  beech,  misprinted  ?  oil  of  brick   1.471 

Oil  of  brick,  distilled  from  spermaceti    (1471 

Oilof  juniper 1-473 

(1.474 
Butter,  cold |1480 

Palmoil    1-475 

Oil  of  rape  seed    1-475 

Naphtha   1.475 

Essence  of  lemon    1 .476 

Uumarabic  (S.G.  =  1.375) 1476 

Oil  of  dill  seed 1-477 

Oilof  thyme 1-477 

Oilolcajeput       \IAK 

Opal  (partly  hydrophanous)     1.479 

Naples  soap 1.479 

Oil  of  mace,  melted    1.481 

•  Tie  8.  0.  of  Newton's  specimen  was  0.913. 


He. 

W. 

Br. 

W. 

Br. 

Br. 

B.Y. 

B.Y. 

B.Y. 

Br. 

W. 

B.Y. 

C. 

Br. 

B.Y. 

Br. 

B.Y. 

Br. 

W. 

B.Y. 

W. 

N. 

B.Y. 

W. 

B.Y. 

Br. 

B.Y. 

Br. 

N. 

C. 

Br. 

W. 

B.Y. 

Br. 

B.Y. 

Br. 

B.Y. 

W. 

B.Y. 

Br. 
W. 
N. 
Br. 
B.Y. 
W. 
C. 

B.Y. 
He. 
Fr. 
N. 
W. 
Br. 
He. 
B.Y. 
Br. 
B.Y. 
Br. 
B.Y. 
Br. 
Br. 
B.Y. 
W. 
B.Y. 
(B.Y. 
Br. 


N. 

Br. 

Br. 

B.Y. 

Br. 

Br. 

B.Y. 

B.Y. 


Oil  of  spearmint     | 

(1.4 
\1.4 


Oil  of  lemon 


.481 
.496 
1.481 
.489 
Carbonate  of  potash  (?)  ....................      1.482 

Oil  of  pennyroyal  ..........................      1  .482 

Ditto     ..................................      1.485 

Linseed  oil  (S.  G.  0.932)  ....................      1.482 

Linseed  oil    ..............................      1  .485 

Ditto     ..................................      1.487 

Oilof  savine  ..............................     1.482 

_..    ,  .     .  (1.4821 

Oilof  juniper  ............................    {  1.491  1 

Sulphate  of  ammonia  and  magnesia    ..........  1.483 

Train  oil   ................................  1.483 

Oil  of  wormwood      ........................  1.4851 

Ditto  ....................................  1.4891 

fl.4S5 

Castor011  ................................  jl.490 

Florence  oil  ..............  .  ...............  1.485 

Oilof  thyme    ............................  1.486 

Oil  of  dill  seed    ..........................  1.487 

Oil  of  feugreek  (?  fenugreek)  ................  1.487 

Ditto     ..................................  1.488 

(  ..............  1.487 

„       ,  1.496 
Camphor  ................  \    ............. 

l(S.G.  =  0996). 

„..    ,, 

Oilof  hyssop  ............................ 

Windsorsoap   ............................     1.487 

Obsidian    ................................     1.488 

Iceland  spar.  ..  .weakest  refraction    .........     11519 

Ditto    ........  strongest  ..................    < 

Ditto     ........  ordinary  index  ..............  1  .6543 

Ditto     .......  extraordinary    .............  1  .4833 

Ditto    ...  .....  (S.G.  =  2.72)  ..............  1.667 

Sulphate  of  magnesia  (?  greatest  refraction)  ....  1.488 

Nut  oil  (perhaps  impure)      .................  1  .490 

Ditto     .................................  1.507 

Oil  of  castor  ..............................  1.490 


1.500 

1.500 

(1.487 


Tallow  (cold)   |  j'' 

Oil  of  carraway  seed  (carui  seminis)     |j' 

Oil  of  marjoram 


i 


483 
,491 
(1.490 
(1.491 

„.,    ,  (1.491 

)il  of  nutmeg {  1.497 

M91 
1.507 

( 1  491 
Oil  of  Angelica   H493 

Beeswax,  cold    1.492 

Bees  wax 1.507 

Ditto  14°  Reaum 1.5123 

Ditto,  melting 1.4503 

Ditto,  boiling    1.4416 

Ditto 1.542 

Ditto  (white  wax,  cold)    1.535 

Sulphate  of  iron,  greatest  refraction    1.494 

.      .  ,  (1.494 

Balsam  of  sulphur j  ,  ,g~ 

Sulphate  of  potash \  j  r,gg 

Honey 1.495 

Rochelle  salt  (mean  green  rays) 1.4985 

Ditto              (mean  red)    1.4929 

Ditto               (tartrate  of  potash  and  soda) 1.515 

Treacle 1.500 

Yolk  of  an  egg  (dry)    1.500 

Oil  of  beech  nut   1.500 

(1.500 

Oil  of  rhodium i  1 .503 

1 1.505 
Glass,  plate  and  crown,  various  specimens : 

Ditto  English  plate 1-500 

Ditto  French  plate 1  504 

Ditto  English  plate  (extreme  red) 1.5133 

Ditto  plate   1.514 


Br. 

B.Y. 

Br. 

B.Y. 

Br. 

Br. 

B.Y. 

N. 

W. 

B.Y. 

Br. 

B.Y. 

Br. 
B.Y. 

B.Y. 

B.Y. 

Br. 

B.Y. 

B.Y. 

B.Y. 

Br. 

B.Y. 

W. 

B.Y. 

C. 

N. 

Br. 

B.Y. 

B.Y. 

Br. 

W. 

B. 

W. 

Br. 

M. 

M. 

N. 

Br. 

He. 

Br. 

Br. 

W. 

B.Y. 

B.Y. 

Br. 

B.Y. 

Br. 

B.Y. 

W. 

B.Y. 

Br. 

B.Y. 

Br. 

B.Y. 

B.Y. 

M. 

M. 

M. 

W. 

W. 

Br. 

B.Y. 

Br. 

W. 

Br. 

B.Y. 

He. 

He. 

Br. 

W. 

B.Y. 

Br. 

Br. 

B.Y. 

Br. 

W. 

W: 
He. 

Bos. 


Part  IV. 


LIGHT. 


571 


tight. 


1  517       w. 

1.525       W. 
1.536       Br. 
1.488       N. 
1.527       Br. 
1.527       Br. 
f  1.528       W. 
)  1.532      B.Y. 
U.549       Br. 
1.529       B.Y. 
1.530      W. 
,  1.531       W. 
11.581       B.Y. 
Si.  586       Br. 
(1.588       B.Y. 
1.531       Br. 
1.552       Br. 
(1.532       B.Y. 
(1.544       Br. 
1.532       C. 
1.532       Br. 
1.544       Br. 
1.532       Br. 
1.533       B.  Y. 
1  1.547)         y 
i  1.565}    RYl 
1.5348     M. 
1.6931     M. 
(1.535       W. 
•j  1.547      Br. 
1  1.550       B.Y. 
1  1.535      W. 
<  1.539       B.Y. 
1  1.560      Br. 
1.535      W. 
f  1  .535      W. 
i  1  .546      B.  Y. 
(1.535      W. 
J  1.549       Br. 
1  1.553      B.Y. 
/  1.535       W. 
\1.539       B.Y. 
1.535      W. 
1.541       B.Y. 
1.545      B.Y. 
1.555      Br. 
1.536      Br. 
1.536       B.Y. 
(1.536      B.Y. 
(1.601       Br. 
1.538       Br. 
1.556      Br. 
(1.538      Br. 
(1.541       B.Y. 
1.540       Br. 
1.542      W. 
1.543      W. 

1.5431     He. 
1.543       Br. 
1.700       Br. 
1.544      Br. 
1.544      B.Y. 
1.545      N. 
1.557      Br. 
(1.545       B.Y. 
(1.557      Br. 
1.545       B.Y. 
1.545       B.Y. 
(1.546      B.Y. 
11.560       Br. 
f  1.546      B.Y. 
11.554       Br. 
1.547      Br. 
1.547      W. 
1.5484    M. 
1.5582    M. 
1.562      W. 
1.562       Br. 
1.563       N. 
(1.568)     „, 
11.575}    C- 

1.525       W. 

1.526       Bos. 

"  A  selenites,  S.  G.  2.252"  

Ditto  crown,  a  prism  by  Dollond,  (extra  red)    ..      1.526       He. 

1.529       Bos. 

Ditto  crown,  another  prism  by  Dollond, 
Ditto  Fraunhofei's    crown,    No.    13, 

(extra  red)     1.5301     He. 
(ray   E,) 
1.5314     Fr. 

Balsam  of  Gilead  

1.532      C. 

Ditto  Fraunhofer's    crown,    No.    9, 
S.  G.2535    

(ray    E,) 
1.5330     Fr. 

Pitch 

(1.533)     w 

'   \  1.536) 
1.534       Br. 

1.538       Bos. 

Ditto  ditto    

1.542       Bos. 

Ditto  St.  Gobin    

1.543       W. 
1.544       Br. 

Brazil  pebble,  (S.  G.  2.62)  

1.545       W. 

Glass  of  phosphorus  (fused  phosphoric  acid).  .  .  . 

1  .550       N. 

Ditto  Fraunhofer's  crown,  M,  S.  G.  2.756,  (ray  £•}     1.5631     Fr. 
Ditto  plate    (S  G  276)                                                1.573       C. 

Glass  of  borax  (fused  borax)  

1.582       Br. 

N.  B.  It  is  probable  that  the  more 
list  are  low  flint  glasses, 

refractive  specimens  of  this 
containing  lead. 
(1.503       B.Y. 

Arragonite,  extraordinary  index  

Ditto,  ordinary     

\1.535      W. 
f  1.503       B.Y. 
J  1.507       Br 

Elemi    

Starch  (dry)  

1  1.510       B.Y. 
....     1  504      B  Y 

1.504       BY 

Arseniate  of  potash  .  . 

1.613      B  Y 

j  1.505       W. 

'    11.507       B.Y. 
1  506       B  Y 

(1.506       Br. 

'    11.507       B.Y. 
f  1.507      W. 
.    <  1'514\    p  v 

Stilbite      

11.516}    RY- 
M.528       Br. 
1  508       Br 

Ditto      

f  1.508       Br. 

Ditto  (after  melting)    

Scammony    

"  U.578       B.Y. 
1  510      B  Y 

Oil  of  mace  

(1.5121     BY 

Gum  Arabic  

'    1  1.526  }     B'Y- 
1  5iu       Br 

Mellite 

Ditto  (not  quite  dry)    

1.513       B  Y 

Ditto     

Jl-5141    w 

Ditto,  least   

'   11.517}    W- 
1.514       Br. 

Box-wood     

Ditto     
"Niter"  (?)  S.  G.  1.9  

1.524      C. 
1.524      N 

Apophyllite,  the  variety  which  exhibits  white  and 

Dantzic  vitriol  (sulphate  of  iron)     .  .  . 

1.515       N 

C    b      t      :  st  °  t      1      t^  f  e/t'  

Nadelstein  from  Faroe  

1.5153     Br 

ivt»    "a     °    j  M"  la>   C<      fe  J     lon  

Mesotype,  least  index    

1.516      Br. 

Dichroite  (iolite)  

Sulphate  of  zinc,  ordinary  refraction  . 
Myrrh    .... 

-••     1.522       Br. 
1517       Br. 
(1.517       B.Y. 

Petroleum  
Sal  gemmae,  S.  G.  2.143  (rock  salt)    

Ditto  (rock  salt)   

'   (1.524       Br. 

Chio  turpentine     

j  1.529       Br. 

Gum  dragon  (Tragacanth)  

(1.575      Br. 
(1.520       Br. 

Glass  of  borax  1,  silex  2  .... 

"   (1.66        W. 
1  522       Br 

Gum  lac,  or  Shell  lac  

(1.52+     W. 
<  1.525      Br 

U.528      B.Y. 
f  1.522       B.Y. 
J  1.532      Br. 

Quartz,  ordinary  refractive  index    

Ditto,     extraordinary  

11.536      W. 
*•  1.544       Eul. 
fl.524       W. 

Amethyst  

Rock  crystal  (double)  

Crystal  of  the  rock  (S.  G.  2.65)  

1  1.557       B.Y. 

Rock  crystal  

4  E2 

Part  IV. 


572 


LIGHT. 


Light. 


Amber 

Ditto,  (S.  G.  1.04) 


Resin 


Guiacum 

Glue,  nearly  hard 

Chalcedony 

Comptonite 

°P'um 


1.547 
1.556 
1.548 
l.552 
.559 
1  .550 
1.553 
1.553 
1  .553 
1.559 


HyposulphAte  of  lime  (mean  red)  ...........      1.5611 

Ditto,  mean  yellow  green    .  .................      1.566 

Dragon's  blood  ...........................      1  .562 

fl.565 
11. 


Horn 


.58— 


Pink,  coloured  glass     ......................  1  .570 

Assafoetida  .............................  1.575 

f  1.576 

Flint  glass  (various  specimens')  ..............  \  1  .578 

11.583 

Ditto,  a  prism  by  Dollond  (extreme  red)     ......  1.584 

Ditto,  (extreme  red)     .....................  1.585 

Ditto,  another  specimen    ....................  1  .586 

Ditto  ..................................  1.590 

Ditto  ...................................  1-593 

Ditto  ....................................  1.594 

Ditto  ....................................  1.596 

Ditto,  a  prism  by  Dollond  (extreme  red)  ........  1.601 

Ditto  ditto,  marked  "  heavy,"  (extreme  red)  ....  1.602 

Ditto,  another  specimen    ....................   1  1  604 

Ditto  ...................................      1-605 

Ditto,  Fraunhofer's  No.  3  (ray  E)   ..........      1.6145 

Ditto,  another  variety  ......................      1.616 

Ditto        ditto  .....................     1-625 

Ditto,  Fraunhofer's  No.  30  (ray  E)  ..........        1.6374 

Ditto         ditto          No.  23  (ray  E)  ............      1.6405 

Ditto        ditto         No.l3(rayE)  ............      1.6420 

Anhydrite,  ordinary  index    ..................      1.5772 

Ditto,  extraordinary     ......................      1.6219 

(1.578 
Gum  ammoniac  .........................   j  j  ^92 

Hyposulphite  of  lime,  least  refraction    ........      1.583 

Ditto,  greatest  ...........................      1  -628 

Balsam  of  styrax  ........................      1  .584 

Emerald    ......  .  .........................      1.585 

f      1.586) 
Benzoin  ................................  |  to  1.596  ) 

1.589 
Oil  of  cinnamon    .......................  1      1  -604  1 

I  to  1.632J 
Tortoise  shell      ...........  ................      1-591 

'  1.593 
Balsam  of  Peru  ...........................   J  1  -597 

11.605 

1.596 

Guiacum   ...............................  J  1  GOO 

\1.619 
Beryl     ..................................      1.598 

(1.60— 
610 

1  1  .627 

1  1  .628 
Ruby  red  glass  ..........................      1.601 

Essential  oil  of  bitter  almonds  ...............      1.603 

Meionite   ................................      1  -606 

Purple  coloured  glass  ......................     1  608 

Resin  of  jalap  ............................      1.008 

Hyposulphite  of  strontia,  least  refraction   ......      1  .60S 

Ditto  ditto  greatest         ........     1.651 

Colourless  topaz     ........................  ,      1.6102 

Bluish  topaz  (cairngorm)      ..................      1.624 

Brazilian  topaz,  ordinary  index   ...............      1  .6325 

Ditto         ditto,  extraordinary  ................      1.6401 

Blue  top*t,  Aberdeen  ......................      1  .636 

Yellow  topaz  .............................      1.638 

Red  topaz  ...............................        1  .652 

Green  coloured  glass    ....................      1.615 


Balsam  of  Tolu. 


Sulphate  of  barvtes,  ordinary      ..............      1.6201 

Ditto,  extraordinary      ......................      1.6302 


W. 

N. 
B.Y. 
B.Y. 
Br. 
B.Y. 
B.Y. 
Br. 
Br. 
B.Y. 
W. 
He. 
He. 
B.Y. 
Br. 
W. 
Br. 
B.Y. 
Br. 
He. 
W. 
He. 
He. 
W. 
Bos. 
Bos. 
Bos. 
Br. 
He. 
He. 
Br. 
Bos. 
M. 
Fr. 
Br. 
Bos. 
Fr. 
Fr. 
Fr. 
Bi. 
Bi. 
B.Y. 
Br. 
He. 
He. 
Br. 
Br. 

W. 
B.Y. 
B.Y. 

Br. 
B.Y. 
Br. 
B.Y. 
W. 
B.Y. 
Br. 
Br. 
W. 
B.Y. 
B.Y. 
Br. 
Br. 
Br. 
Br. 
Br. 
B  Y. 
He. 
He. 
Bi. 
Br. 
Bi. 
Bi. 
Br. 
Br. 
Br. 
Br. 
B.Y. 
Br. 
Bi. 
M. 

Sulphate  of  barytes  

1.6468     M. 

1.6460     He. 
1.6459     He. 
1.6491     He. 

1.643       N. 
1  .646       W. 
1.664       Br. 
(1.624       B.Y. 
11.631       B.Y. 
1  1.641       Br. 
1.625       Br. 
1.634       B.Y. 
1.635       Br. 
1.6429     Bi. 
1.6630     Bi. 
1.644       Br. 
1.647       Br. 
1.653       Br. 
1.657       Br. 
1.661       Br. 
1.703       Br. 
1.668       Br. 
1.668       Br. 
1.685      Br. 
1.67—     He. 
1.67—     He. 
1.89—     He. 
1.678       Br. 
1.695       Br. 
1.701        Br. 
1.715       Br. 
1.724       Br. 
1.729 
1.729       Br. 
1.788       Br. 
1.732       Br. 
1.73.0      He. 
1.7S5       He. 
1.735       Br. 
1.758       Br. 
1.759       Br. 
1.760       Br. 
(1.756       He. 
-{1.761       Br. 
11.811      W. 
1.764      Bf? 
1.768       W. 
1.794       Br. 
(1.768       He. 
\1779       Br. 
1.779       Br. 
1.782       Br. 
1.787       Zei. 
1.792       Br. 
1.80±     He. 
1.8      '    W. 
1.811       W. 
1.813       Br. 
2.084       Br. 
1.866       He. 
1.925       Br. 
1.987       W. 
1.95        W. 
1.961       Br. 
2.015       Br. 
1.958       Ha. 
2.008       B.Y. 
2.04        W. 
2.115       Br. 
2.148       Br. 
1.970       Br. 
1.970       Br. 
2.129      Br. 
(1.980  -    W. 
12.216       Br. 
2.028       Z. 
2.1—       Y. 
2.123      He. 
,2.125       B.Y 
J  2.224       Br. 
12.260      Br. 

Ditto            ditto       ordinary  refraction  (along  the 

A  "  pseudo   topazius"   (S.  G.  4.27)   sulphate  of 

Sulphate  of  barytes  

DUto            ditto     double,  greater  refraction.  ... 

Aloes     

Spargelstein  

Ditto,  Teatest  

Chloruret  of  sulphur  

Ditto,  greatest,  about    

Glass,  lead  1,  flint  2    

Ditto,  greatest   

Glass   lead  3,  flint  4  

Hyposulphite  of  soda  and  silver,  least  refraction  . 

Sapphire,  (white)      

Glass,  lead  1  ,  flint  1  (Zeiher)  

liorate  of  lead,  fused  and  cooled  (extreme  red).  . 

Ditto                    

Gla"   lead  3   flint  1  (by  Zeiher)    

Scaly  oxide  of  iron       

P 

Part  IV. 


L  I  G  H  T. 


573 


Light.          Nitrite  of  lead  (biaxal,  ?  quadro-nitrite)  in  six- 

__  .          sided  prisms,  ordinary  refraction 2.322 

Diamond  (S.  G.  =  3.4)    2.439 

Ditto 2.470 

Ditto  (brown  coloured)    2.487 

Ditto  (examined  by  Rochon) 2  755 

(from  204 

ito...  2.44 


He. 

N. 

Br. 

Br. 

Ro. 

W. 

w. 


Chremate  of  lead 


least  refraction  . 


,2.479 
)  2.500 
1  2.503 


(2.974 
2.500 


Br. 
Br. 
Br. 
Br. 
Br. 
Br. 
Br. 
Br 


Pa-t  IV. 


greatest. 

Octohedrite 

Realgar,  artificial 2.349 

Red  silver  ore 2.564 

Mercury  (probable,  see  Art.  594) 5.829 

In  casting  our  eyes  down  the  foregoing  Table,  we  cannot   but  be  struck  with  the  looseness  and  vagueness       1117 
of  those  results  which  refer  to  bodies  whose  chemical  nature  is  in  any  respect  indeterminate.     The  refractive  Remarks'ou 
indices  assigned  to  the  different  oils,  acids,  &c.  though  no  doubt  accurately  determined  for  the  particular  specimens  the  Table 
under  examination,  are  yet,  as  scientific  data,  deprived  of  most  of  their  interest  from  the  impossibility  of  stating  ofRefracti\e 
precisely  what  was  the  substance  examined.     Most  of  the  fixed  oils  are  probably  (as  appears  from  the  researches  "" 
of  Chevreul)    compounds,   in  very  variable  proportions  of  two  distinct  substances,  a  solid,  concrete   matter, 
(stearine,)  and  a  liquid,  (elaine,)  and  it  is  presumeable,  that  no  two  specimens  of  the  same  oil  agree  in   the 
proportions.     This  is,  probably,  peculiarly  the  case  with  the  oil  of  anise  seed,  which  congeals  almost  entirely 
with  a  very  moderate  degree  of  cold..    An   accurate  reexamination  of  the  refractive  and  dispersive   powers  of 
natural   bodies    of   strictly   determinate   chemical    composition,    and    identifiable  nature,    though  doubtless   a 
task  of  great  labour  and  extent,  would  be  a  most  valuable  present  to  optical  science.     Fraunhofer's  researches 
have  shown  to  what  a  degree  of  refinement  the  subject  may  be  carried,  as  well  as  the  important  practical  uses 
to  which  it  may  be  applied.     The  high  refractive  power  of  oil  of  cassia,   accompanied  by  a  corresponding 
dispersion,  has  led  Dr.  Brewster  to   conceive  the  existence  in  it  of  some  peculiar  chemical  element  not  yet 
cognisable  by  analysis.     The  low  refractions  of  the  oils  of  box-wood  and  ambergris  are  not  less  remarkable. 
It  is  among  the  artificial  salts,  however,  that  the  widest  field  is  open  for  the  application  of  precise  research,  and 
one  in  which  a  rich  harvest  of  important  results  would,  in  all  probability,  amply  repay  the  trouble  of  the  inves- 
tigation, whether  considered  in  an  optical,  a  chemical,  or  a  crystallographical  point  of  view. 


The  fraction  P  = 


'-  1 


where  fi  is  the  refractive  index,  and  s  the  specific  gravity  of  the  medium,  expresses      1118. 

Table  of 


(in  the  doctrine  of  emission)  the  intrinsic  refractive  energy  of  its  molecules,  supposing  the  ultimate  atoms  of  all  intrinsic 
bodies  equally  heavy.     The  following  results  have    been  stated  by  various  authors,  as  its  values  for  todies  most  Refractive 
widely  differing  in  their  chemical  and  mechanical  relations. 

I.     Gases,  taking  the  value  nf  P  for  atmospheric  air  as  unity.     (From  Dial's  Precis  Elementaire,  ii.  224.) 


Oxygen     0.86161 

Air    1.00000 

Carbonic  acid 1.00476 


Azote   1.03408 

Muriatic  gas    1.19625 

Supercarburetted  hydrogen 


Carburetted  hydrogen. .    2.09270 

Ammonia 9.16^51 

Hydrogen    6.61436 


II.    Direct  values  of  P  given  by  thefcrmula. 
Those  marked  Dulong  are  computed  from  the  refractive  indices  ot  Dulcmg  in  the  last  table. 


Tabasheer    0.0976  Brewster. 

Cryolite 0.2742  Brewster. 

Fluor  spar 0.3426  Brewster. 

Oxygen    0.3799  Tlulong. 

f  0.3829     Dulong. 
Sulphate  of  barytes    ..|03979     Newt(fn_ 

Sulphurous  acid  gas  . .  0.44548  Dulong. 

Nitrous  gas 0.44911  Dulong. 

(0.4528  Dulong. 

Air .{0.4530  Biot. 

I 0.5208  Newton. 

Carbonic  acid 0.45372  Dulong. 

Azote 0.4734  Dulong, 

Chlorine 0.48133  Dulong. 

Glass  of  antimony 0.4864  Newton. 

Nitrous  oxide 0.5078  Dulong. 

Phosgen  0.5188  Dulong. 

Selenite  0.5386  Newton. 

Carbonic  oxide  0.5387  Dulong. 

Quartz  0.541 5  Malus. 

f  0.5450      Newton. 

Rock  crystal \  n  CKIC     r> 

(0.6536     Brewster. 

Vulgar  glass    0.5436     Newton. 


Muriatic  acid  glass  ....    0.5514  Dulong. 

Sulphuric  acid    0.6124  Newton. 

,,  ,  1 0.6424  Malus. 

Calcareous  spar |  „  ^  New(on_ 

Sal  gem   0.6477  Newton. 

•Muriate  of  soda 1  .'2086  Brewster. 

Alum    0.6570  Newton. 

Nitric  acid 0.6676  Brewster. 

Borax 0.6716  Newton. 

Niter    0.7079  Newton. 

•Nitre 1.19B2  Brewsler. 

Hydrocyanic  acid    0.7366  Dulong. 

Ruby 0.7389  Brewster. 

Dantzic  vitriul,(sul.iron)  0.7551  Newton. 

Muriatic  ether  (vapour)     0.7552  Dulong. 

Brazilian  topaz    0.7586  Brewster. 

Rain  water 0.7845  Newton. 

Flint  glass  (mean) 0.7986  Brewster. 

Cyanogen    0.8021  Dulong. 

Sulphuretted  hydrogen  .    0.8419  Dulong. 

Gum  Arabic 0.8574  Newton. 

Vapour  of  sulphuret  of 

carbon 0.8743 


Vapour  of  sulph.  ether. .    0.9 1 38 
Protophosphuretted  hydr.  0.96SO 

Ammonia 1.0032 

Rectified  spirits  of  wine  1.0121 

Carbonate  of  potash   ..  1.0227 

Chromate  of  lead   1.0436 

Olefiantgas 1.0654 

*Muriate  of  ammonia..  1.1-90 

Carburetted  hydrogen  1.2201 

Camphor I  .-2551 

Olive  oil 1.2607 

Linseed  oil 1.2819 

Beeswax    IJ308 

Spirit  of  turpentine    ..  1.3222 

Amber     13654 

Octohedrite 1.3816 

Diamond     1 .4566 

Realgar 1.6666 

Ambergris    1.7000 

Mercury  (probable) 2.4247 

Sulphur    2.2000 

Phosphorus 2.8857 

Hydrogen    30953 


Dulong. 

Dulong. 

Dulong. 

Dulong. 

Brewster. 

Brewster. 

Dulong. 

Brewsler. 

Dulong. 

Newton. 

Newton. 

Newton. 

Malus. 

Newion. 

Newton. 

Brewster. 

Newton. 

Brewster. 

Brewster. 

Brewsler. 
Brewster. 

Dulon<;. 


Dulong. 

The  results  marked  with  an  asterisk  in  this  table  have  probably  originated  in  some  miscalculation.     As      1119. 
hydrogen  stands  highest  in  this  scale,  so  it  is  probable  that  fluorine,  should  we  ever  obtain  it  in  an  insulated  Remarks  on 
state,  would  prove  the  lowest.     The  optical  properties  of  tabasheer,  in  all  points  of  view,  are  strange  anomalies,  'his  Table, 

tf  —  1 
It  will  be  observed,  that  the  function only  expresses  the  intrinsic  refractive  power  on  the  hypothesis  of  the 

infinite  divisibility  of  m.ilter,  and  the  equal  gravitating  power  of  every  infinitesimal  molecule.  But  if,  as  modern 
Chemistry  indicates,  material  bodies  consist  of  a  finite  number  of  atoms,  differing  in  their  actual  weight  for  every  dif- 
ferently compounded  substance,  the  intrinsic  refractive  energy  of  the  atoms  of  any  given  medium  will  be  the  product 
of  the  above  function  by  the  atomic  weight.  This  will  aher  totally  the  order  of  media  from  what  obtains  in  the 
foregoing  table.  Thus,  the  weight  of  the  atom  of  hjdrogen  being  the  least,  and  that  of  mercury  one  among  '.he 


574 


LIGHT. 


Light. 


1120. 

Table  of 

Dispersive 

Powers. 


greatest  in  the  chemical  scale,  such  multiplication  will  depress  the  rank  of  the  former,  and  exalt  that  of  the  latter,     Part  IV. 
so  as  to  separate  them  entirely  from  the  proximity  they  now  hold.     A  distinction,  too,  will  require  to  be  regarded  ' 
between  compound  and  simple  atoms.     But  as  these  considerations  are  peculiar  to  the  system  of  emission,  we 
shall  not  prosecute  them  farther  in  detail 

The  dispersive  powers  of  bodies  afford  another  very  interesting  and  distinctive  chaiacter.  Of  these,  Dr. 
Brewster,  in  his  Treatise  on  New  Philosophical  Instruments,  has  given  the  following  extensive  table,  almost 
entirely  from  his  own  observation. 

TABLE  OF  DISPERSIVE  POWERS. 


Column  1  contains  the  name  of  the  medium;  column  2  the  value  of  the  function  - 
S  simply,  8  ft  being  the  difference  of  refractive  indices  of  extreme  red  and  violet  rays. 


;  column  3,  that  of 


Dispersive  Powers. 

3,* 
f~  I 

>^ 

Au- 
thor. 

Dispersive  Powers. 

>*• 

Ip. 

Au- 
thor. 

/»-! 

Chrom.  lead,  greatest  estimated 
Ditto              greatest  exceeds 
Realgar,  melted,  different  kind 
Chrom.  lead,  least  refraction  . 
Realgar  melted  

0.400 
0.296 
0.267 
0.262 
0.255 
0.139 
0.130 
0.128 
0.103 
0.093 
+  0.091 
0.085 
0.074 
0.069 
0.066 
0.066 
0.065 
0.063 
0.062 
0.062 
0.061 
0.060 
0.060 
0.060 
0.060 
0.057 
0.055 
0.054 
0.053 
0.053 
0.0527 
0.052 
0.052 
0.052 
0.051 
0.051 
0.050 
0.050 
0.049 
0.049 
0.049 
0.049 
0.049 
0.048 
0.048 
0.048 
0.048 
0.048 
0.047 
0.046 
0.046 
0.046 

0.770 
0.570 
0.394 
0.388 
0.374 
0.089 
0.149 
0.156 
0.065 
0.058 
+  0.091 
0.058 
0.044 
0.039 
0.041 
0.056 
0.033 
0.037 
0.032 
0.033 
0.037 
0.056 
0.044 
0.032 
0.038 
0.032 
0.028 
0.026 
0.042 
0.029 

0.028 
0.026 
0.032 
0.031 
0.025 
0.024 
0.024 
0.022 
0.024 
0.024 
0.023 
0.023 
0.029 
0.028 
0.028 
0.028 
0.023 
0.022 
0.021 
0.025 
0.032 

B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
Bos. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B 

Oil  brick  

0.046 
0.0457 
0.045 
0.045 
0.045 
0.045 
0.045 
0.045 
0.045 
0.045 
0.044 
0.044 
0.044 
0.044 
0.044 
0.044 
0.044 
0.044 
0.043 
0.043 
0.043 
0.043 
0.042 
0.042 
0.042 
0.041 
0.041 
0.041 
0.041 
0.041 
0.040 
0.040 
0.040 
0.040 
0.040 
0.039 
0.039 
0.038 
0.038 
0.038 
0.037 
0.037 
0.037 
0.037 
0.037 
0.037 
0.037 
0.036 
0.036 
0.036 
0.036 
0.036 

0.021 

0.019 
0.021 
0.023 
0.027 
0.025 
0.024 
0.022 
0.024 
0.018 
0.021 
0.022 
0.022 
0.025 
0.021 
0.020 
0.045 
0.016 
0.024 
0.022 
0.024 
0.020 
0.020 
0.022 
0.022 
0.021 
0.021 
0.021 
0.023 
0.019 
0.031 
0.027 
0.019 
0.023 
0.021 
0.019 
0.056 
0.018 
0.022 
0.013 
0.016 
0.020 
0.022 
0.018 
0.012 
0.020 
0.017 
0.0  IS 
0.019 
0.020 
0.018 

B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 

Flint  glass,  (Boscov.  lowest) 
Nitric  acid  

Oil  lavender   

Balsam  of  sulphur     

Tortoise  shell  

Horn  

Phosphorus    

Canada  balsam  

Balsam  Tolu         

Oil  marjorum  

Carb   lead  greatest   

Nitrous  acid  (?)    

Cajeput  oil  

Oil  hyssop        

Balsam  styrax          

Pink  coloured  glass  

Carb.  lead,  least  refraction   .  . 
Oil  cummin        

Oil  DODDV   . 

Jargon,  greatest  refraction    .  . 

Copal    

Nut  oil      '.   

Burgundy  pitch  

Oil  rosemary  

Glue    

Balsam  capivi     

Stilbite   

Flint  glass,  (Boscov.  greatest)  . 

Oil  peppermint  

Carb.  lime,  greatest  refraction 
Oil  rape  seed  

Oil  fen  ('  fenu)  «reek 

White  of  egg  . 

Oil  dill  seed 

Beryl     

Obsidian  

Ether             

Flint  glass                     

Selenite  

Alum      

Oil  juniper              

Oil  castor  

Oil  chamomile 

Sulphur  copper  

Gum  juniper  

Crown  glass,  very  green    .... 

Carb.  strontia,  greatest  refrac. 

LIGHT. 


575 


Light. 


», 

Au- 

5^ 

Au- 

Dispersive Powers. 

/»-! 

t» 

thor. 

Dispersive  Powers. 

<•-! 

l  P. 

thor. 

0  036 

0  020 

B 

0.031 

0.014 

R, 

Jellyfish  (medusa  ceyuora)  body 
Water            

0.035 
0.035 

0.013 
0.012 

B. 
B 

Apophyllite  (leucocyclite)  .... 

0.031 
0.030 

0.017 
0.016 

He. 
B. 

Aqueous  humour  haddock  eye 

0  035 

0  012 

B 

0.030 

0.014 

B. 

Vitreous  humour  haddock  eye 

0  035 

0  012 

B 

0.030 

0.022 

R. 

0.035 

0.019 

B 

0.029 

0.011 

B. 

Rubellite                 

0  035 

0  027 

B 

Sulph  barytes            

0.029 

0.019 

B. 

Leucite    "!*  

0035 

0.018 

R 

Tourmaline  .  .  .  *  

0.028 

0.019 

B. 

Epidote 
Common    glass,     Boscovich's 

0.035 
0  0346 

0.024 

B. 
Bos 

Crown    glass,    Leith,     (Robi- 
son,)  cited  by  Brewster.  .  .  . 
Carb   strontia   least  refraction 

0.027 
0  027 

0.015 

Rob. 
B. 

0.033 

0.027 

B 

Rock  crystal  

0.026 

0.014 

B. 

Common     glass,     Boscovich  s 

Emerald   

0.026 

0.015 

R. 

lowest,  cited  by  Brewster  .  . 

0.033 
0  033 

0.026 

Bos. 
B 

Carb.  lime,  least  refraction    .  . 
Blue  sapphire     

0.026 
0.026 

0.016 
0.021 

B. 
B 

Chrysolite   

0.033 

0.022 

R 

Bluish  topaz,  cairngorm    .... 

0.025 

0  016 

R. 

0  033 

0  018 

R 

Chrysoberyl              .    . 

0.025 

0.019 

B. 

0.032 

0.012 

R 

Blue  topaz,  Aberdeenshire    .  . 

0.024 

0.025 

B 

0  032 

0  012 

R 

Sulph    stronlia  

0.024 

0  015 

R 

Phosphoric  acid   solid  prism  . 

0  032 

0  017 

R 

Fluor  spar  

0.022 

0.010 

R 

Plate  irlass  .  . 

0.032 

0.017 

B. 

Cryolite  .  . 

0.022 

0.007 

B. 

Part  IV. 


Powers. 


1122. 


Respecting  the  results  in  this  table,  the  remark  applied  to  that  of  refractive  indices  may  be  yet  more  strongly      1121. 
urged.     The  whole   stands  in   need  of  a  radical   reinvestigation.     Those  only,  however,  who  have  had   some  Remarks  on 
experience  of  the  difficulties  in  the  way  of  a  strict  scientific   examination  of  dispersive  powers,  can   appreciate  th?  Table  o( 
either  the  labour  of  such  a  task,  or  the  merit  of  Dr.  Brewster  in  his  researches,  which  we  must  not  be  understood  Dispersive 
as  in  the  slightest  degree  depreciating'  by  this  remark.     But  the  refinements  of  modern   science  are  every  day 
carrying  us  beyond  all  that  could  be  contemplated  in  its  earlier  stages,  and  it  is  matter  of  congratulation,  rather 
than  disappointment,  to  every  true  philosopher,  to  see  his  methods  replaced  by  others  more  powerful,  and  his 
results  rendered  obsolete  by  the   more  exact  conclusions  of  his  successors.     What  is  now  chiefly  wanted  is  a 
knowledge  of  the  whole  series  of  refractive  indices  for  the  several  definite  rays  throughout  the  spectrum,  under 
uniform  circumstances,  and  for  all   media  whose  chemical  and  other  characters  are  sufficiently  definite  and  con- 
stant to  enable  us  to  identify  and  reproduce  them  in  the  same  state,  at  all  times.     The  researches  of  Fraunhofer 
and  Arago  have  shown  that  accuracy  in  the  determination  of  refractive  indices  sufficient  for  the  purpose,  may  be 
attained,  and  we  trust,  therefore,  that  this  great  desideratum  will  not  long  remain  unsupplied. 

To  the  substances  in  the  table  many  important  remarks  apply.  In  general,  high  refractive  is  accompanied  by 
high  dispersive  power ;  but  exceptions  are  endless,  especially  among  the  precious  stones,  of  which  diamond 
affords  a  striking  instance.  Particular  bodies  seem  to  carry  their  dispersive  as  well  as  their  refractive  powers 
with  them  into  their  compounds,  and  that  more  evidently,  because  by  the  peculiar  mode  in  which  the  dispersion 
is  represented,  the  state  of  condensation  is  eliminated.  Thus,  fluorine,  and  even  oxygen,  appear  to  exercise  a 
very  lowering  influence  on  the  dispersive  powers  of  their  compounds,  while  hydrogen,  sulphur,  and  especially 
lead,  act  with  great  energy  in  the  opposite  sense.  The  contrast  between  the  oils  of  ambergris  and  cassia,  is  at  Experiment 
least  as  remarkable  in  point  of  dispersive  as  of  refractive  power.  The  following  experiment  would  seem  to  point  °°  oil  of 
out  the  hydrogen  of  the  latter  oil,  as  the  principle  to  which  its  extraordinary  dispersion  is  due,  and  is  otherwise  cassia- 
instructive,  as  exemplifying  strongly  the  independence  of  the  two  powers  inter  se.  A  stream  of  chlorine  was 
passed  through  oil  of  cassia  till  it  refused  to  act  any  farther.  The  oil  was  at  first  "greatly  deepened  in  colour, 
but  as  the  action  proceeded,  it  changed  to  a  much  lighter  ruddy  yellow,  which  it  retained  till  the  action  was 
complete,  (and  which  in  a  few  days  changed  to  a  fine  rose  red.)  Copious  fumes  of  muriatic  acid  gas  were  given 
off  during  the  whole  process,  indicating  the  abstraction  of  abundance  of  hydrogen,  and  at  length  the  oil  was  con- 
verted into  a  viscous  mass,  drawing  out  into  long  threads,  having  entirely  lost  its  peculiar  perfume,  and  acquired  a 
pungent,  penetrating  scent,  and  an  acrid,  astringent  taste,  totally  unlike  its  former  aromatic  flavour.  It  was  inflam- 
mable, though  less  than  before,  burning  with  a  flame  green  at  the  edges,  indicating  the  presence  of  chlorine.  Its 
refractive  power  was  very  little  diminished.  A  drop  being  placed  in  the  angle  of  two  glass  plates,  and  close  to 
it  a  drop  of  unaltered  oil  of  cassia,  the  spectrum  of  a  line  of  light  was  viewed  at  once  with  the  same  eye  through 
both  the  media.  They  still  formed  a  continuous  line,  the  spectrum  of  the  unaltered  oil  being  more  refracted  by 
only  about  one-fourth  the  breadth  of  that  of  the  altered  specimen.  But  the  dispersive  power  of  the  latter  was 
most  remarkably  diminished,  being  brought  below  not  only  that  of  the  unaltered  oil,  but  even  below  that  of  flint 
glass.  When  the  dispersion  of  the  unaltered  oil  was  corrected  by  flint  glass,  that  of  the  altered  was  found  to  be 
much  more  than  corrected ;  and  when  the  angle  of  the  glass  plates  was  such  that  the  dispersion  of  the  latter  was 
iust  corrected  by  a  prism  of  Dollond's  "heavy"  flint,  whose  refracting  angle  =  about  25°,  the  unconnected 
spectrum  of  the  former  was  about  equal  to  that  of  the  flint  prism.  The  dispersion,  then,  had  been  diminished 
to  half  its  former  amount,  while  the  refraction  had  suffered  hardly  any  appreciable  change.  (October  7,  1825.) 

The   angle   of  complete  polarization  of  a  ray  reflected  at  the  surface  of  a  medium,  affords  a  most  valuable 
character  in  mineralogy,  as  it  gives  at  once  an  approximation  to  the  refractive  index,  sufficient  in  a  great  variety 


1123 


576 


LIGHT. 


Use  of  the 
polarizing 
angle  as  a 
physical 
character. 

Action  of 
crystallized 
surfaces  on 
reflected 
light. 


1124. 

Table  of 
angles  be- 
tween the 
optic  axes 
of  crystals. 


of  cases  to  decide  between  two  substances,  which  might  be  otherwise  confounded  together,  and  inasmuch  as  it  Part  IV. 
can  be  measured  on  any  single  surface  sufficiently  polished  to  give  a  regular  reflexion,  thus  enabling  us  to  apply  v— N^~- 
this  character  to  minute  fragments,  or  to  specimens  set  as  jewels,  or  otherwise  too  precious  to  be  sacrificed ;  to 
opaque  bodies,  and  to  a  variety  of  other  cases  where  a  direct  measure  of  the  refraction  would  be  impracticable. 
It  has  not  escaped  the  acute  and  careful  observation  of  Dr.  Brewster,  that  the  polarizing  angle  on  the  surfaces  of 
crystallized  media  is  not  absolutely  the  same  in  all  planes  of  incidence  ;  and  the  deviation,  though  excessively 
small  when  the  natural  reflexion  is  used,  becomes  very  sensible,  and  even  enormous,  when  the  reflexion  is 
weakened  by  covering  the  surface  with  a  cement  of  a  refraction  approaching  that  of  the  medium,  so  as  to  allow 
only  those  rays  to  reach  the  eye  which  have  penetrated,  as  it  were,  to  some  minute  depth,  and  undergone  some 
part  of  the  action  of  the  crystal  as  such.  The  point  is  among  the  most  curious  and  interesting  in  the  doctrine 
of  reflexion,  and  we  regret  that  our  limits,  as  well  as  the  obscurity  still  hanging  over  it,  and  which  it  will 
require  much  elaborate  research  to  dissipate,  prevent  our  devoting  a  section  to  it,  but  we  must  be  content  to  refer 
the  reader  to  an  excellent  paper  on  the  subject  by  that  Philosopher,  Philosophical  Transactions,  1819. 

The  angles  included  between  the  optic  axes  of  biaxal  crystals  is  a  physical  character  of  the  first  rank,  both 
on  account  of  its  distinctness,  its  extent  of  range,  (indifferently  over  the  whole  quadrant,)  and  its  immediate  and 
intimate  connection  with  the  state  in  which  the  molecules  of  the  crystals  subsist,  and  what  may,  loosely  speaking, 
be  termed  their  structure.  It  is,  however,  a  character  by  no  means  easily  determined  :  both  axes  rarely  lying 
within  one  field  of  view,  capable  of  being  examined  through  natural  surfaces,  and  requiring,  in  almost  all  cases, 
the  production  of  artificial  sections  ;  at  least,  this  is  the  only  safe  way  for  observations  of  the  tints,  for  the 
angles  at  which,  in  a  thin  parallel  plate,  the  several  successive  orders  of  colours  are  produced  in  situations 
remote  from  the  axes,  are  for  the  most  part  far  too  vague  to  lead  to  any  accurate  conclusion  as  to  the  position  of 
these  lines  within  the  plate,  not  to  speak  of  the  sources  of  fallacy  highly  coloured,  or  dichroite,  crystals  obviously 
present.  With  these  considerations  before  us,  we  cannot  but  be  struck  with  surprise  and  admiration  at  the 
unwearied  assiduity,  which  could  produce,  almost  unassisted,  a  table  of  results  so  extensive  and  so  valuable  as 
the  following. 

Table  of  the  Inclinations  of  the  Optic  Axes  in  various  Crystals. 


I.  UNIAXAL  CRYSTALS.     Inclination  =  0. 


Carbonate  of  lime,  (Iceland  spar.) 
Carbonate  of  lime  and  magnesia,  (bitter 

spar.) 

Carbonateof  lime  and  iron,  (.brown  spar.) 
Tourmaline. 
Rubellite. 


Zircon. 
Quartz. 
Oxide  of  Iron. 


Corundum. 

Sapphire, 

Ruby. 

Emerald. 

Beryl. 

Apatite. 

Tangslateof  zinc. 

Titanite. 

Boracite. 

Hyposulphate  of  lime. 


Negative  Class. 

Idocrase,  (Ve&uvian.) 

\Vernerite. 

Mica  from  Kariat. 

Phosphate  of  lead. 

Phosphato-arseniate  of  had. 

Hydrate  of  strontia. 

Positive  Ctas*. 

Apophyllite. 

Sulphate  of  potash  and  iron. 
Superacetate  of  copper  and  lime, 
Unclax&cd. 

Oxysulphate  of  irok. 


Arseniate  of  potash. 
Muriate  of  lio.e. 
Muriate  of  strontia. 
Suhphosphate  of  potash. 
Sulphate  of  nickel  and  copper. 


Hydrate  of  magnesia. 


II.  BIAXAL  CRYSTALS. 


Names  of  crystal*. 

Character  of  th 
principal  axis 
according  to 
Dr.  Brewster's 

system. 

Inclination  o 
optic  axes. 

Names  of  crystals. 

Character  of"  th 
principal  a*i> 
according  in 
Dr.  Hrewsttr's 
system. 

optic  .ixi-i. 

Sulphate  of  nickel,  certain  specimens  .  . 

+ 
+ 

3°     (X 
5     15 

6     56 

5     20 
6       0 
7     24 
11     28 
13     18 
14       0 
18     18 
19     24 
25       0 
27     51 
28       7 
28     42 
30      0 
31       0 
32      0 
34       0 
37       0 
35       8 
37    24 
37     42 
37    40 
38    48 
40      0 
41     42 
42       4 
43     24 
44     28 

+ 
+ 

+ 

-I- 

? 

+ 

+ 

+ 
+ 

+ 
+ 

45°      I)' 
45       (1 
45       8 
•Hi     4'J 
4!)     -42 
•J'J    r50° 
flip      0' 

r>o     o 

51      16 

51     22 
5.')     20 

56     e 

60       0 
62     16 
62     50 
63      0 
65       0 
67       0 
70       1 
70     25 
70    29 
71     20 
79      0 
80      0 
80    30 
81     48 
82      0 
84     19 
94     30 
87     5G 
88     14 
90      0 

Lepi'lolite  

Sulphate  of  magnesia  and  soda    .  .  .  . 

Brazilian  topaz  (Brewsler  and  Biot) 

Talc  ' 

Muriosulphate  of  magnesia  and  iron 
Sulphate  of  ammonia  and  magnesia.  . 

Prussiale  of  potash  (?  Ferrocyanate)  .  . 

+ 

+ 
+ 

+ 
+ 

A  1  -i 

Mica,  various  specimens  examined  by! 
M.Biot  I 

I 

Tartrate  of  potash  and  soda  ........ 

+ 

Stilbite       . 

+ 
+ 

Crystallized  Cheltenham  salts  
Sticciriic  acid,  estimated  at  about 

Sulphate  of  zinc     

LIGHT. 


577 


Lizht. 


Among  crystals  with  one  axis,  Dr.  Brewster  has  enumerated  the  Idocrase,  or  Vesuvian,  and  correctly.     Had     Part  IV. 
he  noticed,  however,  in  the  specimens  examined  by  him  the  very  striking  inversion  of  the  tints  of  Newton's  ^— -v— — 
scale  exhibited  in  the  rings  of  that  now  before  us,  he  would  doubtless  have  made  mention  of  it.     We  insert  here      1125. 
the  scale  of  colours  exhibited  by  a  plate  cut  from  the  specimen  in  question,  fa  fine  large  crystal,)  as  affording  Remarks, 
another  remarkable  case  in  addition  to  that  of  the  hyposulphate  of  lime,  and  the  several  varieties  of  uniaxal  ^erted 

apophyllite  already  mentioned,  of  such  inversion. 

r    J  vesuviar.. 

Table  of  the  tints  exhibited  by  a  plate  of  Vesuvian,  thickness  =  0.11035  inch,  cut  a  little  obliquely  to  a  perpen- 
dicular to  the  axis. 


Angle  of 
Incidence. 

Ordinary  Image. 

Extraordinary  Image. 

«  zz 

Angle  of  Refraction  (. 

+  66°    -f  ' 

No  light  passed     

No  light  passed 

+  66       0 

Brick  red    

+  64       0 

Orange  red 

+  60       0 

Tolerable  orange  pink  

Fine  bluish  green 

+  52       0 

Pale  yellow  pink   

+  47       0 
+  42       0 

Pink,  with  a  dash  of  purple.  . 
Pale  neutral  purple   

Pretty  bright  yellow. 
Good  yellow 

1 

OtO      cflf 

+  37       0 

Bluish  white  

Yellow  less  bright. 

2 

+  30       0 

+   15       0 

Very  pale  yellowish  white.  .  .  . 
Yellowish  white  

Sombre  brownish  yellow. 
Very  sombre  yellow  brown 

"V 

+   10       0 

Yellowish  white  

Almost  totally  extinct 

to 

fi      31    -4- 

+     30 

Yellowish  white  

Very  sombre  purplish  brown 

J 

±00 

Yellowish  white  

Dusky  brownish  Yellow. 

J 

9       0 

Bluish  white  

Rather  dull  yellow 

-    12       0 

Dull  purplish  blue    

Bright  yellow     

1 

+     7     48 

-    16       0 

Ruddy  purple    

Pale  yellow. 

-    19       0 
-   22       0 

Pink,  verging  to  brick  red    .  . 
Yellowish  red    

Imperfect  green. 
Tolerable  bluish  green. 

-    26       0 

-   28       0 

Yellow,  inclining  to  orange  .  . 
Bright  yellow     

Rich  greenish  blue. 
Blue  purple. 

-   28     30 

Bright  yellow     

Neutral  purple  

1 

+  18     10 

-   29       0 

Bright  yellow     

Ruddy  purple. 

-   30       0 

Yellow  green  

Crimson. 

-   32       0 

Good  pink. 

-   35       0 

Greenish  blue    

Orange  pink. 

-   37     30 

Blue  purple  

Pale  yellow. 

-   38     30 

Neutral  purple     

4 

+  24       0 

-   39     15 

Ruddy  purple    

Greenish  yellow. 

-   41     30 

Good  green. 

-   45       0 

Pink  yellow      

Fine  greenish  blue. 

-    47     20 

Yellowish  white 

Blue  purple. 

-    47     30 

•2 

+  28     48 

-   48       0 

Ruddy  purple. 

—   49     30 

Good  pink. 

-    53       0 

Fine  blue  green  

Oranjre  pink. 

-    54       0 

Yellow. 

-    54     — 

No  light  passed     

No  light  passed. 

The  first  ring,  it  will  be  observed,  in  calculating  from  this  table,  is  contracted  beyond  What  is  due  to  the  law 
of  the  sines,  probably  from  the  section  examined  not  passing  precisely  over  their  common  centre,  and  gives  a 
polarizing  power  greater  than  that  deduced  from  the  angles  corresponding  to  n  =  1,  n  =  ^,  n  =  2,  all  which 
agree  in  assigning  41.35  nearly  as  the  measure  of  the  power  in  question.  See  Art.1126. 

It  follows  from  this  series,  that  of  the  two  images  formed  by  double  refraction  in  Vesuvian,  and  other  similar 
crystals,  the  most  refracted  should  be  the  least  dispersed,  a  peculiarity  we  have  not  yet  had  an  opportunity  01 
verifying  by  direct  observation.  It  follows,  however,  immediately  from  the  theory  of  the  rings  above  delivered, 
since  the  smaller  the  diameters  of  the  rings  for  any  coloured  ray,  the  greater  the  separation  of  its  pencils  by 
double  refraction.  Hence,  in  the  present  case,  the  red  rays  will  be  separated  by  a  greater  interval  than  the  violet 
in  the  two  spectra ;  and,  consequently,  the  least  refracted  spectrum  will  be  the  longest.  In  the  variety  of 
apophyllite  exhibiting  white  and  black  rings,  (leitmcyclite)  the  two  dispersions  should  be  almost  exactly  equal, 
and  the  only  difference  between  the  two  spectra  ought  to  consist  in  a  slight  variation  in  the  proportional  breadths 
of  the  several  coloured  spaces  in  them. 

Another  very  important  optical  character  is  the  intensity  of  the  polarizing,  or  doubly  refractive  energy.  This 
may  be  concluded  by  measuring  the  actual  angular  separation  of  the  images;  but  this  is  usually  too  small  to 
VOL.  iv.  4  F 


1126. 


578 


LIGHT. 


Light.  admit  of  being  determined  with  sufficient  precision,  in  such  very  imperfect  specimens  as  are  usually  subjected  to 
^^~^~^  examination  for  the  purpose  of  identification,  and  a  much  better  course  is  to  make  the  tint  developed  at  a  per- 
Polarizing  pendicular  incidence,  by  a  plate  of  given  thickness  in  a  direction  at  right  angles  to  both  the  optic  axes,  the 
powers  con-  object  of  determination.  This  tint  (which  we  shall  term  the  equatorial  tint)  may  be  derived  immediately  from 
observations  of  tints  at  any  angle,  by  the  formula 


Part  IV. 


character 
of  media. 


COS  f 


t       sin  0  .  sin  ff  ' 

where  N  is  the  tint  in  question,  numerically  expressed  as  usual,  and  where  n  is  the  tint,  (also  similarly  expressed) 
developed  at  an  angle  of  incidence  whose  corresponding  angle  of  refraction  is  />,  on  a  plate  whose  thickness  is  t, 
(expressed  in  English  inches  and  decimals)  and  where  6,  &  are  the  angles  made  by  the  ray  in  traversing  the  plate 

with  the  two  axes.     This  value  of  N  is  the  same  with  —  in  the  equation  of  Art.  907.     The  following  list  of  a 

very  few  substances  will  suffice  to  show  the  great  range  the  value  of  N  admits,  and  its  consequent  utility  as  a 
physical  character,  considerations  which  we  hope  will  induce  observers  to  extend  the  list  itself,  as  well  as  to 
give  it  all  possible  exactness. 


UNIAXAL  CRYSTALS. 


For  mean  yellow  rays. 


N  = 

35801 
1246 
851 
470 
312 
109 
101 
41 
33 
3 

*  =  T 

0.000028 
0.000802 
0.001175 
0.002129 
0.003024 
0.009150 
0.009856 
0.024170 
0.0303T4 
0.366620 

Ditto.           3d  variety  .  . 

BIAXAL  CRYSTALS. 


Nitre                       

For  mean 

N  = 

7400 
1900 
1307 
521 
249 

yellow  rays. 

A  =  ¥ 

0.000135 
0.000526 
0.000765 
0.001920 
0.004021 

Heulandite  (white:  —  anele  between  axes  c:  54°  17').. 

1127. 

Useofpola- 
rizeJ  light 
in  detect- 
ing complex 
structures. 


1128. 

Compound 
crystals  of 
nitre. 

Arragonite. 


1129. 

Topaz. 


1130. 

Tesselite. 
rig.  223. 


But  the  phenomena  of  refraction,  reflexion,  and  polarization,  may  not  only  be  applied  by  the  aid  of  these  and 
similar  tables  of  registered  results,  to  the  examination  and  identification  of  substances  in  the  gross,  they  are  also  of 
use  in  detecting  peculiarities  of  structure  in  individual  specimens,  or  in  certain  species  which  would  otherwise 
escape  observation.  The  singular  structure  of  amethyst  has  been  already  explained,  and  a  variety  of  cases  of 
hernitropism  might  be  noticed,  in  which  the  juxtaposition  of  the  parts  is  rendered  evident  by  the  test  of  polarized 
light.  Of  these,  however,  by  far  the  most  curious  and  interesting  are  those  in  which  the  juxtaposed  parts  com- 
bine to  form  a  regular  whole,  and  to  produce  a  species  of  pseudo-crystal,  built  up  as  it  were  of  several  individuals, 
arranged  with  a  regard  to  symmetry,  and  forming  a  structure  of  more  or  less  complication.  Such  instances 
have  been  more  particularly  noticed  in  nitre,  arragonite,  topaz,  apophyllite,  sulphate  of  potash,  analcime,  har- 
motome,  &c. 

The  usual  form  of  the  crystals  of  nitre,  when  large  and  well  developed,  is  the  regular  hexagonal  prism  ;  h 
section  of  this,  cut  at  right  angles  to  the  axis,  is  very  commonly  found  to  consist  of  two  or  more  portions,  in 
which  the  optic  meridians  are  60°  inclined  to  each  other  ;  but  the  plane  of  division  often  intersects  one  of  the 
lateral  faces  of  the  prism,  without  any  visible  external  mark  of  a  breach  of  continuity,  so  that  but  for  the  test  of 
polarized  light,  the  macled  structure  would  never  be  discerned.  The  phenomena  of  arragonite,  in  this  respect, 
are  very  similar  to  those  of  nitre. 

If  a  plate  of  Brazilian  topaz,  cut  at  right  angles  to  the  axis  of  the  rhombic  prism  in  which  it  crysti 
be  examined  by  polarized  light,    it  will  occasionally  be  found  to  consist  of  a  central  rhomb,  surrounded  by  a 
border  in  which  the  optic  meridians  of  the  alternate  sides  are  inclined  at  \  of  a  right  angle  to  that  of  the  centre 
compartment,  and  \  a  right  angle  to  each  other.     In  consequence,  when  such  a  rhombic  plate  is  held  wit! 
its  long  diagonal  in  the  plane  of  primitive  polarization,  two  opposite  sides  of  the  border  appear  bright, 
other  two  black,  and  the  central  compartment  of  intermediate  brightness.     Such  specimens  often  exlnbi 
phenomena  of  dichroism  in  the  central  compartment,  while  the  border  is  colourless  in  all  positions. 

But  it  is  in  the  apophyllite  of  the  variety  named  by  Dr.  Brewster,  Tesselite,  that  this  enclosure  of  one  cry 
in  a  case  as  it  were  of  another,  is  exhibited  in  the  most  regular  and  extraordinary  manner.     In  one  of  t 
ties  of  this  singular  body,  whose  form  is  the  right  rectangular  prism  with  flat  summits,  slices  taken  off  from 
summit  were  found  by  him  to  be  of  uniform  structure ;  but.  when  these  were  detached,  every  subsequent  s 


LIGHT.  579 

Light,     found  to   consist  of  a  rectangular  border  enclosing  no  less  than  nine  several  compartments,  arranged  as  in     Part  IV 
^—  v  — _ '  fig.  223,  and  separated  from  each  other,  and  from  the  border,  by  delicate  lines  or  films  as  there  marked.     Each  ~~-~^-~~. 
of  these  compartments  possesses  its  own  peculiar  crystallographic  structure,  and  polarizes  its  peculiar  tints,  the 
law  of  symmetry  being  observed.     In  some  specimens  the  triangular  spaces  p  q  r  s  were  wanting,  while  in  others 
they  seem  to  have  consisted  of  two  portions,  separated  by  an  imaginary  prolongation  of  the  line  joining  their 
obtuse  angles  with  the  central  lozenge. 

The  terminal  plates,  the  central  lozenge,  and  the  minute  stripes  dividing  the  compartments  from  each  other 
(which  are  sections  of  lamina?  or  films  parallel  to  the  axis  of  the  crystal,  and  running  its  whole  length)  consist 
of  that  uniaxal  variety,  in  speaking  of  which  we  have  used  the  term  leucocyclite,  from  the  whiteness  of  its  rings. 
The  rectangles  R  V,  S  T,  (with  the  exception  of  the  portions  occupied  by  the  lozenge  and  partitions)  consist  of 
a  biaxal  medium,  having  its  axes  34°  inclined  to  each  other,  and  its  optic  meridian  parallel  to  the  axis  of  the 
prism,  and  passing  through  the  diagonals  R  V,  S  T  of  these  rectangles.  The  other  rectangles  are  composed 
of  a  similar  medium,  but  with  its  optic  meridian  at  right  angles  to  the  former,  or  passing  through  the  diagonals 
RT,  S  V. 

A  still  more  remarkable  and  artificial  structure  has  been  observed  by  Dr.  Brewster,  in  a  variety  of  the  Faroe      1131. 
apophyllites  of  a  greenish  white  hue.     When  a  complete  prism  of  this  variety  is  exposed  to  polarized  light,  with  Another 
its  axis  in  45°  of  azimuth,  the  light  being  transmitted  perpendicularly  through  two  opposite  sides,  the  pattern  variety- 
represented  in  fig.  224  is  seen,  in  which  the  central  curvilinear  area  is  red,  and  its  complements  to  the  surround-  F'g-  224. 
ing  rectangle  green.     The  squares  immediately  adjacent  on  either  side  in  the  direction  of  the  axis  are  also  vivid 
red  in  their  centres,  fading  into  white,  while  the  rest  of  the  pattern  consists  in  a  most  brilliant  succession  of  red, 
green,  and  yellow,  bands ,   for  a  coloured  figure  of  which  we  must  refer  the  reader  to  the  original  most  curious 
and  interesting  memoir,  (Edinburgh  Transactions,  vol.  ix.  part  ii.)  where,  as  also  in  the  Edinburgh  Philosophical 
Journal,  vol.  i.  he  will  find  the  phenomena  described  in  full  detail. 

The  sulphate  of  potash  offers  another  very  remarkable  example  of  compound  structure.     This  salt  occurs  in      1132. 
hexagonal  prisms,  and  occasionally  in  bipyramidal  dodecahedrons.     But  besides  these  forms  it  also  occurs  in  Sulphate  o( 
rhombic  prisms  of  1 14°  and  66°.     These  Dr.  Brewster  found  to  have  two  axes,  while  the  hexagonal  prisms  have  Potash- 
but  one  ;  thus  affording  another  instance  of  dimorphism  in  addition  to  those  of  arragonite,  sulphur,  &c.     On  ex- 
amining the  dodecahedrons,  however,  he  found  them  to  consist  of  six  equilateral  triangular  prisms,  of  the  biaxal 
variety,  grouped  together,  and  having  their  optic  meridians  all  converging  to  the  common  axis ;  the  molecules 
being  so  disposed  in   each  opposite  pair  of  individuals  as  to  make  the  angle  between  the  opposite  faces  of  either 
pyramid  (114°)  equal  to  the  obtuse  angle  of  the  rhomboid. 

The  structure  and  mode  of  action  of  the  analcime,  described  by  Dr.  Brewster  in  vol.  x.  of  the  Edinburgh  Tram-  1 133. 
actions,  part  i.  p.  187,  are  so  extremely  singular,  that  it  is  difficult  to  say  whether  it  should  be  regarded  as  a  Analcim*. 
grouped  crystal,  consisting  of  independent  portions  adhering  together,  or  as  a  mass  the  distribution  of  the  ether  in 
whose  parts  is  governed  by  a  general  and  uniform  law  ;  the  latter,  however,  is  probably  the  truth.  The  form  of 
this  crystal  is  the  icositetrahedron,  contained  by  twenty-four  similar  and  equal  trapezia,  and  may  be  regarded  as 
derived  from  the  cube  by  the  truncation  of  each  of  its  angles  by  three  planes  symmetrically  related  to  the  edges 
including  it.  If  we  conceive  from  the  centre  of  this  cube,  (in  its  natural  situation  with  respect  to  the  derived 
figure)  planes  to  pass  through  each  of  the  edges,  and  through  each  of  the  diagonals  of  the  six  faces,  they  will 
divide  the  cube  into  twenty-four  irregular  tetrahedra ;  and  ofthese,  all  the  faces  which  pass  through  edges  of  the 
cube  will  also  pass  through  edges  of  the  derived  figure,  while  those  which  pass  through  diagonals  effaces  of  the 
cube  will  also  pass  through  diagonals  of  the  faces  of  its  derivative,  bisecting  their  obtuse  angles.  Now  it  appears 
from  Dr.  Brewster's  observations,  that  all  the  molecules  situated  in  any  part  of  any  one  of  these  planes  are  devoid 
of  the  power  of  double  refraction  and  polarization  ;  and  that  in  proportion  as  a  molecule  is  distant  from  all  such 
planes,  its  polarizing  power  is  greater.  In  this  respect  it  differs  entirely  from  all  crystals  hitherto  examined, 
every  particle  of  which,  wherever  situated,  so  long  as  they  belong  to  one  and  the  same  crystalline  system,  being 
equally  endued  with  the  polarizing  virtue.  Nor  is  there  a  closer  analogy  between  the  mode  of  action  in  question, 
and  that  of  unannealed  glass  and  similar  bodies  ;  for  in  these  a  change  of  external  form  is  always  accompanied  with 
a  change  of  the  polarizing  powers,  while  in  the  analcime  each  particular  portion,  whether  detached  from  the  mass, 
or  in  its  natural  connection  with  the  adjacent  molecules,  possess  the  very  same  optical  properties.  The  action 
too  of  the  portions  which  possess  a  polarizing  power  is  not  related  to  axes  given  only  in  direction,  and  passing 
through  every  molecule,  but  to  planes  given  both  in  direction  and  in  place,  within  the  mass,  (the  planes  above 
mentioned  ;)  the  tint  developed  at  any  point  of  a  plate  being  as  the  square  of  the  distance  from  the  nearest  of  such 
planes,  and  the  isochromatic  lines  being,  in  consequence,  straight  fringes  of  colour  arranged  parallel  to  the  dark 
bands  marked  out  by  the  intersection  of  such  planes  with  the  plate  examined.  The  phenomena  described  are 
accompanied  with  a  sensible  double  refraction.  The  reader  is  referred  to  the  memoir  already  cited  (which  is  one 
of  the  most  interesting  to  which  we  can  direct  his  attention)  for  further  details  :  and  to  a  work  understood  to  be 
forthcoming  from  the  pen  of  the  eminent  author  here  and  so  often  before  cited,  on  optical  mineralogy,  for  what 
we  are  sure  will  prove  a  treasure  of  valuable  information  on  every  point  connected  with  this  important  application 
of  optical  science 

§  XIV.  On  the  Colours  of  Natural  Bodiei. 

It  was  onr  intention  to  have  devoted  a  considerable  share   ofthese  pages  to  the  explanation  of  such  natural      1134 
phenomena  as   depend  on  optical  principles,  but  the  great  length  to  which  this  essay  has  already  extended,  renders 
it  necessary  to  confine  what  we  have  to  say  on  such  subjects  within  very  narrow  limits,  and  to  points  of  promi- 

4  p  2 


580 


L  I  G  H  T. 


natural 
bodies. 

1135 
Postulates, 

1  1  36. 


1137. 


1  138. 

Cause  of 


1  139. 

Origin  of 

ur*' 


1  140. 

Objections, 


1141. 

Apparent 

steptions 


1142. 

Case  of 
transparent 


nent  importance.     Among  these  there  is  certainly  none  more  entitled  to  consideration  than  the  phenomena  of     Part  IV. 
colour,  as  exhibited  by  natural  objects,  which  strike  us  wherever  we  turn  our  eyes,  and  it  is  impossible  to  pass  in  v—  v*""' 
total  silence  the  theory  devised  by  Newton  to  account  for  them  ;  a  theory  of  extraordinary  boldness  and  subtilty, 
16  'n  wn'cn  oreat  difficulties  are  eluded  by  elegant  refinements,  and  the  appeal  to  our  ignorance  on  some  points  is 
so  dexterously  backed  by  the  weight  of  our  knowledge  on  others,  as  to  silence,  if  not  refute,  objections  which  at 
first  sight  appear  conclusive  against  it.     The  postulates  on  which  this  theory  rests  are  essentially  as  follows  : 

1.  AH  bodies  arc-  porous  ;  the  pores  or  intervals  vacant  of  ponderable  matter,  occupying  a  very  much  larger 
portion  of  the  whole  space  filled  by  the  body,  than  the  solid  particles  of  which  it  essentially  consists. 

2.  These  solid  particles  have  a  certain  size  (and  perhaps  figure)  essential  to  them  as  particles  of  that  particular 
medium,  and  which  cannot  be  changed  by  any  mechanical  action,  or  by  any  means  not  involving  a  change  in  the 
chemical  nature  or  condition  of  the  medium.  They  are,  in  short,  the  ultimate  atoms  ;  to  break  which,  is  to  destroy 
their  essence,  and  resolve  them  into  other  forms  of  matter,  having  other  properties. 

3.  These  atoms  are  perfectly  transparent,  and  equally  permeable  to  light  of  all  refrangibilities,  which,  having 
once  passed  their  surfaces,  is  in  the  act  of  pursuing  its  course  through  their  substances. 

Newton,  indeed,  makes  his  atoms  only  "  in  some  measure  transparent."  But  he  never  refers  to  this  limitation, 
and  his  theory  depends  essentially  on  their  perfect  transparency,  as  is  indeed  obvious  from  his  account  of  opacity, 
which  is  contained  in  the  next  postulate. 

4.  Opacity  in  natural  bodies  arises  from  the  multitude  of  reflexions  caused  in  their  internal  parts. 

jj  js  obvious,  therefore,  that  unless  we  admit  a  cause  of  opacity  in  atoms  different  from  that  which,  on  this 
hypothesis,  causes  it  in  their  aggregates  constituting  natural  bodies,  the  former  cannot  be  otherwise  than  abso- 
lutely pellucid,  since  no  reflexions  can  take  place  where  there  are  no  intervals,  and  no  change  of  medium.  Of 
the  sufficiency  of  this  cause,  either  in  natural  bodies  or  atoms,  however,  we  confess  there  does  appear  to  us  some 
room  for  doubt,  as  it  seems  difficult  so  to  conceive  these  internal  reflexions,  that  the  rays  subjected  to  them  shall 
be  all  and  for  ever  retained,  entangled  as  it  were,  and  running  their  rounds  from  atom  to  atom,  without  a  possi- 
bility of  reaching  the  surface  and  escaping  ;  which,  were  they  to  do,  it  is  evident  that  every  body  so  con- 
stituted, receiving  a  beam  of  light,  would  in  fact  only  disperse  it  in  all  directions  in  the  manner  of  a  self 
luminous  one. 

5.  The  colours  of  natural  bodies  are  the  colours  of  thin  plates,  produced  by  the  same  cause  which  produces  them 
in  thin  lamina  of  air,  glass,  fyc.  viz.  the  interval  between  the  anterior  and  posterior  surfaces  of  the  atoms,  which, 
when  an  odd  multiple  of  half  the  length  of  a  fit  of  easy  reflexion  and  transmission  for  any  coloured  ray  moving 
within  the  medium,  obsti  u  cts  its  penetration  of  the  second  surface,  and  when  an  even,  ensures  it,  (see  Art.  655.)  The 
thickness,  therefore,  of  the  atoms  of  a  medium,  and  of  the  interstices  between  them,  determines  the  colour  they 
shall  reflect  and  transmit   at  a  perpendicular  incidence.     Thus,  if  the  molecules  and  interstices  be  less  in  size 
than  the  interval  at  which  total  transmission  takes  places,  or  less  than  that  which  corresponds  to  the  edge  of  the 
central  black  spot  in  the  reflected  rings,  a  medium  made  up  of  such  atoms  and  interstices  will  be  perfectly  trans- 
parent.    If  greater,  it  will  reflect  the  colour  corresponding  to  its  thickness. 

It  may  be  objected  to  this,  that  all  natural  colours  do  not  of  necessity  find  a  place  in  the  scale  of  tints  of  thin 
plates,  even  those  of  bodies  whose  chemical  composition  is  uniform  ;  but  to  this  we  may  answer,  that  the  colours 
reflected  from  the  first  layer  only  of  molecules  next  the  surface  ought  to  be  pure  tints,  those  from  lower  layers 
having  to  make  their  way  to  the  eye  through  the  upper  strata,  and  thus  undergoing  other  analyses,  by  trans- 
missions and  reflexions  among  the  incumbent  atoms.  Besides  which,  whatever  shape,  we  attribute  to  the  atoms, 
it  is  impossible  that  all  rays  shall  penetrate  them  so  as  to  traverse  the  same  thickness  of  them,  unless  we  regard 
them  as  mere  lamina  without  angles  or  edges,  and  of  enormous  refractive  power.*  The  same  answer  must  be 
made  to  the  objection,  equally  obvious,  that  the  transmitted  tint  ought  to  be  in  all  cases  complementary  to  the 
reflected  one,  and  that  therefore  cases  like  that  of  leaf  gold,  opalescent  glass,  and  infusion  of  lignum  nephriti- 
cum,  all  which  reflect  one  tint  and  transmit  another,  but  in  all  which  this  condition  is  violated,  form  exceptions 
to  the  theory.  But,  in  reality,  the  transmitted  rays  have  traversed  the  whole  thickness  of  the  medium,  and  have 
therefore  undergone,  many  more  times,  the  action  of  its  atoms,  than  those  reflected,  especially  those  near  the 
first  surface,  to  which  the  brighter  part  of  the  reflected  colour  is  due. 

The  infusion  of  lignum  nephriticum  is  a  very  singular  case,  and  its  peculiar  properties  have  been  explained  by 
Dr.  Young,  on  the  supposition  of  minute  particles  of  definite  magnitude  suspended  in  it.  Though  very  truns- 
parent,  it  yet  reflects  a  bluish  green  colour,  while  the  light  transmitted  is  yellow  or  wine-coloured,  in  this  re- 
spect offering  almost  the  exact  converse  of  leaf  gold.  It  is,  however,  no  doubt  a  case  of  opalescence,  and  is 
exactly  imitated  by  certain  yellow  glasses,  in  which  a  very  visible  thin  film  of  opalescent  matter  near  the  surface 
reflects  to  the  eye  a  bluish  green  tint,  while  yet  the  colour  transmitted  has  the  yellow  tint  belonging  to  the  glass. 
The  reflexion  proceeds  from  particles  which  have  nothing  to  do  with  the  transmitted  light. 

But,  in  fact,  the  objection  (as  appears  to  us)  is  not  yet  fully  answered.  Transp  rent  coloured  media  (clear 
liquids  in  which  no  floating  particles  exist,)  have  no  reflected  colour.  When  examined  by  pouring  them  into  an 
opaque  vessel,  blackened  internally  and  filled  to  the  brim,  and  when  the  colourless  reflexion  from  their 
upper  surface  is  destroyed  by  reflexion  in  an  opposite  plane  at  the  polarizing  angle,  it  is  seen  at  once  that 
no  light  is  reflected  from  within  the  medium,  either  near  the  surface,  or  at  greater  depths  ;  and  if  this  mode  of 
examination  be  regarded  as  objectionable,  as  perhaps  destroying  the  internal  as  well  as  external  reflexion,  it  is 
equally  satisfactory  to  observe,  that  the  image  of  a  white  object  reflected  from  the  surface  of  a  fluid  iu  a  black 
opaque  vessel  is  always  purely  white,  whatever  be  the  colour  of  the  reflecting  fluid.  We  are  not  aware  that  the 
objection  so  put  has  been  sufficiently  considered,  or  even  propounded.  To  us  its  weight  appears  considerame, 


Newton  appears  to  nave  been  fully  aware  of  the  necessity  of  taking  this  into  consideration.     Prop.  vii.  book  ii.  Opt.  versvtfintm. 


LIGHT.  981 

Light.      and  we  cannot  but  believe  that  some  other  cause  besides   mere  internal  reflexions  must  interfere  to  prevent  the 
•»v-~»''   complementary  colour  from  reaching  the  eye ;    and  that  absorption,  with  its  kindred  phenomenon,  or  rather  its 
extreme  case,  opacity,  is  not  satisfactorily  accounted  for  in  this  theory,  but  must  rather  be  admitted  as  (at  pre- 
sent,) an  ultimate  fact,  of  which  the  cause  is  yet  to  seek. 

If  this  be  granted,  the  colours  of  all  bodies  may  be  distinguished  into  true,  viz.,  those  which  arise  from  rays 
u  hii-Ji  have  actually  entered  their  substance  and  undergone  their  absorptive  action,  (as  the  colours  of  powders 
of  transparent  coloured  media,  cinnabar,  red  lead,  Prussian  blue,  those  of  flowers,  &c.,)  and  false,  or  superficial, 
or  those  which  originate  obviously  in  the  law  of  interference  ;  thus,  the  variable  colours  of  feathers,  insects'  wings, 
striated  surfaces,  oxidated  steel,  and  a  variety  of  cases  to  which  the  Newtonian  doctrine  strictly  applies,  for  there 
is  no  denying  that  cases  of  colour,  not  merely  superficial,  do  occur,  in  which  the  Newtonian  doctrine,  to  say  the 
least,  is  highly  probable.  To  instance  one  or  two  only.  If  a  few  drops  of  an  extremely  weak  solution  of  Cases  in 
nitrate  of  silver  be  added  to  a  very  dilute  solution  of  hyposulphite  of  lime,  a  precipitate  is  formed  of  an  opales-  which  New. 
cent  whiteness  and  extreme  tenuity.  If  more  of  the  nitrate  be  added,  the  precipitate  increases  in  weight  and  g0"^^ 
aggregation,  and  at  the  same  time  changes  its  colour,  becoming  first  yellow,  then  yellow  brown,  then  a  rich 
orange  brown,  then  a  purplish  brown,  and,  finally,  a  deep  brown  black.  The  precipitate,  meanwhile,  continually 
acquires  density,  and,  finally,  sinks  rapidly  to  the  bottom.  It  is  impossible,  in  this  series,  not  to  trace  the  tints 
of  the  first  order  of  reflected  rings,  produced  by  the  thickening  of  the  minute  particles  in  the  act  of  aggregation, 
but  equally  impossible  not  to  recognise  the  agency  of  a  cause  totally  different,  acting  to  increase  the  opacity  of 
the  compound  by  an  absorptive  action  far  superior  to,  and  independent  of,  the  action  of  the  particles  as  thin 
plates.  The  phenomena  of  Hematine,  described  by  Chevreul  and  cited  by  Dr.  Brewster,  (Encyc.  Edin.  Optics, 
p.  623 ;  see  also  Biot,  Traile  de  Pfiys.  torn  iv.  p.  134,  there  referred  to,)  afford  too  close  an  approximation  to 
the  series  of  tints  of  the  second  order  not  to  authorize  a  presumption  that  the  Newtonian  theory  may  apply  to 
this  case  also.  The  diffused  light  and  blue  colour  of  the  clear  sky,  affords  another  very  satisfactory  instance. 
This  blue  is,  no  doubt,  a  blue  of  the  first  order,  reflected  from  minute  aqueous  particles  in  tht  air.  The  proof 
is,  that  at  74°  distance  from  the  sun,  it  is  completely  polarized  in  a  plane  passing  through  the  sun  s  centre. 

Another  objection,  no  less  obvious,  to  the  Newtonian  doctrine,  has  been  successfully  answered  by  Newton      1144 
himself.     A  change  of  obliquity  of  incidence,  it  may  be  urged,  should  cause  a  change  of  colour,  as  a  plate  of  Another  ob- 
given  thickness  reflects  a  different  tint  at  oblique  and  perpendicular  incidences.     But  this  variation  is  less,  the  jection. 
greater  the  refractive  power  of  the  medium  ;   and  as  the  refractive  power  increases  with  the  density,  that  of  the  Answered- 
"dense  ultimate  atoms  of  bodies  must  be  exceeding  great,  so  that  the  tint  reflected  from  them  will  vary  little  with 
i  change  of  incidence,  (art.  669.)     The  colours  of  oxidated  steel  afford  an  excellent  case  in  point.    The  refractive 
power  of  this  oxide,  though  great,  (2.1),  is,  doubtless,  not  to  be  compared  with  that  of  the  ultimate  atoms  of 
bodies,  yet  the  tints  on  the  surface  of  blued  steel  vary  but  little  with  a  change  of  obliquity.     We  may  add,  too, 
that  the  colour  exhibited  by  any  body  of  sensible  magnitude,  is  in  reality  an  average  of  the  colours  reflected 
from  all  its  molecules  at  all  possible  incidences,  so  that  no  change  of  incidence  ought  to  be  expected  to  affect  it. 

Of  the  extreme  tenuity  of  the  ultimate  molecules  of  bodies,  Newton  seems  to  have  had  but  an  inadequate      1145. 
idea,  as  he  supposed  that  they  might  be  seen  through  microscopes  magnifying  three  or  four  thousand  times.*  Newton's 
We  have  viewed  an  object  without  utter  indistinctness,  through  a  microscope  by  Amici,  magnifying  upwards  of  ideas  of  tht 
three  thousand  times  in  linear  measure,  and  had  no  suspicion  that  the  object  seen  was  even  approaching  to  slze ,°[ tn" 
resolution  into  its  primitive  molecules.     But  it  should  rather  seem  that  Newton  regarded  his  colorific  molecules  {J^gSes 
as  divisible  groitpes  of  atoms  of  a  yet  more  delicate  kind,  and  yet  more  densa^  and  these  again  as  still  furtlier 
resolvable  till  the  last  stage  of  indivisibility  be  reached.     M.  Biot  has  given  a  striking,  and,  we  may  almost  term 
it,  picturesque  account  of  this  doctrine,  in  his  Traile  de.  Physique. 

§  XV.    Of  the  Calorific  and  Chemical  Rays  of  the  Solar  Spectrum. 

It  has  long  been  a  matter  of  everyday  observation,  that  solar  light  exercises  a  peculiar  influence  in  altering      1146. 
the  colours  of  bodies  exposed  to  it,  either  by  deepening  or  discharging  them,  even  when  totally  secluded  from 
air,  and  that  various  metallic  salts  and  oxides,  especially  those  of  silver,  are  speedily  blackened  and  reduced 
when  freely  exposed  to  direct  sunshine,  or  even  to  the  ordinary  light  of  a  bright  day.     Whether  these  effects 
were  owing  to  the  heat  of  the  rays,  or  to  some  other  cause,  remained  long  uninquired.     The  first  step  was 

*  The  passage,  however,  is  in  the  highest   tone  of  a  refined  philosophy,  and,  independent  of  its  theoretic  bearings,  we  extract  it,  as 
indicating  a  scrutinizing  spirit  of  observation  far  beyond  the  age  he  lived  in. 

"  In  these  descriptions  I  have  been  the  more  particular,  because  it  is  not  impossible  uut  that  m  croscopes  may  at  length  be  improved  to  the 
discovery  of  the  particles  of  bodies  on  which  their  colours  depend,  if  they  are  not  already  in  some  measure  arrived  to  that  degree  of  perfection. 
For  if  those  instruments  are  or  can  be  so  far  improved  as  with  sufficient  distinctness  to  represent  objects  five  or  six  hundred  times  bigger 
than  at  a  foot  distance  they  appear  to  our  naked  eyes,  I  should  hope  that  we  might  be  able  to  discover  some  of  the  greatest  of  those  cor- 
puscles. And  by  one  that  would  magnify  three  or  four  thousand  times  perhaps  the)  might  all  be  discovered,  but  those  which  produce 
blackness.  In  the  mean  while  I  see  nothing  material  in  this  discourse  that  may  rationally  he  doubted  of,  excepting  this  position  :  That 
transparent  corpuscles  of  the  same  thickness  and  density  with  a  plate,  do  exhibit  the  same  colour.  And  this  I  would  have  understood  not 
without  some  latitude,  as  well  because  those  corpuscles  may  be  of  irregular  figures,  and  many  rays  must  be  obliquely  incident  on  them,  and 
so  have  a  shorter  way  through  them  than  the  length  of  their  diameters,  as  because  the  Btraitness  of  the  medium  put  in  on  all  sides  within 
such  corpuscles  may  a  litlle  alter  its  motions  or  other  qualities  on  which  the  reflection  depends.  But  yet  I  cannot  much  suspect  the  last 
because  I  have  observed  of  some  small  plates  of  Muscovy  glass  which  were  of  an  even  thickness,  that  through  a  microscope  they  have 


.orpusclcs,  by  reason  of  their  transparency. 


582 


LIGHT. 


Sir   W. 
Herschel. 

Hitter. 


1147. 

Calorific, 
luminous, 
and  chemi- 
cal rays. 


1148. 


Light,  made  by  Scheele,  who  ascertained  that  muriate  of  silver  is  much  more  powerfully  blackened  in  the  violet  rays 
V""V»'/  than  in  any  other  part  of  the  spectrum.  (Traite  de  I'  Air  et  du  Fen,  §  66.)  The  experiments  of  Sir  VV.  llerschel, 
Discoveries  on  tne  heating  power  of  the  several  prismatic  rays,  on  the  other  hand;  which  appeared  in  1800,  showed  satis- 
Sir  W  factorily  that  the  more  refrangible  rays  possess  very  little  heating  power,  the  calorific  effect  being  at  its  maxi- 
mum for  the  extreme  red  rays,  and  even  extending  considerably  beyond  the  limits  of  the  spectrum  in  that 
direction.  This  remarkable  discovery,  which  established  the  independence  of  the  heating  and  illuminating 
effects  of  the  solar  rays,  led  Professor  Ritter,  of  Jena,  in  1801,  to  examine  whether  a  similar  extension  beyond 
the  limits  of  the  visible  spectrum  might  not  also  have  place  in  the  chemical  or  deoxidating  rays,  and  on  exposing 
muriate  of  silver  in  various  points  within  and  without  the  spectrum,  he  found  the  maximum  of  effect  to  lie 
beyond  the  visible  violet  rays,  the  action  being  less  in  the  violet  itself,  still  less  in  the  blue,  and  diminishing 
with  great  rapidity  as  he  proceeded  towards  the  less  refrangible  end.  Dr.  Wollaston  independently  arrived  at 
the  same  conclusion. 

The  solar  rays,  then,  possess  at  least  three  distinct  powers  :  those  of  heating,  illuminating,  and  effecting 
chemical  combinations  or  decompositions,  and  these  powers  are  distributed  among  the  differently  refrangible 
rays,  in  such  a  manner  as  to  show  their  complete  independence  on  each  other.  Later  experiments  have  gone  a 
certain  way  to  add  another  power  to  the  list  —  that  of  exciting  magnetism.  \\  ithout  calling  in  question  the 
accuracy  of  the  observations  which  are  directed  to  establish  this  point,  we  may  be  permitted  to  hope  that  further 
researches  will,  ere  long,  explain  the  causes  of  failure  in  those  numerous  cases  where  such  effects  have  not  been 
produced. 

The  calorific  rays  appear,  from  experiments  of  Berard,  to  obey  the  laws  of  polarization  and  double  refraction, 
All  obey  the  like  those  of  light.     Those  of  interference  could  not  be  made  without  excessive  difficulty.     In  the  case   of  the 
Uwse°P"Cal  cnem'ca'  rays>  tne  same  difficulty  is  not  experienced  ;  and  Dr.  Young,  and  after  him,  by  more  delicate  means, 
M.  Arago,  have  satisfactorily  demonstrated  that  these  conform  to  the  same  laws  of  interference,  whether  po- 
Chemical      larized  or  otherwise,  that  are   obeyed  by  the  luminous  rays  similarly  circumstanced.     Thus,  a  set  of  fringes 
rays  inter-    formed  by  the  interference  of  two  solar  pencils  with  a  common  origin,  being  kept  very  steadily  projected  for  a 
'"  long  time  on  one  and  the  same  part  of  a  sheet  of  paper  rubbed  with  muriate  of  silver,  a  series  of  black  lines 
ones.  became  traced  on  it,  the  intervals  of  which  were  smaller  than  those  of  the  dark  and  luminous  fringes  formed  by 

homogeneous  violet  light. 

1149.  Dr.  Wollaston  having  observed  that  gum  guiacum  is  turned  green  by  exposure  to  solar  light  in  contact  with 
WjllastonV  air,  took  two  specimens  of  paper  coloured  with  a  yellow  solution  of  this  gum  in  alcohol,  and  exposed  one  of 
them  to  air  and  sunshine,  the  other  to  air  in  the  dark.  The  former  was  turned  perceptibly  green  in  five  minutes, 
and  the  change  was  complete  in  a  few  hours,  while  the  latter  was  no  way  discoloured  after  many  months.  He 
then  concentrated  the  violet  rays  on  paper  so  coloured,  by  a  lens,  and  the  change  was  speedily  performed,  while 
in  the  most  luminous  there  was  no  change  of  colo.ir,  and,  iu  the  red  rays,  the  green  colour  was  not  only  not 
produced,  but  when  induced  by  exposure  to  the  violet,  Wcjs  again  destroyed,  and  the  original  yellow  colour 
restored.  This  seems,  however,  to  have  been  merely  an  effect  of  the  heat,  as  the  warmth  from  the  back  of  a 
heated  silver  spoon  discharged  the  green  colour  just  as  effectually. 

Mr.  Faraday  has  observed  that  glass  tinged  purple  with  manganese,  has  its  hue  much  deepened  by  the 
passage  of  solar  light  through  it,  and  that  two  portions  of  the  same  plate,  one  preserved  in  the  dark,  the  other 
exposed  freely,  after  some  time  differ  materially  in  intensity  of  colour. 

The  direct  action  of  solar  light,  or,  possibly,  of  its  heat  also,  produces  otLer  chemical  effects,  such  as  the 
immediate  combination  of  the  elements  of  phosgen,  the  explosion  of  an  atomic  mixture  of  chlorine  and  hydrogen, 
and  other  phenomena,  all  indicative  of  powers  resident  in  this  wonderful  agent,  of  which  we  have  but  a  very 
imperfect  notion  at  present.  The  green  colour  of  plants,  and  the  brilliant  hues  of  flowers,  depend  entirely  on 
it.  Tansies  which  had  grown  in  a  coal  pit,  were  found  totally  destitute  either  of  colour  or  of  their  peculiar  and 
powerful  flavour,  and  the  bleaching  and  sweetening  of  celery  by  the  exclusion  of  light,  is  another  familiar  in- 
stance of  the  same  cause.  How  far  the  differently  coloured  rays  are  concerned  in  these  effects,  has  never  yet 
been  accurately  investigated,  though  attempts  have  been  made  ;  but  we  hope,  from  the  distinguished  ability  of 
an  eminent  individual  who  has  recently  taken  up  this  most  interesting  inquiry,  that  our  stock  of  knowledge  will 
soon  receive  material  accessions. 

We  cannot  close  this  Essay  without  an  expression  of  regret,  that  the  Memoir  of  Professor  Airey,  on  the 
Spherical  Aberration  of  the  Eyepieces  of  Telescopes,  just  on  the  point  of  publication  in  the  Transactions  of  the 
Cambridge  Philosophical  Sociity,  reached  us  too  late  to  allow  of  our  attempting  to  condense  its  valuable  con- 
tents, and  we  can  only  recommend  it  to  the  notice  of  our  readers  in  lieu  of,  and  in  preference  to,  anything  we 
could  ourselves  say  on  that  subject.  A  similar  expression  of  regret  applies  to  the  interesting  "  Theory  of  Sys- 
tems of  Rays,''  by  Professor  Hamilton  of  Dublin,  a  powerful  and  elegant  piece  of  analysis,  communicated  to  the 
Royal  Irish  Academy  in  1824,  and  only  now  in  the  course  of  impression,  but  of  which  enough  has  reached  us, 
by  the  kindness  of  its  Author,  to  make  us  fully  sensible  of  the  benefit  we  might  have  derived  from  its  perusal  at 
an  earlier  period  of  our  undertaking. 


observa 


1150. 

Effect  of 
.ight  on 
purple 
glass. 

1151. 
Other  ef- 
fects of  so- 
lar light. 


1152. 


Slough,  December  12,  1827. 


J.  F.  W.  HERSCHEL 


L  I  G  H  T 


583 


INDEX. 


N.  B.   The  Numbers  are  those  of  the  Articles  at  they  stand  on  the  Margin, 


Light.  Aberration,  of  Light,  10.  Spherical,  for  reflected  rays,  128. 
_-  —  •  Circle  of  least,  151.  156.  Of  a  system  of  surfaces  for  refracted 
rays,  281.  291.  Of  a  thin  single  Irns.  293.  Its  comparative 
amount  in  different  lenses,  807.  Of  lenses  generally,  29T.  Of 
a  system  of  thin  lenses,  SOS.  Its  effect  in  lengthening  or 
shortening  focus,  289.  General  equations  for  its  destruction, 
3!2,  313. 

Aberration,  Chromatic,  explaineJ,  456.  Circle  of  least,  457. 
Principles  of  its  destruction,  459. 

Absorption  of  Light  by  uncrystallized  media,  481,  et  seq.;  by 
crystallized,  1059,  et  seq. 

Achi omaticity,  general  conditions  of,  459. 

Achromatic  refraction,  427 .  448.  Its  general  conditions,  459.  At 
common  surface  of  two  media,  478.  Produced  by  combina- 
tions of  one  medium,  451. 

Achromatic  Telescope,  theory  of,  456,  et  seq. 

Adaptation  of  the  eye  to  different  foci,  356. 

Amethyst,  its  peculiar  structure,  1044. 

A HICI,  his  prismatic  telescope,  453.     His  microscopes,  1145. 

Amplitude  of  an  undulation,  605. 

Analcime,  peculiar  polarization  produced  by,  1 183. 

Analysis  of  solar  light  by  the  prism,  397.  406.  By  coloured 
glasses,  506.  Of  the  colours  of  thin  plates,  644. 

Angle  of  polarization,  831. 

Apertures,  waves  transmitted  through,  631.  Phenomena  of 
diffraction  through,  729.  Of  telescopes,  of  different  forms, 
their  effect,  768. 

Apophyllite,  peculiar  rings  exhibited  by  it>  several  varieties,  915. 
918.  liiaxal,  1130.  Variety  called  Tesselite,  its  structure, 
1130,  1181. 

ARAGO,  M.,  his  mode  of  measuring  refractive  indices,  739.  Mia 
law  of  polarization  by  oblique  transmission,  947.  His  disco- 
very of  the  rotatory  phenomena  in  quartz,  1037.  His  laws  of 
interference  of  polarized  rays,  947. 

Axes  defined,  783.  Optic,  889.  Differ  for  differently  coloured 
rays,  921.  Their  situations  calculated  a  priori,  1008. 

Axes  of  elasticity,  1000.  Polarizing,  Brewster's  theory  of  their 
composition  and  resolution,  1020.  Of  double  refraction,  781. 
Positive  and  negative,  1021.  1032. 

BIOT,  M.,  his  doctrine  of  movable  polarization,  928.  His 
apparatus  described,  929.  His  researches  on  the  rotatory 
phenomena,  1037.  1045.  His  law  of  the  isochromatic  lines 
in  biaxal  crystals,  907.  His  rule  for  determining  the  planes  of 
polarization  within  biaxal  crystals,  1070. 

BLAIR,  Dr.,  his  achromatic  telescopes  with  fluid  object  glasses, 
474. 

Blindness,  its  causes  and  remedies,  360. 

Bow,  coloured  prismatic,  555,  556. 

BREWSTER,  Dr.,  his  law  of  polarization  by  reflection,  831. 
Laws  of  polarization  by  oblique  transmission,  866.  His 
optical  researches  and  discoveries,  passim.  His  theory  of 
polarizing  axes,  1020. 

Brightness,  intrinsic  and  absolute,  29.  See  Photometry.  Of 
Images,  349. 

Calorific  rays  of  the  solar  spectrum,  1 147. 

Camera  obscura,  330. 

Cassia,  oil   of,  its  remarkable  refractive  and  dispersive  powers, 

1117.1121.     Experiment  upon,  1122. 
Catacaustics,  or  Caustics  by  reflexion,  134,  et  seq.     Their  length, 

144.     Determination  of   from   a  given  reflecting  curve,  137. 

Conjugate,  146.    Density  of  rays  in,  160. 
Caustics  by  refraction,  226,  et  seq.    Of  a  plane,  238. 
CHAULNES,  Due  de,  phenomena  observed  by  him,  687. 
Chemical  rays  of  the  spectrum,  1146,  el  seq. 
Chromatics,  395.     Chromatic  aberration.     See  Aberration. 
Circular  polarization,  1037.  "iseq.     Vibrations,  627. 
CLAIRAUT,  his  condition  for  construction  of  achromatic  object 

glasses,  467. 


Coloured  rays  unequally  absorbed  by  media,  486. 

Coloured  rings  an/I  fringes.     See  Rings  and  Fringes. 

Colours  of  natural  bodies  not  inherent,  410.  Newton's  theory 
of  such  colours,  1134,  et  seq.  Of  the  prismatic  spectrum,  424. 
Of  flames,  .521.  Of  thin  plates,  633.  Of  thick  plates,  676. 
Of  mixed  plates,  696.  Of  fibres  and  striated  surfaces,  700. 

Colours,  primary,  Mayer's  hypothesis  respecting,  50i>.  Young's, 
518. 

Colours  polarized  by  crystallized  plates,  884. 

Colours,  periodical,  635,  at  seq.     True  and  false,  1 1 43. 

Composition  and  resolution  of  vibrations,  620.     Of  axes,  1020. 

Cord,  stretched,  analogy  between  its  vibrations  and  those  of  the 
ether, 977.  980.  986. 

Cornea  of  the  eye,  350.  Case  of  malconformation  of,  remedied, 
358,  359. 

Crack  in  a  heated  glass  plate,  its  effect  on  the  polarized  tints, 
1102. 

Crested  fringes  observed  by  Grimaldi,  728. 

Cross,  black,  traversing  tin*  polarized  rings.  Its  form  in  uniaial 
crystals,  911.  In  biaxal,  1092. 

Crystals,  (Jniaxal,  enumerated,  785.  1124.  Law  of  double 
refraction  in,  795.  Biaxal,  table  of  the  inclinations  of  their 
axes,  1124.  Phenomena  of  the  polarized  lemniscates  ex- 
hibited by,  892,  et  stq.  1069,  et  seq.  General  law  of  double 
refraction  in,  1011,  et  seq.  Action  of  heat  on,  1 109.  Positive 
and  negative,  or  attractive  and  repulsive,  803.  942.  How  dis- 
tinguished, 915. 

Crystallised  surfaces,  their  action  on  reflected  light,  1 12S. 

Crystalline  of  the  eye,  352. 

Deflexion  of  light.     Newton's  doctrine  of,  713. 

Depolarization  of  light,  925. 

Depolarizing  axes,  1087. 

Deviation  of  a  ray  a  'ter  any  refraction  in  one  plane,  211.  Mini- 
mum produced  by  a  prism,  216.  Of  tints  from  Newton's  scale 
in  the  polaiized  rings,  915.  1125. 

Diacauslics.     See  Caustics  by  refraction. 

Dichroism,  phenomena  of,  in  uniaxal  crystals,  1064.  In  biaxa), 
1067.  Expressed  by  an  empirical  formula,  1075. 

Dichromatic  media,  499. 

Diffraction  of  light,  706,  et  seq. 

Dilatation  of  rings  at  oblique  incidences,  639.  6R9.  Of  the 
diffracted  fringes  by  approach  of  the  radiant  point,  711.  Of 
glass,  its  effect  in  imparting  the  polarizing  property,  1089.  Of 
jellies,  1094. 

Discs,  spurious,  of  stars,  767. 

Dispersion  of  light,  395,  &c. 

Dispersive  powers  of  media,  425.  Methods  of  determining  them, 
428.431.  435.  A  very  precise  practical  one  for  object  glasses, 
483.  Table  of,  1120.  Of  higher  orders,  446. 

Due  de  Chaulnes,  his  experiment  on  coloured  rings,  687. 

Elastic  forces  of  a  medium  generally  expressed,  998. 

Elasticity,  &\es  of,  1000.     Surface  of,  1004. 

Elliptic,  vibrations  of  ethereal  molecules,  621. 

Emanation,  oblique,  law  of,  43. 

Ether,   its   vibrations  the  (hypothetical)   cause   of  light,  56S 

Frequency  of  its  pulsations,  575.     See  Undulations. 
Extinction  of  light,  481.  11 38. 
Eye,  its  structure,  350.    Change  of  focus,  S56.     Of  fishes,  968. 

See  Vision. 

Field  of  view,  881. 

films,  interrupting,  in  crystals,  phenomena  exhibited  by,  1078, 

et  seq. 

Fits  of  easy  reflexion  and  transmission,  526.  651 
Fixed  lines  in  the  spectrum  described,  418.     Their  utility  in 

optical  determinations.  420. 
Flames,  coloured,  their  phenomena,  590. 
Foci,  general  determination  for  any  curve  by  reflected  rays,  109. 

112.     In  a  sphere,  laS.  250.    Conjugate,  126.    General  inves- 


Index. 


584 


LIGHT. 


Light.  tigalion  of,  for  refracted  rays  in  any  curved  surface,  221.  In 

-_  J  spherical  surface,  239,  el  seq.  For  central  rays,  (fundamental 
equation,)  247.  Of  a  system  of  spherical  surfaces,  258.  257. 
Of  a  system  of  lenses,  268.  Of  thick  lenses,  272.  Of  doubly 
refractive  lenses,  805.  For  oblique  rays,  318,  et  seq.  to  321. 
Aplanatic,  287.  How  conceived  in  the  undulatory  system,  590. 

FRAUNHIIFER,  his  experiments  on  the  spectrum,  436.  On  diffrac- 
tion and  interference,  740. 

FRESNEL,  his  optical  discoveries  and  researches,  passim.  His 
theory  of  transverse  vibrations,  976.  Of  the  diffracted  fringes 
in  shadows,  718-  His  theorem  for  the  resultant  of  two  inter- 
fering rays.  613.  His  analysis  of  the  colours  seen  through  a 
minute  circular  aperture,  731.  His  experiments  on  the  inter- 
ference of  polarized  rays,  954.  957.  His  laws  of  reflexion  of 
polari/.ed  light,  852.  His  theory  of  double  refraction  in 
uniaxal  crystals,  989.  In  biaxal,  997.  His  theory  of  circular 
polarization,  1047. 

Fringes  diffracted,  their  theory,  718.  Their  displacement  by 
interposition  of  a  transparent  plate,  737.  Exterior,  706.  In- 
terior, 726.  Coloured,  seen  between  a  prism  and  a  plane  glass, 
641.  Between  thick  parallel  plates,  688.  Bel  ween  glass  films, 
695.  Produced  by  healing  a  glass  plate,  1099. 

Glass,  flint  and  croim.  Refractive  and  dispersive  indices  of 
their  varieties.  See  Tables,  Art.  1 1 1 6.  1 1 20.  Heated,  pressed, 
or  bent,  its  phenomena,  1086.  1090.  1095.  Unannealed, 
1104. 

Heat,  its  effect  in  changing  colours  of  bodies,  504.  Of  crystals, 
unequal  on  the  two  pencils,  1077.  Effects  of  unequal  heat  on 
glass,  1083.  1095.  On  crystallized  bodies,  their  forms  and 
double  refractions,  1 109. 

Hemitropism,  remarkable  cases  of,  detected  by  polarized  light, 
\  1ST,  el  seq. 

Homogeneous  light,  its  properties,  600.  Purification,  4 12.  In- 
sulation, 503.  Lengths  of  undulations  for  its  several  species, 
576. 

Humours  of  the  eye,  350.  354. 

HUYQENS,  his  law  of  velocity  of  the  extraordinary  ray  in  Iceland 
spar,  787.  His  construction  for  law  of  extraordinary  refrac- 
tion, 806.  Extended  to  biaxal  crystals,  1011. 

Iceland  spar,  phenomena  of,  polarization  and  double  refraction 
exhibited  by,  879,  &c.  Dichroism  of,  106S.  Pyrometrical 
properties  of,  1110.  Interrupted,  phenomena  of,  1080. 

Idiocyclophanous  crystals,  1081. 

Illumination,  formula  for  its  intensity,  44.  47.  Of  the  polarized 
rings  atdifferent  points  of  their  peripheries,  1071. 

Images,  319.  Form  of,  320.  Rule  to  find  their  places,  344. 
Brightness  of,  349.  Formed  within  the  eye,  357. 

Incommensurability  of  coloured  spaces  in  the  spectrum,  441. 

Index  of  refraction,  how  determined,  206.  213.  Wollaston's 
method,  562.  Fraunhofer's,  436.  Arago  and  Fresnel's,  739. 
By  polarizing  angle,  843.  Table  of  its  values,  1 1 16. 

Index  of  transparency,  486. 

Inflexion  of  light,  Newton's  doctrine  of,  71S. 

Intensity  of  light,  its  law  of  diminution,  18.  Its  measure  in  the 
undulatory  doctrine,  578.  Reflected  perpendicularly,  calcu- 
lated, 592. 

Intensity  of  a  polarized  beam  reflected  in  any  plane,  852.  Of 
natural  lights  when  so  reflected,  857.  592.  Of  the  comple- 
mentary pencils  produced  by  double  refraction,  873.  987.  Of 
the  polarized  rings  at  any  points  of  their  periphery,  107 1. 

Interferences  of  rays,  596,  et  seq.  General  investigation  of,  618. 
Young's  fundamental  experiment,  726.  Of  polarized  rays,  946, 
et  leg. 

Interrupting  films,  their  phenomena,  1078. 

Irradiation,  697. 

Isochromatic  lines,  906. 

Jellies,  polarization  of  light  produced  by,  when  dilated  or  com- 
pressed, 1094. 

Least  action,  principle  of  its  use  in  optical  investigations.  SS6. 

Its  general  application,  540.     Its  equivalent  in  the  undulatory 

doctrine,  588.     Application  to  the  theory  of  uniaxal  crystals. 

790. 
Lemniscates,  polarized,  surrounding  the  axes  of  biaxal  crystals 

902.     See  Rings,  Tints,  &c. 
Lenses,  259.     Aplanatic,  304.     "  Crossed,"  305. 
Liquids,  rotatory,  phenomena  produced  by,  1045. 
Longitudinal  and  lateral  aberration,  288. 


MALUS,  his  theory  of  double  refraction,  796  805.  His  discovery 
of  polarization  of  light  by  reflexion,  822. 

MAYEII,  his  hypothesis  of  three  primary  colours,  .M)9. 

Media,  dichromatic,  499. 

Metals,  theiraction  in  polarizing  light  by  reflexion,  815. 

Microscopes,  309.  389. 

MtrsCHERLicH,  M.,  his  researches  on  the  effects  of  heat  on 
crystals,  1 109. 

Modifications  of  light,  80. 

Molecules,  luminous,  their  tenuity,  543.  Their  motion  on  chang- 
ing media  investigated,  528. 

NEWTON,  his  theory  of  light,  526.  Doctrine  of  inflexion  and 
deflexion,  713.  Theory  of  colours  of  natural  bodies,  1134. 
Of  the  size  of  their  particles,  1145. 

Object  glass,  achromatic,  its  theory,  459,  et  seq.  General  equa- 
tion for  destroying  its  aberrations,  465.  Aplanatic,  its  con- 
struction, 468.  470,  &c.  With  separated  lenses,  479.  With 
fluid  lenses,  474. 

Oblique  incidence,  its  effect  on  the  colours  of  thin  plates,  6'9. 
657.  Pencils,  their  foci,  S2 1.32 8.  Reflexion  from  water,  553. 

Opacity,  its  cause  on  Newton's  doctrine,  1 1S8. 

Origin,  of  a  ray  in  the  undulatory  doctrine,  607.  609. 

Periodical  colours,  635,  etseq. 

Periodicity,  law  of,  906. 

Phase  of  an  undulation,  604. 

Photometers,  57.     Photometry,  17,  etseq. 

Piles  of  transparent  plates,  their  phenomena  in  polarized  light, 

869. 

Plagiedral  quartz,  its  rotatory  phenomena,  1012. 
Plane  of  polarization,  828.     Its  change  by  reflexion,  860.     Its 

apparent  rotation  in  quartz.  &c.   1039.     Itsoscillations,  928. 
Plates,  thin,  tl.eir colours,  633, etseq.     Thick, ditto,  676.   Mixed, 
ditto,  696.     Crystallized,  their  phenomena,  936.  (See  Rings.) 
Crossed,  9.SS,  9S9.     Superposition  of.  9  10. 

POISSON,  M.,  his  theorem  for  the  illumination  of  the  shadow  of 
a  small  circular  disc,  and  the  colours  seen  through  a  minute 
aperture,  734.  His  investigation  of  the  intensity  of  reflected 
light,  592. 

Polarization  of  light  generally,  814,  et  seq  Modes  of  effecting, 
819.  Characters  of  a  polarized  ray,  820.  By  reflexion,  821, 
el  seq.  Partial,  847.  By  several  reflexions  in  one  plane,  818. 
By  refraction,  863.  By  several  oblique  transmissions,  863. 
866.  By  double  refraction,  873.  Movable,  Biot's  doctrine  of, 
928.  Explained  on  the  undulatory  doctrine,  993.  Its  princi- 
ples applied  to  the  phenomena  of  biaxal  crystals,  1071. 
Circular,  its  characters,  1049.  How  effected,  1052.  Plane  of, 
its  position  in  the  interior  of  biaxal  crystals,  1070.  Of  sky 
light,  1143. 

Polarised  rings,  surrounding  the  optic  axes  of  crystals,  mode  of 
viewing,  892,  et  seq.  Their  form  in  general,  902.  In  uniaxal 
crystals,  911.  Dependence  of  their  tints  on  law  of  interferences, 
912.  Primary  and  complementary  sets  of,  926.  Explained  on 
hypothesis  of  movable  polarization,  931.  On  undulatory 
hypothesis,  969. 
Polarizing  angle,  Brewster's  law  for  determining,  831.  Its  use  as 

a  physical  character,  1 123. 
Polarizing  energy,  a  physical  character,  1 1 26. 
Poles  of  lemniscates,  902.     Virtual,  in  biaxal  crystals,  924. 
Power  of  a  lens,  262.     Of  a  system  of  spherical  surfaces,  270. 
Magnifying,  374.     Superposition  of  powers,  law  of  in  lenses, 
268. 

Pressure,  its  effect  in  communicating  the  property  of  polariza- 
tion, 1087. 
Principle  of  least  action  applied  to  double  refraction,  790.    Of 

swiftest  propagation,  588. 

Prism,  formulae  for  refraction  through,  198,  etieq.     Of  variable 
refracting  angle,  431,  432.     Analysis  of   light  by,  397.     Tele- 
scopes composed  of  prisms,  453.    Coloured  bow  seen  in,  555. 
Propagation  of  light,  5.     Oersted's  hypothesis  for,  525.     Law  of 

swiftest,  588.     Of  waves  along  canals,  600. 
Punctum  caecum  in  the  eye,  366. 

Quartz^  right  and  left-handed,  1041.  Rotatory  phenomena  in. 
1037.  Double  refraction  of  along  itsaxis,  1048.  Plagiedral,  iu 
phenomena,  1042. 

Radiation  of  light,  5,  et  seq.  Its  law,  72.  Explained  on  ondu- 
latory  doctrine,  578. 

Rays,  calorific,  luminous,  and  chemical,  1147.  Similar  and  dis- 
similar, Mfi.  Their  origins,  607.  Interfering,  theirresultam, 
611.  Polarized,  their  characters,  820. 


Index. 


L  I  G  H  T. 


585 


Light.  Reflecting  forces,  their  intensity,  561.  Distribution,  550,  et  seq. 
-I  '  Reflexion,  law  of,  88.  General  formulas  for,  at  plane  surfaces,  99. 
At  curved  surfaces,  108,  109.  Between  any  system  of  spherical 
surfaces,  301.  Internal  total,  184.  550.  554.  Modification 
impressed  on  light  by  two  such,  1056.  At  common  surface  of 
two  media,  547.  Partial,  explained  on  Newton's  principles, 
544.  Regular  at  rough  or  artificially  polished  surfaces  ex- 
plained, SST.  558.  How  conceived  in  the  undulatory  doctrine, 
581.  At  the  surfaces  of  crystals,  1 1 23.  Of  polarized  light,  its 
laws,  819,  et  seq. 

Refraction,  by  uncrystallized  media,  \l\,ttseq.  Its  law,  189. 
General  formulae  for,  at  plane  surfaces,  198.  Through  prisms, 
203.211.  At  curved  surfaces,  220,  et  seq.  At  common  surface 
of  two  media,  189.  Colourless,  a  case  of,  .478.  Regular,  at 
artificially  polished  surfaces,  explained,  559.  Account  of  in 
undulatory  theory,  586.  595.  628. 

Refraction,  double,  779,  et  seq.     By  what  bodies  produced,  780. 
Its  law  in  uniaxal  crystals,  785.  800.     Produced  by  rock  crystal 
along  its  axis,  1018.     By  compressed  and  dilated  glass,  1107. 
In  uniaxal  crystals,  explained  on  undulatory  doctrine,  989.     In 
biaxa.,  us  general  laws,  101 1.  1014.     Ordinary  and  extraordi- 
nary, relation  of  the  two  pencils,  873. 
Refracting  forces,  their  intensity  and  extent,  561. 
Refractive  power,  intrinsic,  5S5.    Table  of  its  values  in  different 
media,  1118.    Its  connection  with  their  chemical  composition, 
1114. 

Refractive  index,  how  measured,  see  Index.  Table  of  its  values 
for  different  media,  1116.  For  different  homogeneous  rays, 
437. 

Refrangihility  of  different  rays.     See  Chromatics,  Colours,  &c. 
Resultant  of  two  interfering  vibrations,  611.     Of  rays  oppositely 

polarized,  982. 

Retina,  355.     How  affected  by  vibrations  of  ether,  567. 
Rings,  coloured,  seen  between  convex  glasses,  their  colours,  635. 
Breadths,  6.57.     For  different  homogeneous  rays,  644.     Their 
analysis    and  synthesis,  644,    645.      Transmitted,  658.     Ex- 
plained on  the  undulatory  theory,  660      On  the  Newtonian, 
655.     Seen  about  the  images  of  stars  in  telescopes,  766.     Seen 
about  the  poles  of  the  optic  axes  in  crystals,  892.  900.     Law  of 
their  intensity  in  different  points  of  their  circumference,  1071. 
Rotatory  phenomena  of  rock  crystal  and    liquids,    1038.  1040. 
Explained  on  the  undulalory  doctrine,  1057. 

SEEBECK,  Dr.,  his  discovery  of  the  rotatory  property  in  liquids, 
1045.  Of  tlie  effects  of  heat  in  imparting  polarization  to 
glass,  1083. 

Sections,  principal,  of  a  crystallized  plate,  888. 

Soap  bubbles,  colours  reflected  by,  649. 

Solar  light,  its  analysis  by  the  prism,  397.  Its  peculiar  cha- 
racters and  spectrum,  419. 

Spectrum,  prismatic,  3d7.  Fixed  lines  in,  418  ;  secondary,  442; 
tertiary,  446.  Its  distortion  at  extreme  incidences,  450 ; 
subordinate,  452.  Of  first  class,  760  ;  of  second  class,  746  ; 
of  third  class,  761. 

Spheroid  of  double  refraction  in  uniaxal  crystals,  789.  In 
biaxal,  1013. 

Spherometer,  1111. 

Stars,  their  spurious  discs  and  rings,  766,  et  seq. 

Statues,  musical  sounds  produced  by  certain,  a  probable  expla- 
nation of,  1103. 

Strain  of  solids,  ascertained  by  their  polarized  tints,  1090.  State 
of,  in  unequally  heated  glass  plates,  1098. 

Sulphate  of  copper  and  potash,  a  singular  property  of,  1 1 1 1 .  Of 
lime,  action  of  heat  in  altering  its  optical  properties,  1112. 
Of  potash,  singular  structure  of  its  crystals,  1 132. 

Table  of  media  in  their  order  of  action  in  green  light,  443.  Of 
dispersive  powers  of  first  and  second  order  on  a  water  scale, 
447.  Of  the  lengths  of  undulations  of  the  several  homoge- 
neous rays,  575.  756.  Of  the  maxima  and  minima  of  the  ex- 
terior fringes  of  shadows,  720.  Of  colours  seen  by  a  person 


of  defective  vision,  507.  Of  colours  seen  by  diffraction 
through  a  circular  hole,  730.  Of  the  dimensions  of  the 
lemniscates  in  mica,  908.  Of  crystals  whose  optic  axes  differ 
for  different  rays,  (Brewsler,)923.  Of  the  angles  of  rotation  of 
the  several  homogeneous  rays,  1040.  Of  refractive  indices, 
(general,)  1116.  Of  refractive  indices  for  seven  definite  rays, 
(Fraunhofer,)  437.  Of  intrinsic  refractive  powers,  1 1 18.  Of 
dispersive  powers,  (general,)  1120.  Of  angles  between  the 
optic  axes  of  various  crystals,  1124.  Of  polarizing  powers, 
1126. 

Telescopes,  379.  Astronomical,  380.  Galilean,  380.  Hersche- 
lian,  390.  Newtonian,  391.  Prismatic,  453.  Achromatic, 
(see  Achromatic  ) 

Tesselite,  its  singular  structure,  1130,  1131. 

Theories  of  light,  Newtonian,  526.     Undulatory,  563,  et  seq. 

Thick  plates,  colours  of,  676,  et  sey.  Explained  on  undulatory 
system,  678. 

Thin  plates,  colours  of,  633,  et  seq.  Newton's  explanation  of 
them,  651. 

Tint,  its  numerical  measure,  906. 

Tints  of  coloured  media,  vary  with  a  change  of  thickness,  495. 
Of  transmitted  rings  expressed  algebraically,  663.  66*.  Of 
crystallized  plates,  their  law,  886.  906.  Their  dependence  on 
the  thickness  of  the  plate,  905.  Theirdeviation  from  Newton's 
scale,  915.  Singular  succession  of,  exhibited  by  Vesuvian, 
1125.  Of  circular  polarization,  1055. 

Transparency,  on  what  depending,  1 142.  Index  of,  486.  Of 
oiled  paper,  &c.  549. 

Tourmaline,  its  property  of  polarized  light,  817.  Of  absorbing 
one  pencil,  1060. 

Type,  of  the  colour  of  a  medium,  490.  Instanced  in  various 
cases,  498. 

Ultimate  tint  of  an  absorptive  medium,  494. 

Unannealed  glass,  its  optical  properties,  1104. 

Viuhilatiom  of  ether,  574.  Their  lengths  for  homogeneous  rays, 
575.  756.  Their  phases,  604.  Amplitudes,  605.  Propaga- 
tion in  spheroidal  surfaces,  804. 

Undulation,  half  an,  allowance  for  cases  when  required,  966' 
072.717.  Fresnel's  rule  for,  972.  Explained,  &  priori,  983' 

Velocity  of  liglu  9.  IS.  Of  etherial  undulation,  564.  Of  plane 
waves  within  crystals,  1005.  1012.  Of  ordinary  and  extraordi- 
nary ray  on  Huygenian  hypothesis,  787.  Of  luminiferous 
waves  and  of  rays,  distinguished,  813. 

t'esuvian,  its  remarkable  inverted  scale  of  tints,  1125. 

Vibrations  of  ether,  rectilinear,  their  laws,  569.  Resultant  of 
two  interfering,  61 1.  Their  composition  and  resolution,  620  . 
particular  cases,  621.  Elliptic,  621.  Circular,  627. 

Vibration,  its  effect  in  imparting  polarizing  power  to  glass,  109S. 

Vision,  350.     Single,  with  two    eyes,  861.     Double,   361.  363. 
Restoration  of  at  an  advanced  age,  360.    Through  lenses,  &c. 
376.     Of  persons  who  see  only  two  colours,  507.     Oblique 
through  refracting  or  reflecting  surfaces,  341. 
Visual  angle,  376. 

Water,  its  indices  of  refraction  for  seven  definite  rays,  437.    Its 

spectrum,  438. 
Waves  of  light  explained,  573.     Secondary,  583  ;  their  mutual 

destruction,  628.    Transmitted  through  apertures,  631.     Plane, 

their   velocity  and    direction  in   crystals  investigated,  1012. 

Curved,  the  general  equation  of  their  surface  in  biaxal  crystals, 

1013. 
WOLLASTON,  Dr.,  his  determinations  of  refractive  indices,  1115. 

His  researches  on  double  refraction  of  Iceland  spar,  780.     On 

the  chemical  rays,  1147.     Discovery  of  the  fixed  lines  in  the 

spectrum,  418. 

YOUNG,  Dr.,  his  law  of  interference.  See  Interferences.  His 
analogy  between  the  vibrations  of  ether,  and  those  of  a  tended 
cord,  977.  His  optical  discoveries  and  investigations, 


Index. 


roi,.  iv. 


L  i  (i  n  T 


ERRATA  ET  CORRIGENDA. 

'N  3.  Tlie  reader  is  requeued  to  correct  in  advance  the  following  Errata,  and  to  s'rike  out  the  pa«?affe«  here  refcrntr  »r 

/'•ir/e.  I. 'm".                          Error.                                                  Cm-r  rtinn. 

811,  *').  existences,  oxisicm-c. 

do.  "I,  more,  most. 

317,  V.  line.  sine. 

319,  2fi,  as  the  sun's  surface,  at  tin-  sun's  surface 

389.  5  fr.. in  holt.  -  — -  |  " ^—7- I 

3  p\  ?  p  I.   I 

•399,  17,  axis,  axes. 

400.  15,  act.  art. 

401.  15,  PE.  PC. 

402.  30,  R,  ROp.  P,  POp. 
do.  3.-.  p,  Q. 

410,  22  from  b&tt.      dele  "  see  Micrometer,  in  a  subsequent  part  of  tlii.-  Artie).-.' 

414,  44,  dele  "  by  the  writer  of  these  pages." 

415,  8  from  bott.       by  water,  into  water. 

420,  SO,  spectra  of  distortion,  subordinate  spectra. 

428,  17,  secondary,  second. 

481,  S3,  -Rr,  V  v",  R  R',  V  V. 

do.  36,  '    R  N  V,  R'  N  V. 

484,  27,  from  experiments,  from  otjier  experiments. 

.454,  28,  P  H,  1'Q.  ..i   A  B,  I' 15  -  P  Q,  or  A  B. 

461,  11    from  bott.       two  vibrations,  two  rectangular  vibrations. 

47fi.  14  from  bott.       tMe  all  that  relates  to  the  frinjes  on  the-  u  n;.s  of  the  I'n/iil'n  Idn*,  lip utf  founded 

on  a  mistake. 

480,  32,  limits,  limit. 

509,  38,  fails,  falls 

521,  I,  produce.  lo  produce. 

524,        margin,  Art.  925,  polarization,  depolarization 

588,  22,  positive  cla.^s.  attractive  class. 

531,  Ifi,  mid  ns   (nllcms: — With  respect  to  this  third  law,  however.it  must  be  confessed  that  it  appear* 

to  require  a  stricter  examination,  as,  if  admitted  in  its  full  extent,  it  seems 
to  controvert  the  fundamental  principles  of  the  doctrine  of  interference 

5M,  19,     delft  wlint  is  said  about  the  nodal  linos. 

5PO.  32  from  bott.       after  disprove   il,  IHIITI   as  follows:    Instead    of  the   expression  (4,)  Art.   107.%  we 

might  otherwise  assume 

T  =:  (Y  .  cos  2  f!  +  B  .  sin  2  if') .  (y  .  cos  v-  +  b  .  sin  »'), 
and  determining  the  coefficients  accordinirly,  obtain  another  expression  for  the  tint. 


I'l.lt.-  J 


//.•  .  ..//•/.  '.v/  .'?>/.  -'  .In .-i -i 


,l.r  I/If  .  /,;'  ,///•-,-/.     .  /„//.';/.  /*;>-.  /. 


h  i  •:  ii  r, 


I'hil.-  L' 


T 


-:•'.'-, •:-/.••  ../.'.•r'.'/./K;',-'.  /;//,^.//-///y.v,///  f.//</,r,/r,-  .t>/-/ 


LIGHT. 


ful    .'i.'l  .  //'/    ;';V. 


ft,/.. 'If.  .In.  •.':;-. 


CM  .\ 


Ka.35 


//,/.  .'17.  .//'/     •!."/. 


/••„,.  -a. 


fy.39.strt.x33. 


41 


(]  I  [1 \  ) 


'•        K..&.SAKA3.       '*«.  rt'..'*. 


LIGHT. 


I'lalr     I 


.'/ciry 


L  I  G  II  T 


//,/  79  Art.  378 


//;/.  tfC1     III  .:::! 
A 


.',i-m,in  /,ii,(,Mlf  Street 


,  I  G  JI  T . 


A  B C 


I.          A 


Jini 


.   //if. 


^^Z                                                              —  -" 

lu.li.,,, 


\  !:•/,; 


L  I  G  II  T. 


Plate  7. 


////  //.v 


fiii.  ii/  In  /i/./ 


Fifl.  Jiff.  Art.  .iff  6. 


'•  •    \ 


A*. 


V  \\ 


\v 


\       A        Fy.i2z.Art.xio.         \ 


D 


Fy.K5.Art.3Sj. 


t^  *LJ^-^r*sv*J*^-nJ*J^JW*^w^lr^ 


:/ ••>/   -.••';•  l-l- ./..}/, fH 7!r,tSI. I.H.fontl-Stm-t . 


L   I   G  H  T . 


Elate  <"> 


//>/.  /.;/..//•/.  ti./.'i. 


. /' 


LIGHT. 


/•;>/.  i a,.  1,1. 7/1. 


fy.146sirf.7W. 


fig.l-tffJrC.736. 


I'i.l.  II. '  .  Irt. 


r'i,i.  K3.jirt.7W. 


fin./.'.' .  ln.770. 


"  .In. 7  :•:>. 


//„./..././/•/.;;/' 


//,/./.;/. //Y./-/V. 


•'  •/Y'f  J.3fiL»-m.lr 


/•''!/.  i.;i  .V":;  ,  In  7-1:' 


;n.  .  hi. ,-,-;-. 


L  I   G  II  T. 


Plate  1O 


/'/<•/  l.'xf  .  //-/.  777. 


/•iff  /,//.    .Ill  771. 


/•/;/  in:,. 


/•it/.  Hi:'. 


/•ill.  nil.  .///.,--;. 


/'/'/  /i'l<    .  iff  .  777 


«  *  « 

I  I  ,. 


/•///  JSS  .  ///  77 a 


8 


L   I  G  II  T 


/•'/,/. i ;.'>'.  . In., '>,,;;. 


/•/</.  /<>/.  .  //•/.  -A '.'. 


/•/'</.  /,>'•.'..  Iff .,! 


^ 


.//•/. 


. .  trr.  ,i,',i. 


T 


1!.  ,  ,C 

V 

-•;.'&   1> 


ftilili-lir-1,1,1/1:  .  I, •//','/;;/,  M,  //,,'.;..  /;-/l,,/,/i,m .  (•:  /,.-,/,  v/  /lil<riiH'ln-Hini . 


LIGHT 

/•iii.itm.  .li-i  H-.-K. 


I'l.lll-  I-' 


*  *!//• 


/'ill  ////.  .  //•/  .'/: 


fii/./KH.  .///.. <;,'.'. 


fill.  /•/-/  .//•/««. 


O 

VJW 

0 

00   G 

00      B 

,   T 

00     V 
Fi/f  /')<>.   ./!'/.  !/:•!/. 


I.  i  <.-::;  . 


fit?   //;//    Jli  .ltr-{<> 


/•/;/  :•/,-  . 


L  I  G  H  T. 


6tJHK. 


f'ti/. :'.'•.'    ./r/.J/nli 


////.  ;'/'/  .///./A/. r 


////.  ::•/>  .-trt.JlOS 


f?,.  aw 


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