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TEXT-BOOK  OF  PHYSICS 

PROPERTIES   OF   MATTER 


A  TEXT=BOOK  OF  PHYSICS 

BY 

Prof.  J.  H.  POYNTING,  Sir  J.  J.  THOMSON, 

SC.l).,  I-.R.S.,  AND  M.A.,  K.K.S., 

I.ate  Fellow  uf  Trinity  CoIIcrc,  Cambridge  ;  Fellow  of  Trinity  College,  Cambridge  ;  Prof. 

Professor  of  Physics,  liirminghain  of  Experimental  Physics  m  the  University 

University.  of  Cambridge. 

VoI.u^tE  T.     Fifth  Edition,  Revised,  Fully  Illustrated.     Trice  los.  61. 

F»ROF>ERTIES   OF  IVLATTER 

Volume  II.    Fifth  Edition,  Revised,  Fully  Illustrated.    8s.  6d. 

SOUND 

Contents.— The  Nature  of  Sound  and  its  chief  Characteristics.— The  Velocity  of  Sound  in 
Air  and  other  Media.— Reneclion  and  Refraction  of  Sound.— Frequency  and  Pitch  of  Notes.— 
Resonance  and  Forced  Oscillations.— Analysis  of  Vibrations.— The  Transverse  Vibrations  of 
Stretched  Strings  or  Wires.— Pipes  and  other  Air  Cavities.— Rods.— Plates.— Membranes.— 
Vibrations  maintained  by  Heat.— Sensitive  Flames  and  Jets.— Musical  Sand.— The  Superposition 
of  Waves. — Index. 

Volume  III.    Third  Edition,  Fully  Illustrated.     15s. 

MEAT 

Contents. — Temperature.—  Expansion  of  Solids. — Liquids. — Gases.— Circulation  and  Con- 
vection.—Quantity  of  Heat ;  Specific  Heat.— Conductivity.— Forms  of  Energy  ;  Conservation; 
Mechanical  Equivalent  of  Heat.— The  Kinetic  Theory. — Change  of  State  ;  Liquid  Vapour. — 
Critical  Points.— Solids  and  Liquids.— Atmospheric  Conditions  — Radiation.— Theory  of  E.k- 
changes. — Radiation  and  Temperature. — Thermodynamics.  — Isothermal  and  Adiabatic  Changes. 
— Thermodynamics  of  Changes  of  State,  and  Solutions. — Thermodynamics  of  Radiation. — Index. 

Remaining  Volumes  in  Preparation — 
UIGHT;    IVIAGI^ETISlVi:  &  EILiECTMCIT^ir 

In  Crown  8vo,  Handsome  Cloth,  194  pages,  2S.  6d. 
RESEARCHES  ON  THE  PAST  AND  PRESENT  HISTORY  OF 

THE  EARTH'S  ATMOSPHERE 

By  Dr.  THOMAS  LAMB  PHIPSON. 

"  The  book  should  prove  of  interest  to  general  readers,  and  students  of  science." — Nature. 
In  Crown  8vo,  Cloth.     Illustrated  with  Diagrams. 

THE   FORCE  OF  THE  WIND 

By    HERBERT     CHATLEY,    B.Se.   Eng.   (Lond.), 

Lecturer  in  Applied  Mechanics,  Portsmouth  Technical  Institute. 

In  Large  8vo.     Profusely  Illustrated.     8s.  6d.  net. 

WIRELESS  TELEGRAPHY 

By  GUSTAVE  EICHHORN,  Ph.D. 

"Well  written  .  .  .  and  combines  with  a  good  deal  of  description  a  careful  investigation  of 
the  fundamental  theoretical  phenomena." — Nature. 


In  Two  Volumes,  Large  8vo,  Strongly  Bound  in  Plalf-Morocco. 

PHYSICOCHEMICAL  TABLES 

FOR  THE  USE  OF  ANALYSTS,  PHYSICISTS,  CHEMICAL  MANU- 
FACTURERS, AND  SCIENTIFIC  CHEMISTS. 

VOLUME  I.— Chemical  Engineering,  Physical  Chemistry.  Price  24s.  net. 
VOLUME  II. — Chemical  Physics,  I'cre  and  Analytical  Chemistry. 

By  JOHN  CASTELL-EVANS,  F.I.C.,  F.C.S. 

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Metallurgy  at  the  Finsbuty  Technical  College. 

London:  CHARLES   GRIFFIN  &  CO.,  Ltd.,  Exeter  Street,  Strand. 


TEXT-BOOK  OF  PHYSICS 


BY 

J.  H.  POYNTING,  Sc.D.,  F.R.S. 

HON.  Sc.D.  VICTOKIA  UNIVERSITY 

LATE    FELLOW    OF   TRINITY   COLLEGE,    CAMBRIDGE;    MASON    PROFESSOR 
OF   PnYSICS    IN    THE    UNIVERSITY    OF   BIRMINGHAM 

AND 

Sir  J.  J.  THOMSON,  M.A.,  F.RS.,  Hon.  Sc.D.  Dublin 

HON.  D.L.  PRINCETON;  HON.  Sc.D.  VICTORIA:  HON.  LL.D.  GLASGOW 

HON.  Pn.D.  CRACOW 

FELLOW   OF   TRINITY   COLLEGE,   CAMBRIDGE  ;    CAVENDISH   PROFESSOR    OF 

EXPERIMENTAL    PHYSICS    IN    THE    UNIVERSITY   OF   CAMBRIDGE; 

PROFESSOR    OF    NATURAL    PHILOSOPHY    AT   THE 

ROYAL   INSTITDTION 


PROPERTIES  OF  MATTER 


WITH    i58    ILLUSTRATIONS 


FIFTH   EDITION,    CAREFULLY  PiE VISED 


LONDON 

CHARLES  GRIFFIN  AND  COMPANY,  LIMITED 

EXETER  STREET,  STRAND 

1909 

\_AU  r'ights  reserred] 


6? 


c 


nh 


PREFACE. 

The  volume  now  presented  must  be  regardecl  as  tlie  opening 
one  of  a  series  forming  a  Text-Book  on  Physics,  which  the 
authors  are  preparing.  The  second  volume,  that  on  Sound,  has 
already  been  issued,  and  the  remaining  volumes  dealing  with 
Heat,  Magnetism  and  Electricity,  and  Light  will  be  published 
in  succession.      ..- 

As  already-stated  in  the  preface  to  the  volume  on  Sound, 
"  The  Text-Book  is  intended  chiefly  for  the  use  of  students  who 
lay  most  stress  on  the  study  of  the  experimental  part  of 
Physics,  and  who  have  not  yet  reached  the  stage  at  which  the 
reading  of  advanced  treatises  on  special  subjects  is  desirable. 
To  bring  the  subject  within  the  compass  thus  prescribed,  an 
account  is  given  only  of  phenomena  which  are  of  special 
importance,  or  which  appear  to  throw  light  on  other  branches 
of  Physics,  and  the  mathematical  methods  adopted  are  very 
elementary.  The  student  who  possesses  a  knowledge  of 
advanced  mathematical  methods,  and  who  knows  how  to  use 
them,  will,  no  doubt,  be  able  to  work  out  and  remember  most 
easily  a  theory  which  uses  such  methods.  But  at  present  a 
large  number  of  earnest  students  of  Physics  are  not  so 
equipped,  and  the  authors  aim  at  giving  an  account  of  the 
subject  which  will  be  useful  to  students  of  this  class.  Even 
for  the  reader  who  is  mathematically  trained,  there  is  some 
advantage  in  the  study  of  elementary  methods,  compensating 
for  their  cumbrous  form.     They  bring  before  us  more  evidently 


vi  PREFACE 

the  points  at  wliich  the  various  assumptions  are  made,  and  they 
render  more  prominent  the  conditions  under  which  the  theory- 
holds  good." 

In  the  present  volume  the  authors  deal  with  weight,  mass, 
gravitation,  and  those  properties  of  matter  which  relate  chiefly 
to  change  of  form,  such  as  Elasticity,  Fluid  Viscosity,  Surface 
Tension,  Diffusion  and  Solution.  The  molecular  theory  of  matter 
has  necessarily  been  introduced,  inasmuch  as  investigators  have 
almost  always  expressed  their  work  in  terms  of  that  theory. 
But  the  detailed  account  of  the  theory,  especially  as  applied  to 
gases,  will  be  given  in  the  volume  on  Heat,  in  connection  with 
the  account  of  the  phenomena  which  first  brought  it  into 
prominence. 


PEEFACE  TO  FIFTH  EDITION. 

A  FEW  corrections  have  been  made  in  this  edition.  The  authors 
desire  to  thank  the  readers  who  have  kindly  pointed  out  errors 
and  have  enabled  them  to  make  these  corrections. 

J.  H.  P. 

J.  J.  T. 

Jamiary  1909. 


CONTENTS 


CHAP. 

I.    VVEIGIIT   AND   MASS         ....... 

II.    THE   ACCELERATION     OF    GRAVITY.       ITS    VARIATION     AND 

THE  FIGURE   OF   THE   EARTH  .... 

III.  GRAVITATION 

IV.  ELASTICITY 

V.    STRAIN  

VI.    STRESSES.      RELATION    BETWEEN   STRESSES  AND   STRAINS 

VII.    TORSION 

VIII.    BENDING    OF  RODS 

IX.    SPIRAL    SPRINGS 

X.    IMPACT 

XI.    COMPRESSIBILITY    OF   LIQUIDS          .... 
XII.    THE   RELATION    BETWEEN  THE   PRESSURE  AND  VOLUME   01 
A    GAS 

XIII.  REVERSIBLE    THERMAL    EFFECTS    ACCOMPANYING    ALTERA 

TIONS   IN   STRAINS 

XIV.  CAPILLARITY 

XV.   LAPLACE'S   THEORY    OF   CAPILLARITY 

XVI.    DIFFUSION    OF    LIQUIDS 

XVII.    DIFFUSION    OF    GASES 

XVIII.    VISCOSITY    OF   LIQUIDS  ..... 

INDEX         ........ 


PAGE 


7 
28 
53 
62 
68 


85 
103 
109 
116 

124 

131 
135 

173 
182 
196 
205 
225 


PROPERTIES  OF  MATTER. 


CHAPTEK  I. 
WEIGHT  AND  MASS. 

Contents.— Weiglit — Mass — Definition  of  Mass— Mass  proportional  to  Weight  at 
the  same  Point — Constancy  of  Mass — Unit  of  Mass. 

Introductory  Remarks. — Physics  is  the  study  of  the  properties  of 
matter,  and  of  the  action  of  one  portion  of  matter  upon  another,  and 
ultimately  of  the  effects  of  these  actions  upon  our  senses.  The  properties 
studied  in  the  various  branches,  Sound,  Heat,  Light,  and  Magnetism  and 
Electricity,  are  for  the  most  part  easily  classified  under  these  headings. 
But  there  are  other  properties  chiefly  connected  with  changes  in  shape  and 
relative  position  within  a  system  which  are  grouped  together  as  "  General 
Properties  of  INIatter."  Among  these  latter  properties  are  Elasticity, 
Surface  Tension,  Diffusion  and  Viscosity. 

The  most  general  properties  of  matter  are  really  those  studied  in 
Statics  and  Dynamics :  the  relation  between  forces,  when  the  matter 
acted  on  is  in  equilibrium  and  the  motion  of  matter  under  the  mutual 
action  of  the  various  portions  of  a  system.  But  in  Statics  and  Dynamics 
the  recourse  to  experiment  is  so  small,  and  when  the  experimental  foun- 
dation is  once  laid  the  mathematical  structure  is  so  great,  that  it  is  con- 
venient to  treat  these  branches  of  Physics  separately.  We  shall  assume 
in  this  work  that  the  reader  has  already  studied  them,  and  is  familiar 
both  with  the  conditions  of  equilibrium  and  with  the  simpler  types  of 
motion. 

We  shall,  however,  begin  with  the  discussion  of  some  questions  which 
involve  dynamical  considerations.  We  shall  show  how  we  pass  from  the 
idea  of  weight  to  that  of  mass,  and  how  we  establish  the  doctrine  of  the 
constancy  of  mass.  We  shall  then  give  some  account  of  the  measurement 
of  gravity  at  the  surface  of  the  earth,  and  of  the  gravitation  which  is  a 
property  of  all  matter  wherever  situated.  We  shall  then  proceed  to  the 
di^cussion  of  those  properties  of  matter  which  are  perhaps  best  described 
as  involving  change  of  form. 

Weig'ht. — All  matter  at  the  surface  of  the  earth  has  weight,  or  is 
pulled  towards  the  ground.  The  fact  that  the  pull  is  to  the  earth  at 
all  parts  of  its  surface  shows  conclusively  that  it  is  due  to  the  earth. 
Apparent  exceptions,  such  as  the  rising  of  a  balloon  in  air,  or  of  a  cork 
in  water,  are  of  course  explained,  not  by  the  levity  of  the  rising  bodies, 
but  by  the  greater  gravity   of  their  surroundings.     Oommon  experience 

A 


f  PROPERTIES  OF  MATTER, 

with  the  balance  shows  that  the  ratio  of  the  weights  of  two  bodies  is 
constant  wherever  they  are  weighed,  so  long  as  they  are  both  weighed 
at  the  same  point.  Common  experience  shows  too  that  the  ratio  is  the 
same  however  the  bodies  be  turned  about  on  the  scale-pan  of  the  balance. 

The  balance  does  not  tell  us  anything  as  to  the  constancy  of  weight  of 
a  given  body,  but  only  as  to  the  constancy  of  ratio  ;  for  if  the  weights  of 
ditlerent  bodies  varied,  and  the  variation  was  always  in  the  same  ratio,  the 
balance  would  fail  to  indicate  it.  But  here  experiments  with  pendulums 
supplement  our  knowledge.  A  given  pendulum  at  a  constant  temperature 
and  in  a  fixed  position  has,  as  nearly  as  we  can  observe,  the  same  time  of 
8win<'  from  day  to  day  and  from  year  to  year.  This  implies  that  the 
pull  of  the  earth  on  the  bob  is  constant — i.e.,  that  the  weight  at  the 
same  place  remains  the  same. 

This  constancy  of  weight  of  a  body  at  the  same  point  appears  to  hold 
whatever  chemical  or  physical  changes  the  matter  in  it  may  undergo. 
Experiments  have  been  made  on  the  weight  of  sealed  tubes  containing 
two  substances  which  were  at  first  separated,  and  which  were  then 
mixed  and  allowed  to  form  new  chemical  compounds.  The  tubes  were 
weio-hed  before  and  after  the  mixture  of  their  contents.  But  though 
Landolt*  and  Heydweillerf  have  thought  that  the  variations  which  they 
observed  were  real  and  not  due  to  erroi"S  of  experiment,  Sanford  and 
Ray  j  have  made  similar  experiments,  and  considered  that  the  variations 
were  observational  errors.  Where  variations  have  been  observed  they  are 
so  minute  and  so  irregular  that  we  cannot  as  yet  assume  that  there  is  any 
change  in  weight. 

Again,  temperature  does  not  appear  to  affect  weight  to  any  appreciable 
extent.  It  is  extremely  difficult  to  make  satisfactory  weighings  of  a  body 
at  two  different  temperatures.  Perhaps  the  best  evidence  of  constancy  is 
obtained  from  the  agreement  in  the  i-esults  of  diflferent  methods  of 
measuring  liquid  expansion.  In  Dulong  and  Petit's  U-tube  method  of 
determining  the  expansion  of  mercury,  two  unit  columns  have  different 
heights  but  equal  weights,  and  it  is  assumed  that  the  cold  column  would 
expand  into  the  hot  column  without  change  of  weight.  But  in  the 
dilatometer  method  nearly  the  whole  expansion  is  directly  measured,  and 
only  the  small  expansion  jf  the  envelope,  measured  by  assuming  the  expan- 
sion of  mercury,  introduces  the  assumption  of  constancy  of  weight  with 
change  of  temperature.  The  close  agreement  of  the  two  methods  shows 
that  there  is  no  large  variation  of  weight  with  temperature. 

We  may  probably  conclude  that,  up  to  the  limit  of  our  present  powers 
of  measurement,  the  weight  of  a  body  at  a  given  point  is  constant  under 
all  conditions. 

But  when  we  test  the  weight  at  different  points  this  constancy  no 
longer  holds.  The  common  balance  used  in  the  ordinary  way  fails  to  show 
variation,  since  both  pans  are  equally  affected. 

But  very  early  in  the  history  of  the  pendulum,  as  we  shall  show  in  the 
next  chapter,  experiments  proved  that  the  seconds  pendulum  had  different 
lengths  at  different  places,  or  that  the  same  pendulum  had  different  times 
of  swing  at  different  places.  In  other  words,  the  weight  of  the  bob  varied. 
Thus  a  body  is  about  1  in  300  heavier  at  London  than  at  the  Equator. 

*  Zeit.f.  Physik.  Ckem.,  xii.  1,  1894. 

+  Zcil.f.  Physik.,  August  25,  1900,  p.  527. 

J  Phys.  liev.,  v,  1897,  p.  247, 


WEIGHT  AND  MASS.  « 

As  early  as  1662  an  experiment  was  made  by  Dr.  Power*  in  which  a 
variation  of  weight  with  change  of  level  over  the  same  point  was  looked 
for.  A  body  was  weighed  by  a  fixed  balance,  being  first  placed  in  the 
scale-pan  and  then  hung  far  below  the  same  pan  by  a  string.  The 
experiment  was  repeated  by  Hooke,  and  later  by  others,  but  tlie  variation 
was  quite  beyond  the  range  of  observation  possible  with  these  early 
experimenters,  and  the  results  they  obtained  were  due  to  disturbances  in 
the  surroundings.  I'lie  first  to  show  that  the  balance  could  detect  a 
variation  was  von  Jolly  (chap.  iii.  p.  41),  who  in  1878  described  an 
experiment  in  which  he  weighed  a  kilogramme  on  a  balance  5"5  metres 
above  the  floor  and  then  hung  the  kilogramme  by  a  wire  so  that  it  was 
near  the  flooi".  He  detected  a  gain  in  the  lower  position  of  1'5  mgm. 
Later  he  repeated  the  experiment  on  a  tower,  a  5  kgm.  weight  gaining 
more  than  31  mgms.  between  the  top  of  the  tower  and  a  point  21  metres 
below.  More  recently  Richarz  and  Krigar-Menzel  found  a  variation 
in  the  weight  of  a  kilogramme  when  lowered  only  2  metres  (chap,  iii. 

P-^2-)  ... 

The  evidence  then  is  convincing  that  the  weight  of  a  body  varies  from 

point  to  point  on  the  earth's  surface,  and  also  varies  with  its  distance  above 

the  same  point. 

The  question  now  arises — Is  there  any  measurable  quality  of  matter 
which  remains  the  same  wherever  it  is  measured  ?  Experiment  showa 
that  there  is  constancy  in  that  which  is  termed  the  mass  of  matter. 

Mass. — Without  entering  into  any  discussion  of  the  most  appropriate 
or  most  fundamental  method  of  measuring  force,  we  shall  assume  that  wo 
can  measure  forces  exerted  by  bent  and  stretched  spiings  and  similar  con- 
trivances independently  of  the  motion  they  pioduce.  We  shall  assume 
that,  when  a  given  strain  is  observed  in  a  spring,  it  is  acting  with  a  definite 
force  on  the  body  to  which  it  is  attached,  the  force  being  determined  by 
previous  experiments  on  the  spring.  Let  us  imagine  an  ideal  experi- 
ment in  which  a  spring  is  attached  to  a  certain  body,  which  it  pulls 
horizontally,  under  constraint  free  from  friction.  Let  the  spring  be 
always  stretched  to  a  given  amount  as  it  pulls  the  body  along,  so  acting 
on  it  with  constant  force.  Then  all  experiments  and  observations  go  to 
show  that  the  body  will  move  with  the  same  constant  acceleration  wherever 
the  experiment  is  made.  This  constancy  of  accelei-ation  under  a  given 
force  is  expressed  by  saying  that  the  mass  of  the  body  is  constant,  i 
Though  the  experiment  we  have  imagined  is  unrealisable,  actual  experi-  ' 
ments  on  the  same  lines  are  made  for  us  by  good  chronometers.  The 
balance-wheel  of  a  chronometer  moves  to  and  fro  against  the  resistance 
of  the  hair-spring,  and  its  accelei'ation  is  very  accurately  the  same  for  the 
same  strain  of  the  spring  at  the  same  temperature  in  different  lati- 
tudes. The  weight  of  the  balance-wheel  decreases  by  3  in  1000  if  the 
chronometer  is  carried  from  London  to  the  Equator.  If  the  acceleration 
under  given  force  increased  in  the  same  ratio  the  rate  of  the  chronometer 
would  change  by  3  in  2000,  or  by  two  minutes  per  day,  and  the 
chronometer  would  be  useless  for  determinations  of  longitude.  Again,  a 
tuning-fork,  making,  say,  256  vibrations  per  second  at  Paris  at  16°  will 
have  very  accurately  the  same  frequency  at  the  same  temperature  wherever 
tested.  The  same  portion  of  matter  in  the  prongs  has  the  same  acceleration 
for  the  same  strain  and,  presumably,  for  the  same  force  all  the  world  over. 
*  Mackenzie,  The  Laws  of  Gravitation,  p.  2. 


4  PROPERTIES  OF  MATTER. 

This  constancy  of  accelerAtion  of  a  given  body  under  given  force  holds 
true  likewise  whatever  the  nature  of  the  body  exerting  the  force  may  be— 
i.e.,  whether  it  be  a  bent  spring,  a  spiial  spring,  air  pressing,  a  string 
puUing,  and  so  on. 

Further  experiment  shows  that  the  acceleration  of  a  given  body  is 
proportional  to  the  force  acting  on  it.  Thus,  in  a  very  small  vibration  of 
a  pendulum  the  fraction  of  the  weight  of  the  bob  tending  to  restore  it  to 
its  central  position  is  proportional  to  the  displacement,  and  the  simple 
harmonic  type  of  the  motion  with  its  isochronism  shows  at  once  that  the 
acceleration  is  proportional  to  the  displacement,  and  therefore  to  the  force 
acting.  When  a  body  vibrates  up  and  down  at  the  end  of  a  spiral  spring 
we  attain  have  simple  harmonic  motion  with  acceleration  proportional  to 
the  distance  from  the  position  of  equilibrium.  The  variation  in  the  force 
exerted  by  the  spring  is  also  proportional  to  this  distance,  or  acceleration 
is  proportional  to  force  acting.  Indeed,  elastic  vibrations  with  their 
isochronism  go,  in  general,  to  prove  this  proportionality.  If,  then,  we 
accept  the  view  that  we  can  think  of  forces  acting  on  bodies  as  being 
measurable  independently  of  the  motion  which  they  produce — measui\able, 
say,  by  the  strain  of  the  bodies  acting — we  have  good  experimental  proof 
that  a  f^iven  portion  of  matter  always  has  equal  acceleration  under  equal 
force,  and  that  the  accelerations  under  different  forces  are  proportional  to 
the  forces  acting  upon  it. 

We  can  now  go  a  stop  farther  and  use  the  accelerations  to  compare 

different  masses. 

Definition  ofMSiSS.—The  masses  of  bodies  are  p'oportional  to  the  forces 

producing  equal  accelerations  in  them. 

An  equivalent  statement  is,  that  the  masses  are  inversely  as  the 
acceleration  produced  by  equal  forces.  It  follows  from  our  definition  that, 
if  equal  accelerations  are  observed  in  different  bodies,  then  the  masses  are 
proportional  to  the  forces  acting. 

Observation  and  experiment  further  enable  us  to  say  that : 

The  masses  of  bodies  are  2'>^'opoi'tional  to  their  weights  at  the  same  jwint. 
To  prove  this  it  is  only  necessary  to  show  that  all  bodies  have  equal 
acceleration  at  the  same  place  when  acted  on  by  their  weights  alone — to 
show,  in  fact,  that  the  quantity  always  denoted  by  g  is  constant  at  the  same 
place. 

A  very  simple  though  rough  experiment  to  prove  this  consists  in 
tying  a  piece  of  iron  and  a  piece  of  wood  to  the  two  ends  of  a  thread  and 
putting  the  thread  across  a  horizontal  ring  so  that  the  two  weights 
depend  at  the  same  height  above  the  floor.  The  thread  is  now  burnt 
in  the  middle  of  the  ring  and  the  iron  and  wood  begin  to  fall  at  the  same 
in.stant.  They  reach  the  floor  so  nearly  together  that  only  a  single 
bump  is  heard.  If  the  surfaces  presented  to  the  air  are  very  different  the 
air  resistance  may  interfere  with  the  success  of  the  experiment.  But  the 
more  the  air  resistance  is  eliminated  the  more  nearly  is  the  time  of  fall  the 
same.  Thus,  if  a  penny  and  a  sheet  of  paper  are  placed  on  a  board  some 
height  above  the  floor,  and  if  the  board  is  suddenly  withdrawn,  the  penny 
falls  straight  while  the  paper  slowly  flutters  down.  Kow  crumple  up  the 
paper  into  a  little  ball  and  repeat  the  experiment,  when  the  two  reach  the 
ground  as  nearly  as  we  can  observe  together. 

Newton  {Principia,  Book  III.,  Prop.  G)  devised  a  much  more  accurate 
form  of  the  experiment,  using  the  pendulum,  in  which  any  difference  of 


WEIGHT  AND  MASS.  5 

acceleration  wouM   be   cumulative,  and   suspending   in   succession   equal 
weights  of  various  kinds  of  matter.     He  saj's  (Motte's  translation) : 

"It  has  been,  now  of  a  long  time,  observed  by  others,  that  all  sorts  of  heavy 
bodies  (allowance  being  made  for  the  inequality  of  retardation,  which  they 
suffer  from  a  small  power  of  resistance  in  the  air)  descend  to  the  Earth  from 
eqtiul  heights  in  equal  times  ;  and  that  equality  of  times  we  may  distinguish  to 
a  great  accuracy,  by  the  help  of  pendulums.  I  tried  the  thing  in  gold,  silver, 
lead,  glass,  sand,  common  salt,  wood,  water,  and  wheat.  I  provided  two 
wooden  boxes,  round  and  equal.  I  filled  the  one  with  wood,  and  suspended  an 
equal  weight  of  gold  (as  exactly  as  I  could)  in  the  centre  of  oscillation  of  the 
other.  The  boxes  hanging  by  equal  threads  of  eleven  feet,  made  a  couple  of 
pendulums  perfectly  equal  in  weight  and  figure,  and  equally  receiving  the 
resistance  of  the  air.  And  placing  the  one  by  the  other,  I  observed  them  to 
play  together  forwards  and  backwards,  for  a  long  time,  with  equal  vibrations. 
And  therefore  the  quantity  of  matter  in  the  gold  (by  Cor.  1  and  6,  prop.  24, 
book  2)  was  to  the  quantity  of  matter  in  the  wood,  as  the  action  of  the  motive 
force  (or  vis  motrix)  upon  all  the  gold,  to  the  action  of  the  same  upon  all  the 
wood  ;  that  is,  as  the  weight  of  the  one  to  the  weight  of  the  other.  And  the 
like  happened  in  the  other  bodies.  By  these  experiments,  in  bodies  of  the  same 
weight,  I  could  manifestly  have  discovered  a  difference  of  matter  less  than  a 
thousandth  part  of  the  whole,  had  any  such  been." 

Newton  here  uses  "quantity  of  matter"  where  we  should  now  say 
"mass."  Bessel  {Berlin  Ahh.,  1830,  Ann.  Fogg.,  xxv.  1832,  or 
Memoires  relatifs  a  la  Physique,  v.  p.  71)  made  a  series  of  most  careful 
experiments  by  Newton's  method,  fully  confirming  the  conclusion  that 
weight  at  the  same  place  is  proportional  to  mass. 

Constancy  of  Mass. — The  experiments  which  have  led  to  the  con- 
clusion that  weight  at  the  same  place  is  constant  now  gain  another 
significance.  They  show  that  the  mass  of  a  given  portion  of  matter  is 
constant,  whatever  changes  of  position,  of  form,  or  of  chemical  or  physical 
condition  it  may  undergo. 

When  we  "  weigh "  a  body  by  the  common  balance,  say,  by  the 
counterpoise  method,  we  put  it  on  the  pan,  counterpoise  it,  and  then 
replace  it  by  bodies  from  the  set  of  "  weights  "  having  an  equal  weight. 

But  our  aim  is  not  to  find  the  weight  of  the  body,  the  pull  of  the 
earth  on  it.  We  use  the  equality  of  weight  possessed  by  equal  masses  at 
the  same  point  of  the  earth's  surface  to  find  its  mass.  In  buying  matter 
by  weight  we  are  not  ultimately  concerned  with  weight  but  with  mass, 
and  we  expect  the  same  mass  in  a  pound  of  it  whether  we  buy  in  London 
or  at  the  Equator.  A  set  of  weights  is  really  a  set  of  masses,  and  when 
we  use  one  of  them  we  are  using  it  as  a  mass  throvigh  its  weight. 

Unit  of  Mass. — We  can  make  a  definite  unit  of  mass  by  fixing  on 
some  piece  of  matter  as  the  standard  and  saying  that  it  contains  one  unit 
or  so  many  units.  So  long  as  we  are  careful  that  no  portion  of  the 
standard  piece  of  matter  is  removed  and  that  no  addition  is  made  to  it, 
such  a  unit  is  both  definite  and  consistent. 

In  this  country  the  unit  of  mass  for  commercial  purposes  is  the  piece  of 
platinum  kept  at  the  Standards  Ofiice  at  Westminster,  marked  "  P.S. 
1844  1  lb."  and  called  the  Imperial  Avoirdupois  Pound.  But  for  scientific 
purposes  all  over  the  world  the  unit  of  mass  is  the  gramme,  the  one- 
thousandth  part  of  the  mass  of  the  piece  of  platinum-iridivim  called  the 
•'  Kilogramme-International,"  which  is  kept  at  Paris.  Copies  of  this 
kilogramme,  compared  either  with  it  or  with  previous  copies  of  it,  are  now 
distributed  through  the  world,  their  values  being  known  to  less,  perhaps, 


6  PROPERTIES  OF  MATTER. 

than  0-01  mgm.      For  example,  the  copy  in  the  Standards  Office  at  Wes4 
minster  is  certified  to  be 

1-000000070  kgm. 

with  a  prohal)le  error  of  2  in  the  last  place. 

According  to  a  comparison  carried  out  in  1883,  the  Imperial  pound 
contains 

453"5924277  grammes, 

though  Parliament  enacted  in  1878  that  the  pound  contained 

453 •59245  grammes. 

Of  cour.se  one  piece  of  matter  only  can  be  the  standard  in  one  .system  of 
measurements,  and  the  enactment  of  1878  only  implies  that  we  should  us6 
a  diffei-ent  value  for  the  kilogramme  in  England  from  that  used  in  Franca 
The  diflerence  is,  however,  (j^uitenegligible  for  commercial  purposes. 


CHAPTER  II. 

THE  ACCELERATION  OF  GRAVITY.    ITS  VARIATION  AND 
THE  FIGURE  OF  THE  EARTH. 

Contents. — Early  History — Pendulum  Clock — Picard's  Experiments — Huygens' 
Theory — Newton's  Theory  and  Experiments — Bouguer's  Experiments  —  Ber- 
nouilli's  Correction  for  Arc — Experiments  of  Borda  and  Cassini — Eater's  Con- 
vertible Pendulum — Bessel's  Experiments  and  his  Theory  of  the  Reversible 
Pendulum — Repsold's  Pendulum — Yielding  of  the  Support — DefEorges'  Pendulum 
— Variation  of  Gravity  over  the  Earth's  Surface — Richer — Newton's  Theory  of 
the  Figure  of  the  Earth — Measurements  in  Sweden  and  Peru^ — Bouguer's 
Correction  to  Sea-level — Clairaut's  Theorem — Kater  and  Sabine — Invariable 
Pendulum — Airy's  Hydrostatic  Theory — Faye's  Rule — Indian  Survey — Formula 
foig  in  any  Latitude — Von  Sterneck's  Half-second  Pendulums — His  Barymeter 
— Gravity  Balance  of  Threlfall  and  Pollock. 

We  shall  describe  in  this  and  the  following  chapter  the  methods  of 
measuring  two  quantities  ;  the  acceleration  of  falling  bodies  due  to  the 
earth,  at  its  surface  (the  quantity  always  denoted  by  g)  ;  and  the  accelera- 
tion due  to  unit  mass  at  unit  distance  (the  quantity  known  as  the  gravita- 
tion constant  and  denoted  by  G).  The  two  may  be  measured  quite  in- 
dependently, but  yet  they  are  closely  related  in  that  g  is  the  measure  of  a 
particular  case  of  gravitation,  while  G  is  the  expression  of  its  general 
measure.  The  two  together  enable  us  to  find  the  mass  and  therefore  the 
mean  density  of  the  earth. 

The  Acceleration  of  Gravity.* — We  shall  briefly  trace  the  history 
of  the  methods  which  have  been  used  in  measuring  g,  for  in  so  doing  we 
can  set  forth  most  clearly  the  difiiculties  to  be  overcome  and  realise  the 
exactitude  with  which  the  measurement  can  now  be  made.  We  shall 
then  give  some  account  of  the  experiments  made  to  determine  the  varia- 
tions of  gravity  and  the  use  of  the  knowledge  so  gained  to  determine  the 
shape  of  the  earth. 

Early  History. — The  first  step  in  our  knowledge  of  the  laws  of 
falling  bodies  was  taken  about  the  end  of  the  sixteenth  century,  when 
Stevinus,  Galileo,  and  their  contemporaries  were  laying  the  foundations 
of  the  modern  knowledge  of  mechanics.  Stevinus,  the  discoverer  of  the 
Triangle  of  Forces  and  of  the  theory  of  the  Inclined  Plane,  and  Galileo, 

*  A  collection  of  the  most  important  original  papers  on  the  pendulum 
constitutes  vols.  iv.  and  v.  of  Memoircs  relatifs  a  la  Physique.  It  is  prefaced  by  an 
excellent  history  of  the  subject  by  M.  Wolf,  and  contains  a  bibliography.  The  fifth 
volume  of  The  G.  T.  Survey  of  India  consists  of  an  account  of  the  pendulum 
operations  of  the  survey,  with  some  important  memoirs.  In  the  Journal  de 
Physique,  vii.  1888,  are  three  important  articles  by  Commandant  Defforges  on  the 
theory  of  the  pendulum,  concluding  with  an  account  of  his  own  pendulum.  The 
description  given  in  this  chapter  is  based  on  these  works. 


8  PROPERTIES  OF  MATTER. 

the  founder  of  Dynamics,  were  both  aware  that  the  doctrine  then  held  that 
bodies  fall  with  rapidity  pioportional  to  their  weight  was  quite  false,  and 
they  asserted  that  under  the  action  of  their  weight  alone  all  bodies  would 
fall  at  equal  rates.  They  pointed  out  that  the  diflerent  rates  actually 
observed  were  to  be  ascribed  to  the  resistance  of  the  air,  which  has  a 
greater  effect  on  the  movement  of  light  than  of  heavy  bodies  of  e(]ual 
size.  Galileo  made  a  celebrated  experiment  to  verify  this  fact  by  dropping 
bodies  of  different  weights  from  the  top  of  the  Leaning  Tower  of  Pisa, 
and  showing  that  they  reached  the  ground  in  the  same  time.  The  air- 
pump  was  not  yet  invented,  so  that  the  later  verification  by  the  "guinea 
and  feather "  was  not  then  possible.  But  Galileo  did  not  stop  with  this 
experiment,  lie  made  the  progress  of  dynamics  possible  by  introducing 
the  conception  of  equal  additions  of  velocity  in  equal  times  — the  con- 
ception of  uniform  acceleration.  His  first  idea  was  that  a  constant  force 
131-ould  give  equal  additions  of  velocity  in  equal  distances  traversed,  but 
investigation  led  him  to  see  that  this  idea  was  untenable,  and  he  then 
enunciated  the  liypothesis  of  equal  additions  in  equal  times.  He  showed 
that,  by  this  hypothesis,  the  distance  travei^sed  is  proportional  to  the 
square  of  the  time.  Not  content  with  mere  mathematical  deductions, 
he  made  experiments  on  bodies  mov^ing  down  inclined  planes,  and  demon- 
strated that  the  distances  traversed  were  actually  proportional  to  the 
squares  of  the  times — i.e.,  that  the  acceleration  was  uniform.  By  ex- 
periments with  pendulums  falling  through  the  arc  of  a  circle  to  the 
lowest  point,  and  then  rising  through  another  arc,  he  concluded  that  the 
velocity  acquired  in  falling  down  a  slope  depends  only  on  the  vertical 
height  fallen  through  and  not  upon  the  length  of  the  slope,  or,  as  we 
should  now  put  it,  that  the  acceleration  is  proportional  to  the  cosine  of 
the  angle  of  the  slope  with  the  vertical.  He  thus  arrived  at  quite  sound 
ideas  on  the  acceleration  of  falling  bodies  and  on  its  uniformity,  and  from 
his  inclined  plane  experiments  could  have  obtained  a  rough  approxi- 
mation to  the  quantity  we  now  denote  by  g.  But  Galileo  had  no  accurate 
method  of  measuring  small  periods  of  time  in  seconds.  The  pendulum 
clock  was  not  as  yet  invented,  and  he  made  merely  relative  measurements 
of  the  time  intervals  by  determining  in  his  experiments  the  quantity 
of  water  which  flowed  through  a  small  orifice  of  a  vessel  during  each 
interval. 

To  Galileo  we  also  owe  the  foundation  of  the  study  of  pendulum 
vibrations.  The  isochronism  of  the  pendulum  had  been  previously  ob- 
served by  others,  but  Galileo  rediscovered  it  for  himself,  and  showed  by 
further  experiment  that  the  times  of  vibration  of  diflerent  simple  pendu- 
lums are  proportional  to  the  square  roots  of  their  lengths.  He  also  used 
the  pendulum  to  determine  the  rate  of  beating  of  the  pulse  and  recognised 
the  possibility  of  employing  it  as  a  clock  regulator.  He  did  not  publish 
liis  ideas  on  the  construction  of  a  pendulum  clock,  and  they  were  only 
discovered  among  his  papers  long  after  his  death. 

From  Galileo,  therefore,  we  derive  the  conception  of  the  appropriate 
quantity  to  measure  in  the  fall  of  bodies,  the  acceleration,  and  to  him  we 
owe  the  instrument  which  as  a  free  pendulum  gives  us  the  acceleration  of 
fall,  and,  as  a  clock  regulator,  provides  us  with  the  best  means  of  deter- 
mining the  time  of  fall. 

Soon  after  Galileo's  death,  Mersenne  made,  in  1644,  the  first  determi- 
nation of  the  length  of  a  simple  pendulum  beating  seconds,  and  a  littla 


THE  ACCELERATION  OF  GRAVITY.  9 

later  he  suggested  as  a  problem  the  determination  of  the  length  of  a 
simple  pendulum  equivalent  to  a  given  compound  pendulum. 

Pendulum  Clock. — But  it  was  only  with  the  invention  of  the 
pendulum  clock  by  Huygens  in  1G57  that  the  second  became  an  interval 
of  time  measurable  with  consistency  and  ease.  At  once  the  new  clock  was 
widely  used.  Its  rate  could  easily  be  determined  by  star  observalions,  and 
determinations  of  the  length  of  the  seconds  pendulum  by  its  aid  became 
common. 

Picard's  Experiment. — In  1GC9  Picard  determined  this  length  at 
Paris,  using  a  copper  ball  an  inch  in  diameter  suspended  by  an  aloe  fibre 
from  jaws.  This  suspension  was  usual  in  early  work,  the  aloe  fibre  being 
unaffected  to  any  appreciable  extent  by  moisture.  Picard's  value  was 
36  inches  8-|  lines  Paris  measure.  The  Paiis  foot  may  be  taken  as 
■ii^  or  1"0G5  English  feet,  and  there  are  12  lines  to  the  inch,  so  that  the 
length  found  was  39'09  English  inches.  Picard  states  that  the  value  had 
been  fovmd  to  be  the  same  at  London  and  at  Lyons. 

Huygfens'  Theory. — In  1673  Huygens  propounded  the  theory  of  the 
cycloidal  pendulum,  proving  its  exact  isochronism,  and  he  showed  how  to 
construct  such  a  pendulum  by  allowing  the  string  to  vibrate  between 
cycloidal  cheeks.  He  determined  the  length  beating  seconds  at  Paris, 
confii'ming  Picard's  value,  and  from  the  formula  which  we  now  put  in  the 

form  g  =  7rH  he  found    |   the  distance  of    free  fall   in  one   second,  the 

quantity  which  was  at  fii'st  used,  instead  of  the  full  acceleration  we  now 
employ.  His  value  was  15  ft.  1  in.  1^  lines,  Paris  measure,  which  would 
give  ^  =  32-16  Enghsh  feet. 

Huygens  at  the  same  time  gave  the  theory  of  uniform  motion  in  a 
circle  and  the  theory  of  the  conical  pendulum,  and  above  all  in  importance 
he  founded  the  study  of  the  motion  of  bodies  of  finite  size  by  solving 
Mersenne's  problem  and  working  out  the  theory  of  the  compound 
pendulum.  He  discovered  the  method  of  determining  the  centre  of 
oscillation  and  showed  its  interchangeability  with  the  centre  of  suspension. 

Newton's  Theory  and  Experiments.— Newton  in  the  Princijna 

made  great  use  of  the  theory  of  the  pendulum.  He  there  for  the  first 
time  made  the  idea  of  mass  definite,  and  by  his  pendulum  experiments 
{Principia,  sect,  vi..  Book  IL,  Prop.  24),  he  proved  that  mass  is 
proportional  to  weight.  He  used  pendulums  too,  to  investigate  the 
resistance  of  the  air  to  bodies  moving  through  it,  and  repeated  the 
pendulum  experiments  of  Wren  and  others,  by  which  the  laws  of  impact 
had  been  discovered.  But  his  great  contribution  to  our  present  subject 
was  the  demonstration,  by  means  of  the  moon's  motion,  that  gravity  is 
a  particular  case  of  gravitation  and  acts  according  to  the  law  of  inverse 
squares,  the  attracting  body  being  the  earth.  In  Book  III.,  Prop.  4,  he 
calculates  the  acceleration  of  the  moon  towards  the  earth  and  shows  that, 
starting  from  rest  with  this  acceleration,  it  would  fall  towards  the  earth 
15  ft.  1  in.  1^  lines  (Paris)  in  the  first  minute.  If  at  the  surface  of  the 
earth  60  times  nearer  the  acceleration  is  60^  times  greater  the  same 
distance  would  here  be  fallen  through  in  one  second,  a  distance  almost 
exactly  that  obtained  by  Huygens'  experiments. 

In  a  later  proposition  (37)  he  returns  to  this  calculation,  and  now, 
assuming  the  law  of  inverse  squares  to  be  correct,  he  makes  a  more  exact 
determination  of  the  moon's  acceleration,  and  from  it  deduces  the  value 


10  PROPERTIES  OF  MATTER. 

of  gravity  at  the  mean  radius  of  the  earth  in  latitude  45".  Then  by  hiiS 
theory  of  the  variation  of  gravity  with  latitude,  of  which  we  shall  give 
Boine  account  below,  he  finds  the  value  at  Paris.  He  corrects  the  value 
thus  found  for  the  centrifugal  force  at  Paris  and  (in  Pi*op.  19)  for  the 
air  displaced,  which  he  takes  as  fxJjnj  of  the  weight  of  the  bob  used  in  the 
pendulum  experiments,  and  finally  arrives  at  15  ft.  1  in.  1^  lines  (Paris), 
difiering  from  ITuygens'  value  by  about  1  in  7500. 

Boug'Uer's  Experiments. — Though  Newton  was  thus  aware  of  the 
need  of  tiie  correction  for  the  buoyancy  of  the  air,  it  does  not  appear  to 
have  been  applied  again  until  Bouguer  made  his  celebrated  experiments 
in  the  Andes  in  1787.  These  are  especially  interesting  in  i-egard  to 
the  variations  of  gravity,  but  we  may  here  mention  some  important 
points  to  which  Bouguer  attended.  While  his  predecessors  probably 
altered  the  length  of  the  pendulum  till  it  swung  seconds  as  exactly 
as  could  be  observed,  Bouguer  introduced  the  idea  of  an  *'  invariable 
pendulum,"  making  it  always  of  the  same  length  and  observing  how  long 
it  took  to  lose  so  many  vibrations  on  the  seconds  clock.  For  this  purpose 
the  thread  of  the  pendulum  swung  in  front  of  a  scale,  and  he  noted  the 
time  when  the  thread  moved  past  the  centi-e  of  the  scale  at  the  same 
instant  that  the  beat  of  the  clock  was  heard.  Here  we  have  an  elementary 
form  of  the  "  method  of  coincidences,"  to  be  described  later.  He  used, 
not  the  measured  length  from  the  jaw  suspension  to  the  centre  of  the  bob, 
which  was  a  double  truncated  cone,  but  the  length  to  the  centre  of  oscilla- 
tion of  the  thread  and  bob,  and  he  allowed  for  change  of  length  of  his 
measui'ing-rod  with  temperature.  He  also  assured  himself  of  the  coinci- 
dence of  the  centre  of  figure  Avith  the  centre  of  gravity  of  the  bob  by 
showing  that  the  time  of  swing  was  the  same  when  the  bob  was  inverted. 
He  determined  the  density  of  the  air  by  finding  the  vertical  height  through 
which  he  must  carry  a  barometer  in  order  that  it  should  fall  one  line,  and 
he  thus  estimated  the  density  of  the  air  on  the  summit  of  Pichincha  at 
■^-j^  ^  ^^  „  that  of  the  copper  bob  of  his  pendulum.  Applying  these  correc- 
tions to  his  observations  he  calculated  the  length  of  the  seconds  pendulum 
in  vacuo. 

Correction  for  Arc. — In  1747,  D.  BernouilH  showed  how  to  correct 
the  observed  time  of  vibration  to  that  for  an  infinitely  small  arc  of  swing. 
The  observed  time  is  to  a  first  approximation  longer  than  that  for  an 

2 

infinitely  small  arc  in  the  ratio  1  +  -7;  where  a  is  the  amplitude  of  the 

16 

angle  of  swing.      The  correction  has  to  be  modified  for  the  decrease  in 

amplitude  occurring  during  an  observation. 

Experiments  of  Borda  and  Cassini.— The  next  especially  note- 

worthy  experiments  are  those  by  Borda  and  Cassini  made  at  Paris  in  1792 
in  connection  with  the  investigations  to  determine  a  new  standard  of 
length,  when  it  was  still  doubtful  whether  the  seconds  pendulum  might 
not  be  preferable  to  a  unit  related  to  the  dimensions  of  the  earth.  The 
form  of  pendulum  which  they  used  is  now  named  after  Borda.  It  con- 
sisted of  a  platinum  ball  nearly  1^  inches  in  diameter,  hung  by  a  fine  iron 
wire  about  12  Paris  feet  long.  It  had  a  half-period  of  about  two  seconds 
The  wire  was  attached  at  its  upper  end  to  a  knife  edge — the  advantages  of 
a  knife-edge  suspension  having  been  already  recognised — and  the  knife 
edge  and  wire-holder  were  so  formed  that  their  time  of  swing  alone  was  the 
eame  a.«  that  of  the  pendulum.      In  calculating  the  moment  of  inertia, 


THE  ACCEI.ERATION  OF  GRAVITY.  ii 

they  could  therefore  be  left  out  of  account.      At  the  lower  end  the  wire 
was  attached  to  a  shallow  cup  with  the  concavity  downwards,  and  the  ball 
exactly  fitted  into  this  cup,  being  made  to  adhere  to  it  by  a  little  grease. 
The  ball  could  therefore  be  easily  and  exactly  reversed  without  altering 
the  pendulum  length,  and  any  non-coincidence  of  centi-e  of  gravity  and 
centre  of  figure  could  be  eliminated  by  taking  the  time  of  swing  for  each 
position  of  the  ball.     The  pendulum  was  hung  in  front  of  a  seconds  clock, 
with  its  bob  a  little  below  the  clock  bob,  and  on   the  latter  was  fixed  a 
black    paper  with  a  white  X-shaped  cross  on   it.      The  vibrations  were 
watched  through  a  telescope  from  a  short  distance  away,  and  a  little  in 
front  of  the  pendulum  was  a  black  screen  covering  half  the  field.     When 
the  pendulums  were  at  rest  in  the  field  the  edge  of  this  screen  covered 
half  the  cross  and  half  the  wire.     When  the  swings  were  in  progress  the 
times  were  noted  at  which  the  pendulum  wire  just  bisected  the  cross  at  the 
instant  of  disappearance  behind  the  screen.      This  was  a  "  coincidence," 
and,  since  the  clock  bob  made  two  swings  to  one  of  the  pendulum,  the 
interval  between  two  successive  "  coincidences  "  was  the  time  in  which  the 
clock  gained  or  lost  one  complete  vibration  or  two  seconds  on  the  wire 
pendulum.     The  exact  second  of  a  coincidence  could  not  be  determined 
but  only  estimated,  as  for  many  seconds  the  wire  and  cross  appeared  to 
pass  the  edge  together.     But  the  advantage  of  the  method  of  coincidences 
was  still  preserved,  for  it  lies  in  the  fact  that  if  the  uncertainty  is  a 
small  fraction  of  the  interval  between  two  successive  coincidences  the 
error  introduced  is  a  very  much  smaller  fraction  of  the  time  of  vibration. 
For,  suppose  that  the  wire  pendulum  makes  n  half  swings  while  the  clock 
makes  2«  +  2.     If  the  clock  beats  exact  seconds  the  time  of  vibration  of 
the  wire  pendulum  is 


n  \       n 


If  there  is  a  possible  error  in  the  determination  of  each  of  two  successive 
coincidences  of  m  seconds,  or  at  the  most  of  2m  in  the  interval  of  2n  +  2 
seconds,  the  observed  time  might  be 

\        n±m/        ^       n\    ^nj'        \       n^n^j 

In  one  case  Borda  and  Cassini  employed  an  interval  of  2n  =  3000  seconds, 
and  found  an  uncertainty  not  more  than  30  seconds  for  the  instant  of 
coincidence.     Thus 

TO_    30    _      1 

•n?     15U0'     75000 

Now,  as  they  observed  for  about  four  hours,  or  for  five  intervals  in  succes- 
sion, the  error  was  reduced  to  ^.  or  -i-,  >qqq  of  the  value  of  t.  Practically 
the  method  of  coincidences  determined  the  time  of  vibration  of  the 
pendulum  in  terms  of  the  clock  time  with  sufficient  accuracy,  and  the 
responsibility  for  error  lay  in  the  clock.  The  pendulum  was  treated  as 
forming  a  rigid  system,  and  the  length  of  the  equivalent  ideal  simple  pen- 
dulum was  calculated  therefrom.  Corrections  were  made  for  air  displaced, 
for  arc  of  swing,  and  for  vaiiations  in  length  with  temperatvire. 

The  final  value  obtained  was:  Seconds  pendulum  at  Paris  =  4:i0-5593 


12 


PROPJ: IITIES  OF  MATTER. 


.  h 


Qw 


lines  (Paris).     As  the  metre  =  44o-29G  Paris  lines,  this  gives  993-53  mm., 
and,  corrected  to  soa-level,  it  gives  l)90-85  mm. 

Kater's  Convertible  Pendulum. — The  difficulties  in  measuring  the 

length  and  in  calculating  the  moment  of  inertia  of  the  wire-suspended  or 
so-called    .simple    pendulum   led   Prony  in   1800  to  propo.se  a  pendulum 
employing  the  principle  of  interchangeability  of  the  centres  of  oscillation 
and  susfiension.      The  pendulum   was  to   have  two  knife    edges   turned 
inwards  on  opposite  sides  of  the  centre  of  gravity,  so  that  it  could  be 
swung  from  either,  and  was  to  be  so  adjusted  that  the  time  of 
swing  was  the  same  in  both  cases.     The  di;stance  between  the 
knife  edges  would  then  be  the  length  of  the  equivalent  simple 
pendulum.    Prony's  proposal  was  unheeded  by  bis  contemporaries, 
and  the  paper  describing  it  was  only  published  eighty  years  later.* 
In  1811,  Bohnenberger  made  the  same  proposal,  and  again 
in  1817  Captain  Kater  independently  hit  on  the  idea,  and  for 
the  first  time  carried  it  into  practice,  making  his  celebrated 
determination  of  g  at  London  with  the  form  of  instrument  since 
known  as  "  Kater's  convertible  pendulum."     This  pendulum  is 
shown  in  Fig.  1 .     On  the  rod  are  two  adjustable  weights,  w  and  s. 
The  larger  weight  w  is  moved  about  until  the  times  of  swing 
from  the  two  knife  edges  ^-j  k.^  are  nearly  equal,  when  it  is 
screwed  in  position.     Then  s  is  moved  by  means  of  a  screw  to 
make  the  final  adjustment  to  equality.     Kater  determined  the 
time  of  vibration  by  the  method  of  coincidences,  his  use  of  it 
being  but  slightly  difierent  from  that  of  Borda.     A  white  circle 
on  black  paper  was  fastened  on  the  bob  of  the  clock  pendulum ; 
the  convertible  pendulum  was  suspended  in  front  of  the  clock, 
and  when  the  two  were  at  rest  the  tail-piece  t  of  the  former  just 
^h  covered  the  white  circle  on  the  latter  as  viewed  by  a  telescope  a 

^^B||.  ^  few  feet  away.  A  slit  was  made  in  the  focal  plane  of  the 
^^^B|^  eyepiece  of  the  telescope  just  the  width  of  the  images  of  the 
^^^^^  white  patch  and  of  the  pendulum  tail.  A  coincidence  was  the 
instant  during  an  observation  at  which  the  white  circle  was 
quite  invisible  as  the  two  pendulums  swung  past  the  lowest 
point  together.  A  series  of  swings  were  made,  first  from  one 
knife  edge  and  then  from  the  other,  each  series  lasting  over 
four  or  five  coincidences,  the  coincidence  interval  being  about 
500  seconds.  The  fine  weight  was  moved  after  each  series  till 
the  number  of  vibrations  per  twenty-four  hours  only  diflered  by 
a  small  fraction  of  one  vibration  whichever  knife  edge  was  used, 
and  then  the  ditierence  was  less  than  errors  of  observation,  for 
the  time  was  sometimes  greater  from  the  one,  sometimes  greater  from  the 
other.  The  mean  time  observed  when  this  sta<:e  was  reached  was  corrected 
for  amplitude,  and  then  taken  as  the  time  of  the  simple  pendulum  of 
length  equal  to  the  distance  between  the  knife  edges,  this  distance  being 
carefully  measured.  A  correction  Avas  made  for  the  air  displaced  on  tha 
assumption  that  gravity  was  diminished  thereby  in  the  ratio  of  weight  of 
pendulum  in  air  to  weight  of  pendulum  in  vacuo.  The  value  was  then 
corrected  to  sea-level.  The  final  value  of  the  length  of  the  .seconds  pendulum 
at  .sea-level  in  the  latitude  of  London  was  determined  to  be  39'13'J2U  inches.t 

*  Miinoircs  relati/s  a  la  Physique,  iv.  \i.  G5. 

t  The  experiments  are  described   iu  a  paper  in  the  Phil.  Trans,  for  1818  "An 


Fig.  1.— 

Katc'i-'s 

Convertible 

I'eudulutu. 


THE  ACCELERATION  OF  GRAVITY.  13 

Bessel's   Experiments   and   his   Theory   of  the   Reversible 

Pendulum. — In  182G  Jiessel  made  experiments  to  determine  the  length 
of  the  seconds  pendulum  at  Koenigsberg.  He  used  a  wire-suspended 
pendulum,  swung  first  from  one  point  and  then  from  another  point, 
exactly  a  "Tuiseof  Peru"*  higher  up,  the  bob  being  at  the  same  level  in 
each  case.  Assuming  that  the  pendulums  are  truly  simple,  it  will  easily  be 
seen  that  the  difterence  in  the  squares  of  the  times  is  the  square  of  the 
time  for  a  simple  pendulum  of  length  equal  to  the  difference  in  lengths, 
and  therefore  the  actual  length  need  not  be  known.  But  the  practical 
pendulum  departs  from  the  ideal  simple  type,  and  so  the  actual  lengths 
have  to  be  known.  As,  however,  they  enter  into  the  expression  for  the 
difierence  of  the  squares  of  the  times,  with  a  very  small  quantity  as  co- 
efficient, they  need  not  be  known  with  such  accuracy  as  their  diflerences. 
Bessel  took  especial  care  that  this  difference  should  be  accurately  equal  to 
the  toise.  At  the  upper  end,  in  place  of  jaws  or  a  knife  edge,  he  used  a  hori- 
zontal cylinder  on  which  the  wire  wrapped  and  unwrapped.  He  introduced 
corrections  for  the  stiffness  of  the  wire  and  for  the  want  of  rigidity  of 
connection  between  bob  and  wire.  The  necessity  for  the  latter  correction 
was  pointed  out  by  Laplace,  who  showed  that  the  two,  bob  and  wire,  could 
not  move  as  one  piece,  for  the  bob  acquires  and  loses  angular  momentum 
around  its  centre  of  gravity,  which  cannot  be  accounted  for  by  forces 
passing  through  the  centre,  such  as  would  alone  act  if  the  line  of  the  wire, 
produced,  always  passed  through  the  centre.  In  reality  the  bob  turns 
through  a  slightly  greater  angle  than  the  wire,  so  that  the  pull  of  the  wire 
is  now  on  one  side  and  now  on  the  other  side  of  the  centre  of  gravity. 
The  correction  is,  however,  small  if  the  bob  has  a  radius  small  in  comparison 
with  the  length  of  the  wire. 

If  I  is  the  length  of  the  wire,  r  the  distance  of  the  centre  of  gravity  of 
the  bob  from  the  point  at  which  the  wire  is  attached  to  it,  and  k  the  radius 
of  gyration  of  the  bob  about  an  axis  through  the  centre  of  gravity ;  then, 
neglecting  higher  powers  than  k^  the  equivalent  simple  pendulum  can  be 
shown  to  be 

I  +  r     r{l  +  rf 

the  last  term  being  due  to  the  correction  under  consideration.  As  an 
illustration,  suppose  the  bob  is  a  sphere  of  1  inch  radius  and  the  wire 
is  38  inches  long ;  then  the  equivalent  simple  pendulum  in  inches  is 
39  + •010256  + -000102,  and  the  last  term,  1/400000  of  the  whole  length, 
need  only  to  be  taken  into  account  in  the  most  accurate  work. 

Bessel  also  made  a  very  important  change  in  the  air  correction.     The 
effect  of  the  air  on  the  motion  may  be  separated  into  three  parts — 

(1)  The    buoyancy,    the    weight    of    the    pendulum    being    virtually 
decreased  by  the  weight  of  the  air  which  it  displaces. 

(2)  The  flow  of  the  air,  some  of  the  air  moving  with  the  pendulum, 
and  so  virtually  increasing  its  mass. 

account  of  experiments  for  determining  the  length  of  the  pendulum  vibrating 
seconds  in  the  latitude  of  London,"  and  in  a  paper  in  the  Phil.  Trans,  for  1819, 
"  Experiments  for  determining  the  variations  in  the  length  of  the  pendulum 
vibrating  seconds,"  Kater  applies  further  corrections  and  gives  the  above  value. 

*  The  "  Toise  of  Peru  "  was  a  standard  bar  at  the  Paris  Observatory,  6  Paris  feet 
or  about  J949  millimetres  long. 


14  PROPERTIES  OF  MATTER. 

(.3)  The  air  drag,  a  viscous  resistance  which  comes  into  play  between 
the  diflerent  layers  of  air,  moving  at  ditferent  rates,  a  resistance  trans- 
mitted to  the  pendulum. 

As  far  back  as  17SG  Du  Buat  had  pointed  out  the  existence  of  the 
second  effect,  and  had  made  experiments  with  pendulums  of  the  same 
lenijth  and  form,  but  of  different  densities,  to  determine  the  extra  mass  for 
various  shapes.  Bessel,  not  knowing  Du  Buat's  work,  reinvestigated  the 
matter,  and  again  by  the  same  method  determined  the  virtual  addition  to 
the  mass  for  various  shapes,  and  among  others  for  the  pendulum  he  used. 

The  viscous  resistance  was  first  placed  in  its  true  relation  by  Stokes' 
investigations  on  Fluid  Motion  in  1847.  In  pendulum  motion  we  may 
regard  it  as  tending  to  decrease  the  amplitude  alone,  for  the  effect  on  the 
time  of  vibration  is  inappreciable.  We  may  represent  its  effect  by 
introducing  a  term  proportional  to  the  velocity  in  the  equation  of  motion, 
which  thus  becomes 

d  +  rd+fid^O 
The  solution  of  this  is  0  =  Ae  ~  ^cos  j  \  fi  -~t-  a  \ 
where  A  and  a  are  con.stants. 

9 

The  period  is  T  =»    — ^---__  where  v  depends  on  the  viscosity. 

Approximately!  =    —ll  +  —\  or  the   time   is   increased   by  the 

viscosity  in  the  ratio  1  +  -|1 : 1, 

or  since  fi  =  -y^  (nearly),  in  the  ratio  1  +         :  1. 

To  see  the  order  of  this  alteration,  suppose  that  p^  p^  represent  two 
succeeding  amplitudes  on  opposite  sides  of  the  centre — i.e.,  values  for  which 

5' =  0,  or  cos  ( "^ '^    J        "  .7')^^'  then  p,  =  eT  or,  taking  logarithms, 

iogf^'=x=?:? 

Now  in  one  of  Eater's  experiments  the  arc  of  swing  decreased  in 
about  500  seconds  from  1-41°  to  1 -18°,  or  in  the  ratio  MDo  :  1. 

Then /PiV''^  1-195  and  500X=log,M95  =  0-178 

whence  X  =  -000356  and  -^  =  2^J_  =  6  x  1 0-» about. 

In  Borda's  pendulum  the  elFect  was  about  the  same — i.e.,  one  that  ia 
practically  quite  negligible. 


THE  ACCELERATION  OF  GRAVITY.  15 

Bessel  also  used  the  pendulum  to  investigate  afresh  the  correctness  of 
Newton's  proof  that  mass  is  proportional  to  weight,  carrying  out  a  series 
of  experiments  which  still  remain  the  best  on  the  subject.  But  Bessel's 
chief  contribution  to  gravitational  research  consisted  of  his  theory  of  the 
'*  reversible  pendulum."  He  showed  that  if  a  pendulum  were  made 
symmetrical  in  external  form  about  its  middle  point,  but  loaded  at  one 
end,  to  lower  the  centre  of  gravity,  and  provided  with  two  knife  edges, 
like  Kater's  pendulum,  one  very  nearly  at  the  centre  of  oscillation  of  the 
other,  the  length  of  the  seconds  pendulum  could  be  deduced  from  the 
two  times  without  regard  to  the  air  efiects.  Laplace  had  shown  that  the 
knife  edges  must  be  regarded  as  cylinders,  and  not  mere  lines  of  suppoit. 
Bessel  showed,  however,  that  if  the  knife  edges  were  exactly  equal 
cylinders  their  efiect  was  eliminated  by  the  inversion,  and  that  if  they  were 
different  cylinders  their  effect  was  eliminated  by  interchanging  the  knife 
edges,  and  again  determining  the  times  from  each — the  "  erect "  and 
*'  inverted  "  times  as  we  may  conveniently  term  them. 

We  shall  consider  these  vai'ious  points  separately. 

In  the  first  place,  Bessel  showed  that  it  was  unnecessary  to  make  the 
erect  and  inverted  times  exactly  equal.  For  if  Tj  and  T^  be  these  times, 
if  Aj  h^  be  the  distances  of  the  centre  of  gravity  from  the  two  knife  edges, 
and  if  k  be  the  radius  of  gyration  round  an  axis  through  the  centre  of 
gravity,  the  formula  for  the  compound  pendulum  gives 


Multiply  respectively  by  h^,  A^,  subtract  and  divide  by  h^  -  h^  and  w9 
have 

Let  us  put  "'  }      fiii-  =  T^ 

We  shall  term  T  the  computed  time.  We  see  that  it  is  the  time  corresponding 
to  a  length  of  simple  pendulum  Aj  +  h^  It  may  be  expressed  in  a  more 
convenient  form,  thus : 

Let  T2  =  ii-±i:^      and  a^  =  ii_-i2. , 

2  2 

then  Tj-  =  r-  +  «',  T/  ==  t-  -  a^,  and  substituting  in  T^  we  get 

_ A,T,^ -  A3T/ _^,^^^,K  +  h,_ T,^  +  T/ ^ T,^ - T/  h,  +  h. 

Now  h^  +  K  is  measurable  with  great  exactitude,  but  Aj  and  Aj,  and 
therefore  A,  -  A.„  cannot  be  determined  with  nearly  such  accuracy.  The 
method  of  measuring  them  consists  in  balancing  the  pendulum  in  horizontal 
position  on  a  knife  edge  and  measuring  the  distance  of  the  balancing  knife 
edge  from  each  end  knife  edge.  But  the  formula  shows  that  it  is  not 
necessary  to  know  A,  -  h.,  exactly,  for  it  only  occurs  in  the  coefficient  of 
T,-  -  T/,  which  is  a  very  small  fraction  of  T,^  +  T/.  Knowing,  then,  A,  +  A, 
exactly  and  A,  -  h^  approximately,  we  can  conjpute  the  time  corresponding  to 


16 


PROrKRTIES  OF  MATTER. 


h^  +  Aj  from  the  times  in  theei-ect  and  inverted  positions  and  avoid  the  trouble- 
some series  of  trials  which  Kater  made  before  obtaining  exact  equality  for 
them  from  each  knife  edge. 

Now  let  us  consider  the  air  effect.  Take  first  the  erect  position  of  the 
pendulum.  We  may  represent  the  buoyancy  by  an  upward  force  applied 
at  the  centre  of  gravity  of  the  displaced  air,  and  equal  to  its  weight  iwj. 
Let  tliis  centime  of  gravity  be  distant  s  from  the  centre  of  suspension. 

The  mass  of  air  flowing  with  the  pendulum  will  have  no  effective  weight, 
since  it  is  buoyed  up  by  the  surrounding  air.  It  is  merely  an  addition  to 
the  mass  moved  and  serves  to  increase  the  moment  of  inertia  of  the 
pendulum.  Let  us  represent  it  by  the  addition  of  a  term  m'd-  when  the 
pendulum  is  erect. 

Then  we  have  ^^\  - ^^^V  +  ^)  +  "^'^^^ - ^^^^' t  f)  +  "^'^^Yl  +."'^ 


47r- 


MA,  —  ms 


M/i. 


MA, 


Aj"  +  K'        Aj 


K 


-  +  <•  ms      md? 


m 


neglecting  squares   and   products  of  ~  and 


MA, 

^,   since  in    practice  these 

quantities  are  of  the  order  10"*. 

Now  invert  and  swing  from  an  axis  near  the  centre  of  oscillation. 
The  value  of  m  is  the  same,  but  its  centre  of  gravity  may  be  at  a  different 
distance  from  the  new  suspension,  say  s'.  The  air  moving  may  be  different, 
so  that  we  must  now  put  m"d'-  instead  of  m'd'\     We  have  then 


9V 
47r» 


_A/  +  «^_^AJ 


A„ 


+  (c-  ms 

~~ma: 


A, 


+ 


m"<P 
MAT 


If    we    put    hJi.^  =  K-  as    an  approximation  in  the  coefficients  of  the 


m 


small  terms  containing  —  the  computed  time  T  is  given  by 


4«^ 


477- 


A,T,- 


AX 


2\ 


K 


■.h^  +  h.^  +  h  +  ^'-^ '"'' 


ms'     (vi  -  m")d? 


Aj  -  A,        M  M(Aj  -  A,) 


But  if  we  make  the  external  form  of  the 
pendulum  symmetrical  about  its  middle  point, 
so  that  the  two  knife  eilges  are  equidistant 
from  the  centre  of  figure,  then  s  =  s'  and  m 


■  m 


and  JLt=  =  A,  +  A, 


iTT 


Fio.  2. — Effoct  of  cylindrical 
Form  of  Knife  Edge. 


Then  the  air  efTect  is  eliminated  in  the 
computed  time.  It  is  necessary  here  that  the 
barometer  and  thermometer  should  give  the 
same  readings  in  each  observation ;  if  not, 
corrections  must  be  made ;  but,  as  they  will  be 
very  small,  an  exact  knowledge  of  their  value  is 
unnecessary. 

In  investigating  the  effect  of  the  cylindrical 
form  of  the  knife  edges  we  shall  for  simplicity 
suppose  them  each  to  have  constant  curvature, 


THE  ACCELERATION  OF  GRAVITY.  17 

the  radius  of  the  erect  one  being  p^,  that  of  the  inverted  one  p^.  If  C,  Fig.  2, 
is  the  centre  of  curvature  of  the  knife  edge,  0  the  point  of  contact,  G  the 
centre  of  gravity,  then  CG  =h^  +  p^  and  the  work  done  is  the  same  as  if 
G  were  moved  in  a  circle  of  radius  h^  +  p^,  since  the  horizontal  travel  of  G 
does  not  ."fleet  the  amount  of  work.  The  instantaneous  centre  of  motion 
is  the  point  of  contact  0.     The  kinetic  energy  is  therefore 

M(k2  +  0G=)|' 

But  OG2  =  OC=  +  CG=-20CCGcos5 

-pr  +  (Pi  +  ^h)^-Wp.  +  ^:)(l-|)^PProximatcly   -  (^  ^ '^''^  V  U  '  *  "' 

■1  h^^  neglecting  p^h^O-  and  smaller  quantities. 

Then  the  kinetic  energy  is  M(Aj-  +  k")-  . 

The  work  done  from  the  lowest  point  is 

M^  (A,  +  Pi)  (1  -  cos  6)  =  Mg  {h,  +  p,)  ^ 

Hence  the  erect  time  is  given  by  * 

gTlJ>l±l^hfj^L_p,\^^ 
47r*      hi  +  Pi  A,     \       hj 

the  inverted  time  is  given  by 

47r  h,     \       K 

In  the  computed  time  we  may  put  K-  =  hfi^\n  the  coefficient  of  the  small 
quantities  p,  and  p,,  and  therefore 

7-5-73 — I — I —   -^^1  +  ^2  +  7 — r\p2    Pi) 


47r2     47r-\      k^-h^      J  K'K 

Now  interchange  the  knife  edges.  Assuming  that  no  alteration  is 
made  except  in  the  interchange  of  p,  and  p^,  the  computed  time  T'  is 
given  by 

adding  the  two  last  equations  together  and  dividing  by  2, 

47r-         2  '        ' 

*  If  in  simple  harmonic  motion  the  kinetic  energy  at  any  point  is  i«5-  and  the 
work  from  the  centre  of  swing  is  Ihff-,  then  the  periodic  time  is   easily  seen  to  be 


'Wi 


A 


18 


PROPERTIES  OF  MATTER. 


Repsold's  Pendulum. — Bessel  did  not  himself  construct  a  peudulum 
to  fulfil  these  conditions,  but,  after  his  death,  llepsold  in  1860  devised  a  form 
with  interchangeable  knife  edges  and  of  symmetrical 
form  now  known  as  Repsold's  Reversible  Pendulum 
(Fig.  3),  in  which  he  carried  out  Bessel's  suggestions. 
The  stand  for  the  instrument  was,  perhaps  fortunately, 
far  from  sufficiently  firm,  for  as  the  pendulum  swung 
to  and  fro  the  stand  swung  with  it.  Attention  was 
directed  to  the  investigation  of  the  source  of  error. 
Its  existence  was  already  known,  but  its  magnitude 
was  not  suspected  till  Peirce  and  others  showed  how 
seriously  it  might  affect  the  time. 

Yielding"  of  the  Support. — The  centre  of  gravity 
moves  as  if  all  the  forces  acted  on  the  whole  mass 
collected  there,  so  that  if  we  find  the  mass  acceleration 
of  the  centre  of  gravity,  and  subtract  the  weight, 
Mg,  we  have  the  force  due  to  the  support.  Reversing, 
we  have  the  force  on  the  support. 

The  acceleration  of  the  centre  of  gravity  is  h^d 
along  the  arc  and  /i^d'  towards  the  point  of  support. 
Resolving  these  horizontally  and  vertically, 

horizontal  acceleration 


=  h^6  cos  6  -  h^d'  sin  6  =  h^6  approximately ; 

vertical  acceleration 
Fig.  3.— Repsold's  Ee- 
versibie  Pendulum.  =  ^  ^sin  0  +  hJ' COS 6  =  hJd  +  hJ' approximately  ; 

The    Russian    Pen-  '  '  i  i        i  r  j  t 


duluin  used   in  the 
Indian  Survey. 


but   ^=  - 


gh^e 


Then  the  horizontal  force  on  the  stand  is  M^ — ' — 0 

h{  +  K^ 

=  Mcr- — L_     since  K-  =  h,h., 
If  a  is  the  amplitude  of  0,  then  6-  =  -^- {a-  -  0=) 
and  the  vertical  force  upwards,  on  the  pendulum 

Now  in  finding  the  yielding  of  the  stand  we  only  want  the  varying 
part  of  this.     Reversing  it,  the  variation  in  the  force  on  the  stand 

which  is  of  the  second  order  in  0,  and  it  can  be  shown  that  the  effect  on 
the  time  of  swing  is  negligible  in  comparison  with  that  of  the  horizontal 
yielding. 


THE  ACCELERATION  OF  GRAVITY. 


19 


Let  the  yielding  to  a  horizontal  force  be  e  per  dyne.     Let  OC  (Fig.  4) 
be  the  vertical  position,  AG  the  position  when  displaced  through ang'e  0. 


m;?.. 


Then  the  yielding  OA    =e— — l 


ge 


ML 


produce  GA  to  0',  then  OO'  =  OA/0  =e^— J-*7  =  f^i  say, 

or  the  instantaneous  centre  is  raised  cZ,  above  O,  and  the 
centre  of  gravity  is  moving  in  a  circle  of  radius 


Let  the  instantaneous  centre  be  raised  cL  =  e 


when  the  pendulum  is  inverted. 
Hence  the  erect  time  is  given  by 


Fig.  4.— Yielding 
of  the  Support. 


4^  /ii  +  c^j  h^  +  d^ 


the  inverted  time  by 


4»»  h,  +  d. 


"2    '    ^"2 


K  +  ^h 


and  the  computed  time  by  ^-  =  h^  +  h^-\-eM.g,  since  h^d^  =  \dy 

We  see  that  eM^  is  the  horizontal  displacement  of  the  support  due  to 
the  weight  of  the  pendulum  applied  horizontally. 

Defforgfes'  Pendulums. — Starting  from  this  point,  Commandant 
Deftbrges  has  introduced  a  new  plan  to  eliminate  the  effect  of  yielding, 
using  two  convertible  pendulums  of  the  Repsold  type,  of  equal  weight,  of 
different  lengths,  and  with  a  single  pair  of  knife  edges,  which  can  be  trans- 
ferred from  one  to  the  other.  The  ratio  of  /i,  :  A,  is  made  the  same  for 
each. 

Let  the  radii  of  curvature  of  the  knife  edges  be  denoted  by  f),  p^,  let 
Ji  j^h_,  =  \  refer  to  the  first  pendulum,  h\  +  h'.,  =  I.^  refer  to  the  second. 

The  efiect  of  yielding  is  the  Fame  for  each,  increasing  the  length  by  2. 

Let  T  T'  be  their  computed  times, 


then 


and 


K  -  K 


20  PROPERTIES  OF  MATTER. 

7  7     / 

since    -l  =  -i^  the  co-efEcient  of  p^  -  p,  disappears,  and  it  is  not  necessary 

to  interchange  the  knife  edges  on  the  same  penduhim.     Hence   the  pen- 
dulums ai-e  convertible,  and  we  have 

£-^(T^-T'0  =  ?,-;, 

The  United  States  Coast  and  Geodetic  Survey  have  recently  constructed 
a  pendulum  in  which  the  planes  are  on  the  pendulum  and  the  knife  edges 
on  the  support.  The  one  disadvantage  is  the  difficulty  of  so  suspending 
the  pendulum  that  the  same  part  of  the  plane  is  always  on  the  knife  edge, 
but  against  this  is  to  be  set  the  probable  greater  accuracy  of  moasure- 
meiit  of  ^1  + /tj  and  the  freedom  from  the  necessity  of  interchange  of 
knife  edge.  Further,  should  a  knife  edge  be  damaged  it  can  be  reground 
without  affecting  the  pendulum,  whereas  in  the  ordinary  construction 
regrinding  really  alters  the  pendulum,  which  practically  becomes  a 
different  instrument. 


Variation  of  Gravity  over  the  Surface  of  the  Earth. 

Richer. — The  earliest  observation  showing  that  gravity  changes  with 
change  of  place  was  made  by  Richer,  at  the  request  of  the  French 
Academy  of  Sciences,  in  1672.  He  observed  the  length  of  the  seconds 
pendulum  at  Cayenne,  and  returning  to  Paris  found  that  the  same 
pendulum  must  there  be  lengthened  \\  Paris  lines,  12  to  the  inch. 

Newton's  Theory. — This  observation  waited  no  long  time  for  an 
explanation.  Newton  took  up  the  subject  in  the  Princ'qna  (Book  IH., 
Props.  18-20)  and,  regarding  gravity  as  a  terrestrial  example  of  uni- 
versal gravitation,  he  connected  the  variation  with  the  form  of  the 
earth.  He  showed  first  that  if  the  earth  is  taken  as  a  homogeneous 
mutually  gravitating  fluid  globe,  its  rotation  will  necessarily  bring  about  a 
bulging  at  the  Equator,  for  some  of  the  weight  of  the  equatorial  portion 
■will  be  occupied  in  keeping  it  moving  in  its  daily  circle  while  the  polar 
part  has  but  little  of  such  motion.  A  column,  therefore,  from  the  centre 
to  the  surface  must  be  longer  at  the  Equator  than  at  the  Pole  in  order 
that  the  two  columns  shall  pioduce  equal  pressures  at  the  centre.  Assimiing 
the  form  to  be  spheroidal,  the  attraction  will  be  different  at  equal  distances 
along  the  polar  and  equatorial  radii.  Taking  into  account  both  the 
variation  in  attraction  and  the  centrifugal  action  (o^-y-  of  gravity  at  the 
Equator),  Newton  calculated  the  ratio  of  the  axes  of  the  spheroid.  Though 
his  method  is  open  to  criticism,  his  result  from  the  data  used  is  perfectly 
correct,  viz.,  that  the  axes  are  as  230  :  229.  Taking  a  lately  measured 
value  of  1"  of  latitude,  he  found  thence  the  radii,  and  determined  their 
difference  at  17"!  miles.  He  then  found  how  gravity  should  vary  over 
such  a  sphei'oid,  taking  centrifugal  action  into  account,  and  prepared 
a  table  of  tlie  lengths  of  1°  of  latitude  and  of  the  seconds  pendulum 
for  every  5°  of  latitude  from  the  Equator  to  the  Pole.     From  his  table 


THE  ACCELERATION  OF  GRAVITY. 


21 


the  pendulum  length  at  Cayenne,  in  latitude  4°  55',  should  be  1  line 
less  than  at  Paris  in  latitude  48°  50'.  He  assigns  part  of  the  difference 
of  this  from  the  diminution  of  I-^  lines  observed  by  Richer  to  expansion 
of  the  scale  with  higher  temperature  near  the  Equator. 

The  Swedish  and  Peruvian  Expeditions.— Newton's  theory  of 

the  figure  of  the  earth  as  depending  on  gravitation  and  rotation  led  early 
in  the  eighteenth  centuiy  to  measurements  of  a  degree  of  latitude  in  Peru 
and  in  Sweden.  If  the  earth  were  truly  spheroidal,  and  if  the  plumb- 
line  were  eveiywhere  perpendicular  to  the  surface,  two  such  measurements 
would  sutiice  to  give  the  axes  a  and  h,  inasmuch  as  length  of  arc  of  1° 


=  .(r- 


f +  36sin'^^^3G00sinl"  where  6  =  ^^: 
2    J  a 


the  ellipticity  and  X  X' 


are  the  latitudes  at  the  beginnincf  and  end  of  the  arc* 

^^'^e  know  now  that  through  local  variations  in  gravity  the  plumb-line 
is  not  perpendicular  to  a  true  spheroid,  but  that  there  are  humps  and 
hollows  in  the  surface,  and  many  measurements  at  difierent  parts  of  the 
earth  are  needed  to  eliminate  the  local  variations  and  find  the  axes  of  the 
spheroid  most  nearly  coinciding  with  the  real  surface.  But  the  Swedish 
and  Peruvian  expeditions  clearly  proved  the  increase  of  length  of  a  degree 
in  northerly  regions,  and  so  proved  the  flattening  at  the  Poles.  These 
expeditions  have  another  interest  for  us  here  in  that  pendulum  observa- 
tions were  made.  Thus  Maupertuis,  in  the  northern  expedition,  found 
that  a  certain  pendulum  clock  gained  59 "1  seconds  per  day  in  Sweden  on 
its  rate  in  Paris,  while  Bouguer  and  La  Condamine,  in  the  Peruvian 
expedition,  found  that  at  the  Equator  at  sea-level  the  seconds  pendulum 
was  1"26  Paris  lines  shorter  than  at  Paris.  Bouguer's  work,  to  which  we 
have  already  referred,  was  especially  important  in  that  he  determined  the 
length  of  the  seconds  pendulum  at  three  elevations :  (1)  At  Quito,  which 
may  be  regarded  as  a  tableland,  the  station  being  14GG  toises  t  above  sea- 
level  ;  (2)  on  the  summit  of  Pichincha,  a  mountain  rising  above  Quito  to 
a  height  of  2434  toises  above  sea-level;  and  (3)  on  the  Island  of  Inca,  on 
the  river  Esmeralda,  not  more  than  thirty  or  forty  toises  above  sea-level. 
The  Equator  runs  between  Quito  and  the  third  station,  and  they  are  only 
a  few  miles  from  it.  In  space  free  from  matter  rising  above  sea-level  gravity 
might  be  expected  to  decrease  according  to  the  inverse  square  law  starting 
from  the  earth's  centre,  so  that  if  h  is  the  height  above  sea-level  and  r  is 


station. 

Above 

Sea-level 
in  Toises. 

Observed 

Seconds 

Pendulum 

in  Lines. 

Correction 
for  Tem- 
perature. 

Correction 

for 
Buoyancy. 

Corrected 

Seconds 
Pendulum. 

Fraction 
less  than 
at  Sea- 
level. 

Fraction 

jriven  by 

Inverse 

Square 

Law  2  hfr. 

Pichincha    . 
Quito  . 
Isle  of  Inca 

2434 
14G6 

438-70 
438-83 
430-07 

-•05 
-f-075 

-I--04 
-t--05 
-f -06 

438-69 
438-88 
439-21 

TaST 

1 

*  Airy,  "  Figure  of  Earth."  Fncyc.  Met.,  p.  192. 
■|-  The  toise  is  6  Paris  feet,  or  6-395  English  feet. 


22  PROPERTIES  OF  MATTER. 

the  earth's  radius,  the  decrease  should  be  ^hjr  of  the  original  value.  In 
the  table  on  p.  21,  Bouguer's  results  are  given.  In  the  last  column  but 
one  is  the  decrease  observed  at  the  upper  stations,  and  in  the  last  column 
the  decrease  calculated  by  2A/?'. 

It  will  be  seen  that  gravity  decreased  more  slowly  than  by  the  inverse 
square  law.  Centrifugal  force  would  act  in  a  contrary  way,  though,  as 
Bouguer  showed,  by  a  negligible  amount.  The  excess  of  gravity,  as 
observed,  above  its  value  in  a  fi'ee  space  must  therefore  be  assigned  to  the 
attraction  of  the  matter  above  the  sea-level.  Bouguer  obtained  for  the 
value  of  gravity  g^  on  a  plateau  of  height  h,  as  compared  with  its  value  at 
sea -level  g, 

where  I  is  the  density  of  the  plateau  and  A  the  density  of  the  earth. 

This  formula,  now  known  as  Bouguer's  Rule,  seems  to  have  dropped 
out  of  sight  till  it  was  again  obtained  by  Young  in  1819,  but  on  its 
revival  it  was  generally  employed  to  reduce  the  observed  value  at  a  station 
to  the  sea-level  value  in  the  same  latitude. 


Putting  it  in  the  form  ?^ — ^  =  !^/l  _  ?  i\ 

g,  r\       4.  A/ 


3993 
and  using  the  values  at  Quito  and  sea-level,  A=    '.  '  Z 

850 

Bouguer  remarked  that  this  result  sufficed  to  show  that  the  density  of 
the  earth  was  greater  than  that  of  the  Cordilleras,  and  consequently  that 
the  earth  was  neither  hollow  nor  full  of  water,  as  some  physicists  had 
maintained.  We  now  know  that  the  value  of  A  f^o  obtained  is  far  too  great, 
and  shall  see  later  what  is  the  probable  explanation. 

CiairautS  Theorem. — In  1713  Clairaut  published  his  great  treatise, 
Theorie  de  la  Figure  de  la  Terre,  which  put  the  investigation  of  the  figure  of 
the  earth  on  lines  which  have  ever  since  been  followed.  In  this  work  he 
takes  the  surface  of  the  earth  as  a  spheroid  of  equilibrium — i.e.,  such  that  a 
layer  of  water  would  spread  all  over  it,  and  assumes  that  the  internal  density 
varies  so  that  layers  of  equal  density  are  concentric  co-axial  spheroids. 
Denoting  gravity  at  the  Equator,  Pole,  and  latitude  X,  by  g^,  g^,  g\  respec- 
tively, and  putting  in  =  centrifugal  force  at  Equator  jg^  and  e  =  ellipticity  = 
difference  of  equatorial  and  polar  radii  /  equatorial  radius,  he  shows  (1)  that 

where  w  is  a  constant :  (2)  that 
From  (1)  and  (2)  we  get 


5'A  =  5'.|l  +  Uw-ejsin-xj, 


a  result  known  as  Clairaut's  Theorem. 

Laplace  showed  that  the  surfaces  of  equal  density  might  have  any 


THE  ACCELERATION  OF  GRAVITY.  23 

nearly  sphei-ical  form,  and  Stokes  [Math.  Phys.  Papers,  vol.  ii.  p.  104), 
going  further,  showed  that  it  is  unnecessary  to  assume  any  law  of  density 
so  long  as  the  external  surface  is  a  spheroid  of  equilibrium,  for  the  theorem 
gtill  remains  true. 

From  Clairaut's  Theorem  it  follows  that,  if  the  earth  is  an  oblate 
spheroid,  its  ellipticity  can  be  determined  from  pendulum  experiments  on 
the  variation  of  gravity  without  a  knowledge  of  its  absolute  value,  except 
in  so  far  as  it  is  involved  in  m.  And  if  the  theorem  were  exactly 
true,  two  relative  determinations  at  stations  in  widely  different  latitudes 
should  suffice.  But  here  again,  as  with  arc  measurements,  local  variations 
interfere,  and  many  determinations  must  be  made  at  widely  scattered 
stations  to  eliminate  their  effect. 

Kater  and  Sabine.  Invariable  Pendulums.— During  the  last  half 

of  the  eighteenth  century  much  pendulum  work  was  carried  on,  but  hardly 
with  sufficient  accuracy  to  make  the  i-esults  of  value  now,  and  we  may  con- 
sider that  modern  research  begins  with  Kater,  who  constructed  a  number  of 
"invariable  pendulums,"  nearly  beating  seconds,  and  in  shape  much  like 
his  convertible  pendulum  without  the  reverse  knife  edge.  The  principle 
of  "  invariable  pendulum  "  work  consists  in  using  the  same  pendulum  at 
different  stations,  determining  its  time  of  vibration  at  each,  and  correcting 
for  temperature,  air  effect,  and  height  above  sea-level.  The  relative  values 
of  gravity  are  thus  known,  or  the  equivalent,  the  relative  lengths  of  the 
seconds  pendulum,  without  measuring  the  length  or  knowing  the  moment 
of  inertia  of  the  pendulum.  Kater  himself  determined  the  length  of  the 
seconds  pendulum  at  stations  scattered  over  the  British  Islands,  and 
Sabine,  between  1820  and  1825,  carried  out  observations  at  stations 
ranging  from  the  "West  Indies  to  Greenland  and  Spitzbergen.  About  the 
*ame  time  Freycinet  and  Duperry  made  an  extensive  series  ranging  far 
into  the  Southern  Hemisphere,  and  other  observers  contributed  observa- 
tions. Now,  though  different  pendulums  were  used,  these  series  over- 
lapped and  could  be  connected  together  by  the  observations  at  common 
stations  ;  and  Airy  in  1830  (Encyc.  Met.,  "  Figure  of  the  Earth  ")  deduced  a 
value  of  the  ellipticity  of  about  -g^- 

Breaking  down  of  Bouguer'S  Rule. — Subsequent  work  brought 
into  ever-increasing  prominence  the  local  divergences  from  Clairaut's 
formula,  and  it  gradually  became  evident  that  on  continents  and  on  high 
ground  the  value  of  gravity  was  always  less  than  would  be  expected  from 
Clairaut's  formula  when  corrected  by  Bouguer's  rule,  while  at  the  sea 
coast  and  on  oceanic  islands  it  was  greater. 

Indian  Survey. — Thus,  in  the  splendid  series  of  pendulum  ex- 
periments carried  out  in  connection  with  the  Indian  Trigonometrical 
Survey  between  1865  and  1875  [G.  T.  Survey  of  India,  vol.  v.)  the 
variations  were  very  marked.  In  these  experiments,  invariable  pen- 
dulums, Kater's  convertible  and  Eepsold's  reversible  pendulum  were  all 
used,  and  observations  were  made  by  Basevi  and  Heaviside  from  More,  on 
the  Himalayas,  at  a  height  of  15,427  feet,  down  to  the  sea-level.  The  series 
was  connected  with  others  by  swinging  the  pendulums  at  Kew  before 
their  transmission  to  India,  and  very  great  precautions  were  taken  to 
correct  for  temperature,  and  the  air  effect  was  eliminated  by  swinging  in 
a  vacuum.     At  More  the  defect  of  gravity  was  very  marked. 

Airy's  "  Hydrostatic  "  Theory.  Faye's  Rule.— Airy  {PMl.  Trans., 
1855,  p.    101)  had   already   suggested   that   elevated    masses   are   really 


21-  PROPERTIES  OF  MATTER. 

buoyed  up  by  matter  at  their  base  lii^liter  than  the  average ;  that  in  fact 
tliey  float  on  the  licjuid  or  more  probably  viscous  solid  interior  very  much 
as  icebergs  float  on  the  sea.  If  the  high  ground  is  in  equilibrium,  neither 
rising  nor  falling,  we  may  perhaps  regard  the  total  quantity  of  matter 
underneath  a  station  as  being  equal  to  that  at  a  station  at  sea-level 
in  the  same  latitude.     This  hydrostatic  theory  has  led  Faye  to  suggest 

that  the  term in  Bouguer's  rule  should  be  I'eplaced  by  a  term  only 

taking  into  account  the  attraction  of  the  excess  of  matter  under  the 
station  above  the  average  level  of  the  near  neighbourhood,  a  suggestion 
embodied  in  Faye's  rule. 

llecent  work  by  the  American  Survey  {Amer.  Joiirn.  Science,  March 
1896,  G.  R.  Putnam)  has  shown  that  on  the  American  continent  Faye's 
rule  gives  results  decidedly  more  consistent  than  those  obtained  from 
Bouguer's  rule. 

By  a  consideration  of  the  results  obtained  up  to  1880  by  the  pen- 
dulum,   Clarke    {Geodesy,    p.    350)  gives  as  the  value    of  the   ellipticity 

e  =  — -— r,  a  value  almost  coincidintr  with  that  obtained  from  measure- 

2l2-2±  1-5  ° 

ments  of  degrees  of  latitude.  Helmert,  in  1884,  gave  as  the  result  of 
pendulum  work  ,  and  we  may  now  be  sure  that  the  value  differs  very 

little  from . 

300  ^      . 

Helmert  {Theorieen  der  hoheren  Geoddsie,  Bd.  II.  p.  241)  also  gives 

as  the  value  of  g  in  any  latitude  X, 

g^  =  978-00(1  +  0-005310  sin'  \) 

and  this  may  be  taken  as  representing  the  best  results  up  to  the  present. 

Von  Sterneck's  Half-second  Pendulums.— The  labour  of  the 

determination  of  minute  local  variations  in  gravity  was  much  lessened  by 
the  introduction  by  von  Sterneck,  about  1880,  of  half -second  invariable 
pendulums,  and  his  improved  methods  of  observation  have  greatly  in- 
creased the  accuracy  of  relative  determinations  at  stations  connected  by 
telegraph. 

With  half  the  time  of  swing  the  apparatus  has  only  one-fourth  the 
linear  dimensions,  and  it  can  be  made  at  once  more  steady  and  more 
portable.  The  size  of  the  pendulum  being  thus  reduced — it  is  about 
10  inches  long — it  can  without  much  trouble  be  placed  in  a  chamber  which 
can  b3  exhausted  and  which  can  be  maintained  at  any  desired  temperature. 
Each  pendulum  can  therefore  be  made  to  give  its  own  temperature  and  air 
corrections  by  preliminary  observations.  The  form  of  the  pendulum  is 
shown  in  Fig.  5.  The  chief  improvements  in  the  mode  of  observation 
introduced  by  von  Sterneck  consist,  1st,  in  the  simultaneous  comparison 
with  the  same  clock  of  the  swinging  of  two  pendulums  at  two  stations  at 
which  gravity  is  to  be  compared.  For  this  purpose  the  two  stations  are 
connected  by  an  electric  circuit  containing  a  half-seconds  "break  circuit" 
chronometer,  which  sends  a  signil  through  each  station  every  half-second, 
and  thus  clock-rates  are  of  little  importance.  And,  2nd,  the  method  of 
observing  the  coincidences  of  the  pendulum  with  the  chronometer  signals. 
In  the  final  form  this  consists  in  attaching  a  small  mirror  on  the  pendulum 


THE  ACCELERATION  OF  GRAVITY. 


25 


knife  edge  (not  shown  in  Fig.  5,  which  represents  an  earlier  form)  per- 
pendicular to  the  plane  of  vibration  of  the  pendulum,  and  placing  a  fixed 
mirror  close  to  the  other  and  parallel  to  it  whan  the  pendulum  is  at  rest. 


Pia.  5. 


The  chronometer  signals  work  a  relay,  giving  a  horizontal  s  ">ark,  and  this 
is  reflected  into  a  telescope  from  both  mirrors.  When  the  pandulum  is  at 
rest  the  image  of  the  spark  in  both  mirrors  appears  on  the  horizontal 
cross-wire,  and  when  the  pendulum  is  vibrating  a  coinciilenc3  occurs  when 
the  two  images  are  in  this  position.      The  method  admits  of  exceedingly 


26 


PROPERTIES  OF  MATTER. 


accurate  determination.  We  shall  see  later  how  von  Sterneck  used  the 
method  in  fi^ravitation  experiments.  Here  it  is  suflicient  to  say  that  he 
has  used  it  in  many  local  determinations  of  gravity,  nnd  that  his  pendulums 
have  been  used  without  the  simultaneous  method  for  determinations  at 
various  stations  in  both  hemispheres.  The  American  Geodetic  Survey  has 
adopted  very  similar  apparatus  and  methods,  and  it  appears  probable  that 
we  shall  soon  have  a  knowledge  of  the  variation  of  gravity  over  the  surface 
of  tlie  earth  of  a  far  more  detailed  and  accurate  kind  than  could  possibly 
be  obtained  by  the  older  methods. 

Differential  Gravity  Meters. — Before  invariable  pendulums  were 
brought  to  their  present  accuracy  and  portability,  there  was  some  hope 
that  for  relative  determinations  the  pendulum  might  be  superseded  by  a 
statical  measurer  of  gravity  which  would  do  away  with  the  need  for  time 
measurements.  Such  an  instrument  must  essentially  consist  of  a  mass 
supported  by  a  spring,  and  the  variation  in  gravity  must  be  shown  by  the 
alteiation  in  the  spring  due  to  the  alteration  in  the  pull  of  the  earth  on 
the  mass.  The  earlier  instruments  devised  for  the  purpose  need  not  be 
described,  for  they  were  quite  incapable  of  the  accuracy  attained  by 
invariable  pendulums.  The  first  instrument  which  promised  any  real 
success  was  devised  by  von  Sterneck,  and  Ls  termed  by  him  the  Barymeter 
{Mittheilangen  des  K.  K.  Militar-Geog.  Inst.,  Wien,  v.  1885). 

Von  Sterneck's  Barymeter. 

1 0  — A  brass  plate  P  (Fig.  G),  i5U  cm.  x 
20  cm.,  is  balanced  on  a  knife  edge,  s. 
Along  a  diagonal  is  a  glass  tube 
terminating  in  bulbs 0 and  U,  5  cm.  x 
6  cm.,  so  that  in  the  equilibrium 
position  0  is  about  25  cm.  above  U. 
The  tube  and  about  i  of  each  bulb 
is  filled  with  mercury,  and  above  the 
mercury  is  nitrogen.  The  apparatus 
is  adjusted  so  that  at  0°  C.  and  for 
a  certain  value  of  gravity  the  edges 
of  the  brass  plate  are  horizontal 
and  vertical,  a  level  W  showing 
when  this  position  is  attained.  If  now  gravity  were  to  increase,  the 
weight  of  the  mercury  would  be  greater,  and  it  would  tend  to  flow  from  O 
and  compress  the  gas  in  U.  Thus  the  balance  would  tilt  over  to  the  left, 
and  the  tilting  still  further  increasing  the  pressure  on  U,  the  flow 
downwards  is  increased.  The  instrument  can  thus  be  made  of  any 
desired  sensitiveness,  and  its  deflections  can  be  read  by  scale  and  measured 
in  the  usual  way.  To  compensate  for  changes  of  temperature,  a  second 
tube  terminating  in  smaller  bulbs  o  and  u,  each  about  6  cm.  x  3  cm.,  is 
fixed  along  the  other  diagonal.  This  contains  some  mercury,  but  above 
the  mercury  in  u  is  alcohol,  and  only  o  contains  nitrogen.  If  the 
temperature  rises  the  mercury  becomes  less  dense,  and  on  this  account  it 
is  driven  from  U  to  O  in  the  larger  tube,  but  still  more  is  it  driven  in  this 
du-ection  from  the  fact  that  the  increase  of  pressure  of  the  gas  in  U  is 
greater  than  in  O.  Meanwhile,  the  alcohol  in  u  expanding,  drives  the 
mercury  in  the  smaller  tube  into  o,  and  by  .suitable  adjustments  of  volume 
the  two  can  be  made  to  balance  sufficiently  for  such  small  temperature 
variations  as  will  arise  when  the  whole  is  placed  in  a  box  surrounded  with 


Von  Sterneck's  Barymeter, 


THE  ACCELERATION  OF  GRAVITY. 


27 


melting  ice,  and  it  is  thus  that  the  instrument  is  used.  With  this 
instrument  von  Sterneck  could  detect  the  change  in  gravity  in  going  from 
the  cellar  of  a  building  to  a  height  of  25  metres. 

Threlfall  and  Pollock's  Quartz-thread    Gravity  Balance.— 

In  the  ritil.  Trans.,  A.  r.)o,  18131),  p.  215,  Threlfall  and  l^llock  describe  an 
instrument  for  measuring  variations  in  gravity  statically  which  is  both 
accurate  and  portable. 

The  essential  features  of  the  instrument  are  represented  in  Fig.  7. 

A  and  B  are  two  metal  rods  which  can  slide  along  their  common 
axis.  C  is  a  coach-spring  attached  to  A .  H  I  is  a  quartz  thread  30-5  cm. 
long  and  •0038  cm.  in  diameter  stretched  horizontally  between  B 
and  C.  D  is  a  piece  of  gilded  brass  wire  soldered  to  the  quartz  thread. 
Its  weight  is  "018  gm.,  its  length  5%S  cm.,  and  its  centre  of  gravity  is 
a  little  to  one  side  of  the  quartz  thread.  Its  weight  therefore  tends 
to  pull  it  into  the  vertical  position  and  twist  the  quartz.     But  such  a  twist 


^ 


n 


im 


0" 


a 


[r~i 


r\ 


a 


G 


^ 


tr 

Fig.  7. — Tlirelfall  and  Pollock's  Quartz-thread  Gravity  Balance. 

can  be  put  on  the  quartz  thread  by  rotating  the  arm  G,  which  cariies  a 
vernier,  that  D  is  brought  into  the  horizontal  position.  For  this  about 
three  whole  turns  are  required.  The  end  of  D  when  in  the  horizontal 
position  is  on  the  cross-wire  of  the  horizontal  microscope  E.  The  hori- 
zontal position  of  the  brass  wire  is  only  just  stable.  If  it  be  twisted  a  few 
degrees  more  the  point  of  instability  is  reached  and  the  wire  tends  to 
continue  moving  round,  and  would  do  so  but  for  an  arrester.  The  mode 
of  using  the  instrument  consists  in  determiuing  the  twist  put  on  the  quai'tz 
thread  by  the  arm  G  to  bring  it  into  the  horizontal  position.  If  gravity 
increases,  the  moment  of  the  weight  of  D  increases  and  a  greater  twist  is 
required.  To  calibrate  the  insti'ument  the  change  in  reading  of  the  vernier 
on  G  is  observed  in  passing  from  one  station  to  another,  at  both  of  which 
g  is  known — the  two  stations  selected  being  Sydney  and  Melbourne.  Of 
course,  temperature  corrections  are  necessary  both  on  account  of  the  change 
in  length  of  D  and  the  change  in  rigidity  of  the  quartz.  Preliminary 
determinations  of  these  were  made  at  one  station.  For  the  details  of  the 
instrument  and  the  mode  of  using  it  we  refer  the  reader  to  the  original 
account.  It  suffices  here  to  say  that  it  has  given  very  fairly  consistent 
results  at  stations  wide  apart  and  that  it  promises  to  rival  the  invariable 
pendulum. 


CHAPTER  III. 
GRAVITATION. 

CoNTKNTS.* — The   Law   of   Gravitation — The  Gravitation  Constant  and  the  Mean 

Density  of  the  Earth. 

The  full  statement  of  Newton's  Law  of  Gravitation  is  that  any  particle 
of  mass  ]Mj  attracts  any  other  particle  of  mass  M.,  distant  d  from  it  with  a 
force  in  the  line  joining  them  proportional  to  MjM.,/(Z-.  The  evidence  for 
the  law  may  be  briefly  summed  up  as  follows : 

Starting  with  any  single  planet— say  the  earth — and  referring  its 
position  to  a  system,  fixed  relatively  to  the  sun  and  the  distant  stars,  direct 
astronomical  observation  shows  that  it  may  be  described  with  a  close 
approximation  to  the  truth,  as  moving  in  an  ellipse  with  the  sun  in  one 
focus,  at  such  speed  that  the  line  from  the  centre  of  the  sun  to  the  centre 
of  the  planet  sweeps  out  equal  areas  in  equal  times.  This  implies,  as 
Newton  showed,  that  the  acceleration  of  the  planet  is  towards  the  sun  and 
inversely  as  the  square  of  its  distance  from  that  body. 

Now,  comparing  the  diflerent  planet^^,  observation  shows  that  (length  of 
year)-/(niean  distance)'  is  the  same  for  each,  and  from  this  it  follows  that 
the  constant  of  acceleration  is  the  same  for  all,  or  that  at  the  unit  distance 
from  the  sun  they  would  all  have  the  same  acceleration  if  the  law  holding 
for  each  in  its  own  orbit  held  for  it  at  all  distances. 

So  far  this  is  mere  time-geometry,  or  a  description  of  position  and  rate 
of  change  of  position,  and  we  might  have  other  equally  true,  if  less 
convenient,  modes  of  description  referred  to  other  standards,  such  as  the 
epicyclic  geocentric  mode  of  the  ancients,  or  the  practical  mode  in  common 
use  in  which  the  co-ordinates  of  a  planet  are  measured  with  regard  to  some 
observatory,  its  meridian,  and  horizon. 

But  if  we  regard  the  accelerations  as  indicating  forces,  the  different 
methods  of  description  are  no  longer  equivalent.  We  must  select  that 
which  gives  a  system  of  forces  most  consistent  in  itself  and  most  in  accord 
with  our  terrestrial  experience.  Here  the  heliocentric  method,  with  the 
modification  described  hereafter,  is  immensely  superior  to  any  other,  and, 
adopting  it,  we  must  suppose  that  the  accelerations  of  the  planets  indicate 
forces  towards  the  sun,  and  since  the  constant  of  acceleration  is  the  same 
for  all,  that  the  forces  on  equal  masses  are  inversely  as  their  distances 
squared  from  the  sun,  whatever  planets  the  masses  belong  to.     In  other 

*  This  chapter  is  largely  taken  from  TJie  Mean  Density  of  the  Earth,  and  papers 
comnuinicated  to  the  Royal  Institution  and  the  Birmingham  Natural  History  and 
Vbilosophical  Society,  by  J.  H.  Poynting. 


GRAVITATION.  £f) 

words,  the  sun  has  no  favourite  among  its  attendants,  but  pulls  on  each 
pound  of  each  according  to  the  same  rule. 

But  the  assumption  that  the  accelerations  indicate  forces  of  the  kind 
we  experience  on  the  earth,  carries  with  it  the  supposition  of  equality  of 
action  and  reaction,  and  so  we  conclude  that  each  planet  reacts  on  the  sun 
with  a  force  equal  and  opposite  to  that  exerted  by  the  sun  on  the  planet. 
Hence,  each  acts  with  a  force  proportional  to  its  own  mass,  and  inversely 
as  the  square  of  its  distance  away.  If  we  suppose  that  there  is  nothing 
special  in  the  attraction  of  the  sun  beyond  great  magnitude  corresponding 
to  great  mass,  we  must  conclude  that  the  sun  also  acts  with  a  force  propor- 
tional to  its  mass.  But  we  have  just  shown  that  the  force  is  proportional 
to  the  mass  acted  on.  Hence,  we  have  the  force  on  any  planet  proportional 
to  mass  of  sun  x  mass  of  planet /(distance  apart)^. 

Now,  turning  to  any  of  the  smaller  systems  consisting  of  a  primary 
and  its  satellites,  the  shape  of  orbit  and  the  motion  of  the  satellites  agree 
with  the  supposition  that  the  primary  is  acting  with  a  force  according  to 
the  inverse  square  law.  It  is  important  for  our  special  problem  to  note 
here  that  in  the  case  of  the  earth  we  must  include  in  the  term  "  satellite  " 
any  body  at  its  surface  which  can  be  weighed  or  moved. 

We  are  therefore  led  to  conclude  that  the  law  is  general,  or  that  if  we  \ 
have  any  two  bodies,  of  masses  Mj  and  M^,  at  d  distance  apart,  the  force 
on  either  is 

GM,M, 

d^~ 
where  G  is  a  constant — the  constant  of  gravitation. 

The  acceleration  of  one  of  them,  say  M^,  towards  the  other  is   ■    ^  t 

d" 

If  this  conclusion  is  accepted,  we  can  at  once  determine  the  masses  of 
the  various  primaries  in  terms  of  that  of  the  sun  for — 

acceleration  of  satellite  towards  primary  =  G-, '■ — "    primary 

distance  of  satellite' 

and  acceleration  of  primary  towards  sun  =  G-        ^^^'^^ss  ot  sun 


distance  of  primary' 

By  division  G  is  eliminated,  and  we  obtain  the  ratio  of  the  masses  in  terms 
of  quantities  which  may  be  measured  by  observation. 

As  an  illustration,  let  us  make  a  rough  determination  of  the  mass  of 
the  sun  in  terms  of  the  mass  of  the  earth. 

We  may  take  the  acceleration  of  the  moon  to  the  earth  as  approxi- 
mately wji^  y^  d^^^  where  wm  is  the  angular  velocity  of  the  moon  and  (/„  its 
distance  from  the  earth,  and  the  acceleration  of  the  latter  to  the  sun  as 
WE-xc^ii  where  we  is  the  angular  velocity  of  the  earth,  and  d^  its  distance 
from  the  sun.     Let  the  mass  of  the  sun  be  S  and  that  of  the  earth  be  E. 

,  Acceleration  of  Moon  _  w^"  x  cZ  „  _  E  x  d^^ 

Acceleration  of  Earth      we'  x  d^      S  x  d^ ' 

whence     g  =  ^^  -  l^\i^J$^^\'=mm 


30  PROPERTIES  OF  MATTER. 

A  confirmation  of  the  generality  of  the  law  is  obtained  from  the 
perturbations  of  the  planets  from  the  elliptic  orbits  which  we  have  for 
simplicity  supposed  them  to  describe. 

These  perturbations,  in  any  one  planet,  can  at  least  approximately  be 
analysed  into  separate  disturbances,  each  due  to  one  of  its  fellow  planets, 
acting  with  a  foice  inversely  as  the  square  of  its  distance  away,  and  if  we 
assume  this  force  proportional  to  the  mass  of  the  disturber  we  obtain 
another  measure  for  this  mass  in  terms  of  that  of  the  sun. 

The  concoidance  of  the  two  methods  is  as  complete  as  we  could 
expect. 

The  determination  of  the  masses  of  the  different  members  of  our  system 
in  terms  of  that  of  the  sun  enables  us  to  choose  a  still  more  satisfactory 
origin  for  our  system  of  reference  than  the  centre  of  the  sun — viz.,  the 
centre  of  mass  of  the  whole  system.  The  change  is  small,  but  without  it 
we  could  not  account  for  all  the  motions  merely  by  a  set  of  inverse  square 
forces  in  which  action  and  reaction  were  equal  and  opposite. 

We  have  for  simplicity  considered  the  sun  and  planets  as  without 
appreciable  dimensions  as  compared  with  their  distances  apart.  But 
measurement  shows  that  they  are  all  approximately  spheres,  and  the 
attraction  on  a  sphere  with  density  varying  only  with  the  distance  from 
the  centre — i.e., consisting  of  homogeneous  concentric  shells,  if  itis  considered 
as  the  resultant  of  the  attractions  on  the  separate  particles,  all  according 
to  the  same  inverse  square  law,  is  the  same  as  that  on  the  whole  mass 
collected  at  the  centre  of  the  sphere.  Further,  if  the  attraction  is  due,  not 
to  the  attracting  body  as  a  whole  but  to  its  separate  parts,  each  acting,  as 
it  were,  independently  and  according  to  the  same  law,  then  an  attracting 
sphere  acts  as  if  it  were  all  concentrated  at  its  centre.  Since  the  planets, 
with  a  close  approximation,  behave  as  if  they  Avere  merely  concentrated 
masses  at  their  centres,  and  since  the  deviations  from  this  behaviour,  such 
as  the  earth's  precession,  can  all  be  accounted  for  by  their  departure  from 
sphericity,  we  have  strong  presumption  that  the  attraction  is  really  the 
resultant  of  all  the  attractions,  each  element  m^  of  one  body  acting  on  each 
element  m^  of  the  other  with  force  G')7i{nijd?. 

Astronomical  observation  enables  us,  then,  to  compare  the  masses  of 
the  various  members  of  the  solar  system  with  each  other,  and,  by  taking 
into  account  the  sizes  of  the  planets,  to  make  a  table  of  specific  gravities, 
choosing  any  one  as  the  standard  substance.  Thus,  if  we  take  the  earth 
as  standard,  the  mean  specific  gravity  of  the  sun  is  about  0"2.5,  that  of 
Mercury  about  1'25,  that  of  Venus  and  Mars  about  0'9,  and  so  on. 

But  this  does  not  give  us  any  idea  of  the  specific  gravity  in  terms  of 
known  terrestrial  substances  or  any  idea  of  the  masses  in  terms  of  the 
terrestrial  standards,  the  kilogramme  or  the  pound.  It  is  true  that  Newton, 
with  little  more  than  the  astronomical  data  at  his  command,  made  a 
celebrated  guess  on  the  specific  gravity  of  the  earth  in  terms  of  water, 
which  runs  thus  in  Motte's  translation  of  the  Principia  (vol.  ii.  p.  230, 
ed.  1729,  Book  III.,  Prop.  10) :  "  But  that  our  globe  of  earth  is  of  greater 
density  than  it  would  be  if  the  Avhole  consisted  of  water  only,  I  thus  make 
out.  If  the  whole  consisted  of  water  only,  whatever  was  of  less  density 
than  water,  because  of  its  less  specific  gravity,  would  emerge  and  float 
above.  And  upon  this  account,  if  a  globe  of  terrestrial  matter,  covered  on  all 
sides  with  water,  was  less  dense  than  water,  it  would  emerge  somewhere : 
and  the  subsiding  water  falling  back,  would  be  gathered  to  the  opposite 


GRAVITATION.  31 

side.  And  such  is  the  condition  of  our  earth,  which,  in  great  measure,  is 
covered  with  seas.  The  earth,  if  it  was  not  for  its  greater  density,  would 
emerge  from  the  seas,  and  according  to  its  degree  of  levity,  would  be  raised 
more  or  less  above  their  surface,  the  water  and  the  seas  flowing  backwarxls 
to  the  opposite  side.  By  tlie  same  argument,  the  spots  of  the  sun  which 
float  upon  the  lucid  matter  thereof,  are  lighter  than  that  matter.  And 
however  the  Planets  have  been  form'd  while  they  were  yet  in  fluid  masses, 
all  the  heavier  matter  subsided  to  the  centre.  Since,  therefore,  the  common 
matter  of  our  earth  on  the  surface  thereof,  is  about  twice  as  heavy  as 
water,  and  a  little  lower,  in  mines  is  found  about  three  or  four,  or  even  five 
times  more  heavy  ;  it  is  probable  that  the  quantity  of  the  whole  matter  of 
the  earth  may  be  five  or  six  times  greater  than  if  it  consisted  all  of  water, 
especially  since  I  have  before  shewed  that  the  earth  is  about  four  timeg 
more  dense  than  Jupiter." 

It  is  not  a  little  I'emarkable  that  Newton  hit  upon  the  limits  between 
which  the  values  found  by  subsequent  researches  have  nearly  all  lain. 

In  order,  then,  to  complete  the  expression  of  the  law  of  gravitation  we  | 
must  connect  the  celestial  with  the  terrestrial  scale  of  densities.  In  fact,  I 
we  must  do  for  the  masses  of  the  solar  system  that  which  we  do  for  their 
distances  in  the  determination  of  the  solar  parallax,  though  we  cannot 
proceed  quite  so  directly  in  the  former  case  as  in  the  latter  in  connecting 
the  celestial  and  terrestrial  measures.  If  we  could  measure  the  accele- 
ration, say,  of  the  moon,  due  to  any  terrestrial  body  of  known  shape 
and  density — if,  for  instance,  we  knew  the  form  and  extent  of  our 
tidal-wave  and  its  full  lunar  efiect— we  could  at  once  find  the  mass  of 
the  earth  in  terms  of  that  of  the  wave,  or  its  density  as  compared  with 
sea-watei". 

But  at  present  this  cannot  be  done  with  any  approach  to  accuracy,  and 
the  only  method  of  solving  the  problem  consists  in  finding  the  attraction 
between  two  bodies  on  the  earth  of  known  masses  a  known  distance  apart, 
and  comparing  this  with  the  attraction  of  the  earth  on  a  known  mass  at 
its  surface  instead  of  its  attraction  as  a  heavenly  body.  Since  the  law  of 
attraction  is  by  observation  the  same  at  the  surface  of  the  earth  and  at  a 
distance,  we  can  thus  find  the  mass  of  the  earth  in  terms  of  either  of  these 
known  masses. 

To  take  an  illustration  from  an  experiment  hereafter  described,  let  us 
suppose  that  a  spherical  mass  of  20  kilos,  is  attracted  by  another  spherical 
mass  of  150  kilos,  when  the  centres  are  30  cm.  apart  with  a  force  equal  to 
the  weight  of  ^  mgm.  or  s o  o  o^o  o  o  o  ^^  ^^^  weight  of  the  20  kilos,  when 
the  latter  is  on  the  surface  of  the  earth  and  6  x  10^  cm.  from  its  centre, 
we  have  :  (^  ^ 

Mass  of  Earth    1 50000  _.,  ^  ] 

(6  X  loy       '      30^         -^=¥ooooow         -- 

whence  mass  of  earth  =  5  x  10-"  grammes  nearly. 

The  volume  of  the  earth  is  about  9  x  10-''  c.c,  whence  the  mean  density 
of  the  earth  A  is  about  5'5. 

Or,  using  the  experiment  to  give  the  constant  of  attraction,  and 
expressing  the  masses  in  grammes,  the  weight  of  ^  mgm.  or 

AAAo-       Gx  150000x20000 
^30* 


.^2 


PROPERTIES  OF  MATTER. 


Whence,  if  ^  =  98;  G  = 


981  X -00025  x302_   7 
l5U0UUx20UU0    ~1U' 


(nearly). 


Due  South  of 
SuiDinil  onSlope 


I   .? 


2""  StaHon 
Due  We^fof 
First  Station 


A  determination  of  G  completes  the  expression  of  the  law  of 
gravitation. 

This  example  shows  that  the  two  problems,  the  determination  of  the 
gravitation  constant  G  and  the  determination  of  the  mean  density  of  the 
earth  A,  are  practically  one,  inasmuch  as  our  knowledge  of  the  dimensions 
of  the  earth  and  the  acceleration  of  gravity  g  at  its  surface  at  once 
enable  us  to  determine  G  if  we  know  A,  or  to  determine  A  if  we 
know  G. 

The  Methods  of  Experiment. 

These  naturally  fall  into  two  classes.  In  the  one  class  some  natural 
mass  is  selected,    either  a  mountain  or   part   of  the  earth's   crust,  and 

its  mass  and  form  are  more  or  less  accurately 
determined  by  surveys  and  mineralogical 
examination.  Its  attraction  on  a  plumb- 
bob  at  one  side,  or  on  a  pendulum  above  or 
below  it,  is  then  compared  with  the  attrac- 
tion of  the  whole  earth  on  the  same  body. 

In  the  other,  the  laboratory  class  of 
experiment,  a  smaller  mass,  such  as  may 
be  easily  handled,  is  placed  so  as  to  attract 
some  small  suspended  body,  and  this  attrac- 
tion is  measured.  Knowing  the  attracting 
and  attracted  masses,  the  attraction  gives  G. 
Or,  comparing  the  attraction  with  the  attrac- 
tion of  the  earth  on  the  same  body,  we  get  A. 

The  Experiments  of  Boug-uer  in 

Peru. — The  honour  of  making  the  first 
experiments  on  the  attraction  of  teri^estrial 
masses  is  to  be  accorded  to  Bouguer.  He 
attempted  both  by  the  pendulum  experi- 
ments described  in  the  last  chapter,  and  by 
plumb-line  experiments,  to  prove  the  exist- 
ence of  the  attraction  of  mountain  masses  in  the  Andes,  when  engaged  in 
the  celebrated  measurement  of  an  arc  of  the  meridian  in  Peru  about  the 
year  1740.  The  pendulum  experiments  are  sufliciently  described  in  the 
last  chapter. 

In  his  plumb-line  experiments  he  attempted  to  estimate  the  sideway 
attraction  of  Chimborazo,  a  mountain  about  20,000  feet  h'gh,  on  a  plumb- 
line  placed  at  a  point  on  its  side.  Fig.  8  will  show  the  principle  of  the 
method.  Suppose  that  two  stations  are  fixed,  one  on  the  side  of  the 
mountain  due  south  of  the  summit,  and  the  other  in  the  same  latitude, 
but  some  distance  westward,  away  from  the  influence  of  the  mountain. 
Suppose  that  at  the  second  station  a  star  is  observed  to  pass  the  meridian — 
we  will  say,  for  simplicity,  directly  overhead,  then  a  plumb-line  hung 
down  will  be  exactly  parallel  to  the  observing  telescope.  At  the  first 
station,  if  the  mountain  were  away,  it  would  also  hang  down  parallel  to 
the  telescope  when  dii-ectcd  to  the  same  star.  But  the  mountain  pulls  the 
plumb-line  towards   it,  and  changes  the   overhead  point  so  that  the  star 


J 


Fig.  8. — Bouguer's  Plumb-line  Ex- 
periment on  the  Attraction  of 
Cbimborazo. 


GRAVITATION.  33 

appears  to  northward  instead  of  in  the  zenith.  The  method  simply  con- 
sists in  determining  how  much  the  star  appears  to  be  shifted  to  the  north. 
The  angle  of  apparent  shift  is  the  ratio  of  the  horizontal  pull  of  the 
mountain  on  the  plumb- bob  to  the  pull  of  the  earth. 

To  carry  out  the  experiment,  Bouguer  fixed  the  first  station  on  the 
south  slope  of  Chimborazo,  just  above  the  perpetual  snow-line,  and  the 
second  nearly  on  the  same  level,  several  miles  to  the  westward.  He 
describes  {Figure  de  la  Terre,  7th  section)  how  his  expedition  reached  the 
first  station  after  a  most  toilsome  journey  of  ten  hours  over  rocks  and 
snow,  and  how,  when  they  reached  it,  they  had  all  the  time  to  fight  against 
the  snow,  which  threatened  to  bury  theu^  tent.  Nevertheless,  they 
succeeded  in  making  the  necessary  observations,  and  a  few  days  later  they 
were  able  to  move  on  to  the  second  station.  Here  they  hoped  for  better 
things,  as  they  were  now  below  the  snow-line.  But  their  dilficulties  were 
even  gi^eater  than  before,  as  now  they  were  exposed  to  the  full  force  of  the 
wind,  which  filled  their  eyes  with  sand  and  was  continually  on  the  point 
of  blowing  away  their  tent.  The  cold  was  intense,  and  so  hindered  the 
working  of  their  instruments  that  they  had  to  apply  fire  to  the  levelling 
screws  before  they  could  turn  them.  Still  they  made  their  observations, 
and  found  that  the  plumb-line  Avas  drawn  aside  about  8  seconds.  Had 
Chimborazo  been  of  the  density  of  the  whole  earth,  Bouguer  calculated, 
from  the  dimensions  and  distance  of  the  mountain,  that  it  w^ould  have 
drawn  aside  the  vertical  by  about  twelve  times  this,  so  that  the  earth 
appeared  to  be  twelve  times  as  dense  as  the  mountain,  a  result  undoubtedly 
veiy  far  wide  of  the  truth.  But  it  is  little  wonder  that  under 
such  circumstances  the  experiment  failed  to  give  a  good  result,  and  all 
honour  is  due  to  Bouguer  for  the  ingenuity  and  perseverance  which  enabled 
him  to  obtain  any  result  at  all.  At  least  he  deserves  the  credit  of  first 
showing  that  the  attraction  by  mountain  masses  actually  exists,  and  that 
the  earth,  as  a  whole,  is  denser  than  the  surface  strata.  As  he  remarks, 
his  experiments  at  any  rate  proved  that  the  earth  was  not  merely  a  hollow 
shell,  as  some  had  till  then  held  ;  nor  was  it  a  globe  full  of  water,  as  others 
had  maintained.  He  fully  recognised  that  his  experiments  were  mere 
trials,  and  hoped  that  they  would  be  repeated  in  Europe. 

Thirty  years  later  his  hope  was  fulfilled.  Maskelyne,  then  the 
English  Astronomer  Royal,  brought  the  subject  before  the  Royal  Society 
in  1772,  and  obtained  the  appointment  of  a  committee  "  to  consider  of  a 
proper  hill  whereon  to  try  the  experiment,  and  to  prepare  everything 
necessary  for  carrying  the  design  into  execution."  Cavendish,  who  was 
himself  to  carry  out  an  earth-weighing  experiment  some  twenty-five  years 
later,  was  probably  a  member  of  the  committee,  and  was  certainly  deeply 
interested  in  the  subject,  for  among  his  papers  have  been  found  calcula- 
tions with  regard  to  Skiddaw,  one  of  several  English  hills  at  first  con- 
sidered. Ultimately,  however,  the  committee  decided  in  favour  of 
Schiehallion,  a  mountain  near  L.  Rannoch,  in  Peithshire,  3547  feet  high. 
Here  the  astronomical  part  of  the  experiment  was  carried  out  in  1774, 
and  the  survey  of  the  district  in  that  and  the  two  following  years.  The 
mountain  has  a  short  east  and  west  ridge,  and  slopes  down  steeply  on  the 
north  and  south,  a  shape  very  suitable  for  the  purpose. 

Maskelyne,  who  himself  undertook  the  astronomical  work,  decided  to 
work  in  a  way  very  like  that  followed  by  Bouguer  on  Chimborazo,  but 
modified  in  a  manner  suggested  by  him.     Two  stations  were  selected,  one 

C 


34> 


PROPERTIES  OF  MATTER. 


z.l 


rtn  the  south  and  the  other  on  the  north  slope.  A  small  observatory  was 
5rected  first  at  the  south  station,  and  the  angular  distance  of  some  staru 
from  the  zenith,  when  they  were  due  south,  was  most  carefully  measured. 
The  stars  selected  all  passed  nearly  overhead,  so  that  the  angles  measured 
were  very  small.  The  instrument  used  was  the  zenith  sector,  a  telescope 
rotating  about  a  horizontal  east  and  west  axis  at  the  object-glass  end,  and 
provided  with  a  plumb-line  hanging  from  the  axis  over  a  graduated  scale  at 
the  eyepiece  end.  This  showed  how  far  the  telescope  was  from  the  vertical. 
After  about  a  month's  woi'k  at  this  station  the  observatory  was  moved 
to  the  north  station,  and  again  the  same  stars  were  observed  with  the 
zenith  sector.  Another  month's  work  completed  this  part  of  the  ex- 
periment. Fig.  9  will  show  how  the  observations  gave  the  attraction 
due  to  the  hill.  Let  us  for  the  moment  leave  out  of  account  the  curvature 
of  the  earth,  and  suppose  it  flat.  Further,  let  us  suppose  that  a  star  is 
being  observed  which  would  be  directly  overhead  if  no  mountain  existed. 

Then  evidently  at  S.  the  plumb-line  is 
pulled  to  the  north,  and  the  zenith  is 
shifted  to  the  south.  The  star  therefore 
appears  slightly  to  the  north.  At  N. 
there  is  an  opposite  effect,  for  the  moun- 
tain pulls  the  plumb-line  southwards, 
and  shifts  the  zenith  to  the  north  ;  and 
now  the  star  appears  slightly  to  the 
south.  The  total  shifting  of  the  star  is 
double  the  deflection  of  the  plumb-line 
^uh'zci  at  either  station  due  to  the  pull  of  the 
mountain. 

Fig.  D.-Maskelyne's  Plumb-line  Ex-  ,  ,^"*^  *^®  Curvature  of  the  earth  also 
periment  on  Schiehallion.  deflects  the  verticals  at  N.  and  S.,  and 

in  the  same  way,  so  that  the  observed 
shift  of  the  star  is  partly  due  to  the  mountain  and  partly  due  to  the 
curvature  of  the  eaith.  A  careful  measure  was  made  of  the  distance 
between  the  two  stations,  and  this  gave  the  curvature  deflection  as  about  43". 
The  observed  deflection  was  about  55",  so  that  the  effect  of  the  mountain, 
the  difference  between  these,  was  about  12", 

The  next  thing  was  to  find  the  form  of  the  mountain.  This  was  before 
the  days  of  the  Ordnance  Survey,  so  that  a  complete  survey  of  the  district 
was  needed.  When  this  was  complete,  contour  maps  were  made,  giving 
the  volume  and  distance  of  every  part  of  the  mountain  from  each  station. 
Hutton  was  associated  with  Maskelyne  in  this  part  of  the  Avork,  and  he 
carried  out  all  the  calculations  based  upon  it,  being  much  assisted  by 
valuable  suggestions  from  Cavendish. 

Now,  had  the  mountain  had  the  same  density  as  the  earth,  it  was 
calculated  from  its  shape  and  distance  that  it  should  have  deflected  the 
plumb-lines  towards  each  other  through  a  total  angle  of  20-9",  or  14  times 
the  observed  amount.  The  earth,  then,  is  14  times  as  dense  as  the 
mountain.  From  pieces  of  the  rock  of  which  the  mountain  is  composed, 
its  density  was  estimated  as  2i  times  that  of  Avater.  The  earth  should 
have,  therefore,  density  14  x  2|  or  4J-.  An  estimate  of  the  density  of  the 
mountain,  based  on  a  survey  made  thirty  years  later,  brought  the  result 
up  to  5.  All  subsequent  work  has  shown  that  this  number  is  not  very 
far  from  the  truth. 


GRAVITATION.  S5 

An  exactly  similar  experiment  was  made  eighty  years  later,  on  the 
completion  of  the  Ordnance  Survey  of  the  kingdom.  Certain  anomalies 
in  the  direction  of  the  vertical  at  Edinbuigh  led  Colonel  James,  the 
director,  to  repeat  the  Schiehallion  experiment,  using  Arthur's  Seat  as 
the  deflecting  mountain.  The  value  obtained  for  the  mean  density  of  the 
earth  was  about  5  J. 

Repetitions  have  also  been  made  of  the  pendulum  method,  tried  by 
Boucuer  in  the  Andes. 

The  first  of  these  was  by  Carhni,  in  1821.  He  observed  the  length  of 
a  pendulum  swinging  seconds  at  the  Hospice  on  Mont  Cenis,  about  GOOO' 
feet  above  sea-level,  and  so  obtained  the  value  of  gravity  there.  The' 
value  due  to  mere  elevation  above  the  sea-level  was  easily  calculated,  but' 
the  observed  value  was  greater  than  that  calculated  by  about  1  in  5000. 
In  other  words,  the  pull  of  the  whole  earth  was  5000  times  greater  than 
that  of  the  mountain  under  the  Hospice.  Knowing  approximately  the 
shape  of  the  mountain,  and  estimating  its  density 
from  specimens  of  the  rock,  Carlini  found  the 
density  of  the  earth  to  be  about  4^  times  that  of 
water. 

Another  experiment  of  the  same  kind  was 
made  by  Mendenhall,  in  Japan,  in  1880.  Here 
he  determined  the  value  of  gravity  on  the 
summit  of  Fujiyama,  a  mountain  nearly  2|  miles 
high.  He  found  it  greater  than  the  value 
calculated  fi'om  the  increased  distance  from  the 
earth's  centre  by  about  1  in  oOUO,  as  Carlini  had 
done  on_  Mont  Cenis.  Fujiyama,  though  the  ^^^Harto7?riSrtetr''* 
higher,   is   more    pointed   and   less    dense    than 

Mont  Cenis.     Mendenhall  estimated  the  mean  density  of  the  earth  as 
5-77. 

Airy  applied  the  pendulum  to  solve  the  problem  in  a  somewhat  dilTerent 
way,  using,  instead  of  a  mountain,  the  crust  of  the  earth  between  the  top 
and  the  bottom  of  a  mine.  His  first  attempts  were  made  in  1826,  at  the 
Dolcoath  copper  mine,  in  Cornwall.  Here  he  swung  a  pendulum  first  at 
the  surface  and  then  at  the  bottom  of  the  mine.  At  the  point  below  we 
may  consider  that  the  weight  of  the  pendulum  was  due  to  the  pull  of  the 
part  of  the  earth  within  the  sphere  with  radius  reaching  from  the  earth's 
centre  to  the  point  (Fig.  10).  Knowing  the  value  of  gravity  below,  it 
was  easy  to  calculate  what  it  would  have  been  at  the  level  of  the  surface 
had  no  outer  shell  existed,  and  had  the  change  in  value  depended  merely 
on  the  greater  distence  from  the  earth's  centre.  The  observed  value  was 
greater  than  this  through  the  pull  of  the  outer  shell,  and  it  was  hoped 
that  the  difference  would  be  measured  sufficiently  accurately  to  show  how 
much  greater  is  the  mass  of  the  earth  than  that  of  the  crust.  The  first 
attempt  was  brought  to  an  end  by  a  curious  accident.  As  one  of  the 
pendulums  used  was  being  raised  up  the  shaft,  the  box  containing  it  took 
fire,  the  rope  was  burnt,  and  the  pendulum  fell  to  the  bottom.  Two  years 
later  another  attempt  was  made,  but  this  was  brought  to  an  end  by  a 
fall  in  the  mine,  which  stopped  the  pump  so  that  the  lower  station  was 
flooded. 

Many  years  later,  in  1854,  the  experiment  was  again  undertaken  by 
Airy,  this  time  in  the  Harton  coal-pit,  near  Sunderland.    The  method  waa 


S6  PROPERTIES  OF  MATTER. 

exactly  the  same,  a  pendulum  being  swung  above  and  below  the  surface, 
and  the  diminution  in  gravity  above  carefully  determined.  The  experiment 
was  carried  out  with  the  gi-eatest  care  and  in  a  most  thorough  way,  two 
pendulums  being  swung  at  the  same  time — one  above  and  one  below — the 
two  being  interchanged  from  time  to  time.  Several  assistants  were 
occupied  in  taking  the  observations,  which  extended  continuously  night 
and  day  for  about  three  weeks.  Now  gravity  at  the  surface  was  greater 
than  it  would  have  been,  had  no  outer  shell  existed  of  thickness  equal  to 
the  depth  of  the  pit,  by  about  1  in  14,000,  so  that  the  pull  of  the  earth 
was  about  14,000  times  that  of  the  shell.  The  density  of  the  shell  was 
determined  from  specimens  of  the  rocks,  and  Airy  found  the  density  of 
the  earth  about  6|. 

Some  very  interesting  experiments  have  since  been  made  in  a  similar 
way  by  Von  Sterneck  in  silver  mines  in  Saxony  and  Bohemia.  Using  the 
invariable  pendulums  described  in  the  last  chapter  he  obtained  different 
results  with  dilFerent  depths  of  mines,  the  value  of  the  mean  density 
increasing  with  the  increasing  thickness  of  the  shell  used.  This  shows 
very  evidently  that  there  were  sources  of  disturbance  vitiating  the  method. 
Von  Sterneck  found,  on  comparing  his  observations  at  the  two  mines,  that 
the  increase  in  gravity  on  descending  was  much  more  nearly  proportional 
to  the  rise  of  temperature  than  to  the  depth  of  descent.  This  appears  to 
indicate  that  whatever  disturbs  the  regularity  of  gravity  disturbs  also  the 
slope  of  temperature. 

All  the  methods  so  far  described  use  natural  masses  to  compare  the 
earth  with,  and  herein  lies  a  fatal  defect  as  regards  exactness.  We  do  not 
know  accurately  the  density  of  these  masses  and  what  is  the  condition  of 
the  surrounding  and  underlying  strata.  We  can  really  only  form  at  the 
best  rough  guesses.  Indeed,  the  experiments  might  i-ather  be  turned  the 
other  way  about,  and  assuming  the  value  of  the  mean  density  of  the  earth, 
we  might  measui-e  the  mean  density  of  the  mountain  or  strata  of  which 
the  attraction  is  measured. 

The  Cavendish  Experiment. 

We  turn  now  to  a  different  class  of  experiment,  in  which  the  attracting 
body  is  altogether  on  a  smaller  scale,  so  that  it  can  be  handled  in  the 
laboratory.  The  smallness  of  the  attraction  is  compensated  for  by  the 
accuracy  with  which  we  know  the  size  and  mass  of  the  attracting  body. 

The  idea  of  such  an  experiment  is  due  to  the  Rev.  John  Michell,  who 
completed  an  apparatus  for  the  purpose  but  did  not  live  to  experiment 
with  it. 

Michell's  plan  consisted  in  suspending  in  a  narrow  wooden  case  a 
horizontal  rod  G  feet  long,  with  a  2-incli  sphere  of  lead  hung  at  each  end 
by  a  short  wire.  The  suspending  wire  for  the  rod  was  40  inches  long. 
Outside  the  case  were  two  lead  spheres  8  inches  in  diameter.  These  were 
to  be  brought  up  opposite  the  suspended  splieres,  one  on  one  side,  the 
other  on  the  other,  so  that  their  attractions  on  those  spheres  should  con- 
spire to  turn  the  rod  the  same  way  round.  Now  moving  each  large  sphere 
on  to  the  other  side  of  the  case  so  as  to  pull  the  suspended  sphere  with 
equal  force  in  the  opposite  direction,  the  rod  should  turn  through  twice  the 
angle  which  it  would  describe  if  the  spheres  were  taken  altogether  away. 
Hence  half  this  angle  would  give  the  twist  due  to  the  attractions  in  one 


GRAVITATION.  37 

position  alone.  Knowing  the  torsion  couple  of  the  suspending  wire  for  a 
given  angle  of  twist  and  the  length  of  the  rod,  the  attracting  force  would 
be  calculable.  To  find  the  torsion  couple,  Michell  proposed  to  set  the  rod 
vibrating.  From  its  moment  of  inertia  and  time  of  vibration  the  couple 
could  be  found. 

Neglecting  all  corrections,  the  mathematics  of  the  method  may  be 
reduced  to  the  following  : 

Let  the  two  suspended  balls  have  mass  m  each,  the  two  attracting  balls 
mass  M  each.  Let  the  rod  have  length  2a  and  with  the  suspended  balls 
moment  of  inertia  I ;  let  d  be  the  distance  apart  cf  the  centres  of  attracting 
and  attracted  balls,  and  let  0  be  the  angle  through  which  the  attraction 
twists  the  rod. 

If  /i  is  the  toision  couple  per  radian  twist,  and  G  the  gravitation 


constant,  then 

The  time  of  vibration 
whence,  eliminating  /i, 


„     2GM«irt. 


27r  Jllix, 


47r^ie    ^   2GMma. 

N"  d' 


Now  we  may  obtain  another  equation  containing  G  by  expressing  the 
acceleration  of  gravity  in  terms  of  the  dimensions  and  density  of  the 
earth, 

where  r  is  the  radius,  0  the  circumference,  and  A  the  density  of  the 
earth.  Eliminating  G  between  the  last  two  equations  and  putting  for 
glir^  the  length  of  the  seconds  pendulum  L — a  useful  abbreviation — we 

find 

^     3     L     Mma     N%  '>  ■ 

A  =  -  X  —  X X  — -' 

4     C        d'        10 

where  all  the  terms  on  the  right  hand  are  known  or  may  be 
measured. 

On  Michell's  death  the  apparatus  which  he  had  collected  for  his 
experiment  came  into  the  possession  of  Prof.  Wollaston,  who  gave  it  to 
Cavendish,  Cavendish  determined  to  carry  out  the  experiment,  with 
certain  modifications  ;  but  he  found  it  advisable  to  make  the  greater  part 
of  the  apparatus  afresh,  though  closely  following  Michell's  plan  and 
dimensions. 

The  actual  work  was  done  in  the  summer  of  1797  and  the  following 
spring  tf  1798* 

He  selected  for  the  experiment,  according  to  Baily,  an  outhouse  in  his 
garden  at  Clapham  Common,  and  within  this  he  appears  to  have  constructed 
an  inner  chamber  to  contain  the  apparatus,  for  he  states  that  he  "resolved 
to  place  the  apparatus  in  a  room  which  should  remain  constantly  shut,  and 
to  observe  the  motion  of  the  arm  from  without  by  means  of  a  telescope," 
in  order  that  inequalities  of  temperature  and  consequent  air  currents  within 
the  case  should  be  avoided. 

*  Experiments  to  determiue  the  density  of  the  earth.  Phi'.  Irai.s.,  Ixxxviii,, 
1798. 


38 


PROPERTIES  OF  MATTER. 


The  torsion  rod  h  h  (Fig.  11,  reduced  from  the  figure  in  Cavendish's 
paper)  was  of  deal,  G  feet  long,  strengthened  by  a  silver  wire  tying  the  ends 
to  an  upright  m  g  in  the  middle.  The  two  attracted  balls  x  x  were  lead, 
2  inches  in  diameter,  and  hung  by  short  wires  from  the  ends  of  the  rod. 

The  torsion  wire  was  o'd\  inches  long,  of  silvered  copper,  and  at  first  of 
such  cross  section  as  to  give  a  time  of  oscillation  about  15m.  This  was 
soon  changed  for  one  with  a  time  of  oscillation  about  7m. 

The  position  of  the  rod  was  determined  by  a  fixed  scale  on  ivory  divided 
to  -^(jth.  inch  near  the  end  of  the  arm,  the  arm  itself  carrying  a  vernier  of 
five  divisions.  This  was  lighted  by  a  lamp  outside  the  room,  and  was 
viewed  through  a  telescope  passing  through  a  hole  in  the  wall. 

The  torsion  case  was  supported  on  four  levelling  screws.     The  attracting 


n:[- 


Fig.  11. — Cavendish's  Apparatus,  h  h,  torsion  rod  hung  by  wire  I  /;;  x  x, 
attracted  balls  hung  from  its  ends;  W  VV,  attracting  masses  movable 
round  axis  P.     T  T,  telescopes  to  view  position  of  torsion  rod. 


masses,  lead  spheres  12  inches  in  diameter,  WW,  hung  down  from  a  cross 
bar,  being  suspended  by  vertical  copper  rods.  This  bar  could  be  rotated 
by  ropes  passing  outside  the  room  round  a  pin  fixed  to  the  ceiling  in  the 
continuation  of  the  torsion  axis. 

The  masses  were  stopped  when  |-  inch  from  the  case  by  pieces  of  wood 
fastened  to  the  wall  of  the  building.  When  the  masses  were  against  the 
stops  their  centres  were  8'85  inches  from  the  central  line  of  the  case. 

The  method  of  experiment  was  somewhat  as  follows :  Tlie  torsion  rod 
was  never  at  rest,  and  the  centi'e  of  swing  was  taken  as  the  position  in 
which  it  would  be  if  all  disturbances  could  be  eliminated.  This  centre  of 
swing  was  determined  from  three  succeeding  extremities  of  vibration  when 
the  attracting  masses  WW  were  against  the  stops  on  one  side.  They  were 
then  swung  round  so  as  to  come  against  the  stops  on  the  other  side  of  the 
attracted  masses,  and  the  now  centre  of  swing  was  observed.  In  a 
particular  experiment  the  dill'erence  between  the  two  centres  was  about 
sij  scale  divisions.  The  time  of  vibration  was  observed  from  several  suc- 
cessive passages  past  the  centre  of  swing,  the  value  obtained  in  the  saniQ 


GRAVITATION.  39 

experiment  being  about  427  sees.,  and  the  masses  were  then  moved  back 
to  their  first  position,  giving  a  second  value  for  the  deflection. 

In  computing  the  results  various  coirections  had  to  be  introduced  into 
the  equivalents  of  the  simple  formula;  which  have  been  given  above. 
Taking  the  attraction  formula, 

a  correction  had  to  be  made,  because  the  attracting  masses  were  not  quite 
opposite  those  attracted,  as  the  suspending  bar  was  a  little  too  short. 
Then  allowance  was  necessary  for  the  attraction  on  the  torsion  rod,  and  a 
negative  correction  had  to  be  applied  for  the  attraction  on  the  more 
distant  ball.  The  copper  suspending  rods  were  also  allowed  for,  and  a 
further  correction  was  made  for  the  change  in  attraction  with  change  of 
scale  reading— i.e.,  for  change  of  distance  between  attracting  and  attracted 
masses.  This  correction  was  proportional  to  the  deviation  from  the  central 
position,  and  may  be  regarded  as  an  alteration  of  yn. 

As  to  the  case,  it  would  evidently  have  no  effect  when  the  rod  was 
central,  but  it  was  necessary  to  examine  its  attraction  when  the  rod  was 
deflected.  Cavendish  found  that  in  no  case  did  it  exceed  1/1170  of  the 
attraction  of  the  masses,  and  therefore  neglected  it. 

Tui-ning  now  to  the  vibration  formula, 

this  was  correct  when  the  masses  were  in  the  "  midway"  position — i.e.,  in 
the  line  perpendicular  to  the  torsion  rod.  But  when  they  were  in  the 
positive  or  negative  position,  the  variation  in  their  attraction,  as  the  balls 
approached  or  receded  from  them,  made  an  appreciable  alteration  in  the 
value  of  the  restoring  couple,  and  thus  virtually  altered  [i.  The  time  had 
therefore  to  be  reduced  by  2/185  of  its  observed  value  where  S  was  the 
deflection  in  scale  divisions  due  to  the  change  of  the  masses  from  midway 
to  near  position. 

But  it  is  to  be  observed  that,  if  the  weights  were  moved  from  one  near 
position  to  the  oti^er,  and  the  time  of  vibration  was  taken  in  either 
position,  then  the  same  correction  having  to  be  applied  to  fi  in  both 
formulae,  it  might  be  omitted  from  both. 

In  all,  Cavendish  obtained  twenty-nine  results  with  a  mean  value  of 

D  =  5-448  ±-033. 

By  a  mistake  in  his  addition  of  the  results,  pointed  out  by  Baily,  he 
gave  as  the  mean  5'48. 

Repetitions   by   Reich,   Baily  and   Cornu  and  Bailie.— His 

experiment  has  since  been  repeated  several  times.  Reich  made  two 
experiments  in  Germany  by  Cavendish's  method,  obtaining  in  1837  a 
value  5'49,  and  about  1849  a  value  5'58.  In  England  it  was  repeated 
by  Baily  about  1841  and  1842.  Baily's  experiment  excited  great  attention 
at  the  time,  and  the  result  obtained,  5'674,  was  long  supposed  to  be  very 
near  indeed  to  the  truth.  But  certain  discrepancies  in  the  work  gradually 
impaired  confidence  in  the  final  result,  and  in  1870  MM.  Cornu  and 
Bailie,  in  France,  undertook  a  repetition,  with  various  improvements  and 
refinements.     In  planning  out  their  own  work  they  succeeded  in  detecting 


40 


PROPERTIES  OF  MATTER. 


probably  the  chief  source  of  error  in  Daily's  work.  They  have  as  yet  only 
given  an  interim  result  of  aboiit  5-5,  and  liave  shown  that  Baily's  work, 
if  properly  interpreted,  should  bring  out  a  not  very  difl'erent  result.  Their 
final  conclusion  is  still  to  be  published. 

Boys'S  Cavendish  Experiment. — Tn  the  Philosophical  Transactions 
for  18'J5  (vol.  li^i),  A.  p.  1)  is  an  account  of  a  determination  of  the  gravita- 
tion constant  carried  out  with  the  greatest  care  by  Prof.  Boys.  He  had 
discovered  a  method  of  drawing  exceedingly  fine  quaitz  fibres  and  had 

found  them  exceedingly 
strong  and  true  in  their 
elastic  properties.  They  are 
therefore  pre-eminently  ap- 
plicable in  torsion  experi- 
ments where  small  forces  are 
to  be  measured.  Using  a 
quartz  fibre  as  the  torsion 
wire  in  a  Cavendish  appara- 
tus, he  was  able  to  reduce 
the  attracted  Aveight  and 
the  whole  apparatus  and  yet 
reduce  the  diameter  of  the 
suspending  fibre  so  far  that 
the  sensitiveness  was  as  great 
as  in  earlier  experiments. 
At  the  same  time  the  small- 
ness  of  the  apparatus  allowed 
it  to  be  kept  at  a  much  more 
uniform  temperature,  and 
the  disturbances  due  to  con- 
vection air  currents  were 
much  lessened.  These  dis- 
turbances had  much  troubled 
the  earlier  workers.  In  Fig. 
12  is  a  diagrammatic  repre- 
sentation of  the  apparatus. 
The  attracted  masses  mm 
were  of  gold,  one  pair  0-2 
inch,  another  pair  0'25  inch 
in  diameter.  The  torsion 
rod  N  was  0-9  inch 
and 


Fig.  12. — Diagrammatic  Ktpreseritatioii  of  a  Section  of 
Boys's  Apparatus. 


long 


was 
was  itself  a  mu-ror  in 
which  the  reflection  of  a  scale  distant  about  23  feet,  and  divided  to  50ths 
of  fin  inch,  was  viewed.     The  quartz  fibre  was  17  inches  long. 

The  attracting  masses  MM  were  lead  balls  4|  inches  in  diameter.  Had 
the  masses  all  been  on  one  level,  as  in  the  original  arrangement,  with  such 
a  short  torsion  rod  the  attracting  masses  wotild  have  attiacted  both  gold 
balls  nearly  equally.  To  avoid  this.  Boys  had  one  attracting  and  one 
attracted  mass  at  one  level  and  the  other  two  at  a  level  six  inches  below. 
The  balls  mm  were  hung  from  the  torsion  rod  by  quartz  fibres  inside  a 
tube  about  ]  h  inches  diameter.  The  atti'actinjr  masses  MM  were  hung 
from  the  revolving  lid  of  a  concentric  tubular  case  about  10  inches  in 
fliametci'.     These  masses  were  ariangod  in   tie   position  in   which    they 


GRAVITATION.  41 

exerted  the  maximum  couple  on  the  gold  balls  first  in  one  direction  and 
then  in  the  opposite.  The  deflection  varied  from  o51  to  u77  divisions, 
according  to  the  balls  used  and  the  times  of  vibration  from  188  to  242 
seconds.  The  apparatus  was  moat  exactly  constructed  and  measured,  and 
the  results  were  very  concordant. 

The  final  value,  probably  the  best  yet  obtained,  was : 

G  =  G-G57G  X  lO^^jwhence  A=_5j)270 

Braun'S  Experiment  {Denhschrift.  der  Math.  Nat.  Classe  der  Kais. 
Ahad.  Wien.  1896.  Bd.  Ixiv.). — In  189G  Dr.  Braun  published  an  account 
of  an  experiment  carried  out  by  him.  He  used  the  torsion-rod  method, 
and  though  his  apparatus  was  considerably  larger  than  that  of  Boys,  it 
was  still  much  smaller  than  that  of  Cavendish,  Reich  or  Baily.  The 
rod  was  about  24  cm.  long  and  was  suspended  from  a  tripod  by  a  brass 
torsion  wire  nearly  one  metre  long  and  0*055  mm.  in  diameter.  The 
whole  torsion  arrangement  was  under  a  glass  receiver,  about  a  metre  high 
and  30  cm.  in  diameter,  resting  on  a  flat  glass  plate.  The  receiver  could  be 
exhausted  and  in  the  later  experiments  the  pressure  was  about  4  mm.  of 
mercury  and  the  disturbances  due  to  air  cuirents  were  very  greatly 
reduced.  The  attracted  masses  at  the  end  of  the  rod  were  gilded  brass 
spheres  each  weighing  about  54  gms.  Round  the  xipper  part  of  the 
receiver,  and  outside  it,  was  a  graduated  metal  ring  which  could  be 
revolved  about  the  axes  of  the  toi-sion  wire ;  from  this  were  suspended, 
about  42  cm.  apart,  the  two  attracting  masses.  Two  pairs  were  used,  one 
a  pair  of  bi-ass  spheres  about  five  kgms.  each,  the  other  a  pair  of  iron 
spheres  filled  with  mercury  and  weighing  about  nine  kgms.  each. 

Special  arrangements  had  to  be  used  to  determine  the  position  of  the 
rod  by  means  of  a  mirror  fixed  on  its  centre,  the  beam  being  reflected 
down  through  the  bottom  of  the  plate.  The  time  of  vibration  was  about 
1275  sees.     The  result  obtained  was  very  near  to  that  of  Boys,  viz.  : 

G  =  G-G578GxlO-8;  A  =  5-52725 

A  result  very  nearly  the  same  has  recently  been  obtained  by  von 
Eotvos  {Wied.  Ann.  59,  189G,  p.  354),  but  he  has  not  yet  completed  the 
work. 

Wilsing's  Experiment. — About  1886,  Dr.  Wilsing,  of  Potsdam, 
devised  a  modified  form  of  Cavendish's  experiment,  in  which  a  sort  of 
double  pendulum  is  used — i.e.,  one  with  a  ball  below  and  another  at  a 
nearly  equal  distance  above  the  suspension.  The  pendulum  is  then  in  a 
very  sensitive  state,  and  a  very  small  horizontal  force  pulls  it  through  a 
large  angle. 

It  is  then  just  like  a  toi'sion  balance,  but  with  a  vertical  instead  of  a 
horizontal  rod.  If  weights  are  brought  up,  one  to  pull  the  upper  ball  to 
one  side  and  the  other  to  pull  the  lower  ball  to  the  other  side,  the 
pendulum  twists  round  slightly.  From  the  observed  twist  and  the  time 
of  swing  the  attraction  can  be  measured  and  compared  with  the  pull  of 
the  earth.     Wilsing  found  that  the  earth  had  a  mean  density  of  5'579. 

Experiments  with  tlie  Common  Balance. 

Von  Jolly's  Experiment. — In  1878  and  in  1881  Professor  von  Jolly 
described  a  method  which  he  had  devised.     He  had  a  balance  fixed  at  the 


42 


PROPERTIES  OF  MATTER. 


top  of  a  tower  in  Munich,  and  from  the  scale-pans  hung  wires  supporting 
two  other  scale-pans  at  the  bottom  of  the  tower  (21  metres  below). 
Imagine  that  two  weights  are  balanced  against  each  other  at  the  top  of 
the  tower.  If  one  is  now  brought  down  and  put  in  the  lower  scale-pan  on 
the  same  side  it  is  nearer  the  centre  of  the  earth,  and,  therefore,  heavier. 
Von  Jolly  found  a  gain  of  about  32  milligrammes  in  5  kilogrammes.  He  now 
built  up  a  large  lead  sphere  under  the  lower  pan,  a  yard  in  diameter,  so  that 
its  attraction  was  added  to  that  of  the  earth.  The  gain  on  transferring 
the  weight  from  the  upper  to  the  lower  pan  now  came  out  to  about  half  a 
milligramme  more,  so  that  the  attraction  of  the  sphere  was  this  half  milli- 
gramme. The  earth's  attraction  was  about  10,000,000  times  that  of  the 
sphere,  and  its  density  was  calculated  to  be  5"G*J. 


Fig.  13.— EicharB  and  Krigar-Merzel'B  Experiment. 

Experiment  of  Richarz  and  Krig-ar-Menzel.— An  experiment 

very  much  like  that  of  Yon  Jolly  in  piinciple  has  been  carried  out  by 
Drs.  Richarz  and  Krigar  Menzel  at  Spandau,  near  Berlin  (Abhand.  der 
K'onigl.  Preuss  Akad.  Berlin,  1898),  A  balance  with  a  beam  23  cm. 
long  was  supported  at  a  height  above  the  floor,  and  from  each  end 
were  suspended  two  pans,  one  near  the  beam  the  other  near  the  floor, 
more  than  two  metres  lower.  Fig.  13.  In  principle  the  method  was  as 
follows  :  Spherical  gilded  or  platinised  copper  weights  wore  used,  and  to 
begin  with  these  were  placed,  say,  one  in  the  right-hand  top  pan,  the  other 
in  the  left-hand  bottom  pan.  Suppcse  that  in  this  position  they  exactly 
balanced.  The  weights  were  then  moved,  the  right-hand  one  into  the 
right  lower  pan,  when  it  gained  weight  through  the  increase  of  gravity 
with  a  descent  of  over  two  metres ;  the  left-hand  one  into  the  left  upper 
pnn,  when  it  lost  weight  thi-ough  the  ascent  of  the  same  amount.  The 
result  after  corrections  was  that  the  right-hand  pan  appeared  heavier  by 
1"24:53  mgm.,  half  this  being  due  to  the  change  in  position  of  a  single 
kilogramme. 


GRAVITATION.  43 

A  lead  parallelepiped  was  now  built  up  of  separate  blocks,  between  the 
upper  and  lower  pans,  2  metres  high  and  2'1  metres  square,  horizontally, 
with  passages  for  the  wires  suspending  the  lower  pans.  The  weighing 
of  the  kilogrammes  was  now  repeated,  but  the  attraction  of  the  lead, 
which  was  reversed  when  a  weight  was  moved  froai  bottom  to  top,  was 
more  than  enough  to  make  up  for  the  decrease  in  gravity,  and  the  right- 
hand  now  appeared  lighter  on  going  through  the  same  operation  by 
0*1211  mgm. ;  whence  the  attraction  of  the  lead  alone  made  a  dilFei'ence 
of  1-36G4  mgm.  This  is  four  times  the  attraction  of  the  lead  on  a  single 
kilogramme.  Knowing  thus  the  pull  of  a  block  of  lead  of  known  form  and 
density  on  the  kilogramme  at  a  known  distance,  and  knowing  too  the  pull 
of  the  earth  on  the  same  kilogramme,  viz.,  10"  mgm.,  the  mean  density  of 
the  earth  could  be  found. 


The  final  result  was  : 


G  =  6-G85xl0-» 

A  =  5-505 


Poynting"'S  Experiment. — The  method  of  using  the  balance  in  this 
experiment  will  be  gathered  from  Fig.  14.  A  B  are  two  lead  weights 
about  50  lb.  each,  hanging  down  from  the  ends  of  a  very  large  and  strong 
balance  inside  a  protecting  wood  case.  M  is  a  large  lead  sphere,  weighing 
about  350  lb.,  on  a  turn-table,  so  that  it  can  move  round  from  under  A  till 
it  comes  under  B.  The  distance  between  the  centres  of  M  and  A  or  M 
and  B  is  about  one  foot.  When  under  A,  M  pulls  A,  and  so  increases  its 
weight.  When  moved  so  as  to  come  under  B  the  increase  is  taken  from 
A  and  put  on  to  B.  The  balance  is  free  to  move  all  the  time,  so  that  it 
tilts  over  to  the  B  side  an  amount  due  to  double  the  attraction  of  M 
on  either,  m  was  a  balance  weight  half  the  mass  of  M,  but  at  double  the 
distance.  Before  this  was  used  it  was  found  that  the  movement  of  M 
tilted  the  floor,  and  the  balance,  which  was  a  very  sensitive  level,  was 
afiected  by  the  tilt. 

To  observe  the  deflection  due  to  the  alteration  in  weight,  a  mirror  was 
connected  with  the  balance  pointer  by  the  "  double  suspension  "  method, 
due  to  Lord  Kelvin,  and  shown  in  Fig.  15. 

With  the  suspension  the  mirror  turned  through  an  angle  150  times  as 
great  as  that  turned  through  by  the  balance  beam.  In  the  x-oom  above 
was  a  telescope,  which  viewed  the  reflection  of  a  scale  in  the  mirror,  and 
as  the  mirror  turned  round  the  scale  moved  across  the  field  of  view.  The 
tilt  observed  meant  that  the  beam  turned  through  rather  more  than  1", 
and  that  the  weight  moved  nearer  to  the  mass  by  about  y  ^Vo  ^^  ^^  inch. 
The  weight  in  milligrammes  producing  this  tilt  had  to  be  found.  This  was 
done  virtually  (though  not  exactly  in  detail)  by  moving  a  centigiamme 
rider  about  1  inch  along  the  beam,  which  was  equivalent  to  adding  to  one 
side  a  weight  of  about  ^0  milligramme.  The  tilt  due  to  the  transfer  was 
observed,  and  was  found  to  be  very  nearly  the  same  as  that  due  to  the 
attraction,  so  that  the  effect  of  moving  M  round  from  A  to  B  was 
equiv^alent to  increasing  B  by  -^o  milligramme,  or  ^^oooovQ  ^^  ^^^ previous 
weight.  The  pull  on  either  is  half  this.  In  other  words,  the  earth  pulled 
either  about  100,000,000  times  as  much  as  the  mass  M,  and  the  earth, 
which  is  20,000,000  times  as  far  away,  would  at  the  same  distance  have 
exerted  400,000,000,000,000  times  100,000,000  times  the  pull,  and  is, 
therefore,  so  many  times  heavier.     Thus  we  find  that  the  earth  weighs 


4>4 


PROPERTIES  OF  MATTER. 


about  1-25  X  10-^  lb.  In  obtaining  the  attraction  of  M  on  A  or  B,  the 
attraction  on  the  beam  had  to  be  eliminated.  This  was  done  by  moving 
the  masses   A  B   into  the  positions  A'  B'  one  foot  higher,  and  finding 


Fig.  14. —  Poynting's  Experimect.  A  U,  weights,  each  about  50  lb.,  hanging  from 
the  two  anas  of  balance.  M,  attracting  mass  on  turn-table,  movable  so  as  to 
come  under  either  A  or  B.  m,  balancing  maos.  A'  B',  second  positions  for  A 
and  B.  In  this  position  the  attraction  of  M  on  the  beam  and  suspending  wirps 
is  the  same  as  before,  so  that  the  difference  of  attr.iction  on  A  and  B  in  tlie 
two  positions  is  due  to  the  difference  in  distance  of  A  and  B  only,  and  thus  the 
attraction  on  the  beam,  &c.,  is  eliminated. 

the  attraction  in  this  position.  The  difference  was  due  to  the  chinge 
in  A  and  B  alone,  for  the  attraction  on  the  beam  remained  the  Siime 
throughout. 

The  final  result  was — 

G  =  G-G984xlO« 
A  =  5-i934 


GRAVITATION. 


45 


Experiments  on  the  Qualities  of  Gravitation. 

The  Rang-e  of  Gravitation. — The  first  question  which  arises  is, 
whether  the  law  of  gravitation  holds  down  to  the  minutest  masses  and 
distances  which  we  can  deal  with.  All  our  observations  and  experiments 
go  to  show  that  it  holds  throughout  the  long  range  fiom  interplanetary 


Microscope  stage 


^radu^ 


^gUllUa. 


I      Mirror 


J 


Varies  u^crkon^ 
in  dcishpot 

Fig.  15 — Double  Suspension  Mirror  (half  sizeX 

distances  down  to  the  distances  between  the   attracting   bodies   in   the 
laboratory  experiments  described  above. 

The  first  step  in  the  descent  from  celestial  spaces  is  justified  by  the  fact 
that  the  acceleration  of  gravity  at  the  earth's  surface  agrees  with  its  value 
on  the  moon,  as  attracted  by  the  earth.  The  further  step  downward 
appears  to  be  justified  by  the  fair  agreement  of  the  results  obtained  by  the 
various  forms  of  Cavendish,  balance,  and  pendulum  experiments  on  the 
mean  density — expeinments  which  have  been  conducted  at  distances  varying 
from  feet  down  to  inches.  Where  the  law  ceases  to  hold  is  yet  a  matter  for 
experiment  to  determine.  When  bodies  come  into  what  we  term  "  contact," 
the  adhesion  may  possibly  still  be  due  to  gravitation,  according  to  the  inverse 
square  law,  though  the  varying  nature  of  the  adhesion  in  different  cases 
seems  to  point  to  a  change  in  the  law  at  such  minute  distancesT 


4-6  PROPERTIES  OF  MATTER. 

Gravitation  not  Selective.— it  might  be  possible  that  some  matter 
is  attracted  more  tlum  in  proponion  to  its  mass  and  some  less.  The  agree- 
ment of  astronomical  observations  with  deductions  from  the  general  law  is 
not  perfectly  decisive  as  to  this  possibility,  for  there  might  be  such  a 
mixture  of  dillerent  kinds  of  matter  in  all  the  planets  that  the  general 
average  attraction  was  in  accordance  with  the  law  though  Hot  the  attraction 
on  each  individual  kind.  A  supposition  somewhat  of  this  description  is 
required  in  an  explanation  which  has  been  given  of  the  formation  of 
comets'  tails,  some  matter  in  the  comet  being  supposed  to  be  acted  on  by 
the  sun,  not  by  the  ordinary  law  but  by  a  repulsion.  This  explanation  is, 
however,  now  generally  abandoned,  an  electrical  origin  of  the  tails  being 
regarded  as  more  probable. 

But,  with  regard  to  ordinary  terrestrial  matterj  l^ewton's  hollow 
pendulum  experiments  {Principia,  Book  III.,  Prop.  C)  repeated  with  more 
detail  and  precision  by  Bessel  {Versuche  iiber  die  Kraft,  niit  welcher  die 
Erde  Kijrper  von  verschiedenerBeschaffenheit  anzeiht,  Abhand.  der  Berl. 
Ak.  1830,  p.  41;  or  Memoires  relatifs  d,  la  Physique,  tome  v.  pp.  71- 
133)  prove  that  the  earth  as  a  whole  is  not  selective.     Still,  the  results 


Fig.  16. — raramagnetic  Sphere  placed  Fig.  17. — Diamagnetic  Sphere  placed 

in  a  previously  Straight  Field.  in  a  previously  Straight  Field. 

might  just  conceivably  be  due  to  an  average  of  equal  excesses  and  defects. 
But  again  we  may  quote  the  various  mean  density  experiments,  and  especially 
those  made  by  Baily,  in  which  a  number  of  different  attracting  aod  attracted 
substances  have  been  used  with  nearly  the  same  results. 

Gravitation  not  Affected  by  the  Medium. — When  we  compare 

gravitation  with  other  known  forces  (and  those  which  have  been  most 
closely  studied  are  electric  and  magnetic  forces)  we  are  at  once  led  to 
inquire  wdiether  the  lines  of  gravitative  fo';ce  are  always  straight  lines 
radiating  fi-om  or  to  the  mass  round  which  they  centre,  or  whether,  like 
electric  and  magnetic  lines  of  force,  they  have  a  preference  for  some  media 
and  a  distaste  for  others.  We  knov/,  for  example,  that  if  a  magnetic 
sphere  of  iron,  cobalt  or  manganese  is  placed  in  a  previously  straight  field, 
its  permeability  is  greater  than  the  air  it  replaces,  and  the  lines  of  force 
crowd  into  it,  as  in  Fig.  16.  The  magnetic  action  is  then  stronger  in  the 
presence  of  the  sphere  near  the  ends  of  a  diameter  parallel  to  the  original 
course  of  the  lines  of  force,  a-ad  the  lines  are  deflected.  If  the  sjihere  be 
diamagnetic,  of  water,  copper,  or  bismuth,  the  permeability  being  less 
than  that  of  air,  there  is  p,a  opposite  effect,  as  in  Fig.  17,  and  the  field  is 
weakened  at  the  ends  of  a  diameter  parallel  to  the  lines  of  force,  and  again 
the  lines  aie  deflected.  Similarly,  a  dielectric  body  placed  in  an  electric 
field  gatheis  in  the  lines  of  force,  and  makes  the  field  where  the  lines  enter 
and  leave  stronger  than    t  was  before. 


\ 


GRAVITATION.  47 

If  we  enclose  a  magnet  in  a  hollow  box  of  soft  iron  placed  in  a 
magnetic  field,  the  lines  of  force  are  gathered  into  the  iron  and  largely 
cleared  away  from  the  inside  cavity,  so  that  the  magnet  is  screened  from 
external  action. 

Astronomical  observations  are  not  conclusive  against  any  such  effect  of 
the  medium  on  gravitation,  for  the  medium  intervening  between  the  sun 
and  planets  approaches  a  vacuum,where  so  far  we  have  no  evidence  for 
variation  in  quality,  even  for  electric  and  magnetic  induction.  In  the  case 
of  the  earth,  too,  its  spherical  form  might  render  observation  inconclusive, 
for  just  as  a  sphere  composed  of  concentric  dielectric  shells,  each  with  its 
surface  uniformly  electrified,  would  have  the  same  external  field  in  air, 
whatever  the  dielectric  constant,  if  the  quantity  of  electrification  within 
were  the  same,  so  the  earth  might  have  the  same  field  in  air  whatever  the 
varying  quahty  of  the  underlying  strata  as  regards  the  transmission  of  the 
action  across  them,  if  they  were  only  suitably  arranged. 

But  common  experience  ^__,,^  b 

might  lead   us  at   once  to  ^^^^^."''^'''''^ 

say  that  there  is  no  very  --"l5t^^^'^'^--i>^^^^'''' 

considerable  efiect   of   the  .-^^^•^^"■^^^^TT^^^^^-^^^^^^ — -j^^^^^^^^^^^ 

kind  with  gravitation.   The  .,^<:^^^^^^^^^ir^^^c^--'" ' '^^^^^^^^^^^'^ 

evidence  of  ordinary  weigh-     ^^^^^^^'.       ^^" /Z — 

ings  may,  perhaps,  be  re-  ^ — ■ 

jected,    inasmuch   as   both  ^"'*°''^^^^^=^^n~---^^C:/"-----^^ 

sides    will   be    equally   af-  """^^^^^>^>.^  '^ 

fected   as  the    balance    is  ^'''*"^~~-^--~__ 

commonly    used.       But    a    -c,        m     i-a    i.     «   •  i         •,.•        «  ii„ 

•^  Fig.    18. — Effect    of    interposition    of  more  permeable 

spring  balance  should  show  Medium  in  radiating  Field  of  Force. 

if  there  is  any  large  effect 

when  used  in  different  positions  above  different  media,  or  in  different 
enclosures.  And  the  ordinary  balance  is  used  in  certain  experiments  in 
which  one  weight  is  suspended  beneath  the  balance  case,  and  surrounded, 
perhaps,  by  a  metal  case,  or,  perhaps,  by  a  water-bath.  Yet  no  appreciable 
variation  of  weight  on  that  account  has  yet  been  noted.  Nor  does  the 
direction  of  the  vertical  change  rapidly  from  place  to  place,  as  it  would 
with  varying  permeability  of  the  ground  below.  But  perhaps  the  agreement 
of  pendulum  results,  whatever  the  block  on  which  the  pendulum  is  placed, 
and  whatever  the  case  in  which  it  is  contained,  gives  the  best  evidence 
that  there  is  no  great  gathering  in,  or  opening  out  of  the  lines  of  the 
earth's  force  by  different  media. 

Still,  a  direct  experiment  on  the  attraction  between  two  masses  with 
different  media  interposed  was  well  worthy  of  trial,  and  such  an  experiment 
has  been  carried  out  by  Messrs.  Austin  and  Thwing.*  The  effect  to  be 
looked  for  will  be  understood  from  Fig.  18.  If  a  medium  more  permeable 
to  gravitation  is  interposed  between  two  bodies,  the  lines  of  force  will 
move  into  it  from  each  side,  and  the  gravitative  pull  on  a  body,  near  the 
interposed  medium  on  the  side  away  from  the  attracting  body,  will  be 
increased. 

The  apparatus  they  used  was  a  modified  kind  of  Boys's  apparatus 
(Fig.  19).  Two  small  gold  masses  in  the  form  of  short  vertical  wires,  each 
•4  gm.  in  weight,  were  arranged  at  different  levels  at  the  ends  virtually  of 
a  torsion  rod  8  mm.  long.     They  are  represented  in  the  figure  by  the  two 

*  Physical  Review,  v.  1897,  p.  294. 


48 


PROPI'.RTIES  OF  MATTER. 


thickenings  on  the  suspending  fibre.  The  attracting  masses  MjM,  were  lead, 
each  about  1  kgm.  These  were  first  in  the  positions  shown  by  black  lines  in 
the  figure,  and  were  then  moved  into  the  positions  shown  by  dotted  lines. 
The  attraction  was  measured  first  when  merely  the  air  and  the  case  of  the 
instrument  intervened,  and  then  when  various  slabs,  each  3  cm.  thick,  10 
cm.  wide  and  29  cm.  high,  were  interposed.  With  screens  of  lead,  zinc, 
mercury,  water,  alcohol  or  glycerine,  the  change  in  attraction  was  at  the 
most  about  1  in  500,  and  this  did  not  exceed  the  errors  of  experiment. 
That  is,  they  found  no  evidence  of  a  change  in  pull  with  change  of  medium. 
If  such  change  exists,  it  is  not  of  the  order  of  the  change  of  electric  pull 

with  change  of  medium, 
but  something  far  smaller. 
It  still  remains  just  pos- 
sible, however,  that  there 
are  variations  of  gravita- 
tional permeability  compar- 
able with  the  vaiiations  of 
magnetic  pei-meability  in 
media  such  as  water  and 
alcohol. 

Gravitation  not  Di- 
rective. —  Yet  another 
kind  of  eflfect  might  be  sus- 
pected. In  most  crystalline 
substances  the  physical  pro- 
perties are  different  along 
different  directions  in  a 
crystal.  They  expand  dif- 
ferently, they  conduct  heat 
differently,  and  they  trans- 
mit light  at  different  speeds 
in  different  directions.  We 
might  then  imagine  that 
the  lines  of  gravitative  force 
spread  out  from,  say,  a  crys- 
tal sphere  unequally  in  dif- 
ferent directions.  Some 
years  ago  Dr.  Mackenzie*  made  an  experiment  in  America,  in  which  he 
sought  for  direct  evidence  of  such  unequal  distribution  of  the  lines  of 
force.  He  used  a  form  of  apparatus  like  that  of  Professor  Boys  (Fig.  12), 
the  attracting  masses  being  calc  spar  spheres  about  2  inches  in  diameter. 
The  attracted  masses  in  one  experiment  were  small  lead  spheres  about 
^  gm.  each,  and  he  measured  the  attraction  between  the  crystals  and  the 
lead  when  the  axes  of  the  crystals  were  set  in  various  positions.  But  the 
variation  in  the  attraction  was  merely  of  the  order  of  error  of  experiment. 
In  another  experiment  the  attracted  masses  were  small  calc  spar  crystal 
cylinders  weighing  a  little  more  than  I  gm-  each.  But  again  there  was  no 
evidence  of  variation  in  the  attraction  with  variation  of  axial  direction. 

Practically  the  same  problem  was   attacked    in   a   different   way   by 
Poynting  and  Gray.t     They  tried  to  find  whether  a  quartz  crystal  sphere 

*  Physical  Review,  ii.  1895,  p.  321. 
t  PhU.  Trans.,  192,  1899,  A.  p.  245. 


•■'o""-. 

1      S      .^-,1 

1         .'  H--    I 

\          \ 

/          •'               ( 

'     m'    •■O-. 

[yb/        \ 

\ 

•..  ( 

)    / 

"\ 

( 

■ 

y 

■\ 

/ 

V 

\ 

1    A^ 

!       /'/'      : 

''\"v 

11  1 

1 

3 

c— 

1 

:::■    - 

- 

^ 

.    1 

Fig.  19. — Experiment  on  Gravitative  Permeability 
(Austin  and  Thwing). 


GRAVITATION.  49 

had  any  directive  action  on  another  quartz  crystal  sphere  close  to  it,  whether 
they  tended  to  set  with  their  axes  parallel  or  crossed. 

It  may  easily  be  seen  that  this  is  the  same  problem  by  considering 
what  must  happen  if  there  is  any  diflerence  in  the  attraction  between  two 
such  spheres  when  their  axes  are  parallel  and  when  they  are  crossed. 
Suppose,  for  example,  that  the  attraction  is  always  greater  when  their  axes 
are  parallel,  and  this  seems  a  reasonable  supposition,  inasmuch  as  in 
straightforward  crystallisation  successive  parts  of  the  crystal  are  added  to  the 
existing  crystal,  all  with  their  axes  parallel.  Begin,  then,  with  two  quartz 
crystal  spheres  near  each  other  with  their  axes  in  the  same  plane,  but 
perpendicular  to  each  other.  Remove  one  to  a  very  great  distance,  doing 
work  against  their  mutual  attractions.  Then,  when  it  is  quite  out  of  range  of 
appreciable  action,  turn  it  round  till  its  axis  is  parallel  to  that  of  the  lixed 
crystal.  This  absorbs  no  work  if  done  slowly.  Then  let  it  return.  The 
force  on  the  return  journey  at  every  point  is  greater  than  the  force  on  the 
outgoing  journey,  and  more  work  will  be  got  out  than  was  put  in.  When 
the  sphere  is  in  its  first  position,  turn  it  round  till  the  axes  are  again  at 
right  angles.  Then  work  must  be  done  on  turning  it  through  this  right 
angle  to  supply  the  diflerence  between  the  outgoing  and  incoming  works. 
For  if  no  work  were  done  in  the  turning,  we  could  go  through  cycle  after 
cycle,  always  getting  a  balance  of  energy  over,  and  this  would  appear  to 
imply  either  a  cooling  of  the  crystals  or  a  diminution  in  their  weight,  neither 
supposition  being  admissible.  We  are  led  then  to  say  that  if  the  attraction 
with  parallel  axes  exceeds  that  with  crossed  axes,  there  must  be  a  directive 
action  resisting  the  turn  from  the  crossed  to  the  parallel  positions.  And 
conversely,  a  directive  action  implies  axial  variation  in  gravitation. 

The  straightforward  mode  of  testing  the  existence  of  this  directive 
action  would  consist  in  hanging  up  one  sphere  by  a  wire  or  thread,  and 
turning  the  other  round  into  various  positions,  and  observing  whether  the 
hanging  sphere  tended  to  twist  out  of  position.  But  the  action,  if  it  exists, 
is  so  minute,  and  the  disturbances  due  to  air  currents  are  so  great,  that  it 
would  be  extremely  difiicult  to  observe  its  efiect  directly.  But  the  prin- 
ciple of  forced  oscillations  may  be  used  to  magnify  the  action  by  turning 
one  sphere  round  and  round  at  a  constant  rate,  so  that  the  couple  would 
act  first  in  one  direction  and  then  in  the  other  alternately,  and  so  set  the 
hanging  sphere  vibrating  to  and  fro.  The  nearer  the  complete  time  of 
vibration  of  the  applied  couple  to  the  natural  time  of  vibration  of  the 
hanging  sphere,  the  greater  would  be  the  vibration  set  up.  This  is  well 
illustrated  by  moving  the  point  of  suspension  of  a  pendulum  to  and  fro  in 
gradually  decreasing  periods,  when  the  swing  gets  longer  and  longer  till 
the  period  is  that  of  the  pendulum,  and  then  decreases  again.  Or  by  the 
experiment  of  varying  the  length  of  a  jar  resounding  to  a  given  fork,  when 
the  sound  suddenly  swells  out  as  the  length  becomes  that  which  would 
naturally  give  the  same  note  as  the  fork.  Now,  in  looking  for  the  couple 
between  the  crystals,  there  are  two  possible  cases.  The  most  likely  is  that 
in  which  the  couple  acts  in  one  way  while  the  turning  sphere  is  moving 
from  parallel  to  crossed,  and  in  the  opposite  way  during  the  next  quarter  ■ 
turn  from  crossed  to  parallel.  That  is,  the  couple  vanishes  four  times 
during  the  revolution,  and  this  we  may  term  a  quadrantal  couple.  But  it 
is  just  possible  that  a  quartz  crystal  has  two  ends  like  a  magnet,  and  that 
like  poles  tend  to  like  directions.  Then  the  couple  will  vanish  only  twice 
.11  a  revolution,  and  may  be  termed  a  semicircular  couple.     Both  were 

D 


50  PROPERTIES  OF  MATTER. 

looked  for,  but  it  is  enough  now  to  consider  the  possibility  of  the  quadrantal 

couple  only. 

The  mode  of  working  will  be  seen  from  Fig.  20.  The  hanging  sphere, 
•9  cm.  in  diameter  and  1  gm.  in  weight,  was  placed  in  a  light  aluminium 
wire  cage  with  a  mirror  on  it,  and  suspended  by  a  long  quartz  fibre  in  a 
brass  case  with  a  window  in  it  opi)Osite  the  mirror,  and  surrounded  by  a 
double-walled  tinfoiled  wood  case.     The  position  of  the  sphere  was  read  in 


^f  To    Accujni 


tneUned   Mirror   [p 

'Gf      (J 


?????^?^ 


•d* 


"^^^ 


\i 


w5 


IE 


(3 


Fig.  '20.  — Experimeut  on  directive  Action  of  one  Quartz  Crystal  on  another. 


the  usual  way  by  scale  and  telescope.  The  time  of  swing  of  this  little 
sphere  was  120  seconds. 

A  larger  quartz  sphere,  6 "6  cm.  diameter  and  weighing  400  gms.,  was 
fixed  at  the  lower  end  of  an  axis  which  could  be  turned  at  any  desired  rate 
by  a  regulated  motor.  The  centres  of  the  spheres  were  on  the  same  level 
and  5-9  cm.  apart.  On  the  top  of  the  axis  was  a  wheel  with  20  equidistant 
marks  on  its  rim,  one  passing  a  fixed  point  every  11*5  seconds. 

It  might  be  expected  that  the  couple,  if  it  existed,  would  have  the 
greatest  efiect  if  its  period  exactly  coincided  witli  the  1 20-second  period  of 
the  hanging  sphere — i.e.,  if  the  larger  sphere  revolved  in  210  seconds.  But 
in  the  conditions  of  the  experiment  the  vibrations  of  the  small  sphere  were 
very  much  damped,  and  the  forced  oscillations  did  not  mount  up  as  they 
would  in  a  freer  swing.     The  disturbances,  which  were  mostly  of  au  im- 


GRAVITATION. 


51 


pulsive  kinfl,  continiinlly  set  the  hanging  sphere  into  large  vibration,  and 
these  might  easily  be  taken  as  due  to  the  revolving  sphere.  In  fact, 
looking  for  the  couple  with  exactly  coincident  periods  would  be  something 


PkrwdL  VZ5 

Fia.  21. — Upper  Curve  a  regular  Vibration. 
Disturbance  dying  away. 


Lower  Curve  a 


Sfj 


like  trying  to  find  if  a  fork  set  the  air  in  a  resonating  jar  vibrating  when 
a  brass  band  was  playing  all  round  it.  It  was  necessary  to  make  the 
couple  period,  then,  a  little  difi'erent  from  the  natural  120-second  period, 
and  accordingly  the  large  sphere  was  revolved  once  in  230  seconds,  when 
the  supposed  quadrantal  couple  would  have  a 
period  of  115  seconds. 

Figs.  21  and  22  may  help  to  show  how 
this  tended  to  eliminate  the  disturbances. 
Let  the  ordinates  of  the  curves  in  Fig.  21 
represent  vibrations  set  out  to  a  horizontal 
time  scale.  The  upper  curve  is  a  regular 
vibration  of  range  ±  3,  the  lower  a  disturbance 
beginning  with  range  ±1(*.  The  first  has 
period  1,  the  second  period  l"2r).  Now,  cutting 
the  curves  into  lengths  equal  to  the  period  of 
the  shorter  time  of  vibration,  and  arranging 
the  lengths  one  under  the  other,  as  in  Fig.  22, 
it  will  be  seen  that  the  maxima  and  the 
minima  of  the  regular  vibration  always  fall  at 
the  same  points,  so  that,  taking  7  periods,  and 
adding  up  the  ordinates,  we  get  7  times  the 
range,  viz.,  ±21.  But  in  the  disturbance  the 
maxima  and  minima  fall  at  different  points, 
and  even  with  7  periods  only  the  range  is 
from  +  16  to  -  13,  or  less  than  the  range  due 
to  the  addition  of  the  much  smaller  regular 
vibration. 

In  the  experiment  the  couple,  if  it  existed, 
would  very  soon  establish  its  vibration,  which  would  always  be  there,  and 
would  go  through  all  its  values  in  115  seconds.     An  observer,  watching 


7^^ 


Fio.  22.— Results  of  Superposi. 
tion  of  Lengtbj  of  Curves  iu 
Fig.  21  equal  to  tlie  Period  of 
the  regular  one. 


bi  PROPERTIES  OF  MATTER. 

the  wheel  at  the  top  of  the  revolving  axis,  gave  the  time  signals  every  1 1  -5 
seconds,  regulating  the  speed  if  necessary,  and  an  observer  at  the  telescope 
gave  the  scale  reading  at  every  signal,  that  is,  10  times  during  the  period. 
The  values  were  arranged  in  10  columns,  each  horizontal  line  giving  the 
readings  of  a  period.  The  experiment  was  carried  on  for  about  2^  hours 
at  a  time,  covering,  say,  80  periods.  On  adding  up  the  columns,  the 
maxima  and  minima  of  the  couple  efiect  would  always  fall  in  the  same  two 
columns,  and  so  the  addition  would  give  80  times  the  swing,  while  the 
maxima  and  minima  of  the  natural  swings  due  to  disturbances  would  fall 
in  different  columns,  and  so,  in  the  long  run,  neutralise  each  other.  The 
results  of  different  days'  work  might,  of  course,  be  added  together. 

There  always  was  a  small  outstanding  effect  such  as  would  be  produced 
by  SI  quadrantal  couple,  but  its  effect  was  not  always  in  the  same  columns, 
and  the  net  result  of  observations  over  about  350  periods  was  that  there  was 
no  115 -second  vibration  of  more  than  1  second  of  arc,  while  the  disturbances 
were  sometimes  50  times  as  great.  The  semicircular  couple  required  the 
turning  sphere  to  revolve  in  115  seconds.  Here,  want  of  symmetry  in  the 
apparatus  would  come  in  with  the  same  effect  as  the  couple  sought,  and 
the  outstanding  result  was,  accordingly,  a  little  larger.  But  in  neither  case 
could  the  experiments  be  taken  as  showing  a  real  couple.  They  only  showed 
that,  if  it  existed,  it  was  incapable  of  producing  an  effect  greater  than  that 
observed.  Perhaps  the  best  way  to  put  the  result  of  the  work  is  this :  Imagine 
the  small  sphere  set  with  its  axis  at  45°  to  that  of  the  other.  Then  the 
couple  is  not  greater  than  one  which  would  take  5J  hours  to  turn  it 
through  that  45°  to  the  parallel  position,  and  it  would  oscillate  about  that 
position  in  not  less  than  21  hours. 

The  semicircular  couple  is  not  greater  than  one  which  would  turn  from 
crossed  to  parallel  position  in  4|  hours,  and  it  would  oscillate  about  that 
position  in  not  less  than  17  hours.  Or,  if  the  gravitation  is  less  in  the 
crossed  than  in  the  parallel  position,  and  in  a  constant  ratio,  the  difference 
is  less  than  1  in  16,000  in  the  one  case  and  less  than  1  in  2800  in  the  other. 

We  may  compare  with  these  numbers  the  difference  of  rate  of  travel 
of  yellow  light  through  a  quartz  crystal  along  the  axis  and  perpendicular 
to  it.     That  difference  is  of  quite  another  order,  being  about  1  in  170. 

Other  possible  Qualities  of  Gravitation. — Weight  might  con- 
ceivably change  with  temperature,  but  experiments  *  show  that  if  there  is 
any  change  it  is  probably  less  than  1  in  10^'  of  the  weight  per  1°  C. 

It  is  possible  that  weight  might  change  when  the  bodies  weighed  enter 
into  chemical  combination.  Many  experiments  have  been  made  to  detect 
such  a  change,  the  most  extensive  and  exact  by  Landolt.t  At  first  it 
appeared  as  if  in  sdme  cases  a  diminution  of  weight  occurred  on  combina- 
tion, but  ultimately  the  effect  was  traced  to  an  expansion  of  the  containing 
vessel  through  the  heat  developed.  The  vessel  did  not  return  at  once  to 
its  original  volume  on  cooling  and  so  there  was  a  slight  increase  in  the 
buoyancy  of  the  air  in  the  weighing  after  combination.  The  experiments 
show  that  the  change,  if  it  exists,  is  too  small  to  measure. 

No  research  yet  made  has  succeeded  in  showing  that  gravitation  is 
related  to  anything  but  the  masses  of  the  attracting  and  the  attracted 
bodies  and  their  distance  apart.  It  appears  to  have  no  relation  to  physical 
or  chemical  conditions  of  the  acting  masses  or  to  the  intervening  medium. 

*  Poynting  &Phillips,P.;?.,S'.,A  76,  1905,  p.  445;  Southern8,P.ii.5.,A78, 1906,  p.  392. 
f  Landolt,  Preuss.  Ak.  Wiss.  Berlin,  Sitz.  Ber.,  viii.  1906,  p.  266,  and  xvi.  1908, 
March  19.     References  to  other  work  are  given  in  the  first  paper. 


CHAPTER  IV. 
ELASTICITY. 

Contents. — Limits  of  Elasticity — Elastic  after  effect — Viscosity  of  Metals  and 
Elastic  Fatigue — Anomalous  Effects  of  first  Loading  a  Wire — Breakir/g  Stress. 

In  this  chapter  we  shall  consider  changes  in  the  conformation  of  solid 
bodies  and  the  connection  between  these  changes  and  the  forces  which 
produce  them. 

Many  of  the  points  with  which  we  shall  have  to  deal  are  well 
illustrated  by  the  simple  case  of  a  vertical  metal  wire  the  upper  end  of 
which  is  fixed  while  the  lower  end  carries  a  scale-pan.  If  we  measure 
the  increments  of  elongation  of  the  wire  when  different  weights  are 
placed  in  the  scale-pan  and  plot  our  results  as  a  curve  in  which  the 
abscissae  are  the  elongations  of  the  wire — i.e.,  the  extension  of  the  wire 
divided  by  its  unstretched  length,  and  the  ordinates  the  stretching  weight 
(inclusive  of  the  weight  of  the  scale-pan)  divided  by  the  area  of  cross 
section  of  the  unstretched  wire,  we  obtain  results  similar  to  those  shown 
in  Fig.  23  (from  A  History  of  the  Theory  of  Elasticity  and  of  the  Strength 
of  Materials),  which  represents  the  results  of  experiments  made  by  Professor 
Kennedy  on  a  bar  of  soft  steel. 

The  first  part  of  the  curve — when  the  stretching  force  per  unit  area  is 
less  than  a  certain  value,  is  a  straight  line — i.e.,  up  to  a  certain  point  the 
elongationjs  proportional  to  the  load  per  unitj-rea  of  cross  section,*  and 
up  to  this  point  we  find  that  whenjwe  remoWthe  weight  from  the  scale- 
pan  jthe  stretched  wire  shortens  u ntil  its  length  is  the  same  as  it  was 
before  the  weights  were  put  ^on_  (the  elongations  in  this  stage  are  so 
small  thaton  the  scale  of  Fig.  23  this  part  of  the  curve  is  hardly  distinguish- 
able from  the  axis  AB).  When,  however,  we  get  beyond  a  certain 
point  B  on  the  curve — i.e.,  when  the  stretching  foi-ce  per  unit  area  is 
greater  than  the  value  represented  by  AB,  the  curve  becomes  bent,  and 
we  find  on  removing  the  weights  that  the  wire  does  not  return  to  its 
original  length ,  but  is  permanently  lengthened,  and  is  Bai3~to^  have 
axx^mredrpermahentjet. 

The  range  of  elongations  over  which  the  wire,  when  unloaded,  recovers 
its  original  length,  is  called  the  range_of  perfect  elasticity;  when  we 
go  beyond  this  range  we  are  said  to  exceed  the  elastic  limit. 

*  This  seems  to  be  only  approximately  true  for  certain  kinds  of  iron.  {A  Hiitory 
of  the  Theory  of  Maatioity  and  of  the  Strength  of  Materials.  Todhunter  and  Tcarsou, 
Vol.  i.  p.  893. 


54 


PROPERTIES  OF  MATTER 


After  passing  the  point  represented  by  B  a  stage  is  readied  where  the 
extension  becomes  very  large.  Tlie  scale-pan  runs  rapidly  down  and  the 
wire  looks  as  if  it  wei-e  about  to  break.     By  far_the  greater  part  of  this 


extension  is  _permanent,  and  the  wire,  after  passing  the  state  represented 
by  C,  is  not  able  to  sustain  as  great  a  pull  as  before  without  sufiering 
further  elongation ;  this  is  shown  by  the  bending  back  of  the  curve.  The 
place  C  where  this  great  extension  begins  is  called  the  yield-point ;  it 
seems  to  be  always  fm^ther  along  the  curve  than  the  elastic  limit  B. 


S 


\A. 


H  ExZcfhsioiv. 

Fig.  23.— Elongation  of  a  Sti etched  Wire. 


The  part  of  the  incrcnient  of  elongation  which  disappears  on  the 
removal  of  the  stretchiui;  weiirht,  between  the  elastic  limit  and  the  yield- 
point,  is  proportional    to   the  stretching;_weight,  and   the  ratio    of  this 


ELASTICITY.  55 

movement  to  tlie  stretching  weight  per  unit  area  is,  according  to  the 
experiments  of  Professor  Kennedy,  the  same  as  that  within  tlio  limits  of 
perfect  elasticity  {see  Todhunter  and  Pearson's  History  of  Elasticity, 
p.  889). 

After  passing  the  yield-point  the  elongation  increaees  very  rapidly 
with  the  load,  and  at  this  stage  the  wire  is  plastic,  the  elongation 
depending  upon  the  time  the  stretching  force  acts.  The  extension  rapidly 
increases  and  the  area  rapidly  contracts  until  the  breaking- point  E  is 
reached.  The  apparent  maximum  for  the  load  per  unit  area  shown  in 
Fig.  23  is  due  to  the  contraction  of  the  area,  so  that  the  pull  per  unit  area 
of  the  stretched  wire  is  no  longer  represented  even  approximately  by  the 
ordinates.  About  the  point  D  the  wire  begins  to  thin  down  or  flow 
locally,  so  that  its  cross  section  is  no  longer  uniform,  some  parts  being  now 
smaller  than  the  rest. 

The  portion  GHG'  of  the  curve  represents  the  effect  of  unloading 
and  reloading  at  a  point  G  past  the  yield  point.  We  see,  from  the  shape 
of  this  portion  of  the  curve,  that  the  limit  of  perfect  elasticity  for  this 
permanently  stretched  wire  has  been  extended  beyond  the  yield-point  of 
the  wire  before  it  was  permanently  stretched.  The  range  between  the 
limit  of  perfect  elasticity  and  the  breaking-point  is  very  different  for 
different  substances  ;  for  ductile  substances,  such  as  lead,  it  is  considerable, 
while  for  b}'ittle  ones,  such  as  glass,  it  is  evanescent. 

We  are  thus  from  our  study  of  the  loaded  wire  led  to  divide  the 
phenomena  shown  by  substances  acted  upon  by  forces  into  two  divisions — 
qne^  divisiQn_in  which  the  solid  recovers  its  original  form  after  the 
removal  of  the  forces  which  deformed  it,  the  other  division  in  which  a 
permanenJL-clLange  is  produced  by  the  application  of  the  force!  Even, 
within  the  limits  of  perfect  elasticity  different  bodies  show  distinct 
differences  in  their  behaviour.  Some  recover  their  form  immediately 
after  the  removal  of  the  force,  while  others,  though  they  recover  it 
ultimately,  take  considerable  time  to  do  so.  Thus  a  thread  of  quartz  fibre 
will  recover  its  shape  immediately  after  the  removal  of  the  tensional 
and  torsional  forces  acting  upon  it,  while  a  glass  fibre  may,  if  the  forced 
have  been  applied  for  a  considerable  time,  be  several  hours  before  it 
regains  its  original  condition.  This  delay  in  recovering  the  original 
condition  of  the  substance  is  called  the  elastic  after-effect ;  it  may  be 
conveniently  studied  in  the  case  of  the  torsion  of  glass  fibres. 

Take  a  long  glass  fibre  and  fasten  to  it  a  mirror  from  which  a  spot  of 
light  is  reflected  on  to  a  scale,  twist  the  fibre  about  its  axis  and  keep  it 
twisted  for  a  considerable  time.  Then  remove  the  twisting  couple :  the 
spot  of  light  will  at  once  come  back  a  considerable  distance  towards  its  old 
position,  but  will  not  reach  it,  and  the  rest  of  the  journey  will  be  a  slow 
creep  towards  the  old  position,  and  several  hours  may  elapse  before  the 
journey  is  completed.  The  larger  the  initial  twist  and  the  longer  the 
time  for  which  it  was  applied  the  greater  is  the  tempoi-ary  deflection  of 
the  spot  of  light  from  its  original  position. 

The  general  shape  of  the  curve  which  represents  the  relation  between 
the  displacement  of  the  zero — i.e.,  the  displacement  of  the  position  of  the 
spot  of  light — and  the  time  which  has  elapsed  since  the  removal  of  the 
twist,  is  shown  in  Fig.  24.  In  this  cui-ve  the  ordinates  represent  the 
displacement  and  the  abscissse  the  time  since  the  removal  of  the  twist. 
The  altitude  PN,  when  the  abscissa  ON  is  given,  depends  upon  the 


56 


PROPERTIES  OF  MATTER. 


magnitude  of  the  initial  twist  and  the  time  for  which  it  was  applied ;  the 
curve  is  steep  at  first  but  gets  flatter  and  flatter  as  the  time  increases. 
The  longer  the  initial  twist  is  applied  the  more  slowly  does  the  zero 
approach  its  original  position.  Yery  complicated  movements  of  the  zero 
may  occur  if  the  fibre  has  been  twisted  first  in  one  direction  and  then 
in  the  opposite  for  a  considerable  number  of  times.  The  general  features 
of  this  phenomenon  will  be  illustrated  by  the  following  simple  case.  Suppose 
that  immediately  after  the  removal  of  the  first  twist,  whose  after-effect, 
if  it  were  alone,  would  be  represented  by  the  curve  (I),  Fig.  24,  a  second 
twist  in  the  opposite  direction  is  applied  for  a  time  represented  by  ON  and 
then  removed.  Suppose  that  the  deflection  of  the  zero  due  to  this  twist 
alone  is  represented  by  the  dotted  curve  (II)  (as  the  twist  is  in  the  opposite 


2V  K 

Pig.  24. — Curve  showing  the  Elastic  After-effect  in  a  Twisted  Glass  Thread. 

direction,  the  ordinates  represent  negative  deflections).  Then  if  we  can 
superpose  the  efiects,  the  displacement  of  the  zero  at  a  time  NK  after  the 
removal  of  the  second  twist  will  be  represented  by  the  differences  between 
the  ordinates  KR,  KS  of  the  two  curves.  The  ordinate  of  the  second  curve 
may  be  above  that  of  the  first  at  the  time  tlie  second  twist  is  removed,  and 
yet,  as  the  curve  is  very  steep  just  after  the  removal  of  the  twist, 
curve  (II)  may  drop  down  so  quickly  as  to  cut  the  first,  as  shown  in  the 
figure.  Thus  in  this  case  we  should  have  the  following  effects:  immediately 
after  the  removal  of  the  second  twist  there  would  be  a  displacement  of 
the  zero  in  the  direction  of  the  last  applied  twist,  the  spot  of  light  would 
then  creep  back  to  the  zero  but  would  not  stay  there,  but  pass  through 
the  zero  and  attain  a  maximum  deflection  on  the  other  side  ;  it  would  then 
creep  back  to  the  zero  and  would  not  again  pass  through  it.  In  this 
way,  by  superposing  twists  of  difierent  signs,  we  can  get  very  complicated 
movements  of  the  zero,  which  are  a  source  of  trouble  in  many  instruments 
which  depend  upon  the  torsion  of  fibres.     With  quartz  fibres  the  residual 


ELASTICITY. 


57 


effect^is^exceedingly  smaLl,  and  this  is  one  of  the  chief  causes  which  make 
their  use  so  valuable.  The  residual  after-effect  in  glass  is  a  cause  of 
trouble  in  thermometry,  each  change  of  temperature  causing  a  temporary 
change  in  the  zero. 

The  magnitude  of  the  elastic  after-effect  seems  to  increase  very  greatly 
when~  there  is^  a  want  of  homogeneity  in  the 
constitution  of  the^  body.  In  the  most  homo- 
geneous bodies  we  know,  crystals,  it  is  exceedingly 
small,  if  it  exists  at  all,  while  it  is  very  large  in 
glass  which  is  of  composite  character,  being  a 
mixture  of  different  silicates ;  it  exists  in  metals, 
although  not  nearly  to  the  same  extent  as  in 
glass.  A  similar  dependence  upon  want  of 
uniformity  seems  to  characterise  another  similar 
effect — the  residual  charge  of  dielectrics  {see 
volume  on  Electricity  and  Magnetism),  the  laws 
of  which  are  closely  analogous  to  those  of  the 
I  elastic  after-effect. 

The  phenomenon  of  elastic  after-effect  may 
be  illustrated  by  a  mechanical  model  similar  to 
that  shown  in  Fig.  25. 

A  is  a   spring,  from  the  end,   B,    of   which 
another    spring    0    is    suspended,   carrying  a 


Fig.  25. 


damper    D,    which    moves    in    a    very    viscous 

liquid.     If  B  is  moved  to  a  position  B'  and  kept 

there  for  only  a  short  time,  so  short  that  D  has 

not  time  to  move  appreciably  from  its  original 

position,  then  when  B  is  let  go  it  will  return  at 

once  to  its  original  zero,  for  D  has  not  moved,  so 

that  the  conditions  are  the  same  as  they  were 

before  B  was  displaced.     If,  however,  B  is  kept 

in  the  position  B'  for  a  long  time,  D  wiU  slowly  move  off  to  a  position  D', 

such  that  D'  is  as  much  below  B'  as  D  was  below  B.     If  now  B'  is  let  go 

it  will  not  at  once  return  to  B,  for  in  this  position  the  spring  between  B 

and  D  is  extended,  B  will  slowly  move  back  towards  its  old  zero,  and  will 

only  reach  it  when  the  slow  moving  D'  has  returned  to  D. 

Viscosity  of  Metals  and  Elastic  Fatigue.—  If  two  vertical  wires, 

one  made  of  steel  and  the  other  of 


zmc. 


O' 


FlO.  26. 


are  of  the  same  length  and 

diameter,  and  carry  vibration  bars 
of    the    same    diameter,   then   if 

these  bars  are  set  vibrating  the  vibrations  die  away,  but  at  very  different 
rates :  the  steel  wire  will  go  on  vibrating  for  a  long  time,  but  the  zinc 
wire  will  come  to  rest  after  making  only  a  small  number  of  vibrations. 
This  decay  in  the  vibrations  of  the  wire  is  not  wholly  nor  even  mainly 
due  to  the  resistance  of  the  air,  for  this  is  the  same  for  both  wires ;  it  is 
due  to  a  dissipation  of  energy  taking  place  when  the  £arts  of  a  metal  wire_ 
are  in  relative  motion,  and  may,  from  analogy  with  the  case  of  liquids 
and  gases,  be  said  to  be  due  to  the  viscosity  of  the  metal.  We  can 
see  that  elastic  after-effect  would  cause  a  decay  in  the  vibrations  of 
the  wire.  For  suppose  0,  Fig.  26,  represents  the  original  zero — i.e.,  the 
place  where  the  force  acting  on  the  system  vanishes,  then  if  the  wire  is 


5d 


PROPERTIES  OF  MATTER. 


displaced  to  A  and  then  let  go  the  new  zero  will  be  at  0',  a  point  between 
A  and  0 ;  thus  the  force  will  tend  to  stop  the  vibration  as  soon  as  the 


.<>• 


^ 


Fig.  27. 


I'iG.  2«. 


wire  passes  0' — sooner,  that  is,  than  it  would  do  if  there  were  no  after- 
effect. Again,  when  the  wire  is  on  the  other  side  of  0,  the  zei'o  will  be 
displaced  by  the  elastic  after-effect  to  O",  a  point  between  0  and  B,  and 
thus  again  the  force  tending  to  stop  the  vibration  will  begin  to  act  sooner 

than  it  would  if  there  were  no 
elastic  after-effect.  "We  can  see  the 
same  thing  from  the  study  of  the 
model  in  Fig.  25,  for  some  of  the 
kinetic  energy  will  be  converted  into 
heat  by  the  friction  between  the 
viscous  fluid  and  the  damper  D. 

Lord  Kelvin  discovered  a  remark- 
able property  of  the  viscosity  of 
metals  which  he  called  elasticj^aligue. 
He  found  that  if  a  wTre^  were~kept 
vibrating  almost  continuously  the 
rate  at  which  the  vibrations  died 
away  got  greater  and  greater ;  in 
fact,  the  wire  behaved  as  if  it  got 
tired  and  could  only  with  difiiculty 
keep  on  vibrating.  .  If  the  wire 
were  given  a  rest  for  a  time  it 
recovered  itself,  and  the  vibrations 
for  a  short  time  after  the  rest  did 
not  die  away  nearly  so  rapidly  as 
they  had  gone  just  before  the  rest 
jPjq  29  began.     Muir  {Proc.  Roy.  Soc,  Ixiv. 

p.    337)    found   that   a   metal    wire 
recovered  from  its  fatigue  if  it  were  warmed  up  to  a  temperature  above 

100°  c. 

Anomalous  Effects  on  first  Loading*  a  Wire.— The  extension  pro- 
duced by  a  given  load  placed  on  a  wire  for  the  first  time  is  not  in  general 
quite  the  same  as  that  produced  by  subsequent  loading ;  the  wire  requires 


ELASTICITY. 


59 


to  be  loaded  and  unloaded  several  times  before  it  gets  into  a  steady  state. 
The  first  load  after  a  rest  also  gives,  in  general,  an  irregular  result.  It 
seems  as  if  straining  a  wire  produced  a  change  in  its  structure  from  which 
it  did  not  recover  for  some  time. 

Great  light  will  probably  be  thi'own  on  this  and  the  other  effects  we 
have  been  considering  by  the  examina- 
tion by  the  microscope  of  sections  of 
the  metals.  Wli^n  examined  in  this 
way  it  is  found  that  metals  possess  a 
structure  coarse  enough  to  be  easily 
rendered  visible.  Figs.  27,  28,  29 
show  the  appearance  under  the  micro- 
scope of  certain  metals.  It  will  be 
seen  from  these  figures  that  in  these 
metals  we  have  aggregates  of  crystals 
of  very  great  complexity— the  linear 
dimension  of  these  aggregates  is  some- 
times a  considerable  fraction  of  a 
m'llimetre.  These  large  aggregates 
arc  certainly  altered  by  large  strains. 
Thus  Ewing  and  E,osenhain  (Proc. 
Roy.  Soc,  xlv.  p.  85)  have  made  the 

very  interesting  discovery  that  when  a  metal  is  strained  past  its  yield- 
point  thereis  a  slipping  of  the  crystals,  which  build  ujg^  the  aggregates 
along  tbeiFpTanes  of  cleavage.  The  appearance  of  a  piece  of  iron  after 
straining"  past  the  yield-point  is  shown  in  Fig.  30 ;  the  markings  in 
the   figure   are   due   to  the   steplike  structure  of  the  aggregates  caused 


Fio.  30. 


Before  straining. 


,     ,      ,  y  I  I  I  I  I  I  I  - 


After  straining. 
Fia.  31. 

by  the  slipping  past  each  other  during  the  strain  of  the  crystals  in 
the  aggregates,  as  in  Fig.  31.  Plasticity  may  tlius  be  regarded  as  the 
yielding,  or  rathei^  slipping  past  each  other  of  the-jrystals  of  the  large 
J^ggJ^egatesjwhich  the^microscope  shows  exist  in  metals. 
'^In  harmony  with  this  view  is  the  observation  of  McConnel  and  Kidd 
{Proc.  Hoy.  S'oc,  xliv.  p.  331)  that  ice  in  mass  is  plastic  when  consisting  of 
crystals  irregularly  arranged.  In  later  experiments  {Proc.  Roy.  Soc,  xlix. 
p.  323),  McConnel  found  that  a  single  crystal  of  ice  is  not  plastic  under 
pressure  applied  along  the  optic  axis,  but  that  it  does  yield  under  pressure 


6o 


PROPERTIES  OF  MATTER 


inclined  to  the  axis,  as  if  there  were  slipping  of  the  planes  perpendicular 
to  the  axis. 

If  there  is  a  general  change  in  these  aggregates  under  large  strains  it 
is  possible  that  there  are  soine_aggregates,\vhich  are  unstable  eijough  to 
be  broken  up  by  smaller"^trains,  and  that  the  first  application  is  accom- 
panied byXbreaking^up^f  some^fjbhe^nore  unstable  groups,  so  that  the 
structure  of  thejm^al_jsJligMlx^^"g^^  ^®  ^^  *^^®°  understand  the 
irregularities^ observed  when  a  wire  is  first  loaded  and  also  the  existence 
of  the  elastic  after-eflect.  Indeed,  it  would  seem  almost  inevitable  that 
any  strain  among  such  irregular-shaped  bodies  as  those  shown  in  Fig.  28 
would  result  in  some  of  them  getting  jammed,  and  thus  becoming  exposed 
to  very  great  pressures,  pressures  which  might  be  sufficient  to  break  up 
some  of  the  weaker  aggregates,  and  thus  give  relief  to  the  system.  The 
existence  of  such  a  structure  as  that  shown  in  Fig.  28  causes  us  to 
wonder  whether,  if  a  succession  of  very  accurate  observations  of  the 
elastic  properties  of  a  metal  were  made,  the   results  would   not   difler 

from  each  other  by  more  than  could  be  accounted 
for  by  the  errors  of  experiment. 

The  term  viscosity  is  often  used  in  another 
sense  besides  that  on  p.  57.  We  call  a  substance 
viscous  if  it  cannotresist  the  application  or  a 
small  force  actingj^oi^a  long  time.  Thus  we  call 
prtcE"vIscous  because,  if  given  a  sufiiciently  long 
time,  it  will  flow  like  water ;  and  yet  pitch  can 
sustain  and  recover  from  a  considerable  force  if 
this  acts  only  for  a  short  time.  Fig.  32  shows 
the  way  in  which  some  very  hard  pitch  has 
flowed  through  a  vertical  funnel  in  which  it  has 
been  kept  in  the  Cavendish  Laboratory  for  nine 
years.  In  an  experiment,  due  to  Lord  Kelvin, 
pieces  of  lead  placed  upon  a  plate  of  pitch  found 
in  course  of  time  their  way  through  the  plate. 
Many  substances,  however,  show  no  trace  of 
viscosity  of  this  kind,  for  the  existence  of  sharp 
impressions  on  old  coins,  the  preservation  of 
bronze  statues  and  the  like,  show  that  metals  can 
sustain  indefinitely  (or  at  any  rate  so  nearly 
indefinitely  that  no  appreciable  change  can  be  detected  after  thousands  of 
years)  their  shape  even  under  the  application  of  small  forces. 

Breaking-  of  Wires  and  Bars  by  Tension.— The  following  table, 

due   to  Wertheim,  gives  the  load  in  kilogrammes  per  square  millimetre 
necessary  to  break  wires  of  difierent  substances : 


Fio.  82. 


Lead  . 

.       21 

Copper 

.     40-3 

Tin      . 

.       2-5 

Platinum     . 

.     341 

Gold   . 

.     27 

Iron     . 

.     Gl 

Silver  . 

.     29 

Steel  Wire  . 

.     70 

Zinc    . 

.     128 

The  process  of  drawing  into  wire  seems  to  strengthen  the  material, 
and  the  finer  the  wire  the  greater  is  the  pull,  estimated  per  unit  araa  of 
cross  section,  required  to  break  it.  This  is  shown  in  the  following  table 
given  by  Baumeister  (Wiedemann,  Annakii,  xviii.  p.  C07) : 


Material. 


Swedish  Iron 


»> 
i» 
n 

j> 
Brass 

a 
n 


>» 
n 

>> 


ELAS 

Ticny. 

Diameter  of  wire            ^""  ^"  kilogramrres 
x^iaiiietci  ui  wiiB                          ^^^^   required 

^"  ™°"-                        to  break  the  wire. 

.     -72 

.      Gi 

•50 

83 

•30 

96 

•25 

94 

•15 

98 

■10 

123 

•75 

76 

.     ^25 

.      98 

.     •lO 

.      98 

61 


The  efl'ect  of  temperature  on  the  pull  required  to  break  a  wire  is  com- 
plex. Iron  wire  shows  several  maxima  and  minima  between  15°  C. 
and  400°  C.  (Pisati,  Bend.  Ace.  Lincei.  1876,  76);  the  strength  of  copper, 
on  the  other  hand,  steadily  diminishes  as  the  temperature  increases. 

The  strength  of  a  material  is  sometimes  very  seriously  affected  by  the 
addition  of  only  a  small  quantity  of  another  substance.  Thus  Sir  William 
Roberts-Austen  found  that  gold,  to  which  2  per  cent,  of  potassium  had 
been  added,  could  only  sustain  1/12  of  the  weight  required  to  break 
pure  gold.  In  the  case  of  steel,  the  addition  of  small  quantities  of  carbon 
to  the  iron  increases  the  strength.  The  microscopical  examination  of  the 
structure  of  metals,  such  as  is  shown  in  Figs.  27—30,  may  be  expected  to 
throw  a  good  deal  of  light  on  effects  of  this  kind.  In  this  way  it  has  been 
shown  that  the  foreign  substance  is  sometimes  collected  between  the 
aggregates  of  the  crystals  of  the  original  metals  forming  a  weak  kind  of 
mortar,  and  thus  greatly  reducing  the  strength  of  the  metal.  In  other 
cases,  such  as  steel,  a  carbide  is  formed,  and  the  appearance  of  a  section 
of  the  steel  under  the  microscope  shows  that  the  structure  is  much 
finer  than  in  pure  iron.  It  would  seem  from  Sir  William  Roberts-Austen's 
experiments  that  the  addition  to  gold  of  a  metal  of  greater  atomic  volume 
than  the  gold  diminishes,  while  a  metal  of  smaller  atomic  volume  increases 
the  strength. 


CHAPTER  V. 


STRAIN. 


Contents. —  Homogeneous    Strain  —  Principal    Axes    of    Strain  —  Pure    Strain  — 
Elongation — Dilation  or  Compression — Contraction — Shear — Angle  of  Shear. 


"When  a  body  changes  in  shape  or  size  it  is  said  to  be  strained,  and  the 
deformation  ot  the  bodyis  called  strain. 

HomOgreneOUS  Strain. — We  shall  restrict  ourselves  to  the  most  simple 
class  of  strain  to  which  bodies  can  be  subjected  ;  this  is  when  any  two  lines 
which  are  equal  and  parallel  before  straining  remain  equal  and  parallel 
after  straining.     This  kind  of  strain  is  called  homogeneous  strain. 

Thus  by  a  homogeneoua  strain  a  parallelogram  is  strained  into  another 
parallelogram,   though  its  area  and  the 
altered    by   straining ; 


angle 
parallel    planes   strain 
B 


between  its  sides  may  be 
into   parallel    planes,    and 


Fig.  33. 

parallelepipeds  into  parallelopipeds.  Figures  which  are  similar  before 
straining  I'emain  similar  after  the  strain. 

It  follows  from  the  definition  of  homogeneous  strain  that  the  ratio  of 
the  length  of  two  parallel  lines  will  be  unaltered  by  the  strain.  Let  AB 
and  CD  (Fig.  33)  be  two  parallel  lines.  Let  the  ratio  of  A B  to  CD  be  m  :  n. 
Then,  if  m  and  n  be  commensurable,  we  can  divide  AB  and  CD  respectively 
into  Nm  and  N«,  equal  parts  each  equal  to  a.  Then,  as  before  straining  all 
these  parts  are  equal  and  parallel,  they  will  remain  so  after  a  homogeneous 
strain.  Thus  AB,  after  straining,  will  consist  of  Nm  and  CD  of  Nn  parts, 
each  equal  to  a  ;  and  the  ratio  of  the  strained  lengths  is  m  :  w,  the  same 
as  that  of  the  unstrained  lengths.  If  m  and  n  are  not  commensurable  we 
can  deduce  the  same  result  in  the  usual  way  by  the  method  of  limits. 

From  this  result  we  can  at  once  prove  that  a  sphere  is  strained  into  an 
ellipsoid,  and  that  three  mutually  perpendicular  diameters  of  the  sphere 


STRAIN. 


63 


strain  into  three  conjugate  diameters  of  the  ellipsoid.  As  some  of  our 
readers  may  not  be  familiar  with  solid  geometry,  we  shall  confine  our 
attention  to  strains  in  one  plane  and  pi'ove  that  a  circle  is  strained  into 
an  ellipse;  the  reader  who  is  acquainted  with  solid  geometry  will  not 
have  any  difiiculty  in  extending  the  method  to  the  case  of  the  sphere. 
Let  ABA'B'  (Fig.  34)  be  a  circle,  centre  0,  which  strains  into  aba'b', 
corresponding  points  on  the  two  figures  being  denoted  by  corresponding 
letters.     Let  P  be  a  point  on  the  circle,  PL  and  PM  parallel  to  CA 


Fm.  34. 

and  CB  respectively;   let  these  lines  on  the  strained  figure  be  denoted 
by  pi,  pm. 

Thus,  since  the  ratio  of  parallel  lines  is  not  altered  by  the  strain 

PL^pZ 

CA     ca  _ 

PM  _pm 
CB'^cb 


But  since  P,  A,  B  are  on  a  circle  whose  centre  is  0 

1 


PL2     PM^ 


hence 


ca-      CO- 


OT p  is  on  an  ellipse  of  which  ca  and  cb  are  conjugate  diameters.  Thus 
a  circle  is  strained  into  an  ellipse,  and  two  diameters  at  right  angles  to 
each  other  in  the  circle  strain  into  two  conjugate  diameters  of  the  ellipse. 
Now  there  are  two,  and  only  two,  conjugate  diameters  of  an  ellipse 
(unless  the  ellipse  degenerates  into  a  circle)  which  are  at  right  angles  to 
each  other.  Hence  there  are  two,  and  only  two,  diameters  at  right  angles 
to  each  other  before  straining  which  remain  at  right  angles  after  the  strain. 
Now,  though  in  general  these  diameters  will  not  have  the  same  direction 


64 


PROPERTIES  OF  MATTER. 


after  straining  as  they  had  before,  yet  we  shall  not  be  introducing  any 
real  limitation  on  the  strain  in  so  far  as  it  affects  the  forces  called  into  play 
by  elasticity  if  we  suppose  they  retain  the  same  direction  after  straining 
as  before.  For,  suppose  OA,  OB  (Fig.  35),  are  the  unstrained  directions, 
Oa,  Ob,  the  strained  ones,  we  can  make  Oa,  06  coincide  with  OA,  OB  by 
rotating  the  strained  system  as  a  rigid  body  through  the  angle  AOa. 
This  rotation  as  a  rigid  body  will  not  involve  any  relative  motion  of  the 
parts  of  the  system,  and  so  will  not  call  into  play  any  forces  depending 
upon  the  elasticity  of  the  system ;  if,  then,  as  is  at  present  the  case,  our 
object  is  to  investigate  the  connection  between  these  forces  and  the  strains, 
we  may  leave  the  rotation  out  of  account. 

The  three  directions  at  right  angles  to  each  other  which  remain  at  right 
angles  to  each  other  after  sti\aining  are  called  the  principal  axes  of  strain. 
If  these  axes  have  the  same  direction  after  straining  as  before,  the  strain 

is  said  to  be  a  pure^trainj 
if  it  requires  a  rotation  to 
make  the  principal  axes 
after  straining  coincide 
with  their  position  before 
the  strain,  the  strain  is 
said  to  consist  of  a  pure 
strain  and  a  rotation. 

Thus  the  most  general 

homogeneous    strain    may 

be  resolved  into  extensions 

(regarding    a   compression 

/j  as    a   negative    extension) 

Fig.  35.  along   three    directions    at 

right  angles  to  each  other, 
fake  these  directions  as  the  axes  of  x,  y,  z  respectively,  then  if  a  line  of 
unit  length  parallel  to  the  axis  of  x  has,  after  the  strain,  a  length  1+e ; 
one  parallel  to  the  axis  of  ?/  a  length  1  +/;  and  one  ♦parallel  to  the  axis  of  » 
a  length  1  +  g',  e,  f,  g  are  called  the  principal  elongations.  If  e  =/=  g, 
then  a  sphere  strains  into  a  sphere,  or  any  figure  into  a  similar  figure, 
the  strained  figure  being  an  enlarged  or  diminished  copy  of  the  unstrained 
one.  These  cases,  which  are  called  uniform  dilatation  or  compression, 
involve  changes  in  size  but  not  in  shaped 

A  cube  whose  sides  were  parallel  to  the  axes  before  straining  and  one 
unit  in  length  becomes  after  straining  a  rectangular  parallelepiped,  whose 
edges  are  1  +  e,  1  +/,  1+9'  respectively,  and  whose  volume  is  (1+e)  (1  +f) 
(1  +g).  If,  as  we  shall  suppose  all  through  this  chapter,  the  elongations 
e,  y,  g  are  such  small  fractions  that  the  products  of  two  of  them  can  be 
neglected  in  comparison  with  e,  f,  or  g,  the  volume  of  the  parallelepiped 
is  l+e+Z+gr. 

Hence  the  increase  of  unit  volume  due  to  the  strain  is  e+f+g.  This 
is  called  the  cubical  dilatation .     We  shall  denote  it  by  I,  and  we  have 

l-'e+f+g. 

If  the  strain  is  a  uniform  dilatation  e=f=g,  and  therefore 

CO  that  in  this  case  the  cubical  expansion  is  three  times  tb»  linear  elongation. 


STRAIN. 


65 


Resolution  of  a  Homogfeneous  Strain  into  Two  Strains,  one  of 
which  chang-es  the  Size  but  not  the  Shape,  while  the  other 
changfes  the  Shape  but  not  the  Size. 

Let  us  consider  the  case  of  a  strain  in  one  plane.  Let  OA,  OB  (Fig.  30) 
be  the  principal  axes  of  strain.  Let  P  be  the  initial  position  of  a  point,  P'  its 
position  after  the  strain.  Then  if  e,fixve  the  elongations  parallel  to  OA  and 
OB,  '£,  and  rj  the  displacements  of  P  parallel  to  OA  and  OB  respectively, 

;  =  eON  =  i  (e  +/)0N  +  i(e  -/)0N, 
„  =/0M  =  h{e  +/)0M  -  i(e  -/)0M. 

From  these  expressions  wc  see  that  we  may  regard  the  strain  e,  f 
as  made  up  of  a  uniform  ^ 
dilatation  equal  to  -v(e+/), 
together  with  an  elongation 
i(e  -/)  along  OA,  and  a  con- 
traction ^(e  -/)  along  OB. 
Thus  the  strain  superposed 
on  the  uniform  dilatation  con- 
sists of  an  expansion  along 
one  of  the  principal  axes  and 
an  equal  contraction  along 
the  other.  This  kind  of  strain 
does  not  alter  the  size  of  the 
body ;  for  if  a  is  the  elonga- 
tion along  OA  and  the  con- 
traction   along    OB,    then    a 

square  whose  sides  are  one  unit  in  length  and  parallel  to  the  principal 
axes  becomes  a  rectangle  whose  sides  are  1  -1-  o-,  and  1  -  o-  respectively  ;  the 
area  of  this  lectangle  is  1  -  ff",  or  since  we  neglect  the  square  of  o-  the  area 
is  unity,  and  thus  is  not  altered  by  the  strain.  A  strain  which  does  not 
alter  the  size  is  called  a  shear.  Thus  any  strain  in  one  plane  can  be 
resolved  into  a  uniform  dilatation  and  a  shear. 

We  have  considered  a  shear  as  an  extension  in  one  direction  and  an 
equal  compression  in  a  direction  at  right  angles  to  this ;  there  is,  however, 
another  and  more  usual  way  of  considering  a  shear,  which  may  be  deduced 
as  follows : 

Let  OA,  OB  (Fig.  37)  be  the  axes  along  which  the  extension  and 
contraction  take  place.  Let  OA  ==  OB  =  OA'  =  OB'  =  1 ,  so  that  before 
straining  ABA'B'  is  a  square  ;  let  this  square  after  straining  be  represented 
by  aha'b',  which  will  be  a  parallelogram. 

Since  Oa  =  1  +  <r 

06  -  1  -  <r 

ah' =  2 +  2(7-' 
=  2 

as  we  suppose  that  o-  is  so  small  that  its  square  may  be  neglected.  Thus 
a6  =  AB.  Hence  we  can  move  aha'h'  as  a  rigid  body  and  place  it  so  that  a6 
coincides  with  AB,  as  in  Fig.  38.  Then,  since  the  area  of  aba'h'  is  equal  to 
that  of  ABA'B',  when  the  figures  are  plactd  so  as  to  have  one  side  in  common 

E 


(i6 


PROPERTIES  OF  MATTER. 


they  will  lie  between  the  same  parallels.  Thus,  if  a"6"  be  the  position  of  a'V 
when  ah  is  made  to  coincide  with  AB,  a"b"  (Fig.  38)  will  lie  along  A'B' ; 
hence,  except  with  regard  to  the  rotation,  the  expansion  along  AO  and  the 


Fig.  37. 

contraction  along  OB  is  equivalent  to  the  strain  which  would  bring  ABA'B' 
into  the  position  ABa"6".  But  we  see  that  this  could  be  done  by- 
keeping  AB  fixed  and  sliding 
every  point  in  the  body  par- 
allel to  AB  through  a  distance 
proportional  to  its  distance 
from  AB.  We  can  illustrate 
this  kind  of  strain  by  a  pack 
of  cards  lying  on  the  table, 
with  their  ends  in  vertical 
■^  planes ;  now  slide  the  cards 
forward,  keeping  the  lowest 
one  at  rest  in  such  a  way 
that  the  ends  are  still  flat 
although  the  planes  are  no 
longer  vertical ;  each  card 
will  havebeen  moved  forwards 
through  a  distance  propor- 
tional to  its  distance  from  the 
lowest  card .  The  angle  A'Ba" 
tliroiigh  which  a  line  is  dis- 
placed which  to  begin  with  is  perpendicular  to  AB  is  called  the  angle  of 
shear.  The  plane  of  the  shear  is  a  plane  parallel  to  tbe_  direction  of 
motion  anci  at  riglit  angles  to  theirxecTpIane. 

'  The  relation  between  y — tlie  circular  measure  of  the  angle  of  shear — and 
the  elongation  a  along  OA,  and  the  contraction  it  along  OB  can  be  found  as 
follows.  Before  the  rotation  making  ah  coincide  witli  Al),  ha!  makes 
with  BA'  the  angle  B^6  ;  to  make  ah  coincide  with  AB  (Fig.  ^7)  the  system 
has  to  be  rotated  through  the  angle  Bp6,  so  that  after  the  rotation  ha  will 


Fig.  33. 


STRAIN.  67 

make  with  BA'  the  angle  B^^  +  Bpb.  Now  by  the  figure,  Bqh  =  Rph,  hence 
the  angle  of  shear  is  2  L  liqb  =  2  ^apA.  If  Am  is  perpendicular  to  ap  (Fig.  37), 
then,  since  the  angle  apA  is  by  hypothesis  small,  its  cii'cular  measure 

Km _  A(xsin45       Aa_ 

^Tp^  TaojW  "ao""' 

hence  d,  the  circular  measure  of  the  angle  of  shear,  =  2<r. 

If  e  and  /  are  the  extensions  along  two  principal  axes  in  the  genei'al 
case  of  homogeneous  strain  in  two  dimensions,  we  see  from  p.  G5  that  this 
strain  is  equivalent  to  a  vmiform  dilatation  ^  (e  +/)  and  to  a  shear  the 
circular  measure  of  w^hose~angle  Is  e  —f,  '~ 


CHAPTER  VI. 
STRESSES.     RELATION  BETWEEN  STRESSES  AND  STRAINS. 

Contents. — General  Considerations — Hooke's  Law — Work  required  to  produce  any 
Strain — Rectangular  Bar  acted  upon  at  Right  Angles  to  its  Faces. 


^71. 


In  order  that  a  body  may  be  strained  forces  must  act  upon  it.  Consider  a 
small  cube  in  the  middle  of  a  sti-ained  solid,  and  suppose  for  a  moment  that 
the  external  forces  are  confined  to  the  surface  of  this  solid.  Then  the  forces 
which  strain  this  cube  must  be  due  to  the  action  exerted  upon  it  by  the 
surrounding  matter.  These  forces,  which  are  due  to  the  action  of  the 
molecules  outside  the  cube  on  those  inside,  will  only  be  ajjpreciable  at 
molecular  distances  from  the  surface  of  the  cube,  and  may  therefore 
without  appreciable  error  be  supposed  to  be  confined  to  the  surface.     The 

most  general  force  which  can 
act  on  a  face  ABCD  of  the 
cube  may  be  resolved  into 
three  components,  one  at  right 
angles  to  ABCD,  the  other 
two  components  in  the  plane 
of  ABCD,  one  parallel  to  AB, 
the  other  to  BC :  similarly 
over  the  other  faces  of  the 
cube  we  may  suppose  similar 
forces  to  act.  These  forces 
are  called  stresses ;  the  com- 
ponent at  right  angles  to  a 
face  is  called  a  normal  stress, 
the  component  parallel  to  the 
Fig.  39.  face  a  tangential  stress.     The 

intensity  oi  any  component  of 
the  stress  is  the  amount  of  the  component  over  the  face  divided  by  the 
area  of  the  face.  We  shall  for  brevity  leave  out  the  word  "  intensity^' 
and  speak  of  it  simply  as  the  stress.  The  dimensions  of  a  stress  are  those 
of  a  force  divided  by  an  area  or  M/LT-.  It  is  measured  in  dynes  per 
square  centimetre ;  on  the  C.G  S.  system  of  units  the  pressure  of  the 
atmosphere  is  about  10"  units  of  stress. 

When  we  know  the  stresses  over  three  planes  meeting  at  a  point  O 
(Fig.  40)  we  can  determine  the  stresses  on  any  other  plane  through  O.  For 
let  OABC  be  a  very  small  tetrahedi-on,  AOB,  BOC,  COA  being  the  pianos 
over  which  we  know  the  stresses,  and  ABC  being  parallel  to  the  plane  across 
which  we  wish  to  determine  the  stress.  Then  as  this  tetrahedron  is  in 
equilibrium  under  the  action  of  forces  acting  on  its  four  faces,  and  as  we 


/>k^- y. 


STRESSES. 


fin 


1 


know  the  forces  over  three  of  the  faces,  OAB,  OBC,  OCA,  we  can 
determine  the  force,  and  hence  the  stress,  on  the  fourth.  We  need  not 
take  into  account  any  external  forces  which  are  proportional  to  the  volume 
on  which  they  act,  for  the  forces  due  to  the  stresses  are  proportional  to  the 
area  of  the  faces,  that  is,  to  the  square  of  the  linear  dimensions  of  the 
tetrahedron,  while  the  external  forces  are  proportional  to  the  cube  of  the 
linear  dimensions,  and  by  making  the  linear  dimensions  of  the  tetrahedron 
exceedingly  small  we  can  make  the  eflect  of  the  volume  forces  vanish  in 
comparison  with  that  of  the  surface  forces. 

The  stresse§_in  a  strained  solid  constitute  a  system  of  forces  which  are 
in  equilibrium  at  each  part  of  the  solid  with  the  external  forces  acting"o"n 
the  solid!     If  we  call  the  external  forces  the  load,  then  if  a  load  W  pro- 
duces  a  system  of  stresses  P, 
and  a  load   W   a   system   of 
stresses  P',  then  when  W  and 
W  act  together  the  stresses 
will  be  P  +  P'  if  the  deforma- 
tion produced  by  either  load 
is  small. 

Hooke's  Law.— The  fun- 
damental law  on  which  all 
applications  of  mathematics 
to  elasticity  are  based  is  due 
to  Hooke,  and  was  stated  by 
him  in  the  form  ut  tensio  sio 
vis,  or,  in  modern  phraseology, 

that  tjif  gfraiy^s  are  propor- 
tional to  the  loads.  The  truth 
ot  this  law,  when  the  strains 
do  not  exceed  the  elastic  limit 
(see  p.  53),  has  been  verified 
by  very  careful  experiments 
on  most  materials  in  common 

use.     Another  way  of  stating  Fio.  40. 

Hooke's  Law  is  that  if  a  load 

W  produces  a  strain  S,  and  a  load  W  a  strain  S',  then  a  load  W  +  W  will 
produce  a  strain  S  +  S'.  Hence,  it  follows  from  the  last  article  that  if  a 
system  of  stresses  P  correspond  to  a  system  of  strains  S,  and  a  system  of 
stresses  P'  to  a  system  of  strains  S',  then  a  system  of  stresses  P  -|-  P'  will 
correspond  to  a  system  of  strains  S  -|-  S'.  Hence,  if  we  know  the  stress 
corresponding  to  unit  strain,  we  can  find  the  stress  corresponding  to  a 
strain  of  any  magnitude  of  the  same  type.  Thus,  as  long  as  Hooke's  law 
holds  good,  the  stress  and  strain  will  be  connected  by  a  relation  of  the 
form 

Stress  =  c  X  strain 


where  c  is  a  quantity  which  does  not  depend  either  upon  the  stress  or  the 
strain.  It  is  called  a  modulus  of  elasticity.  Thus,  if  the  strain  corresponds 
to  a  change  in  size  but  not  in  shape,  then  the  stress  is  a  uniform  pressure, 
and  the  strain  the  diminution  in  volume  of  unit  volume  of  the  unstrained 
substance ;  in  this  case  c  is  called  the  modulus  of  elasticity  of  bulk,  or 
jnore  frequently  the  bulk  modulus.     Again,  il"  the  strain  is  a  shear  wliich 


w 


>{k 


70 


PROPERTIES  OF  MATTER. 


.-^< 


^  .  alters  the  shape  but  not  the  size,  the  strain  is  measured  by  the  angle  of 

^-r"^^^  shear  and  the  stress  by  the  tangential  force  per  unit  area,  which  must  bo 
applied  to  produce  this  sliear.  In  this  case  c  is  called  the  modulus  of 
rigidity.  If  we  stretch  a  wire  by  a  weight,  the  stress  is  the  weight  divided 
by  theiuea  of  cross  section  of  the  wire,  the  strain  is  the  increase  of  length 
in  unit  length  of  the  wire,  and  in  this  case  c  is  called  Young's  modulus. 
Since  we  can  reduce  the  most  general  system  of  homogeneous  strain  to 
a  uniform  expansion  or  contraction  and  a  system  of  shears  (see  p.  65)  it 
follows  that  if  we  know  the  behaviour  of  the  body  (1)  when  its  size  but  not 
its  shape  is  changed,  and  (2)  when  its  shape  but  not  its  size  is  changed,  we 
can  determine  its  behaviour  under  any  homogeneous  strain.  This  is  true 
when,  and  only  wlien,  the  properties  of  the  substance  are  the  sjime  in  all 
directions,  so  that  a  uniform  hydrcstatic  pressure  produces  no  change  in 


L  ^f  JV 
Fig.  41. 

shape,  and  the  tangential  stress  required  to  produce  a  given  angle  of  shear 
is  independent  of  the  plane  of  the  shear.  This  statement  is  equivalent  to 
saying  that  it  only  requires  two  moduli — i.e.,  the  bulk  modulus  and  the 
modulus  of_iigiclitv.  to  tix  the  elastic  behaviour  of  the  substance,  so  that  all 
other  moduli,  such  as  Young's  modulus,  must  be  expressible  in  terms  of 
these  two. 

Work  required  to  produce  any  Strain.— The  result  for  the  most 

general  case,  and  the  method  by  wliich  it  can  be  obtained,  may  be  illus- 
trated by  considering  the  work  required  to  stretch  a  wire.  Let  us  suppose 
that  the  load  is  added  so  gradually  that  the  scale-pan  in  which  the  weighta 
are  placed  never  acquires  an  appreciable  velocity,  so  that  none  of  the  work 
done  is  converted  into  kinetic  energy,  but  all  is  spent  in  stretching  the 
wire.  When  this  is  the  case,  the  weight  in  the  scale- pan  when  in  any 
position  never  exceeds  by  more  than  an  infinitesimal  amount  the  weight 
required  to  stretch  the  wire  to  that  position. 

Let  the  straight  line  AB,  Fig.  41,  represent  the  relation  between 
the  weight  in  the  scale-pan  and  the  extension  of  the  wire,  the 
weight  being  the  ordinate  and  the  extension  the  abscissa ;  let  OA  repre- 


STRESSES.  71 

eent  the  unstretched  length  of  the  wire.  Consider  the  work  done 
in  stretching  the  wire  from  L  to  M,  where  L  and  M  are  two  points  veiy 
near  together.  The  force  will  be  approximately  equal  to  PL ;  thus 
the  woi-k  done  in  stretching  from  L  to  M  will  be  PL  x  LM — i.e.,  the 
area  PLMQ' ;  similarly,  the  work  done  in  stretching  the  wire  from  M  to  N 
will  be  represented  by  the  area  QMNIl',  and  thus  the  work  spent  in 
stretching  the  wire  from  OA  to  OG  will  be  repiesented  by  the  sum  of  the 
little  rectangular  areas ;  but  when  these  rectangular  areas  are  veiy  small, 
their  sum  is  equal  to  the  area  ABO,  and  this  equals  iBO  x  AC — i.e.,  one- 
half  the  final  weight  in  the  scale-pan  x  extension  of  the  wire.  Let  a  be 
the  area  of  cross  section  of  the  wire  and  I  the  length,  then  EC  ■=«  x  stress 
and  AC  =  ^  X  strain.  Thus  the  work  done  in  stretching  the  wire  is  equal 
to  al  X  I  strain  x  stress.  Now  al  is  the  volume  of  the  wire,  hence  the 
energy  in  each  unit  volume  of  the  wire  is  -J  strain  x  stress.  Though  we 
have  considered  a  special  case,  it  will  be  seen  that  the  method  is  of  general 
application,  and  that  the  result  will  hold  whenever  Ilooke's  law  is  true. 

We  have  considered  two  ways  of  regarding  a  shear  :  one  where  the 
paiticles  of  the  body  were  pushed  forward  by  a  tangential  force  as  is 
represented  in  Fig.  38.  In  this  case  the  work  done  on  unit  volume,  which 
is  the  energy  possessed  by  the  sheared  body,  is 

where  T  is  the  tangential  force  per  unit  area  and  d  the  angle  of  shear. 

The  other  way  of  regarding  a  shear  is  to  consider  it  as  an  extension  in 
one  dii'ection  combined  with  an  equal  contraction  in  a  direction  at  right 
angles  to  the  extension.  Let  e  be  the  magnitude  of  the  extension  or 
contraction,  P  the  pull  per  unit  area  producing  the  extension  ;  this  is  equal 
to  the  push  per  unit  area  producing  the  contraction.  Considering  unit 
volume  of  the  strained  body,  the  work  done  by  the  pull  is  |  Pe,  and  that 
by  the  push  is  also  ^  Pe;  hence  the  energy  per  unit  volume  is  ^  Pe  +  ^  Pe  =  Pe, 
but  this  energy  is  also  equal  to  J  Td,  hence 

Pe  =  lT0. 

But  we  know  (p.  67)  that  d  =  2e,  hence 

P  =  T. 

Hence  the  pull  or  push  per  unit  area  in  the  one  way  of  considering  a 
shear  is  equal  to  the  tangential  stress  per  unit  area  which  occurs  m  tEe 
other  way. 

it  n  IS  the  coefficient  of  rigidity,  then  by  the  definition  of  n  given  on 
p.  70, 

T  =  n6 

hence  P  =  2ne 

P 

or  6  =  — 

2n 

Rectangular  Bar  acted  on  by  Forces  at  Right  Angles  to  its 

Faces. — Let  ABCDEFGH,  Fig.  42,  be  a  rectangular  bar  Let  the 
faces  CDEF,  ABGH  be  acted  on  by  normal  pulls  equal  toP  per  unit  area, 
the  faces  A  BCD,  EFHG  by  normal  pulls  equal  to  Q  per  unit  area,  and  the 
faces  DEGB,  CFHA  by  normal  pulls  equal  to  R  per  unit  area.     We  shall 


72 


PROPERTIES  OF  MATTER. 


proceed  to  find  the  deformation  of  the  bar.  Considering  the  bar  .^s  made 
up  of  rectangular  pai-allelopipeds,  with  their  faces  parallel  to  the  bar,  we  see 
that  these  Avill  all  be  in  equilibrium,  whether  they  are  in  the  interior  of  the 
bar  or  whether  some  of  their  faces  are  on  the  surface  of  the  bar,  if  the 
normal  stresses  parallel  to  AC,  CD,  DE  are  respectively  equal  to  P,  Q,  R, 
and  if  there  are  no  tangential  stresses.  Each  of  these  parallelepipeds  will  be 
subject  to  the  same  stresses,  and  will  therefore  be  strained  in  the  same 
way.  Let  e,/,  g  be  the  extensions  parallel  to  P,  Q,  R  respectively.  Con- 
sider for  a  moment  what  the  strains 
would  be  if  the  stress  P  acted  alone :  P 
would  produce  an  extension  proportional 
to  P  in  the  direction  of  P ;  let  us  call 
this  XP;  it  would  also  produce  contraction 
proportional  to  P  in  any  direction  at 
right  angles  to  P ;  and  if  the  properties 
of  the  strained  substances  were  the  same 
in  all  directions,  then  the  contractions 
would  be  the  same  in  all  directions  at 
right  angles  to  P ;  let  these  contractions 
"^"be  yuP.  Then  when  P  acts  alone  the 
extensions  parallel  to  P,  Q,  R  respectively 
are  XP,  -  juP,  -  /iP  ;  similarly  when  Q 
acts  alone  the  extensions  in  these  directions 
are  -/iQ,  XQ,  -juQ,  and  when  R  acts 
alone  the  extensions  are  -  fxR,  -  /iR,  XR ; 
consequently  when  these  stresses  act  simul- 
taneously we  have 


e  =  XP  -  yuQ  -  ^iR" 
/=-^P  +  XQ-^R 
g=  -^iP-^Q  +  XR 


(1) 


Now  we  have  seen  (p.  70)  that  the 

elastic    pi-operties   of  the   substance   are 

completely  defined  if  we  know  the  bulk 

rio.  42.  modulus,  which  we  shall  denote  by  k,  and 

the   modulus   of  rigidity  which  we  shall 

denote  by  n.     Hence  we  must  be  able  to  express  X  and  ^  in  terms  of  n 

and  k.     We  proceed  to  do  this.     If  we  apply  a  uniform  tension  to  each 

side  of  the  bar  equal  to  P  the  dilatation    of  unit  volume  is  equal  to  Vjk, 

by  the  definition  of  k ;  but  in  this  case  the  dilatation  is  uniform  in  all 

directions,  and  the  linear  dilatation  is  one-third  of  the  volume  dilatation 

— i.e.,  it  is  equal  to  Tj^k. 

p 
Hence,  when  P  =  Q  =  R,  e=/=gr  =  _-, 

ok 

hence,  from  equations  (1)  --  =  X  -  2/t. 

ok 

Let  us  now  shear  the  body  in  the  plane  of  PQ — i.e.,  put  Q  =  -  P  and 
R  =  0.     In  this  case  e=  -/=  P/2n  (see  p.  71) ;  hence  by  equations  (1) 


2^-'+'- 


STRESSES.  73 


l>ink 


g  =  l{^-a{V  +  q)\ 


If  the  bar  is  prevented  from  contracting  laterally, 

hence  Q  =  R  = , 

1  —  ff 

P/  2er2 

80  that  e  =  —  1  -  - — 

-  '  o  \       1  -  c 


/  ^, 


3\?i      o^/        'J7^^ 

Yoiing''s  Modulus. — A  very  important  case  is  that  of  a  bar  acted  on 
by  a  pull  parallel  to  its  length,  while  no  forces  act  at  right  angles  to  the 
length.     In  this  case  Q  =  R  =  0,  and  we  have 

e  =  XP,/=  -/zP,r/=-yuP. 

But  in  this  case  the  stress,  divided  by  the  longitudinal  strain,  is  called 
Young's  modulus  ;  hence,  if  we  denote  Young's  modulus  by  q,  we  have, 

This  equation  gives  Young's  modulus  in  terms  of  the  bulk  modulus  and 
the  rigidity. 

PoiSSOn's  Ratio. — Poisson's  ratio  is  defined  to  be  the  ratio  of  the 
lateral  contractiPiTto  the  longitudinal  extension  for  a^ar  acted  on  by  a 
stress  parallel  to  its  length.     If  we  denote  it  by  <r,  then  by  this  definition 

f 
9-= -^,  when  Q  =  R  =  0.  __  _      ^.7 

Thus  (T  =  "  =  —r—, — -^  . 
\     2{U  +  n) 

Since  n  is  a  positive  quantity,  we  see  from  this  expression  that  c  must 
be  less  than  1/2.  According  to  a  molecular  theory  worked  out  by  Cauchy 
and  Poisson,  o-,  for  all  non-crystalline  substances,  is  equal  to  1/4.  The 
determinations  of  or  given  in  the  table  of  elastic  constants  on  p.  102  do 
not  lend  much  support  to  this  view. 

Bar  stretched  long-itudinally,    with  its   Sides   fixed.— The 

equations  (1)  may  be  written 

«=lfp-<T(Q  +  Il) 

q\ 

9\ 


74 


PRO  PR  [{TIES  OF  MATTER. 


B 


Hence  the  elongation  is  less  than  if  the  sides  of  the  bar  were  free  in 

the  ratio  of  1  -  , — -  to  1.     In  the  case  of  a  steel  bar  for  which  o-  =  -268 
i.  —  a 

the  elongation  if  the  sides  were  fixed  would  be  about  4/5  of  the  elonga- 
tion when  the  sides  are  free. 

Determination  of  Young^'s  Modulus. — A  simple  way  of  measuring 

Young's  modulus  for  a  wire  of  which  a  considerable  length  is 
available  is  the  following  :  Fix  as  long  a  length  of  the  wire 
AB,  Fig.  4;-i,  as  is  available  firmly  to  a  support.  Another 
wire,  CD,  which  need  not  be  of  the  same  material,  hangs  from 
the  same  suppoit  down  by  the  side  of  the  first  wire.  CD 
carries  a  millimetre  scale,  the  length  of  the  scale  being  parallel 
to  the  wire ;  a  weight  is  attached  to  the  end  of  this  wire  to 
keep  it  straiglit.  A  vernier  is  attached  to  the  wire  AB  and 
moves  against  the  scale  fixed  to  the  wire  CD.  The  wire  AB 
carries  a  scale-pan  into  which  various  weights  can  be  placed. 
By  reading  the  vernier  when  different  weights  are  on  the 
scale-pan  we  get  the  vertical  depression  of  a  fixed  point  on  the 
vernier,  that  is  of  a  known  point  on  the  wire,  produced  by  a 
given  weight.  Let  this  depression  be  e,  when  the  weight  in 
the  scale-pan  is  increased  by  W.  Measure  the  length  of  the 
wire  between  the  fixed  support  and  the  point  of  attachment  to 
the  vernier ;  let  this  be  I,  then  the  elongation  per  unit  length 
is  til.  If  w  is  the  cross  section  of  wire,  then  the  stress  which 
produces  this  elongation  is  W/'w,  so  that,  as  Young's  modulus 
is  stress  divided  by  strain,  it  is  equal  to 


To  determine  the  cross  section,  the  most  accurate  way  is  to 
weigh  a  known  length  of  the  wire,  first  in  air  and  then  in 
water.  The  difierence  of  the  weighings  in  grammes  will  be 
the  volume  of  the  wire  in  cubic  centimetres,  and  if  we  divide 
the  volume  by  the  length  we  get  the  cross  section.  Preliminary 
measurements  should  have  been  taken  with  a  screw  gauge  to 
see  that  the  wire  was  uniform  in  section.  It  is  advisable  to 
load  and  unload  the  wire  several  times  before  making  the  final 
measurements.  This  serves  to  straighten  the  wire,  and  avoids 
the  anomalous  results  which,  apart  from  straightening,  are 
obtained  when  a  wire  is  loaded  for  the  first  time  after  a  rest. 
Fig.  43.  We  owe  the  following  improvements  of  this  method  to  Mr. 

G.  F.  C.  Searle.  Two  brass  frames,  CD,  (d'\y.  Fig.  44,  hang  from 
the  lower  ends  of  the  wires  and  support  the  two  ends  of  a  sensitive  level  L. 
One  end  of  the  level  is  pivoted  to  the  frame  CD  by  the  pivots  H,  the 
other  end  of  the  level  rests  upon  the  end  of  a  vertical  screw  S  working  in 
a  nut  attached  to  the  frame  CD'.  The  two  links,  K,  K',  prevent  the 
frames  from  twisting  relatively  to  each  other  about  a  vertical  axis,  but  freely 
allow  vertical  relative  motion.  When  these  links  are  horizontal  the  two 
wires  are  parallel  to  each  other.  A  mass  M  and  a  pan  P  hang  from  the 
lower  ends  of  the  frames,  and  the  weights  M  and  P  are  suflicient  to 
straighten  the  wires.     The  connections  between  the  wires  and  the  frames 


STRESSES. 


75 


are  made  by  the  swivels  F,  into  which  the  ends  of  the  wires  are  soldered. 
The  swivels  prevent  the  torsion  o?  the  wire.  The  head  of  the  screw  is 
divided,  say,  into  100  parts,  while  the  pitch  of  the  screw  may  be  "5  mm.; 
thus  each  division  on  the  head  corresponds  to  1/200  mm.  The 
measui'ements  are  made  in  the  following  w^ay :  Adjust  the  screw  so  that 
one  end  of  the  bubble  is  at  zero;  if  a  weight  be  placed  in  the  pan  P  the 


Fig,  44. 


wire  A'  is  stretched,  and  the  bubble  moves  towards  H ;  bring  the  bubble 
back  to  zero  by  turning  the  screw ;  the  distance  through  which  the  screw 
is  moved  is  equal  to  the  extension  of  the  wire. 

When  the  substance  for  which  Young's  modulus  is  to  be  determined 
is  a  bar  and  uot  a  wire,  the  extensions  obtained  by  any  practicable  weight 
would  be  too  small  to  be  measured  in  the  way  just  desciibed.  In  this  case 
Ewing's  extensometer  may  be  used.  This  instrument  is  represented  in 
Fig.  45.  A  is  the  rod  whose  extension  is  to  be  measured,  B  and  C 
are  pieces  attached  to  A  by  set  screws  about  the  axes  of  which  they 
revolve;  the  arm  B'  fixed  to  B  ends  in  a  rounded  point  P,  which  fits 
into  a   V-shaped  slot  cut   transversely  across   the    end   of   the  piece  C 


76 


PROrERTIES  OF  MATTER. 


'  \ 


Si 


Thus,  when  the  rod  A  is  stretched,  the  point  P  acts  as  a  fulcrum,  and 
Q,  the  opposite  end  of  C,  moves  down  through  a  distance  proportional 
to  the  extension  between  the  axes  of  the  set  screws.  The  displacement  of 
Q  is  PQ/OP  times  the  extension  of  the  bar.  This  displacement  is  observed 
by  a  microscope  which  is  attached  to  the  bar  B,  and  sights  an  object 
at  Q.  The  displacement  is  measured  by  means  of  a  mici'ometer  scale  en- 
graved on  glass  in  the  eye-piece  of  the  microscope ;  extensions  of  1/20,000 
of  a  centimetre  are  readily  measured  in  this  way.  There  is  a  fine  screw, 
with  a  divided  head  between  B'  and  the  point  P.  This  serves  to  bring  Q 
into  a  convenient  position  for  sighting,  and   also  to  determine  what  is 


-.^"C 


Fig.  46. 

the  absolute  amount  of  extension  corresponding  to  a  division  of  the 
eye-piece  scale ;  for  if  we  know  the  pitch  of  the  screw  we  know  the  dis- 
placement of  Q  when  the  screw-head  is  turned  through  one  revolution ; 
if  we  find  how  many  divisions  of  the  micrometer  scale  this  corresponds 
to  we  can  at  once  standardise  the  scale.  The  pull  is  applied  to  the  bar 
by  means  of  a  small  testing  machine. 

Optical  Measurement  of  Young-'s  Modulus. — Michelson's  method 

of  interference  fringes,  produced  by  the  aid  of  semi-transparent  mirrors, 
gives  a  very  delicate  way  of  measuring  small  extensions. 

The  principle  of  the  method  is  shown  in  Fig.  46.  A  and  B  are  plane 
plates  of  very  carefully  worked  glass  of  the  same  thickness.  One  surface  of 
A  is  coated  with  a  tliin  film  of  metal,  preferably  platinum.  The  platinum 
may  be  deposited  on  the  glass  by  placing  the  glass  near  a  platinum 
cathode  in  an  exhausted  tube,  and  sending  a  current  from  an  induction 


STRESSES. 


77 


coil  through  the  tube.  The  platinum  sputters  from  the  terminal  and  is 
deposited  on  the  glass.  This  film  is  so  thin  as  to  be  semi-transparent ;  it 
allows  part  of  the  light  to  pass  through  it.  Suppose  a  beam  of  light, 
starting  from  S,  falls  on  the  plate  A,  some  of  it  is  reflected  from  the 
upper  surface  of  the  plate,  and  after  being  reflected  from  the  mirror  C 
returns  and  passes  out  of  the  plate  A  and  enters  the  eye  at  E ;  another 
part  of  the  beam  passes  through  the  plate  A,  is  reflected  at  D,  returns  to 
the  plate  A,  where  it  is  reflected  to  E.  Even  when  the  difierence  of  path 
is  great,  if  A  and  B  are  very  truly  plane  and  of  the  same  thickness  the  first 
part  of  the  beam  from  Swill  interfere  with  the  second  part  and  produce  inter- 
ference bands.  If  the  distance  between  one  of  the  mirrors  and  the  plate  A  is 


D 


E 


Fig.  4«, 


altered,  the  bands  are  shifted ;  an  alteration  of  the  distance  through  1/4  of 
a  wave-length  will  make  the  dark  bands  and  light  bands  interchange 
their  position ;  by  observing  the  position  of  the  bands  we  can  measure 
movements  of  the  mirror  amounting  to  1/50  of  the  wave-length  of  sodium 
light,  or  say  a  millionth  of  a  centimetre.  To  apply  this  method  to  the 
determination  of  Young's  modulus  we  keep  one  of  the  mirrors  fixed  while 
the  other  is  cari'ied  by  the  wire  whose  extension  we  wish  to  measure. 
Since  we  can  measure  accurately  in  this  way  very  small  extensions  we  are 
able  to  use  comparatively  short  wires,  and  so  have  all  the  conditions  of 
the  experiment  under  much  better  control  than  when  a  long  wire  is 
used.  This  method  has  been  used  by  Mr.  Shakespear  at  the  Cavendish 
Laboratory.  He  has  also  used  the  method  described  on  p.  43  for  multi- 
plying the  small  movements  of  the  pointer  of  a  balance,  to  multiply  the 
movement  due  to  the  extension  of  a  wire. 

Other  methods  of  detej'mining  q  will  be  given  in  the  chapter  on  the 
Bending  of  Rods. 


CHAPTER  Vli. 
TORSION. 

Contents. — Torsion  of    Circular  Tabes  and   Hods— De  St.  Venanf  s  Researches— 
ytatical  and  Dyuumical  Methods  of  Measuring  Jiigidity. 


Torsion  of  a  thin  Cylindrical  Tube  of  Circular  Section.— The 

case  of  a  thin  cylindrical  tube  of  circular  section  tixed  at  one  end  and 
twisted  by  a  couple  whose  axis  is  the  axis  of  the  tube,  admits  of  a  very 
simple  solution.  We  can  prove  that  each  cross-section  of  the  tube  made 
by  a  plane  at  right  angles  to  the  axis  is  twisted  as  a  rigid  body  in  its  own 
plane  through  an  angle  proportioned  to  its  distance  from  the  fixed  end, 
and  that  there  is  no  displacement  of  any  point  in  the  tube  either  radially 
or  longitudinally.  The  last  result  follows  at  once  from  the  symmetry  of 
the  tube  about  its  axis ;  for  from  the  symmetry,  if  the  radial  displace- 
ment is  outwards  at  one  part  of  the  section  it  will  be  outwards  at  every 

point,  so  that  there  would 
be  a  swelling  of  the  tube  ; 
reversing  the  couple  ap- 
plied to  the  tube  would, 
however,  reverse  the  dis- 
placement (since  we  sup- 
pose Hooke's  Law  to 
hold) ;  hence  a  couple  in 
one  direction  would  cause 
the  tube  to  swell,  while 
one  in  the  opposite  direc- 
tion would  cause  it  to 
contract ;  it  is  evident, 
however,  that  whether 
the  tube  swells  or  con- 
tracts under  a  twist  about  its  axis  cannot  depend  upon  the  direction  of  the 
twist,  hence  we  conclude  that  there  is  no  radial  displacement.  Similar 
reasoning  will  show  that  the  longitudinal  displacement  must  also  vanish. 

We  shall  now  show  that  the  tube  will  be  in  equilibrium  when  each 
cross  section  is  twisted  as  a  rigid  body  through  an  angle  proportional  to 
the  distance  of  the  section  from  the  fixed  end. 

For  suppose  ABCDEFGH  is  a  rectangular  parallelopiped  cut  out 
of  the  tube  before  the  twist  was  applied,  suppose  the  distance  between 
the  planes  ABCD,  EFGH  is  d,  and  let  k  be  the  distance  of  the  plane 
EFGH  from  the  fixed  end  of  the  tube.  Then,  since  the  angle  through 
which  each  section  is  twisted  is  proportional  to  its  distance  from  the  fixed 
end,  if  0  bo  the  angle  through  which  the  section  at  unit  distance  from  tho 
fixed  end  is  twisted,  the  rotation  of  EFGH  is  A'0,  and  that  of  ABCD 
is  {h->rd)  (p.  If  a  is  the  radius  of  the  tube,  and  if  t,  its  thickness,  is  small 
compared  with  a,  each  point  in  EFGH  will  be  moved  through  a  distance 


Torsion. 


19 


ak(p,  and  each  point  of  ABUD  through  a  distance  a  {k  +  d)  ^,  hence 
After  the  twist  the  shape  of  the  parallelofiiped  ABODE FGH  will  be 
dmilar  to  EFGHA'B'C'D',  where  A  A' =  BB' =  CO"  =  DD' =  acZf  Hence 
the  deformation  of  the  elements  will  be  a  shear  of  which  the  angle 
of  shear  =  AA'/AE  =  rt0.  The  t;ingential  stress  T  will  therefore  be  na(p. 
Hence  the  stresses  on  the  elements  will  be  as  shown  in  Fig.  47, 
horizontal  tangential  stresses  equal  to  T  on  the  faces  A  BCD,  EFGH,  and 
vertical  tangential  stresses  equal  to  T  on  the  faces  ABEF,  ODHG.  As  0 
is  uniform  for  all  parts  of  the  tube  these  stresses  are  constant  throughout 
the  tube,  and  therefore  each  portion  of  the  interior  will  be  in 
equilibrium  under  these  stresses.  To  find  the  condition  for  equilibrium 
under  the  external  couple,  consider  a  portion  ABOD,  Fig.  48,  cut  from 
the  tube;  this  portion  is  in  equilibrium  under  the  action  of  the  tangential 
stress  T  on  its  cross  section,  and  the  external 
couple  whose  moment  we  shall  suppose  is  C.  For 
equilibrium  the  moment  of  the  tangential  stresses 
round  the  axis  must  equal  C.  The  moment  of  the 
tangential  stresses  is,  however,  T  x  area  of  cross- 
section  of  tube  X  radius  of  tube,  which  is  equal  to 


hence  we  have  C  =  n(p27raH 


(1) 


which  gives  the  rate  of  twist  <p  when  the  external 
couple  is  known. 

Case  of  a  Solid  Rod  of  Circular  Section.— 

We  can  regard  the  rod  as  made  up  of  a  series  of 

tubes,  and  hence  from  the  preceding  investigation 

we  see  that  each  cross-section  of   the  rod  will  be 

twisted  as  a  rigid  body  through  an  angle  proportional 

to   its   distance   from    the    fixed  extremity.*     The 

couple  C  required  to  twist  the  rod  will  be  the  sum  of  the  couples  required 

to   twist  the   tubes  of  which  it   is  built   up,   or  in  the  notation  of  the 

integral  calculus, 


Fig.  48. 


c 


ra 
=  27r?t^  /  r 


'dr 


nence 


if  a  is  the  radius  of  the  solid  cylinder.     If  *  is  the  angle  through  which 
the  lower  extremity  of  the  rod  is  twisted  and  I  the  length  of  the  rod,  then 


Thus  the  coiiple_jrequired  to^twist  the  lower  ^nd  of  the  bar  through  a 
given  angle  varies  directly  as  the  fourth  powerof  the  radius_and^nversely 
as  tlie  length'oFthe  bar.     If  instead  of  aTbar  we  have  a  thick  tube  whose 

*  For  if  the  cross-sections  of  the  different  tubes  were  twisted  through  different 
angles,  so  as  to  shear  one  tube  past  the  next,  there  would  be  twisting  couples  acting 
on  the  inner  parts  of  the  tube,  and,  since  the  outside  of  the  rod  is  free,  nothing  to 
balance  these  on  the  outside. 


80 


PROPERTIES  OF  MATTER. 


inner  radius  is  b  and  outer  radius  a,  the  couple  (J  required  to  twist  its 
lower  extremity  through  an  angle  *  is  given  by  the  equation 


* 


G  =  i7rnj{a'-y] 


The  work  required  to  twist  the  cylinder  through  an  angle  #  can  be  shown 

by  a  method  exactly  similar  to 
tuat  given  on  p.  71  to  be  equal 
to  iC* ;  hence  in  the  case  of  a 
solid  rod  the  energy  is 

V 

The  volume  of  the  rod  is  Wa-, 
hence  the  mean  eneigy  stored  up 
in  unit  volume  of  the  rod  is  \na-(i?. 
When  the  cross-section  of  the 
bar  is  not  a  circle  the  problem 
becomes  much  more  difficult.  It 
has,  however,  been  solved  by  St. 
Venantfor  a  considerable  number 
of  sections  of  different  shapes, 
including  the  ellipse,  the  equilateral  triangle  and  the  square  with  rounded 
corners.  In  every  case  except  the  circle  a  cross  section  made  by  a  plan© 
at  right  angles  to  the  axis  does  not  remain  a  plane  after  twisting  but  is 
buckled,  part  of  the  section  l^eing  convex  and  part  concave.  In  these 
cases  there  is  a  longitudinal  displacement  of  the  particles, 
some  moving  up  and  others  down.  The  longitudinal 
movement  is  the  same  for  all  particles  that  were  originally 
in  a  straight  line  parallel  to  the  axis  of  the  cylinder.  We 
can  see  in  the  following  way  that  there  must  be  longitudinal 
displacements  of  the  particles  and  find  the  direction  of  the 
displacement.  Let  us  take  the  case  when  the  section  is 
an  ellipse ;  then,  if  each  section  were  rotated  round  the 
axis  without  any  longitudinal  displacement,  the  stress  in 
each  section  at  any  point  P  would  be  at  right  angles  to 
the  line  joining  O  to  that  point.  Thus,  if  Fig.  49 
represent  the  section  of  an  elliptic  cylinder,  twisted  in  the 
direction  represented  by  the  arrow,  the  fixed  end  of  the  cylinder  being 
below  the  plane  of  the  paper  and  the  twist  applied  to  the  end  above  the 
paper,  the  stress  in  the  section,  if  there  were  only  rotation,  would  be  at 
right  angles  to  OP ;  now,  if  P  is  a  point  on  the  ellipse,  the  tangent  to  the 
ellipse  will  not  be  at  right  angles  to  OP  except  at  the  extremities  of  the 
axes ;  hence  in  general  the  stress  would  have  a_component__along  the 
normaL  to  the  cylinder.  Since,  however,  the  sides  of  the  cylinder  are 
supposed  to  be  free  and  not  acted  upon  by  forces,  there  cannot  be 
equilibrium  unless  the  stre.ss  along  the  normal  to  the  cylinder  vanishes ; 
hence  there  must  be  some  other  displacements  which  will  produce  a  stress 
to  balance  the  normal  component  of  the  stress  at  right  angles  to  OP. 
This  component  is  directed  outwards  in  the  quadrants  AB,  A'B',  inwards 
in  the  quadrants  BA',  B'A  ;  hence  the  additional  stress  must  be  directed 


TORSION. 


81 


inwards  in  the  quadrants  AB,  A'B',  and  outwards  in  the  quadrants  BA', 
B'A.  Now  suppose  PQRSTUVW,  Fig.  50,  represents  a  paiallelopiped 
cut  from  the  quadrant  AB,  the  faces  PQRS,  TUVW  being  at  riglit  angles 
to  the  axis  of  the  cylinder  and  the  latter  nearer  to  the  fixed  end,  the  faces 
Y 


I       *^  If 


■•^^  Fig.  52. 


PQTU,  RSVW  being  at  right  angles  to"  OP ;  then  there  must  be  a  stress 
in  the  plane  PQRS  directed  from  R  to  Q ;  but  if  there  is  a  stress  in  this 
direction  there  must  be  a  stress  in  RSVW  parallel  to  RV,  otherwise  the 
parallelepiped  would  be  set  in  rotation  and  could  not  be  in  equilibrium. 
Now  the  stress  in  RW  parallel  to  RV  implies  either  that  the  longitudinal 
displacement  in  the  direction  RV  is  greater  than  that  in  the  same 
direction  in  the  face  PQTU — i.e.,  that 
the  longitudinal  displacement  increases 
as  we  recede  from  the  axis  or  else  that 
the  longitudinal  displacement  in  the 
opposite  direction  VR  is  less  than  that 
in  the  face  TPQU — i.e.,  that  the  longi- 
tudinal displacement  diminishes  as  we 
recede  from  the  axis.  But  as  the 
longitudinal  displacement  vanishes  at 
the  axis  itself,  it  seems  clear  that  it 
must  increase  as  we  i-ecede  from  the 
axis ;  hence  we  conclude  that  the 
longitudinal  displacement  is  in  the 
direction  RV — i.e.,  towards  the  fixed 
end  of  the  cylinder.  In  the  quadrant 
B'A'  the  tangential  stress  at  right 
angles  to  OP  has  a  component  along  Fig.  53. 

the  outward  normal,  hence  the  longi- 
tudinal displacement  is  again  towards  the  fixed  end  of  the  cylinder.  In 
the  other  quadrants  BA',  B'A  the  tangential  stress  has  a  component  along 
the  inward  normal,  and  in  this  case  the  longitudinal  displacement  will  be 
in  the  opposite  direction — i.e.,  aicay  from  the  fixed  end  of  the  cylinder. 
Along  the  axis  of  the  ellipse  there  is  no  longitudinal  displacement.  In 
Figs.  51,  52,  53,  taken  from  De  St.  Venant's  paper,  the  lines  of  equal 
longitudinal  displacement  are  given  in  Fig.  51,  when  the  cross  section  of 
the  cylinder  is  an  ellipse,  in  Fig.  52,  when  it  is  an  equilateral  triangle, 
and  in  Fig.  53,  when  it  is  a  square.  The  dotted  lines  represent 
displacements  towards  the  fixed  end  of  the  cylinder,  the  full  lines 
displacements  away  from  it.  The  direction  of  twist  is  indicated  by  the 
axrows.     It  will  be  seen  that  in  all  cases  the  displacement  is  towards  the 


82 


PROPERTIES  OF  MATTER. 


fixed  end  or  away  from  it,  according  as  the  component  of  the  tangential 
stress  at  right  angles  to  OP  along  the  normal  to  the  boundary  is  directed 
to  the  outside  or  inside  of  the  cylinder.  The  reason  for  this  we  saw 
when  we  considered  the  elliptic  cylinder. 

The  appearance  of  cylindeis  under  considerable  twist  is  shown  in 
Fig.  54;  this  case  can  be  realised  by  twisting  a  rubber  spring  of  elliptic  or  rect- 
angular section  and  observing  the  distortion  of  lines  drawn  on  the  spring. 

In  the  case  of  the  elliptic  cylinder,  De  St.  Venant  showed  that  the 
longitudinal  displacement  lo  reckoned  positive  when  towards  the  fixed  end 
of  the  cylinder  at  a  point  whose  co-ordinates  referred  to  the  principal 
axes  of  the  ellipse  are  x,  y  is  given  by  the  equation 


W  =  (j) 


.«' 


b- 


a'  +  b 
where  a  and  h  are  the  semi-axes  of  the  ellipse,  and  ^  the  rate  of  twist, 


Fig.  55. 


Thus  the  lines  of  equal  longitudinal  displacement  are  rectangular  hyper- 
bolas with  the  axes  of  the  ellipse  for  asymptotes. 

The   couple   C  required  to  produce  a  rate  of  twist  ^  was  shown  by 
De  St.  Venant  to  be  given  by  the  equation 


C  =  ??^7 


a^b 


3A3 


a'  +  b^ 


In  the  case  of  a  thin  strip  of  elliptic  section  where  h  is  small  compared 
with  a  this  equation  is  approximately 

C  =  oiipTrab' 

Let  us  compare  this  with  the  couple  C  required  to  produce  the  same 
rate    of  twist  in  a  wire  of  circular  section,  the  area  of  the  cross-section 
being  the  same  as  that  of  the  strip.    If  r  is  the  radius  of  the  cross-sectio?, 
then  (see  p.  79)  —h, 


60  that 


C^^2ab^ 


TORSION. 


8S 


Now,  as  the  areas  of  the  cross-sections  are  the  same 

vr°  =  Trab  ^ 


hence 


0  ^26 
C      a 


thus,  as  b  is  very  small  compared  with  a,  0  is  small  compared  with  C. 

Thus,  if  we  use  the  torsion  to  measure  small 

couples,    the    strip   will    be    vpry   much   more 

sensitive    than    the  circular   wire.     Strips  of 

fhin    metal   are    employed    in    some   delicate 

torsion  balances. 

The  greatest  strain  was  shown  by  De  St, 
Venant  to  be  in  the  parts  of  the  boundary 
nearest  the  axis — i.e.,  the  extremities  of  the 
minor  axis  in  the  case  of  the  elliptic  cylinder 
and  the  middle  points  of  the  sides  iji  the  case 
of  the  triangular  cylinder. 

The  stress  vanishes  at  a  projecting  corner, 
as,  for  example,  at  angles  of  the  triangle  and 
square.  On  the  other  hand,  it  becomes 
infinite  at  an  internal  angle,  such  as  is  shown 
in  Fig.  55.  These  should,  therefore,  be 
avoided  in  shafts  subject  to  torsion,  or  if  they 
have  to  be  used  the  angle  should  be  rounded 
oft: 

Determination  of  the   Rigridity   by 

Twisting'. — The  coeflicient    of  rigidity  n   is 
frequently  determined  by  means  of  equation, 

(see  p.  79)  which  gives  the  relation  between" 
the  couple  C  required  to  twist  a  circular  rod 
of  radius  a  and  length  I  and  the  angle  <I> 
through  which  the  rod  is  twisted  by  the 
couple.  The  ratio  of  the  couple  to  the  angle 
may  be  determined  (1)  statically ;  (2)  dyna- 
mically. 

In  the  statical  method  a  known  couple  is 
applied  to  the  wire  or  rod  by  an  arrangement 
such  as  that  shown  in  Fig.  56,  and  the  angle 

through  which  a  pointer  or  mirror  attached  to  the  wire  is  deflected  is 
measured.  This  gives  C  and  $,  and  if  we  measure  a  and  I,  the  preceding 
equation  gives  n. 

In  the  dynamical  method  for  determining  the  rigidity,  the  wire  whose 
rigidity  is  to  be  determined  hangs  vertically,  and  carries  a  vibration  bar 
of  known  moment  of  inertia.  If  this  bar  is  displaced  from  its  position 
of  equilibrium  it  vibrates  isochronously,  and  the  time  of  its  vibration 
can  be  determined  with  great   accuracy.     The  torsional  couple  tending 


84  PROPERTIES  OF  MATTER. 

to  bring  the  bar  back  to  its  position  of  equilibrium  when  it  is  displaced 
through  an  angle  *  is  equal  to 

hence,  if  MK^  is  the  moment  of  inertia  of  the  bar,  the  time  T  of  a  complete 
vibiation  is  given  by 

SkMKH 
hence  n=     ,..,, 

J-    it 

This  experiment  is  easily  made  and  T  can  be  measured  very  accurately. 
The  values  of  n  found  by  this  method  are,  as  a  rule,  higher  than  those 
found  by  the  statical  method.  Both  methods  are  open  to  the  objection 
that,  as  a  occurs  to  the  fourth  power,  if  we  make  an  error  of  1  per  cent, 
in  the  determination  of  a  the  use  of  the  formula  will  lead  to  an  error  of 
4  per  cent,  in  the  determination  of  oi.  Again,  the  use  of  wire  in  the 
determination  of  elastic  constants  is  objectionable,  as  the  process  of  wire- 
drawing seems  to  destroy  the  homogeneity  of  the  metal,  the  outer  layers 
differing  fi-om  the  inner.  Unless  the  material  is  homogeneous  it  is  not 
justifiable  to  use  the  equation  of  page  79,  and  any  abnormality  in  the 
outer  layers  would  seriovisly  affect  the  torsion,  as  it  is  in  these  layers  that 
the  strain  is  greatest.  The  values  of  n  for  all  metals  are  found  to  decrease 
as  the  tempei^ature  increases.     (Horton,  Froc.  Roy.  Soc.  73,  p.  334.) 


CHAPTER  VIII. 
BENDING   OF   RODS. 

Contents. — Bar  bent  into  a  Circular  Arc — Energj'  in  Bar — Bar  Loaded  at  one  End — 
Depression  of  End-  Bar  Loaded  in  Middle,  Ends  Tree — Bar  Loaded  in  Middle, 
Ends  clamped— Vibration  of  Loaded  Bars — Elastic;  Curves — .Stability  of  Loaded 
Pillar — Young's  Modulus  determined  by  Flexure — Table  of  Moduli  of  Elasticity. 

By  a  rod  in  this  chapter  we  mean  a  bar  of  uniform  material  and  cross- 
section  whose  length  is  great  compared  with  its  transverse  dimensions. 
We  shall  suppose  that  such  a  bar  is  acted  on  by  two  couples,  equal  and 
opposite,  applied  at  the  two  ends  of  the  rod,  the  plane  of  the  couples 
passing  through  the  centres  of  gravity  of  all  the  cross-sections  of  the  rod, 
and  intersecting  the  cross-sections  in  a  line  which  is  an  axis  of  symmetry 
of  the  cross-section.  Let  the  couples  act  so  that  the  upper  part  of  the  bar 
is  extended  while  the  lower  part  is  compressed.  There  will,  therefore,  be 
a  part  of  the  bar  between  the  top  and  the  bottom  which  is  neither 
extended  nor  compressed.  This  part  of  the  bar  is  called  the  neutral 
surfa^,  and  the  section  of  it  by  the  plane  of  the  couple  is  called  the 
neutral  axis.     Let  us  suppose  the  bar  divided  into  thin  filaments  parallel 


Fia.  67. 

to  its  length.  We  shall  now  proceed  to  show  that  the  bar  will  be  in 
equilibrium  if  each  filament  above  tlie  neutral  surface  is  extended,  each 
filament  below  that  surface  compressed,  the  extension  or  compression 
being  proportional  to  the  distance  of  the  filament  from  the  neutral 
surface,  the  filaments  being  extended  or  compressed  as  they  would  be  if 
the  sides  of  the  filament  were  free  from  stress ;  so  that  if  P  is  the  tension 
and  e  the  elongation,  V  =  qe  whei^e  q  is  Young's  modulus. 

First  consider  the  equilibrium  of  any  filament ;  the  strain  is  a  uni- 
form extension  or  contraction,  according  as  the  filament  is  above  or  below 
the  neutral  surface.  Tlie  strain  will,  therefore,  be  a  uniform  longitudinal 
tension  or  compression,  there  will  be  no  shearing  stresses  and  no  stresses  \ 
at  right  angles  to  the  length  of  the  bar ;  all  these  statements  hold  whether 
the  filament  abuts  on  the  surface  or  not.  As  the  only  foi'ces  acting  on 
the  filament  are  at  right  angles  to  its  ends,  and  are  equal  and  opposite, 
the  filament  will  be  in  equilibrium.  Thus  each  internal  portion  of  the 
bar  is  in  equilibrium,  and  the  bar  as  a  whole  will  be  in  equilibi'ium  if  the 
stresses  due  to  the  strain  are  in  equilibrium  with  the  external  forces. 

Suppose  that  the  bar  is  cut  at  C,  and  that  EFGH  (Fig.  58)  represents  a 
cross-section  of  the  bar,  0  being  the  centre  of  gravity  of  the  section  ;  then  the 
forces  acting  on  the  portion  OA  (Fig.  57)  of  the  bar  are  the  external  couple, 


86 


PROPERTIES  OF  MATTER. 


'iS^QAj-^ 


whose  moment  we  shall  take  to  be  C  and  the  stresses  acting  across  the' 
cross  section.  Thus  the  condition  for  equilibrium  is  that  the  stresses  across 
this  section  should  be  equivalent  to  a  couple  in  the  plane  of  bending  whose 
moment  is  C,  Now  the  tension  acting  on  the  cross-section  of  a  filament 
at  P  is  equal  per  unit  area  to  qe  where  e  is  the  elongation  of  the  filament. 
Now  e  is  proportional  to  PN  if  ON  is  perpendicular  to  the  plane  of 
bending  and  PN  perpendicular  to  ON  ;  let  e  =  aPN.  Thus  the  force  acting 
on  tlie  filament  parallel  to  the  length  of  the  rod  is  ^.a.PNw  where  w  is 
the  cross-section  of  the  filament,  and  the  forces  on  all  the  filaments  into 
which  the  bar  may  be  supposed  to  be  divided  must  be  together  equivalent 
to  a  couple  of  moment  C  in  the  plane  of  bending.  The  conditions  for  this 
are  (1)  that  the  resultant  force  should  vanish  ;  (2)  that  the  moment  of  the 
forces  about  OM,  which  is  perpendicular  to  ON,  should  be  zero ;  and  (3) 
^ui_  that  the  moment  of  the  forces  about 

ON  =  C.  All  these  conditions  can  be 
fulfilled  if  OM,  ON  are  the  principal 
axes  of  the  cross-section. 

For  the  resultant  force  is  S^a.PN.w 
where  ZqaPN.o)  denotes  the  sum  of 
the  product  ga.PN.w  for  all  the  fila- 
ments ;  this  vanishes  since  2PNw  =  0, 
0  being  the  centre  of  gravity  of  the  cross- 
section.  The  moment  of  these  foi-ces 
about  OM  is  equal  to  E^-aPN.PMw; 
this  vanishes  since  SPN.PM  =  0,  as 
OM,  ON  are  principal  axes.  The  mo- 
ment of  the  tension  about  ON  is 
25^aPN*w  ;  this  is  equal  to  qaAk-  if  A^ 
is  the  moment  of  inertia  of  the  cross- 
section  about  ON.  Hence  the  tensions 
will  be  in  equilibrium  with  the  external 
forces  if  qaAk'  =  G. 
To  find  a,  let  us  consider  the  deformation  of  a  rectangle  ABCD  (Fig.  59) 
in  the  plane  of  bending,  AB  being  a  portion  of  the  neutral  axis.  Let 
A'B'C'D'  be  the  strained  configuration  of  this  rectangle  ;  then,  since  there 
is  no  shear,  the  angles  at  A'  and  B'  will  be  right  angles,  and  C'A',  D'B' 
will  be  normals  to  the  curve  into  which  the  neutral  axis  is  bent ;  if  these 
normals  intersect  in  0,  then  O  is  the  centre  of  curvature  of  the  neutral 
axis.     We  have  from  the  figure 

C'D'C'O 
A^     ATO 

But  A'B'  =  AB,  since  the  neutral  axis  is  not  altered  in  length  by  the 
bending,  and  AB  -  CD  ; 

CD'  -  CD     A'C 


Fio.  68. 


hence 


CD 


A'O 


But  if  e  is  tne  elongation  along  CD,  e  ■■ 


CD'  -  CD 


CD 


Kence 


A'C  _  A'C  _  AC 
'  A'O         p  a 


approximately, 


BENDING  OF  RODS. 


87 


where  p  is  the  radius  of  curvature  of  the  neutral  axis  at  A. 

the  previous  notation  e  =  a.AC,  so  that  a  =  -  • 

P 

Since  qaAk-  -»  0,  we  have  q =  0  ;  or,  p  =  q — - 

P  0 


But  with 


B 


Thus  the  radius  of  curvatiu-e  of  the  neutral  axis  is  constant,  so  that  the 
neutral  axis  is  a  circle.  ' 

The  fact  that  a  thin  bar  C\ 

or  lath  is  bent  into  a  circle 
by  the  application  of  two 
couples  is  often  utilised 
for  the  pur  pose  of  drawing 
circles  of  large  radius. 

The  bending   of    the  ^ ^ 

bar  will  be  accompanied    J 

by  a  change  in  the  shape  J 

of  the  cross  section.  The 
elongation  of  the  upper 
filaments  will  be  accom- 
panied by  a  lateral  con- 
traction equal  to  a  times 
the  elongation  where  a  is 
Poisson's  ratio  (see  p.  73), 
while  the  shortening  of 
the  lower  filaments  will 
beaccompaniedbyalateral  Fio.  59. 

expansion.      Thus  the 

shape  of  the  cross-section  supposed  to  be  originally  a  rectangle  will  after 
the  bending  be  as  represented  in  PQLM  (Fig.  60). 

Suppose   LM   is   the   line  where  the   neutral   surface  cuts  the  cross 
section,  then  the  lateral  contraction  of  PQ  is  equal  to 

LM  -  PQ 
LM 

and  the  longitudinal  extension  is  equal  to 

P 
LM  -  PQ       QM  .      ,  . , 

hence    — ^  ^r      —  ^ ^    ■'  - 

LM  p 

but  if  LP,  MQ  intersect  in  O',  then  ^^^"^^  =  ^ 


^^""^  Tw  -- 


.r 


But  LO'  is  equal  to  the  radius  of  curvature  of  the  neutral  surface  in 
the  plane  at  right  angles  to  the  length  of  the  rod.  If  this  is  denoted  by 
p'  we  have 

<rp'  =  p 


Thus  the  ratio  of  the  two  curvatures  is  equal  to  Poisson's  ratio. 


88 


PROPERTIES  OF  MATTER. 


Energy  in  the  Bar. — Consider  one  of  the  filaments  into  which  the 
bar  was  supposed  (p.  85)  to  be  divided.  Thus,  if  e  is  the  elongation  in 
this  filament,  I  the  length  of  the  filament  (which  is  equal  to  the  length  of 
the  bar),  w  the  area  of  its  cross-section,  the  energy  in  the  filament  is  by 
p.  71, 


Iqe 


^U. 


But  e  =  a.PN; 


hence  the  energy  in  the  filament  is  J^a-PN-w?. 

The  energy  in  the  bar  is  the  sum  of  the 
energies  in  tlie  filaments,  and  is  thus 
^qaH-LVWut ;  but  SPN^w  =  Ak\ 

and  a  =  l/p  where  p  is  the  radius  of  curva- 
ture of  the  natural  axis,  and  thus  the 
energy  is  equal  to  ^qAkH/p-. 

Again,  qaAk"  =  C,  where  0  is  the  couple 
applied  to  the  bar, 

hence  the  energy  =  |C- 

P_ 
=  half  the  product  of  the  couple  and 
the  angle  between  the  tangents  at  the 
^  extremity  of  the  bar.  This  result  could 
be  deduced  at  once  by  the  method  already 
given. 

Rod  bent  by  a  Weigrht  applied  at  one  End.— In  the  case  just 


FiQ.  60. 


I* 


Fia  61. 

considei'ed  the  stresses  in  the  bar  were  entirely  normal ;  in  this  case,  how- 
ever, we  see  that  for  equilibrium  the  normal  stresses  must  be  accompanied 
by  tangential  ones.  For,  suppose  ACB,  Fig.  61,  represents  the  bar,  the 
weight  being  applied  at  B  while  A  is  fixed ;  consider  a  section  through  C 
made  by  a  plane  at  right  angles  to  the  length  of  the  bar.  Then  the 
portion  CB  of  the  bar  must  be  in  equilibrium  under  the  action  of  the 
stresses  across  the  section  at  C  and  the  weight  W  at  the  end  of  the  bar ; 
thus  the  stresses  across  C  must  be  equivalent  to  a  vertically  upward  force 


BENDING  OF  RODS. 


89 


W  and  a  couple  whose  moment  is  W.BC  :  there  must  be,  therefore,  tangential 
stresses  acting  across  the  section  whose  resultant  is  a  force  W  acting 
upwards.  We  shall  show,  however,  that  if  the  lateral  dimensions  of  the 
bar  are  very  small,  then,  except  quite  close  to  the  end  15,  the  tangential 
stresses  will  be  very  small  compared  with  the  normal  stresses.  For  let 
EFGH  represent  a  section  of  the  bar,  O  the  centre  of  the  section,  and  ON 
an  axis  at,  right  angles  to  the  plane  of  bending.  Then,  if  A  is  the  area  of 
the  cioss-section,  T  the  average  tangential  stress  over  the  area 

TA  =  W 

Let  N  represent  the  normal  stress  at  a  point  P,  dw  a  small  area  round  P, 

then    since     these     normal 

stresses  are  equivalent  to  a    ^  -^ 

couple  whose  moment  round 

ON  is  W.BC,  we  have 


/ 


N.PN(Zw  =  W.BO. 


Thus   the    average    normal  0 
stress  must  be  of  the  order 
of  magnitude 

W.BO 

Ad 


^ 


Fig.  62. 


where  c?  is  a  quantity  comparable  with  the  depth  of  the  bar.     Hence, 

W 

since  —  =  T,  the  magnitude  of  N  is  comparable  with  T  x  BO/tZ,  so  that  if  the 

distance  of  the  section  from  the  end  is  large  compared  with  the  lateral 
dimensions  of  the  bar,  the  normal  stresses  will  be  very  large  compared 
with  the  tangential  ones.  In  the  subsequent  work  we  shall  confine  our 
attention  to  the  effect  of  the  normal  stresses,  but  this  must  be  regarded  as 
an  approximation  only  applicable  to  very  thin  rods.  Let  Fig.  62 
represent  a  small  rectangular  parallelopiped  cut  out  of  the  bar,  the  faces 
EFGH,  E'F'G'H'  being  at  right  angles  to  the  length  of  the  bar,  while  the 
faces  FF'H'H,  EE'GG'  are  parallel  to  the  plane  of  bending,  then  the 
actual  state  of  stress  may  be  thus  described.  The  normal  stresses  are 
confined  to  the  faces  EFGH,  E'F'G'H',  there  being  no  normal  stresses 
over  the  other  faces ;  there  are  tangential  stresses  on  the  faces  FF'HH', 
EE'GG',  and  also  on  the  faces  GG'HH'  and  EE'FF',  but  there  are  no 
tangential  stresses  over  the  faces  EFGH,  E'F'G'H'. 

We  may  proceed  to  find  the  bending  of  the  rod  produced  by  the 
weight  at  its  end  in  the  following  way.  Suppose  PQRS  (Fig.  62a)  represents 
a  portion  of  a  rod  bent  as  on  p.  85,  by  couples  of  moment  C  acting  at  its 
ends,  then  the  stresses  in  the  bar  are  such  as  to  cause  a  couple  with 
moment  C  to  act  across  PQ  and  a  couple  whose  moment  is  C  to  act  across 
the  section  ES.  The  stresses  which  produce  these  couples,  as  we  have 
seen  on  p.  87,  coirespond  to  a  state  of  strain  such  that  the  central  axis  of 
the  portion  of  the  bar  is  bent  into  a  circle  whose  radius  p  is  given  by  the 
equation 

q—=0. 
9 


90 


PROPERTIES  OF  MATTER. 


Now  suppose  that  PQRS,  instead  of  being  a  portion  of  a  bar  acted  on 
by  a  couple,  is  a  portion  of  one  acted  on  by  a  force  at  the  end  A :  then 
ne^^lectin",  for  the  reasons  given  above,  the  tangential  stresses  across  the 
sec°tion,  the  stresses  are  equivalent  to  a  couple  W.  AN  across  the  section  PQ 
and  a  couple  W.AM  across  the  section  US,  and  as  AN  and  AM  differ  but 

little  from  AL  whei^e  L  is 

/'  // , 


N\     L      \M- 


Fia.  C2a. 


the  middle  point  of  MN, 
we  may  regard  the  ends  , 
of  PQRS  as  being  acted, 
on  by  equal  and  opposite  ' 
couples  whose  moment  is 
W  AL.    Hence,  by  what  we  have  just  seen,  the  central  axis  of  PQRS  will  be 
bent  into  the  arc  of  a  circle  whose  radius  p  is  given  by  the  equation 

g.^  =  W.AL; 

hence  when  thej)ar  is^ac^ted  jnijjy^aweight  ajaplied  at  one  end,  the  neutral 
'^s^'the^bar  is  bent  into  a  curve  such  that  the  radius  of  curvature  at  a 
^Hnl.'v^rles  ihvei-seTy^agjfche_clistah^ of ;the_^oint  from  the^nd  to  which 
the  weigTiFisli^plied.  ^        ..    ^ 

Depression  of  the  Bar;  Ang-le  between  Tangfents  at  two 
Points  on  the  neutral  Axis.— Suppose  Pig.  63  represents  the  curved 


Fig.  68. 


position  of  the  neutral  axis.*  Suppo.se  RS  are  two  points  near  together 
on  the  neutral  axis,  then  the  angle  between  the  tangents  at  R  and  S  is 
equal  to  RS/p  where  p  is  the  radius  of  curvature  of  RS  ;  but  1/p  is  equal 
to  W.Altxjq.Ak-,  hence  Ao  the  angle  between  the  tangents  at  R  and  S  is 
equal  to 

W 


q.A/c 


AR.RS 


*  Though  this  figure  shows  for  clearness' sake  considerable  curvature,  yet  it  must 
be  remembered  that  in  all  these  investigations  we  are  only  dealing  with  cases  in 
which  the  bending  is  very  slight  and  the  neutral  axis  consequently  nearly  straight. 


BENDING  OF  RODS.  gi 

or,  in  the  notation  of  the  differential  calculus,  if  s  — AR,  we  have 

^^=,^>  (1) 

hence  S,  the  angle  between   the  tangents  at  A  and  P,  is  given  by  the 
equation 


a 


=  i-~  AP2 


Suppose  the  tangent  at  P  cuts  the  vertical  through  A  in  the  point  T, 
we  shall  proceed  to  find  an  expression  for  AT.  Let  the  tangents  at  K,S 
cut  the  vertical  line  through  A  in  the  points  ;M,N,  then,  remembering  that 
these  tangents  are  veiy  nearly  horizontal,  we  have  approximately,  if  Ao  is 
the  angle  between  the  tangents  at  R  and  S, 

MN  =  AR.A6=^-f  c/a  by  (1) 


AP  

J  oAk'  8  X  a  Alt'  ^  ' 


NowAT  =  .MN^^^^,_.      ^^^^^ 

If  the  end  B  of  the  bar  is  clamped  so  that  the  tangent  is  horizontal, 
then  the  distance  between  A  and  the  point  where  the  vertical  through  A 
cuts  this  tangent  will  be  the  vertical  depression  of  A  produced  by  the 
weight  W ;  hence,  if  d  be  this  depression,  we  have  by  (3) 

z*^-       \  d  =  --^,-A^'  (4) 


yQ.\>Y^i5iof^-      1^^215: 


Thus  the  vertical  depression  of  the  end  is  proportional  to  the  weight, 
to  the  cube  of  the  length,  and  inversely  proportional  to  the  moment  of 
inertia  of  the  cross  section  about  an  axis  through  its  centre  at  right  angles 
to  the  plane  of  bending ;  it  is  also  inversely  proportional  to  the  value  of 
Young's  modulus  for  the  material  of  which  the  bar  is  made. 

Since  the  depression  is  proportional  to  the  weight,  the  energy  stored 
in  the  bar  is  equal  ^Wd,  and  this  by  equation  (4)  is  equal  to 

We  shall  now  proceed  to  find  the  depression  PM  (Fig.  64)  of  any  point 
P  on  the  bar  below  the  horizontal  tangent  at  B.  Let  the  tangent  to  the 
central  axis  at  P  cut  the  vertical  line  through  A  in  the  point  T,  and  let  the 
horizontal  line  through  P  cut  this  line  at  O ;  then  the  vertical  depression 
of  Pis 

PM  =  AN-AT-TO 

Now  TO  =  PO  X  angle  the  tangent  at  P  makes  with  the  tangent  at 
B,  and  since  PO  is  approximately  equal  to  AP,  and  the  tangent  at  A 
makes  with  the  tangents  at  P  and  B  angles  whose  circular  measures  are 


92 


PROPERTIES  OF  MATTER. 


respectively   W.A'Py^qAk'   and   W.AB72^A^^   (by    equation    (2)),     we 
have 

AP    W 

TO  =  ^^  iJ2_(aB-  -  AP=) 

2qAk-  ' 


By  equation  (3)  we  have 


Thus 


Hence 


AN  =  — ^  AB» 

3^AP 

W 
AT-^         ,    AP-^ 

3gA/fc- 


FiG.  64. 

PM-  ^  rAB«-AP3_AP(AB^-AP0^  ^  .^..  ^.m- 
?AF\         3  2  /     ^1 


W  BP-r3AP  +  2BP) 
\  6 


qkJi 


I 


-     t 


:t  fAf--'rr/^.-i^^Os 


(f) 


f/^/'^ 


^-  fJP'f 


Fig.  G5. 


Let  us  now  find  what  would  be  the  depression  of  A  if  the  weight  W 
were  applied  at  P.  In  this  case  AP  would  be  straight,  and  if  AN, 
Fig.  65,  is  the  depression  of  A, 

AN  =  PM  +  AP  X  angle  which  tangent  at  P  makes  with  the  horizoctal 
Now  by  (4) 

W 
PM  =  :^,BF 

and  by   (2)   the    angle   the   tangent  at  P  makes  with  the  horizontal   i; 
equal  to 

W 


2qAk' 


BP^ 


hence 


BENDING  OF  RODS. 

qAk^        I   3  +-2"j 
=  _^BP^'  f3AP+2BP 


93 


} 


(6) 


Comparing  equations  (5)  and  (6)  we  see  that  the  depression  at  P_when Jha 
loa4_js^ap£lied__atjL Jsjbjie_same^^  at  A  when  the  load  is 

applied^^^     In  the  case  we  have  just  been  considering  one  of  the  points^ 
IS  at  the  end' of  the  rod.     The  theorem,  however,  is  a  general  one  and 
holds  wherever  the  points  A  and  P  may  be.  ' 

The  relation  between  the  depression  and  the  weight  given  by  equa- 
tion (4)  gives  us  a  means  of  determining  q  by  measuring  the  flexure  of 
a  beam.  In  experiments  made  with  this  object,  however,  it  has  been 
more  usual  to  use  the  system  considered  in  the  next  paragraph,  that  of  a 
beam  supported  at  the  ends  and  loaded  in  the  middle. 


Beam  Supported  at  the  Ends  and  Loaded  in  the  Middle.— The 

ends  of  the  beam  (Fig.  66)  are  supposed  to  rest  on  knife  edges  in  the  same 
horizontal  line.  The  tangent  at  0,  the  middle  point,  is  evidently  hori- 
zontal, and  the  pressure  on  each  of  the  supports  is  W/2.  Considering  now 
the  portion  AC  of  the  rod,  it  has  the  tangent  at  C  horizontal,  and  it  is  acted 
upon  by  a  vertical  force  equal  to  W/2  at  A.  The  conditions  are  the  same 
as  for  a  rod  of  length  AC  clamped  at  C  and  acted  on  by  a  vertical  force 
W/2,  the  case  just  treated ;  hence  by  equation  (4)  d,  the  vertical  distance 
between  A  and  0,  is  given  by  the  equation 


d: 


3)^t»^&  S&l  OVi. 


W     AC^ 

'IqKk'     3 

w 


A^Kk"" 


K& 


Rod  Clamped  at  both  Ends  and  Loaded  in  the  Middle.— 

Suppose  AB  is  a  rod  loaded  at  C,  its  middle  point,  and  clamped  at  the 
ends  A  and  B,  which  are  supposed  to  be  in   the   same  horizontal  line. 


^ 


94  PROPERTIES  OF  MATTER. 

The  action  of  the  supports  A,  B  on  the  rod  will  be  equivalent  to  a  vertical 
force  and  a  couple.  The  magnitude  of  the  vertical  force  is  evidently  W/2  if 
W  is  the  weight  at  C.  We  can  find  the  value  of  the  couple  r  as  follows . 
By  the  action  of  the  force  W/2  alone  the  tangent  to  the  neutral 
axis  at  A  would  make,  with  the  tangent  at  0,  an  angle  whose  circular 
measure  is 

W     AC^ 
2yA^-    2 

But  since  the  tangent  at  A  is  parallel  to  the  tangent  at  C,  the  couple 
must  bend  the  bar  so  that  if  it  acted  alone  the  tangent  at  A  would  make  with 
that  at  C  an  angle  equal  and  opposite  to  that  just  found.  Through  a  couple 
r  applied  to  the  bar  the  tangents  at  A  and  C  wovdd  make  with  each  other 
an  angle  whose  circular  measure  is  p- 


hence 


or 

To  find  the  depression  of  the  middle  point,  we  consider  the  efiect  of  the 
foi"C"  W/2,  and  the  couple  r  separately.  In  consequence  of  the  action 
of  the  force  W/2,  the  middle  point,  0  would  by  equation  (4)  be  depressed 
below  the  line  AB  by 

W     AC3 

The  couple  r  would  bend  the  bar  into  a  circle  whose  radius  p  is  qAk^Jr. 
This  would  raise  the  point  C  above  A  by 

AC2  -i 


r 

qAk- 

AC 

f 

W     AC= 
2qAk'     2 

r 

qAk^ 

AO 

r=i 

W.AC. 

•  ,^rw 


2p 


.      ,  rAC-_    W     AC» 

*-^-'  ^  2p:f~2^aFT~  *'  ^ 

The  depression  of  C  when  both  the  force  and  the  couple  act  is  therefore 

W     AC^  _    W     AC^ 
2qA/c'  ~Y~      'JqAk'  T 

-     W  _^  WAB'  ( 

24qAk-  U)iVyA^-)    D  e  \5  V'  e-  3  3  )  0  n. 

The  depression^of  the  middle  point  of  the  bar  when  the  ends  are  fixed  is 
thus  only  l^Tof  the  depression  ot  the  same  bar  when  the  encls7\re  free — 
""Vibration  of  Loaded  Bars.— Since  the  deflection  of  the  bar  is  in  all 
cases  proportional  to  the  deflecting  weight,  a  bar  when  loaded  will  execute 
isochronous  vibrations,  the  time  of  a  complete  vibration  being  equal  to 

where  M  is  the  mass  of  the  load  and  ^  the  force  required  to  produce  unit 


BENDING  OF  RODS. 


95 


depression.  From  the  preceding  investigations  we  see  that  fi=p.qkh-IP 
where  I  is  the  length  of  the  bar  and  p  a  numerical  factor,  which  is  equal 
to  S  when  the  weight  is  applied  at  the  end  of  the  bar,  to  48  when  the 
weight  is  applied  at  the  middle  point  of  a  bar  with  its  ends  free,  and  to 
192  when  the  load  is  applied  to  the  middle  point  of  a  bar  with  its  ends 
clamped. 

To  take  a  numerical  example.  Let  us  suppose  we  have  a  steel  bar 
30  cm.  long,  2  cm.  bioad,  and  "2  cm.  deep,  loaded  at  the  end  with  a  mass 
of  100  grammes.  Then  since  for  steel  3^  =  2-139  x  10'^,  and  in  this  case 
M  =  100,  p  =  3,  Z  =  30,  A  =  -4,  h-  =  \  (•1)==0033,  we  find  by  substituting 
in  the  formula  that  the  time  of  vibration  is  about  \  of  a  second. 

To  take  another  case,  suppose  a  man  weighing  70  kilogrammes  stands 
on  the  middle  of  a  wooden  plank  4  metres  long,  30  cm.  wide,  and  4  deep, 
supported  at  its  ends,  what  will  be  the  time  of  swing  ?  For  wood  we  may 
take  17  =  10";  putting  ;>  =  48,  M-7xl0\  ^  =  4  x  10^  A  =  120,  h?  =  \{2y 
=  1'33,  we  find  that  the  time  of  swing  is  about  '5  seconds. 


A^ 


B 


FlQ.  68. 


'^  Elastic  Curve. — Let  us  now  consider  a  case  like  that  of  a  bow 
where  the  force  is  parallel  to  the  line  joining  the  ends  of  the  bar.  Con- 
sider the  equilibrium  of  the  portion  CB  (Fig.  68)  under  the  stresses  at  C, 
and  the  tension  T  in  the  string  at  B. 

Thus  the  stresses  across  C  must  be  equivalent  to  a  couple  T.CN  and  a 
force  T,  CN  being  the  perpendicular  from  C  on  the  line  of  action  of  the 
force.  Confining  our  attention  to  the  couple,  we  see  that  if  p  is  the  radius 
of  curvatures  at  0  of  the  neutral  axis  of  the  rod, 


^:^=T.CN 


(1) 


where  q  is  Young's  modulus  for  the  rod,  A^^,  the  moment  of  inertia  of 
the  cross-section  of  the  rod  about  an  axis  through  its  centre  at  right 
angles  to  the  plane  of  bending.  From  equation  (7)  we  see  that  1/p  is  propor- 
tional to  CIST;  hence  the  curve  into  which  the  central  axis  is  bent  is  such 


Fig.  69. 


96 


PROPERTIES  OF  MATTER. 


that  th«  reciprocal  of  the  radius  of  curvature  at  any  poii:.t  \s  proportioual 
to  the  distance  of  the  point  from  a  straight  line.  Curves  having  this 
property  are  called  elastic  curves  or  elasticas ;  curves  such  as  those  shown 
in  Fig.  69  are  included  in  this  family ;  they  may  be  produced  by  taking 
a  flexible  metal  ribbon,  such  as  a  watch-spring,  and  pushing  the  ends 
too-ether.  One  of  these  curves  is  of  especial  importance — ^viz.,  the  one 
where  the  distance  of  any  point  on  the  bent  rod  from  the  line  of  action  of 
the  force  is  very  small.  We  shall  show  that  this  curve  is  tlie  path  of  a 
point  near  the  centre  of  a  circle  when  the  circle  rolls  on  a  straight  line. 
To  prove  this  it  is  only  necessary  to  show  that  the  reciprocal  of  the  radius 
of  curvature  of  this  path  is  proportional  to  the  distance  from  the  straight 
line  which  is  the  path  of  the  centre  of  the  circle.  Let  us  suppose  that  the 
circle  rolls  with  uniform  angular  velocity  w  along  the  straight  line.  JUit 
C  be  the  centre  of  the  circle,  P  any  position  of  the  moving  point,  G  the 
point  of  contact  of  the  circle  with  the  line  along  which  it  rolls,  PN  the 
perpendicular  on  GO.  Then  if  v  be  the  velocity  of  the  point,  p  the  radius 
of  curvatui-e  of  the  path, 


—  =  acceleration  of  P  along  the  normal  to  its  path  (8) 

p 

Now  sinco  the  circle  rolls  on  the  line  without  slipping  the  velocity  of  G  is 

zero.  Hence  the  system  is  turning  about 
"^^^  -^        ~"^  G,  so  that  the  velocity  at  P  is  at  right 

angles  to  PG  and  equal  to  wPG ;! 
hence  PG  is  the  normal  to  the  path 
and 

v  =  w.PG.  ^ 


Now  the  acceleration  of  P  is  equal  to 
the  acceleration  of  0  plus  the  accelera- 
tion of  P  relative  to  0 ;  since  0  moves 
uniformly  along  a  straight  line  the 
'acceleration  of  0  is  zero,  and  since  P 
describes  a  circle  round  0,  the  accelera- 
tion of  P  relative  to  C  is  equal  to  w^GP 

and  is  along  PC.     Thus  the  acceleration  of  P  along  the  normal  to  its  path 

is  equal  to 


and  we  have  therefore  by  (8) 


w'CPcosCPG 


iii 


=^PG2 


=  w-CPcosCPG 


or 


l^CPcosCPG 
P 


PG^ 


Since  the  angle  PGC  is  very  small,  the  angle  CPG  is  very  nearly  equal  to 
the  angle  PCN,  and  PG  is  very  nearly  equal  to  «,  the  radius  of  the  rolling 
circle ;  hence  approximately 


l^CPcosPON 


a- 


CN 
a' 


BENDING  OF  RODS. 


97 


Thus    l/p  is  proportional  to  the  distance   of  P    from    the  straight  line 
described  by  0. 
From  the  equation 


we  see  that 


a 


"T~ 


The  shape  of  the  curve  is  shown  in  Fig.  71.      The  distance  between 


two  points  of  inflection,  that  is,  between  two  points,  such  as  A  and  B,   ^ 
where  Ifp  vanishes,  is  equal  to  ira. 

Stability  of  a  loaded  Pillar.— The  preceding  result  at  once  gives 
us  the  condition  that  a  vertical  pillar  with  one  end  fixed  vertically  in  the 
ground  should  not  bend  when  loaded  with  a  weight  W — i.e.,  the  condition 
that  the  pillar  should  be  stable.      For,  suppose  the  pillar  bends  slightly, 
assuming  the  position  AB,  Fig.  72,  then  AB  is  an  elasticaand  B  must  be  a 
point  of  inflection,  while,  since  A  is  fixed  vertically  in  the  ground,  the  tangent 
at  A  is  parallel  to  the  line  of  action  of  the  force.     The  distance — measured 
parallel  to  the  base-lines — between  a  point  of  inflec- 
tion and  the  point  where  the  tangent  is  parallel  to  B 
the  base-line  is  half  the  distance  between  two  points 
of  inflection,  and  is,  therefore,  equal  to  ^ttcs,  or,  sub- 
stituting the  value  of  a,  to 

V     ^ 

where  W  is  the  weight ;  hence,  in  order  that  the 
pillar  should  be  able  to  bend,  I,  the  length  of  the 
pillar,  must  not  be  less  than 


^     W 


or,  in  order  to  avoid  bending, 


W 


w 


w 


(9) 


If  the   cross-section  of  the   pillar  is  a  circle  of 


Thus  the  weight  which  a 


■irW 


Fig.  72. 


i-adius  6,  then  AJc'  =  ^ivb^,  

vertical  pillar  can  support  without  becoming  unstable 

IS  proportionaLtojthe  fourth  power  of  the  radius  and 

inversely  proportional  to  the_sc[uare  of  the  length  of  the  pillar^    To  take 

ar"special    case,  let  us  consider  a  steel  knitting-needle,  20  cm.  long  and 

•1  cm.  in  radius  and  take  q  =  2-U  x  10^^     We  find  W  less  than  1-04  x  10" 

- — i.e.,  less  than  about  1056  grammes. 

If  the  rod,  instead  of  being  fixed  at  one  end,  is  pressed  between  two 

G 


/<r^ 


98 


PROPERTIES  OF  MATTER. 


supports  so  that  the  ends  are  free  to  bend  in  any  direction,  Fig.  73,  the 
ends  must  be  points  of  inflection,  the  distance  between  which  is  -na  or 


hence 


KsJqAk'/Wl 


in  the  limiting  case  when  the  pillar  can  bend.     Hence  for  stability 


W< 


(10) 


In  the  case  where  both  ends  are  fixed  (as  in  Fig.  74),  the  tangents  at 
B  B 


Fig.  73. 


Fig.  74. 


the  ends  must  be  parallel  to  the  line  of  action  of  the  force,  and  there  must 
be  two  points  of  inflection  at,  6andc;  hence  the  distance  between  the  ends 
is  twice  the  distance  between  two  points  of  inflection,  so  that 

l=2iva 


9     fq^h;' 


Hence  for  stability 


w. 


-q£ 


AA^ 


(11) 


Comparing  (9)  and  (11),  we  see  that  a  rod  with  both  ends  fixed  will, 
without  buckling,  support  a  weight  sixteen  times  greater  than  if  one  end 
were  free. 

Since  a  pillar  can  only  support  without  buckling  a  finite  weight,  and  as 
this  weight  diminishes  as  the  length  of  the  pillar  increases,  it  follows  that 
a  pole  of  given  cro.ss-section  would,  if  high  enough,  begin  to  bend  under  its 
owu  weight,  so  that  there  is  a  limit  to  the  height  of  a  vertical  pillar  or 


BENDING  OF  RODS.  99 

tree  of  given  cross-section.  Suppose  VV  is  the  weight  of  the  pillar,  and 
suppose  as  an  approximation  that  the  problem  is  the  same  as  if  the  weight 
were  applied  at  the  middle  point  of  the  pillar,  then  if  I  is  the  length  of 
the  pillar  we  see  from  (9)  that 

I' 


or 

A   more   accurate   investigation,    which    requires   the   aid    of    higher 
mathematics,  shows  that  the  accurate  relation  is 


Let  us  take  the  case  of  a  pine  tree  of  uniform  circiilar  section  from  top  to 
bottom,  let  the  diameter  of  the  tree  be  15  cm.  For  deal  g'=10",  and 
taking  the  specific  gravity  of  deal  as  '6,  we  have 

„    7-84  X  10"  X  15^ 
^e  get  "<—7. — 7rT\ — T7, — 

^  -6x981x16 

;<2  7xl0'cm. 

Thus  the  height  of  the  tree  cannot  exceed  about  27  metres. 

Determination   of   Young-'s    Modulus   by   Flexure. — Young's 

modulus  is  often  determined  by  measuring  the  deflection  of  a  beam  supported 
at  both  ends  and  loaded  in  the  middle.  If  d  is  the  depression  of  the  middle 
of  the  bar,  then  (see  p.  93) 

d  =  — ^_  AB' 

48<7Ay5;- 

where  W  is  the  load,  AB  the  length  of  the  bar,  q  Young's  modulus,  Ak'' 
the  moment  of  inertia  of  the  cross-section  of  the  bar  about  an  axis  through 
the  centre  of  gravity  of  the  section  at  right  angles  to  the  plane  of  bending. 
The  value  of  d  can  be  determined  by  fixing  a  needle  point  to  the  middle 
of  the  bar,  and  observing  through  a  microscope  provided  with  a  micrometer 
eyepiece  the  depression  of  the  beam  when  loaded  in  the  middle  with  various 
weights.  Another  method  of  measuring  d  is  by  means  of  a  very  carefully 
made  screw,  the  end  of  which  is  brought  into  contact  with  the  bar;  by 
measuring  the  fraction  of  a  turn  through  which  the  head  of  the  screw- 
must  be  turned  to  renew  the  contact  after  the  bar  has  been  loaded  we  can 
determine  the  value  of  d  corresponding  to  given  loads.  The  most  accurate 
method,  how^ever,  would  be  an  optical  one,  in  which,  by  Michelson's  method, 
interference  fringes  are  produced  by  the  interference  of  light  reflected 
from  two  mirrors,  one  of  which  is  fixed  while  the  other  is  attached  to  the 
middle  point  of  the  bar.  By  measuring  the  displacement  of  the  fringes 
when  the  load  is  put  on  we  could  determine  d,  and  the  method  is  so 
delicate  that  the  displacenaents  corresponding  to  very  small  loads  gould  b© 


100 


PROPERTIES  OF  MATTER. 


Another  method,  due  to  Konig,  consists  in  measuring  the  angle  through 
which  the  free  ends  of  the  bar  are  bent.  The  method  is  represented  in 
Fig.  75.  AB  is  the  rod  resting  on  two  steel  knife  edges  S,,  S,.  The  mirrors 
Pp  P,,  which  are  almost  at  right  angles  to  the  rods,  are  rigidly  attached  to 
it.  The  vertical  scale  S  is  reflected  first  from  the  mirror  P^,  then  from 
the  mirror  Pj,  and  then  read  through  the  telescope  F.  The  Aveight  is 
applied  to  the  knife  edge  r,  which  is  exactly  midway  between  the  knife 
edges  Sp  S^.  On  looking  tlii'ough  the  telescope  we  find  one  of  the  divisions 
of  the  scale  coinciding  with  the  cross  wires;  on  loading  the  beam  another 
division  of  the  scale  will  come  on  the  cross  wire,  and  by  measuring  the 
distance  between  these  divisions  we  can  determine  the  angle  ^>  through 
which  each  free  extremity  of  the  bar  has  been  bent.     For,  let  us  follow 

n 


Fig.  75. 

the  ray  backward  from  the  telescope ;  when  tlio  minor  P,  is  twist<^d 
through  an  angle  ^,  the  point  where  the  reflected  ray  strikes  the  mirror 
P,  is  shifted  through  a  distance  2cZ0,  where  d  is  the  distance  between  the 
mirrors;  thus,  if  the  light  reflected  from  P,  were  parallel  to  its  original 
direction,  the  scale  reading  would  be  altered  by  2d(f>,  but  the  light  reflected 
from  Pg  is  turned  through  an  angle  40  ;  this  alters  the  scale  reading  by 
4D0  where  D  is  the  distance  of  the  scale  S  from  the  mirror  P^j,  hence  v,  the 
total  alteration  in  the  scale  reading,  is  given  by 

'v  =  {-ld  +  \J))(p 


Tluis 


but.  (see  p.  91) 


^     2d  +  4D 
W       P 


f  = 


2.qAk'  8 

where  I  is  the  distance  between  the  knife-edges. 

Thus  knowing  v  we  can  determine  q.  The  advantage  of  this  method  is 
that  V,  the  alteration  in  the  scixle  reading,  may  be  made  very  much  greater 
than  the  depression  of  the  middle  of  the  bar. 


BENDING  OF  RODS. 


101 


The  following  cotiveniont  method  for  cletLTiniuiug  hoth  a  ;iiul  q  for  ;i 
wire  was  given  by  G.  F.  0.  Searle  in  the  Philosophical  Matjazine,  Feb.  IDOO. 
AB,  CD  (Fig.  7())  are  two  e<pial  bras.s  bars  of  square  section,  the  wire 
under  observation  is  lirnily  secured  by  passing  throiigli  horizontal  holes 
drilled  through  the  centres  G,  G'  of  the  bars.  The  system  can  be  suspended 
by  two  pai-allel  torsionless  strings 
by  means  of  hooks  attached  to  the 
bars.  If  now  the  ends  B  and  D 
are  made  to  approach  each  other 
through  equal  distances  and  are 
then  set  free  the  bars  will  vibrate 
in  a  horizontal  plane.  To  a  tirst 
approximation  the  centres  G  and  G' 
remain  at  rest,  so  that  the  action 
of  the  wire  on  the  bar,  and  therefore 
of  the  bar  on  the  wire,  is  a  pure 
couple  ;  the  wire  will,  therefore,  be 
bent  into  a  horizontal  circle  and 
the  couple  will  be  qA^k-jp.  Here 
q  is  Young's  modulus,  Ak-  the 
moment  of  inertia  of  the  cross- 
section  of  the  wire  about  an  axis 
through  the  centre  of  gravity  at 
right  angles  to  the  plane  of  bending, 
p  the  radius  of  curvature  of  the 
wire,  which  is  equal  to  1/2(1}  if  I  is  the  length  of  the  wire  and  (p  the  angle 
through  which  each  bar  is  twisted.  Hence,  is  K  if  the  moment  of  inertia 
of  CD  about  a  vertical  axis  through  G,  we  have 


Fio.  76, 


^d'(t> qAk- 2(7  AF 


de 


I 


hence,  if  T,  is  the  time  of  vibration, 

T  =9^    /J5 


Kl 

.qAk' 


(12) 


The  bars  are  now  unhooked  from  the  strings  and  one  clamped  co  a  shelf, 
so  that  the  wire  is  vertical ;  if  we  make  the  wire  execute  torsional  vibra 
tions,  and  T,  is  the  time  of  vibration, 


V   7r?ia 


(13) 


(see  p.  84),  n  being  the  coefficient  of  rigidity  and  a  the  radius  of  ice  wire 
As  the  wire  is  of  circular  section, 


hence  by  (12)  and  (13)  we  have 


Ak'  =  '^; 


n     T,« 


102 


PROPERTIES  OF  MATTER. 


TABLE  OF  MODULI  OF  ELASTICITY. 

The  values  of  the  moduli  of  elasticity  vary  so  much  with  the  treatment  a  meta 
has  received  in  wire-drawing,  rolling,  annealing,  and  so  on,  that  whenever  thej 
are  required  ior  a  given  specimen  it  is  necessary  to  determine  them,  if  any  degree 
of  accuracy  is  required.  The  following  table  contains  the  limits  within  which 
determinations  of  the  moduli  of  different  metals  lie.  They  are  taken  from  the 
results  of  experiments  by  Wertheim,  Kiewiet,  Lord  Kelvin,  Pisati,  Baumeister, 
Mallock,  Cornu,  Everett,  and  Katzenelsohn.  The  values  are  given  in  C.G.S.  units, 
n  is  the  rigidity,  q  Young's  modulus,  k  the  bulk  modulus,  and  <t  Poissou's  ratio. 


% 


cr- 


3  K-3-n 


nAO" 

g/ion 

A/IO" 

<r 

Aluminium    , 

2-38— 3 -36 

7-4 

-13 

Brass     . 

3-44— 4-03 

9-48- 10-75 

10-2- 10-85 

•226— -469 

Copper . 

3-5— 4-5 

10-3— 12-8 

17 

•25  — -35 

Delta-Metal  . 

3-6 

9-1 

10 

— 

Glass     . 

1-2- 2-4 

5-4- 7-8 

3-4-4-2 

-20— -26 

Gold      . 

3-9— 4-2 

f  5 -48  (drawn)) 
\    8  (rolled)    / 

— 

•17 

Iron  (cast)     . 

3-5- 5-3 

9-8—16 

9-7—14-7 

•23— ^31 

Iron  (wrought) 

6 -6-7 -7 

17—20 

— 

— 

Lead     . 

•18 

•5-1-8 

3-7 

•375 

Phosphor  Bronze  . 

3-6 

9-8 

— 

— 

Platinum 

6-6~7-4 

15     17 

— 

•16 

Silver    . 

2-5— 2-6 

7-0— 7-5 



•37 

Steel 

7-7— 9-8 

18—29 

14-7-19 

•25— ^33 

Tin 

1-5 

4-2 





Zinc 

3-8 

8-7 

— 

•20 

CHAPTER  IX. 


SPIRAL  SPRINGS. 


Contents. — Flat  Springs — Inclined  Springs — Angular  deflexion  of  Free  End 
on  Loading — Vibrations  of  Loaded  Spring. 


^, 


The  theories  of  bending  and  twisting  have  very  important  applications 
to  the  case  of  spiral  springs.  By  a  spiral  spring  we  mean  a  uniform  wire 
or  ribbon  wound  round  a  circular  cylinder  in  such  a  way  that  the  axis  or 
the  wire  makes  a  constant  angle  with  the  generating  lines  of  the  cylinder. 

The  first  case  we  shall  consider  is  that  of  a  spiral  spring 
made  of  uniform  wire  of  circular  cross-section,  and  wound 
round  the  cylinder  so  that  the  plane  of  the  wire  is  everywhere 
approximately  perpendicular  to  the  axis  of  the^ylinder — i.e.,  a 
"tlat"  spring.  Let  us  suppose  that  such  a  spring  is  hung 
with  its  axis  vertical,  and  that  a  weight  W,  acting  along  the 
axis  of  the  cylinder,  is  applied  to  an  arm  attached  to  the 
lower  end  of  the  spring. 

Considering  the  equilibrium  of  the  portion  CP  of  the 
spring,  the  stresses  over  the  cross- section  P  must  be  in  equili- 
biium  with  the  force  W  at  C,  and  hence  these  stresses  must 
be  equivalent  to  a  tangential  force  W  acting  upwards,  and  a 
couple  whose  moment  is  Wa  and  whose  axis  coincides  with  the  I 
axis  of  the  wire  at  P,  a  being  the  radius  of  the  cylinder  on 
which  the  axis  of  the  wire  lies.  If  the  diameter  of  the  wire  is 
very  small  compared  with  a  we  may,  by  the  principles  ex- 
plained on  p.  89,  neglect  the  effects  of  the  tangential  force  in 
!comparison  with  that  of  the  couple  and  consider  the  couple 
alone.  This  couple  is  a  toi-sional  couple  and  is  constant  all 
along  the  wire ;  it  will  produce,  therefore,  a  uniform  rate  of 
twist ;  if  ^  is  the  rate  of  twist,  b  the  radius  of  the  wire,  and 
n  its  coefficient  of  rigidity,  then  we  have  (see  p.  79), 

Wa  =  |7rw6*0. 

Now  suppose  that  we  have  a  series  of  arms  of  length  a 
attached  to  the  wire  at  right  angles,  the  free  ends  of  these 
arms  all  being  in  the  axis  of  the  cylinder.  Then,  if  P,  Q  are 
two  points  near  together,  the  effect  of  the  twisting  is  to 
increase  the  vertical  distance  between  the  ends  of  the  arms 
attached  to  P,  Q  respectively  by  PQ  x  af,  and  since  a  and  ^  are  constants  | 
this  result  will  hold  whatever  the  distance  between  P  and  Q.  Suppose  Q  is  at 
the  fixed  and  P  at  the  free  end  of  the  spring,  then  the  increase  in  the 
vertical  distance  between  the  arm  attached  to  P  and  Q  will  be  the  vertical 
depression  of  the  weight  W  ;  in  this  case  PQ  =  /,  the  length  of  wire  in  the 
spring 


W 


Fig. 


^ 


hence,  if  d  is  the  depression  of  W, 


Vs 


s'- 


104  PROPERTIES  OF  MATTEB 


:P^i^ 


d=l xaxf 


Thus  d  varies  directly  as  the  area  of  the  cross-section  of  the  cylinder 

and  inversely  as  the  square  of  the  area  of  the  cross-section  of  the  wire.     We 

see  that  the  depression  of  the  weiglit  is  the  same  as  the  displacement  of 

the  extremity  of  a  horizontal  arm  of  length  a  attached  to  the  end  of  the 

Y  same  length  of  wire  when  pulled  out  straight  and  hung  vertically,  the  end 

,  of  the  horizontal  arm  being  acted  on  by  a  horizontal  force  equal  to  W  at 

aU'^.     right  angles  to  the  arm. 

'  To  take  a  numerical  example  :  suppose  we  have  a  steel  spring  300  cm. 

long  wound  on  a  cylinder  3  cm.   in  diameter,  the  diameter  of  the  wire 
being  '2  cm. 

w  =  8xl0",  a  =  l-5,  b  =  -l. 

If  this  spring  is  loaded  with  a  kilogramme  so  that  W  =  981  x  10^,  the 
depression  d  will  be  given  by 

^_600x981xl0^x(l-5)' 
7rx8x  lO'^xlO-* 

=  5  cm.  approximately. 

Energry  in  the  Spring". — Q,    the   energy  stored  in  the  spring,  is 
(see  p.  80)  given  by  the  equation 

Q  =  iTrnlby 

2Wa 


But  ^  = 

thus  Q 


Trnb* 

Trnb* 


This  result  illustrates  the  theorem  proved  on  p.  71. 

Spring-s  inclined  at  a  finite  Ang-le  to  the  horizontal  Plane.— 

The  flat  spring,  as  we  have  just  seen,  acts  entirely  by  torsion ;  in  inclined 
springs  however,  bending  as  well  as  torsion  comes  into  play.  Let  the  axis  of 
the  spring  make  a  constant  angle  a  with  the  horizontal.  Let  the  spring 
(Fig.  78)  be  stretched  by  a  weight  "W  acting  along  the  axis  of  the  cylinder 
on  which  the  spring  is  wound.  Then,  considering  the  equilibrium  of  the 
portion  AP  of  the  spring,  and  neglecting  as  before  the  tangential  stresses 
at  P,  we  see  that  the  stresses  at  P  must  be  equivalent  to  a  couple  whose 
moment  is  Wa,  and  whose  axis  is  PT,  the  horizontal  tangent  to  the 
cylinder  at  P.  This  couple  may  be  resolved  into  two : — one  with  the 
moment  \V«cosa  and  axis  along  the  wire  PQ,  tending  to  twist  the  spring ; 
the  second,  having  the  moment  Wasina  and  its  axis  PN  at  right  angles  to 
the  plane  of  the  spring  at  P  tending  only  to  bend  the  spring.  Now  the 
twisting  couple  Wacosa  will  produce  a  rate  of  twist  <p  given  by 

_  Wacosa 


SPIRAL  SPRINGS. 


105 


/\ 


\) 


where  C  is  a  quantity  depending  on  the  shape  and  size  of  the  cross- 
feection  of  the  spring.  When  the  spring  is  a  circular  wire  of  radius  6,  we 
have  seen  that  C  =  77672.  The  couple  Wasina  will  bend  the  spring  and 
will  alter  the  inclination  of  the  tangents  at  two  neighbouring  points  PQ  by 


Wa^ino  .  PQ 


^^T-^ 


where  D  =  A^*,  the    moment   of  inertia  of  the  area  of  the  cross-section 
of  the  wire  of  the  spi'ing  about  an  axis  through 
its  centre  of  gravity  at  right  angles  to  the  plane 
of  bending. 


Let    us    now 


changes  on  the  radial   arms 
spi'ing- 


consider   the    effect    of    these 

which    we  imagine 

fixed    to    the  spi-ing.     Let  us  first  consider  the 

vertical  displacements  of  the  ends  of  the  arms  at 

two  neighbouring  points  PQ.     Taking  first    the 

torsion,  the  relative  motion  of  the  ends  is  PQ^a, 

but    in   consequence   of   the   inclination    of   the 

spring   this   relative   motion    is   inclined    at   an 

angle  a  with  the   vertical    so   that   the  relative 

vertical  motion  is 

PQ.Wa''cos''a 
PQfwpcosa  = p 

Thus,  if  I  be  the  length  of  the  wire  in  the 
spring,  the  vertical  displacement  of  the  end  of 
the  spring  due  to  torsion  is 

ZWa^cos^a 

Now  consider  the  effect  of  the  bending  on  the 
vertical  motion  of  the  ends  of  the  rods  at  PQ. 
In  consequence  of  the  bending,  the  relative 
motion  is  in  a  plane  making  an  angle  a  with  the 
horizontal  plane  and  is  equal  to 


Wasina 
qD 


PQa 


L<-^-- 


«.*>■ 


To  get   the   vertical   component  of  this  we  must  multiply  by  sina,  and 
we  see  that  the  vertical  displacement  due  to  bending  is 


PQ 


Wa-sin'^a 


or  for  the  whole  spring 


qj) 

IWa-sin^a 
—qB 


Thus  the  total  vertical  displacement  is 

^fcos-a 
i.-J 


Y^^i"  *  c  a 


3)  >   S^<^ld'=ev^^G 


i 


ma- 


i  nG 


sin^a\ 


106  PROPERTIES  OF  MATTER. 

In  addition  to  the  vertical  displacement  there  will  be  an  angular  dis- 
placement of  the  pointer  at  the  end  of  the  bar  which  we  may  calculate  as 
follows.     First  take  the  torsion.     The  arm  at  Pis  twisted  relatively  to  the 

arm  at  Q  through  an  angle  in  a  plane  making  an  angle  -  -  a  with  the 

horizontal  plane  equal  to  PQ  x  f ;  the  angular  motion  in  the  horizontal 
plane  is,  therefore, 

PQ  X  ^  X  cos  I  ^  -  a  I 
or  p^WasinacoPa 

And  the  direction  is  such  that  as  we  proceed  along  the  spring  the  arms  are 
rotated  in  the  direction  in  which  the  spring  is  wound,  so  that  this  angular 
movement  due  to  the  toi-sion  is  such  as  to  tend  to  coil  up  the  spring. 
The  angular  deflection  due  to  torsion  for  the  whole  spring  is,  therefore, 

^.Wasinacosa 

Let  us  now  consider  the  angular  deflection  due  to  bending.  The  arm  at 
P  is  bent  relatively  to  that  at  Q  through  an  angle 

.Wasina 


PQ- 


q\y 


in  a  plane  making  an  angle  a  with  the  horizontal  plane  ;  projecting  this 
angle  on  the  horizontal  plane  the  relative  angular  motion  in  this  plane  of 
the  two  arms  is 

p^.  Wasinacosa . 
^         qD         * 

thus  the  angular  deflection  due  to  bending  for  the  whole  length  of  the 
spring  is 

iWosinacosa 

"     qiy 

The  deflection  in  this  case  is  in  the  opposite  dii'ection  to  that  due  to  the 
torsion,  and  is  such  as  to  tend  to  uncoil  the  spring.  The  total  angular 
deflection  is  thus 

H.^rixovv-t^i  lAV«sinacosa|-i-i-^ 

in  the  dii-ection  tending  to  coil  up  the  spring.  The  angular  deflection  is 
thus  proportional  to  sin  a  cos  a  and  is  greatest  when  a  =  3r/4.  The  deflection 
tends  to  coil  up  the  spring  or  uncoil  it  according  as 

if  the  spring  is  very  stifi'to  resist  bending  in  its  own  plane,  it  will  coil  up 
under  the  action  of  the  weight ;  if,  on  the  other  hand,  it  is  very  stiff  to 
resist  torsion,  it  will  uncoil.  This  is  exemplified  by  the  two  springs 
shown  in  Figs.  79,  80.    The  first,  which  is  made  of  strip  metal,  with  the  shon'' 


SPIRAL  SPRINGS. 


107 


dimension  in  the  plane  of  bending,  is  very  weak  to  resist  bending,  and  so 
tends  to  uncoil  when  stretched,  while  the  second,  which  is  also  made  of  a 
strip  of  metal,  but  with  the  long  side  in  the  plane  of  bending,  is  very  stiff 
to  resist  bending,  and  so  tends  to 
coil  up  when  stretched.  In  the 
case  of  a  circular  wire  of  radius  h 


C: 


hh^ 


so  that 

1-   1 

«C      qT> 

For  metals  q  is  greater  than  2?i,  so 
that 

L-i. 

?lU      qD 

is  positive,  and  thus  a  spring  made 

up 


tends   to  coil 


^V 


of   circular   wire 
when  extended. 

Vibrations    of    a    Loaded 

Spring". — We  can  use  the  up  and 
down  oscillations  of  a  fiat  spiial 
spring  to  determine  the  coefficient 
of  rigidity  of  the  substance  of  which 
the  spring  is  made.  Let  us  take 
the  case  of  a  fiat  spiral  spring  made 
of  wire  of  circular  cross-section ; 
then,  if  the  spring  is  extended  a 
distance  x  from  its  position  of  equi- 
librium, the  potential  energy  in  the 
spring  is  (see  p.  104)  equal  to 

X-  (1) 


/ 


TT? 


Ud" 


where  n  is  the  coefficient  of  rigidity, 

h  the  radius  of  cross-section  of  the  'I  I 

wire,  a  the  radius  of  the  cylinder  ^^^-  79-  Fig.  8Q 

on  which  the  spring  is  wound,  and 

I  the  length  of  the  spring.     If  the  end  of  the  spring  is  loaded  with  a  mass 

M,  the  kinetic  energy  of  this  mass  is  equal  to 

'dx\ 
di 


IM 


The  spring  itself  is  moving  up  and  down,  so  that  there  will  be  some  kinetic 
energy  due  to  the  motion  of  the  spring.  To  a  first  approximation  the 
vertical  motion  of  a  point  on  the  spring  is  proportional  to  its  distance  from 
the  fixed  eud,  so  that  the  velocity  at  a  distance  s  from  the  fixed  end  will  be 

s  dx 


108  PROPERTIES  OP  MATTER. 

If  p  is  the  mass  of  unit  length  of  the  spring,  the  mass  of  au  element  of 
length  ds  is  pds  and  its  kinetic  energy  is 

,    /dxVs' 

''[dt!  r^' 

fntegrating  this  expression   from   s  =  o  to  s  =  Z,  we  find  that  the  kinetic 
energy  of  the  spring  is 


^dt 
or  if  m  he  the  mass  of  the  spring 


H\dt) 

hence  the  total  kinetic  energy  is  equal  to 

Since  the  sum  of  the  kinetic  and  potential  energy  is  constant 


\  3j\dtJ        Ua^ 

is  constant,  hence  differentiating  with  respect  to  t  we  have 

This  equation  represents  a  periodic  motion,  the  time  T  of  a  complete 
vihration  being  given  by  the  equation 

T  =  2^A/M+m73 
^  ivnb'ma' 

When  T  has  been  determined  n  can  be  found  by  this  equation. 

Angular  Oscillations.* — We  can  prove  in  a  similar  way  that  T, 
the  time  of  vibration  of  a  suspended  bar  about  the  vertical  axis,  is  given 
by  the  equation 

where  MP  is  the  moment  of  inertia  of  the  bar  about  the  vertical  axis  and 
q  Young's  modulus  for  the  wire,  by  measuring  T,  we  can  determine  q. 

*  Avrton  and  Ferrv.  Proc.   R.S.,  vol.    xxxvi,,   p.    311  ;    Wilbeiforce,  Pliil.   Mag., 
Oct.  1894. 


CHAPTER   X. 


IMPACT. 


Contents. — Co-efficient  of  Restitution — Newton's  Experiments — Hodgkinson'g 
Experiments — Example  of  Collision  of  Kaihvay  Carriages — Hertz's  Investiga- 
tions— Table  of  Coefficients. 


Co-efficient  of  Restitution.— An  interesting  class  of  phenomena 
depending  on  the  elasticity  of  matter  is  that  of  collision  between  elastic 
bodies.  The  laws  governing  these  collisions  were  investigated  by  Newton 
and  his  contemporaries,  who  nsed  the  following  method.  The  colliding 
bodies  were  spherical  balls  suspended  by  strings  in  the  way  shown  in 
Fig.  81  ;  the  balls,  after  falling  from  given  heights,  struck  against 
each  other  at  the  lowest  point,  and  after  rebounding  again  reached  a 
certain  height.  By  measuring  these  heights  (and  allowing,  as  Newton 
did,  for  the  resistance  of  the  aii)  the  velocities  of  the  balls  before  and 
after  collision  can  be  determined.  New- 
ton in  this  way  showed  that  when  the 
collision  was  direct — i.e.,  when  the  rela- 
tive velocities  of  the  two  bodies  at  the 
instant  of  collision  was  along  the  common 
normal  at  the  point  of  impact — the 
relative  velocity  after  impact  bore  a 
constant  ratio  to  the  relative  velocity 
before  impact  —  the  relative  velocity 
being,  of  course,  reversed  in  direction. 
Thus,  if  u,  V  are  the  velocities  of  the 
bodies  before  impact,  u  being  the  velocity 
of  the  more  slowly  moving  body,  while  , 
U,  V  are  the  velocities  after  impact,  i 
then 

U-V  =  e(v-M)  (1) 

,-    -      ,  Fig.  81. 

where  e  is  a  quantity  called  the  co- 
efficient of  restitution,  and  Newton's  experiments  showed  that  e  depended 
only  on  the  materials  of  which  the  balls  were  made,  and  not  on  the  masses 
or  relative  velocities^  ATseries  of  experiments  were  made  by  Hodgkinson, 
tHeTesults  ofwhich  were  in  general  agreement  with  Newton's.  Hodgkinson 
found,  however  {Report  of  British  Associatio7i,  1834),  that  when  the  initial 
relative^yelocity  was  very  large  e  was  smaller  than  it  was  with  moderate 
velocity. 

Tincent*  has  shown  that  the  coefficient  of  restitution  is  given  by  the 
equation  6  =  6,,-  bu,  where  u  is  the  velocity  of  approach  and  e^  and  b  are 
constants. 

Equation  (1)  and  the  equation 

mM  +  Mv  =  mU  +  MV  (2) 

•  Vincent,  Proceedings  Cambridge  Philosophical  Society,  vol.  x  p,  332. 


\ 
I 


no  PROPERTIES  OF  MATTER. 

which  expresses  that  the  momentum  of  the  system  of  two  bodies  is  not 
altered  by  the  impact,  vi  and  M  being  the  masses  of  the  bodies,  are  sufficient 
to  determine  U,  V ;  solving  equations  (1)  and  (2)  we  find 

m  +  M        m  +  M^         ' 

m  +  M        m  +  M^         ' 
Hence  we  have 

^    ■  3,niV'+my'=^hmu'  +  mv'-m-e'),^^(v-uy  (3) 

Thus  the  kinetic  energy  after  impact  is  less  than  the  kinetic  energy 
before  impact  by 

Thus,  if  e  is  unity  there  is  no  loss  of  kinetic  energy.  In  all  other  cases 
there  is  a  finite  loss  of  kinetic  energy,  some  of  it  being  transformed  during 
the  collision  into  heat ;  a  small  part  only  c-f  it  may  in  some  cases  be  spent 
in  throwing  the  balls  into  vibration  about  theLr  figures  of  equilibrium. 

Collision  of  Railway  Carriagres. — To  get  a  clearer  idea  of  what 
goes  on  when  two  elastic  balls  impinge  against  each  other,  let  us  take  the 
case  of  a  collision  between  two  railway  carriages  running  on  frictionless 
rails,  each  carriage  being  provided  with  a  bufler  spring.  When  the 
carriages  come  into  collision,  the  first  efiect  is  to  compress  the  springs,  the 
pressure  Avhich  one  spring  exerts  on  another  is  transmitted  to  the  carriages, 
and  the  momentum  of  the  carriage  that  was  overtaken  increases,  while 
that  of  the  other  diminishes ;  this  goes  on  until  the  two  carriages  are 
moving  with  the  same  velocity,  when  the  springs  have  their  maximum 
compression  and  the  pressure  between  them  is  a  maximum.  The  kinetic 
energy  of  the  carriages  is  now  less  than  it  was  before  impact  liy 

and  this  energy  is  stored  in  the  springs.  The  springs  having  reache<i 
their  maximum  compression  begin  to  expand,  increasing  still  further 
the  momentum  of  the  front  carriage  and  diminishing  that  of  the  carriage 
in  the  rear.  This  goes  on  until  the  springs  have  legained  their  original 
length,  when  the  pressure  between  them  vanishes  and  the  carriages 
separate.  There  is  now  no  strain  energy  in  the  springs,  and  the  kinetic 
energy  in  the  carriages  after  the  collision  has  ceased  is  the  same  as  it 
was  before  it  began. 

The  reader  who  is  acquainted  with  the  elements  of  the  differential 
calculus  will  find  it  advantageous  to  consider  the  analytical  solution  of 
the  problem,  which  is  very  simple.  Let  .t,  y  be  the  coordinates  of  the 
centres  of  gravity  of  the  first  and  second  carriages  respectively,  ^t,  ^  the 
strength  of  the  springs  attached  to  these  Girriages  (by  the  strength  of 
a  spring  we  mean  the  force  required  to  produce   upit  ej^tensioft  of  th© 


IMPACT.  ni 

spring),  E,  T]  the  compressions  of  these  springs,  and  P  the  pressure  between 
them;  then  we  have 

h^  =  P,  ^ir}  =  P 

X-  y  =  constant  —  (^  +  ?;) 
The  solution  of  these  equations  is 

r  =  {v-  u]b)—- sinwi 

^         ^  M  +  m 


where  w  =./- '^^^ ;    u   and   v   are    the    initial    velocities   of   tlie 

V  11  + fj!    Mm    ' 

carriages,  and  t  is  measured  from  the  instant  when  the  collision  began. 

m'^  =      "'      {mu  +  M^}  -  -^  (u  -  w)cosa,« 
dt     M  +  7«^  ^     M  +  «i  ^ 

k  M^  =     ^^     [mw  +  Mv]  +  ^^^(»  -  w)coso>< 

Thus  the  springs  have  their  maximum  compression  when  -J-'  =  _ii,  i.e.^ 

CLL        etc 

when  (j)t  =  7r/2,  or  f  =  —  ;   at  this  instant  the  energy  stored  in  the  first 


P2  f/ 


9 — ^=*i'u-Mr— ! — T  ^ 

^  fM     ^^         ^fji  +  ^ili  +  m 


while  the  energy  in  the  second  spring  is  equal  to 


h^n 


^i^=U^-uy-J^,- 


Mm 


fi      ^  '         '  n  +  /i'  M  +  m 

At  the  instant  of  greatest  compression  the  amounts  of  energy  stored  I 
in  the  two  springs  are  inversely  as  the  strengths  of  the  springs. 

The   springs    regain    their   original   length   and    the    collision    ceases 
when  P  =  0 — i.e.,  when  wt  =  iT,  or 


TT  _  /    M?M     yit  +  ^ 


this  is  the  time  the  collision  lasts.  We  see  that  it  increases  as  the  masses 
of  the  carriages  increase  and  diminishes  as  the  strengths  of  the  springs 
increase.  It  is  independent  of  the  relative  velocity  of  the  carriages 
before  impact. 

In  the  case  of  the  collision  between  elastic  bodies  the  elasticity  of  the 
material  serves  instead  of   the  springs  in  the  preceding  example.     Th© 


112 


PROPERTIES  OF  MATTER. 


bodies  when  they  come  into  collision  flatten  at  the  point  of  contact  so 
that  the  bodies  have  a  tinite  area  in  common.  In  the  neighbourhood  of 
this  area  each  body  is  compressed ;  the  compression  attains  a  maximum, 
then  diminishes  and  vanishes  when  the  bodies  separate.  The  theory  of 
the  collision  between  elastic  bodies  has  been  worked  out  from  this  point 
of  view  by  Hertz  (see  Collected  Papers,  English  Tran.slation,  p.  140),  who 
finds  expressions  for  the  area  of  the  surface  in  contact  between  the 
colliding  bodies,  the  duration  of  the  contact  and  the  maximum  pre.ssure. 
The  duration  of  contact  of  two  equal  spheres  was  proved  by  llertz  to 
be  equal  to 

2-9432R:/?5SiZZr 

V      S{v  -  u)f 

where  R  is  the  radius  of  either  of  the  spheres,  s  the  density  of  the 
sphere,  q  and  <t  respectively  Young's  modulus  and  Poisson's  ratio  for 
the  substance  of  which  the  spheres  are  made.  Hamburget  has  measured 
the  time  two  spheres  are  in  contact  by  making  the  spheres  close  an 
electric  circuit  whilst  they  are  in  contact  and  measuring  the  time  the 
current  is  flowing.  The  results  of  his  experiments  are  given  in  the 
following  table.  They  relate  to  the  collision  of  brass  spheres  1"3  cm. 
in  radius: 


Relative  Velocity  in  cm.  per  sec. 

7-37 

12-29 

19-21 

29-5 

Duration  of  collision  (calculated) 
„         „          (observed)  . 

•000185 
•000196 

•000167 
•000173 

•000153 
■000157 

•000140 
•000148 

The  duration  of  the  impact  is  several  times  the  gravest  time  of  vibra- 
tion of  the  body.  In  order  to  start  such  vibrations  with  any  vigour 
the  time  of  collision  would  have  to  be  small  compared  with  the  time 
of  vibration.  We  conclude  that  only  a  small  part  of  the  energy  is  spent 
in  setting  the  sphejes  in  vjbiration. 

As  an  example  of  the  order  of  magnitude  of  the  quantities  involved 
in  the  collision  of  spheres  we  quote  the  results  given  by  Hertz  for  two 
steel  spheres  2  5  cm.  in  radius  meeting  with  a  relative  velocity  of  1  cm. 
per  second.  The  radius  of  the  surface  of  contact  is  'OlS  cm.  The  time 
of  contact  is  '00038  seconds.  The  maximum  total  pressure  is  2'-t7 
kilogrammes  and  the  maximum  pressure  per  unit  area  is  7300  kilogrammes 
per  square  centimetre. 

In  this  theory  and  in  the  example  of  the  carriages  with  springs  we 
have  supposed  that  the  work  dene  on  the  springs  is  all  stored  up  as 
available  potential  energy  and  is  ultimately  reconverted  into  kinetic 
energy,  so  that  the  total  kinetic  energy  at  the  end  of  the  impact  is  the 
SJime  as  at  the  beginning.  Tliis  is  the  case  of  the  impact  of  what  are 
called  perfectly  elastic  bodies,  for  which  the  coefficient  of  restitution  is 
equal  to  unity.  In  other  cases  we  see  by  equation  (3)  that,  instead  of  the 
whole  work  done  on  the  springs  being  reconverted  into  kinetic  energy, 
only  the  constant  fraction  e"  of  it  is  so  reconverted,  the  rest  being  ulti- 
mately converted  into  heat.  Now  our  study  of  the  elastic  properties  of 
bodies  has  shown  many  examples  in  which  it  is  impossible  to  convert  the 
energy  due  to  strain  into  kinetic  energy  and  the  kinetic  energy  back 
again  into  energy  due  to  strain  without  dissipation.  We  may  mention 
the  phenomena  of  elastic  fatigue  or  viscosity  of   metals  (see  page  57), 


IMPACT.  lis 

as  exemplified  by  the  torsional  vibrations  of  a  metal  wire,  where  the 
successive  transformations  of  the  energy  were  accompanied  by  a  con- 
tinued loss  of  available  energy.  Again,  the  elastic  after-effect  would 
prevent  a  total  conversion  of  strain  energy  into  mechanical  energy. 
For  example,  if  we  load  a  wire  up  to  a  certain  point,  and  measure 
the  extension  corresponding  to  any  load,  then  gradually  unload  the 
wire,  if  the  straining  has  gone  beyond  the  elastic  limit  the  extensions 
during  unloading  will  not  be  the  same  as  during  loading;  and  in  this  case 
there  will  in  any  complete  cycle  be  a  loss  of  mechanical  energy  proportional 
to  the  area  included  between  the  curves  for  loading  and  unloading.  The  per- 
centage loss  in  this  case  would  depend  upon  the  intensity  of  the  maximum 
stress;  if  this  did  not  strain  the  body  beyond  its  elastic  limit  there  would 
be  no  loss  from  this  cause,  while  if  the  maximum  strain  exceeded  this  limit 
the  loss  might  be  considerable.  This  may  be  the  reason  why  the  value 
of  e  diminishes  as  the  relative  velocity  at  the  moment  of  collision  increases, 
for  Hertz  has  shown  that  the  maximum  pressure  increases  with  the 
relative  velocity  being  proportional  to  the  2/r)ths  power  of  the  velocity, 
while  it  is  in(lependent  of  the  size  of  the  balls.  Thus  the  greater  the 
relative  velocity  the  more  will  the  maximum  pressure  exceed  the  elastic 
limit  and  the  larger  the  amount  of  heat  produced.  In  addition  to 
the  loss  of  energy  by  the  viscosity  of  metals  and  hysteresis  there  is 
in  many  cases  of  collision  permanent  deformation  of  the  surface  in 
the  neighbourhood  of  the  surface  of  contact.  This  is  very  evident 
in  the  case  of  lead  and  brass.  The  harder  the  body  the  greater  the  value 
of  e.  We  can  see  the  reason  for  this  if  we  remember  that  the  hardness 
of  a  body  is  measured  by  the  maximum  stress  it  can  suffer  without 
being  strained  beyond  the  elastic  limit,  while  any  strain  beyond  the 
elastic  limit  wotdd  increase  the  amount  of  heat  produced  and  so  diminish 
the  value  of  e. 

When  we  consider  the  various  ways  in  which  imperfections  in  the 
elastic  property  can  prevent  the  complete  transformation  of  the  energy  due 
to  strain  into  liinetic  energy  and  vice  versa,  it  is  somewhat  surprising  that 
the  laws  of  the  collision  of  imperfectly  elastic  bodies  are  as  simple  as 
Newton's  and  Hodgkinson's  experiments  show  them  to  be,  for  these  laws 
expi-ess  the  fact  that  in  the  collision  a  constant  fraction,  e^,  of  the  initial 
kinetic  energy  is  converted  into  heat,  and  that  this  fraction  is  independent 
of  the  size  of  the  spheres  and  only  varies  very  slowly  with  the  relative 
velocity  at  impact.  For  example,  Hodgkinson's  experiments  show  that 
when  the  relative  velocity  at  impact  was  increased  threefold  the  value  of  « 
in  the  case  of  the  collision  between  cast-iron  spheres  only  diminished  from 
•69  to  "59.  A  series  of  experiments  on  the  impact  of  bodies  meeting  with 
very  small  relative  velocities  would  be  very  interesting,  for  with  small 
velocities  the  stresses  would  diminish,  and  if  these  did  not  exceed  those 
corresponding  to  elastic  limits  some  of  the  causes  of  the  dissipation  of 
energy  would  be  eliminated,  and  it  is  possible  that  the  value  of  e  might 
be  considerably  increased. 

We  find,  too,  from  experiment  that  bodies  require  time  to  recover  even 
from  small  strain,  so  that,  if  the  rise  and  fall  of  the  stress  is  very  rapid, 
there  may  be  dissipation  of  energy  in  cases  where  the  elastic  limit  for 
slowl)'  varying  forces  is  not  overstepped. 

Hodgkinson  gives  the  following  formula  for  the  value  of  e^B?  when  two 
different  bodies  A  and  B  collide,  in  terms  of  the  values  of  e^A  ^or  the 

H 


114 


PROPERTIES  OF  MATTER. 


collision  between  two  bodies  each  of  material  A  and  e^B,  the  value  for  the 
collision  between  two  bodies  each  of  material  B. 


*AB  — 


and  he  finds  this  formula  agrees  well  with  his  experiments. 

The  following  considerations  would  lead  to  a  formula  giving  c^b  ^^ 
terms  of  e^A  ^^^^  ^bb-  Hertz  has  shown  that  the  displacements  of  the 
bodies  A  and  B  in  the  direction  of  the  common  normal  to  the  two  surfaces 
over  which  the  bodies  touch  are  proportional  to 

Lz:^' and  1^5.' 


^AA 

^^BB 

9, 

9, 

1 

-f-1 

9i 

92 

9x  9i 

where  <Tp  a.^  are  the  values  of  Poisson's  ratio  for  the  bodies  A  and  B  and 
5",,  q^  the  values  of  Young's  modulus.  Now  the  stresses  are  equal,  so  that, 
assuming  that  the  quantities  of  work  done  on  the  two  bodies  are  in  the 
ratio  of  the  displacements,  then,  if  E  is  the  whole  work  done, 


^-^E 


1  -  a.: 


E 


+ 


1- 


"i  and 


izV  +  i^^ 


9i  92  9i  92 

will  be  the  amounts  done  on  the  two  bodies.  Now  the  first  body  converts 
1  -  e^A  fii^d  the  second  1  -  e-^^  of  this  work  into  heat ;  hence  the  energy 
converted  into  heat  will  be 


9x  92_ 


and  this  must  equal 


e- 


AA 


9i  92 

(l-e\B)E 


E 


^  BB ■ 


hence 


*  AB  = 


1-: 


1-, 


The  following  table  of  the  values  of  e  is  taken  from  Hodgkinson's 
Report  to  the  British  Association,  1834  : 


Dast-iron  balls     . 

•66 

Clay     . 

Oast-iron — lead  . 

•13 

Clav — soft  brass 

Cast-iron — boulder  stone     . 

•71 

Glass  , 

Boulder  stone— brass  . 

■62 

Cork    . 

Boulder  stone — lead    . 

•17 

Ivory  . 

Boulder  stone — elm     . 

.    •oe 

Lead — glass 

Elm  balls     .... 

•60 

,Soft  brass — glass 

Soft  brass  (16  pt.  Cu.  and  1  pt. 

tin 

)     -36 

Bell  metal — glass 

Bell  metal  (16  pt.  Cu.  and  4  pt. 

tin 

)     -59 

Cast-iron — glass 

Lead    .         .         .         .         • 

.     ^20 

Lead  -ivory 

Lead — elm  .... 

.      ^41 

Soft  brass— ivory 

Elm — »oft  brass  . 

.     -52 

Bell  metal — ivorj 

17 
16 
94 
65 
81 
25 
78 
87 
91 
44 
78 
77 


IMPACT.  113 


The  case  where  a  permanent  deformation  is  produced  has  recently  been 
investigated  by  Vincent  {Proceedivgs  Cambridge  Philosophical  Society, 
vol.  X.  p.  332).  The  case  taken  is  that  of  the  indentation  produced  in  lead 
or  pai-affin  by  the  impact  of  a  steel  sphere.  He  finds  that  the  volume  of 
the  dent  is  proportional  to  the  energy  of  the  sphere  just  before  impact ; 
that  during  the  impact  {i.e.,  while  the  lead  is  flowing)  the  pressure  between 
the  sphere  and  the  lead  is  constant  and  varies  from  6  x  10**  to  13  x  10'*  dynes 
per  square  centimetre  f<ir  different  specimens  of  lead  ;  for  parafiin  the 
corre=!ponding  pressure  is  about  10"  dynes  per  square  centimetre. 


CHAPTEH  XL 
COMPRESSIBILITY  OF  LIQUIDS. 

Contents.— Changes  in  Volume  of  a  Tube  unrlcr  Internal  and  External  Tresvure — 
Measurements  of  Corapressibilitv  of  Liquids  by  methods  of  Jamin,  Regnault, 
Buchanan  and  Tait,  Aniagnt-  Comi)resfibility  of  Water— Effects  of  Temperature 
and  Pressure— Compressibility  of  Mercury  and  other  Liquids— Tensile  Strength 
of  Liquids. 

The  fact  that  water  is  compressible  under  pressure  was  establishctl  in  17G2 
by  Canton,  and  since  then  measurements  of  the  changes  of  volume  of 
liquids  under  pressure  have  been  made  by  many  physicists. 

The  problem  is  one  beset  with  experimental  ditficulties,  some  of  which 
may  be  illustrated  by  considering  the  case  of  a  liquid  inclosed  in  a  vessel 
such  as  a  thermometer;  when  pressure  is  applied  to  the  liquid,  the 
depression  of  the  liquid  in  the  stem  will  be  due  partly  to  the  contraction 
of  the  liquid  under  pressure  and  partly  to  the  expansion  of  the  bulb  of  the 
thermometer.  In  order,  then,  to  be  able  to  determine  from  the  depression 
of  the  liquid  the  compressibility  of  water  we  must  be  able  to  estimate  the 
alteration  in  volume  of  the  tube  under  pressure.  We  shall  therefore 
consider  in  some  detail  the  alteration  in  volume  of  a  vessel  subject  to 
internal  and  external  pressui-e.  We  shall  take  the  case  of  a  long  cylindrical 
tube  with  flat  ends  exposed  to  an  external  pressure  p^  and  an  internal 
pressure  jo^.  The  strain  in  such  a  cylinder  has  been  shown  by  Lame 
to  be  (1)  a  radial  displacement  p  given  by  the  equation 

p  =  A?-  +  - 
r 

where  r  is  the  distance  of  the  point  under  consideration  from  the  axis  of 
the  cylinder  and  A  and  B  constants,  and  (2)  an  extension  parallel  to  the 
axis  of  the  cylinder. 

The  radial  displacement  p  involves  an  elongation  along  the  radius  equal 
to  dpjdr  and  an  elongation  at  right  angles  to  p  in  the  plane  at  right  angles 
to  the  axis  of  the  cj'linder  equal  to  pjr.  Let  the  elongations  along  the 
radius,  at  right  angles  to  it  and  to  the  axis  of  the  cylinder,  and  along  the 
axis  be  denoted  by  e,  f,  g  respectively,  and  let  P,  Q,  E,  be  the  normal 
stresses  in  these  directions ;  then  by  equation  (1 ),  p.  72,  we  can  easily  prove 

'^. '^.''  -n     ft  .  4n\     .  /,      2n\ 


where  k  is  the  bulk  modulus  and  n  the  coefficient  of  rigidity. 


0) 


r 


COMPRESSIBILITY  OF  LIQUIDS.  117 


Since  e=  f  and  f=^ 

dr  r 

we  have  e  =  A  -  - ,        /=  A  +  — 

Thus  the  radial  stress  is  equal  to 


o 


2/.A  +  =^lA-^Vf^-??V 


n 


•6\         1- )      \        3/ 

If  a  and  h  are  respectively  the  internal  and  external  radii  of  the  tube, 
then  when  r  =  a  the  radial  stress  is  equal  to  —p^  and  when  r  =  h  the  radial 
stress  is  equal  to  —  ^j,  hence  we  have 

-,.=  2M4f(A-??).(.-|'),  (2) 

The  whole  force  parallel  to  the  axis  tending  to  stretch  the  cylinder  i.-> 

Trd-p^  -  nb-p^ 

hence  the  stress  in  this  direction  is  equal  to 

na-po  -  7ri7?i 
7r{b''  -  a-) 

The  stress  parallel  to  the  axis  is,  however,  equal  to 


(.4)..(.--).A 


hence  we  have 


From  (2),  (;3)  and  (4)  we  get 

and  ^  =  2i^(¥-^f''"-P^^    |  (^"^ 

Since   the  radial  displacement  is  Ar+— ,  the  internal  volume  of  the 

tube  when  sti-ained  is        n(a  +  Aa  +  —]'l{l+g)  A/9( 

•■'*-  where  Hs  the  length  of  the  tube;  hence,  retaining  only  the  first  powers 

v-''^,         of  the  small  quantities  A,  B  and  g,  we  b-ave,  if  cv^  is  the  change  in  the 
internal  volume, 


-Lf 


-  ;'•     /.  ^A.\ 


118  PROPERTIES  OF  MATTER. 

and  if  3y,  is  the  change  in  the  external  volume, 

I     b'  -a-     k^ 0-  -  a'      n     J 

Methods  of  Measuring"  Compressibility  of  Liquids. — There  are 

two  cases  ot"  s^jecial  iinportaiice  in  the  determination  of  the  compressibility 
of  fluids  :  the  first  is  when  the  internal  and  external  pressures  are  equal ; 
in  this  case  Pq=j>v  '^^^  ^®  have 

TTClH 

T 


^^1=  --jrPo 


Thus  the  diminution  of  the  volume  is  independent  of  the  thickness  of  the 
walls  of  the  tube.  Some  experimenters  have  been  led  into  error  by  supposing 
that,  if  the  walls  of  the  tube  were  very  thin,  there  would  be  no  appreciable 
diminution  in  the  volume  of  the  tube.  If  the  vessel  had  been  filled  with 
liquid  which  was  subject  to  the  pressure  p^,  the  diminution  in  the  volume 
of  the  liquid  would  be  Tra"lpJK,  where  K  is  the  bulk  modulus  of  the  liquid. 
The  diminution  of  volume  of  the  liquid  minus  that  of  the  vessel  is 
therefore 


'^"^^E"^) 


thus   by  experiments  with   equal    pres.sui-es   inside   and  out,   which   was 
Regnault's  method,  we  determine 

11 
K    k 

so  that  to  deduce  K  we  must  know  k. 

Another  method,  used  by  Jamin,  waste  use  internal  pressure  only,  when 
the  apparent  change  in  the  volume  of  the  liquid  is  the  sum  of  the  changes 
of  volumes  of  the  liquid  and  of  the  inside  of  the  vessel.  Jamin  thought 
that  he  determined  the  change  of  volume  of  the  vessel  by  placing  it  in  an 
outer  vessel  full  of  water  and  measuring  the  rise  of  the  water  in  a  gradu- 
ated capillary  tube  attached  to  this  outer  vessel ;  by  subtracting  this  change 
in  volume  from  the  apparent  change  he  thought  he  got  the  change  in 
volume  of  the  liquid  without  requiring  the  values  of  the  elastic  constants 
of  the  material  of  which  the  vessel  is  made.  A  little  consideration  will 
show,  however,  that  this  is  not  the  case.  Let  cv  be  the  change  in  the 
volume  of  the  liquid,  Sr,  the  change  in  the  internal  volume,  h\  that  in  the 
external  volume;  it  is  oi'^  that  is  measured  by  the  rise  of  liquid  in  the 
capillary  tube  attached  to  the  vessel  containing  the  tube  in  which  the 
liquid  is  compressed. 

Observations  on  the  liquid  inside  the  tube  give 

Sv  +  Svj 
if  we  subtract  Jamin's  correction  we  get 

Sv  +  h\  -  Svj 
substituting  the  values  of  h\  and  Sr^  when  7?j  =  o  we  find 

h  +  Si',  -di\  =  ov-  — -^-5-  and  dv  =  — ^ 


COMPRESSIBILITY  OF  LIQUIDS. 


119 


f  1      V 

■-.--]  the  same 


Hence,  after  applying  Jamin's  correction,  we  get  naHpJ  t>  ~  r 

quantity  as  was  determined  by  Regnault's  method,  so  that  to  get  K  by 
Jamin's  method  we  require  to  know  k. 

The  apparatus  used  by  Rcgnault  in  his  experiments  on  the  comf>r66si- 
bility  of  Hquids  (Memoires  de  VInstitut  de  France, 
vol.  xxi.  p,  429)  was  similar  to  that  represented  in 
Fig.  82.  The  piezometer  was  filled  with  the  liquid 
whose  compressibility  was  to  be  measured,  the 
greatest  care  being  taken  to  get  rid  of  air-bubbles, 
The  liquid  reached  up  into  the  graduated  stem  of  the 
piezometer,  the  volume  between  successive  marks  on 
the  stem  being  accurately  known.  The  piezometer 
was  placed  in  an  outer  vessel  which  was  filled  with 
water  and  the  whole  system  placed  in  a  large  tank 
filled  with  water,  the  object  being  to  keep  the 
temperature  of  the  system  constant.  The  tubes 
shown  in  the  system  were  connected  with  a  vessel 
full  of  compressed  air,  the  pressure  of  which  was 
measured  by  a  carefully  tested  manometer ;  the 
tubes  were  so  arranged  that  by  turning  on  the 
proper  taps  pressure  could  be  applied  (1)  to  the 
outside  of  the  piezometer  and  not  to  the  inside ;  (2) 
simultaneously  to  the  outside  and  the  inside ;  (3)  to 
the  inside  and  not  to  the  outside.  The  piezometer 
used  by  Regnault  was  in  the  form  of  a  cylindrical 
tube  with  hemispherical  ends.  For  simplicity  let 
us  take  the  case  (represented  in  the  figure)  of  a  piezometer  in  the  form  of 
a  cylinder  with  flat  ends,  to  which  the  foregoing  investigation  applies. 
If  Wp  Wj,  W3  are  the  apparent  diminution  in  the  volume  of  the  liquid  in 
the  three  cases  respectively,  the  pressure  being  the  same,  we  have  by  the 
preceding  theory 

ira?hH  /I  ,  1\ 


Fig.  82. 


(t) 


b) 


Hence  Wj  +  Wj  =  wg 

a  relation  by  which  we  can  check  to  some  extent  the  validity  of  the 
theoretical  investigation.  Such  a  check  is  very  desirable,  as  in  this  investi- 
gation we  have  assumed  that  the  material  of  which  the  piezometer  is  made 
is  isotropic  and  that  the  walls  of  the  piezometer  are  of  unifoi-m  thickness, 
conditions  which  are  very  difficult  to  fulfil,  while  it  is  important  to 
ensure  that  a  failure  in  any  one  of  them  has  not  been  sufficient  to 
appreciably  impair  the  accuracy  of  the  theoretical  investigations.     Regnault 

in  his  investigations  adopted  Lamp's  assumption  that  Poisson's  ratio  ia 

3 
equal  to  1/4 ;  on  this  assumption  n  =  -A;,  so  that  the  measurement  of  Wj 

0 


120 


PROPERTIES  OF  MATTER. 


gives  the  value  of  Jc,  and  then  the  measurement  of  w^  the  value  of  K,  the 
bulk  modulus  for  the  liquid.  This  was  the  method  adopted  by  Regnault. 
It  is,  however,  open  to  objection.  In  the  first  place,  the  determinations 
which  have  been  made  of  the  value  of  Poisson's  ratio  for  glass  range  from 
•33  to  "22,  instead  of  the  assumed  value  "25,  while,  secondly,  the  equation  by 
which  k  is  determined  from  measurements  of  Wj  is  obtained 
on  the  assumption  of  perfect  uniformity  in  the  material 
which  it  is  difficult  to  verify.  It  is  thus  desirable  to 
determine  k  for  the  material  of  which  the  piezometer  is 
made  by  a  separate  investigation,  and  then  to  determine  the 
compressibility  of  the  liquids  by  using  the  simplest  relation 
obtained  between  the  apparent  change  in  volume  of  the  liquid 
and  the  pressure ;  this  is  when  the  inside  and  outside  of  the 
piezometer  are  exposed  to  equal  pressures.  The  most  direct, 
and  probably  the  most  accurate,  way  of  finding  k  for  a  solid  is 
to  measure  the  longitudinal  contraction  under  pressure.  An 
arrangement  which  enables  this  to  be  done  with  great 
accuracy  is  described  by  Amagat  in  the  Journal  de  Physique, 
Series  2,  vol.  viii.  p.  359.  The  method  was  first  used  by 
Buchanan  and  Tait.  Another  method  of  determining  k  for 
a  solid  is  to  make  a  tube  of  the  solid  closed  by  a  graduated 
capillary  tube  as  in  Fig.  83.  The  tube  and  part  of  the 
capillary  being  tilled  with  water,  a  tension  P  is  applied  to  the 
tube,  the  tube  stretches  and  the  internal  volume  increases,  the 
increase  in  volume  being  measured  by  the  descent  of  the  liquid 
in  the  capillary  tube ;  if  v  is  the  original  internal  volume,  Su 
the  increase  in  this  volume,  then  we  see  by  the  inve.stigation, 
p.  12,  that 

£y^P 
V      '6k 

If  we  have  found  k,  then  K  can  be  found  by  means  of  the 
piezometer. 

If  we  can  regard  the  compressibility  of  any  liquid,  say 
mercury,  as  known,  the  most  accurate  way  of  finding  tha 
compressibility  of  any  other  liquid  would  be  to  fill  the 
piezometer  first  with  mercury,  and  determine  the  apparent 
change  of  volume  Avhen  the  inside  and  outside  of  the 
piezometer  are  exposed  to  the  same  pressure ;  then  fill  the 
piezometer  with  the  liquid  and  again  find  the  apparent  change 
in  volume.  We  shall  thus  get  two  equations  from  which  we 
can  find  the  value  of  K  for  the  liquid  and  k  for  the  piezometer. 

Results   of  Experiments. — The   results  of  experiments  made   by 
diflerent  observers  on  the  compressibility  of  water  are  given  below. 

Regnault.* — Temperature  not  specified  ;  pressures  from  1  to  10  atmo- 
spheres— 

compressibility  per  atmosphere  =  0.0000 18. 


Fio.  83. 


*  Mimoircs  de  Vlnstitut  de  France,  vol.  xxi.  p.  429. 


COMPRESSIBILITY  OF  LIQUIDS. 


121 


GKASSI.* 

PAGLIANIand  VICENTINI.t 

RONTGEN  and  SCHNEIDER.  J 

Temp. 

Compressibility 
pel'  atmosxjheie. 

Temp. 

Compressibility 
per  atmosphei.. 

Temp. 

Compressibility 
per  atmosphere. 

00 
1-5 
4-0 

max.  density 
pt. 

10-8 
13-4 
18-0 
25.0 
34-5 
430 
530 

503x10-'' 

515 

499 

480 
477 
462 
456 
453 
412 
441 

0.0 
2-4 
15-9 
49-3 
61-1 
C6-2 
77-4 
99.2 

503x10-'' 

496 

450 

403 

389 

389 

398 

409 

0  0 

9-0 

18-0 

512x10-7 

4S,1 

462 

Tait§  has  found  that  the  efiect  of  temperature  and  pressure,  for 
temperatures  between  6°  0.  and  15°  0.  to  pressures  from  150  to  500 
atmospheres,  may  be  represented  by  the  empirical  formula 

IiLlJ'  =  0-0000489  -  0-00000025^  -  0.0000000067o 

where  v  is  the  volume  at  f  C.  under  the  pressure  of  p  atmospheres  and  v 
the  volume  at  f°  under  one  atmosphere.    Thus  the  compressibility  diminishes 
as  the  pressure  increases. 

The  numbers  given  above,  from  Grassi's  experiments,  indicate  that 
water  has  a  maximum  compressibility  at  a  temperature  between  0°  and  4:°  C: 
this  result  has  not,  however,  been  confirmed  by  subsequent  observers.  The 
I'esults  of  Pagliani  and  Vicentini  indicate  a  minimum  compressibility  at 
a  temperature  between  60°  and  70°  0. 

The  results  of  various  observers  on  the  compressibilit}'  of  mercury  are 

given  in  the  following  table : 

Com  pressibility 
per  atmosphere. 


Observer. 


Colladon  and  Sturm  H         , 

Aime^     .  .  .         . 

Regnault** 

Amaury  and  Descamp>  ft 

JLait^.^      .  .  .  . 

Amagat§§ 

De  Metzllll 

Mean        .         .         .         . 


35-2x10-' 
39-0x10-' 
35-2x10-^ 
38-6x10-^ 
36-0x10-'' 
39-0x10-' 
37-4x10-' 

37-9x10-' 


The  compressibility  of  mercury,  like  that  of  most  fluids,  increases  as  the 

*  Gras.si,  Annalcs  de  C'himic  ct  de  Physique  [3],  31,  p.  437,  1851. 

f  Pagliani  and  Vicentini,  Nuovo  Cimento  [3],  16,  p.  27,  1884. 

X  Rontgen  and  Schneider,  Wicd.  Ann.,  33,  p.  644,  1888. 

§  Tait,  Properties  of  Matter,  1st  ed.  (1885),  p.  190. 

II  Colladon  and  Sturm,  Ann.  de  Chimie  et  de  Physique,  36,  p.  137,  1827. 

«I  Airae,  Annah.f  de  Chimie  et  de  Physique  [3],  8,  p.  268.  1843. 

**  Eegnault,  Memoires  de  I'lnstitut  de  France,  21,  p.  429,  1847. 

tt  Araaury  and  Descarops,  Compt.  Rend.,  68,  p.  1564,  1869. 


+t 


Tait,  Challenger  Report,  vol.  ii.  part  iv. 


§§  Amagat,  Journal  de  Physique  [2],  8,  p.  203,  1889. 
nil  De  Metz,  Wicd.  Ann.,  47,  p.  731,  1892. 


122 


PROPERTIES  OF  MATTER. 


temperature  increases.     According  to  De  Metz,  the  compressibility  at  t"  0. 
is  given  by 

37-4  X  10-' +  87-7  xlO-"« 

The  compressibilities  of  a  number  of  liquids  of  frequent  occurrence  are 
given  below. 


Fluid. 

Conipressiliility  per 
atinospliere. 

Temp. 

Observer. 

Sea-water    , 
Ether  . 

11     •         • 
Alcohol 

11              •         I 
11              •         I 

Methyl  alcohol 

Turpentine  . 

11          • 
Chloroform  . 
Glycerine     . 
Olive  oil 
Carbon  bisulphid 

11               II 
Petroleum    . 

e 

436x10-7 
1156x10-7 
1110x10-7 
828x10-7 
959x10-7 
828x10-7 
913x10-7 
682x10-7 
779x10-7 
625x10-7 
252x10-7 
486x10-7 
539x10-7 
038x10-7 
650x10-7 
745x10-7 

17-5° 
0° 
0° 
0° 

17-5' 
7-3° 

13-5" 
0° 

18-6 
8-5- 
0° 
0° 
0° 
17° 
0° 

19-2' 

Grassi 
Quincke 

Grassi 
Quincke 

Grassi 

fj  ■ 
Quincke 

11 

Grassi 

Quincke 

» 

II 

II 

II 

» 

waters 


V 


Quincke's  paper  is  in  Wiedemanns  Anndlen,  19,  p.  401,  1883.  Eeferences  to 
the  papers  by  the  other  observers  have  already  been  given.  An  exten- 
sive series  of  investigations  on 
the  compressibility  of  solutions 
has  been  made  by  Rontgen 
and  Schneider  {Wied.  Ann.,  29, 
p.  16 5,  and  81,  p.  1000),  who  have 
shown  that  the  compressibility 
of  aqueous  solutions  is  less 
than  that  of  water.  For  the 
details  of  their  results  we  must 
refer  the  leader  to  their  paper. 

Tensile   Streng-th  of 

Liquids. — Liquids  from  which 
the  air  has  been  carefully  ex- 
pelled can  sustain  a  considerable 
pull  without  rupture.  The  best 
known  illustiation  of  this  is 
the  sticking  of  the  meicury  at 
the  top  of  a  barometer-tube. 
If  a  barometer-tube  filled  with 
mercury  be  carefully  tilted  up 
to  a  vertical  position,  the  mer- 
cury sometimes  adheres  to  the 
top  of  the  tube,  and  the  tube  remains  filled  with  mercury,  although  the 
length  of  the  column  is  gieater  than  that  which  the  normal  barometric 
pressure  would  support,  and  the  extra  length  of  mercury  is  in  a  state  of 
tension.      Another  method  of  showing  that  liquids  can  sustain  tension 


water  vapour 


COMPRESSIBILITY  OF  LIQUIDS. 


123 


without  rupture  is  to  use  a  tube  like  that  in  Fig.  84,  filled  with  water  and 
the  vapour  of  water,  and  from  which  the  air  has  been  eai*efully  expelled 
by  boiling  the  water  and  driving  the  air  out  by  the  steam.*  If  the  water 
occupies  the  position  indicated  in  the  figure,  the  tube  mounted  on  a  board 
may  be  moved  rapidly  forward  in  the  direction  of  the  arrow,  and  then 
brought  suddenly  to  rest  by  striking  the  board  against  a  table  without  the 
water  column  breaking,  although  the  column  must  have  experienced  a 
considerable  impulsive  tension.  If  the  column  does  break,  a  small  bubble 
of  air  can  generally  be  observed  at  the  place  of  rupture,  and  until  this 
bubble  has  been  removed  the  column  will  break  with  great  ease.  On  the 
removal  of  the  bubble  by  tapping,  the  column  can  again  sustain  a  con- 
siderable shock  without  rupture. 

Professor  Osborne  Reynolds  used  the  following  method  for  measuring 
the  tension  liquids  would  stand  without  breaking.      ABCD,  Fig.  85,  is  a 
glass   U-tube,  closed  at  both  ends,  containing  air-free  liquid  ABO   and 
vapour  of  the  liquid  CD.      The  tube  is  fixed  to  a  board  and  whirled  by  a 
lathe  about  an  axis  O  a  little  beyond  the  end  A  and  perpendicular  to  the 
plane  of  the  board.    If  CE  is  an  arc  of  an  circle  with  centre  O,  then  when 
the  board  is  rotating  the  liquid  EA  is  in  a  state  of  tension, 
the   tension   increasing   from   E    to    A,    and    being    easily 
calculable   if   we   know   the    velocity   of  rotation.     By   this 
method  Professor  Osborne  Reynolds  found  that  water  could 
sustain  a  tension  of  72 "5  pounds  to  the  square  inch  without 
rupture,  and  Professor  Worthington,  using  the  same  method, 
found  that  alcohol  could  sustain  116  and  strong  sulphuric 
acid  173  pounds  per  square  inch.     This  method  measures  the 
stress  liquids  can  sustain  without  rupture.     Berthelot  has 
used  a  method  by  which  the  strain  is  measured.     The  liquid 
freed  from  air  by  long  boiling  nearly  filled  a  straight  thick- 
walled  glass  tube,  the  rest  of  the  space  being  occupied  by  the 
vapour  of  the  liquid.     The  liquid  was  slightly  heated  until  it 
occupied  the  whole  tube;  on  cooling,  the  liquid  continued  for 
some  time  to  fill  the   tube,    finally    breaking   with   a   loud 
metallic  click,  and  the  bubble  of  vapour  reappeared :    the  length  of  this 
bubble  measured  the  extension  of  the  liquid.     M.  Berthelot  in  this  way 
got  extensions  of  volume  of  1/120  for  water,  1/93  for  alcohol,  and  1/51)  for 
ether.     Professor  Worthington  has  improved  this  method  by  inserting  in 
the  liquid  an  ellipsoidal  bulb  filled  with  mercury  and  provided    with  a 
narrow  graduated  capillaxy  stem  ;  when  the  liquid  is  in  a  state  of  tension 
the  volume  of  the  bulb  expands  and  the  mercury  sinks  in  the  stem ;  from 
the  amount  it  sinks  the   tension  can  be  measured.     The  extension  was 
measured   in  the  same  way  as  in  Berthelot's  experiments.    In  this  way 
Professor  Worthington  showed   [Phil.  Trans.  A.  1892,  p.  355)  that  the 
absolute    coeflicient    of   volume    elasticity   for   alcohol    is   the   same   for 
extension  as  for  compression,  and  is  constant  between  pressures  of   + 12 
and  — 17  atmospheres. 

*  Dixon  and  Joly  (Pldl.  Trans.  B.  1895,  p.  568)  have  shown  that  air  or  other  gases 
held  in  solution  do  not  afltect  these  experiments.  The  bojling  is  probablj^  efficacious 
ooly  in  removing  bubbles  or  free  gases, 


0 


Fia.  85. 


CHAPTER  XII. 

THE  RELATION  BETWEEN  THE  PRESSURE  AND  VOLUME 

OF  A  GAS. 

Contents— Boyle's  Law— Deviations  from  Bovle's  Law— Fegnault's  Experiments— 
Amagat's  Experiments — Experiments  at  Low  rre.s»urcs — Van  der  Waals'  Equation. 

In  this  chapter  we  shall  confine  ourselves  to  the  discussion  of  the  relation 
between  the  pressure  and  the  volume  of  a  gas  when  the  temperature  is 
constant  and  no  change  of  state  takes  place ;  the  liquefaction  of  gases 
will  be  dealt  with  in  the  volume  on  Heat. 

The  relation  between  the  pressure  and  the  volume  of  a  given  mass  of 
gas  was  first  stated  by  Boyle  in  a  paper  communicated  to  the  Royal  Society 
in  1061.  The  experiment  which  led  to  this  law  is  thus  described  by  him. 
"  We  took  then  a  long  glass  tube,  which  by  a  dexterous  hand  and  the  help 
of  a  lamp  was  in  such  a  manner  crooked  at  the  bottom,  that  the  part 
turned  up  was  almost  parallel  to  the  rest  of  the  tube,  and  the  orifice  of 
this  shorter  leg  of  the  siphon  (if  I  may  so  call  the  whole  instrument)  being 
hermetically  sealed,  the  length  of  it  was  divided  into  inches  (each  of  which 
was  subdivided  into  eight  parts)  by  a  straight  list  of  paper,  which,  con- 
taining those  divisions,  was  carefully  pasted  all  along  it.  Then  putting  in 
as  much  quicksilver  as  served  to  fill  the  arch  or  bended  part  of  the  siphon, 
that  the  mercury  standing  in  a  level  might  reach  in  the  one  leg  to  the 
bottom  of  the  divided  paper  and  just  to  the  same  height  or  horizontal  line 
in  the  other,  we  took  care,  by  frequently  inclining  the  tube,  so  that  the 
air  might  freely  pass  from  one  leg  into  the  other  by  the  sides  of  the 
mercury  (we  took,  I  say,  care),  that  the  air  at  last  included  in  the  shorter 
cylinder  should  be  of  the  same  laxity  with  the  rest  of  the  air  about  it. 
This  done,  we  began  to  pour  quicksilver  into  the  longer  leg  of  the  siphon, 
which  by  its  weight  pressing  up  that  in  the  shoi-ter  leg  did  by  degi-ees 
strengthen  the  included  air,  and  continuing  this  pouring  in  of  quicksilver 
till  the  air  in  the  shorter  leg  was  by  condensation  reduced  to  take  up  but 
half  the  space  it  possessed  (I  say,  possessed  not  filled)  before,  we  cast  our 
eyes  upon  the  longer  leg  of  the  glass,  on  which  was  likewise  pasted  a  list 
of  paper  carefully  divided  into  inches  and  parts,  and  we  observed  not 
without  delight  and  satisfaction  that  the  quicksilver  in  that  longer  part 
of  the  tube  was  29  inches  higher  than  the  other  .  .  .  the  same  air 
being  brought  to  a  degree  of  density  about  twice  as  great  as  that  it  had 
befoi-e,  obtains  a  spring  twice  as  strong  as  formeily."  Boyle  made  a  series 
of  measurements  with  greater  compressions  until  he  had  reduced  the 
volume  to  one  quarter  of  its  original  value,  and  obtained  a  close  agreement 
between  the  pressure  observed  and  "  what  that  pressure  should  be  according 
to  the  hypothesis  that  supposes  the  pressures  and  expansions*to  be  in 
reciprociil  proportions."  Although  Mariotte  did  not  state  the  law  until 
fourteen  years  after  Boyle  had  published  his  discovery,  "  the  hypothesis 

*  Or  volumes,  in  modern  Euglisli. 


THE  PRESSURE  AND  VOI,UME  OF  A  GAS.  125 

that  supposes  the  pressures  and  expansions  to  be  in  reciprocal  proportions  " 
is  often  on  the  Continent  called  Mariotte's  Law. 

If  V  is  the  volume  of  a  given  mass  of  gas  and  p  the  pressure  to  which 
it  is  subjected,  then  Boyle's  Law  states  that  when  the  temperature  is 
constant 

pv  =  constant. 

Anotlier  way  of  stating  this  law  is  that,  if  p  is  the  density  of  a  gas  under 
pressure  p, 

p  =  'Rp, 

where  R  is  a  constant  when  the  temperature  is  constant.  Later  researches 
made  Vjy  Charles  and  Gay-Lussac  have  shown  how  R  varies  with  the 
temperature  and  with  the  nature  of  the  gas.  These  will  be  described  in 
the  volume  on  Heat ;  it  will  suffice  to  say  here  that  the  pressure  of  a  perfect 
gas  is  given  by  the  equation 

;>  =  KNT, 

where  T  is  the  absolute  temperature,  N  the  number  of  molecules  of  the  gas 
in  unit  volume,  and  K  a  constant  which  is  the  same  for  all  gases. 

From  the  equation  pv  =  c  we  see  that  if  Aj),  Av  are  corresponding  incre- 
ments in  the  pressure  and  volume  of  a  gas  whose  temperature  is  constant, 
then 

or  -  — ^  =Pi 

Av 

but  the  left-hand  side  is  by  definition  the  bulk  modulus  of  elasticity, 
hence  the  bulk  modulus  of  elasticity  of  a  gas  at  a  constant  temperature  is 
equal  to  the  pressure. 

The  work  required  to  diminish  the  volume  of  a  gas  by  Av  is  pAv  ;  the 
work  which  has  to  be  done  to  diminish  the  volume  from  v^  to  v^  is  there- 
fore 


j  pdvy 


or,  since  by  Boyle's  Law  p  =  c/v,  when  the  temperature  is  constant,  we 
see  that  in  this  case  the  work  is 

c     -dv  =  c  log,-i  =;;  Vjlog,'^ , 
J    V  ^j  ^« 

where  />,  is  the  pressure  when  the  volume  is  w,. 

Deviations  from  Boyle's  Law. — The  first  to  establish  in  a  satis- 
factory manner  the  existence  in  some  gases,  at  any  rate,  of  a  departure  from 
Boyle's  Law  was  Despretz,  who,  in  1827,  enclosed  a  number  of  different 
gases  in  barometer-tubes  of  the  same  length  standing  in  the  same  cistern. 
The  quantity  of  the  different  gases  was  adjusted  so  that  initially  the  mercury 
stood  at  the  same  height  in  the  diflerent  tubes ;  pressure  was  then  applied 
to  the  mercury  in  the  cistern,  so  that  mercury  was  forced  up  the  tubes. 
It  was  then  found  that  the  volumes  occupied  by  the  gases  were  no  longer 


126 


PROPERTIES  OP  MATTER. 


equal,  the  volumes  of  carbonic  acid  and  ammonia  were  less  than  that  of 
air,  -while  that  of  hydrogen  was  greater.  This  showed  that  some  of  the 
gases  did  not  obey  Boyle's  Law;  it  left  open  the  question,  however,  as  to 
whether  any  gases  did  obey  it.  The  next  great  advance  was  made  by 
llegnaultf  who  in  1847  settled  the  question  as  to  the  behaviour  of  certain 
gases  for  pressures  between  1  and  about  30  atmospheres,  llegnault's 
method  was  to  start  with  a  certain  quantity  of  gas  occupying  a  volume  v 
in  a  tube  sealed  at  the  upper  end,  and  with  the  lower  end  opening  into  a 
closed  vessel  full  of  mercury,  and  then  by  pumping  mercury  up  a  long 
mercury  column  rising  from  the  closed  vessel  to  increase  the  pressure  until 
the  volume  was  halved .  By  measuring  the  difference  of  height  of 
mercuiy  in  the  column  and  in  the  tube  the  pressure  required  to  do  this 
could  be  determined.  Air  under  this  pressure  was  now  pumped  into  the 
closed  tube  until  the  volume  occupied  by  the  gas  was  again  v  ;  mercury 
was  again  pumped  up  the  column  until  the  volume  had  again  been  halved 
and  a  new  reading  of  the  pressure  taken ;  air  was  pumped  in  again  until 
the  volume  was  again  7;,  and  then  the  pressure  increased  again  until  the 
volume  was  halved.  In  this  way  the  values  oi  pv  at  a  series  of  different 
pressures  could  be  compared.  The  results  are  shown  in  the  following 
table 


,  -p^  is  given  in  millimetres  of  mercury,  /)„?'„  is  the  value  of  pv  at  the 
pressure  given  in  the  table,  p^^  the  value  at  double  this  pi-essure.  The 
experiments  were  made  at  temperatures  between  T  C.  and  10°  C. 


AIR. 

NITROGEN. 

CARBONIC  ACID. 

HYDROGEN. 

Vo 

'PoVollh'Vl 

Po 

PoVoIPiVi 

Po 

PoVolPiVy 

Po 

PoVolp^y^ 

73S72 

1-001414 

753-96 

1-001012 

764-03 

1-007597 

2068-20 

1-002709 

1159-43 

1-001074 

1414-77 

1-012313 

4219-05 

1-003336 

2159-22 

1-001097 

2164-81 

1-018973 

2211-18 

0-998.^.84 

6770-15 

1-004286 

3030-22 

1-001950 

3186-13 

1-028494 

3989-47 

0-996961 

9336-41 

1-006366 

4953-92 

1-002952 

4879-77 

1-045625 

5845-18 

0-9961-21 

11472-00 

1-005619 

5957-96 

1-003271 

6820-22 

1-066137 

7074-96 

0-994697 

7294-47 

1-003770 

8393-68 

1-084278 

9147-61 

0-993258 

8628-54 

1-004768 

96-20-06 

1-099830 

10361-88 

0-992327 

9767-42 

1-005147 

10981-42 

1-006456 

It  will  be  seen  from  these  figures  that  between  pi'essures  of  from  about 
1  to  30  atmospheres  the  product  pv  constantlydiminishes  for  air,  nitrogen, 
andjcarbonic  acid,  as  the  pressure  increasesTTFie  diminution  being  most 
marked  forcarbonicacid  ;  on  the  other  hand  in  h3'drogen  pv  increases  with 
The  pressure.  JNatterer^who  in  1850  published  the  results  of  experiments 
on  the  relation  between  the  pressure  and  volume  of  a  gas  at  very  high 
pressure,  showed  that  after  passing  certain  pressures  ^w  for  air  and  nitrogen 
begins  to  increase,  so  that  pv  has  a  minimum  value  at  a  certain  pressure; 
after  passing  this  pres.sure  air  and  nitrogen  resemlile  hydrogen,  and  pv 
continually  increases  as  the  pressure  increases.  This  result  was  confirmed 
by  the  researches  of  Amagat  and  Cailletet.  Eaoii  of  these  physicists  worked 
at  tlie  bottom  of  a  mine,  and  produced  their  pressures  by  long  columns  of 
ineicury  in  a  tube  going  up  the  shaft  of  the  mine.  Amagat's  tube  waa 
300  metres  long,  Cailletet's  250.  Amagat  found  that  the  minimum  value 
ol  pv  between  18°  and  22°  C.  occurred  at  the  following  pressures: 

*  Mdmoires  de  V Institut  de  France,  vol.  xxi.  p.  329, 


THE  PRESSURE  AND  VOLUME  OF  A  GAS. 


127 


Nitrogen  . 

50  metres  of  mercury 

Oxygen     . 

.     100 

Air   . 

65            „             „ 

Carbon  monoxide      50  metres  of  mercury. 
Marsh  gas  ,120  „  „ 

Ethylene  .       65  „ 


The  results  of  his  experi- 
ments arc  cxhil)ited  in  the  fol- 
lowing figures  ;  the  ordinates  are 
the  values  oi pv,  and  the  abscissse 
the  pressure,  the  unit  of  pressure 
being  the  atmosphere,  which  is 
the  pressure  due  to  a  column  of 
mercury  7 GO  mm.  high  at  0°  C, 
and  at  the  latitude  of  Paris. 
The  numbers  on  the  curves  indi- 
cate the  temperature  at  which 
the  experiments  were  made.  It 
will  be  noticed  that  for  nitrogen 
the  pressure  at  which  pv  is  a 
minimum  diminishes  as  the  tem- 
perature increases,  so  much  so 
that  at  a  temperature  of  about 
100°  0.  the  minimum  value  of 
pv  is  hardly  noticeable  in  the 
curve.  This  is  shown  clearly  by 
the  following  results  given  by 
Amagat : 


Fia.  86.— Ethylene 


IT-r  C. 

30-1°  C. 

50-4°  C. 

75-5°  C. 

lOO-r  C. 

p 

pv 

po 

pv 

pv 

pv 

30  metres 

2745 

2875 

3080 

3330 

3575 

60       „            ... 

2740 

2875 

3100 

3360 

3610 

100       „            ... 

2790 

2930 

3170 

3445 

3695 

200       „            ... 

3075 

3220 

3465 

3750 

4020 

320       „            ... 

3525 

3675 

3915 

4210 

4475 

Amagat  extended  his  experiments  to  very  much  higher  pressures,  and 
obtained  the  results  shown  in  the  following  table ;  the  temperature  was 
15"  C,  andpv  was  equal  to  1  under  the  pressure  of  1  atmosphere: 


p  (in  atmospheres). 

Air. 

J>0 

Nitrogen. 
pv 

Oxygen. 
pv 

Hydrogen. 
pv 

750 

1-650 

1-6965 

_ 

„ 

1000 

1-874 

2-032 

1-735 

1-688 

1500 

2-563 

2-644 

2-238 

2-016 

2000 

3-132 

3-2-20 

2-746 

2-322 

2500 

3-672 

3-787 

3-235 

2-617 

3000 

4-203 

4-338 

3-705 

2-892 

\ 


A  question  of  considerable  importance  in  these  experiments,  and  one 
(vhich  we  have  hardly  sufficient  information  to  answer  satisfactorily,  arises 
from  the  condensation  of  gas  on  the  walls  of  the  manometer,  and  possibly 
a  penetration  of  the  gas  into  the  f-ubstance  of  these  walls.     It  is  well  known 


128 


PROPERTIES  OF  MATTER. 


Fig.  87.— Nitrogen. 


that  when  we  attempt  to  exhaust  a  glass  vessel  a  considerable  amount  of 
gas  comes  off  the  glass,  and  if  the  vessel  contains  pieces  of  metal  the 
difficulty  of  getting  a  vacuum  is  still  further  increased,  as  gas  for  some  time 
continues  to  come  from  the  metal.     Much  of  this  is,  no  doubt,  condensed  on 

the  surface,  but  when  we 
remember  that  water  can 
be  forced  through  gold  it 
seems  not  improbable  that 
at  high  pressure  the  gas 
may  be  forced  some  dis- 
tance into  the  metal  as 
well  as  condensed  on  its 
surface. 

Boyle's  Law  at  Low 

Pressures.  —  The  diffi- 
culty arising  from  gas  com- 
ing off  the  walls  of  the 
manometer  becomes  spe- 
cially acute  when  the  pres- 
sure is  low,  as  here  the 
deviations  from  Boyle's  Law  are  so  small  that  any  trifling  error  may 
completely  vitiate  the  experiments.  This  is  probably  one  of  the  reasons 
why  our  knowledge  of  the  relation  between  the  pressure  and  volume  of 
g:vses  at  low  pressures  is  so  unsatisfactory,  and  the  results  of  different 
experiments  so  contradictory.  According  to  Mendeleeff,  and  his  result  has 
been  confirmed  by  'Fuchs,  pv  for  air  at  pressures  below  an  atmosphere 

diminishes  as  the  pressure 
diminishes,  the  value  of  pv 
changing  by  about  3-5  per 
cent,  between  the  pressure 
of  7G0  and  14  mm.  of 
mercury.  If  thi?  is  the 
case,  then  pv  for  air  has  a 
maximum  as  well  as  a  mini- 
mum value.  On  the  other 
hand,  Amagat,  who  made 
a  series  of  very  careful 
experiments  at  low  pres- 
sures, was  not  able  to  detect 
any  departure  from  Boyle's 
^  Fig.  88.-Hydrogen,  Law.     According  to  Bohr, 

and  his  result  has  been 
confirmed  by  Baly  and  Bamsay,  the  law  connecting  p  and  v  for  oxygen 
changes  at  a  pressure  of  about  '75  mm.  of  mercury.  It  has  been 
suggested  that  this  is  due  to  the  formation  of  ozone.  The  recent 
investigations  by  Lord  Rayleigh  on  the  relation  between  the  pressure  and 
volume  of  gases  at  low  pressures  do  not  show  any  depai-ture  from 
Boyle's  Law  even  in  the  case  of  oxygen. 

The  results  of  Amagat's  experiments  are  in  fair  accordance  with 
the  relation  between  p  and  v,  arrived  at  by  Van  der  Waals  from 
the  Kinetic  Theory  of  Gases.  This  relation  is  expressed  by  the 
equation 


THE  PRESSURE  AND  VOLUME  OF  A  GAS. 


129 


(p  +  ^^{v-b)  =  UT 


v  here  a,  h,  II  are  constants  and  T  is  the  absolute  temperature.  Thus  p  in 
Boyle's  equation  is  replaced  by  p  +  a/v-  and  vhy  v  —  b.  The  term  a/v-  or 
a|0-,wherejois  the  density,  arises  from  the  attractions  between  the  molecules 
of  the  gas ;  this  attraction  assists  the  outside  pressure  to  diminish  the 
volume  of  the  gas.  If  we  imagine  the  gas  divided  by  a  plane  into  two 
portions  A  and  B,  then  ap^  is  the  attraction  of  A  on  B  per  unit  area  of  the 
plane  of  separation  ;  it  is  the  quantity  we  call  the  intrinsic  pressure  in  the 


pv 


Fio.  89. 

theory  of  Capillarity  (see  chap.  xv).  The  v  of  Boyle's  Law  is  replaced  by 
v  —  b.  Since  the  molecules  are  supposed  to  be  of  a  finite  although  very 
small  size,  only  a  part  of  the  volume  "occupied"  by  the  gas  is  taken  up 
by  the  molecules,  and  the  actual  volume  to  be  diminished  is  the  difference 
between  the  space  "occupied  "  by  the  gas  and  that  filled  by  its  molecules; 
b  is  proportional  to  the  volume  of  a  molecule  of  the  gas. 
Van  der  Waals'  equation  may  be  written  : 


ht){'-l 


RT 


so  that  if 


we  have 


pv  =  p  and  -  =p 


■x. 


V 


(2/  +  ax){l-b.v)  =  nT 


Thus,  if  the  temperature  is  constant,  the  curve  which  represents  the  rela- 
tion between  pv  and  p  is  the  hyperbola 

{y  +  ax)  (1  -  bx)  =  constant. 


130  PROPERTIES  OF  MATTER. 

The  asymptotes  of  this  hyperbola  are  y  +  ax  =  o,  1  — &a;  =  0.  There  is  a 
minimum  value  of  pv  at  the  point  P  (Eig.  89)  where  the  tangent  is  horizontal. 
The  value  of  x  at  this  point  is  easily  shown  to  be  given  by  the  equation 

a{\-hxf  =  hllT. 

If  &RT/a  is  less  than  unity  there  is  a  ])ositive  value  of  x  given  by  this 
equation.  This  corresponds  to  the  minimum  value  for  pv  in  the  cases  of  air, 
nitrogen,  and  carbonic  acid.  We  see,  too,  from  the  equation  that  as  T 
increases  x  diminishes,  that  is,  the  pressure  at  which  the  minimum  value 
oi  pv  occui-s  is  lower  at  high  temperatures  than  at  low.  This  agrees  with 
the  results  of  Amagat's  experiments  on  nitrogen.  When  T  gets  so  large 
that  6RT/a  is  unity  x  —  Q;  at  all  higher  temperatures  it  is  negative — i.e., 
P  is  to  the  left  of  the  vertical  axis,  there  is  thus  no  minimum  value  of  pv, 
and  the  gas  behaves  like  Lydro^en  in  that^^7  continually  increases'  ys  the 
pressure  increaisea. 


CHAPTER  XIII. 

REVERSIBLE  THERMAL  EFFECTS  ACCOMPANYING 
ALTERATIONS  IN  STRAINS. 

Contents. — Application  of  Thermodynamics — Ratio  of  Adiabatic  to  Isothermal 

Elasticity. 


^C'. -.>-  -  c 


If  the  coefficients  of  elasticity  of  a  substance  depend  upon  the  tempera' 

ture  an  alteration  in  the  state  of  strain  of  a  body  will  be  accompanied 

by  a  change  in  its  temperature.     If  the  body  is  stifler  at  a  high  tern-  .,. 

perature  than  at  a  low  one,  then,  if  the  strain  is  increased,  there  will    ,      JCi^L 

be  an  increase  in  the  temperature  of  the  strained  body,  while  if  the  body  ^^  .,  /  >  / 

is  stifter  at  a  low  temperature  than  at  a  high  one,  there  will    be  a  fall 

in  the  temperature  when  the  strain  is  increased.     Thus,  if  the  changes  in 

strain  in  any  experiment  take  place  so  rapidly  that  the  heat  due  to  these 

changes  has  not  time  to  escape,  the  coefficients  of  elasticity  determined 

by  these  experiments  will  be  larger  than  the  values   determined    by   a 

method  in  which  the  strains  are  maintained  constant  for  a  sufficiently  long 

time  for  the  temperature  to  become  uniform ;  this  follows  from  the  fact 

that  the  thermal  changes  which  take  place  when  the  strains  are  variable 

are  always  such  as  to  make  the  body  stiffer  to  resist  the  change  in  strain. 

In  those  experiments  by  which  the  coefficients  of  elasticity  are  determined 

by  acoustical  methods — i.e.,  by  methods  which  involve  the  audible  vibration 

of  the  substance  {see  Sound,  p.  125) — the  heat  will  not  have  time  to  dijBfuse, 

and  we  should  expect  such  methods  to  give  higher  values  than  the  statical 

ones  we  have  been  describing.     When  we  calculate  the  ratio  of  the  two 

coefficients  we   find   that   the    theoretical   difference   is  far  too   small  to 

explain  the  considerable  excess  of  the  values  of  the  constants  of  elasticity 

found  by  Wertheim  by  acoustical  methods  over  those  found  by  statical 

methods. 

We  can  easily  calculate  by  the  aid  of  Thermodynamics  the  thermal 
effects  due  to  a  change  of  strain.  To  fix  our  ideas,  suppose  we  have  two 
chambers,  one  maintained  at  a  temperature  T,,,  the  other  at  the  tempera- 
ture Tj*,  these  temperatui-es  are  supposed  to  be  absolute  temperatures,  and  ' 
T(,  to  be  less  than  Tj.  Let  us  suppose  that  we  have  in  the  cool  chamber  a 
stretched  wire,  and  that  we  increase  the  elongation  e  by  oe  ;  then  if  P  is  the 
tension  required  to  keep  the  wire  stretched,  the  work  done  on  the  wire  is 

PaZae 

where  a  is  the  area  of  the  cross-section  and  I  the  length  of  the  wire.  Now 
transfer  the  wire  with  its  length  unaltered  to  the  hot  chamber,  and  for 
simplicity  suppose  the  thermal  capacity  of  the  wire  exceedingly  small,  so 
that  we  can  neglect  the  amount  of  heat  required  to  heat  up  the  wire ; 
if  the  stiffness  of  the  wire  changes  with  temperature  the  tension  P* 
required  to  keep  it  stretched  will  not  be  the  same  as  P.     Let  the  wire 


132  PROPERTIES  OF  MATTER. 

contract  in  the  hot  chamber  until  its  elongation  diminishes  by  5e,  then  the 
work  done  by  the  wire  is 

Valde, 

Now  transfer  the  wire  with  its  length  unaltered  back  to  the  cold  chamber  , 
it  will  now  be  in  the  same  state  as  when  it  started.  The  work  done  by 
the  wire  exceeds  that  done  on  it  by 

hence  the  arrangement  constitutes  a  heat  engine,  and  since  it  is  evidently 
reversible  it  must  obey  the  laws  of  such  engines.  These  engines  work 
by  taking  heat  oH  from  the  hot  chamber  and  giving  oh  out  in  the  colder 
cliamber,  and  from  the  Second  Law  of  Thermodynamics  we  hava 

m_hh_m-^h 

T       T      T  -  T 

J-,  J-o  -Lj  J-0 

Now  by  the  Conservation  of  Energy 

^H  -  ch  =  mechanical  work  done  by  the  engine 
«=(F-P)aZae; 

hence  SA  =  T}— — p— 'a  Ide 

-L|  ~  J-o 

^TM\:dce       . 

e  constant 


-01    grj, 


Now  Ih  is  the  amount  of  heat  given  out  by  the  wire  when  the  elongation 
is  increased  by  ce,  and  al  is  the  volume  of  the  wire ;  hence  the  mechanical 
equivalent  of  the  heat  given  out  per  unit  volume,  when  the  elongation  is 
measured  by  ^e,  is  equal  to 

\  0  J-  /  e  constant 

If  this  heat  is  prevented  from  escaping  from  the  wire  it  will  raise  the 
temperature,  and  if  cQ  is  the  rise  in  temperature  due  to  the  elongation 
^CjWe  see  that 

se=TA|Ei"''"""'xj«  (1) 

JKp 

where  p  is  the  density  of  the  wire,  K  its  specific  heat,  and  J  the  mechanical 

,  equivalent  of  heat.     We  see  that   this  expression  proves  the  statement 

made  above,  that  the_temperature  change  which  takes  place  on  a  change  in 

the  strain  is  alwavs~such  as^tomake^  tlic  body  .stiJjkaJto  resist  tiie  change. 

•^,  We  can  readily  obtain  ruiotho?   expression  tor  t6.  wl.ich  is  often  more 

/  convenient  than  tliat  just  given.     In  Uiat  I'ormuia  we  liave  the  expression 

(5P/6T)e  constant.     Now.   suppose  tliat.    instead  of  keeping  «  constant  all 

througli,  we  first  allow  the  body  to  expand  under  constant  tension ;  if  w  is 

the    coefficient    of   linear    expansion    for    tioat.    and    3T    the   change    in 

temperature,    the   increase    in    the    elongation    is    u)^~T ;    now   keep    the 

temperature  constant,  and  diminish  tJie  tension  until   the  shortening  due 


THERMAL  EFFECTS  ACCOMPANYING  STRAINS. 


133 


fco  the  diminution  in  tension  just  compensates  for  the  lengthening  due  to 
the  rise  in  temperatures.  In  order  to  diminish  the  elongation  by  wiJT  we 
must  diminish  the  tension  by  qwoT  where  q  is  Young's  modulus  for  the 
wiie,  hence 


OP 


hence  by  equation  (1) 


SP 

aT 


e  coustant 


=  -qui 


But  qle  is  the  additional  tension  SP  required  to  produce  the  elongation  he, 
hence  the  increase  in  temperature  Id  produced  by  an  increase  of  tension 
2P  is  given  by  the  equation 

(2) 


JKp 


Equations  (1)  and  (2)  are  due  to  Lord  Kelvin. 

Dr.  Joule  {Phil.  Trans,  cxlix.  1859,  p.  91)  has  verified  equation  (2)  by 
experiments  on  cylindrical  bai's  of  various  substances,  and  the  results  of  his 
experiments  are  given  in  the  following  table.  The  changes  in  temperature 
wei-e  measured  by  thermo-electric  couples  inserted  in  the  bars. 


T 

P 

u 

K 

«P 

5(9 

observed. 

59 
calculated. 

Iron     . 
Hard  steel  . 
Cast  iron     . 
Copper 

286-3 

274-7 
282-3 
274-2 

7-5 

7-0 

6-04 

8-95 

1-24x10-5 
1-23x10-5 
1-11x10-5 
1-7182x10-5 

-110 
-102 
-120 
-095 

1-09  xlO» 
1-09x109 
1-10x109 
1-08x109 

-  -1007 

-  -16-20 
-•1481 
--174 

-•107 
-•125 
--115 
-  -154 

A  qualitative  experiment  can  easily  be  tried  with  a  piece  of  india- 
rubber.  If  an  indiarubber  band  be  loaded  sufficiently  to  produce  a 
considerable  extension  and  if  it  be  then  warmed  by  bringing  a  hot  body 
near  to  it,  it  will  contract  and  lift  the  weight ;  hence  the  indiarubber  gets 
stiffer  by  a  rise  in  temperature  ;  by  the  rule  we  have  given,  it  ought  to 
increase  in  temperature  when  stretched,  since  by  so  doing  it  becomes 
stift'er  to  resist  stretching.  That  this  is  the  case  can  easily  be  verified  by 
suddenly  stretching  a  rubber-band  and  then  testing  its  temperature  by 
placing  it  against  a  thermopile,  or  even  between  the  lips,  when  it  will  be 
found  perceptibly  warmer  than  it  was  before  stretching. 

We  can  easily  calculate  what  efiect  the  heat  produced  will  have  on  the 
apparent  elasticity  if  it  is  not  allowed  to  escape.  The  modulus  of  elasticity, 
when  the  change  in  strain  takes  place  so  rapidly  that  the  heat  has  not 
time  to  escape,  is  often  called  the  adiabatic  modulus. 

Ratio  of  Adiabatic  to  Isothermal  Elasticity.— Suppose  we  take 

the  case  of  a  wire,  and  suppose  the  tension  increased  by  ^P,  if  the  heat 
does  not  escape  the  increase  ce  in  the  elongation  will  be  due  to  two  causes 
— one  from  the  increase  in  the  pull,  the  other  from  the  increase  in  the 
temperature.     The  first  part  is  equal  to  hVjq,  where  q  is  Young's  modulus 


134- 


PROPERTIES  OF  MATTER. 


for  steady  strain  ;  the  second  part  is  equal  to  cdo>  where  ?0  is  the  change  in 
temperature,  w  the  coeflScient  of  linear  expansion  ;  lience 


but  by  equation  (2) 


hence 


ae 


or 


le 

9 

+  10^6; 

ao- 

=r    - 

u.T„gP 
JKp' 

i  = 

1 

JKp 

^e 

1 

w-T^ 

hP 

9 

JKp 

But  if  2'  is  the  adiabatic  "  Young's  Modulus," 

~  q      JKp 

It  follows  from  this  equation  that  Ifq'  is  always  less  than  Ifq  or  7 


(3) 


is  always  greater  than  g,  as  we  saw  from  general  reasoning  must  be 
^Ke^case!  By  equation  p3)  we  can  calculate  the  value  of  q'/q.  The 
results  for  temperature  lo°C  are  given  in  the  following  table,  taken  from 
Lord  Kelvin's  article  on  "  Elasticity  "  in  the  Enct/clopcedia  Bi'itannica  : 


Substance. 


Zinc 

Tin 

Silver 

Copper 

Lead 

Glass 

Iron 

Platinum 


p 

'     K 

7-008 

•0927 

7-404 

-0514 

10-369 

•0557 

8-933 

•0949 

11-215 

•0293 

2-942 

•177 

7-553 

•1098 

21-275 

•0314 

fa) 


•0000249 

•000022 

•000019 

■000018 

•000029 

•0000086 

•000013 

•0000086 


7/ion 


4 

7 

12 

1 

6 

18 

16 


•56 

■09 

•22 

•20 

•74 

-02 

-24 

-7 

(flq  deduced 
from  equat.  3. 


_  -008 
[•00362 
•00315 
.  ^00325 
: -00310 
.  -000600 
[•00259 
[•00129 


Thus  we  see  that  in  the  case  of  metals  q'  is  not  so  much  as  1  per 
cent,  greater  than  q.  In  Wertheim's  experiments,  however,  the  excess 
of  q  determined  by  acoustical  methods  over  q  determined  by  statical 
methods  exceeded  in  some  cases  20  per  cent.  This  discrepancy  has  never 
been  satisfactorily  acco'inted  for. 


CHAPTER  XIV. 


CAPILLARITY. 


Contents. — Surface  Tension  and  Surface  Energy — Rise  of  Liquid  in  a  Capillary 
Tube — Relation  between  Pressure  and  Curvature  of  a  Surface — Stability  of 
Cylindrical  Film — Attractions  and  Repulsions  due  to  Surface  Tension — Methods 
of  Measuring:  Surface  Tension — Temperature  Coefficient  of  Surface  Tension — 
Cooling  of  Film  on  Stretching — Tension  of  very  Thin  Films — Vapour  Pressure 
over  Curved  Surface— Effects  of  Contamination  of  a  Surface. 

There  are  many  phenomena  which  show  that  liquids  behave  as  if  they 
were  enclosed  in  a  stretched  membrane.  Thus,  if  we  take  a  piece  of  bent 
wire  with  a  flexible  silk  thread  stretching  from  one  side  to  the  other  and 
dip  it  into  a  solution  of  soap  and  water  so  as  to  get  the  part  between  the 
silk  and  the  wire  covered  with  a  film  of  the  liquid,  the  silk  thread  will  be 
drawn  tight  as  in  Fig.  90,  just  as  it  would  be  if  the  film  were  tightly 


Fig.  90. 


I'lG.  i,i. 


Fig.  92. 


stretched  and  endeavouring  to  contract  so  that  its  area  should  be  as  smaR 
as  possible.  Or  if  we  take  a  framework  with  two  threads  and  dip  it  into 
the  soap  and  water,  both  the  threads  will  be  pulled  tight  as  in  Fig.  91,  the 
liquid  again  behaving  as  if  it  were  in  a  state  of  tension.  If  we  take  a 
ring  of  wire  with  a  liquid  film  upon  it  and  then  place  on  the  film  a  closed 
loop  of  silk  and  pierce  the  film  inside  the  loop,  the  film  outside  will  pull 
the  silk  into  a  circle  as  in  Fig.  92.  The  efiect  is  again  just  the  same  as  it 
would  be  if  the  films  were  in  a  state  of  tension  trying  to  assume  as  small 
an  area  as  possible,  for  with  a  given  circumference  the  circle  is  the  curve 
whii  ii  has  the  largest  area ;  thus,  when  the  silk  is  dragged  into  the  circular 
form,  tlie  aTrrt  of  the  film  outside  is  as  small  as  possible. 

Another  method  of  illustrating  the  tension  in  the  skin  of  a  liquid  is 
to  watch  the  changes  in  shape  of  a  drop  of  water  forming  quietly  at  the 
end  of  a  tube  before  it  fi.nally  breaks  away.     The  observation  is  rendered 


136 


PROPERTIES  OF  MATTER. 


much  easier  if  the  water  drops  are  allowed    to    form    in   a   mixture    of 
paraffin  oil  and  bisulphide  of  carbon,  as  the  drops  are  larger  and  form 
more  gradually.     The  shape  of  the  drop  at  one  stfige  is  shown  in  Fig.  93. 
If  we  mount  a  thin  indiarubber  membrane  on  a  hoop  and  suspend 


Fio.  93. 


Fio.  94. 


it  as  in  Fig.  94,  and  gradually  fill  the  vessel  with  water  and  watch  the 
changes  in  the  shape  of  the  membrane,  these  will  be  found  to  correspond 
closely  to  those  in  the  drop  of  w^ater  falling  from  the  tube ;  the  stage 
corresponding  to  that  immediately  preceding  the  falling  away  of  the  drop 
is  especially  interesting ;  a  very  marked  waist  forms  in  the  membrane  at 
this  stage,  and  the  water  in  the  bag  falls  rajiidl}^  and  looks  as  if  it  were 

going  to  burst  away  ;  the 
membrane,  however, 
reaches  another  figui-e  of 
equilibrium,  and  if  no 
more  water  is  poured  in 
remains  as  in  Fig.  94. 

Again,  liquids  behave 

as  if  the  tension  in  their 

outer  layers  was  different 

Fio.  95.  for  different  liquids.    This 

may  easily  be  shown  by 
covering  a  white  flat-bottomed  dish  with  a  thin  layer  of  coloured  water 
and  then  touching  a  part  of  its  surface  with  a  glass  rod  which  has  been 
dipped  in  alcohol ;  the  liquid  will  move  from  the  part  touched,  leaving  the 
white  bottom  of  the  dish  dry.  This  shows  that  the  tension  of  the  water  is 
greater  than  that  of  the  mixtuie  of  alcohol  and  water,  the  liquid  being 
dragged  away  from  places  where  the  tension  is  weak  to  places  where  it  ia 


strong. 


CAPILLARITY. 


137 


VA.lvi 


There  is  one  very  important  difference  between  the  behaviour  of  ordinaiy 
stretched  elastic  membranes  and  that  of  liquid  films,  for  while  the  tension 
in  a  membrane  increases  with  the  amount  of  stretching,  the  tension  in 
a  liquid  film  is  independent  of  the  stietching,  provided  that  tliis  is  not  so 
great  as  to  reduce^  the  thickness  of  the  film  below  about  five  millionths 
of  a  centimetre.  This  can  be  shown  by  the  following  experiment :  bend 
a  piece  of  wire  into  a  closed  plane  curve  and  dip  this  into  a  solution  of 
soap  and  water  so  as  to  get  it  covered  with  a  film,  then  hold  the  wire  in 
a  nearly  vertical  position  so  as  to  allow  the  liquid  in  the  film  to  drain 
down ;  this  will  cause  the  film  to  be  thinner  at  the  top  than  at  the 
bottom  ;  the  difierence  in  thickness  is  very  apparent  when  the  film  gets 
thin  enough  to  show  the  colours  of  thin  plates,  yet  though  the  film  is  of 
very  uneven  thickness  the  equilibrium  of  the  film  shows  that  the  tension 
is  the  same  throughout,*  for  if  the  tension  in 
the  thin  part  were  greater  than  that  in  the 
thick,  the  top  of  the  film  would  di'ag  the 
bottom  part  up,  while  if  the  tension  of  the 
thick  part  were  greater  than  that  of  the  thin 
the  lower  part  of  the  film  would  drag  the  top 
part  down. 

Definition  of  Surface  Tension.— Sup- 
pose that  we  have  a  film  stretched  on  the 
framework  ABOD,  Fig.  96,  of  which  the  sides 
AB,  BC  and  AD  are  fixed  while  CD  is 
movable ;  then,  in  order  to  keep  CD  in 
equilibrium,  a  force  F  must  be  applied  to  it 
at  right  angles  to  its  length.  This  force  is 
required  to  balance  the  tensions  exerted  by 
each  face  of  the  film ;  if  T  is  this  tension, 
then 

2T.CD-F;  Fig.  96. 

the  quantity  T  defined  by  this  eqviation  is  called  the  surface  tension  of  the 
liquid ;  for  water  at  18°C.  it  is  about  73  dynes  per  centimetre. 

Potential  Energ-y  of  a  Liquid  arising-  from  Surface  Tension. — 

If  we  pull  the  bar  CD  out  through  a  distance  x,  the  Avork  done  is  F.«,  and 
this  is  equal  to  the  increase  in  the  potential  energy  of  the  film,  but 
Fa?  =  2T.CDa:;  =  Tx  (increase  of  area  of  film).  Thus  the  increase  in  the 
potential  energy  of  the  film  is  equal  to  T  multiplied  by  the  increase  in  area, 
so  that  in  consequence  of_surface  tension  a  liquid  will  possess  an  amount  of 
potential  energy  equal  to  the  product  of  the  surface  tension  of  the  liquid  and 
the  area  of  the  surface.  Starting  from  this  result  we  can,  as  Gauss  showed,  y 
deduce  the  consequences  of  the  existence  of  surface  tension  from  the  I 
principle  that  when  a  mechanical  system  is  in  equilibrium  the  potential  J 

energy  is  a  minimum.  Suppose  that  we  take,  as  Plateau  did,  two  liquids  of 
the  same  density,  say^oil  and  a  mixture  of  alcohol  and  water,  and  consider 
the  equilibrium  of  a  mass  of  oil  in  the  mixture.  Since  the  density  of  the 
oil  is  the  same  as  that  of  the  surrounding  fluid,  changes  in  the  shape  of  the 
mass  will  not  affect  the  potential  energy  due  to  gravity ;  the  only  change 

*  If  the  film  is  vertical  the  tension  at  the  top  is  very  slightly  greater  than  that  at 
the  bottom,  so  as  to  allow  the  difference  of  tension  to  balance  the  exceedingly  small 
weight  of  the  film 


138  PROPERTIES  OF  MATTER. 

in  the  potential  enei-gy  will  be  the  change  in  the  energy  dne  to  surface 
tension,  and,  by  the  principle  just  stated,  the  oil  will  assume  the  shape  iu 
which  this  potential  energy  is  a  minimum — i.e.,  the  shape  in  which  the 
area  of  the  surface  is  a  minimum.  The  sphere  is  the  surface  which  for  a 
given  volume  has  the  smallest  surface,  so  that  the  drops  of  oil  in  the  liquid 
will  be  spherical.  This  experiment  can  easily  be  tried,  and  the  spherical 
form  of  the  drops  is  very  evident,  especially  if  the  oil  is  made  more 
distinct  by  the  addition  of  a  little  iodine. 

If  a  drop  of  liquid  is  not  surrounded  by  fluid  of  the  same  density, 
but  is  like  a  drop  of  mercury  on  a  plate  which  it  does  not  wet,  then  any 
change  in  the  shape  of  the  drop  will  affect  the  potential  energy  due  to 
gravitation  as  well  as  that  due  to  surface  tension,  and  the  shape  of  the 
drop  will  be  determined  by  the  condition  that  the  total  potential  energy  is 
to  be  as  small  as  possible  ;  if  the  drop  is  very  large,  the  potential  energy  f 
due  to  the  surface  tension  is  insignificant  compared  with  that  due  to  gravity, 

and  the  drop  spreads  out 
flat  so  as  to  get  its  centre  of 
gravity  low,  even  though 
this  involves  an  increase  in 
the  potential  energy  due  to 
the  surface-tension.  If,  how- 
ever, the  drop  is  very  small, 
the  potential  energy  due  to  gravity  is  insignificant  in  comparison  with 
that  due  to  surface-tension,  and  the  drop  takes  the  shape  in  which  the 
potential  energy  due  to  surface-tension  is  as  small  as  possible  ;  this  shape, 
as  we  have  seen,  is  the  spherical,  and  thus  surface-tension  will  cause  all 
very  small  drops  to  be  spherical.  Dew-drops  and  rain-drops  are  very 
conspicuous  examples  of  this ;  other  examples  are  afforded  by  the 
manufacture  of  spherical  pellets  by  the  fall  of  molten  lead  from  a  shot 
tower  and  by  the  spherical  form  of  soap-bubbles.  We  shall  show  later  on 
that  if  the  volume  of  liquid  in  a  drop  is  the  same  as  that  of  a  sphere  of 
radius  a  the  liquid  will  i-emain  very  nearly  spherical  if  a^  is  small  compai'ed 
with  T/gp  where  T  is  the  surface-tension  and  p  the  density  of  the  liquid. 
Thus,  in  the  case  of  water,  where  T  is  about  73,  drops  of  less  than  2  or  3 
millimetres  in  radius,  will  be  approximately  spherical. 

Another  important  problem  which  we  can  easily  treat  by  the  method  of 
energy  is  that  of  the  spreading  of  one  liquid  over  the  surface  of  another. 
Suppose,  for  example,  we  place  a  drop  of  liquid  A  on  another  liquid  B 
(Fig.  97),  we  want  to  know  whether  A  will  spread  over  B  like  oil  over 
water,  or  whether  A  will  contract  and  gather  itself  up  into  a  drop.  The 
condition  that  the  potential  energy  is  to  be  as  small  as  possible  shows  that 
A  will  spread  over  B  if  doing  so  involves  a  diminution  in  the  potential 
energy;  while,  if  the  spreading  involves  an  inci  "i^o  in  the  potential 
energy,  A  will  do  the  reverse  of  r^  •  ■,idir<j  and  will  gath'.^r  itself  up  in  a 
drop.  Let  us  consider  tl-e  change  in  the  potential  eneigy  due  to  an 
increase  S  in  the  area  of  contact  of  A  and  B  where  A  is  a  flat  drop.  We 
have  three  surface-tensiri^"  to  consider:  that  of  the  surface  of  contact 
between  A  and  the  air,  which  we  shall  call  Tj ;  that  of  the  surface  of 
contact  between  B  a'.d  tne  air,  which  we  shall  call  T, ;  and  that  of  the 
surface  of  contact  of  A  and  B,  which  we  shall  call  T^^.  Kow  wnon  we 
increase  the  surface  of  contact  between  A  and  B  by  S  we  increase  the 
energy  due  to  the  surface-tension  between  these  two  fluids  by  Tj,  x  S,  we 


CAPILLARITY,  139 

increase  that  due  to  the  surface-tension  between  A  and  the  air  by  T,  x  S 
and  diminish  that  due  to  the  surface-tension  between  B  and  the  air  by 
Tg  X  S.     Hence  the  total  increase  in  the  potential  energy  is 

<,7^  cLi     and  if  this  is  negative  S  will  increase — i.e.,  A  will  spread  over  B  ;  the  con- 
^  1^1^,     dition  for  this  to  be  negative  is  that 

so  that  if  this  condition  is  fulfilled  the  liquid  A  will  spread  out  into  a  thin 
film  and  cover  B,  and  there  will  be  no  place  where  three  Ii(]uid  surfaces 
meet.  If,  on  the  other  hand,  any  one  of  the  tensions  is  less  than  the  sum 
of  the  other  two — i.e.,  if  we  can  construct  a  triangle  whose  sides  are 
proportional  to  Tj,  T^  and  Tj^,  then  a  drop  of  one  liquid  can  exist  on  the 
surface  of  the  other,  and  we  should  have  the  three  liquid  surfaces  meeting 
at  the  edge  of  a  drop.  The  triangle  whose  sides  are  proportional  to 
T„  Tj,  T,2  is  often  called  Neumann's  triangle ;  the  experiments  of 
Quincke,  Marangoni  and  Van  Mensbrugghe,  show  that  for  all  the  liquids 
hitherto  investigated  this  triangle  cannot  be  drawn,  as  one  of  the  tensions 
is  always  greater  than  the  sum  of  the  other  two,  and  hence  that  there  can 
be  no  position  of  equilibrium  in  which  three  liquid  surfaces  meet. 
Apparent  exceptions  to  this  are  due  to  the  fouling  of  the  surface  of  one  of 
the  liquids.  Thus,  when  a  drop  of  oil  stands  on  water,  the  water  surface 
is  really  covered  with  a  thin  coating  of  oil  which  has  spread  over  the 
surface ;  or  again,  when  a  drop  of  water  stands  on  mercury,  the  mercury 
surface  is  greasy,  and  the  grease  has  spread  over  the  water.  Quincke  has 
shown  that  a  drop  of  pure  water  will  spread  over  the  surface  of  pure 
mercury. 

Though  three  liquid  surfaces  cannot  be  in  equilibrium  when  there  is  a 
line  along  wHich  all  three  meet,  yet  a  solid  and  two  liquid  surfaces  can  be 
in  equilibrium  ;  this  is  shown  by  the  equilibrium  of  water  Or  of  mercury 
in  glass  tubes  when  we  have  two  fluids,  water  (or  mercury),  and  air, 
both  in  contact  with  the  glass.  The  consideration  of  the  condition  of 
equilibrium  in  this  case  naturally  suggests  the  question  as  to  whether 
there  is  anything  corresponding  to  surface-tension  at  the  surface  of 
separation  of  two  substances,  one  of  which  is  a  solid.  Though  in  this  case 
the  idea  of  a  skin  in  a  state  of  tension  is  not  so  easily  conceivable  as  for  a 
liquid,  yet  there  is  another  way  of  regarding  surface-tension  which  is  as 
readily  applicable  to  a  solid  as  to  a  liquid.  We  have  seen  that  the 
existence  of  surface-tension  implies  the  possession  by  each  unit  area  of  the 
liquid  of  an  amount  of  potential  energy  numerically  equal  to  the  surface- 
tension  :  we  may  from  this  point  of  view  regard  surface-tension  as  surface 
energy.  There  is  no  difficulty  in  conceiving  that  part  of  the  energy  of  a 
solid  body  may  be  proportional  to  its  surface,  and  that  in  this  sense  the 
body  has  a  surface-tension,  this  tension  being  measured  by  the  energy  per 
unit  area  of  the  surface. 

Let  us  now  consider  the  equilibrium  of  a  liquid  in  contact  with  air  and 
both  resting  on  a  solid,  and  not  acted  upon  by  any  forces  except  those  due 
to  surface-tension.     Suppose  A,  Fig.  98,  represents  the  solid,  B  the  liquid, 
C  the  air,  FG  the  surface  of  separation  of  liquid  and  air,   ED  the  sur- 
face of  the  solid.     Let  the  angle  FGD  be  denoted  by  Q  ;  this  angle  is 


140 


PROPERTIES  OF  MATTER. 


called  the  angle  of  contact  of  the  liquid  with  the  solid.  Let  the  surface 
of  separation  FG  come  into  the  position  F'G'  parallel  to  FG.  Then  if  FG 
represented  a  position  of  equilibrium,  the  potential  energy  due  to  surface- 
tension  must  be  a  minimum  in  this  position,  so  that  it  will  be  unafi'ected 


D 


Fio.  98. 


by  any  small  displacement  of  the  substances ;  thus  the  potential  energy 
must  not  be  altered  by  the  displacement  of  FG  to  F'G'.  This  displace- 
ment of  the  surface  causes  B  to  cover  up  a  long  strip  of  the  solid,  the 
breadth  of  the  strip  being  GG'.  Let  S  be  the  area  of  this  strip.  Then 
if  Tj,  Tj  and  Tj^  are  respectively  the  surface-tensions  between  A  and  C,  B 
and  0,  and  A  and  B,  the  changes  in  the  energy  due  to  the  displacement  are  : 

(1)  An  increase  Tj^S  due  to  the  increase  S 
in  the  surface  between  A  and  B. 

(2)  An  increase  T^S  cos  0  due  to  the 
increase  S  cos  0  in  the  surface  between  B 
and  C. 

(3)  A  diminution  T,S  due  to  the  diminu- 
tion S  in  the  surface  between  A  and  C. 

Hence  the  total  increase  in  the  energy  is 

S(T,,  +  T,cos0-T,) 

and   as   this   must   vanish   when    we    have 
equilibrium  we  have 

T„-|-T,  cos0  =  T,; 


or 


T  -T 

cos  0  =  ^ ^ 

T 


Thus,  if  T,  is  greater  than  Tj^,  cos  0  is 
positive  and  0  is  less  than  a  right  angle ;  if 
Tj  is  less  than  Tjj,  cos  0  is  negative,  and  0  is 

greater  than  a  right  angle ;    mercury  is  a 

case  of  this  kind,  as  for  this  substance  0  is 

Fig.  9^.  about    140°.     The   angle   0   is   termed   the 

angle  of  contact.     Since  cos  0  cannot  exceed 

unity,  the  greater  of  the  two  quantities  Tj  or  Tj^  must  be  less  than  the 

sum  of  the  other  two.     If  this  condition  is  not  fulfilled  the  liquid  B  will 

spread  over  the  surface  A. 

Rise  of  a  Liquid  in  a  Capillary  Tube. — We  can  apply  the  result  we 

have  just  obtained  to  find  the  elevation  or  depression  of  a  liquid  in  a  tube 
which  it  does  not  wet  and  with  which  it  has  a  finite  angle  of  contact. 

Suppose  h  is  the  height  of  the  fluid  in  the  tube  above  the  horizontal 
surface  of  the  fluid  outside,  when  there  is  equilibrium  ;  and  suppose  that 
r  is  the  radius  of  the  tube  at  the  top  of  the  fluid  column.     Let  T,  be  the 


V  =  TT}*^  -  -TTJ-'' 


2        ,        Trtj 

3 


hence  A+-?'  =  — * 

If  6  is  greater  tlian  a  right  angle  h  is  negative,  that  is,  the  level  of  the 
liquid  in  the  tube  is  lower  than  the  horizontal  surface ;  this  is  strikingly 
shown  by  mercury,  but  by  no  other  fluid.  The  angle  of  contact  between 
mercury  and  glass  was  measured  by  Gay  Lussac  by  causing  mercury  to 
flow  up  into  a  spherical  glass  bulb ;  when  the  mercury  is  in  the  lower  part 
of  the  bulb  the  surface  near  the  glass  will  be  very  much  curved ;  as  the 
mercury  rises  higher  in  the  bulb  the  curvature  will  get  less ;  the  surface 
of  the  mercury  at  different  levels  is  represented  by  the  dotted  lines  in 
Fig.  100.  There  is  a  certain  level  at  which  the  surface  will  be  horizontal; 
at  this  place  the  tangent  plane  to  the  sphere  makes  with  a  horizontal  plane 
an  angle  equal  to  the  supplement  of  the  angle  of  contact  between  mercury 
and  glass.     A  modification  of   this  method  is   to  make  a  piece  of  clean 


r 


CAPILLARITY.  141 

surface-tension  between  the  tube  and  air,  T,  that  between  the  liquid  and  air 
and  Tjjthat  between  the  tube  and  the  liquid.  Then,  if  there  is  equilibrium, 
a  slight  displacement  of  the  fluid  up  the  tube  will  not  alter  the  potential 
energy.  Suppose  then  that  the  fluid  rises  a  short  distance  x  in  the  tube, 
thus  covering  an  additional  area  ^ttvx  of  the  tube,  and  diminishing  the  area 
of  the  tube  in  contact  with  the  air  by  tliis  amount.  This  increases  the 
potential  energy  due  to  surface-tension  by  2Trrx{T^^  —  T,). 

The  increase  in  the  potential  energy  due  to  gravity  is  the  work  done 
(1)  by  lifting  the  mass  td"  x  p  x  x,  where  p  is  the  density  of  the  liquid, 
against  gravity  through  a  height^:^this  is  equal  to  gphirr^x ;  and  (2)  by    -t  "*'* 
lifting  the  volume?;  of  the  meniscus  through  a  height  x — this  work  is  equal 
to  (jpiw. 

Hence  the  total  increase  in  potential  energy  is 

277-?-a;(T,j  -  T,)  +  gphnrx  +  gpvx^ 
and  as  this  must  vanish  we  have 

;.  +  JL  =  2(T,-T„), 

irT'  gpr 

but  if  d  is  the  angle  of  contact,  we  have  just  proved  that 

Tj  cos  0  =  T,  -  T„ 

,  ,^v      2T.COS0 

hence  n  +  — -  =  — • 

Trt"         gpr 

When  the  fluid  wets  the  tube  6  is  zero  and  cos  0  =  1.  If  the  meniscus 
is  SO  small  that  it  may  be  regarded  as  bounded  by  a  hemisphere,  v  is  the 
difference  between  the  volume  of  a  hemisphere  and  that  of  the  circum- 
scribing cylinder — i.e., 


142 


TROPERTIES  OF  MATTER. 


plate  glass  dipping  into  mercury  rotate  about  a  horizontal  axis  until  the 
surface  of  the  meicury  on  one  side  of  the  plate  is  flat ;  the  angle  made  by 
the  glass  plate  with  the  horizontal  is  then  the  supplement  of  the  angle  of 
*contact  between  mercury  and  glass. 

The  angle  of  contact  between  mercury  and  glass  varies  very  widely 
under  different  circumstances;  thus  the  meniscus  of  the  mercury  in  a 
thermometer  may  not  be  the  same  when  the  mercury  is  rising  as  when  it 
is  falling.  We  should  expect  this  to  be  the  case  if  tlie  mercury  fouls  the 
glass,  for  in  this  case  the  mercury  when  it  falls  is  no  longer  in  contact 
with  clean  glass  but  with  glass  fouled  by  mercury,  and  we  should  expect 
the  angle  of  contact  to  be  very  different  from  that  with  pure  glass.  Quincke 
found  that  the  angle  of  contact  of  a  drop  of  mercury  on  a  glass  plate 
steadily  diminished  with  the  time;  thus  the  angle  of  contact  of  a  freshly 
formed  drop  was  148'^  55',  and  this  steadily  diminished,  and  after  two  days 


Fig.  100. 


Fig.  101. 


was  only  137°  14';  on  taj)ping  the  plate  the  angle  rose  to  141°  19',  and 
after  another  two  days  fell  to  140°. 

If  we  force  mercury  iip  a  narrow  capillary  tube  and  then  gradually 
diminish  the  pressure,  the  mercury  at  first,  instead  of  falling  in  the  tube, 
adjusts  itself  to  the  diminished  pressure  by  altering  the  curvature  of  its 
meniscus,  and  it  is  only  when  the  fall  of  pressuie  becomes  too  large  for  such 
an  adjustment  to  be  possible  that  the  mercui-y  falls  in  the  tube  ;  the  con- 
sequence is  that  the  fall  of  the  mercury,  instead  of  being  continuous,  takes 
place  by  a  series  of  jumps.  This  effect  is  illustrated  by  the  old  experiment 
of  bending  a  piece  of  capillai-y  tubing  into  a  U-tube  (Fig.  101),  pouring 
mercury  into  the  tube  until  it  covers  the  bend  and  stands  at  some  height 
in  either  leg  of  the  tube ;  if  the  tube  is  vertical,  the  mercury  can  be  made 
by  tapping  to  stand  at  the  same  height  in  both  legs  of  the  tube.  Now  slowly 
tilt  the  tube  so  as  to  cause  the  mercury  to  run  up  the  left  leg  of  the  tube ; 
if  the  tube  is  slowly  brought  back  to  the  vertical,  the  mercury  will  be 
found  to  stand  at  a  higher  level  in  the  left  leg  of  the  tube  than  in  the 
right,  while  the  meniscus  will  be  flatter  on  the  left  than  on  the  right. 
Tliis  principle  explains  the  action  of  what  are  called  Jamin's  tubes,  which 
aie  simply  capillary  tubes  containing  a  large  number  of  detached  drops  of 
liquid ;  these  can  stand  an  enormous  difference  of  pressure  between  the 
ends  of  the  tube  without  any  appreciable  movement  of  the  drops  along  the 


CAPILLARITY. 


143 


tube.     Thus,  suppose  that  AB,  CD,  EF  (Fig.  102)  represent  three  consecu- 
tive drops  along  the  tube,  then  in  consequence  of  the  diflerent  curvatures 


A       B 


C        D         E       F 

FiQ.  102. 

of  AB  at  A  and  B  the  pressure  in  the  air  at  A  will  be  greater  than  that  at  B, 
while  the  pressure  at  G  will  be  greater  than  that  at  D,  and  so  on ;  thus 
each  drop  transmits  a  smaller  pressure  than  it  receives ;  if  we  have  a  largo 
number  of  drops  in  the  tube  the  difterence  of  pressure  at  the  ends  arising 
in  this  way  may  amount  to  several  atmospheres. 

Relation  between  Pressure  and  Curvature  of  a  Surface.— If 

we  have  a  curved  liquid  surface  in  a  state  of  tension  the  pressure  on  the 
concave  side  of  the  surface  must  be  greater  than  that  on  the  convex ;  we 
shall  proceed  to  find  the  relation  between  the  diflerence  of  pressure  on  the 
two  sides  and  the  curvature  of  the  surface. 

Let  the  small  portion  of  a  liquid  film,  represented  in  Fig.  103  by  ABCD 
where  AB  and  CD  are  equal  and  parallel  and  at  right  angles  to  AD  and  BO , 
be  in  equilibrium  under  the  surface  tension  and  a  diflerence  of  pressure  p 
between  the  two  sides  of  the  film.  When  a  system  of  forces  acting  on  a 
body  are  in  equilibrium  we  know  by  Mechanics  that  the  algebraical 
sum  of  the  work  done  by  these  forces  when  the  body  sufiers  a  small  dis- 
placement is  zero.  Let  the  film  ABCD  (Fig.  103)  be  displaced  so  that 
each  point  of  the  film  moves  outwaid  along  the  normal  to  its  surface  through 

a  small  distance  x,  and  let  A'B'C'D'  be  the 
displaced  position  of  ABCD.  Then  the 
work  done  by  the  pressure  is  equal  to 

p  X  area  ABCD  x  x  ; 

the  work  done  against  the  surface  tension 
is  T  X  increase  in  area  of  the  surface  ;  and 
since  a  film  has  two  sides  the  increase  in 
the  area  of  the  film  is  twice  the  difference 
between  the  areas  A'B'C'D'  and  the  area 
ABCD,  Hence  the  work  done  against  sur- 
face tension  is  equal  to 

2T  X  (area  A'B'C'D'  -  area  ABCD) 

Hence  by  the  mechanical  principle  referred  to 

p  X  area  ABCD  xx  =  2T(area  A'B'C'D'  -  area  ABCD) 

if  we  are  considering  a  drop  of  water  instead  of  a  film  we  musi  write  T 
instead  of  2T  in  this  equation. 

Spherical  Soap-bubble. — In  this  case  ABCD  will  be  a  portion  of  a 
spherical  surface  and  the  normals  AA',  BB',  CC,  DD'  will  all  pass  through 
O,  the  centre  of  the  sphere.  Let  R  be  the  radius  of  the  sphere,  then  by 
similar  triangles 


(1) 


A'B'  =  AB^'  =  ABM + 


OA 


\ 


s) 


B'C'  =  BC^'  =  Bc/n-^) 


144 


PROPERTIES  OF  MATTER. 


The  area        A'B'C'D'  =  A'B'.B'C'  =  AB.  Bc/n- ^Y 


=  AB.BC[1  +  ^ 


'IX 


as  we  suppose  r/E-  is  so  small  that  its  square  can  be  neglectecf. 


Hence 


Fia.  104. 

area  A'B'C'D'=  area  A  BCD 


Fio.  105. 


(-S) 


(2) 


substituting  this  value  for  the  area  A'B'O'D'  in  equation  (1)^  the  equation 
becomes 

4T 


so  that  the  pressure  inside  a  spherical  soap-bubble  exceeds  the  pressure 
outside  by  an  amount  which  is  inversely  proportional  to  the  radius  of  the 
bubble. 

General  Case  of  a  Curved  Soap-bubble.— If  the  element  of  tha 

film  ABCD  forms  a  portion  of  a  curved  surface,  we  know  from  the  theory 
of  such  surfaces  that  we  can  find  two  lines  AB,  BC  at  right  angles 
to   each   other   on    the   surface   such    that   the   normals   to    the   surface 


CAPILLARITY.  145 

at  A  and  B  intersect  in  O,  while  those  at  B  and  0  intersect 
iu  a  point  O'.  The  lines  AB,  BC  are  said  to  be  elements  of  the 
jurves  of  Principal  Curvature  of  the  surface,  and  AO  and  BO'  are  called 
the  Radii  of  principal  curvature  of  the  surface.  We  must  now  distinguish 
between  two  classes  of  surfaces.  In  the  first  class,  which  includes  spheres 
and  ellipsoids,  the  two  points  0  and  0'  are  on  the  same  side  of  the  surface, 
and  the  surfaces  are  called  synclastic  surfaces;  in  the  second  class,  which 
includes  surfaces  shaped  like  a  saddle  or  a  dice-box,  O  and  0'  are  on 
opposite  sides  of  the  surface ;  and  the  surfaces  are  called  anti-clastic 
surfaces.  We  shall  consider  these  cases  separately,  and  take  first  the 
case  of  synclastic  surfaces.  In  this  case  (Fig.  104)  we  have  by  similar 
triangles 

A'B'  =  AB— — -  =  ABn-|-— jifRifl  the  radius  of  principal  curvature  OA. 
Similarly  B'C  =  BCf  1  -^  ^  j  if  R'  is  the  radius  of  principal  curvature  O'B. 
Hence  area  A'B'C'D' =  area  ABCD^l  +|^  i^'^^) 

»areaABCD(l+.(^+^)) 

as  we  suppose  tc/R,  x/K  both  so  small  that  we  can  neglect  the  product  of 
these  quantities  in  comparison  with  their  first  powers.  Substituting  this 
value  for  the  area  A'B'O'D'  in  equation  (1)  we  get 


^'^■.R 


bk)  <'> 


Let  us  now  take  the  case  of  an  anti-clastic  surface,  represented  in 
Fig.  105.     In  this  case  we  have 

a'b'.ab(i  +  -) 

hence  area  A'B'O'D' =  area  ABOd/i +a^/i  -  lA^ 

Substituting  this  value  of  the  area  A'B'CD'  in  equation  (1)  we  get 


We  can  include  (3)  and  (4)  in  the  general  formula 

if  we  make  the  convention  that  the  radius  of  curvature  is  to  be  taken  as 
positive  or  negative  according  as  the  corresponding  centre  of  curvatuie 

a, 


146 


PROPERTIES  OF  MATTER. 


is  on  the  side  of  the  surface  where  the  pressure  is  grea'".esfe  or  on  the 
opp)site  side. 

When  a  soap  film  is  exposed  to  equal  pressures  on  the  two  sides  7)  =  0, 
and  we  must  therefore  have 

K     IV       ' 

In  this  case  the  curvature  in  any  normal  section  must  be  equal  on  1  op{  o  ite 
to  the  curvature  in  the  normal  section  at  right  angles  to  the  first.     By 


Fio.  106. 

stretching  a  film  on  a  closed  piece  of  wire  and  then  bending  the  wire  we 
can  get  an  infinite  number  of  surfaces,  all  of  which  possess  this  property; 
we  can  also  get  surface?  with  this  property  by  forming  a  film  between  the 
rims  of  two  funnels  open  at  the  end,  as  in  Fig.  lOG.  By  moving  the 
funnels  i-elati\ely  to  each  other  we  get  a  most  interesting  series  of 
surfaces,  all  of  which  have  their  principal  curvatures  equal  and  opposite. 


\ 


Tia.  107. 


If  the  film  is  in  the  shape  of  a  surface  of  revolution — i.e.,  one  which  can  be 
traced  out  by  making  a  plane  curve  rotate  about  a  line  in  its  plane — we 
know  from  the  geometry  of  such  surfaces  that  (Fig.  107) 


E  =  PO 


R'  =  PG 


where  0  is  the  centre  of  curvature  of  the  plane  curve  at  P,  and  G  the 
point  where  the  normal  at  P  cuts  the  axis  AG  about  -whidi  the  curve 
rotates. 


CAPILLARITY.  U7 

if  the  pressures  on  the  two  sides  of  the  film  are  equal  we  must  have 
PO  -  -  PG. 

The  only  curve  with  this  property  is  the  catenary,  the  curve  in  which 
a  uniform  heavy  string  hangs  under  gravity,  and  thirf,  therefore,  is  the 
shape  of  the  cross-section  of  a  soap  film  forming  a  surface  symmetrical 
about  an  axis,  when  the  prf  ssures  on  the  two  sides  are  equal. 

Stability  of  Cylindrical  Films.— Let  us  consider  the  case  of  a 
symmetrical  film  whose  surface  approaches  closely  that  of  a  right  circular 
cylinder.  Let  EPF  be  the  curve  which  by  its  rotation  about  the  straight 
line  AB  generates  the  surface  occupied  by  the  film.  EPF  will  not  differ 
much  from  a  straight  line,  and  PG,  the  normal  at  P,  will  be  very  nearly 


J- 


equal  to  PN  Avhere  PN  is  at  right  angles  to  AB.  Hence,  if  R  is  the 
radius  of  curvature  at  P  and  p  the  constant  diti'erence  of  pressure  between 
the  inside  and  outside  of  the  film,  we  have 

Let  9/  be  the  height  of  P  above  the  straight  line  EF  and  a  the  distance 
between  the  lines  EF  and  AB,  then 

PN  =  a-f-2/ 

and  as  y  is  very  small  compared  with  a  we  have  approximately 

PJST     a    a^ 

Substituting  this  value  of  1/PN  in  equation  (1)  we  get 

R     2T     a     a'     a^['^^     [2T     a)  \      a'  ^^ 

if  y'  is  the  distance  of  P  from  a  horizontal  line  at  a  distance 

below  EF.  Since  the  film  is  very  nearly  cylindrical,  p  is  very  nearly 
equal  to  2T/a,  so  that  the  distance  between  this  line  and  EF  will  be 
very  small. 

Hence  we  see  from  equation  (2)  that  the  reciprocal  of  the  radius  of 
curvature  at  a  point  on  the  curve  is  proportional  to  the  distance  of  the 
point    from    a    straight    line.      Now   we   saw   (p.  96 )    that    the    path 


a' 


148 


PROPERTIES  OF  MATTER. 


described  by  a  point  fixed  near  to  the  centre  of  a  circle  when  the  circle 
rolls  on  a  straight  line  possesses  this  property,  hence  we  conclude  that  the 
cross-section  of  a  nearly  cylindrical  film  is  a  curve  of  this  kind.  The  curve 
possesses  the  following  properties  :  it  cuts  the  straight  line,  which  is  the  path 
of  the  centre  of  the  circle,  in  a  series  of  points  separated  by  half  the 
circumference  of  the  rolling  circle,  its   greatest   distance   from  this  line 


p 

^^- 

i 

^  *  -.  » 

M 


L 

Fia.  109, 


is  equal  to  the  distance  of  the  point  from  the  centre  of  the  rolling 
circle,  while  the  reciprocal  of  the  radius  of  curvature  at  a  point  is  pro- 
portional to  its  distance  from  this  line. 

Let  us  now  consider  what  is  the  pressure  in  a  nearly  cylindrical 
bubble  with  a  slight  bulge.  Let  us  suppose  that  the  length  of  the  bubble 
is  less  than  the  distance  between  two  points  where  the  curve  which 
generates  the  surface  crosses  the  path  of  the  centre  of  the  rolling 
circle.  The  section  of  the  bubble  must  form  a  part  of  this  curve. 
Let  A  and  0,  Fig.  109,  be  the  ends  of  the  bubble  APC,  the 
section  of  the  film.  Let  the  dotted  line  denote  the  completion  of  the 
curve  of  which  APC  forms  a  part.    Then  if  p  is  the  excess  of  pressure 


^'"-. 


''  > 


M 


K 


FlQ.  110. 


inside  the   bubble  over  the   outside   pressure  and  P  any  point  on  the 
curve, 


'=-(^p^) 


where  p  is  the  radius  of  curvature  of  the  curve  at  P.     Now  if  we  take 
P  at  Q,  a  point  where  the  curve  crosses  its  axis  1//3  =  0,  hence 


P=-- 


2T 


CAPILLARITY. 


149 


Now  if  the  film  were  straight  between  A  and  0  the  excess  of  pressure 
/?'  would  be  given  by  the  equation 

,_  2T 
^     AM* 

As  QK  is  less  than  AM,  p  is  greater  than  p,  hence  the  pressure  in  the 
film  which  bulges  out  is  greater  than  the  pressure  in  the  straight  film. 
We  can  prove  in  the  same  way  that  in  a  film  that  bends  in,  as  in  Fig.  110, 
if  the  distance  between  the  ends  is  less  than  the  dibtance  between  the 


^  ,-)'/     points  Q  and  Q'  on  the  curve ,  that  is,  if  the  length  of  the  film  is  less  than 
C      half  the  circumference  of  its  ends,  the  pressure  is  less  than  the  pressure 
— ^-^T-  \  in  the  straight  film. 

A   •  -,,.     If   the  distance   between  the  ends  of   the   film  is  greater   than  half 
^^  jA  the  circumference  of  the  ends  of  the  film  these  conditions  are  reversed. 
t  ^T^' '     For  let  Fig.  1 1 1  repre- 
^       sent    such  a  film    bending 

in  ;  as  before,  the  excess  of 

pressure  p  will  be  given  by 

the  equation 

2T 

P  =  . 


fi 


QK 

where  Q  is  the  point  where 
the  curve  of  the  film  crosses 
its  axis.  If  the  film  were 
straight  between  A  and  C, 
p',  the  excess  of  pressure, 
would  be  given  by  the 
equation 

2T 
^'"AM 


f 

-V — ~ V- 

B 

r 

N 

A 

E 

FiQ.  112, 


Since  in  this  case  AM  is  greater  than  QK,  p  is  less  than  p.  Hence 
the  pressure  in  the  film  which  bends  in  is  greater  than  that  in  the 
straight  film.  In  a  similar  way  we  can  prove  that  in  this  case  the 
pressure  in  a  film  which  bulges  out  is  less  than  the  pressure  in  a  straight 
film.  Hence  we  arrive  at  the  result  that,  iTiheJen^h  of_the  ^Im  is 
less_than  half  the  circumferenc_e  of  its  end,  the  pressure_in  afi^m 
that  bulges  out  ia' greater  than  thatjn_a_jlm  which_bends  in,  while 


150 


PROPERTIES  OF  MATTER. 


if  the  length  ^f^the  film  is  greater  than  its  semi-circumference  the 
pressure^in  the  film  that  bulges  put  is  less  than  the  jpressure^in  one 
that  bends  in.  Mr.  Boys  has  devised  a  very  beautiful  experiment  which 
illustrates   this   point.     The   arrangement   is   represented   in    Fig.    112. 

A  and  J3  are  pieces  of  glass 
tubing  of  equal  diameter  com- 
municating with  each  other 
through  the  tube  C  ;  this  com- 
munication can  be  opened  or 
closed  by  turning  the  tap.  E 
and  F  are  pieces  of  glass  tubing 
of  the  same  diameter  as  Aj  they 
are  placed  vertically  below  A 
and  B  respectively.  The  distance 
between  A  and  E  and  B  and  F 
be   altered    by    laising    or 


can 


lowering  the  system  ABC.  First 
begin  with  this  distance  less 
than  half  the  circumference  of 
the  glass  tube,  Fig.  113,  close 
the  tap  and  blow  between  A  and 
E  a  bubble  which  bulges  out, 
and  between  B  and  F,  one  that  bends  in.  Now  open  the  tap ;  they  will 
both  tend  to  straighten,  air  going  from  the  one  at  A  to  help  to  fill  up 
that  at  B,  showing  that  the  pressure  in  the  one  at  A  is  greater  than  ia 
that  at  B.     Now  repeat  the  experiment   after   increasing   the   distance 

between  A  and  E  and  B  and  F  to 
more  than  half  the  circumference  of 
the  tube.  We  now  find  on  opening 
the  tap  that  the  film  which  bulges 
out  is  blown  out  still  more,  while  the 
one  that  bends  in  tends  to  shut  itself 
up,  showing  that  air  has  gone  from 
B  to  A  or  that  now  the  pressure  at 
B  is  greater  than  that  at  A. 

It  follows  from  this  result  jthat 
the  equilibrium  of  a  cylindrical  filin. 
is  unstable  whenTitsTength  is  greater 
tjian  its  circumference,  while  shorter 
^ms.are  stabled 

For  let  us  consider  the  equilibrium 
of  a  cylindrical  film  between  two 
equal  fixed  discs,  A  and  B,  Fig.  115, 
and  consider  the  behaviour  of  a 
movable  disc  C  of  the  same  size  placed 
between  them.  Suppose  the  length 
of  the  film  is  less  than  its  circum- 
ference and  that  C  is  midway  between  A  and  B ;  move  C  slightly  towards 
B,  then  the  film  between  B  and  C  will  bulge  out  while  that  between  A 
and  C  will  bend  in.  As  the  distance  between  each  of  the  films  is  less  than 
half  the  circumference  the  pressure  in  the  film  which  bulges  out  will  be 
greater  than  in  that  which  bends  in,  thus  C  will  be  pushed  back  to  its 


Fig.  114. 


CAPILLARITY. 


151 


c 
Fio.  115. 


original  position  and  the  equilibrium  will  be  stable.  If  C  is  not  midway 
between  AB  but  nearer  to  J3  than  to  A,  then  even  if  AC  is  greater  than 
the  semi-circumference  so  that  when  C  is  pushed  towards  B  the  pressure 
in  AC  is  greater  than  when 
the  film  is  straight,  yet  it  is 
easy  to  prove  that  the  excess 
of  pressure  in  BC  is,  in 
consequence  of  its  greater 
curvature,  greater  than  that 
in  AC,  so  that  0  is  again 
pushed  back  to  its  old  posi- 
tion and  the  film  is  again  a. 
stable. 

Suppose   now    that    the 
distance  between  A  and  B  is  gi^eater  than  the  circumference  of  the  film, 
and  that   C,   originally  midway  between  A  and  B,  is  slightly  displaced 
towards  B.     CB  will  bulge  out  and  CA  will  bend  in;    as  the  length  of 
each  of  these  films  is  gi^eater 
than   the   semi-circumference 
of   the   film   the    pressure  in 
BC  will  be  less  than  that  in 
AC,    and    C    will    be   pushed 
still  further  from  its  original 
position  and  the  equilibrium 
will   be  unstable.      The  film 
will     contract     at    one    pait 
and  expand  in  another  until 
its  two  sides  come  into   con- 
tact and  the  film  breaks  up 
into   two    separate    spherical 
portions. 

These  results  apply  to  fluid 
cylinders  as  well  as  to  cylin- 
drical films.  Such  cylinders 
are  unstable  whentheir  length 
IS  grea{er_than  "tEeir  circum- 
ference. Examples  of  this 
unstability  are  afibrded  by  the 
breaking  up  of  a  liquid  jet 
into  drops.  The  development 
of  inequalities  in  the  thickness 
of  the  jet  is  shown  in  Figs. 
116  and  117  taken  from  instan- 
taneous photogx*aphs.  Thelittle  ij'io.  116. 
drops  between   the   big  ones 

are  made  from  the  nari-ow  necks  which  form  before  the  jet  finally 
breaks  up.  Another  instance  of  this  instability  is  afforded  by  dipping 
a  glass  fibre  in  water,  the  water  gathers  itself  up  into  beads.  A  very 
beautiful  illustration  of  the  same  efiect  is  that  of  a  wet  spider's  web, 
shown  in  Fig.  118,  when  again  the  water  gathers  itself  up  into  spherical 
beads. 

If  the  Auid  is  very  viscous  the  effect  of  viscosity  may  counterbalance 


152 


PROPERTIES  OF  MArPER. 


the  instability  due  to  surface  tension ;  thus  it  is  possible  to  get  long  thin 


threads  of  treacle  or  of  molten  glass 

Force  between  two  Plates 


ntid  quartz. 

due  to  Surface-tension.— Let  A 

and  13  ( Fig.  11  'J)  be  two  parallel  plates 
sepai-ated  by  a  film  of  water  or  some 
liquid  which  wets  them;  then,  if  d 
is  the  distance  between  the  plates 
and  D  the  diameter  of  the  area  of 
the  plate  wet  by  the  liquid,  the 
radii  of  curvature  at  the  free  sur- 
face of  the  liquid  are  approximately 
-  d/2  and  D/2,  hence  the  pressure 
inside  the  film  is  less  than  the 
atmospheric  pressure  by 


2T 


Fig.  117. 


---1 
d     DJ 


or  if  cZ  is  very  small  compared  with 


D  the  difference  of  pressure  is  approximately 


2T 


Now  the  plate  A  is  pressed  towards  B  by  the  atmospheric   pressure 
and  away  from  B  by  a  pressure  which  is  less  than  this  by  2T{d  ;  hence,  if 


Fig.  118, 


A  is  the  area  of  the  plate  wet  by  the  film,  the  force  urging  A  towards 

T^  o  2AT 

B  IS  .   ,  .  . 

d 
The  force  varies  inversely  as  the  distance  between  the  plates ;    thus, 


B 


Fig.  119. 


if  a  drop  of  water  is  placed  between  two  plates  of  glass  the  plates  are 
forced  together,  and  this  still  further  increases  the  pull  between  the  plates 
as  the  area  of  the  wetted  sui-face  increases  while  the  distance  between  the 
plates  dimiuishos. 


CAPILLARITY. 


153 


1 


V 


IV 


0 


s. 


Attractions  and  Repulsions  of  small  Floating:  Bodies.— Small 

bodies,   such   as   straw   or   pieces   of   cork,    floating    on    the    surface    of 

a  liquid  often  attract  each  other  and   cuUec^t  together  in   clusters;    this 

occurs  when  the  bodies  are^all  wet  by  the  liquid,  and  also  when^ione  of 

them  are  wet  ;^  one  body  is  wet  and  one  is  not  wet  they  repel  each  other 

when  they  come  close  together.     To  investigate  the  theory  of  this  eflect, 

let  us  suppose  that  A  and  B  are  two  parallel  vertical  plates  immersed  in  a 

liquid  which  wets  both  of  them,  the  liquid  will  stand  at  a  higher  level 

between  the  plates  than  it  does  outside.     We  shall  begin  by  showiBg  that 

the  horizontal    force    exerted   on  a 

plate  by  a  meniscus  such  as  PRQ,  ^ 

UVW   is   the   same   as   the    force 

which    would    be    exerted    if    the  u 

meniscus  were  done  away  with  and 

the  liquid    continued    horizontally 

up  to  the  surface  of  the  plate.     For 

consider  the  water  in  the  meniscus 

PQR;   it  is  in   equilibrium   under 

I ;  the    horizontal   tension    at    P,   the 

>A  vertical    tension   at    Q,    the  ^  force 
exerted  by  the  plate  on  the  liquid, 

'^' the  vertical  liquid  pressure  over 
PR,  and  the'  pressui-e  of  the  atmo- 
sphere over  PQ.  The  resultant  A 
pressure  of  the  atmosphere  over 
PQ,  which  we  shall  call  tt,  in  the 
horizontal  direction  is  equal  to  the 
pressure  which  would  be  exerted  on 
QR,  the  part  of  the  plate  wet  by  Fia.  120. 
the  meniscus,  if  this  were  exposed 

directly  to  the  atmospheric  pressure  without  the  intervention  of  the 
liquid.     The  horizontal  forces  acting  from  left  to  right  on  the  meniscus 

^  «-  —  T- force  exerted  by  plate  on  meniscus,  j  —  "*=^  ->^<»x>-^-»^  \rrz,^ 

Since  the  meniscus  is  in  equilibrium  the  horizontal  forces   must   be  in   ^ilf-' 
equilibrium  ;    hence  ^.  - 

force  exerted  by  meniscus  on  plate  =  T  —  tt, 

but  this  is  precisely  the  force  which  would  be  exerted  if  the  meniscus  were 
done  away  with  and  the  horizontal  surface  of  the  liquid  prolonged  to  meet 
the  plate.  Hence,  as  far  as  the  horizontal  forces  are  concerned,  we  may 
suppose  the  surfaces  of  the  liquid  flat,  and  represented  by  the  dotted  lines 
in  Fig.  120.  Considering  now  the  forces  acting  on  the  plate  A,  the  pulls 
exerted  by  the  surface-tension  at  R  and  U  are  equal  and  opposite  ;  on  the  left 
the  plate  is  acted  on  by  the  atmospheric  pressure,  on  the  right  by  the  pressure 
in  the  liquid.  Now  the  pressure  in  the  liquid  at  any  point  is  less  than  the 
atmospheric  pressiu-e  by  an  amount  proportional  to  the  height  of  the  point 
above  the  level  of  the  undisturbed  liquid  ;  thus  the  pressure  on  A  tending 
to  push  it  towards  B  is  greater  than  the  pressure  tending  to  push  it  away 
from  B,  and  thus  the  plates  are  pulled  together. 

Now  suppose  neither  of  the  plates  is  wet  by  the  liquid — a  case  repre- 


154 


PROPERTIES  OP  MATTER. 


sented  in  Fig.  121.  "We  can  prove,  as  before,  that  we  may  suppose  the 
fluid  to  be  prolonged  horizoutiilly  to  meet  the  plates.  The  force  tending 
to  push  the  plate  A  towards  13  is  the  pressure  in  the  liquid,  the  force 


B 


r-\ 


Tig.  121. 


V 


Fig.  122. 


r~ 


tending  to  piish  it  away  is  the  atmospheric  pressure.  Now  the  pressure 
at  any  point  in  the  liquid  is  greater  than  the  atmospheric  pressure  by  an 
amount  proportional  to  the  depth  of  the  point  below  the  undisturbed 
surface  of  the  liquid ;  hence,  the  pressure  tending  to  push  A  to  B  will  be 

greater  than  that  tending  to  push  it  away  from 
O  13,    so   that   the    plates    will   again   appear   to 

attract  each  other. 

Now  take  the  case  where  one  plate  is  wet 
by  the  liquid  while  the  other  is  not.  The 
section  of  the  liquid  surface  will  be  as  in  Fig. 
122,  the  curvature  of  the  surface  being  of  one 
sign  against  one  plate,  and  of  the  opposite  sign 
against  the  other.  When  the  plates  are  a 
considerable  distance  apait,  the  surfaces  of  the 
liquid  will  be  like  that  shown  in  Fig.  122; 
between  the  plates  there  is  a  flat  horizontal 
surface  at  the  same  level  as  the  undisturbed 
liquid  outside  the  plates ;  in  this  case  there  is 
evidently  neither  attraction  nor  repulsion  between 
Fio.  12a  the   plates.      Now  suppose   the  plates    pushed 

nearer  together,  this  flat  surface  will  diminish, 
and  the  last  trace  of  it  will  be  a  horizontal  tangent  crossing  the  liquid. 
Since  the  curvature  changes  sign  in  passing  from  A  to  B,  there  must  be  a 
place  between  A  and  B  where  it  vanishes,  and  when  the  curvature 
vanishes,  the  pressure  in  the  liquid  is  equal  to  the  atmospheric  pressure ; 
this  point,  at  which  the  tangent  crosses  the  surface,  must  be  on  the 
prolongation  of  the  free  surface  of  the  liquid.  Now  suppose  that  the 
plates  are  so  near  together  that  this  tangent  ceases  to  bo  horizontal,  and 
the  liquid  takes  the  shape  shown  in  Fig.   123.     We  can  show,  by  the 


CAPILLARITY. 


155 


method  given  on  p.  153,  that  the  action  on  the  plate  A  of  the  meniscus 
inside  A  is  the  same  as  if  the  meniscus  were  removed  and  the  liquid 
surface  stretched  horizontally  between  the  plates,  the  surface-tension  in 
this  surface  being  equal  to  the  horizontal  comjJonent  of  the  surface  tension 
at  the  point  of  inflection.  Now  consider  the  plate  A ;  it  is  pulled  from 
B  by  the  surface-tension  and  towards  it  by  only  the  horizontal  component 
of  this.  The  force  pulling  it  away  is  thus  greater  than  the  other,  and  the 
plates  will  therefore  I'epel  each  other.  If  the  plates  are  pushed  very  near  \ 
together  so  that  the  point  of  inflection  on  the  surface  gets  suppressed  the 
liquid  may  rise  between  the  plates  and  the  repulsiou  be  replaced  by  an 
attraction. 


Methods  of  Measuring'  Surface-tension. 
By  the  Ascent  of  the  Liquid  in  a  Capillary  Tube.— A  finel> 

divided  glass  scale  is  placed  in  a  vertical  position  by  means  of  a  plumb 

line,    the    lower    end   of    the    scale 

dipping  into  a  vessel  V,  which  contains 

some   of    the   liquid    whose    surface 

tension    is   to   be    determined.     The 

capillary  tube  is  prepared  by  drawing 

out  a  piece  of  carefully  cleaned  glass 

tube  until  the  internal  diameter  is 

considerably  less  than  a  millimetre ;  |V_y' 

the   bore  of  the   tube  should    be   as 

uniform  as  possible,  for  although  the 

height  to  which  the  fluid  rises  in  the 

capillary   tube  depends  only  on  the 

radius  of  the  tube  at  the  top  of  the 

meniscus,  yet  when  we  cut  the  tube 

at  this  point  to  determine  its  radius, 

if  the  tube  is  of   uniform  bore,   no 

error  will  ensue  if  we  fail  to  cut  it  at 

exactly  the  right  place.     Attach  the 

capillary  tube  to    the  scale    by  two 

elastic  bands,  and  have  a  good  light 

behind  the  scale.     Dip  the  capillary 

tube  in  the  liquid,  and  it  will  rush 

up  the  tube ;  then  raise  the  capillary 

tube,  keeping  its  end  below  the  fluid 

in  V.     This  will  make  the  meniscus 

sink  in  the  tube  and  ensure  that  the 

tube  above  the  meniscus  is  wetted  by 

the   liquid.      Now   read    off  on   the 

scale  the  levels  of   the   liquid  in  V 

and  the  capillary  tube,  and  the  dif-  Fig.  124. 

erence  of  levels  will  give  the  height 

to  which  the  liquid  rises  in  the  tube.     To  measure  r,  the  radius  of  the 

tube  atthe  level  of  the  meniscus,  cut  the  capillary  tube  carefully  across  at 

this  point  and  then  measure  the  internal  radius  by  a  good  microscope  with 

a  micrometer  scale  in  the  eyepiece.     If  the  section,  when  observed  in  the 

microscope,  is  found  to  be  far  from  ciicular,  the  experiment  should  be 


156 


PROPERTIES  OF  MATTER. 


repeated  with  another  tube.     The  surface  tension  T  is  determined  by  the 
equation  (p.  141), 


T  =  ip«( hr+  —  )  where  p  is  the  density  of  the  fluid. 

If  the  angle  of  contact  is  not  zero  a  knowledge  of  its  value  is  required 
before  T  can  be  determined  by  this  method. 

By  the  Measurements  of  Bubbles  and  Drops.— This  method  is 

due  to  Quincke.     The  theory  is  as  follows;  suppose  that  A13,  Figs.  125 


Fia.  125. 

and  126,  represents  the  section  of  a  large  drop  of  meixury  on  a  horizontal 
glass  plate  or,  when  turned  upside  down  a  large  bubble  of  air  under  a  glass 
plate  in  water.  Let  a  central  slab  be  cut  out  of  the  drop  or  bubble  by  two 
parallel  vertical  planes  unit  distance  apart,  and  suppose  that  this  slab  is 
cut  in  half  by  a  vertical  plane  at  right  angles  to  its  length ;  consider  the 
equilibrium  of  the  portion  of  this  slab  above  the  horizontal  section  BC  of 
gi-eatest  area  in  the  case  of  the  drop,  and  below  it  in  the  case  of  the  bubble. 


Fig.  126. 

The  horizontal  forces  acting  on  the  upper  portion  are  the  surface  tension 
T,  and  the  horizontal  pressures  acting  over  the  flat  section  ADEC  and  the 
curved  surface.  If  the  drop  is  so  large  that  the  top  may  be  considered  as 
plane  there  will  be  no  change  of  pressure  as  we  pass  from  the  air  just  above 
the  surface  of  the  drop  to  the  mercury  just  below  it ;  *  in  this  case  the 
difference  in  the  horizontal  components  of  the  pressure  over  ADEC  and 
the  pressure  of  the  atmosphere  over  the  curved  surface  is,  since  AD  is 
unity,  equal  to  v    ^x^«^...u«_  _    Pi  hi 


ypT>W 


i^^\- 


>\.j-i-a ^'-'^ 


b£./H^.f^^ 


As  this  must  be  balanced  by  the  surface  tension  over  AD  we  must  have 

T  =  ypJ)W  (1) 

By  considering  the  equilibrium  of  the  portion  ABFGHD  of  the  drop  we 
have  T(l  +  cos  w)  -  igph'  (2) 

where  h  is  the  thickness  of  the  bubble  or  drop,  and  w  the  angle  of  contact 
at  F  between  the  liquid  and  the  plate.     From  equation  (2)  we  have 

•  If  the  drops  are  not  large  enough  for  this  assumption  to  be  true,  a  correction 
has  to  be  applied  to  allow  for  the  difference  in  pressure  on  the  two  sides  of  the 
surface  through  A. 


CAPILLARITY. 


157 


4Tcos»- 


h?  = 


99 


Thus  the  thickness  of  all  large  drops  or  bubbles  in  a  liquid  is  independent  of 
tlie  size  of  the  drops  or  bubbles.  By  measuring  either  DE  or  h,  and  using 
equation  (1)  or  (2)  we  can  determine  T.  In  the  case  of  bubbles  it  is  more 
convenient  to  use,  instead  of  a  flat  piece  of  glass,  the  concave  surface  of  a 
large  lens,  as  this  facilitates  greatly  the  manipulation  of  the  bubble.  In 
this  case,  if  we  use  equation  (2),  we  must  remember  that  h  is  the  depth  of  the 
bottom  of  the  bubble  below  the  horizontal  plane  through  the  circle  of 
contact  of  the  liquid  with  the  glass.  Thus,  in  Fig.  127,  A  is  equal  to  NE  and 
not  to  AE.  It  is  more  convenient  to  measure  AE  and  then  to  calculate  NE 
from  the  radius  of  curvature  of  the  lens  and  the  radius  of  the  circle  of 
contact  of  the  glass  and  the  liquid.     Determinations  of  the  surface  tension 


N 


E 

Fig.  127. 

of  liquids  by  this  method  have  been  made  by  Quincke,  Magie,  and  Wilbei- 
f  orce.*  Magie  used  this  method  to  determine  the  angle  of  contact,  as  it  is 
evident  from  equations  (1)  and  (2)  that 

(t)         h 
cos-  =  -= 

2     ^/2DE 


By   this   method   Magie    {Phil.   Mag.,    vol.   xxvi. 
following  values  for  the  angle  of  contact  with  glass : 


1888)    found    the 


Angle  zero. 
Ethyl  alcohol  . 
Methyl  alcohol 
Chloroform 
Formic  acid 
Benzine  . 


Angle  finite. 
Water  (?) 
Acetic  acid 
Turpentine 
Petroleum 
Ether    . 


u 

small 

20° 

17° 

26° 

16° 


Determination  of  the  Surface-tension  by  Means  of  Ripples.— 

The  velocity  with  which  waves  travel  over  the  surface  of  a  liquid  depends 
on  the  surface-tension  of  the  liquid.  The  relation  between  the  velocity  and 
surface-tension  may  be  found  as  follows :  Let  Fig.  128  represent  the  section 
of  a  harmonic  wave  on  the  surface  of  the  liquid,  the  undisturbed  level  of 
the  liquid  being  xy.  If  gravity  were  the  only  force  acting,  the  increase  in 
vertical  pressure  at  N  due  to  the  disturbance  produced  by  the  wave  would 
be  equal  to  gpP^,  when  p  is  the  density  of  the  liquid. 

The  surface  tension  will  give  rise  to  an  additional  normal,  and  therefore 

T 

approximately  vertical,  pressure  equal  per  unit  area  to  :g  ,  where  II  is  the 


*  See  foot-note  on  opposite  page. 


il 


^'^^^-'^ 


/?' 


i58 


PROPERTIES  OF  MATTER. 


radius  of  curvature  of  the  section  of  the  wave  by  the  plane  of  the  paper; 
the  radius  of  curvature  in  the  normal  plane  at  right  angles  to  the  plane  of 
the  paper  is  infinite.  Now  if  the  amplitude  of  the  wave  is  very  small 
compared  with  the  wa\'e  lenirth,  the  wave  curve  may  be  generated  by  a 
point  fixed  to  a  circle  rolling  m  a  straight  line  ;  the  amplitude  is  equal , 
to  the  distance  of   the    point    from   the    centre   of   the  circle,  and  the 

P 


Fio.  128. 

wave  length  is  equal  to  the  circumference  of  the  rolling  circle.  The  line 
xy  is  the  path  of  the  centre  of  the  rolling  circle,  Now  we  saw  (^p  9G)  that 
for  such  a  curve 

R      a- 

where  a  is  the  radius  of  the  rolling  circle ;  but  if  X  is  the  wave  length 
27ra  =  X,  so  that 

1  _47r^PN 

R        X^ 

Thus  the  pressure  at  N,  due  both  to  gravity  and  surface-tension,  is 


hence  we  see  that  the  effects  of  surface-tension  are  the  same  as  if  gravity 
were  increased  by  47r-T/X-p.  Now  the  velocity  of  a  gravity  wave  on  deep 
water  is  the  velocity  a  body  would  acquire  under  gravity  by  falling 
vertically  through  a  distance  X/47r,  where  X  is  the  wave  length— i.e.,  the 
velocity  is  JgX/'Jn.  Hence  v,  the  velocity  of  a  wave  propagated  under 
the  influence  of  surface-tension  as  well  as  gravity,  is  given  by  the  equation 

The  velocity  of  propagation  of  the  wave  is  thus  infinite  both  when 
the^jvave  length  is  zero  and  when  it  is  infinite ;  it  is  proportional  toThe 
square  root  of  an  expression  consisting  of  the  sum  of  two  terms  whose 
product  is  constant.  It  follows  from  a  well-known  theorem  in  algebra 
that  the  expression  will  be  a  minimum  when  the  two  terms  are  equal. 
Thus  the  velocitj^f  propagation  of  the^  waves  wjUbe  least  when 

4,r-T 
9- 


X-t 


or  when 


=  27r\/- 


in  this  case  the  velocity  is  equal  to 


;x 


CAPILLARITY. 


159 


In  the  case  of  water,  for  which  T  =  75, 


X  =  1'7  cm.,  and  v  =  23  cm. /sec. 


Fig.  129. 


Hence  no  waves  can  travel  over  the  surface  of  water  with  a  smaller 
velocity  than  23  cm.  per  second.  For  any  velocity  greater  than  this  it 
is  possible  to  find  a  wave  length  X  such  that  waves  of  this  length  will 
travel  with  the  given  velocity.  Waves  whose  lengths  are  smaller  than 
that  corresponding  to  the  minimum  velocity  are  called  "ripples,"  those 
whose  lengths  exceed  this  value  "  waves."  A  wave  is  propagated  chiefly 
by  gravity,  a  ripple  chiefly  by  surface  tension. 

The  velocity  of  a  '•  wave  "  increases  as  the  wave  length  increases,  while 
that  of  a  "  ripple  "  diminishes.  Interesting  examples  of  the  formation 
of  ripples  are  furnished  by  the  standing  patterns  often  seen  on  the  surface 
of  running  water  near  an  obstacle,  such  as  a 
stone  or  a  fishing-line.  Thus,  let  AB  represent 
a  stone  in  a  stream  running  from  right  to  left, 
the  disturbance  caused  by  the  flow  of  the  water 
against  the  stone  will  give  rise  to  ripples  which 
travel  up  stream  with  a  velocity  depending  upon 
their  wave  length.  Close  to  the  stone  the 
velocity  of  the  water  is  zero,  so  that  the  ripples 
travel  rapidly  away  from  the  stone.  When, 
hov^^ever,  we  get  so  far  away  from  the  stone,  say 
at  P,  that  the  velocity  of  the  water  is  greater 
than  23  cm. /sec,  it  is  possible  to  find  a  ripple  of 
such  a  wave  length  that  its  velocity  of  propagation 
over  the  water  is  equal   to  the  velocity   of  the 

stream,  the  ripple  will  be  stationary  at  P,  and  will  form  there  a  pattern  of 
crests  and  hollows.  As  the  velocity  of  the  water  increases  as  we  recede 
from  the  stone  the  ripples  which  appear  stationary  must  get  shorter  and 
shorter  in  wave  length,  and  thus  the  crests  in  the  pattern  will  get  neai-er 
and  nearer  together  as  we  proceed  up  stream.  We  see  that  the  condition 
that  the  pattern  should  be  formed  at  all  is  that  the  velocity  of  the  stream 
must  exceed  23  cm. /sec.  Fig.  129  is  taken  from  a  photograph  of  the 
ripples  behind  a  stone  in  running  water.  A  similar  explanation  applies  to 
the  pattern  in  front  of  a  body  moving  through  the  liquid. 

Lord  Rayleigh  was  the  first  (Phil.  Mag.,  xxx.  p.  38G)  successfully  to 
apply  the  measurement  of  ripples  to  the  determination  of  the  surface- 
tension,  and  his  method  was  used  by  Dr.  Dorsey  {Phil.  Mag.,  xliv.  p.  3G9) 
to  determine  the  surface-tension  of  a  large  number  of  solutions.  Lord 
Rayleigh's  method  is  to  generate  the  ripples  by  the  motion  of  a  glass  plate 
attached  to  the  lower  prong  of  an  electrically  driven  tuning-fork,  and 
dipping  into  the  liquid  to  be  examined.  To  render  the  ripples  (which  for 
the  theory  to  apply  have  to  be  of  very  small  amplitude)  visible,  light  reflected 
from  the  surface  is  brought  to  a  focus  near  the  eye  of  the  observer.  On 
account  of  the  rapidity  with  which  all  phases  of  the  waves  are  presented 
in  succession  it  is  necessary,  in  order  to  see  the  waves  distinctly,  to  use 
intermittent  illumination,  the  period  of  the  illumination  being  the  same 
as  that  of  the  waves.  The  illumination  can  be  made  intermittent  by  placing 
in  front  of  the  source  of  light  a  piece  of  tin  plate  rigidly  attached  to  the 
prong  of  a  tuning-fork,  acd  so  arranged  that  once  on  each  vibration  the 


160  PROPERTIES  OF  MATTER. 

li<yht  is  intercepted  by  the  interposition  of  the  plate.  This  fork  is 
in  unison  with  the  one  dipping  into  the  liquid.  It  is  driven  electro- 
magnetically,  and  the  intermittent  current  furnished  by  this  fork  is 
used  to  excite  the  vibrations  of  the  dipping  fork.  By  this  means  the 
ripples  can  be  distinctly  seen,  the  number  between  two  points  at  a  known 
distance  apart  counted,  and  the  wave-length  X  determined.  If  r  is  the 
time  of  vibration  of  the  fork  vt  =  \, 

and  since  <^  =  K-  +  ^r— 

T_    X^    _g\* 
p  ~  2^      47r* 

an  equation  from  which  T  can  be  determined.  The  second  term  in  this 
expression  is  in  these  experiments  small  compared  with  the  first. 

Determination   of  Surface  Tension  by   Oscillations   of   a 

Spherical  Drop  of  Liquid. — When  the  drop  is  in  equilibrium  under 
surface-tension  it  is  spherical ;  if  it  is  slightly  deformed,  so  as  to  assume 
any  other  form,  and  then  left  to  itself,  the  surface-tension  will  pull  it 
back  until  it  again  becomes  spherical.  When  it  has  reached  this  state 
the  liquid  in  the  drop  is  moving,  and  its  inertia  will  carry  the  drop  through 
the  spherical  form.  It  will  continue  to  depart  from  this  form  until  the 
surface-tension  is  able  to  overcome  the  inertia,  when  it  is  again  pulled  back 
to  the  spherical  form,  passes  through  it  and  again  returns ;  the  drop  will 
thus  vibrate  about  the  spherical  shape.  We  can  find  how  the  time  of 
vibration  depends  upon  the  size  of  the  drop  by  the  method  of  dimensions, 
and  the  problem  forms  an  excellent  example  of  the  use  of  this  method. 
Suppose  the  drop  free  from  the  action  of  gravity,  then  t,  the  time  of 
vibration  of  the  drop,  may  depend  upon  a  the  radius,  p  the  density,  and 
S  the  surface-tension  of  the  liquid  ;  let 

«  =  CaVS^ 

where  0  is  a  numerical  constant  not  depending  upon  the  units  of  mass, 
length,  or  time.  The  dimensions  of  the  left-hand  side  are  one  in  time, 
none  in  length,  and  none  in  mass,  which,  adopting  the  usual  notation,  we 
denote  by  [T]^  [L]"  [M]" ;  the  right-hand  side  must  therefore  be  of  the 
same  dimensions.  Now  a  is  of  dimensions  [T]«  [L]'  [M]" ;  p,  [T]"  [L]-^  [Mji ; 
and  S,  since  it  is  energy  per  unit  area,  [T]-^  [L]"  [M]^ ;  hence  the  dimen- 
sions of  aY^'  are,  [T]"-'  [L]-^^+''  [M]-+^.  As  this  is  to  be  of  the  dimensions 
of  a  time,  we  have 

-22  =  1,  -3y  +  x  =  0,   y  +  z  =  0 

therefore  x  =  ^,  y  =  ^,  z=  -| 

So  that  t,  the  time  of  vibration,  varies  as  Vpa'ys ;  i.e.,  it  varies  as  the 
square  root  of  the  mass  of  the  drop  divided  by  the  surface-tension;  a  more 
complete  investigation,  involving  considerable  mathematical  analysis,  shows 

that  t  =  -^/y  P     ^  where  t  is  the  time  of  the  gravest  vibration  of  the  drop. 

The  reader  can  easily  calculate  the  time  of  vibration  of  a  drop  of  any  size 
if  he  remembers  that  the  time  of  vibration  of  a  drop  of  water  25  cm.  in 
radius  is  very  nearly  1  second.     The  vibrations  of  a  sphere  under  surface- 


CAPILLARITY. 


161 


tension  can  easily  be  followed  by  the  eye  if  a  large  spherical  drop  of  water 
is  formed  in  a  mixture  of  petroleum  and  bisulphide  of  carbon  of  the  same 
density.  Lenard  (Wiedemann's  Annalen,  xxx.  p.  209)  applied  the 
oscillation  of  a  drop  to  detei-mine  the  surface-tension  of  a  liquid.  He 
determined  the  time  of  vibration  by  taking  instantaneous  photographs  of 
the  drops,  and  from  this  time  deduced  the  surface-tension  by  the  aid  of 
the  preceding  formuhe. 

Determination  of  Surface-tension  by  the  Size  of  Drops.— The 

surface-tension  is  sometimes  measured  by  determining  the  weight  of  a  drop 
of  the  liquid  falling  from  a  tube.  If  we  treat  the  problem  as  a  statical 
one  and  suppose  that  the  liquid  wets  the  tube  from  which  it  falls,  then 
just  on  the  point  of  falling  the  drop  below  the  section  ^i5  (Fig.  130)  is  to  be 
regarded  as  in  equilibiium  under  the  surface-tension  acting 
upwards,  the  weight  of  the  drop  acting  downwards,  the 
pressure  of  the  air  on  the  surface  of  the  drop  acting  upwards, 
and  the  pressure  in  the  liquid  acting  downwards  across  the 
gection  AB.  If  a  is  the  radius  of  the  tube,  T  the  surface- 
tension,  then  the  upward  pull  is  2naT.  If  we  suppose  at 
the  instant  of  falling  that  the  drop  is  cylindrical  at  the  end  of 
the  tube,  the  pressure  in  the  liquid  inside  the  drop  will  be 
greater  than  the  atmospheric  pressure  by_T/a  (see  p.  145), 
Hence  the  effect  of  the  atmospheric  pressure  over  the  surface 
of  the  drop  and  the  fluid  pi-essure  across  the  section  AB  is  a 
downwards  force  equal  to  iraP'Tja  or  TraT.  Hence,  if  w  is  the  weight  of 
the  drop  we  have,  equating  the  upwards  and  downwards  forces, 

27raT  =  to  +  iraT ;  or  TraT  =  w. 

The  detachment  of  the  drop  is,  however,  essentially  a  dynamical  pheno- 
menon, and  no  statical  treatment  of  it  can  be  complete.  We  should  not 
therefore  expect  the  preceding  expression  to  accord  exactly  with  the  results 
of  experiment.  Lord  Eayleigh*  finds  the  relation  3-8aT  =  to  to  be  sufficiently 
exact  for  many  purposes.  Most  observers  who  have  used  this  method 
seem  to  have  adopted  the  relation  27raT  =  tw,  a  formula  which  gives  little 
more  than  half  the  true  surface-tension ;  the  error  comes  in  by  neglecting 
the  change  of  pressure  inside  the  drop  produced  by  the  curvature  of  its 
surface. 

Wilhelmy's  Method.t— This  consists  in  measuring  the  downward  pull 
exerted  by  a  liquid  on  a  thin  plate  of  glass  or  metal  partly  immersed  in  the 
liquid  ;  the  liquid  is  supposed  to  wet  the  plate.  The  pull  can  be  readily 
measured  by  suspending  the  plate  from  one  of  the  arms  of  the  balance  and 
observing  the  additional  weight  which  must  be  placed  in  the  other  scale-pan 
to  balance  the  pull  on  the  plate  when  it  is  partially  immersed  in  the  liquid, 
allowance  being  made  if  necessary  for  the  effect  of  the  water  displaced.  If 
I  is  the  length  of  the  water-line  on  the  plate,  T  the  surface-tension,  then  i! 
the  liquid  wets  the  plate  the  downward  pull  due  to  surface-tension  is  T^. 

Method  of  Detachment  of  a  Plate.-— Some  observers  have  deter- 
mined the  surface-tension  of  liquids  by  measuring  the  pull  required  to  drag 
a  plate  of  known  area  away  from  the  surface.  The  theory  of  this  method 
resembles  in  many  respects  that  by  which  we  determined  the  thickness  of 
^  drop  or  air  bubble  {see  p.  156).     Let  us  take  the  case  of  a  rectangular 

*  Lord  Rayleigh,  Phil.  Mag.,  48,  p.  321. 

t  Glazebrook  and  Shaw,  Practical  Physics,  oh.  vii.  §  X.  L 


7'/- 


ktcr^ 


162  PROPERTIES  OF  MATTER. 

plate  being  pulled  away  from  the  surface,  and  let  the  figure  represent  a 
section  by  a  plane  at  right  angles  to  the  length  of  the  rectangle.  Considering 
the  equilibrium  of  the  portion  whose  section  is  EBCF,  and  whose  length 
perpendicular  to  the  paper  is  unity,  the  horizontal  forces  acting  upon  it 
are  :  (1)  the  forces  due  to  surface-tension— i.e.,  2T  acting  from  left  to  right ; 
(2)  the  atmospheric  pressure  on  the  curved  surface  BG  acting  from  right  to 

left,  which  is  equal  to  Tld 
J? n  where    11   is   the    atmos- 

pheric pressure  and  d  is 
the  height  of   the  lower 

■j?         ^  surface  of  the  plate  above 

Tig.  131.  the   undisturbed  level   of 

the   liquid ;    and    (3)  the 
fluid  pressure  acting  across  the   surface   EF   from   left   to   right.     The 
pi-essure  in  the  liquid  at  F  i?,  equal  to  IT,  and  therefore  the  resultant  fluid 
pressure  across  EF  is  equal  to  Hd-^gpd',  where  p  is  the  density  of  the, 
liquid.     Hence,  equating  the  components  in  the  two  directions,  we  have      "^^^'^f 

2T  +  ncZ  -  lopd'  =  Ud,  ordr^  =  ~  ^ 

ffP 

Now  the  fluid  pressure  just  below  the  surface  is  less  than  the  atmospheric 
pressure  by  gpd,  hence  the  upward  pull  P  required  to  detach  an  area  of  the 
plate  equal  to  A  is  equal  to  Agpd,  and  substituting  for  d  its  value,  we  find 

p  =  2aVi>7 

Jaegfer's  Method. — In  this  method  the  least  pressure  which  will  force 
bubbles  of  air  from  the  narrowjorifice  ofa'capillary  tube  dipping  into_the 
n^mdlsjmeasured.  The  pressure  ina  spherical  cavity  exceeds  the  pressure 
outsidenoy"2T7a~where  a  is  the  radius  of  the  sphere,  hence  the  pressure 
required  to  detach  the  bubble  of  air  exceeds  the  hydrostatic  pressure  at  the 
orifice  of  the  tube  by  a  quantity  proportional  to  the  surface-tension.  This 
method,  which  was  used  by  Jaeger,  is  a  very  good  one  when  relative  and 
not  absolute  values  of  the  surface-tension  are  required  ;  when,  for  example, 
we  want  to  find  the  variation  of  surface-tension  with  temperature. 

The  following  are  the  values  of  the  suiface-tension  at  0°  C,  and  the 
temperature  coefficients  of  the  surface-tension  for  some  liquids  of  frequent 
occurrence.     The  surface-tension  at  t°  0.  is  supposed  to  be  equal  to  T^  -  fit. 

.  "lly 

.  -087 

.  -132 

.  -379 

.  -152 

The  surface-tension  of  salt  solutions  is  generally  greater  than  that  of 
pure  water.  If  T„  is  the  surface-tension  of  a  solution  containing  n  gramme 
equivalents  per  litre,  T„  the  surface-tension  of  pure  water  at  the  same 
temperature,  Dorsey*  has  shown  that  T„  =  T^.  +  lxn,  where  R  lias  the 
following  values— NaCl  (1-53);  KCl  (1-71);  h{Nsi.jGO^)  (200);  i(K,CO^ 
(1-77);  ](ZnS0J(l-8G). 

♦  Dorsej,  Phil.  Mag.,  44,  1897,  p.  369. 


Liquid 
Ether  (C,H,„0) 
Alcohol  (C,H,0)        . 
Benzene  (CgHg) 

To 
.     19-3 

.     25-3 

.     30-6 

Mercury  . 
Water      . 

.  527-2 
.     75-8 

CAPILLARITY.  l63 

On  the  Effect  of  Temperature  on  the  Surface-tension  of 

LiQ,uids. — The  surface-tension  of  all  liquids  diminishes  as  the  temperature 
increases.  This  can  be  shown  in  the  case  of  water  by  the  following 
experiment :  A  pool  of  water  is  formed  on  a  horizontal  plate  of  clean 
metal ;  powdered  sulphur  is  dusted  over  the  surface  of  the  water  and  heat 
applied  locally  to  the  under  surface  of  the  metal  by  a  fine  jet.  On  the 
application  of  the  heat  the  portion  of  the  water  immediately  over  the  flame  is 
rapidly  swept  clear  of  the  sulphur ;  this  is  due  to  the  greater  tension  in 
the  cold  liquid  outside  pulling  the  sulphur  away  against  the  feebler  tension 
in  the  warmer  water. 

Eotvos  {Wied.  Ann.  27,  p.  448)  has  pointed  out  that  for  many  liquids 
d{Tv^)/dl  is  equal  tu-  2'1,  being  independent  of  tlie  nature  of  the  liquid  and 
lEi  tem2)erature;  here  T  is  the  sni-face-tension  of  the  liquid,  v  the  "molecular 
\  volume" — i.e.,  the  molecular  weight  divided  by  the  density — and  t  the 
temperature.  It  is  clear  that,  if  we  assume  that  d{Tv^)ldt  has  this  value 
for  a  liquid  whose  density  and  surface-tension  at  difterent  temperatures 
are  known,  we  can  determine  the  molecular  weight  of  the  liquid.  The 
^  method  has  been  applied  for  this  purpose,  and  some  interestmg  results 
have  been  obtained  ;  for  example,  water  is  a  liquid  for  which  Eotvos'  rule 
does  not  hold,  if  we  suppose  the  molecular  weight  of  water  to  be  18. 
If,  however,  we  assume  the  molecular  weight  of  water  to  be  36 — i.e.,  that 
each  molecule  of  water  has  the  composition  2II2O,  then  Eotvos'  rule  is 
found  to  hold  at  temperatures  between  100°  and  200°  0. ;  below  the  lower 
of  these  temperatures  the  molecular  weight  would  have  to  be  taken  as 
greater  than  3G  in  order  to  make  Eotvos'  rule  apply.  Hence,  Eotvos  con- 
cluded that  the  molecules  of  water,  or  at  any  rate  the  molecules  of  the 
surface  layers,  have  the  composition  211,0  above  100°  C,  while  below  that 
temperature  they  have  a  still  more  complicated  composition. 
It  follows  that  if  Eotvos'  rule  is  true, 

Tv3  =  2-1  {t,-t) 

where  ^j  is  some  constant  temperature,  which  can  be  determined  if  we 
know  the  value  of  T  and  v  at  any  one  temperature  ;  t^  is  the  temperature  at 
which  thfi  siyfapft-t.pnsinT^  vfinishegj  it  is  therefore  a  temperature  which 
probably  does  not  difler  much  from  the  critical  temperature;  the  values  of 
i,  for  ether,  alcohol,  water^  are  roughly  about  180°,  295",  560°  0.  Their 
critical  temperatures  are  estimated  by  Van  der  Waals  to  be  190°,  256°, 
390°  C. 

Cooling  due  to  the  Stretching'  of  a  Film.— Since  the  surface- 
tension  changes  with  the  temperature,  any  changes  in  the  area  of  a  film  will, 
as  they  involve  work  done  by  or  against  surface-tension,  be  accompanied  by 
thermal  changes.  We  can  calculate  the  amount  of  these  thermal  changes 
if  we  can  imagine  a  little  heat  engine  which  works  by  the  change  of 
surface-tension  with  temperature.  A  very  simple  engine  of  this  kind  is  as 
follows :  Suppose  that  we  have  a  rectangular  framework  on  which  a  film 
is  stretched,  and  that  one  of  the  sides  of  the  framework  can  move  at  right 
angles  to  its  length.  Let  the  mass  of  the  framework  and  film  be  so  small 
that  it  has  no  appreciable  heat  capacity.  Suppose  we  have  a  hot  chamber 
and  a  cold  chamber,  maintained  respectively  at  the  absolute  temperatures 
0,  and  d^,  where  0,  and  d.^  are  so  near  together  that  the  amount  of  heat 
I'equired  to  raise  the  body  from  6.^  to  B^  is  small  compared  with  the 
thermal  eflect  due  to  change  of  area      Let  us  place  the  film  in  the  hot 


164  PROPERTIES  OF  MATTER. 

chamber,  and  stretch  it  so  that  its  area  increases  by  A,  then  take  it  out 
of  the  hot  chamber  and  place  it  in  the  cold  one,  and  allow  the  film  to 
contract  by  the  amount  A ;  the  film  has  thus  recovered  its  original  area. 
Let  it  be  now  placed  again  in  the  hot  chamber.  If  the  surface-tension  of 
the  film  when  in  the  cold  chamber  is  greater  than  when  in  the  hot,  then 
the  film  when  contracting  may  be  made  to  do  more  work  than  was 
required  to  stretch  it,  so  that  there  will  be  a  gain  of  work  on  the  cycle  ; 
the  process  is  plainly  reversible,  so  that  the  film  and  its  framework  and 
the  two  chambers  constitute  a  reversible  engine.  Hence,  if  Hj  is  the 
heat  absorbed  in  the  hot  chamber,  H^  that  given  out  in  the  cold,  both 
being  measured  in  mechanical  units,  we  have  by  the  Second  Law  of 
Thermodynamics, 

5]  =  5?  =  ^^1  ~  ^»  (1) 

0j  03  0,  -  f>3 

If  Tgj,  Tg.,  are  respectively  the  surface-tensions  at  the  temperatures  6, 
and  02,  then  the  work  done  in  stretching  the  film  =  2T0^A,  while  the  work 
done  by  the  film  when  conti^acting  is  2Te2A,  hence  the  mechanical  work 
gained  =  2(Te,-T0j) A.  By  the  principle  of  the  Conservation  of  Eoergy 
the  mechanical  work  gained  must  equal  the  difference  between  the 
mechanical  equivalents  of  the  heat  taken  from  the  hot  chamber  and  given 
up  to  the  cold  ;  hence 

H,-H,  =  2(T,,-T,0A 

and  from  (1)  H,  =  2d,A^^^^^^ 

0j     0j 


If  /3  is  the  temperature  coefiicient  of  T,  then 

/3  =  ' 


1  _  T^i  -  Tg 


6,-0, 

hence  H,=  -20,A/3 

Thus  Hj   is  positive  when  j3  is  negative,    so  that  when    the  surface- 

=>  \j,<^  -vv^  tension  gets  less  as  the  temperature  increases,  heat  must  be  applied  to  the 

^  film  to  keep  the  temperatvu-e  constant  when  it  is  extended — i.e.,  the_film 

ilJLaft  to  itself  will  cool  wlien_jDulled_oiit.     This  is  an  example  of  the   rule 

given  on  page  1^2THat  the  temperature  change  which  takes  place  is  such 

\!!  as  to  make  the  system  stifl'er  to  resist  extension.     For  water  /3  is  about 

>   -^'  f  T/550,  so  rhat  the  mechanical  equivalent  of  the  heat  required  to  keep 

the  temperature  constant    is    about  half    the  woi^k    done    in    stretching 

the  film. 

Surface-tension  of  very  thin  Films.— The  fact  that  a  vertical 

soap  film  when  allowed  to  drain  shows  different  colours  at  different 
places  and  is  yet  in  equilibrium  shows  that  tho^  thickness  of  the  film 
may  vary  within  wide  limits  without  any  subsbmtial  change  in  the 
euifnce-tension.  The  connection  between  the  thickness  of  the  fifm 
ana  tlie  surtace-tension  was  investigated  by  llucker  and  Reinold.*  The 
method  used  is  represented  diagrammatically  in  Fig.  132.  Two  cylindrical 
films  were  balanced  against  each  other,  and  one  of  them  was  kept  thick  by 
passing  an  electric  current  up  it ;  this  keeps  the  film  from  draining,  the 

*  JUicker  and  Reinold,  Phil.  Trans.   177.  part  ii.  p.  627,  1886. 


CAPILLARITY. 


16*5 


L.i. 


1 


other  film  was  allowed  to  drain,  and  a  difference  of  surface-tension   was 
indicated   by  a  bulging  of  one  of  the  cylinders  and  a  shrivelling  of  the 
other.     When  films  are  fiist  formed  the  value  of  their  surface-tension  is 
very  irregular  ;  but   lliicker  and  Reinold  found  that,  if  they  were  allowed 
to  get  into  a  steady  state,  then  a  direct  comparison  of  the  surface-tension 
overarange  of  thickness  extending  from  ISoOyu.yii  {j-i.jx  is  10""  cm.)  down  to 
the  stage  of  extreme  tenuity,  when  the  film  shows  the  black  of  the  first  order 
of  Newton's  scale  of  colour, 
showed      DO      appreciable 
change  in  surface-tension, 
althougli,  had  the  ditlerence 

amounted   to   as   much  as  I  ^  j>-  U 

one- half  per  cent.,  Reinold 
and  Riicker  believed  they 
could  have  detected  it.  A 
large  number  of  determina- 
tions of  the  thickness  of 
the  black  films  were  made, 
some  by  determining  the 
electrical  i-esistance  and 
then  deducing  the  thick- 
ness, on  the  assumption 
that  the  specific  resistance 
is    the    same    as   for    the 


T 


7T 

i 


% 


_y 


* 


"A 


B 


V 


A 


liquid  in   bulk,  others  by  Fig.  132. 

determining  the  retarda- 
tion which  a  beam  of  light  suffers  on  passing  through  the  film,  and 
assuming  the  refraction  index  to  be  that  of  the  liquid  in  mass :  all  these 
determinations  gave  for  the  thickness  of  the  black  films  a  constant  value 
a^out^2  n-ii.  At  first  sight  it  appears  as  if  the  surface-tension  suffered 
no^  change  until  the  thickness  is  less  than  12  /x.yu.  The  authors  have 
shown,  however,  that  this  is  not  the  right  interpretation  of  their  results, 
for  they  find  that  the  black  and  coloured  parts  of  the  film  are  separated 
by  a  sharp  line  showing  that  there  is  a  discontinuity  in  the  thickness. 
In  extreme  cases  the  rest  of  the  film  may  be  as  much  as  250  times  thicker 
than  the  black  part  with  which  it  is  in 

contact.     The  section  of  a  film  showing    r  [~  | 

a  black  part  is  of  the  kind  shown  in 

Fig.    133.     The  stability  of  the    film  Fig.  133. 

shows  that  the    tension    in    the    thin 

part  is  equal  to  that  in  the  thick.  It  is  remarkable  that  in  these  films 
there  are  never  any  parts  of  the  film  with  a  thickness  anywhere  between 
12  yu./i.  and  something  between  45  and  95  fx.fi.;  films  whose  thicknesses 
are  within  this  range  are  unstable.  This  is  what  would  occur  if  the 
surface-tension  first  begins  to  diminish  at  the  upper  limit  of  the  unstable 
thickness,  and  after  diminishing  for  some  time,  then  begins  to  increase  as 
the  thickness  of  the  film  gets  less,  until  at  12  ^.^.  it  has  regained  its 
original  value ;  after  this  it  increases  for  some  time,  and  then  diminishes 
indefinitely  as  the  thickness  of  the  film  gets  smaller  and  smaller.  The 
changes  in  surface-tension  are  represented  graphically  by  the  curve  in 
Fig.  134,  where  the  ordinates  represent  the  surface-tension  and  the 
libscissae  the  thickness  of  the  film.     For  suppose  we  have  ft  film  thinning,  it 


■S 


166  PROPERTIES  OP  MATTER. 

will  be  in  equilibrium  until  the  upper  part  gets  the  thickness  corresponding 
to  the  point  P  on  the  curve  ;  as  the  tension  now  gets  less  than  in  tlie  thicker 
part  of  the  film,  the  thicker  parts  pull  the  thin  part  away,  and  would  cei'tainly 
break  it,  were  it  not  that  alter  the  film  gets  thinner  than  at  R  the  tension  in- 
creases until,  wlien  the  film  reaches  the  thickness  corresponding  to  Q,  the 
tension  is  the  same  as  in  the  thick  film,  and  there  is  equilibrium  between  the 
thick  and  the  thin  pieces  of  the  film.  This  equilibrium  would  be  stable, 
for  if  the  film  were  to  get  thinner  the  tension  would  get  greater,  and  the 
film  would  contract  and  thicken  again,  while  if  it  got  thicker  the  tension 
would  fall  and  the  film  would  be  pulled  out  until  it  regained  its  original 
thickness.  Thus  all  the  films  which  are  in  contact  with  thick  films  must 
have  the  constant "tluckness  corresponding  to  (J.  The  equilibrium  at  W, 
when  the  tension  has  the  same  value  as  at  Q,  is  unstable,  for  any 
extension  of  the  film  lowers  the  tension,  and  thus  makes  the  film  yield 
more  readily  to  the  extension.     The  region  between  R  and  P  is  unstable, 


^  Thickness 

Fig.  134. 

SO  is  that  between  T  an^J2^ The  region  TR  would  be  stable,  but  would 

loe  very  difficult  to  realise.  If  we  start  with  a  thick  film  and  allow  it  to 
thin,  the  only  films  of  thickness  less  than  that  at  P  which  will  endure  will 
be  those  whose  thickne.S3  is  constant  and  equal  to  the  thickness  at  Q. 
Johannot  (Phil.  Mag.,  47,  p.  501,  1899)  has  recently  shown  that  a  black 
film  of  oleate  of  soda  may  consist  of  two  portions,  one  having  a  thickne.ss 
of  12  f^i.^,  the  other  of  6  ^i./i.  In  this  case  there  mu.st  bs  another  dip 
between  S  and  R  in  the  curve  representing  the  relation  between  surface- 
tension  and  thickne.ss. 

Vapour  Pressure  over  a  Curved  Surface. — Lord  Kelvin  was  the 

first  to  show  that  in  consequence  of  suiface-tenision  the  vapour  pressure  in 
equilibrium  with  a  curved  surface  is  not  the  same  as  the  pressure  of  the 
vapour  in  equilibrium  with  a  flat  one.  We  can  see  from  very  genei-al 
considerations  that  this  must  be  the  case,  for  when  water  evaporates  from 
a  flat  surface  there  is  no  change  in  the  area  of  the  surface  and  therefore 
no  change  in  the  potential  energy  due  to  surface-tension ;  in  the  case  of  a 
curved  surface,  however,  such  as  a  spheric;il  diop,  when  water  evaporates 
there  will  be  a  diminution  in  the  area  of  the  surface  and  therefore  a 


Capillarity. 


167 


1 


dimihtition  in  tlie  potential    energy  due   to   surface-tension.     Thus  the 
surface-^tension  will  promote  evaporation  in  this  case,  aa  evaporation   is  '-i  H 
accompanied  by  a  diiuinuticni  iu  the  potential  enern;3\     Thus  evaporation  ^ 
will  go  on  further  from  a  spherical  drop  than  from  a  plane  surface  ;  that  is, 
thepressure  of  the  water  vapour  in  equilibrium  with  the  spherical  drop  is 
greater  than  for  the  plane  area.  " 

Lord  Kelvin's  determination  of  the  effect  of  curvature  on  the  vapour 
pressure  is  as  follows :  Let  a  fine  capillary 
tube  be  placed  in  a  liquid,  let  the  liquid  rise 
to  A  in  the  tube,  and  let  B  be  the  level  of  the 
liquid  in  the  outer  vessel.  Then  there  must 
be  a  state  of  equilibrium  between  the  liqiiid 
and  its  vapour  both  at  A  and  B,  otherwise 
evaporation  or  condensation  would  go  on  and 
the  system  would  not  attain  a  steady  state. 
Let  p  p  be  the  pressures  of  the  vapour  of  the 
liquid  at  B  and  A  respectively,  h  the  height 
of  A  above  B, 

p=p'  + pressure  aue  to  a  column   of  vapour 
whose  height  is  h 


^p  +  gah, 


(1) 


where  a  is  the  density  of  the  vapour.  If  r  is 
the  radius  of  the  surface  of  the  liquid  at  A, 
then  T  being  the  surface-tension, 

2T  ca  difference  of  pressure  on  the  two  sides  of 
r  the  meniscus. 

Now  the  pressure  on  the  liquid  side  of  the 
meniscus  is  equal  to  11  -  gph  where  p  is  the 
density  of  the  liquid  and  11  the  pressure  at 
the  level  of  the  liquid  surface  in  the  outer 
vessel ;  the  pressure  on  the  vapour  side  of  the  meniscus  is  11  -  gah ;  thus 
the  difference  of  pressxu-es  is  equal  to  g[p  —  a)h,  so  that 

—  =  9{p-<^V^* 
r 


FiQ.  135. 


or 


gah  = 


2T 


Hence  by  equation  (1) 


p  =p- 


r  p  -  ff 

2T    <T 
r 


t 


—  a 


*  In  the  investigation  of  the  capillary  ascent  in  tubes  given  on  p.  Ill,  c  is  neglected 
in  comparison  with  p. 

f  The  formula  in  the  text  gives  the  value  for^'-^  when  this  is  small  compared 
with  p ;  the  general  equation  for  p'  may  be  proved  to  be  (neglecting  a  in  comparison 

with  p)  ^     P'     -  2T.  1 

log^  =  — 


P 


R^ 


where  9  is  the  absolute  temperature  and  R  the  constant  in  the  equation  for  a  perfect 
gas — i.e.,  pv  =  ^Q. 


l68 


PROPERTIES  OF  MATTER. 


hence  the  equilibrium  vapour  pressure  over  the  concave  hemispherical 
surface  is  less  than  that  over  a  plane  surface  at  the  same  temperature  by 
2T<r/(f)  -  (T)r.  We  may  write  this  as  io(T/{p  —  a)  where  w  is  the  amount  by 
which  the  pressure  below  the  curved  surface  is  less  than  that  below  the  plane. 
If  the  shape  of  the  liquid  surface  had  been  convex,  like  that  of  a  dewdrop, 
instead  of  concave,  the  pressure  below  the  curved  surface  of  liquid  would  be 
greater  than  that  in  the  plane  sui  face  instead  of  being  less,  and  the  pressure 
of  the  water  vapour  over  the  surface  would  be  greater  than  that  over  a  plane 
surface.  It  can  be  shown  that  if  an  external  pressure  w  were  applied  to  a  plane 


— >. 


is 


n- 


Fio.  136. 

surface  the  vapour  pressure  would  be  increased  by  wa/p  (see  J.  J.  Thomson, 
Applications  of  Dynamics  p.  171).  Unless  the  drops  are  exceedingly  small, 
the  effect  of  curvature  on  tl>e  vapour  pressure  is  inappreciable  ;  thus  if  the 
radius  of  the  drop  of  water  is  one-thousandth  part  of  a  millimetre  the 
change  in  the  vapour  pressure  only  amounts  to  about  one  part  in  nine 
hundred.  As  the  effect  is  inversely  proportional  to  the  radius,  it  in- 
creases rapidly  as  the  size  of  the  drop  diminishes,  and  for  a  drop  1  yu./i 
in  radius  the  vapour  pressure  over  the  drop  when  in  equilibrium  would  be 
more  than  double  that  over  a  plane  surface.  Thus  a  drop  of  this  size 
would  evaporate  rapidly  in  an  atmosphere  from  which  water  would  condense 
on  a  plane  surface.  This  has  a  very  important  connection  with  the 
phenomena  attending  the  formation  of  rain  and  fog  by  the  precipitation  of 
water  vapour.  Suppose  that  a  drop  of  water  "had  to  grow  from  an 
indefinitely  small  drop  by  pi^ecipitation  of  water  vapour  on  its  surface; 
since  the  vapour  pressure  in  CLjuilibrium  with  a  very  small  drop  is  much 


CAPILLARITY.  169 

greater  than  the  normal,  the  drop,  unless  placed  in  a  space  in  which  the 
water  vapour  is  in  a  very  supersaturated  condition,  will  evaporate  and 
diminish  in  size  instead  of  being  the  seat  of  condensation  and  increasing  in 
radius.  Thus  these  small  drops  would  be  unstable  and  would  qtiiclcly 
disappear.  Hence  it  would  seem  as  if  this  would  be  an  insuperable  difficulty 
to  the  formation  of  drops  of  rain  or  cloud  if  these  drops  have  to  pass 
through  an  initial  stage  in  which  tlieir  size  is  very  small.  Aitken  has 
shown  that  as  a  matter  of  fact  these  drops  are  not  formed  under  ordinary 
conditions  when  water  and  water  vapour  alone  are  present,  even  though 
the  vapour  is  considerably  oversaturated,  and  that  for  the  formation  of 
rain  and  fog  the  presence  of  dust  is  necessary.  As  the  water  is  deposited 
around  the  particles  of  dust,  the  drops  thus  commence  with  a  finite  radius, 
and  so  avoid  the  difficulties  connected  with  their  early  stages.  The  effect 
of  dust  on  the  formation  of  cloud  can  be  shown  very  easily  by  the  following 
experiment.  A  and  B  are  two  vessels  connected  with  each  other  by  a 
flexible  pipe  ;  when  B  is  at  the  upper  level  indicated  in  the  diagram  the  globe 
A  is  partly  filled  with  water  ;  if  the  vessel  B  is  lowered  the  water  runs  out  of 
A,  the  volume  of  the  gas  in  A  increases,  and  the  cooling  caused  by  the 
expansion  causes  the  region  to  be  oversaturated  with  water  vapour.  If  A  is 
filled  with  the  ordinary  dusty  air  from  a  room,  a  cloud  is  formed  in  A 
whenever  B  is  lowered ;  this  cloud  falls  into  the  water,  carrying  some  dust 
with  it ;  on  repeating  the  process  a  second  time  more  dust  is  carried  down, 
and  so  by  continued  expansions  the  air  can  be  made  dust  free.  We  find 
that,  after  we  have  made  a  considerable  number  of  expansions,  the  cloud 
ceases  to  be  formed  when  the  expansion  takes  place ;  that  the  absence  of 
the  cloud  is  due  to  the  absence  of  dust  can  be  proved  by  admitting  a  little 
dust  through  the  tube ;  on  making  the  gas  expand  again  a  cloud  is  at  once 
formed. 

It  was  supposed  for  some  time  that  without  dust  no  clouds  could  be 
formed,  but  it  has  been  shown  by  C.  T.  E.  Wilson  that  gaseous  ions  can 
act  as  nuclei  for  cloudy  condensation  if  the  supersatTiration  exceeds  a 
certain  value^  and  he  has  also  shown  that  if  perfectly  dust-free  air  has  its 
volume  suddenly  increased  1*4  time  a  dense  cloud  is  produced.  Though 
dust  is  not  absolutely  essential  for  the  formation  of  clouds,  yet  the 
conditions  under  which  clouds  can  be  foimed  without  dust  are  very 
exceptional,  inasmuch  as  they  require  a  very  considerable  degree  of  super- 
saturation. 

Movement  of  Camphor  on  Water. — If  a  piece  of  camphor  is 

scraped  and  the  shavings  allowed  to  fall  on  a  clear  water  surface  they 
dance  about  with  great  vigour.  This,  as  Marangoni  has  shown,  is  due  to 
the  camphor  dissolving  in  the  water,  the  solution  having  a  smaller  surface- 
tension  than  pure  water  ;  thus  each  little  patch  of  surface  round  a  particle 
of  camphor  is  surrounded  by  a  film  having  a  stronger  surface-tension  than  its 
own,  it  will  therefore  be  pulled  out  and  the  surface  of  the  water  near  the 
bit  of  camphor  set  in  motion.  For  the  movements  to  take  place  the 
surface-tension  of  the  water  suiface  must  be  greater  than  that  of  the 
camphor  solution  ;  if  the  surface  is  greasy  the  surface-tension  is  less  than 
that  of  pure  water,  and  may  be  so  much  reduced  that  it  is  no  longer  sufficient 
to  produce  the  camphor  movements.  Lord  Rayleigh  has  measured  the 
thickness  of  the  thinnest  film  of  oil  which  will  prevent  the  motion  of  the 
camphor ;  the  thickness  was  determined  by  weighing  a  drop  of  oil  which 
was    allowed    to   spread    over   a  known  area.      He    found  that  to  stop 


170 


PROPERTIES  OF  MATTER. 


the  camphor  movements  (which  involved  a  reduction  of  the  surface-tension 
by  about  28  per  cent.)  ala3-orof  oil  2  ju^ithick  was  required  (1  fij^  =  10"' cm.), 
and  that  Avith  thinner  films  the  movements  were  still  perceptible.  This 
thickness  is  small  compared  with  12  ^/t  tlie  thickness  found  by  llUcker 
and  Reinold  for  black  films,  but  it  must  be  remembered  that  the  surface 
which  stops  the  camphor  movements  is  still  far  from  acting  as  a  surface 
of  oil ;  the  surface-tension,  though  less  than  that  of  water,  is  greater  than 
that  of  oil.  The  manner  in  which  the  tension  of  a  contaminated  water 
surface  varies  with  the  amount  of  contamination  has  been  investigated 
by  Miss  Pockels  and  also  by  Lord  Ptayleigh  {Phil.  Mag.,  48,  p.  321).  Miss 
Pockels  determined  the  surface-tension  by  measuring  the  force  required  to 
detach  a  disc  of  known  area  from  the  surface ;  Lord  Rayleigh  used 
Wilhelmy's  method.  The  amount  of  contamination  was  varied  by  confining 
the  greased  surface  between  strips  of  glass  or  metal  dipping  into  the  water; 
by  pulling  these  apart  the  area  of  the  greased  surface  was  increased  and 
therefore  the  thickness  of  it  diminished,  while  by  pushing  them  together 
the  thickness  could  be  increased. 

The  way  in  which  the  surface-tension  is  aflTected  by  the  thickness  of  the 
layer  of  grease  is  shown  by  the  curve  (Fig.  137)  given  by  Lord  Rayleigh. 


d 
o 

a> 
H 

O 

ca 


XJl 


/^ 


Thickness  of  Oil  Film 


Fia.  137, 


In  this  curve  the  ordinates  are  the  values  of  the  surface-tension,  the  abscissae 
the  thicknesses  of  the  oil  film ;  both  of  these  are  on  an  arbitrary  scale.  It 
will  be  seen  that  no  change  in  the  surface-tension  occurs  until  the  thickness 
of  the  oil  film  exceeds  a  certain  value  (about  l/x.yu);  at  this  stage  the  surface- 
tension  begins  to  fall  rapidly  and  continues  to  do  so  until  it  reaches  the 
thickness  corresponding  to  the  point  C  (about  2.i.i.f^.)  ;  this  is  called  the  can^- 
plior  ])oint,  being  the  thickness  required  to  stop  the  movemt  nts  of  the  cam- 
phor particles.  After  passing  this  point  the  variation  of  the  surfac-tension 
with  the  thickness  of  the  film  becomes  much  less  rapid.  Loril  llayleigh 
gives  reasons  for  thinking  that  the  thickness  lyu./t  is  equal  to  the  diameter 
of  a  molecule  of  oil! 


CAPILLARITY, 


171 


Thus,  when  the  amount  of  contamination  is  betweeti  the  limits  corre- 
sponding to  a  thickness  of  the  surluce  layer  bclwcLii  1  fi.fx  and  the  .sinullest 
thickness  requumTtd  gTve  the  surface-tension  ot  oil,  any  diminution  in  the 
contammation  such  as  would  be  produced  by  an  extension  ot  tlio  surtace 
would  result  inan  increase  in  the  surface-tension.  This  is  a  principle  of 
greatlmportance ;  it  seems  first  to  have  been  clearly  stated  by  Marangoni. 
Suppose  we  push  a  strip  of  metal  along  a  surface  in  this  condition,  the  metal 
will  heap  up  the  grease  in  front  and  scrape  the  surface  behind,  thus  the 
surface-tension  behind  the  strip  will  be  greater  than  that  in  fi'ont,  so  that 
the  strip  will  be  pulled  back;  there  will  thus  be  a  force  resisting  the  motion 
of  the  strip  due  to  the  variation  of  the  surface-tension.  This  is  Marangoni's 
explanation  of  the  phenomenon  of  superficial  viscosity  discovered  by  Plateau. 
Plateau  found  that  ii  i  vibrating  body  such  as  a  compass-needle  was 
disturbed  from  its  position  of  equilibiium  and  then  allowed  to  return  to  it 
(1)  with  its  surface  buried  beneath   the  surface  of  the  liquid,  (2)  with 


Fro.  138. 

its  face  on  the  surface  of  the  liquid,  then  with  certain  liquids,  of  which 
water  was  one,  the  time  taken  in  the  second  case  is  considerably  greater 
than  that  in  the  first.  We  see  that  it  must  be  so  if  the  surface  of  the 
liquid  is  contaminated  by  a  foreign  substance  which  lowers  its  surface- 
tension. 

Calming"  of  Waves  by  Oil. — Similar  considerations  will  explain  the 
action  of  oil  in  stilHng  troubled  waters.  Let  us  suppose  that  the  wind 
acts  on  a  portion  of  a  contaminated  sui'face,  blowing  it  forward ;  the 
motion  of  the  surface  film  will  make  the  liquid  behind  the  patch  cleaner 
and  therefore  increase  its  surface-tension,  while  it  will  heap  up  the  oil  in 
front  and  so  diminish  the  surface-tension ;  thus  the  pull  back  will  be 
greater  than  the  pull  forward,  and  the  motion  of  the  surface  will  be  i 
retarded  in  a  way  that  could  not  occur  if  it  were  perfectly  clean.  The 
oiled  surface  acts  so  as  to  check  any  relative  motion  of  the  various  parts ' 
of  the  surface  layer  and  so  prevents  any  heaping  up  of  the  water.  It  is 
these  heaps  of  water  which,  under  the  action  of  the  wind,  develop  into  a 
high  sea ;  the  oil  acts  not  so  much  by  smoothing  them  down  after  they 
have  grown  as  by  stifling  them  at  their  birth. 

A  contaminated  surface  has  a  power  of  self-adjustment  by  which  the 
surface-tension  can  adjust  itself  within  fairly  wide  limits;  a  film  of  such  a 
liquid  can  thus,  as  Lord  Rayleigh  jsoints  out,  adjust  itself  so  as  to  be  in 
equilibrium  under  circumstances  when  a  film  of  a  pure  liquid  would  have 
to  break.  Thus,  to  take  the  case  of  a  vertical  film,  if  the  surface-tension 
were  absolutely  constant,  as  it  is  in  the  case  of  a  pure  liquid  when  the  film 


172  PROPERTIES  OF  MATTER. 


is  not  too  thin,  this  film  would  break,  since  there  would  be  nothing  to 
balance  the  weight  of  the  film.  If,  however,  the  film  were  dirty,  a  very 
slight  adjustment  of  the  amount  of  dirt  at  diflerent  parts  of  the  surface 
would  be  suflicient  to  produce  a  distribution  of  surface-tension  which  would 
ensure  equilibrium.  It  is  probably  on  this  account  that  films  to  be  durable 
have_tojiajiiode  of  a  mixture_of  substances,~such  as  soap  and  water. 

Collision  of  Drops. — If  a  ]et~of  water  be  turned  nearly  verticjilly 
upwards  the  drops  into  which  it  breaks  will  collide  with  each  other;  if  the 
water  is  clean  the  drops  will  rebound  from  each  other  after  a  collision,  but 
if  a  little  soap  or  oil  is  added  to  the  water,  or  if  an  electrified  rod  is  held 
near  the  jet,  the  drops  when  they  strike  will  coalesce  instead  of 
rebounding,  and  in  consequence  will  grow  to  a  much  larger  size.  This  can 
be  made  very  evident  by  allowing  the  drops  to  fall  on  a  metal  plate ;  the 
change  in  the  tone  of  the  sound  caused  by  the  drops  striking  against  the 
plate  when  an  electrified  rod  is  held  near  the  jet  is  very  remarkable. 

The  same  thing  can  be  shown  with  two  colliding  streams.  If  two 
streams  of  pure  water  stiike  against  each  other  in  dust-free  air,  as  in 
Fig.  138,  they  will  rebound;  if  an  electritied  rod  is  held  neai-,  however, 
they  coalesce. 


CHAPTER  XV. 
LAPLACE'S  THEORY  OF  CAPILLARITY. 

Contents, — Intrinsic  Pressure  in  a  Fluid — "Work  required  to  move  a  Particle  from 
the  Inside  to  the  Outside  of  a  Liquid— Work  required  to  produce  a  new  Licjuid 
Surface— Effect  of  Curvature  of  surface— Thickness  at  which  Surface-tension 
changes  effect  of  abruptness  of  transition  between  two  Liquids  in  contact. 

Laplace's  investigations  on  surface-tension  throw  so  much  light  on  this 
subject,  as  well  as  on  the  constitution  of  liquids  and  gases,  that  no  account 
of  the  phenomena  associated  with  surface-tension  would  be  complete  without 
an  attempt  to  give  a  sketch  of  his  theory.  Laplace  started  with  the  assump- 
tion that  the  forces  between  two  molecules  of  a  liquid,  although  very  intense 
when  the  distance  between  the  molecules  is  very  small,  diminish  so  rapidly 
when  this  distance  increases  that  they  may  be  taken  as  vanishing  when  the 
distance  between  the  molecules  exceeds  a  certain  value  c :  c  is  called  the 
range  of  molecular  action.     We  shall  find  that  we  can  obtain  an  explana- 

^^  ..„ 

1 

B  ^ 

^__^ i . 

A 
Fio.  189. 

tion  of  many  surface-tension  phenomena  even  although  we  do  not  know 
the  law  of  force  between  the  molecules.  Let  the  attraction  of  an  infinite 
flat  plate  of  the  fluid  bounded  by  a  plane  surface  on  a  mass  m  at  a  point 
at  a  distance  z  above  the  surface  be  ma-^iz)^  where  o-  is  the  density  of  the 
fluid ;  in  accordance  with  our  hypothesis  i\/{z)  vanishes  when  z  is  greater 
than  c.  It  is  evident,  too,  that  ina-^(z)  will  be  the  attraction  at  a  point  ou 
the  axis  of  any  disc  with  a  flat  face  whose  thickness  ,is  greater  than  c  and 
whose  diameter  is  greater  than  2c.  •    •/,'■-.•  t_    - 

Suppose  we  imagine  a  fluid  divided  into  two  portions  A  and  5  by  a 
plane  ;  let  us  find  the  pull  exerted  on  Bhy  A.  Divide  B  up  into  thin  layers 
whose  thickness  is  dz  ;  then  if  z  is  the  height  of  one  of  these  layers  above  the 
surface  of  separation  the  force  on  unit  area  of  the  layer  is  equal  to  a\l/{z)adz  ; 

CO 

hence  the  pull  of  ^  on  6  per  unit  area  is  equal  to  cr  I  \p[z)dz, 

0 

c 

which,  since  \^{z)  vanishes  when  2>c,is  the  same  as  (T  /  ■^{z)dz. 

0 

This  pull  between  the  portions  A  and  B  is  supposed  to  be  balanced  by  a 
pressure  called  the  "intrinsic  pressure,"  which  we  shall  denote  by  K.  K  then 


CO 

is  equal  to  or  I  ^{z)dz 


174  PROPERTIES  OF  MATTER 

We  shall  find  that  the  phenomena  of  capillarity  require  us  to  suppose 
that,  in  the  case  of  water,  the  intrinsic  pressure  is  very  larg^e.  amounting 
on  the  lowest  estimate  to  several  thousand  atmospheres.  We  may  remark 
in  passing  that  the  intrinsic  pressure  plays  a  very  important  part  in 
Van  der  Waals'  Theory  of  the  Continuity  of  the  Liquid  and  Gaseous  States ; 
it  is  the  term  ajv-  which  occurs  in  his  well-known  equation 

[p  + 1')(^  -  ^')  =  RT  {see  p.  129) 

We  see,  too,  at  once  from  the  preceding  investigation  that  K  is  equal 
to  the  tensile  strength  of  the  liquid,  so  that  if  the  common  supposition 
that  liquids  are  as  "  weak  as  water,"  and  can  only  bear  very  small  tensile 
stresses  without  ruptvire,  were  true,  Laplace's  theory,  which,  as  we  have 
seen,  requires  liquids  to  possess  great  tensile  strength,  would  break  down 
at  the  outset.  We  have  seen,  however,  p.  122,  that  the  rupture  of 
liquids  under  ordinary  conditions  gives  no  evidence  as  to  the  real  tensile 
strength  of  the  liquids,  for  it  was  shown  that  when  water  and  other 
liquids  are  carefully  deprived  of  gas  bubbles — in  fact,  when  they  are 
not  broken  before  the  tension~is  applied — they  can  stand  a  tension  of  a 
great  many  atmospheres  without  rupture  ;  thus  on  this  point  the  properties 
of  liquids  are  in  accordance  with  Laplace's  theory. 

There  is  another  interpretation  of  K  given  by  Dupre  which  enables 
us  to  form  an  estimate  of  its  value.  Consider  a  film  of  thickness  A 
(where  A  is  small  compared  with  c)  at  the  top  of  the  liquid ;  the  work 
required  to  pull  unit  of  area  of  this  film  oft'  the  liquid  and  remove  it 
out  of  the  sphere  of  its  attraction  is  evidently 


or  KA 

0 

Thus  the  work  required  to  remove  unit  volume  of  the  liquid  and 
scatter  it  through  space  in  the  form  of  thin  plates  whose  thickness  is 
small  compared  with  the  range  of  molecular  atti-action  is  K.  Now  the 
work  required  to  take  one  of  these  films  and  still  further  disintegrate 
it  until  each  molecule  is  out  of  the  sphere  of  action  of  the  others  will 
be  small  compared  with  the  work  required  to  tear  the  film  ofl;'  the  surface 
of  the  liquid ;  hence  K  is  the  work  required  to  disintegrate  unit  volumes 
of  the  Kquid  until  its  molecules  are  so  far  apart  that  they  no  longer 
exert  any  attraction  on  each  other  ;  in  other  words,  it  is  the  work  required 
to  vaporise  unit  volume  of  the  gas.  In  the  case  of  water  at  atmospheric 
temperature  this  is  about  GOO  thermal  units  or  600  x  4:-2  x  10'  =  25'2  x  10* 
mechanical  units ;  or  since  an  atmosphere  expressed  in  these  units  is  10® 
this  would  make  K  equal  to  about  2."). 000  atmospheres.* 

Work  required  to  move  a  Particle  from  the  Inside  to  the 

Outside  of  a  Fluid. — Consider  the  force  on  a  particle  P  at  a  depth  z 
below  the  surface ;  the  force  due  to  the  stratum  of  fluid  above  P  will  bo 
balanced  by  the  attraction  of  the  stratum  of  thickness  z  below  P ;  thus 
the  force  acting  on  P  will  be  that  due  to  a  slab  of  liquid  on  a  particle  at 

*  Van  der  Waals  gives  the  following  value  of  K  deduced  from  his  equation: 
water  10,500-10,700,  ether  1300-1430  alcohol  2100-2400,  carbon  bisulphide  2900-2890 
atmospheres. 


-/ 
^;^      '     LAPLACE'S  THEORY  OF  CAPILLARITY.  175 

a  distance  z  above  its  surface— i.e.,  ma^{z).  Hence  the  work  done  in 
bringing  the  particle  to  the  surface  is 

00  V 

m  I  a^{z)dz  =  m(K/o-) ; 

as  an  equal  amount  of  work  will  be  required  to  take  the  particle  from  the 
surface  out  of  the  range  of  molecular  attraction,  the  total  amount  of  work 
required  is  thus  2?rt(K/(7). 

ITence,  ijHi^rticlejxiovingjwith  a  velocity  v  towardsthe^urface  starts 
from  a  depth  greater  than  c  it  cannot  cross  the  surface  unliss 

■kmv  > ■  or  v  >  —   • 

L  o- 2 


In  the  case  of  water,  for  which  a  =  \  and  K  on  the  preceding  estimate 
is  25,000  atmospheres  or  2-5  x  10"-,  we  see  that  a  particJe^waiilxLxiQt_eross 
the  surface  unless  its  velocity  w^ere  greater  than  3-2  x  10^.  The  average 
velocity  of  a  molecule  of  water  vapour  at  0°  C.  is  about  G  x  lO*,  so  that  if 
the  water  contained  molecules  of  water  vapour  it  would  only  be  those 
possessing  a  velocity  considerably  greater  than  the  mean  velocity,  which 
would  be  able  to  escape  across  the  surface. 

Work  required  to  produce  a  new  Liquid  Surface.  —Let  us  con- 
sider the  amount  of  work  required  to  separate  the  two  portions  A  and  B 
into  which  a  plane  G  divides  the  liquid.  Dividing  B  up,  as  before,  into 
slices  parallel  to  the  interface,  then  the  work  done  in  removing  the  slice, 
whose  thickness  is  dz  and  whose  height  above  the  plan©  is  »,  is  per  unit 
of  area  equal  to  -^^^  ^-    , 

00  00  fl  ,1      1 

er'dz  I  xp^xjdx  =  a'dzv^ii  V  =   j  \l/{x)dx  ILi.,.AJ^ 

hence  the  work  required  to  remove  the  whole  of  the  liquid  B  standing  on     '•v^ 
unit  area  away  from  -4  is  /  a^vdz ; 

0 

integrating  this  by  parts  we  see  that  it  is  equal  to 


Ai> 


w  £«-(-£-%. 


J'"- 


crzv       -   I  (TZ-^dz 


dv 


.^iM^  ^•''N'ow  the  term  within  brackets  vanishes  at  both  limits,  and  --=  -  ^{z), 


hence  the  work  required  is 


For  this  amount  of  work  we  have  got  2  units  of  area  of  new  surface, 
hence  the  energy  corresponding  to  each  unit  of  area  {i.e.,  the  surface- 
tension),  which  we  shall  denote  by  T  is  given  by  the  equation 


Iz  ^l) 


176 


PROPERTIES  OF  MATTER. 


Young,  at  the  beginning  of  the  century,  showed  how  from  T  and  K  it 
was  possible  to  calculate  the  range  of  molecular  forces.  He  did  this  by 
assuming  a  particular  value  for  the  force,  but  his  argument  is  appliciible 
even  when  we  leave  the  force  undetermined. 

If  \p{z)  is  always  positive,  then,  since  c  is  the  greatest  value  of  z  for 
which  \p{z)  has  a  finite  value,  we  see  from  equation  (1)  that 


0C5 

\iT-c  I  \p{z)dz 


:icK 


hence 


c> 


2T 
K 


For  water  T  =  75,  and  if  we  take  K  =  25,000  atmospheres  =  25  x  10'",  then 
the  above  relation  shows  that  c>(]xlO"^     In  this  way  we  can  get  an 


FiQ.  140. 

inferior  limit  to  the  range  of  molecular  action.  This  method,  which  was 
given  by  Young,  was  the  first  attempt  to  estimate  this  quantity,  and  it 
s-eems  to  have  been  quite  overlooked  until  attention  was  recently  called  to 
it  by  Lord  Rayleigh. 

It  is  instructive  to  consider  another  way  of  finding  the  expression  for 
the  surface-tension.  Consider  a  pomFP  inside  a  liquid  sphere  (Fig.  140)T 
Then,  if  /-*  is  at  depth  d,  below  the  surface,  greater  than  c,  the  forces  acting 
on  it,  due  to  the  attraction  of  the  surrounding  molecules,  are  in 
equilibrium ;  if  the  distance  of  P  below  the  surface  is  less  than  c,  then 
to  find  the  force  on  F  describe  a  sphere  with  radius  c  and  centre  F,  and 


LAPLACE'S  THEORY  OF  CAPILLARITY. 


177 


the  force  on  P,  acting  towards  the  centre  of  the  larger  sphere,  will  be 
equal  to  the  attraction  which  would  be  exerted  on  P  by  a  quantity  of  the 
fluid  placed  so  as  to  fill  BA  CD,  the  portion  of  the  sphere  whose  centre  is 
P,  which  is  outside  the  larger  sphere.  This  portion  may  be  regarded  aa 
consisting  of  two  parts — (1)  the  portion  above  the  tangent  plane  at  A,  the 
point  on  the  lai'ge  sphere  nearest  to  /*,  and  (2)  the  lenticular  portion 
between  this  plane  and  the 

sphere.     Now  the  attraction  A 

of  the  portion  above  the 
tangent  plane  is  the  same  as 
that  of  a  slab  of  the  liquid 
extending  to  infinity  and 
having  the  tangent  plane  for 
its  lower  face,  for  the  por- 
tions of  liquid  which  have 
to  be  added  to  the  volume 
ADEF  to  make  up  this  slab 
are   at   a    greater    distance  Fig.  141. 

from  P  than   c,  and  so  do 

not  exert  any  attraction  on  matter  at  P.  Thus,  if  AP  =  z,  the  attraction 
of  AFDE  on  unit  mass  at  P,  using  the  previous  notation,  is  a-^{z) ;  the 

attraction  of  the  lenticular  portion  at  P  can  be  shown  to  be— ;/'(^)  where 

E  is  the  radius  of  the  liquid  sphere.  Hence  the  total  force  at  P  acting 
on  unit  mass  in  the  direction  AP  \^  equal  to 


'^^(^)  +  g'^(^) 


(3) 


Consider  now  the  equilibrium  of  a  thin  cylinder  of  the  fluid,  the  axis  of 
the  cylinder  being  PA  ;  divide  this  cylinder  up  into  thin  discs,  then  if  dz 
is  the  thickness  of  a  disc,  z  its  distance  from  A  and  a  the  area  of  the  cross- 
section  of  the  cylinder,  the  force  acting  on  this  disc  is  equal  to 


|<rV(.)-.^^(.)|a(^« 


This  force  has  to  be  balanced  by  the  excess  of  pressure  on  the  lower  face 
of  the  disc  over  that  on  the  upper  face ;  this  excess  of  pressure  is,  if  p 

represents  the  pressure,  equal  to    a—dz; 

dz 

hence,  equating  this  to  the  force  acting  on  the  disc,  we  get 

f^=^^M^)+a--^-^{z) 

Thus  the  excesa_of  pressure  at  a  point  at  a  distance  c,  below  A  over  the 
pressure  at^  is  equal  to 


/  cr^^{z)dz  +   /  (r~\l>{z)dz 


2T 
or  with  our  previous  notation         K  +  ^  . 


178 


PROPERTIES  OF  MATTER 


The  pressure  has  the  same  value  at  all  points  whose  depth  below  the 
surface  is  greater  than  c.  The  term  2T/R  represents  the  excess  of 
pressure  due  to  the  curvatui'e  of  the  surface  ;  we  obtained  the  same  value 

by  a  difTerent  process  on 
p.  145.  If  the  surface  of 
the  liquid  sphere  had  been 
concave  instead  of  convex, 
an  inspection  of  the  figure 
shows  that  to  obtain  the 
force  on  P  we  should 
have  to  subtract  the  attrac- 
tion due  to  the  lenticular 
portion  from  the  attrac- 
tion due  to  the  portion 
ADE  instead  of  adding 
it ;  this  would  make  the 
pressure  at  a  point  in  the 
mass  of  the  fluid  less  than 
that  at  a  point  in  the 
fluid  but  close  to  the 
surface  by  2T/R. 

Thickness  at  which 
the  Surface  -  tension 

Chang'eS. — We  can  determine  the  point  at  which  the  surface-tension 
begins  to  change  by  finding  the  change  of  pressure  which  takes  place  as 
we  cross  a  thin  film.  Let  Fig.  143  represent  the  section  of  such  a  film, 
bounded  by  spheres ;  if  the  thickness  of  the  film  is  small,  the  radii  of  these 
spheres  may  be  taken  as  approximately  equal.  Let  P  be  a  point  in  the  film, 
ABP  a  line  at  right  angles  to  both  surfaces,  then  the  investigation  just 
given  shows  that  if  AP  =  z,  BP  =  z',  the  force  on  unit  mass  at  P  is  equal  to 

when  R  is  the  radius  of  one  of  the  films.  We  see,  too,  from  the  last  paragraph 
that  the  pressure  at  B  must  be  greater  than  that  at  A  by 

t  t  t 

f{amz) + qm]dz  -  J  WW)  -  i^w)W = ~fzuz)dz 


¥lG.  142. 


where  t  is  equal  to  A  B,  the  thickness  of  the  film.  Hence,  from  the  for- 
mula (p.  145)  for  the  difference  of  pressure  inside  and  outside  a  soap 
bubble,  we  may  regard 


^^ 


-2  J  ^W)dz 


as  the_^urface-tension  of  a  film  of  thickness  t.  Since  yl{z^  vanishes  when 
g_is  greater  than  c,  the  suija^e-tension_win_reach  a  constant_value  when  t 
Is  as  great  as  c ;  hence  c.  the  rangeofmolccular  action.  ITtLe  thickness  "of  a 


LAPLACE'S  THEORY  OF  CAPILLARITY.  179 

film  when  the  surface-tension  begins  to  fall  off.     When  t  is  less  than  c  we 
eee  from  the  preceding  expression  that,  T  being  the  surface-tension, 


4 


J  Now  if  T  is  represented  by  a  curve  like  Fig.  134,  cTVldt  is  zero  down  to  P, 
y^  positive  from  P  to  R,  negative  from  R  to  T,  and  positive  again  for  all 

^>  thinner  films;   hence,  since  the   force  of   a  slab  is  attractive  when  ^  is 

positive,   repulsive  when   ;//  is  negative,   this  would   imply,  on  Laplace's 

theory,  that  the  molecular  forces  due  to  a  slab  of  lifxiud  at  a  point  outside 
_  y^ ,  Q^re  atjirst  attrdctioii&4.  then,  as  the  p oint  gets  near ei^ie  slab,  tliey  change 
C    to  repulsions,  and  change  again  " 

to  attractions  as  the  point  ap- 

pi-oaches  still  nearer  to  the  slab. 

If  t  is  so  small  that  ^^{t)  can  be 

regarded    as    constant,    we    see 
jjL^f^ZcX^  that  T  will  vary  as  {-,  so  that 

ultimately    the    surface  -  tension 

will  diminish  very  rapidly  as  the 

film  gets  thinner. 

On  the  Effect  of  the  Ab- 
ruptness of  Transition  be-  fig.  143 
tween  two   Liquids  on  the 

Surface-tension  of  their  Interface. — Laplace  assumed  that  the  range  I 
of  molecular  forces  was  the  same  for  all  bodies,  and  that  at  equal  distances 
the  force  was  proportional  to  the  density  of  the  substance.  This  implies 
that  the  function  \p{z)  is  the  same  for  all  bodies.  This  hypothesis  is 
certainly  not  general  enough  to  cover  all  the  facts;  it  is  probably,', 
however,  sufficiently  general  to  give  the  broad  outlines  of  capillary 
phenomena.  Let  us  calculate  on  this  hypothesis  the  surface-tension 
between  two  fluids  A  and  13.  Let  o-j  cr^  be  the  densities  of  these  fluids; 
then  to  separate  a  sphere  whose  area  is  S  from  the  liquid  A  requires  the 
expenditure  of  work  equal  to 


|S 


00 
cr,^  /  z\p{z)dz  {see  p.  175) 


Let  us  make  a  spherical  hole  of  equal  size  in  B.     To  do  this  will  require 
the  expenditure  of  an  amount  of  work  equal  to 


iSff//  z-^{z)dz 


Let  us  place  the  sphere  A  in  the  hole  in  B,  and  let  the  fluids  come  into 
contact  imder  their  molecular  forces ;  during  this  process  the  amount  of 
woi'k  done  by  these  forces  is 


09 


\ 


180  PROPERTIES  OF  MATTER. 

Hence  the  total  expenditure  of  work  required  to  produce  an  area  S  of 
interface  of  A  and  B  is 

CO  OO  00 

iS,7,-  fz^|.{z)dz  +  ^S<t/  fz-^{z)dz  -  8a,(T,  jzi{z)dz 

0  0  0 

OO 

^m^,-a,yjz-^{z)dz 
o 

But  tins  work  is  by  definition  equal  to  T^uS  where  T^n  is  the  surface- 
tension  between  A  and  B ;  hence  we  see  that  T^b  =  (^x  -  o-^)"^*  where 

CO 

C  =  ^  fz^z)dz 

is  a  constant  for  all  substances.  This  result  is  not  a  complete  representa- 
tion of  the  surface-tension,  for  if  it  were  there  would  always  be  surface- 
tension  between  liquids  of  different  densities,  so  that  two  such  liquids 
could  not  mix ;  it  would  also  require  that  the  surface-tension  between 
fluids  of  equal  density  should  be  zero,  and  that 


V  Tab  =  V  Tac  +  v  T( 


CB 


where  Tab,  Tao  and  Tcb  are  respectively  the  surface -tensions  between  fluids 
A  and  B,  A  and  C,  and  B  and  C  respectively.  None  of  these  results  are  in 
accordance  with  experiment.  Let  us,  however,  on  the  assumption  that  the 
surface-tension  is  represented  by  an  expression  of  this  kind,  calculate 
(following  Lord  RayJeigh)  the  effect  of  making  the  transition  between 
A  and  B  more  gradual ;  we  can  do  this  by  supposing  that  we  have  between 
A  and  B  a  layer  of  a  third  fluid  C  whose  density  is  the  arithmetical  mean 
between  the  densities  of  A  and  E;  then  Tac  =  i  'i\B  =  TcB-  Hence,  though 
now  wc  have  two  surfaces  of  sepai'ation  instead  of  one,  the  energy  per  unit 
area  of  each  is  only  one  quarter  of  that  of  unit  area  of  the  original  suiface ; 
hence  the  total  energy  due  to  surface-tension  is  only  one  half  of  the 
energy  when  the  traiisition  was  more  abrupt.  By  making  the  transition 
Betwesfi  A  and  B  still  Uiore'^  gradual  by  interposing  «  liquids  whose 
densities  are  in  arithmetical  progress,  we  reduce  the  energy  due  to  surface- 
tension  to  y(n+  1)  of  its  original  value.  Thus  we  conclude  that  any  dimi- 
nution in  tlie  abruptness  will  diminish  the  energy  due  to  surface  tension. 
This  result  may  have  important  bearings  on  the  nature  of  chemical  action 
between  the  surface  layers  of  liquids  in  contact,  for  if  a  layer  of  a  chemical 
compound  of  A  and  B  were  interposed  between  A  and  B  the  transition 
between  A  and  B  Avould  be  less  abrupt  than  if  they  were  directly  in  contact, 
and  therefore  the  potential  energy,  as  far  as  it  results  from  surface-tension, 
would  be  less.  Chemical  combination  between  A  and  B  would  result  in  a 
diminution  of  this  potential  energy.  Now  anything  that  tends  to  increase 
the  diminution  in  potential  energy  resulting  from  the  chemical  combina- 
tion promotes  the  combination;  the  forces  that  give  rise  to  surface- 
tension  would,  therefore,  tend  to  promote  the  chemical  combination.  Thus, 
in  the  chemical  combination  between  thin  layers  of  liquid  there  is  a  factor 
present  which  is  absent  or  insignificant  in  the  case  of  liquids  in  bulk,  and 


LAPLACE'S  THEORY  OF  CAPILLARITY.  181 

we  may  expect  that  chemical  combination  between  thin  layers  of  liquids 
might  take  place  even  though  it  were  absent  in  ordinary  cases, 

Similar  considerations  would  lead  us  to  expect  changes  in  the  strength  of  a 
solution  near  the  surface  whenever  the  surface-tension  of  the  solution  depends 
upon  its  strength :  if  the  surface-tension  increased  with  the  strength  there 
would  be  a  tendency  for  the  salt  to  leave  the  surface  layers,  while  if  the 
surface-tension  diminished  as  the  strength  of  the  solution  increased  the 
salt  would  tend  to  get  to  the  surface,  so  that  the  surface  layers  would  be 
stronger  solutions  than  the  bulk  of  the  liquid.  The  concentration  or 
dilution  of  the  surface  layers  would  go  on  until  the  gradient  of  the 
osmotic  pressures  resulting  from  the  variation  in  the  strengtlis  of  different 
layers  is  so  great  that  the  tendency  to  make  the  pressure  equal  just 
balances  the  efiects  due  to  surface-teiision. 


CHAPTER  XVI. 
DIFFUSION   OF  LIQUIDS. 

Contents. — Gen6l-al  Law  of  Diffusion — Methods  of  determining^  the  Co-efficictit  of 
Diffusion — Difl'iision  through  Membranes.  Osmosis — Osmotic  Pressure^ Vapour 
Pressure  of  a  (Solution — Elevation  of  the  Boilirg-jioint  of  Solutions — Depression 
of  the  Freezing-point— Dissociation  of  Electrolytes. 


If  two  liquids  are  left  in  contact  with  each  other  and  are  free  from  the 
action  of  external  forces,  then  if  they  can  mix  in  any  proportion  they  will 
of  themselves  go  on  mixing  until  the  whole  mass  is  uniform  in  composi- 
tion. This  process  may  be  illustrated  by  taking  a  vertical  glass  tube  and 
filling  the  lower  part  with  a  strong  solution  of  a  coloured  salt,  such  as 
copper  sulphate.  On  the  top  of  this  clear  water  is  poured  very  slowly 
and  carefully,  so  as  not  to  give  rise  to  any  currents  in  the  liquid.  The 
coloured  part  will  at  first  be  separated  from  the  clear  by  a  shai"ply  marked 
surface,  but  if  the  vessel  is  left  to  itself  it  will  be  found  that  the  upper 
part  will  become  coloured,  the  colour  getting  fainter  towards  the  top, 

while  the  colour  in  the  lower  part  of  the 
tube  will  become  fainter  than  it  was  origin- 
ally. This  change  in  colour  will  go  on  until 
ultimately  the  whole  of  the  tube  is  of  a 
uniform  colour.  There  is  thus  a  gradual 
transference  of  the  salt  from  the  ^places 
Avhere  the  solution  is  strong  to  those  where 
it  is  weak  and  of  water  in  the  opposite 
directiqUj^and  eqixilibrium  is  not  attained 
untiljthe^strength  of  the  solution  is  uniform. 
This  process  is  called  difl'usion.  In  liquids 
it  is  an  exceedingly  slow  process.  Thus,  if 
the  tube  containing  the  copper  sulphate 
solution  were  a  metre  long  and  the  lower 
half  were  filled  with  the  solution,  the  upper 
half  with  pure  Avater,  it  would  take  con- 
siderably more  than  ten  years  before  the 
mixture  became  approximately  uniform  ;  if  the  height  of  the  tube  were  a 
centimetre,  it  would  take  about  ten  hours,  th^_time  required  being 
proportional  to  the  square  of  the  lengtli^f  the  tube. 

The  first~systcmatic  experTmeiffcs  on  diffusion  were  made  by  Graham  in 
1851.  The  method  he  used  was  to  take  a  wide-necked  bottle,  such  as  is 
Bhown  in  Fig.  144,  and  fill  it  to  within  a  short  distance  of  the  top  with 
the  salt  solution  to  be  examined;  the  bottle  was  then  carefully  filled  up' 
with  pure  water  pi-essed  from  a  sponge  on  to  a  disc  of  cork  floating  on  the 
top  of  the  solution ;  the  bottle  was  placed  in  a  larger  vessel  filled  with 
pure  water  to  about  an  inch  above  the  top  of  the  bottle.  This  was  left 
undistui'bed  for  several  days,  and  then  the  amount  of  salt  which  had 
escaped  from  the  battle  into  the  outer  vessel  was  determined.  Graham 
was  in  this  way  able  to  show  that  solutions^of   the  same  stx'cugth  of 


» 


Fio.  Ut. 


DIFFUSION  OF  LIQUIDS. 


183 


di5erent_substajicesj[iflus^^        diQerent  velocities ;  that  solutions  of_tIie  j.'i^Jt 
same  salt  of  dUerent  stre^tE£_dittuse'a"wItLveIocities  proportionalto  the  3.  l.*tf 


8tj;engthX~thatjthe~rate  ofdiffusion  increased  with  the  temperatureTand  -^  ■  /vt 
that  the^-oportion^ftvvo  salts  iji  admixture  was  altered  by  diffusionj  and  *^  (o^J 
that  in  some  cases  a  decomposition  or  separation  of  the  constituents  of 
complicated  salts,  such  as  bisulphate  of  potash  and  potash  alum,  could  bo 
brought  about  by  diflfusion.  Though  Graham's  experiments  proved  many 
important  and  interesting  properties  of  diffusion,  they  did  not  lead  to 
sufficiently  definite  laws  to  enable  us  to  calculate  the  state  of  a  mixture  at 
any  future  time  from  its  state  at  the  present  time.  This  step  was  made 
by  Fick,  who,  guided  by  Fourier's  law  of  the  conduction  of  heat — the 
diffusion  of  temperature — enunciated  in  1855  the  law  of  difl'usion,  which 


3  4 

Ce)'cU.inet3'es 

Fig.  145. 


6  T  r'    1 


has  been  abundantly  verified  by  subsequent  experiments.  Fick^sjaw  may 
be  stated  as  follows  :  Imagine  a  mixture  of  salt  and  water  arranged  so  that 
layers  of  equal  density  are  horizontal.  Let  the  state  of  the^mixture 
be  such  that^n  the  layer  at^_a  height  ^^^jibiaveA^  fixed  plane  there  are 
n  grammes  of  salt  per  cubic  centimetre;  then^  ajsross  unit  area  of^Jjiis 

plane  R— -  grammes  of  salt  will  pass  in  unit  time  from  the  side  on  which  i 

the  soluFion  is  strongerjtojthat_on  which  it  is  weaker.  R  is^called  the 
SlfiusiyityjofjEHe^subSaiice ;  it  depends  on  the  nature  of  the  salt  and  the 
solvent,  on  the  temperature,  and  to  a  slight  extent  on  the  strength  of 
the  solution!  This  law  is  analogous  to  Fourier's  law  of  the  conduction  of 
heat,  and  the  same  mathematical  methods  which  give  the  solution  of  the 
thermal  problems  can  be  applied  to  determine  the  distribution  of  salt 
through  the  liquid.  The  curves  in  Figs.  145  and  146  represent  the  solution 
of  two  impoi'tant  problems.  The  first  represents  the  diffusion  of  salt  from 
a  saturated  solution  into  a  vertical  column  of  water,  the  surface  of  separa- 
tion being  initially  the  plane  x  =  o.  The  ordinates  represent  the  amount 
of  salt  in  the  solution  at  a  distance  from  the  original  surface  of  separation 
represented  by  the  abscissse.  The  times  which  have  elapsed  since  the 
commencement  of  diffusion  are  proportional  to  the  squares  of  the  numbers 


184 


PROPERTIES  OF  MATTER. 


on  the  gurve ;  thus^  if  the  first  curve  represents  the  state  of  things  after 
time  T,  the  second  represente  it  after  a  time  2-^,  the  third  after  a  time 
o-'T,  and  so  on ;  for  the  same  ordinate  the  abscissa  on  curve  2  is  twice 
that  on  curve  1,  on  curve  3  three  times  that  on  curve  1,  and  so  on;  thus 
the_time  required  for  diffusion  through  a  given  length  is  proportional  to 
the  square    of  the  ^        ^^' 

Collected  Papers,  vol. 
T  =  25,700  seconds,  for 
chlorK 


length 


The  curves   are   copied  from  Lord  Kelvin's 
111.   p.   432 :    for   copper   sulphate,  through    water 


e  througli  water 


sugar 


through 


w'ater  T=  17,100,  and  for  soclium 


T  =  5390.     The  second  figure.  Fig.  146,  represents 


the  difl'usion  when  we  have  initially  a  thin  layer  of  ealt  solution  at  the  bottom 
of  a  vertical  vessel,  the  rest  of  the  vessel  being  filled  with  pure  water ;  the 
ordinates  represent  the  amount  of  salt  at  a  distance  from  the  bottom  of 
the  vessel  represented  by  the  abscissae.      The  times  which  have  elapsed 

since  the  commencement  are 
proportional  to  the  squares  of 
the  numbers  on  the  curves. 

By  stirring  up  a  solution 
of  a  salt  Avith  pure  water  we 
bring  thin  layers  of  the  solvent 
and  of  the  salt  near  together ; 
as  the  time  required  for  diffus- 
ing through  a  given  distance 
varies  as   the   square  of   the 
distance,   the    time    required  \ 
for    the    salt    and    water    to 
become  a  uniform  mixture  is 
greatly  diminished  by  drawing 
out  the  licjuid  into  these  thin  ( 
layers    by    stirring,    and    as  f 
much  diftusion  will  take  mueE  M^<"^^ 
in   a   fe^w^econds    as    would    ) 


1  IG.   ii6. 


take  place  in  asmany  hours 
withoujfc^thejnixing.  We  can  see  in  a  general  way  why  the  time  required 
will  be  proportioiial  to  the  square  of  the  thickness  of  the  layers ;  for  if  we 
halve  the  thickness  of  the  layers  we  not  only  halve  the  distance  the  salt 
has  to  travel  but  we  double  the  gradient  of  the  strength  of  the  solution, 
and  thus  by  Fick's  law  double  the  speed  of  diffusion  ;  thus,  as  we  halve 
the  distance  and  double  the  speed,  the  time  required  is  reduced  to  one 
quarter  of  its  original  value. 

Methods  of  Determining;  the  Coefficient  of  Diffusion.— If  we 

know  the  original  distribution  of  the  salt  through  the  water  and  the  value  of 
li,  we  can,  by  Fourier's  mathematical  methods,  calculate  the  distribution  of 
salt  after  any  interval  T  ;  conversely,  if  we  know  the  distribution  after  this 
interval,  we  can  use  the  Fourier  result  to  determine  the  value  of  R. 
Thus,  if  we  have  any  means  of  measuring  the  amount  of  salt  in  the 
different  parts  of  the  solution  at  successive  intervals,  we  can  deduce  the 
value  of  R.  It  is  not  advisable  to  withdraw  a  sample  from  the  solution 
and  then  determine  its  composition,  as  the  withdi-awal  of  the  .'^ample 
might  produce  currents  in  the  liquids  whose  effects  might  far  outweigli  any 
due  to  pure  diffusion  ;  it  is,  therefore,  necessary;  to  sample  the  composition 
of  the  solution  when  vi  situ,  and  this  has  been  done  by  measuring  some 
physical  property  of  the  solution  yhich^-aries  in_aJkiiown  way  with  the 


DIFFUSION  OF  LIQUIDS. 


185 


strength  of  the  solution.  In  Lord  Kelvin's  method  the  specific  gravity  is 
the  pi-operty  investigated  :  theTower  halFoTaTvertical  vessel  is  filled  with 
a  solution,  the  upper  half  with  pure  water .  C  lass  beads  of  difl'erent  densities 
are  placed  in  the  solution  ;  at  first  they  float  at  the  junction  of  the  solution 
and  the  water,  but  as  diflfusion  goes  on  they  separate  out,  the  heavier  one? 
sink  and  the  lighter  ones  rise.  By  noting  the  position  of  the  beads  ol 
known  density  we  can  get  the  distribution  of  salt  in  the  solution,  and 
thence  deduce  the  value  of  R.  The  objection  to  the  method  is  that  air 
bubbles  are  apt  to  form  on  the  beads  when  salt  will  crystallise  out  on  them, 
and  thus  alter  their  buoyancy.  In  the  case  of  sugar  solutions  the  strength 
ot  the  diHerent  layers  can  be  determined  by  the  rotation  of  the  £lane  of 
polarisation^  H.  F.  Weber  verified  Fick's  law  in  the  case  of  zinc  sulphate 
solution  b^jneasuring  the  electromotivejforce^ietween  two^malgamated 


jr^            ~^ 

\                        / 

If                                   // 

Fio,  147. 

zinc^plateg,;  he  had  previously  determined  how  the  electromotive  force 
depends  on  the  strength  of  the  solutions  in  contact  with  the  plates.  The 
diffusion^of  differen^salts  was  compared  byj^ong  ( Wied.  Ann.  9,  p.  Gl  3)  by  the 
metliod  shown  in  Fig.  147.  A  stream  of  pure  water  flows  through  the  bent 
tube,  a  wide  tube  fastened  on  to  the  bent  tube  establishes  commvinication  with 
the  solution  in  the  beaker  ;  after  the  water  has  flowed  through  the  bent  tubts 
for  some  time  the  amount  of  salt  it  carries  over  in  a  given  time  becomes 
constant.  As  the  water  in  the  tube  is  continually  being  renewed,  while  the 
strength  of  the  solution  inthe  beaker  may  be  regarded  as  constant,  since  in 
the  experiments  only  a  very  small  fraction  of  the  salt  is  carried  over,  the 
gradient  of  concentration  in  the  ueck  will  be  proportional  to  the  strength  of 
the  solution;  so  that  the  amount  of  salt  carried  ofl'  by  the  stream  of  water 
in  unit  ~time  is  proportionaj^o  the  product  of  the  diffuslvi:ty:_aml,J/he 
.strength  of  the  .solutionr  By  measuring  the  amount  of  salt  carried  over  by 
the  stream  in  unit  time  the  diffusivities  of  difl'erent  salts  can  be  compared. 
As  a  re.-ult  of  these  experiments  it  has  been  found  that  as  a  general  rule 
the_Jn^herjUie_el^cUjc^^  oL  a^  solution  of  a  ^alt  Jhe  more 

i^pidly  does  the  salt  diffuse.  The  relative  values  of  the  diffusivity  for  some 
of  the  commoneF  salts  and  acids  are  given  in  the  table  on  p.  18G.  The 
solutions  contain  the  same  number  of  gramme  equivalents  per  litre,  and 
the  numbers  in  the  table  aie  proportional  to  the  number  of  molecules  of 
the  salt  which  cross  unit  surface  in  unit  time  under  the  same  gradient  of 
strength  of  solution. 


186 


PROPERTIES  OF  MATTER. 


cjiibstance. 

Substance. 

KCl 

803 

KI 

NH^Cl 

G89 

Nal 

:NaCl      . 

GOO 

NH^NO, 

LiCl 

541 

KNO3      . 

KCy 

707 

JSTaNOj    . 

BaCl, 

450 

LiNOj     . 

SnCi; 

432 

BaN,Oc  . 

CaCl, 

429 

SrNA    • 

MsCl,     . 

392 

(NHJ^SC), 

COCl, 

300 

Na.,SO,    . 

NiCl, 

304 

MgSO,    . 

KBr 

811 

ZnSO,     . 

NH^Br 

029 

CuSO,     . 

NaBr 

509 

MnSO,    . 

'T?«(:a-t-«'t 

Va  1  u  p  s 

^l-y.ffas 

.,U 

f     (5ai«e    lYo.    ^"1- 

H.'jjl/ts      ^v 

These  niimbei 

s  show 

that 

as  a  general  rule  the  salts  w 

bich  diffus 

\^    , 


823 

072 
680 
007 
524 
512 
050 
552 
724 
078 
348 
332 
316 
298 

the  most 

rapidly  are  those  whose  solutions  have  the  highest  electrical  conductivity. 
The  absolute  values  of  the  diffusivity  for  a  large  number  of  substances  have 
been  determined  by  Schuhmeister  {Wien.  Akacl.  79,  p.  003)  and  Scheffer 
{Chem.  Ber.  xv.  p.  788,  xvi.  p.  1903).  The  largest  value  of  the  diffusivity 
found  by  Scheffer  was  for  nitric  acid  ;  the  diffusivity  varied  with  the 
concentration  and  with  the  temperature;  for  very  dilute  solutions  at 
90°  0.  it  was  2  x  10j^^_{cm.)7sec. — i.e.,  if  the  strength  of  solution  varied  by 
one  per  c«nt.  in  1  cm.  the  amount  of  acid  crossing  unit  area  in  one  second 
would  be  about  one  five-millionths  of  the  acid  in  1  c.c.  of  the  solution. 
For  solutions  of  NaCl  the  diffusivity  was  only  about  one  half  of  this  value. 
Graham  found  that  the  velocity  of  diffusion  of  ISTaCl  through  gelatine  was 
about  the  same  as  through  water. 

Diffusion  throug^h  Membranes.    Osmosis.— Graham  was  led  by 

his  experiments  on  diffusion  to  divide  substances  into  two  cla.sses — crystal- 
loid and  colloid.  The  crystalloids,  which  include  mineral  acids  andTalts, 
and^which  as  a  rule  canbe~obtame<L  in  defimte^crystallihe  forms,  ^fluse 


much  moi'e>apidtythan  the  substances  called  by  Graham  collo^ds^udLAS 

are  amorphous  and^ow  no  signs 


starch,  glass,  which 


|0 


If 


the  gumsTaTbumen,  _^_ ^ 

of  crystallisation^  The^x-ystalloids  when  dissolved  m  water  change  in  a 
marked  degree  its  properties :  for  example,  they  dimihisli  tlie  vapour 
pressure,  lower  the  freezing-  and  raise  the  boiling-point.  Colloidal  sub- 
staiices,jwJaen_^issolye4jiIJva^^  any  effects^pf  this  kind, 

in  fact,  many^coIIoHlarsolutions  seem~tobeirtHe  more  than  mechanical 
mixtures,  the  colloid  in  a  very  finely  divided  state  being  suspended  in  the 
fluid.  The  properties  of  solutions  of  this  class  are  very  interesting ;  the 
particles  move  in  the  electric  field,  in  some  cases  as  if  they  were  positively, 
in  others  as  if  they  were  negatively,  charged.  The  addition  of  a  trace  of 
acid  or  alkali  is  often  sufficient  to  produce  precipitation.  The  reader  will 
find  an  account  of  the  properties  of  these  solutions  in  papers  by  Picton 
and  Linder  {Journal  of  Chemical  Society,  vol.  70,  p.  508,  1897  ;  vol.  01, 
p.  148,  1892);  Stoeckl  and  Vanino  {Zeitschrift  f.  Phys.  Chem.,  vol.  30, 
p.  98,  1899)  ;  Hardy  (Proceedings  of  Poyal  Society,  00,  p.  110  ;  Journal  of 
Physiology,  24,  p.  288).  Colloidal  substances  when  mixed  with  not  too 
much  water  form  jellies ;  the  structure  of  these  jellies  is  sometimes  on  a 
sufficiently  coarse  scale  to  be  visible  under  the   microscope   (see  Hardy, 


DIFFUSION  OF  LIQUIDS. 


187 


Proceedings  Royal  Society,  6G,  p.  95,  1900),  and  apparently  consists  of  a 

more  or  less  solid  framework  tlirough   which    the    liquid   is  dispersed. 

Through   many  of    these  jellies   crystalloids  are  able  to  difi'use  with    a 

velocity  approaching  that  through  pure  water ;  the  colloids,  on  the  other 

hand,  are  stopped  by  such  jellies.    Graham  founded  on  this^  method  for  the 

separation  of  crystalloids  and  colloidsj^called  dialy.sis.  In  this  method  a  film  of 

a    colloidal    substance,   such  as   parchment   paper 

(paper  treated  with  sulphuric  acid)  or  a  piece  of 

bladder  is  fastened  round  the  end  of  a  glass  tube, 

the  lower  end  of  the  tube  dipping  in  water  which  is 

frequently  changed,  and  the  solution  of  cryfetalloids 

and  colloids  is  put  in  the  tube  above  the  parchment 

paper.     The  crystalloids  difluse  through  into  the 

water,  and  the  colloids  remain  behind ;  if  time  be 

given  and  the  water  into   which  the   crystalloids 

■diffuse  be  kept  fresh,  the  crystalloids  can  be  entirely 

separated  from  the  colloids. 

The  passage  of  liquids_through  films  of  this 
kind_ia_jcallei  osmosis.  The  first  example  of  it 
seems  to  have  been  observed  by  the  Abbe  Nollet, 
in  1748,  who  found  that  wdien  a  bladder  full  of 
alcohol  was  immersed  in  water,  the  water  entered 
the  bladder  more  rapidly  than  the  alcohol  escaped, 
so  that  the  bladder  swelled  out  and  almost  burst. 
If,  on  the  other  hand,  a  bladder  containing  water 
was  placed  in  alcohol  the  bladder  shrank. 

The  motion  of  fluids  throvigh  these  membranes 
can  be  observed  with  very  simple  apparatus :  all 
that  is  necessary  is  to  attach  a  piece  of  parchment- 
paper  firmly  on  the  end  of  a  glass  tube,  the  upper 
portion  of  which  is  drawn  out  into  a  fine  capillary 
tube.  If  this  tube  is  filled  with  a  solution  of  sugar 
and  immersed  in  pure  water,  the  top  of  the  liquid 
in  the  capillary  part  of  the  tube  moves  upwards 
with  sensible  velocity,  showing  the  entrance  of 
water  through  the  parchment-paper.  Graham 
regarded  this  transport  of  water  throughT  the 
membrane  as  due  to  this  colloidal  substance^being 
able  to  hold  more  wateFln^ombination  when  in 
contact  with  pure  waterjthan  when  in^contsct  wjtb 
ar~salt  solution ;  thus,  when  the  hydration  of  the 
membrane^orresponding  to  the  side  next  the  water 
extends  to  the  side  next  the  solution,  the  membrane  Fio.  148. 

cannot  hold  all  the  Avater  in  combination,  and  some 

of  it  is  given  up ;  in  this  way  water  is  transported  from  one  side  of  the 
membrane  to  the  other. 

Membranes  of  parchment-paper  or  bladder  are  permeable  by  crysta-lloids 
as  well  jis  by  wateri  There  are  otTier  memBranes,  however,  whicETwhile 
permeable  to  water  are  impermeable  to  a  large  number  of  salts ;  these 
membranes  are  called  semi-permeable  membi-anes.  One  of  these,  which 
has  been  extensively  used,  is  the^elatinous  precipitate  of  ferrocyanide  of 
copper,  which   is  produced  when  copper  sulphate  and  potassium  ferro- 


E^ 


188 


PROPERTIES  OF  MATTER. 


cyanide  come  into  contact.  This  jrccipitate  is  mechanically  exceedingly 
Weak,  but  Pfeffer  made  serviceable  membranes  by  precipitating  it  in  the 
pores  of  a  poi-ous  pot.  If  such  a  pot  is  filled  with  a  very  dilute  solution 
of  copper  sulphate  and  immersed  in  one  of  ferrocyanide  of  potassium  the 
two  solutions  will  diffuse  into  the  walls  of  the  pot,  and  where  they  meet 
the  gelatinous  precipitate  of  ferrocyanide  of  copper  will  be  formed ;  in  this 
way  a  continuous  membrane  may  be  obtained.  For  details  as  to  the  pre- 
cautions which  must  be  taken  in  the  preparation  of  these  membranes  the 
reader  is  referred  to  a  paper  by  Adie  {Froceedings  of  Chemical  Society/, 
lix.  p.  344).  If  a  membrane  of  this  kind  be  deposited  in  a  porous  pot 
fitted  with  a  pressure  gauge,  as  in  Fig.  148,  and  the  pot  be  filled  with  a 
dilute  solution  of  a  salt  and  immersed  in  pure  water,  water  will  flow  into 

the  pot  and  compress  the  air  in  the  gauge,  the 
pi-essure  in  the  pot  increasing  until  a  definite 
pressure  is  reached  depending  on  the  streugth 
of  the  solution.  When  this  pressure  is' 
I'eached  thei-e  is  equilibrium,  and  there  is  no 
further  increase  in  the  volume  of  water  in- 
the  pot. 

Osmotic  Pressure. — Thus  the  flow  of 

water  through  the  membrane  into  the 
stronger  solution  can  be  prevented  by  apply- 
ing to  the  solution  a  definite  pressure ;  this 
pressure  is  called  the  osmotic  pressure  of  the 
solution.  It  is  a  quantity  of  fundamental 
importance  in  considering  the  properties  of 
the  solution,  as  many  of  these  properties, 
such  as  the  diminution  in  the  vapour  pres- 
sure, and  the  lowering  of  the  freezing-point, 
are  determinate  as  soon  as  the  osmotic 
pressure  is  known. 

The  work  done  when  a  volume  v  of 
water  passes  across  a  semi  permeable  mem- 
brane from  pure  water  into  a  solution  where 
the  osmotic  pressure  is  P  is  equal  to  Pv. 
For,  let  the  solution  be  enclosed  in  a  vertical 
■Water  tube  closed  at  the  bottom  by  a  semi-permeable 
membrane,  then  when  there  is  equilibrium 
the  solution  is  at  such  a  height  in  the  tube 
that  the_pressure^^aiLjthe_inembrane  due  to 
the  head  of  the  solution  is  equal  to  the 
osmotic  pressure.  When  the  system  is  m 
equilibrium  we  know  by  Mechanics  tMt  the  total  work  done  during  any 
small  alteration  of  the  system  must  be  zero.  Let  this  alteration  consist  in 
a  volume  v  of  water  going  through  the  semipermeable  membrane.  This 
will  raise  the  level  of  the  solution,  and  the  work  done  against  gravity  is 
the  same  as  if  a  volume  v  of  the  solution  were  raised  from  the  level  of  the 
membrane  to  that  of  the  top  of  the  liquid  in  the  tube.  Thus  the  work  done 
against  gravity  is  vgph,  where  h  is  the  height  of  the  solution  in  the  tube 
and  p  the  density  of  the  solution ;  since  the  pressui^e  due  to  the  head  of 
solution  is  equal  to  the  osmotic  pressure,  gph  =  F.  Hence  the  work  done 
against  gravity  by  this  alteration  is  Pv,  and  since  the  total  work  done 


'-: 


■ScUUtfifl 


Membrane 


Fio   149. 


DIFFUSION  OF  LIQUIDS. 


189 


must  be  zero,  the  work  done  on  the  liquid  when  it  crosses  the  membrane 
must  be  Py. 

The  values  of  the  osmotic  pressures  for  different  solutions  was  first 
determined  by  Pfefler,*  who  found  the  very  remarkable  result  that  for 
weak  solutions  Avhich  do^  not  conduct  electricity  the  osmotic  pressure  is 
equal  to  the  gaseous  pressure  which  would  be  exerted  by  the  molecules  of 
the  salt  if  these  were  jnJ^the'gasebusYtate  and  occupying  a  volume  equal 
to_that  of  the  solvent  in  which  the  salt  is  dissolved.  Thus,  if  1  gramme 
equivalent  of  the  salt  were  dissolved  in  a  litre  of  water  the  osmotic  pressure 
would  be  about  2'2  atmospheres,  which  is  the  pressure  exerted  by  2 
grammes  of  hydrogen  occupying  a  litre.  Pfeffer's  experiments  showed 
that  approximately,  at  any  rate,  the  osmotic  pressure_wa8,  like  the^pregsure 


of  a  gas,  proportional  to  the  absolute  temperature.  If  the  cell  is  placed  in 
another  solution  instead  of  pure  water,  water  will  tend  to  run  into  the  cell 
if  the  osmotic  pressure  of  the  solution  in  the  cell  is  greater  than  that  of 
the  solution  in  which  it  is  immersed,  while  if  the  osmotic  pressure  in  the 
cell  is  less  than  that  outside  the  volume  of  water  in  the  cell  will  decrease ; 
if  the  osmotic  pressure  is  the  same  inside  and  outside  there  will  be  no 
change  in  the  volume  of  the  water  inside  the  cell.  Solutions  which  have 
the  same  osmotic  pressure  are  called 
method 


of 


pressure  are  called  isotonic  solutions.      A  convenient 
finH^ing  the  strengtEs  of  solutTons  or  dillerent  salts  which  are 

He  showed  that  the  membrane  lining 


4. 

I 


isotonic  was  invented  by  De  Vries 
the  cell-wall  of  the  leaves  of  some  plants,  such  as  Tradescantia  discolor, 
Curcuma  ruhricaidis,  and  Begonia  manicata,  is  a  semi- permeable  membrane, 
being  permeable  to  water  but  not  to  salts,  or  at  any  rate  not  to  many 
salts.  The  contents  of  the  cells  contain  salts,  and  so  have  a  definite  osmotic 
pressure.  If  tb.ese  cells  are  placed  in  a  solution  having  a  greater  osmotic 
pressure  than  their  own,  water  will  run  fi-om  the  cells  into  the  solution, 
the  cells  will  shrink  and  will  present  the  appearance  shown  in  Fig.  1 50  b. 
Fig.  150  a  shows  the  appearance  of  the  cells  when  surrounded  by  water; 
the  weakest  solution  which  produces  a  detachment  of  the  cell  will  be 
approximately  isotonic  with  the  contents  of  the  cell.     In  this  way  a  series 

*  PfefTer,  Osmotischc  Untcrsuchungcn,  Leipzig,  1877. 
f  De  Vries,  Zcit.  f.  Physik.  Chemie,  ii.  p   415 


190 


PROPERTIES  OF  MATTER. 


WaXer  vapour. 


of  solutions  can  be  prepared  which  are  isotonic  with  each  other.  De  Vrie» 
found  that  for  noi\-electroly tes  isotonic  solutions  contained  in  each  unit  of 
volume  a  weight  ofjhe  salt  proportional  to  the  molecular^weight ;  in  other 
wor3s,  thatjsptpnic^solutions  of  non-electrolytes  contain  the  same  number 
of  molecules  of  the  salt.  ThisTs^iiother  instance  of  the  analogy  between 
osmotic  pressure  an^  gaseous  pressure,  for  it  is  exactly  analogous  ta 
Avogadro's  law,  that  when  the  gaseous  pressures  are  the  same  all  gasea 
at  the  same  tempei'ature  contain  the  same  number  of  molecules  per  unit 
volume.  Although  the  direct  measurements  on  osmotic  pressure  hitherto 
made  may  seem  a  somewhat  slight  base  for  the  establishment  of  such  an 
important  conception,  an  immense  amount  of  experimental  woik  has  been 
done  in  the  investigation  of  such  phenomena  as  the  lowering  of  the  vapour 
pressure,  the  raising  of  the  boiling-  and  the  lowering  of  the  freezing-point 
produced  by  the  solution  of  salts  in  water.  The  conception  of  osmotic 
pressm*e  enables  us  to  calculate  the  magnitude  of  these  effects  from  the 

strength   of  the  solution ; 
M^nxhra^  ^j^g  agreement  between  the 

values  thus  calculated  and 
the  values  observed  is  so 
\faier  '■apour  closB  ES  to  furnish  strong 
evidence  of  the  truth  of 
this  conception. 

Vapoup  Pressure  of 

a  Solution.— The  change 
in  the  vapour  pressure  due 
to  the  presence  of  salt  in 
the  solution  can  be  calcu- 
lated by  the  following 
method  due  to  Van  t' 
Hoff:  Suppose  the  salt 
solution  A,  Fig.  151,  is 
divided  from  the  pure  water  5  by  a  semi-permeable  membrane — i.e.,  one 
which  is  permeable  by  water  and  not  by  the  salt ;  transfer  a  small 
quantity  of  water  whose  volume  is  v  from  ^1  to  B  by  moving  the 
membrane  from  right  to  left.  If  IT  is  the  osmotic  pressure  of  the  solu- 
tion the  work  required  to  effect  this  transference  is  Hv  ;  now  let  a  volume 
V  of  water  evaporate  from  B  and  pass  as  vapour  through  the  membrane  into 
the  chamber  ^1  and  there  condense.  If  Y  is  the  volume  of  the  water  vapour, 
Ip  the  excess  of  the  vapour  pressure  of  the  water  over  B  above  that  over  A^ 
the  work  done  in  this  process  is  cpY.  The  process  is  clearly  a  reversible  one, 
and  hence  by  the  Second  Law  of  Thermodynamics,  since  the  temperatures 
of  the  two  chambers  are  the  same,  there  can  be  no  loss  or  gain  of  mechanical 
work.  Thus,  since  the  work  spent  in  one  part  of  the  cycle  must  be  equal 
to  that  gained  in  the  other,  we  have 


Solution, 


WaXcr 


'  Memhrano 

Fia.  151. 


Suppose  p  is  the  vapour  pressure  over  the  water,  let  V  be  the  volume 
occupied  at  atmospheric  pressure  IT,,  by  the  quantity  of  water  vapour  which 
at  the  pressure  p  occupies  the  volume  V ;  then  by  Boyle's  Law, 

n„v'=pV 


DIFFUSION  OF  LIQUIDS. 


191 


80  that 


but  for  water  vapour  vfV'  =  1/1200,  hence 

lp_  n     1 


p     Uo  1200 


WfXter     vcLpotKr 
cuui.    atr 


SoUxti^n,^ 


Water    vapour 
oTUi     air 


Wojt^r 


The  osmotic  pressure  in  a  solution  of  1  gramme  equivalent  per  litre  ,    ,  c^ 

of  a  salt  which  Joes  not  dissociate  when  dissolved  is  about  '22  atmospheres  ; 
thus  for  such  a  solution 

8p_   22 
p     12U0 

or  the  vapour  pressure  over  the  solution  is  nearly  2  per  cent,  less  than 
over  pure  wat^  ,^ 

IftEe  surface  of  the  solution  is  subjected  to  a  pressure  equal  to 
the  osmotic  pressure  the  vapour  pressure  over  the^  solution  will  increase 
and^  wilT  be  equal  to  the 
pressure  over  pure  water. 
For  let  Fig.  152  represent  a 
vessel  divided  by  a  dia- 
phragm permeable  only  by 
water  and  by  water  vapour, 
and  let  the  salt  solution  in 
A  be  subject  to  a  pressure 
equal  to  the  osmotic  pres- 
sure. Under  this  pressure 
the  liquids  will  be  in  equi- 
librium,  and   there  will  be 

no  flow  of  water  across  the  diaphragm.  If  the  vapour  pressure  of  the 
water  is  greater  than  that  of  the  salt  solution,  then  water  vapour  from  B 
will  go  across  the  diaphragm  and  will  condense  on  A  •  this  will  make 
the  solution  in  A  weaker  and  reduce  the  osmotic  pressure.  Since  the 
external  pressure  on  A  is  now  greater  than  its  osmotic  pressure,  water 
will  flow  from  ^  to  ^  across  the  diaphragm ;  thus  there  would  be  a 
continual  circulation  of  water  round  the  system,  which  would  never  be 
in  equiUbrium.  As  this  is  inadmissible,  we  conclude  that  the  vapour 
pressure  of  the  water  is  not  greater  than  that  of  the  solution ;  similarly  if 
it  were  less  we  could  show  that  there  would  be  a  continual  circulation  in 
the  opposite  direction ;  in  this  wa}'^  we  can  show  that  the  vapour  pressure 
of  the  solution  when  exposed  to  the  osmotic  pressure  is  equal  to  that  of 
pure  water.  This  is  an  example  of  the  theorem  proved  in  J.  J.  Thomson's 
Applications  of  Bi/namics  to  Physics  and  Chemistry,  p.  171  (see  also  Poyn- 
ting,  Phil.  Mag. ,'sii.  p.  39),  that  if  a  pressure  of  n  atmosphei^es  be  apphed 
to  the  surface  of  a  liquid  the  vapour  pressure  of  the  liquid,  p,  is  increased 
by  Ip,  where 


Memhrcuna 


Fig.  152. 


hp 
V 


n 


density  of  the  vapour  at  atmospheric  pressure 
density  of  the  liquid 


Raising"  of  the  Boiling-point  of  Solutions.— The  determina- 
tion of  the  vapour  pressure  is  attended  with  considerable  difficulty,  and  it 


192 


PROPERTIES  OF  MATTER. 


B 


Solution, 


Water 


is  much  easier  to  measure  the  effect  of  salt  on  the  boiling-  or  freezing-point 
of  the  solution. 

Let  A  and  B  be  vessels  containing  respectively  salt  solution  and  pure 
water,  separated  by  a  semi-permeable  membrane,  and  let  the  temperatures 
of  the  vessels  be  such  that  the  vapour  pressure  over  the  solution  is  the 
same  as  that  over  pure  water.     Let  0  be  the  absolute  temperature  of  the 

water,  d  +  cd  that  of  the  solu- 
tion. Now  suppose  a  volume 
■y  of  water  flows  from  B  to  A 
across  the  diaphragm  ;  if  11  is 
the  osmotic  pressure  of  the 
solution,  mechanical  work  llu 
will  be  done  in  this  operation. 
Let  this  quantity  of  water  be 
evaporated  from  A  and  pass 
through  the  walls  of  the 
diaphragm  and  condense  in 
B.  As  the  vapour  pressui-es 
are  the  same  in  the  two 
cases,  no  mechanical  work  is 
gained  or  spent  in  this  opera- 
tion. The  system  is  now  in 
its  original  state,  and  the  operation  is  evidently  a  reversible  one,  so  that 
we  can  apply  the  Second  Law  of  Thermodynamics.  Now  by  that  law  we 
have 

Heat  taken  from  the  boiler  Heat  given  up  in  the  refrigerator 

Absolute  temperature  of  boiler     Absolute  temperature  of  refrigerator 

Mechanical  work  done  by  the  engine 
Difierence  of  the  temperatures  of  boiler  and  refrigerator. 

In  our  case  the  mechanical  work  done  is  Uv.  The  heat  given  up  in  the 
refrigerator  is  the  heat  given  out  when  a  volume  v  of  water  condenses 
fiom  steam  at  a  temperature  d;  if  X  is  the  heat  given  out  when  unit  mass 
of  steam  condenses  and  a  the  density  of  the  liquid,  the  heat  given  out  in 
the  refrigerator  is  "Kav  ;  hence  by  the  Second  Law  we  have 


Fio.  153. 


Xo-y     TIv 
—  = —    or 

9        od 


dd 
d 


n* 
x^ 


Let  us  apply  this  to  find  the  change  in  the  boiling-point  produced  by 
dissolving  1  gramme  equivalent  of  a  salt  in  a  litre  of  water ;  here  II 
is  22  atmospheres,  or  in  C.G.S.  units  22  x  10^  X  is  the  latent  heat  of 
steam  in  mechanical  units — i.e,  536  x  4-2  x  10',  <t  is  unity,  and  0  =  373; 


hence 


m  = 


373x22xl0« 


53Gx4-2x  10^ 


=  '37  of  a  degree. 


The  experiments  of  Raoult  and  others  on  the  raising  of  the  boiling- 
point   of  solutions  of  organic  salts  which  do  not  dissociate  have  shown 

*  The  heat  given  out  or  taken  in  by  the  volume  of  water  when  going  from  one 
chamber  to  the  other  is  negligible  in  comparison  with  that  required  to  vaporise  the 
water 


DIFFUSION  OF  LIQUIDS. 


193 


( 


that  the  amount  of  the  rise  in  the  boiling-point  is  almost  exactly  '37  of 
a  degree  for  each  gramme  equivalent  per  litre,  a  result  which  is  strong 
confirmation  of  the  truth  of  the  theory  of  osmotic  piessure. 

Lowering-  of  the  Freezingrpoint  of  Solutions.— A  similar  in- 
vestigation enables  us  to  calculate  the  depression  of  the  freezing-point 
due  to  the  addition  of  salt.  Let  yl,  B  (Fig.  15-1)  represent  two  vessels 
separated  by  a  semi-permeable  membrane,  A  containing  the  salt  solution 
at  its  freezing-point  and  B  pure  water  at  its  freezing-point.  Let  a  volume 
V  of  water  pass  across  the  seuii- permeable  membrane  from  B  to  A  ;  if  11  is 
the  osmotic  pressure  of  the  solution,  mechanical  work  \lv  will  be  gained  by 
this  process.  Let  this  quantity  of  water»be  frozen  in  A,  the  ice  produced 
taken  from  A  placed  in  1>,  and  there  melted.  The  system  has  now  returned 
to  its  original  condition,  and  the  process  is  plainly  reversible  ;  hence  we  can 


Mertxhrtxnjt 


Va/er   vof)  0  ur_  ^ 


SoluXicn, 


_  Water   vapou^ 


WcUer 


Fig.  154. 


apply  the  Second  Law  of  Thermodynamics.  If  d  is  the  absolute  tempera- 
ture of  the  freezing-point  of  pure  water,  6  -  cO  that  of  the  freezing-point 
of  the  solution,  if  X  is  the  latent  heat  of  water,  and  o-  its  density ;  the 
heat  taken  from  the  hot  chamber  B  at  the  temperature  6  is  Xcxv ;  hence 
by  the  Second  Law  we  have 

Thus  in  the  case  of  water  for  which  0  =  273,  X  =  80  x  4-2  x  10^  a=l  and 
when  the  strength  of  the  solution  is  1  gramme  equivalent  per  litre, 

n  =  22xl0'';  hence  30  =  1-79°. 

This  has  been  verified  by  Raoult  in  the  case  of  solutions  of  organic 
salts  and  acids.  The  result  of  the  comparison  of  theory  with  experi- 
ment for  a  variety  of  solvents  is  shown  in  the  following  table : 


Solvent 

Acetic  acid     .         . 
Formic  acid    . 
Benzene 
Nitro-benzene 
Ethylene-dibromide 


Lowering  of  freezing  point  for  organic  salts, 
1  gramme  molecule  dissolved  in  a  litre 

Observed  Calculated 


3-9 
2-8 
4-9 
7-05 
11-7 


3 
2 
5 
6 
11 


'88 

•8 

•1 

■9 

•9 

N 


194  PROPERTIES  OF  MATTER. 

Dissociation  of  Electrolytes.— The  preceding  theory  gives  % 
satisfactory  account  of  the  efl'ecb  upon  the  boiling-  and  freezing-points 
produced  by  organic  salts  and  acids  when  the  osmotic  pressure  is 
calculated  on  the  assumption  that  it  is  equal  to  the  gaseous  pressure 
which  Avould  be  produced  by  the  same  weight  of  the  salt  if  it  were 
gasified  and  confined  in  a  volume  equal  to  that  of  the  solvent.  When, 
however,  mineral  salts  or  acids  are  dissolved  in  water,  the  eflect  on  the 
boiling-  and  freezing-points  produced  by  oi  gramme  equivalents  per  litre  is 
greater  than  that  produced  by  the  same  number  of  gramme  equivalents  of 
an  organic  salt,  although  if  the  osmotic  pressure  were  given  by  the  same 
rule,  the  effects  on  the  freezing-  and  boiling-points  ought  to  be  the  same 
in  the  two  cases.  The  osmotic  pressure  then  in  a  solution  of  a  mineral 
salt  or  acid  is  greater  than  in  one  of  equivalent  strength  (i.e.,  one 
for  which  n  is  the  same)  of  an  organic  salt  or  acid ;  this  has  been 
verified  by  direct  measurement  of  the  osmotic  pressure  by  the  methods 
of  Pfefler  and  De  Vries.  This  increase  in  the  osmotic  pressure  ia 
explained  by  Arrhenius  as  being  due  to  a  partial  dissociation  of  the 
molecules  of  the  salts  into  their  constitutents ;  thus  some  of  the 
molecules  of  KaCi  are  supposed  to  split  up  into  separate  atoms  of 
Na  and  01.  Since  by  this  dissociation  the  number  of  individual 
particles  in  unit  volume  is  increased,  the  osmotic  pressure,  if  it  follows 
the  law  of  gaseous  pressure,  Avill  also  be  increased.  According  to 
Arrhenius,  the  atoms  of  Na  and  01  into  which  the  molecule  of  the  salt 
is  split  are  charged  respectively  with  positive  and  negative  electricity, 
which,  as  they  move  under  electric  forces,  will  make  the  solution  a 
conductor  of  electricity.  In  this  way  he  accounts  for  the  fact  that 
those  solutions  in  which  the  osmotic  pressure   is    abnormally  large   are 

I  conductors  of  electricity,  and  that,  as  a  rule,  the  greater  the  conduc- 
tivity the  greater  the  excess  of  the  osmotic  pressure.  This  view,  of 
v,'hich  an  account  will  be  given  in  the  volume  on  Electricity,  has  been 
very  successful  in  connecting  the  various  properties  of  solutions. 

Though  the  osmotic  pressure  plays  such  an  important  part  in  the 
theory  of  solution,  there  is  no  generally  accepted  view  of  the  Avay  in  Avhich 
the  salt  produces  this  pressure.  One  view  is  that  the  salt  exists  in  the 
interstices  between  the  molecules  of  the  solvent  in  the  state  corresponding 
to  a  perfect  gas.  If  the  volume  of  these  interstices  bore  a  constant 
proportion  to  the  volume  of  the  solvent,  then,  whatever  this  ratio  may  be, 
we  should  get  the  ordinary  relation  between  the  quantity  of  salt  and 
the  osmotic  pressure  to  which  it  gives  rise.  For,  suppose  p  is  the 
pressure  of  the  gaseous  salt,  v  the  volume  of  the  interstices,  V  the  volume 
of  the  solvent ;  then  if  a  semi-permeable  membrane  be  pushed  so  that  a 
volume  hV  of  water  passes  through  it,  and  11  is  the  osmotic  pressure, 
then  the  work  done  is  n5V;  but  if  ^v  is  the  diminution  in  the  volume 
I  of  the  interstices,  the  work  done  is  j)lv ;  hence 

UhY=pdv 

But  if  the  volume  occupied  by  the  intei'stices  bears  a  constant  ratio  to 
that  of  the  solvent 

V      V 

where  V  is  the  volume  of  the  solvent ;  hence 


DIFFUSION  OF  LIQUIDS.  196 

nV=;;y    or    II -^J; 

that  is,  the  osmotic  pressure  is  the  same  as  if  the  gaseous  salts  occupied 
the  whole  volume  of  the  solvent. 

Another  view  {see  Poynting,  JPhil.  Mag.  42,  p.  289)  is  that  the 
phenomenon  known  as  osmotic  pressure  arises  from  the  molecules  of  salt 
clinging  to  the  molecules  of  the  water,  and  so  diminishing  the  mobility  and 
therefore  the  rate  of  diflusion  of  the  latter.  Thus,  suppose  we  have  pure 
water  and  a  salt  solution  separated  by  a  semi-permeable  membrane,  since  the 
water  molecules  in  the  solution  are  clogged  by  the  salt  they  will  not  be  able 
to  pass  across  the  membrane  as  quickly  as  those  from  the  pure  water,  and 
there  will  be  a  flow  of  water  across  the  membrane  from  the  pure  water 
to  the  solution.  Poynting  shows  that  the  mobility  of  the  molecules  of 
a  liquid  is  increased  by  pressure,  so  that  by  applying  a  proper  pressure 
to  the  solution  we  may  make  the  mobility  of  the  molecules  of  water  in 
it  the  same  as  those  of  the  pure  water,  and  in  this  case  there  will  be  no 
flow  across  the  membrane ;  the  pressure  required  is  the  osmotic  pressure. 
Poynting  shows  that  this  view  Avill  explain  the  properties  of  inorganic 
Baits  if  we  suppose  that  each  molecule  of  salt  can  completely  destroy  the 
mobility  cf  one  molecule  of  water. 


CHArTER  XVII. 
DIFFUSION  OF  GASES. 

Contents. — Co-eflicient  of  Diffusion — Diffusion  of  Vapours— Explanation  of  DilTu- 
sion  on  Kinetic  Tiieory  of  Gases — Effects  of  a  Terforatcd  Diaphrapra — I'a.-sage  of 
Gases  through  Porous  Bodies — Thermal  Effusion — Atmolvsis — Passage  of  (Jasea 
through  Indiarubber,  Liquids,  Hot  Metals — Diff'usion  of  iletals  through  Metal. 

Ip  a  mixture  of  two  gases  A  and  B  is  confined  in  a  vessel  the  gases 
will  mix  and  each  will  ultimately  be  uniformly  difl'used  through  tlie  vessel 
as  if  the  other  were  not  present.  If  they  are  not  uniformly  mixed  to 
begin  with,  there  will  be  a  flow  of  the  gas  A  from  the  places  where  the 
density  of  A  is  great  to  those  where  it  is  small.  The  law  of  this  diflfusion 
is  analogous  to  that  of  the  conduction  of  heat  or  to  the  difTusion  of  liquids 
and  may  be  expressed  mathematically  as  follows :  Suppose  the  two  gases 
are  arranged  so  that  the  layers  of  equal  density  are  horizontal  planes,  and 
let  p  be  the  density  of  A  at  a  height  x  above  a  fixed  horizontal  plane ;  then 
in  unit  time  the  mass  of  A  which  passes  downward  through  unit  area  of  a 
horizontal  plane  at  a  height  x  is  proportional  to  the  gradient  of  p  and  is 

equal  to  K-^  where  K  is  the  interdifFusity  of  the  gases  A  and  B.     The 
^  dx ^- ■ 

value  of  K  has  been   measui^ed   by  Loschmidt*  and  Obermayerf  for  a 

considerable  number  of  pairs  of  gases.     The  method  employed  by  these 

observers  was  to  take  a  long  vertical  cylinder  separated  into  two  parts  by  a 

disc  in  the  middle.     The  lower  half  of  the  cylinder  was  filled  with  the 

heavier  gas,  the  upper  half  with  the  lighter.     The  disc  was  then  removed 

with  great  care  so  as  not  to  set  up  air  cm-rents,  and  the  gases  were  then 

allowed  to  diffuse  into  each  other ;  after  the  lap^e  of  a  certain  time  the 

disc  was  replaced  and  the  amount  of  the  heavier  gas  in  the  upper  half  of 

the  cylinder  determined.     From  this  the  value  of   K  was  determined  on 

the   assumption  (which  is  probably  only  approximately  true)   that    the 

value  of  K  does  not  change  when  the  pi-oportions  of  the  two  gases  ai-e 

altered.     WaitzJ  used  a  diffei-ent  method  to  determine  the  coefficient  of 

interdifiusion  of  air  and  carbonic  acid ;  beginning  with  the  carbonic  acid 

below  the  air  he  measured  by  means  of  Jamin's  interference  refractometer 

the  refractive  index  of  various  layers  after  the  lapse  of  definite  intervals  of 

time  ;  from  the  refractive  index  he  could  calculate  the  proportion  of  air  and 

carbonic  acid  gas,  and  was  thus  able  to  follow  the  course  of  the  diffusion. 

iHe  found  that  the  coefficient  of  diffusion  depended  to  some  extent  on  the 

jproportion  between  the  two  gases,  the  values  of  K  at  atmospheric  pressure 

fat  0°  C.  varying  between  •1288  and  'ISCG  cm. -/sec.     The  values  found  by 

Loschmidt  and  v.  Obermayer  are  given  in  the  following  table.     They  are 

for  76  cm.  pressure  and  0°  C: 

•  Loschmidt,  Wien.  Berichte,  61,  p.  367,  1870,  62,  p.  463,  1870. 

J  Obermayer,  Wien.  Berichte,  81,  p.  162,  1880. 
Wailz,  Wiedemann  t  Annalai,  17,  p.  201,  1882. 


DIFFUSION  OF  GASES. 


197 


Gases. 


CO, 

-N,0 

CO. 

-CO. 

CO, 

-0,  . 

CO, 

-  Air 

CO, 

-CH, 

CO, 

-H, 

CO, 

-C,H, 

CO- 

-0. . 

CO- 

-H,  . 

CO 

-C.,fT, 

so, 

-H,  . 

0,- 

H,    . 

0.- 

N,    . 

0,- 

Air   . 

H,- 

■Air  . 

H,- 

■CH, 

H,- 

■N,0 

H.- 

-O.H, 

LOSCUMIDT. 

VON  Obkhmaybk. 

K  cm.7sec. 

Kcm.'^sec. 

•09831 

.     •09166 

•14055 

.     -13142 

•14095 

.     -13569 

•14231 

.     -13433 

•15856 

,     -14650 

•55585 

•53409 

•10061 

•18022 

.     -18717 

•G4223 

•64884 

•11639 

•48278 

•721 G7 

•66550 

-17875 

.     -17778 

•63405 

•62544 

•53473 

—            , 

-45933 

—            , 

-48627 

H,-0,H, 

We  may,  perhaps,  gain  some  idea  of  the  rapidity  of  diffusion  by  saying 
that  the  rate  of  equalisation  in  composition  of  a  mixture  of  hydrogen  and 
air  is  about  half  that  of  the  equalisation  of  temperature  in  copper. 

As  an  example  of  the  rate  at  which  diffusion  goes  on  we  may  quote  the 
result  of  an  experiment  by  Graham  on  the  diffusion  of  CO^  into  air. 
Carbonic  acid  was  poured  into  a  vertical  cylinder  57  cm.  high  until  it  filled 
one- tenth  of  the  cylinder.  The  upper  nine-tenths  of  the  vessel  was 
filled  with  air  and  the  gases  were  left  to  diffuse.  They  were  found  to  be 
very  approximately  uniformly  distributed  throughout  the  cylinder  after 
the  lapse  of  about  two  hours.  As  the  time  taken  to  reach  a  state  of 
approximately  uniform  distribution  is  proportional  to  the  square  of  the 
length  of  the  cylinder,  if  the  cylinder  were  only  one  centimetre  long 
approximately  uniform  distribution  would  be  attained  after  the  lapse  of 
about  two  seconds. 

The  intei'diffusity  is  inversely  proportional  to  the  pressure  of  the 
mixed  gas ;  it  increases  with  the  temperature^  According  to  the'  experi- 
ments of  Losclimidt  and  v.  Obermayer  it  is  proportional  to  0"  where  0  is 
the  absolute  temperature  and  n  a  quantity  which  for  different  pairs  of 
gases  varies  between  1^75  and  2. 

Diffusion  of  Vapours. — The  case  when  one  of  the  diffusing  gases 
is  the  vapour  of  a  liquid  is  of  special  importance,  as  it  is  on  the  rate 
of  diffusion  that  the  rate  of  evaporation  depends.  The  methods  which 
have  been  employed  to  measure  the  rate  of  diffusion  of  the  vapour  of  a 
liquid  consist  essentially  in  having  some  of  the  liquid  at  the  bottom  of  a 
cylindrical  tube  and  directing  a  blast  of  vapour-free  gas  across  the  mouth 
of  the  tube.  When  the  blast  has  been  blowing  for  some  time  a  uniform 
gradient  of  the  density  of  the  vapour  is  established  in  the  tube^  the  value 
of  this  is  hjl  where  h  is  the  maximum  vapour  pressure  of  the  liquid  at  the 
temperature  of  the  experiment  and  I  the  distance  of  the  surface  of  the 
liquid  from  the  mouth  of  the  tube.     The  mass  of  vapour  which  in  unit 


198  PROPERTIES  OF  MATTER. 

time  flows  out  of  the  tube — {i.e.,  the  amount  of  the  liquid  which  evaporates 
in  unit  time  and  which  can  therefore  be  easily  measured),  is  K.S/l  where  K 
is  the  difTusivity  of  the  vapour  into  the  gas;  as  ^  is  known  we  can  readily 
determine  K  by  this  method.  A  few  of  the  results  of  experiments  made 
by  Stefan*  and  Winkelmannt  are  given  in  the  following  table : 


Valuk  of  K  in 

cm.7sec.  AT 

0°C. 

AND 

760  mm. 

Pressube. 

Hj-drogen, 

Air, 

Carbonic  acid. 

Water- vapour 

.     •G87 

•198 

•131 

Ether 

.     -296 

•0775 

•0552 

Carbon-bisulphide 

.     -309 

•0883 

•0629 

Benzol 

.     ^294 

•0751 

•0527 

Methyl-alcohol     . 

.     -5001 

•1325 

•0880 

Ethyl-alcohol 

.     -3800 

•0994 

•0G93 

Explanation  of  Diffusion  on  the  Kinetic  Theory  of  Gases.— 

The  kinetic  theory  according  to  which  a  gas  consists  of  a  great  number  of 
individual  particles  called  molecules  in  rapid  motion,  affords  a  ready  ex- 
planation of  diffusion.  Siipjiose  Ave  have  two  layers  A  and  B  in  a  mixture 
of  gases  and  that  these  layers  are  separated  by  a  plane  0.  Let  there  be 
more  molecules  of  some  gas  y  in  A  than  in  B,  then  since  the  molecules  are 
in  motion  they  will  be  continually  crossing  the  plane  of  separation,  some 
going  from  A  to  B  and  some  from  B  to  A,  but  inasmuch  as  the  molecules 
of  7  in  A  are  more  numerous  than  those  in  B,  more  will  pass  from  A  to  B 
than  from  B  to  A.  Thus,  A  will  lose  and  B  gain  some  of  the  gas  y ;  this 
will  go  on  until  the  quantities  of  y  in  unit  volumes  of  the  layers  A  and  B 
are  equal,  when  as  many  molecules  will  pass  from  A  to  B  as  from  B  to  A, 
and  thus  the  equality,  when  once  established,  will  not  be  disturbed  by  the 
motion  of  the  molecules.  It  follows  from  the  kinetic  theory  of  gases 
(see  Boltzmann,  Vorlesungen  ilher  Gastheorie,  p.  91)  that,  if  there  ai^e  n 
molecules  of  y  in  unit  volume  of  B,  n  +  hi  in  a  unit  volume  of  A  at  a 
distance  2cc  from  that  in  B,  and  if  x  be  measured  at  right  angles  to  the 
plane  separating  the  layers,  then  the  excess  of  the  number  of  molecules 
of  y  which  go  across  unit  area  of  C  from  A  to  B  over  those  which  go  from 

A  to  B  is  equal  to  •3502Xc — ,  where  X  is  the  mean  free  path  of  the  molecules 

dx 

of  y  and  c,  their  average  velocity  of  translation  ;  the  quantity  Xc  is  evidently 
proportional  to  the  diffusity. 

Now  c  only  depends  upon  the  temperattire,  being  proportional  to  the 
square  root  of  the  absolute  temperature,  while  X  is  inversely  proportional 
to  the  densit}',  and  if  the  density  is  given  it  does  not,  at  least  if  the 
molecules  are  regarded  as  hard  elastic  spheres,  depend  upon  the  tempera- 
ture. If  the  pressure  is  given,  then  the  density  will  be  inversely,  and 
X  therefore  directly  pi'oportional  to  the  absolute  temperature.  Thus,  on 
this  theory  the  coeliicient  of  diflusion  should  vary  as  ei  where  0  is  the 
absolute  temperatute.  The  experiments  of  Loschmidt  and  von  Obermayer 
seem  to  show  that  it  varies  somewhat  more  rapidly  with  the  temperature. 

Another  method  of  regarding  the  process  of  diffusion,  which  for  some 
purposes  is^f  g'reat  utility,  is  as  follows :  The  diffusion  of  one  gas  A 
through  another  B  when  the  layers  of  equal  density  are  at  right  angles  to 

*  Stefan,  Wicn.  Alad.  Bcr.,  ef),  p.  823,  1872. 

t  Winkelmann,  ]Vkd.  Ann.,  22,  pp.  1  aud  152,  1884. 


DIFFUSION  OF  GASES.  19f) 

the  axis  x  may  be  regarded  as  due  to  a  current  of  the  gas  A  moving 
parallel  to  the  axis  of  x  with  a  certain  velocity  ic  through  a  current  of  B 
streaming  with  the  velocity  v  in  the  opposite  dii^ection.  To  move  a  current 
of  one  gas  through  another  requires  the  application  of  a  force  to  one  gas  in 
one  direction  and  an  equal  force  to  the  other  gas  in  the  opposite  direction. 
This  force  will  be  proportional  (1)  to  the  relative  velocity  ti  +  v  of  the  two 
currents,  (2)  to  the  number  of  molecules  of  A  per  unit  volume,  and  (3)  to 
that  of  the  molecules  of  B.  Let  it  then  per  unit  volume  of  gas  be  equal 
to  Aj3  p^p^  (u  +  v),  where  Aj^  is  a  quantity  depending  on  the  nature  of  the 
gases  A  and  B,  but  not  upon  their  densities  nor  upon  the  velocity  with  which 
they  are  streaming  through  each  other ;  pj  and  p^  are  respectively  the 
densities  of  the  gases  A  and  B — i.e.,  their  masses  per  unit  volume.  Hence, 
to  sustain  the  motion  of  the  gases  a  force  Ajg  p^  p^  (u  +  v)  parallel  to  x  must 
act  on  each  unit  of  volume  of  A  and  an  equal  force  in  the  opposite 
direction  on  each  unit  volume  of  B.  These  forces  may  arise  in  two  ways; 
there  may  be  external  forces  acting  on  the  gases,  and  there  may  also  be 
forces  arising  from  variations  in  the  partial  pressures  due  to  the  two 
gases.  Let  Xj,  X^  be  the  external  forces  per  unit  mass  acting  on  the  gases 
A  and  B  respectively,  and  Pi,p.^  the  partial  pressui-es  of  the  gases  A  and  B 
respectively.  Considering  the  forces  acting  parallel  to  x  on  unit  volume 
of  A,  the  external  force  is  Xjp,,  and  the  force  due  to  the  variation  of  the 
partial  pressure  is  -  dp^jdx ;  hence  the  total  force  is  equal  to  —  dpjdx  +  Xjpj, 
and  as  this  is  the  force  driving  A  through  B  we  have 


"Tfi  +  ^iPi  =  ^i2Pi/'2(^  +  ^)  0) 


dx 
similarly,  _  J^4.X_,p2=  -  A„p,p,(w  +  v)  (2) 

Let  us  consider  the  case  when  there  are  no  external  forces  and  when 
the  total  pressui'e  p^  +2^2  ^^  constant  throughout  the  vessel  in  which 
difiusion  is  taking  place.  In  this  case  the  number  of  molecules  of  A 
which  cross  unit  ai-ea  in  unit  time  must  equal  the  number  of  molecules  of 
B  which  cross  the  same  area  in  the  same  time  in  the  opposite  direction. 
Let  this  number  be  q ;  then  if  n^  n^  are  respectively  the  numbei'3  of 
molecules  of  A  and  B  per  unit  volume, 

q  =  n^u  =  n^v 
If  wij,  mj  are  the  masses  of  the  molecules  of  A  and  B  respectively 

hence  A^^p^p.^ii  +  v)  =  A  ^^m^m.^n^  +  n.^q 

Now  n^  +  n^  is  proportional  to  the  total  pressure,  and  aa  this  is 
constant  throughout  the  volume,  ?i,  +  n^  will  be  constant.  Putting  X  =  0  in 
equation  (1)  and  writing  N  for  n^  +  n^,  we  get 

1  dp. 


9=  - 


Aypi^m^  dx 


Now  ^3  =  1^ 


n. 


200  PROPERTIES  OF  MATTER. 

where  v„  is  the  number  of  molecules  of  a  gas  in  unit  volume  at  a  standard 
pressure  2^0  i 

hence  (7  =  -  - — ^ -^ 

Now  q  is  the  number  of  molecules  of  A  pnssinc:  unit  suiface  in  unit 
time  and  dvjdx  is  the  gradient  of  the  number  per  unit  volume  ;  hence, 
from  the  definition  of  K,  the  interdifiusity,  given  on  p.  lUC,  we  see 

or  if  P  is  the  total  pressure 

K=        '        /M' 


m^n.^K^\no 


w  0 


Thus,  if  A, 3  is  constant,  K  varies  inversely  as  P,  and  directly  as  {pjn^''. 
Since  the  pressure  of  a  given  number  of  molecules  per  unit  volume  is 
proportional  to  the  absolute  temperature,  K,  if  Aj^  is  constant,  varies 
directly-  as  the  square  of  the  absolute  temperature. 

We  can  determine  A,2  if  we  know  the  velocity  acquired  by  one  of  the 
gases  A  when  acted  upon  by  a  known  force.  Suppose  that  the  gas  A  is 
uniformly  distributed,  so  that  dp^jdx  =  0,  and  that  when  acted  upon  by  a 
known  force  it  moves  through  B  with  a  velocity  ?t ;  svxppose,  too,  that  B  is 
very  largely  in  excess  and  is  not  acted  upon  by  the  force,  we  have  then  v 
very  small  compared  with  u,  and  from  equation  (1)  we  have 

A   -^ 

Thup,  if  we  know  ?«,  the  velocity  acquired  under  a  known  force  X,  we  can 
find  A,,,  and  hence  K,  the  diffusivity.  This  result  is  of  great  importance 
in  the  theory  of  the  difiusion  of  ions  in  electrolytes,  and  Nernst  haa 
developed  an  electi^olytic  theory  of  diffusion  in  fluids  on  this  basis. 
Another  important  application  of  this  result  is  to  determine  X  from 
measurements  of  K  and  ic.  Thus,  to  take  an  example,  if  the  particles  of 
the  gas  A  are  charged  with  electricity  and  placed  in  an  electric  field  of 
known  strength,  the  force  X  will  depend  upon  the  charge  ;  hence,  if  in  this 
case  we  measui^e  (as  has  been  done  by  Townsend)  the  values  of  K  and  ic, 
we  can  deduce  the  value  of  X,  and  hence  the  charge  carried  by  the 
particles  of  A . 

On  the  Obstruction  offered  to  the  Diffusion  of  Gases  by  a 

perforated  Diaphrag'm. — if  a  perforated  diaphragm  is  placed  across  a 
cylinder  it  does  not  diminish  the  difi'usion  of  gases  in  the  cylinder  in  the 
ratio  of  the  area  of  the  openings  in  the  diaphragm  to  the  whole  area  of 
the  diaphragm,  but  in  a  much  smaller  degree,  for  the  eftect  of  the  per- 
foration is  to  make  the  gradient  in  the  density  of  the  gases  in  the  neigh- 
bourhood of  the  hole  greater  than  it  would  have  been  if  the  diaphragm 
had  been  removed,  and  therefore  the  flow  through  the  hole  greater  than 
through  an  equal  area  when  there  is  no  diaphragm.  Thus,  to  take  a  case 
investigated  by  Dr.  Horace  Brown  and  Mr.  Escombe  {Proceedings  Royal 
Society,  vol.  07,  p.  121),  suppose  we  have  OO3  in  a  cylinder,  and  place 
across  the  cylinder  a  disc   wet  with  a  solution   of  caustic   alkali   which 


DIFFUSION  OF  GASES.  201 

absorbs  the  CO^,  so  that  the  density  of  the  COj  next  the  disc  is  zero. 
Then  if  p  is  the  density  of  the  CO,  at  the  top  of  the  cylinder,  the  density 
gradient  is  p/l  where  I  is  the  distance  between  the  disc  and  the  top  of  the 
cylinder,  so  that  the  amount  of  00^  absorbed  by  unit  area  of  the  disc 
it  will  be^^g/Z^ where  k  is  the  diffusivity  of  CO,  through  itself.  Now  suppose, 
L."  insteadofa  disc  extending  completely  across  the  cylinder,  we  have  a  much 
smaller  disc  of  radius  a,  then  at  the  disc  the  density  of  the  CO^  will  be 
zero,  but  it  will  recover  its  normal  value  p  at  a  distance  from  the  disc 
proportional  to  a ;  thus  the  gradient  of  density  in  the  neighbourhood  of 
the  disc  will  be  of  the  order  p/a  and  not  p/l,  and  the  amount  of  CO, 
absorbed  by  the  disc  will  be  proportional  to  k  (p/a)  ttot — i.e..  will  be 
proportional  to  a;  so  that  the  absorption  of  the  CO,  will  only  diminish  as  I 
the  radius  of  the  disc  and  not  as  the  area.  This  was  verified  by  Brown 
and  Escombe,  and  it  has  very  important  applications  to  the  passage  of 
gases  thi-Qugh  the  openings  in  the  leaves  of  plants. 

Passag'e  of  Gases  through  Porous  Bodies.— There  are  ^hr^ 

^ocesses_by^ jvhicli  gas  may^jass  through  a^olid__perforated  by  a  series  of 
holesor  canals ;  the  size  of  the  holes  or  pores  determining  the  method  by 
which  the  gas  escapes.  If  the  plate  is  thin  and  the  pores  are  not 
exceedingly  fine,  the  gas  escapes  by  what  is  called  effusion  ;  this  is  the 
process  by  which  water  or  air  escapes  from  a  vessel  in  which  a  hole  is 
bored.  The  rate  of  escape  is  given  by  Torricelli's  theorem,  so  that  the  Y^  t  ^ 
velocity  with  which  a  gas  streams  through  an  aperture  into  a  vacuum  is  ^ 

proportional  to  the  square  root  of  the  quotient  of  the  pressure  of  the  gas 
by  its  density,  and  thus  for  different  gases  under  the  same  pressure  the 
velocity  will  vary  inversely  as  the  square  root  of  the  density  of  the  gas. 
Bunsen  foiinded  on  this  result  a  method  of  finding  the  density  of  gases.  oC^ 

This  case,  strictly  speaking,  is  not  one  of  diflusion  at  all,  but  merely  the 
flow  of  the  gas  as  a  whole  through  the  apeiLure.  If  the  gas  is  a  mixture 
of  different  gases  its  composition  will  i\ot  be  altered  when  the  gas  passes 
through  an  aperture  of  this  kind. 

The  second  method  is  the  one  which  occurs  when  the  holes  are  not  too 
fine,  and  when  the  thickness  of  the  plate  is  large  compared  with  the 
diameter  of  the  holes.  In  this  case  the  laws  are  the  same  as  when  a  gas 
flows  through  long  tubes;  they  depend  on^the  viscosity  of  the  gas,  and  are 
discussed  in  the  chapter  relating  to  that  property  of  bodies.  No  change 
in  the  composition  of  a  mixture  of  gases  is  produced  when  the  gases  are 
forced  through  apertures  of  this  kind  ;  this  is  again  a  motion  of  the  gas 
as  a  whole,  and  not  a  true  case  of  diflusion.  The  third  method  occurs 
when  the  pores  are  exceedingly  fine,  such  as  those  found  in  plates  of 
meerschaum,  stucco,  or  a  plate  of  graphite  prepared  by  squeezing  together 
powdered  graphite  until  it  forms  a  coherent  mass.  In  this  case,  when  we  [ 
have  a  mixture  of  two  gases,  each  finds  its  way  through  the  plate  / 
independently  of  the  other,  and  the  composition  of  the  mixture  is  in  I 
general  altered  by  the  passage  of  the  gas  through  the  plate.  The  laws  / 
governing  the  passage  of  gases  through  pores  of  this  kind  were  investi- 
gated by  Graham,  who  found  that  the  volume  of  the  gas  (estimated  at  a 
standard  pressure)  passing  through  a  porous  ^late  was  directly  propor- 
tional  to  the  difference  of  the  pressures  of  the  gas  on  the  two  sides,  and 
inversely  proportional  to  the^qiiare  jroot  of  f  he^molecular  weight  of  J.he 
gas^  Thus  for  the  same  diflerence  of  pressure  hydrogen  was  found  to 
escape  through  a  plate  of  compressed  graphite  at  four  times  the  rate  of 


^ 


202 


propi:rties  of  matter. 


oxygen.  Thus,  if  we  have  mixtures  of  equal  volumes  of  hydrogen  and  oxygen 
and  allow  them  to  pass  through  a  porous  diaphragm,  since  the  hydrogen 
gets  througli  at  four  times  the  rate  of  the  oxygen,  the  mixture,  after  pass- 
ing through  the  plate,  will  be  much  richer  in  hydrogen  than  in  oxygen. 
The  rate  of  diffusion  can  be  measured  by  an  instrument  of  the  following 
kind  (Fig.  155)  :  A  porous  plate  is  fastened  on  the  top  of  a  tube  which  can 
be  used  as  a  barometer  tube.  A  vessel  for  holding  the  gas  being  attached 
to  the  upper  part  of  the  tube,  this  and  the  space  above  the  mercury  are 
exhausted  ;  gas  at  a  definite  pressure  is  then  let  into  the  vessel,  and  the 
rate  at  which  it  passes  through  the  diaphragm  into  the  vacuum  over  the 

mercury  is  measured  by  the   rate   of 
depression  of  the  mercury  column. 

The  laws  of  diffusion  of  gases 
through  fine  pores  are  readily  explained 
by  the  Kinetic  Theory  of  Gases ;  for  if 
the  pores  are  so  fine  that  the  molecules 
pass  through  them  without  coming 
into  collision  with  other  molecules,  the 
rate  at  which  the  molecules  pass  through 
will  be  proportional  to  the  average 
velocity  of  translation  of  the  molecules. 
According  to  the  Kinetic  Theory  of 
Gases  this  average  velocity  is  inversely 
proportional  to  the  square  root  of  the 
molecular  weight  of  the  gas  and  directly 
proportional  to  the  square  root  of  the  ab- 
solute temperature.  Hence  at  a  given 
temperature  the  velocity  with  which 
the  gas  streams  through  the  apertures 
will  be  inversely  proportional  to  the 
square  root  of  the  molecular  weight ; 
this  is  the  result  discovered  by  Graham. 

Thermal  Effusion.  — The  same 

reasoning  will  explain  another  pheno- 
menon sometimes  called  thermal  effu- 
sion. Suppose  we  have  a  vessel  divided 
nto  two  portions  by  a  porous  diaphragm ;  let  the  pressures  in  the  two 
portions  be  equal  but  their  temperatures  different,  then  gas  will  stream 
from  the  cold  to  the  hot  part  of  the  vessel  through  the  diaphragm.  For 
since  the  pressures  are  equal  the  densities  in  the  two  paits  of  the  vessel 
are  inversely  proportional  to  the  absolute  temperatures  while  the  velocities 
are  directly  proportional  to  the  square  roots  of  the  absolute  temperatures. 
Hence  the  number  of  molecules  passing  from  the  gas  through  the 
diaphragm,  which  is  proportional  to  the  product  of  the  density  and  tho 
velocity,  will  be  inversely  proportional  to  the  square  root  of  tho  absolute 
temperature;  thus  more  gas  will  pass  from  the  cold  side  than  from  the 
hot,  and  there  will  be  a  stream  of  gas  from  the  cold  to  the  hot  portion 
throu<ih  the  dianhrafjm. 

Atmolysis. — The  diffusion  of  gases  through  porous  bodies  was  applied 
by  Graliam  to  produce  the  separation  of  a  mixtui^e  of  gases ;  this 
separation    was   called    by    him   atmolYsis,    and    to  efVect  it  he  used  an 


instrument  of  the  kind  shown  in  'Fiix.  15G.     A  long  tube  made  from  the 


Fig.  15». 


DIFFUSION  OF  GASES. 


203 


stems  of  clay  tobacco-pipes  is  fixed  by  means  of  corks  in  a  glass  or 
metal  tube.  A  glass  tube  is  inserted  in  one  of  the  end  corks,  and  is 
connected  with  an  air-pump  so  that  the  annular  space  between  the 
tobacco-pipes  and  the  outer  tube  can  be  exhausted.  The  mixed  gases 
whose  constituents  have  to  be  separated  is  made  to  flow  through  the  clay 
pipes.  Some  of  tlie  gases  escape  through  the  walls  and  can  be  pumped 
away  and  collected  while  the  rest  flows  on  through  the  tube.  In  the  gas 
which  passes  through  the  walls  of  the  tube  there  is  a  greater  proportion 
of  the  lighter  gas  than  there  was  in  the  mixture  originally,  while  in  the 
gas  which  flows    along  the  tube   there  is   a  greater  proportion  of   the 


Fig,  156. 


heavier  constituent.  If  the  constituents  of  the  mixture  differ  much  in 
density  a  considerable  separation  of  the  gases  may  be  produced  by  this 
arrangement. 

Passagre  of  a  Gas  through  India-rubber. — The  fact  that  gases 

can  pass  through  thin  india-rubber  Avas  discovered  in  1831  by  Mitchell,  who 
found  that  india-rubber  toy-balloons  collapsed  sooner  when  inflated  with 
carbonic  acid  than  with  hydrogen  or  air,  and  sooner  with  hydrogen  than 
air.  The  subject  was  investigated  by  Graham,  who  gave  the  following 
table  for  the  volumes  of  difierent  gases  which  pass  through  india-rubber 
in  the  same  time  : 


CO 
Air 
CH. 


1 

11 3 
1-149 

2-148 


0. 
CO, 


2-556 

5-5 

13-585 


The  speed  with  which  the  gases  pass  through  the  rubber  increases 
very  rapidly  with  its  temperature.  There  is  no  simple  relation  between 
these  volumes  and  the  densities  of  the  gas  as  there  is  in  the  case  of 
diffusion  through  a  porous  plate,  and  the  mechanism  by  which  the  gases 
effect  their  passage  is  pi'obably  quite  different  in  the  two  cases.  The 
passage  of  gases  through  rubber  seems  to  have  many  points  of  resem- 
blance to  the  passage  of  liquids  through  colloidal  membranes  such  as 
parchment-paper  or  bladder.     The  rubber  is  able  to  absorb  and  retain  a 


certain  amount  of_carbonic  acid  gas,_this^^inount  increasing  with  the 
pressure  of  the  gas  in  contact  with  the  surface  of  the  rubber.  Thus  the 
layers^  of^'ubber  next  the  CO^  first  get  saturated  with  the  gas,  and  this 
state  of  saturation  gets  transmitted  from  layer  to  layer  ;  but  as  on  the 
other  side  of  the  sheet  of  rubber  the  pressure  of  the  CO.,  is  less,  the  outer 
layei's  cannot  retain  the  whole  of  their  COg  so  that  some  of  the  gas 
gets  free. 

Passag-e  of  a  Gas  through  Liquids.— This  is  probably  analogous 

to  the  last  case  ;  the  gases  which  are  most  readily  absorbed  by  the  liquid 
are  those  which  pass  through  it  most  rapidly. 


204 


PROPERTIES  OF  MATTER. 


Passag'e  of  Gases  throug-h  red-hot  Metal.— Deyille  and  Troost 

found  that  hydrogen  passed  readily  tlirough  red-liot  platinum  and  iron. 
No  gas  besides  liydrogen  is  known  to  pass  through  platinum.  Troost 
found  that  oxygen  diil'used  through  a  red-hot  silver  tube  ;  quartz  is  said 
to  be  penetrable  at  high  temperatures  by  the  gases  from  the  oxy hydrogen 
flame. 

Diffusion  of  Metals  through  Metals.  —  Daniel!  showed  that 

mercury  diffused  through  lead,  tin,  zinc,  gold,  and  silver,  Henry  proved 
the  dif fusion  of  mercury  through  lead  by  a  very  striking  experiment :  he 
took  a  bent  piece  of  lead  and  placed  the  lower  part  of  the  shorter  arm  in 
contact  with  mercury  ;  after  the  lapse  of  some  time  he  found  that  the 
mercury  trickled  out  of  the  longer  arm.  He  also  showed  the  diffusion  of 
two  solid  metals  thi'ough  each  other  by  depositing  a  thin  layer  of  silver 
on  copper  ;  when  this  was  heated  the  silver  disappeared,  but  on  etching 
away  the  copper  surface  silver  was  found.  A  remarkable  series  of  ex- 
periments on  the  diffusion  of  metals  through  lead,  tin  and  bismuth  has  been 
made  by  Sir  W.  Roberts- Austen*;  his  results  are  given  in  the  following 
table.     K  is  the  diffusivity  : 


DifYusing  Metal. 

Solvent.              Temperature. 

K  cm.7sec. 

Gold 

Lead 

492°      ... 

3-47  X  10-5 

9?                                    *  •  • 

>»               •• 

492°     ... 

3-55  X  10-5 

Platinum 

»> 

492°     ... 

1-90x10-5 

,,                 *  *  * 

» 

492^     ... 

l-96xl0-» 

Gold 

5J 

555       ... 

3-G9xl0-5 

J)                ... 

Bismuth 

555        ... 

5  23x10-5 

,,                 •  •  • 

Tin 

555       ... 

5-38x10-5 

Silver 

>>             •  • 

555 

4-77x10-5 

Lead 

5> 

555 

3-68x10-5 

Gold 

Lead 

550       ... 

3-69  X  10-5 

Illiodium 

5> 

550       ... 

3-51x10-5 

It  will  be  seen  from  these  results  that  the  rate  of  difl^usion  of  gold 
through  lead  at  about  500°  is  considerably  greater  than  that  of  sodium 
chloride  through  water  at  18°  C.  Sir  W.  Roberts-Austen  has  lately  shown 
that  there  is  an  appreciable  diffusion  of  gold  through  solid  lead  kept  at 
ordinary  atmospheric  temperatures. 

•  Roberts-Austen, .PAtZ.  Trans.  A.,  1896,  p.  393. 


CHAPTER  XVIII. 
VISCOSITY  OF  LIQUIDS.      '""^ 

Contents.— Definition  of  Viscosity — Flow  of  Liquid  through  Capillary  Tube— Flow 
of  Gas  throiisrh  Capillary  Tube— Metliods  of  Measurement  of  Co-efficients  of 
Viscosity — Effect  of  Tcrupcrature  and  Trcssurc  on  Viscosity  of  Liquids — Vis- 
cosity of  Solutions  and  Mixtures— Lubrication — Explanation  of  Viscosity  of 
Gases  on  Kinetic  Theory — Mean-free  Path— Efi'ects  of  Temperature  and  Pressure 
on  Viscosity  of  Gases— Viscosity  of  Gaseous  Mixtures— Eesistance  to  Motion 
of  a  Solid  through  a  Viscous  Fluid. 

A  FLUID,  whether  liquid  or  gaseous,  when  not  acted  on  by  external 
forces,  moves  like  a  rigid  body  when  in  a  steady  state  of  motion.  When 
in  this  state  there  can  be  no  motion  of  one  part  of  the  liquid  relative 
to  another ;  if  such  relative  motion  is  produced,  say  by  stirring  the 
liquid,  it  will  die  away  soon  after  the  stirring  ceases.     Thus,  for  example, 

I  when  a  stream  of  water  flows  over  a  fixed  horizontal  plane,  since  the 
top  layers  of  the  stream  are  moving  while  the  bottom  layer  in  contact 
with  the  plane  is  at  lest,  one  part  of  the  stream  is  moving  relatively 
to  the  other,  but  this  relative  motion  can  only  be  maintained  by  the 
action  of  an  external  foi'ce  which  makes  the  pressure  increase  as  we  go 
up   stream.     If    this  force  were   withdrawn  the   whole   of   the  stream 


A . B 

C  D 

Fig.  157. 

would  come  to  rest.  The  slowly  moving  liquid  near  the  bottom  of 
the  stream  acts  as  a  drag  on  the  more  rapidly  moving  liquid  near  the  top, 
and  there  are  a  series  of  tangential  forces  acting  between  the  horizontal 
layers  into  which  we  may  suppose  the  stream  divided ;  thus  the  foi  ce 
acting  along  a  surface  such  as  AB  tends  to  retard  the  more  rapidly 
moving  liquid  above  it  and  accelerate  the  motion  of  the  liquid  below 
it ;  it  thus  tends  to  equalise  the  motion,  and  if  there  were  no  external 
forces  These^  tangential  ^stresses  ^would  soon  reduce  the  fluid  to  rest. 
The  property  of  a  liquid  whereby  it  resists  the  relative  motion  of  its 
parts  iscalled  viscosityT  TheTawof  tEis  viscous  resistance  was  formu- 
lated by  JSewton  {^frincijna,  Lib.  II.,  Sec.  9).  It  may  be  stated  as 
follows :  Suppose  that  a  stratum  of  liquid  of  thickness  c  is  moving 
horizontally  from  left  to  right  and  that  the  horizontal  velocity,  which 
is  nothing  at  CD,  increases  uniformly  with  the  height  of  the  liquid, 
and  let  the  top  layer  be  moving  with  the  velocity  V ;  then  the 
tangential  stress  which  may  be  supposed  to  act  across  each  unit  of  a 
surface  such  as  AB  is  proportional  to  the  gradient  of  the  velocity — i.e., 
to  Y/c — and  tends  to  stop  the  relative  motion,  the  tangential  stress  on  the 
liquid  below  AB  being  fiom  left  to  right,  that  on  the  liquid  above  AB 
from  right  to  left.  The^ratio  of  the  stress  to  the  velocity^radient  is  called 
the  co-efiicient  of  viscosity  of  the  fluid  ;  we  shall  denote  it  by  the  sym- 
bol i{.  The  viscosity  may  be  defined  in  terms  of  quantities,  which  may  be 
directly  measured  as  follows :  "  The  viscosity  of  a  substance  is  measured 


206  PROPERTIES  OF  MATTER. 

by  the  tangential  force  on  unit  area  of  either  of  two  horizontal  planes 
at  unit  distance  apart,  one  of  which  is  tixed,  while  the  other  moves 
with  the  unit  of  velocity,  the  spa^e  between  being  filled  with  the  viscous 
substance"  (Maxwell's  Them-y  of  Hcat).j'/)  'V^^-S'. 

It  will  be  seen  that  there  is  a  close  analogy  between  the  vLscous 
stress  and  the  shearing  stress  in  a  strained  elastic  soUd.  If  a  stratum 
of  an  elastic  solid,  such  as  that  in  Fig.  157,  is  strained  so  that  the  hori- 
zontal displacement  at  a  point  P  is  proportional  to  the  height  of  P 
above  the  plane  CD,  the  tangential  stress  is  equal  to  n  x  (gradient  of 
the  displacement)  where  n  is  the  rigidity  of  the  substance.  The  viscous 
stress  is  thus  related  to  the  velocity  in  exactly  the  same  way  as  the 
shearing  stress  is  related  to  the  displacement.  This  analogy  is  brought 
out  in  the  method  of  regarding  viscosity  introduced  by  Poisson  and 
Maxwell.  According  to  this  view,  a  viscous  liquid  is  regarded  as  able 
to  exert  a  certain  amount  of  shearing  stress,  but  is  continually  breaking 
down  under  the  influence  of  the  stress.  \Ye  may  crudely  represent 
the  state  of  things  by  a  model  formed  of  a  mixture  of  matter  in 
states  A  and  B,  of  which  A  can  exert  shearing  stress  while  B  cannot, 
while  under  the  influence  of  the  stress  matter  is  continually  passing 
from  the  state  A  to  the  state  B.  If  the  rate  at  which  the  shear 
disappears  from  the  model  is  proportional  to  the  shear,  say  X0,  where 
0  is  the  shear,  then,  when  things  are  in  a  steady  state,  the  rate  at 
which  unit  of  volume  of  the  substance  is  losing  shear  must  be  equal 
to  the  rate  at  which  shear  is  supplied  to  it.  If  ^  is  the  horizontal 
C  displacement  of  a  point  at  a  distance  x  from  the  plane  of  reference,  then 

A  d  =  —.     The  rate  at  which  shear  is  supplied  to  unit  volume  is  dd/dt  or J^ : 

■•je  dx  ^^  —!—      dx  dV 

— but  d^\dt  is  equal  to  «,  the  horizontal  velocity  of  the  particle,  hence  the 

rate  at  which  the  shear  is  supplied  is  dvjdx.     Thus,  in  the  steady  state, 

dx 

If  n  is  the  coeflScient  of  rigidity,  the  shear  0  will  give  a  tangential 
stress  equal  to  nQ  or 

n  dv 

X  dx. 
If  q  is  the  coefficient  of  viscosity,  the  viscous  tangential  stress  is  equal  to 

dv  -  '--^  .  -  ^„-;.-x:-?:a*A 

Hence,  if  the  viscous  stress  arises  ^ora_the  rigidity  ofthe_substance|^ 

7J  =  W/X. 

The  quantity  1/X  is   called    the  time  of   relaxation    of   the   medium ;    it 
measures  the  time  taken  by  the  shear  to  disappear  from  the  substance 
when  no  fresh  shear  is  supplied  to  it. 
-V  This  view  of  the  viscosity  of  liquids  is  the  one  that  naturally  suggests 

itself  when  we  approach  the  liquid  condition  by  starting  from  the  solid 
state ;  if  we  approach  the  liquid  condition  by  starting  from  the  gaseous 
state  we  ai-3  led  (see  p.  218)  to  regard  viscosity  as  analogous  to  diflfusion 


VISCOSITY  OF  LIQUIDS. 


«o7 


And.  as  ai'ising  from  the  movement  of  the  molecules  from  one  part  of  the 
Eubstance  to  another.  This  jioint  of  view  will  be  considered  later. 

Flow  of  a  Viscous  Fluid  through  a   Cylindrical  Capillary 

Tube. — When  the  fluid  is  driven  through  the  tube  by  a  constant 
dillerence  of  pressure  it  settles  down  into  a  steady  stsite  of  motion  such 
that  each  particle  of  the  fluid  moves  parallel  to  the  axis  of  the  tube, 
'provided  that  the  velocily  of 
the  fluid  through  the  tube  does 
not  exceed  a  certain  value  de- 
pending on  the  viscosity  of  the 
liquid  and  the  radius  of  the 
tube.  The  relation  between 
the  diflTerence  of  pressure  at 
the  beginning  and  end  of  the 
tube  and  the  quantity  of  liquid 
flowing  through  the  tube  in 
unit  time  can  be  determined  as 
follows : 

Let  the  cross-section  of  the 
tube  be  a  circle  of  radius  OA  =  a, 
let  V  be  the  velocity  of  the  fluid 
parallel  to  the  axis  of  the  tube 
at  a  point  P  distant  r  from  this 
axis.  Then  dv/dr  is  the  gradient 
of  the  velocity,  and  the  tangen- 
tial stress  due  to  the  viscosity 
is  rjdv/dr:  this  stress  acts  paralltl 

to  the  axis  of  the  tube.  Consider  the  portion  of  fluid  bounded  by  two 
coaxial  cylinders  through  P  and  Q  and  by  two  planes  at  right  angles  to 
the  axis  of  the  tube  at  a  distance  Az  apart.  Let  r,  r  +  Ar  be  the  radii  of 
the  cylinder  through  P  and  Q  respectively.     The  tangential  stress  due  to 

viscosity  acting  in  the  direction  to  diminish  v  is  at  P  equal  to  n— ;    the 

dr 

area  of  the  surface  of  the  cylinder  through  P  included  between  the  two 

planes  is  2TrrAz,  hence  the  total  stress  on  this  surface  is 

2nt]r-—Az 
dr 

Similarly  the  stress  acting  on  the  surface  of  the  cylinder  through  Q 
ipcluded  between  the  two  planes  is 

2nr,!y^  +  ^(r'^]Ar\Az  IT.    ^^ 

(  dr     dr\  dr  J      J 

and  this  acts  in  the  direction  to  increase  v;  hence   the  resultant  stress 


Fio.  1.08. 


-'V 


r  A^^ 


tending  to  increase  v  is  equal  to 


27rr]—(r—-]ArAz 
dr\  dr I 


'J/y.  yol^^ 


-6tc-*£.< 


Besides  these  tangential  forces  there  are  the  pressures  acting  over  the 
plane  ends  of  the  ring;  if  11  denote  the  pressure  gradient — i.e.,  the 
increase    of  pressure  per  unit  length   in  the  direction   of   v.   then   the 


208  PROPERTIES  OF  MATTER. 

effect  of  the  pressui'es  over  the  ends  of  the  ring  is  equivalent  to 
a  force  2TrrAr.llAz  tending  to  dimini.sh  v.  Since  the  motion  is  steady  there 
is  no  change  in  the  momentum  of  the  fluid,  hence  the  force  tending  to 
diminish  v  must  be  equal  to  that  tending  to  increase  it ;  we  thus  get 

dr\  dr J 

Now  since  the  liquid  is  moving  parallel  to  the  axis  of  the  tube  the 
pressure  must  be  the  same  all  over  a  cross-section  of  the  tube;  hence 
IT  does  not  depend  upon  r.  Again,  v  must  be  the  same  for  all  points 
at  the  same  distance  from  the  axis,  if  the  fluid  is  incompressible,  for  if 
V  changed  as  we  moved  parallel  to  the  axis  down  the  tube,  the  volume  of 
liquid  flowing  into  the  ring  through  P  and  Q  would  not  be  the  same  as 
that  flowing  out.  Since  IT  does  not  depend  upon  r,  and  the  left-hand  side 
of  equation  (1)  does  not  depend  upon  anything  but  r,  we  see  that  n  must 
be  constant ;  hence,  integrating  (1),  we  get   • 

dr      ~ 
where  C  is  a  constant ;  we  have  therefore 

dr  r 

Integrating  again  we  have 

VV  =  |>-n4-0  1ogr-t-C'  (2) 

where  C  is  another  constant  of  integration.  Since  the  velocity  is  not 
infinite  along  the  axis  of  the  tube — i.e.,  when  r  =  0,  0  must  vanish.  To 
determine  C  we  have  the  condition  that  at  the  surface  of  the  tube 
the  liquid  is  at  rest,  or  that  there  is  no  slipping  of  the  liquid  past 
the  walls  of  the  tvxbe.  This  has  been  doubted ;  indeed,  Helmholtz  and 
Piotrovvski  thought  that  they  detected  finite  efl'ects  due  to  the  slipping 
of  the  liquid  over  the  solid.  Some  very  careful  experiments  made  by 
Whethani  seem  to  show  that  under  any  ordinaiy  conditions  of  flow  no 
appreciable  slipping  exists,  at  least  in  the  case  of  liquids.  We  shall 
assume  then  that  ■?;  =  0  at  the  surface  of  the  tube — i.e.,  v/hen  r  —  a;  this 
condition  reduces  equation  (2)  to 

,v  =  |(r2-a2)n  (3) 

Now  if  pj  is  the  pressure  where  the  liquid  enters  the  tube,  p^  the 
pressure  where  it  leaves  it,  I  the  length  of  the  tube, 


11=  - 


{P^  -  2^2) 


the  negative  sign  is  taken  because  the  pressure  gradient  was  taken 
positive  when  the  pressure  increases  in  the  direction  of  v.  Substituting. 
this  value  for  11,  equation  (3)  becomes 

rjv==^-^p{d--r^)  (4) 


VISCOSITY  OF  LIQUIDS. 


20& 


The   volume   of  liquid  Q  which  passes  in  unit  time  across  a  section  of 


the  tube 


a 

-f 


2Trrvdr. 


OlT) 


(5) 


This  is  the  law  discovered  by  Poiseuille  for  the  flow  of  liquids  through 
capillary  tubes.  We  see  that  the  quantity  flowing  through  such  a  tube 
isj)roportional  to  the  square  of  the^areaTofcross-section  of  the  tube^ 

When  the  liquid  flows  through  the  capillary  tube  from  a  large  vessel, 
as  in  Fig.  159,  the  pressui'e  p^  at  the  orifice  A  of  the  capillary  tube 
diflers  slightly  from  that  due  to  the  head  of  the  liquid  above  A,  for  this 


B 


Fio.  159. 


head  of  liquid  has  not  merely  to  drive  the  liquid  through  the  capillary 
tube  against  the  resistance  due  to  viscosity,  it  has  also  to  communicate 
velocity  and  therefore  kinetic  energy  to  the  liquid,  so  that  part  of  the 
Eead  is  used  to  set  the  liquid  in  motion.  We  can  calculate  the  cor- 
rection due  to  this  cause  as  follows ;  let  h  be  the  height  of  the  surface 
of  the  liquid  in  the  large  vessel  above  the  outlet  of  the  capillary  tube,  p 
the  density  of  the  liquid ;  then  if  Q  is  the  volume  of  the  Hquid 
flowing  through  the  tube  in  unit  time,  the  work  done  in  unit  time  is 
equal  to  gpIiQ.  This  work  is  spent  (1)  in  driving  the  liquid  through  the 
capillary  tube  against  viscosity,  and  this  part  is  equal  to  (^J, -/>,)  Q  ^^ 
j3,  and  p^  are  the  pressures  at  the  beginning  and  end  of  the  capillary  tube 
(2)  in  giving  kinetic  energy  to  the  liquid.  The  kinetic  energy  given  to 
the  liquid  in  unit  time  is  equal  to 


h 


/"' 


xvx  2Trrdr 


Vrt  * 


'(. 


where  v  is  the  velocity  of  exit  at  a  distance  r  from  the  axis  of  the  capillary 
tube.  If  we  assume  that  the  distribution  of  velocity  given  by  equation  (4) 
holds  right  up  to  the  end  B  of  the  tube,  then  by  the  help  of  the  equation  (5) 


we  have 


.    /    s- 


210  PR0P]:RTIES  OF  MATTER. 

Substituting  this  value  in  the  integral  we  find  that  the  kinetic  ehebgy 
possessed  by  the  fluid  issuing  from  the  tube  in  unit  time  is  pQ^Tr-a*; 
hence,  equating  the  -work  spent  in  unit  time  to  the  kinetic  enei'gy  gained 
plus  the  work  done  in  overcoming  the  viscous  resistance,  we  have 

77  a 
or  9l>[h--^]=2\-2^, 

Thus  the  head  which  is  spent  in  overcoming  the  viscous  resistance  is  not  h, 


but  h- 


■K-a^g 


This  correction  has  been  investigated  by  Hageubach,*  Couette,t  and 
"Wilberforce,t  and  has  been  shown  to  make  the  results  of  experiments 
agree  more  closely  with  theory.  It  is  probably,  however,  not  quite  accu- 
rate on  account  of  the  assumption  made  as  to  the  distribution  of  velocity 
at  the  orifice. 

Viscosity  of  Gases. — The  viscosity  of  gases  may  be  measured  in 
the  same  way  as  that  of  liquids,  but  the  case  of  a  gas  flowing  through  a 
capillaiy  tube  differs  somewhat  from  that  investigated  on  p.  208,  where 
the  liquid  was  supposed  incompressible  and  the  density  constant ;  in  the 
case  of  the  gas  the  density  will,  in  consequence  of  the  variation  in 
pressure,  vary  from  point  to  point  along  the  tube.  Using  the  notation  of 
the  previous  investigation,  instead  of  v  being  constant  as  we  move  parallel 
to  the  axis  of  the  tube,  the  fact  that  equal^masses  pass  each  cross-section 
requires  pv  to  be  constant  as  long  as  we  keep  at  a  fixed  distance  from  the 
xxis'of  the^libe.  Since  p  is  propoitional  to />,  where  p  is  the  pressure  of 
the  gas,  we  may  express  this  condition  by  saying  that  ^>w  must  be 
independent  of  z  where  2;  is  a  length  measured  along  the  axis  of  the  tube. 
Thus,  since  p  varies  along  the  tube,  v  will  not  be  constant  as  z  changes ; 
this  variation  of  v  will  introduce  relative  motion  between  parts  of  the  gas 
at  the  same  distance  from  the  axis  of  the  tube,  and  will  give  rise  to 
viscous  forces  which  did  not  exist  in  the  case  of  the  incompressible  liquid. 
We  shall,  however,  neglect  these  for  the  following  reasons  :  if  Y„  is  the 
greatest  velocity  of  the  fluid,  the  gradient  of  velocity  along  the  tube  is  of 
the  oi'der  "Vjl,  where  I  is  the  length  of  the  tube ;  the  gradient  of  velocities 
across  the  tube  is  of  the  order  V^/a,  where  a  is  the  radius  of  the  tube ;  as 
a  is  very  small  compared  with  I,  the  second  gradient,  and  therefore  the 
viscous  forces  due  to  it  are  very  large  compared  with  those  due  to  the  first, 
Weshall  therefore  neglect  the  effect  of  thefirat  gradient.  On  this  supposition 

equation  (1)  still  holds,  and,  since  11  =  — ,  we  have 

dz 

—  I  ^^-  -17=  -^ 
dr\  dr)  dz 


• 


Hagenbach,  Poggendorff's  Annalen,  109,  p.  SS5. 
t  Conette,  AnnaJcs  de  Chimic  ct  de  Physique,  [6],  21,  p.  433 
X  Wilberforce,  Philosophical  Magazine,  (5)  31,"p.  407. 


VISCOSITY  OF  LIQUIDS.  2ii 

Of,  reo'arding  ^;  as  constant  over  a  cross-section  of  the  tube,  we  have 


dr\     dr 
Since  pv  is  independent  of  z,  we  see 


that  ^^'  is  constant  and  equal  to 
dz 


Fig.  160. 
Solving  the  differential  equation  in  the  same  way  as  that  on  p.  208.  re  get 


J.  J.. 


212  PROPERTIES  OF  MATTER. 

and  if  Y^  is  the  volume  entering,  V,  that  leaving  the  tube  per  second,  w» 
have 

Measurement  of  the  Coefficient  of  Viscosity.— The  viscosity  ij 

has  must  frequently  been  determined  by  measurements  of  the  rate  of  flow 
of  the  iluid  through  cajiillary  tubes.  An  apparatus  by  which  this  can  be 
done  is  shown  in  Fig.  IGO.  G  is  a  closed  vessel  containing  air  under 
pressure;  the  pressure  in  this  vessel  is  kept  constant  by  means  of  the  tube 
I>,  which  connects  G  with  a  Mariotte's  bottle ;  the  pressure  in  G  is  always 
that  due  to  a  column  of  water  whose  height  is  the  height  of  the  bottom  of 
the  air  tubes  in  the  Mariotte's  bottle  above  the  end  of  the  tube  U.  The 
glass  vessel  abcdef,  in  which  de  is  a  capillary  tube,  contains  the  fluid  whose 
coefficient  of  viscosity  is  to  be  determined ;  this  vessel  communicates  with 


i^ 


•[^  "       , ,  "A^^Jf^^-^ 


Fig.  161.  FiG.  162. 

G  by  means  of  the  tube  LKI ;  the  pressure  acts  on  the  liquid  in  ahcdef, 
and  causes  it  to  flow  through  the  capillary  tube  from  left  to  right ;  two 
marks  are  made  at  b  and  c,  and  the  volume  between  these  marks  is 
carefully  determined.  Let  us  call  it  V ;  then,  if  T  is  the  time  the  level  of 
the  liquid  takes  to  fall  from  h  to  c,  Q  =  V/T.  The  area  of  cross-section  of 
the  tube  has  to  be  determined  with  great  care,  and  precautions  must  be 
taken  to  prevent  any  dust  getting  into  the  capillary  tube.  As  the 
viscosity  varies  very  rapidly  with  the  temperature,  it  is  necessary  to 
maintain  the  temperature  constant;  for  this  purpose  the  vessel  aZ^cc^e/ is 
placed  in  a  bath  filled  with  water. 

With  an  apparatus  of  this  kind  Poiseuille's  law  can  be  verified,  and 
the  viscosity  determined.  It  is  found  that,  although  Poiseuille's  law  holds 
with  great  exactness  when  the  rate  of  flow  is  slow,  yet  it  breaks  down 
when  the  mean  velocity  Q/7ra^  exceeds  a  certain  value  depending  on  the 
size  of  the  tube  and  the  viscosity  of  the  liquid.  This  point  has  been 
investigated  by  Osborne  Reynolds,  who  finds  that  the  state  of  flow  we 
have  postulated  in  deducing  Poiseuille's  law — i.e.,  that  theJig[iud_moves  in 
straight  linesparallel  to  the  axis  of  the  tube — cannot  exist  when  the  niean 
vetocity  exc&eusjTmt^  the  steady  flow  is  then  replaced  by  an 

irregular  turbulent  motion71;he  particles  of  liquid  moving  from  side  to  side 
of  the  tube.  This  is  beautifully  shown  by  one  of  Reynolds'  experiments. 
Water  is  made  to  flow  through  a  tube  such  as  that  shown  in  Fig.  IGl,  anda 
little  colouring  matter  is  introduced  at  a  point  at  the  mouth  of  the  tube  :  if 
the  velocity  is  small  the  coloured  water  forms  a  straight  band  parallel  to 
the  axis  of  the  tube,  as  in  Fig.  161 ;  when  the  velocity  is  increased  this  band 
becomes  sinuous  and  finally  loses  all  defiuiteness  of  outline,  the  colour 
filling  the  whole  of  the  tube,  as  in  Fig.  1G2.    Reynolds  conckided  from  his 

(experiments   that  the^ steady^motion  cannot  exist  if  the  mean  velocity  is 
greater  than  1000  rj/fM  where  t]  is  the  viscosity,  p  the  density  oFthe  Tiquid, 
and  a  tlie  radius  ofThe  tube.    The  units  are  centimetre,  gramme  and  second. 
Measurements  of  the  viscosity  of  fluids  both  liquid  and  gaseous,  have  been 


VISCOSITY  OF  LIQUIDS. 


213 


made  by  detei'mining  the  couple  which  must  be  apph'ed  to  a  cylinder  to 
keep  it  fixed  when  a  coaxial  cylinder  is  rotated  with  uniform  velocify,  the 
space  between  tlie  cylinders  being  filled  with  the  liquid  whose  viscositr 
has  to  be  determined.  This  method  has  been  used  by  Couette  and  Mallock. 
The  theory  of  the  method  is  as  follows :  the  particles  of  the  fluid  will 
describe  circles  round  the  common  axis  of  the  cylinders.  Let  PQ  be  points 
on  a  radius  of  the  cylinders; 
after  a  time  T,  let  P  come  to  P', 
Q  to  Q',  let  OP'  produced  cut  QQ' 
in  Q".  Then  the  velocity  gradient 
at  P  will  be  equal  to  {Q'Q"IT)^ 
P'Q"  ;  if  w  is  the  angular  velocity 
with  which  the  particle  at  P  de- 
scribes its  orbit,  w  +  3w  that  of  the 
particle  at  Q,  then  Q'Q"  =  OQ'cwT. 
liBt  OP  =  r,  OQ  —  r  +  cr,  then  since 
P'Q"  =  cr  the  velocity  gradient  at 

P  is  (r  +  h')^^,  or  when  h'  is  very 

07' 

small,  r-~  ;   hence  the  tangentiol 
dr 

stress  acting  on  unit  area  of  the 

surface  at  P  is  nr—-.    Now  consider 
dr 


Fig.  1G3. 


the  portion  of  liquid  bounded  by 

coaxial  cylinders  through  P  and  R  and  by  two  parallel  planes  at  right 
angles  to  the  axes  of  the  cylinders  and  at  unit  distance  apart.  This 
annulus  is  rotating  with  constant  angular  velocity  round  the  axis  of  the 
cylinders,  hence  the  moment  about  this  axis  of  the  forces  acting  upon  the 
annulus  must  vanish.  Now  the  moment  of  the  forces  acting  on  the  inner 
face  of  this  annulus  is 

o        d(o       o      -.db)  -  ^^' 

Znri]y^r  =  ^nr]r*-z- 
,  dr  dr 

and  this  must  be  equal  and  opposite  to  the  moment  of  those  acting  on  the 
outer  surface  of  the  cylinder;  now  E  may  be  taken  anywhere;  hence  we 
see  that  this  expression  must  be  constant  and  equal  to  the  moment  of  the 
couple  acting  on  unit  length  of  the  outer  cylinder,  which  is,  of  course,  equal 
and  opposite  to  the  moment  of  that  on  the  inner.    Let  us  call  this  moment 


r,  then 

Integrating  this  equation  we  find 


2nrjr^^=r 
dr 


to)=  — 


4iTrT]r'- 


+  0 


■where  C  is  a  constant.  If  the  radii  of  the  inner  and  outer  cylinders  are 
a  and  b  respectively,  and  if  the  inner  cylinder  is  at  rest  and  the  outer  one 
rotates  with  an  angular  velocity  Q,  then  since  w  =  0,  when  r  =  a,  and  w  =  i2 


when  r  =  b,  we  find 


07  O 

0-  -a^ 


214  PROPERTIES  OF  MATTER. 

Hence,  if  we  measure  r  for  a  given  velocity  il,  we  can  deduce  the  value  of  tf. 
This  case  presents  the  same  peculiarities  as  the  flow  of  a  viscous  liquid 
through  a  capillary  tube ;  the  law  expressed  by  the  preceding  equation  is 
only^beyed  when  Ci,  is  less  than  a  cei'tain  critical  valua  When  £1  exceeds 
this  value  the  motion  of  the  fluid  becomes  turbulent,  and  for  values  of  Q, 
just  above  this  value  the  relation  between  r  and  il  becomes  irregular ;  it 
becomes  regular  again  when  i2  becomes  considerably  greater ,  but  r  is  no 
longer  proportional  to  il,  but  is  of  the  form  ai^  4-  0Qi-  where  a  and  ft  are 
constants.      These  facts  are  well  shown  by  the  curve  given  in  Fig.  164, 


/ 

/ 

/ 


/ 


I 
I 
I 
t 
r 

-' B 


XI 


Fio.  164. 

which  represents  the  results  of  Oouette's*  experiments  on  the  viscosity  of 
water.  The  abscissae  are  the  values  of  Q,  and  the  ordinates  the  values 
of  r/li.  The  instability  set  in  at  B  when  the  outer  cylinder  made  about 
one  revolution  per  second ;  the  radii  of  the  cylinders  were  14*64  and  14'39 
cm.  respectively. 

This  method  can  be  applied  to  determine  the  viscosity  of  gases  as  well 
as  of  liquids. 

Method  of  the  Oscillating'  Disc. — Another  method  of  determining 
T},  Avhich  has  been  used  by  Coulomb,  Maxwell,  and  O.  E.  Meyer,  is  that  of 
measuring  the  logax-ithmic  decrement  of  a  horizontal  disc  vibrating  over  a 
fixed  parallel  disc  placed  at  a  short  distixnce  away,  the  space  between  the 
discs  being  filled  with  the  liquid  whose  viscosity  is  required.    The  viscosity 

•  Couette,  Annales  de  Chimie  et  de  Physique  [6],  21,  p,  433. 


VISCOSITY  OF  LIQUIDS. 


215 


of  the  liquid  gives  rise  to  a  couple  tending  to  retard  the  motion  of  the 
disc  proportional  to  the  product  of  the  angular  velocity  of  the  disc  and 
the  viscosity  of  the  ligmd :  the  calculation  of  this  couple  is  somewhat 
difficult.     We  shall  refer  the  reader  to  the  solution  given  by  Maxwell 


1Z00 


iBoa 


woa 


I 


uoo 


900-^ 


700- 


500  i 


300 


GO  eo  70 

TeTTiper^ttiine 

Fig.  166. 


{Collected  Paj-ers,  vol.  ii.  p.  1).     This  method,  as  well  as  the  preceding  one, 
can  be  used  for  gases  as  well  as  for  liquids. 

Among  other  methods  for  measuring  rj  we  may  mention  the  determina- 
tion of  the  logarithmic  decrement  for  a  pendulum  vibrating  in  the  fluid 
(Stokes) ;  the  logarithmic  decrement  of  a  sphere  vibrating  about  a  diameter 


216 


PROPERTIES  OF  MATTER. 


iu  an  ocean  of  the  fluid ;  the  logarithmic  decrement  of  a  hollow  sphere 
filled  with  the  liquid  and  vibrating  about  a  diameter  (Helmholtz  and 
Piotrowski,  Helmholtz  Collected  Papers,  vol.  i.  p.  j  72). 

Temperature  Coefficient  of  Viscosity. — In  all  experiments  on 

viscosity  it  is  necessary  to  pay  great  attention  to  the  measurement  of  the 
temperature,  as  the  coefficient  of_yiscosity^  of  liquids  dim^ishes  vei-y 
rsyDicUy^as  ilie_jtemperature  jnerei^  This  is  shown  by  the  curve  (Fig. 
Ibojiaken  from  the  paper  by  Thorpe  and  Rodger  [Phil.  Trans.,  1894,  A. 
Part  ii.  p.  397),  which  shows  the  relation  between  the  viscosity  of  water 
and  its  temperature.  It  will  be  seen  that  the  viscosity  of  water  at  80°  C. 
is  onl}'  about  one-third  of  its  value  at  10°  0.  Thorpe  and  Rodger,  who 
determined  the  co-efficients  of  viscosity  of  a  large  number  of  liquids,  found 
tlie  formula  given  by  Slotte,  ?;  =0/(1  +  ii)",  where  rj  is  the  co-efficient  of 
viscosity  at  the  temperature  t  and  U,  h  and  n  are  constants  depending  on 
the  nature  of  the  liquid,  was  the  one  that  agreed  best  with  their  experi- 
ments.    For  water  they  found  that 

•017941 


(1  +  •02312001- 


biXZ 


where  t  is  the  temperature  in  degrees  Centigrade. 

The  following  table,  taken  from  Thorpe  and  Rodger's  paper  {Phil. 
Trans.,  A.  1894,  p.  1),  gives  the  value  of  i^  in  C.G.S.  units  for  some  liquids 
of  frequent  occurrence.  The  table  gives  the  value  of  the  constants  (J,  6,  n 
in  Slotte's  formula 


,  =  C/(l+5i)» 

Substance 

0 

h 

n 

Bromine 

•012535 

■008935 

1-4077 

Chloroform  . 

■007006 

•006316 

1-8196 

Carbon  tetrachloride 

•013466 

•010521 

1-7121 

Carbon  bisulphide 

•004294 

•005021 

1-6328 

Formic  acid 

•029280 

•016723 

1-7164 

Acetic  acid  . 

•016867 

•008912 

2-0491 

Ethyl  ether 

•002864 

•007332 

1-4644 

Benzene 

•009055 

•011963 

1-5554 

Toluene 

•007684 

•008850 

1-6522 

Methyl  alcohol     . 

•008083 

•006100 

26793 

Ethyl  alcohol       .     .    . 

•017753 

•004770 

4-3731 

Propyl  alcohol 

•038610 

•007366 

3-9188 

Butvl  alcohol : 

6°  to  52°           .... 

•051986 

■007194 

4-2452 

52° to  114°       

•056959 

■010869 

32150 

Inactive  amyl  alcohol : 

0°to40° 

•085358 

■008488 

4-3249 

40°  to  80°        ...         . 

•093782 

•012520 

3-3395 

80°  to  128° 

•152470 

•026540 

24618 

Active  amyl  alcohol : 

0°to35° 

•111716 

■009851 

4-3736 

35°  to  73° 

•124788 

•015463 

3-2542 

73°  to  124° 

•147676 

•127583 

2  0050 

Allyl  alcohol 

•021736 

•009139 

2-7925 

Nitrogen  peroxide        .... 

■005267 

■007098 

1-7349 

VISCOSITY  OF  LIQUIDS.  217 

"Warburg  found  that  rj  for  mercury  at  17"2°  is  equal  to  •016329.  A  later 
determination  by  Umani  (iY?fo?'.  Cim.  [4]  .3,  p.  151)  gives  »;  =  "01577at  10°. 

Tlie  value  of  t;  for  liquid  carbonic  acid  is  very  small,  being  at  15°  only 
1/14-6  of  that  of  water. 

Effect  of  Pressure  on  the  Viscosity. — The  viscosity_pf  water 

diminishes  slightly  under_jiicreased  pressure,  while~that  ofbenzol  and 
ether^increases^ 

Viscosity  of  Salt  Solutions. — A  large  number  of  experiments 
have  been  made  on  the  viscosity  of  sohitions,  but  no  simple  laws  con- 
necting the  viscosity  with  the  strength  of  the  solution  have  been  arrived 
at.  In  some  cases  the  viscosity  of  the  solution  is  less  than  that  of  water, 
and  in  many  cases  the  viscosity  of  the  solution  is  a  maximum  for  a  particular 
strength. 

Viscosity  of  Mixtures. — Here  again  no  general  results  have  been 
arrived  at,  although  considerable  attention  has  been  paid  to  this  subject. 
In  many  cases  the  viscosity  of  a  mixture  of  two  liquids  A,  B  is  less  than 
that  calculated  by  the  foi-mula 


7  = 


a  +  b 


where  rix,  Vb  ^^^  respectively  the  viscosities  of  A  and  B,  and  a,  h  are  the 
volumes  of  A  and  B  in  a  volume  a  +  b  of  the  mixture. 

Lubrication. — When  the  surfaces  of  two  solids  are  covered  with  oil 
or  some  other  lubricant  they  are  not  in  contact,  and  the  friction  between 
them,  which  is  much  less  than  when  they  are  in  contact,  is  due  to  fluid 
friction.  The  laws  of  fluid  friction  discussed  in  this  chapter  show  that, 
iFweTiave  two  parallel  planes  at  a  distance  d  apart,  the  interval  between 
them  being  filled  with  a  liquid,  then  if  the  lower  plane  is  at  rest  and 
the  upper  one  moving  parallel  to  the  lower  one  with  the  velocity  V, 
if  V  is  not  too  great  there  is  a  retarding_tang^ntial  force  acting  on_the 
moving  plane,  and  equal  per  unifarea  to  TfYjd,  where  17  is  a  quantity 
called  the  coefficient  of^viscosity^oFThe  liquid.  If  we  regard  this  as  a 
frictional  force  acting  on  the  moving  plate  we  see  that  the  friction  would 
depend  upon  the  velocity,  and  would  only  depend  upon  the  pressure  between  \ 
the  bodies  in  so  far  as  the  pressure  afiected  the  thickness  of  the  liquid  \ 
layer  and  the  viscosity  of  the  lubricant. 

The  laws  of  friction,  when  lubricants  are  used,  are  complicated,  depending 
largely  upon  the  amount  of  lubrication.  When  the  lubricant  is  present 
in  sufficiently  large  amounts  to  fill  the  s-paces  between  the  moving  parts 
the  friction  seems  to  be  proportional  to  the  relative  velocity  of  these  parts. 
When  the  supply  of  lubiicant  is  insufficient,  part  of  it  collects  as  a  pad 
between  the  moving  parts,  as  in  Fig.  166;  here  the  lower  surface  is  at 
rest  and  the  upper  one  rotating  from  left  to  right.  Professor  Osborne 
Reynolds*  has  shown  that,  as  the  breadth  and  thickness  of  this  pad 
depend  upon  the  pressure  and  relative  velocity,  it  would  be  possible  to  get 
friction  proportional  to  the  pressure  and  independent  of  the  relative 
velocity,  even  when  the  friction  was  entirely  caused  by  the  viscosity  of  a 
thin  layer  of  liquid  between  the  moving  parts. 

Viscosity  of  Gases. — Gases  possess  viscosity,  and  the  forces  called 
into  play  by  this  property  are,  as  in  the  case  of  liquids,  proportional  to 
the  velocity  gradient ;  in  fact,  the  definition  of  viscosity  given  on  p.  205, 
*  Reynolds,  Phil.  Trans.,  1886,  pt.  i.  p.  157. 


218  PROPERTIES  OF  MATTER. 

applies  to  gases  as  well  as  to  liquids.  The  most  remarkable  property  of 
the_viscosity^_gases  is  that  within  jvvide  limits  of  pressure  the  viscosity 
is  indepemTent  of  the  pressure,  being  under  ordinary  circumstances  the 
same  at  a  pressure  of  a  few  millimetres  of  mercury  as  at  atmospheric 
pressure.  This  is  known  as  Maxwell's  Law,  as  it  was  deduced  by  Maxwell 
from  the  Kinetic  Theory  of  Gases ;  it  has  been  verified  by  numerous 
experiments.      Boyle  has  some  claim  to  be   regarded   as  the   discoverer 


of  this  law,  for  about  1660  he  experimented  on  the  effect  of  diminishing 
the  pressure  on  the  vibrations  of  a  pendulum,  and  found  that  the  vibrations 
died  away  just  as  quickly  when  the  pressure  was  low  as  when  it  waj 
high.  This  law  follows  very  readily  from  the  view  of  viscosity  supplied 
by  the  Theory  of  Gases.  Thus,  suppose  we  have  two  layers  of  gas  A 
and  B  at  the  same  pressure,  and  that  A  has  a  motion  as  a  whole  from 
left  to  right,  while  B  is  either  at  rest  or  moving  more  slowly  than  A  in 
this  direction.  According  to  the  Kinetic  Theory  of  Gases,  molecules  of 
the  gas  will  be  continually  crossing  the  plane  separating  the  layer  A  from 


B 


Fig.  167. 

the  layer  B.  Some  of  these  molecules  will  cross  the  plane  from  A  to  B 
and  an  equal  number,  since  the  pressure  of  the  gas  remains  uniform,  from 
B  to  A.  The  momentum  parallel  to  the  plane  of  those  which  leave  A 
and  cross  over  to  B  is  greater  than  that  of  those  which  replace  them 
coming  over  from  Bto  A  ;  thus  the  layer  A  is  continually  losing  momentum 
^  while  the  layer  B  is  gaining  it.  The  effect  is  the  same  as  if  a  force  parallel 
to  the  plane  of  separation  acte3^on_  the  layer  ^17  so  as  to  tend  to 
stop  theliibtion  from  left  to  right,  while  an  equal  and  opposite  force  acted 
on  B,  tending  to  increase  its  motion  in  this  direction ;  these  forces  are 
the  viscous  forces  we  have  been  discussing  in  this  chapter.  If  the  distri- 
bution of  velocity  remains  the  same,  the  magnitude  of  these  forces  will 
be  proportional  to  the  number  of  molecules  which  cross  the  plane^of  sepa- 
ration in  unit  time. 

The  molecules  are  continually  striking  against  each  other,  the  average 
free  run  between  two  collisions,  called  the  mean  free  path  of  the  molecules, 
being  extremely  smalTj  only  about  10^  cm.  tor  air,  at  txtmosplieric 
pressure.  This  free  path  varies,  however,  inversely  as  the  pressure,  and  at 
the  extremely  low  pressures  which  can  be  obtained  with  modern  air-  pumps 


VISCOSITY  OF  LIQUIDS. 


219 


can  attain  a  length  of  several  centimetres.  When  one  molecule  strikes 
against  another  its  course  is  deflected,  so  that,  although  it  is  travelling  at 
a  great  speed,  it  makes  but  little  progress  in  any  assigned  direction.  The 
consequence  of  this  is  that  the  molecules  which  cross  in  unit  time  the 
plane  of  separation  between  A  and  B  can  all  be  regarded  as  coming  from 
a  thin  layer  of  gas  next  this  plane,  a  definite  fraction  of  the  molecules 
in  this  layer  crossing  the  plane.  The  longer  the  free  pfithj)f  the  molecules 
the  thicker  the^  layer,  the 


\ 


tliickhess  being  directly 
proportional  to  the  mean 
free  path.  If  n  is  the 
number  of  molecules  per 
unit  volume  and  t  the 
thickness  of  the  layer, 
the  number  of  molecules 
which  in  unit  time  cross 
unit  area  of  the  plane 
separating  A  and  B  will 
be  proportional  to  nt. 
Let  us  consider  the  efiect 
on  this  numbsr  of  halvini? 
the  pressure  of  the  gas. 
This  halves  n  but  doubles 
l<  ;  t  is  proportional  to  the 
(free  path,  which  varies 
inversely  as  the  pressure, 
hence_the_prQdjict_2i4_and 
therefore  the  .jviscosity, 
'  This 

the 


remains  unaltered. 
l(Ii~'until 


rasoning 


J 


Milliont-hs  of  an  Atmospheric  • 

Fig.  168. 


thickness  of  the  layer  from 
which  the  molecules  cross 
the  plane  of  separation 
gets  so  large  that  the  layer 
reaches  to  the  sides  of 
the  vessel  containing  the 
gas.  When  this  is  the 
case  no  further  diminu- 
tion^in  Jihe  ^essure  can 
increase  t,  and  as  n  dimin- 
ishes as  the  pressure 
diminishes,   the   product 

nt  and,  therefore,  the  viscosity,  will  fall  as  the  pressure  falls.  Thus  in  a 
vessel  of  given  size  the  viscosity  remains  unaffected  by  the  pressure  until 
the  pressure  reaches  a  certain  value,  which  depends  upon  the  size  of  the 
vessel  and  the  nature  of  ^theTg^fl  wheinhis  pressure  is  passed  the 
viscosity  diniinisTies  rapidly  with  the  pressure.  This  is  shown  very  clearly 
by  the  curves  in  Fig.  168,  based  on  experiments  made  by  Sir  William 
Orookes  {Phil.  Trans.,  172,  pt.  ii.  387).  In  these  curves  the  ordinates 
represent  the  viscosity  and  the  abscisste  the  pressure  of  the  gas. 

The  diminution  in  viscosity  at  low  pressures  is  well  shown  by  an  incan- 
descent electric  lamp  with  a  broken  filament.     If  this  be  shaken  while  the 


220 


PROPERTIES  OF  MATTER. 


lamp  is  exhausted  it  will  be  a  long  time  befoi-e  the  oscillations  die  away; 
if,  however,  air  is  admitted  into  the  lamp  through  a  crack  made  with  a 
file  the  oscillations  when  started  die  away  almost  immediately. 

Another  reason  why  the  effects  of  viscosity  are  less  at  very  low  pressures 
than  at  higher  ones  is  the  slipjiing  of  the  gas  over  the  surface^of  t^e  solids 
with^  which  it  is  in  contact.  In  the  case  of  liquids,  no  effects  due  to  slip 
have  been  detected.  Kundt  and  Warburg*  have,  however,  detected  such 
effects  in  gases  even  up  to  a  pressure  of  several  millimetres  of  mercury. 
TheJaHZ-oL^slip  {see  Maxwell,  "  Stresses  in  a  Rarefied  Gas,"  Phil.  Trans., 
187)  may  be  expressed  by  saying  that  the  motion  in  the  gas  is  the  same 
as  if  a  certain  tjiickness  L  were  cut  off  the  solids,  and  that  the  gas  in 
contact  vvithtliisjiew  siw-face^were  at  restV  Tbis  thickness  L  is  propor- 
liionaTjtoJLhejgiean  free  path  of  the  molecules  of  the^as.  ^SccordTng  to 
the  experiments  of  Kundt  and  Warburg  it  is  equal  to  twice  the  free 
path ;  hence,  as  soon  as  the  free  path  gets  comparable  with  the  distance 


between  the  solids  in  the  gas. 


of 


the  slip 
the  same 


the  gas  over 
direction    as 


these  solids  will 
a    reduction  in 


produce   appreciable   eflects   in 
viscosity. 

Mean  Free  Path. — If  we  know  the  value  of  the 
calculate  the  mean  free  path  of  the  molecules  of  a  gas : 
late,  from  the  principles  of  the  Kinetic  Theory  of  Gases,  the  rate  at  which 
momentum  is  flowing  across  unit  area  of  the  plane  A,  B,  Fig,  167,  we  find 


viscosity  we  can 
for  if  we  calcu- 


that  it  is  equal  to 


•350cpX^ 


where  v  is  the  velocity  of  the  stratum  at  a  height  x  above  a  fixed  plane, 
X  is  the  mean^'free  path,  p  the  density  of  the  gas,  c  the  "  velocity  of  mean 
square"  (this  can  be  calculated  from  the  relation  ;j  =  |-pc- where  jt>  is  the 
pressure  in  the  gas).  The  rate  of  flow  of  momentum  across  unit  area 
is  equal  to  the  tangential  stress  at  the  plane  AB  ;  hence,  if  rj  is  the  viscosity 
of  the  ga«,  T]  — '  350cpX.  Let  us  calculate  from  this  equation  the  value 
of  X  for  air ;  taking  for  the  viscosity  at  atmospheric  pressure  and  at 
15°  C.  j;  =  1-9x10'^,  p  at  pr-essure  10®  and  temperature  15°  C, 
1'2G  X  10"^,  we  get  c  =  i-yy  x  lU*,  and  X  =  -00001  cm.  At  the  pressui-e  of  a 
millionth  j)f_juwitmospherejthejnean  free  path  in  air  is  10  cm. 

TEe^values^oF^/Tor^Tfewof  the  most  important  gases  are  given  in 
the  following  table  ;  the  temperature  is  about  15°  C.  These  numbers 
are  given  by  0.  E.  Meyer ;  they  are  deduced  from  his  own  experiments 
on  the  viscosity  of  air  by  the  method  of  the  oscillating  disc  and  the  expe- 
riments made  by  Graham  on  the  relation  between  the  rates  of  flow  of 
different  gases  thi'ough  capillai-y  tubes : 


Gaa 

T)  X  lOJ 

Gas 

>)X10< 

Air  . 

.     1-9 

Sulphuretted  hydrogen 

.     1-3 

Hydrogen 

.       -93 

Ilydrocliloric  acid    . 

.     1-56 

Marsh-gas 

.     1-2 

Carbonic  acid  . 

.     1-6 

Water-vapour  . 

■975 

Nitrons  oxide  (NoO) 

.     1-6 

Ammonia 

.    1-os 

Methyl  ether    .   "     . 

.     1-02 

Carbonic  oxide 

.     l-8i 

Methyl  chloride 

.     116 

Ethylene  . 

,     1-09 

Cyanogen 

.     1-07 

Nitrogen  . 

.     1-84 

Sulphurous  acid  (SO.,) 

.     1-38 

Oxygen     . 

.     2-12 

Ethyl  chloride 

.     1-05 

Nitric  oxide  (NO) 

.     1-86 

Chlorine  .        .        . 

.     1  41 

Pugg.  Ann.,  155,  p.  357. 


VISCOSITY  OF  LIQUIDS. 


221 


Effect  of  Temperature  upon  the  Viscosity  of  Gases.— Increase 

_of  temgeiatuie  has  opposite  eliccts  on  the  viscosities  of  liquids  and  oFgases, 
for  while,  as  we  have  seen,  it  diminishes  tliejviscosity  of  liquids  it  increases 
that  of  gases.  If  ??  is  the  coeiHcient  of  viscosity,  and  if  this  is  assumed 
to  Fo  proportional  to  T"  where  T  is  the  absolute  temperature,  then,  according 
to  Lord  ilayleigh's*  experiments,  we  have  the  following  values  for  n : 


Air       . 

Oxygen 
Hydrogen 
Helium 
Areon  . 


n 

•754 
•782 
•681 
•681 
•815 


1113 

128-2 
72-2 
72-2 

150-2 


The  values  of  c  relate  to  a  formula  suggested  by  Sutherland,  according 

T*  ... 

to  which  7j  =  a n^  ;   thus,  at  very  high  temperatures,  if  this  relation 

'       1+c/T' 

is  true,  jj  would  vary  as  the  square  root  of  the  absolute  temperature. 
According  to  Koch,t  the  viscosity  of  mercury  vapour  varies  much  more 
rapidly  with  the  temperature  than  that  of  any  other  known  gas.  He 
concluded  from  his  experiments  that  for  this  gas  rj  =  aT''".  The  results 
given  above  for  helium  and  argon,  both,  like  mercury  vapour,  monatomic 
elements,  show  that  a  rapid  variation  with  temperature  is  not  a  necessary- 
characteristic  of  monatomic  gases.  Loi-d  Eayleigh  found  that  the  viscosity 
of  argon  was  1-21,  and  of  helium  0"9G  that  of  air. 

Coefficient  of  Viscosity  of  Mixtures.— Graham  made  an  extensive 
series  of  experiments  on  the  coeiiicients  of  viscosity  of  mixtures  of  gases 
by  meisuring  the  time  taken  by  a  known  volume  of  gas  to  flow  through 
a  capillary  tube.  He  found  that  for  mixtures  of  oxygen  and  nitrogen,  and 
of  oxygen  and  carbonic  acid,  the  rate  of  flow  through  the  tubes  of  thej 
mixture  was  the  arithmeticaLmean  rate  of  the  gases  mixed  ;  with  mixtures 
containing  hydrogen  the  results  were  very  diflferent ;  how  difterent  is  shown 
by  the  following  table,  which  gives  the  ratio  of  the  transpiration  time  of 
the  mixtures  to  that  of  pure  oxygon  : 

Hydrogen  and  Carbonic  Acid. 

4434 
5282 
5880 
7488 
8179 
8790 
8880 
8960 
900 

It  will  be  seen  from  this  table  that,  while  the  addition  of  5  per  cent, 
of  air  to  pure  hydrogen  alters  the  time  of  eflusion  by  about  20  per  cent., 
the  mixture  of  half  hydrogen,  half  air,  has  a  time  of  effusion  which  only 
diflfei's  from  that  of  pure  air  by  about  8  per  cent.  Thus  the  addition  of 
hydrogen  to  air  has  little  influence  on  the  viscosity,  while'~tITe~ addition 
of~air  to  hydrogenrliaE^n  enormous  influence. 

ResistaHct^cTarSolttlrTTrovli^^^  a  Viscous  Fluid.— When 

a  solid  moves  through  a  fluid  the  portions  of  the  fluid  next  the  solid  are 

*  Kayleicb,  Proc.  Boy.  Soc,  66,  p.  68. 
t  Koch,^Wicd.  Ann.,  19,  p.  587. 


100 

0 

97-5   . 

2-5 

95 

5 

90 

..   10 

75   . 

..   25 

50 

..   50 

25 

..   75 

10 

90 

0 

..   100 

Uydrogen 

and  Afr. 

4321 

100 

0 

4714 

95 

5 

5157 

90 

.    10 

5722 

75    .. 

25 

6786 

50 

50 

7339 

25 

.   75 

7535 

10 

90 

7521 

5 

95 

•7470 

0 

.   100 

222  PROPERTIES  OP  MATTER. 

moving  with  the  same  velocity  as  tlie  solid,  while  the  portions  of  the  fluid  at 
some  distance  ofi"  are  at  rest.  The  movement  of  the  solid  thus  involves 
relative  motion  of_the  fluid ;  the  viscosity  oT  the  fluid  resists  tins  motion, 
so  thatThereTs  a  force  acting  on  the  solid  tending  to  resist  its  motion. 

Sir  George  Stokes  has  shown  tliat  in  the  case  of  a  sphere  moving  with 
a  very  small  uniform  velocity  V  through  the  fluid  the  force  resisting  the 
motion  is  equal  to  67r>;ftV  where  a  is  the  radius  of  the  sphere,  ri  the 
viscosity  oT~the  fluid  through  which  it  is  falling.  Consider  now  the  case 
of  a  sphere  falling  through  a  viscous  fluid  ;  just  after  starting  from  rest  the 
velocity  will  be  small  and  the  weight  of  the  sphere  will  be  greater  than 
tne  viscous  resistance;  the  velocity  of  the  sphere,  and  therefore  the 
i-esistance,  will  increase  until  the  resistance  is  equal  to  the  weight  cf  the 
sphere.  When  this  velocity,  which  is  called  the  critical  velocity,  is  reached, 
the  forces  acting  on  the  sphere  will  be  in  equilibriuna^  and  Hie  sphere  \vill 
fa]ljyvathjjmiiorm^elocit^^  the  terminal  velocity. 

Since  the  effective  weightoFThe^here  is  equal  to~i7r<r(p  -  o-)^/3,  where  p  is 
the  density  of  the  sphere  and  o-that  of  the  liquid  through  which  it  is  moving, 
if  V  is  the  terminal  velocity, 

6 

or  ^2gAp^  ^j^ 

so  that  the  terminal  velocity  is  proportional  to  the  square  of  the  radius 
of  the  sphere.  In  the  case  of  a  drop  of  water  falling  through  air  for  which 
ij  =  1"8  X  10"*,  we  find,  if  the  radius  "of  the  drop  is  1/100  of  a  millimetre, 
"V'=  l'2cm^sec.  This  result  explains  the  slow  rate  at^which  clouds  con- 
sisting of  fine  drops  of  water  fall.  Since  ri  is  independent  of  the  pressure, 
the  terminal  velocity  in  a  gas  will,  since  c  in  this  case  is  small  compared 
"with  p,  be  independent  of  tJ^^'p^^essure. 

As  an  applicjxtion  of  this  formula  we  may  mention  that  the  size  of  small 
drops  of  water  has  been  determined  by  measuring  the  rate  at  which  they 
fell  through  air ;  from  this  the  value  of  the  radius  can  be  determined  by 
equation  (1).  The  expression  for  tlie  resistance  experienced  by  the  sphere 
falling  through  the  viscous  liquid  is  obtained  on  the  supposition  that  the 
motion  of  the  liquid  is  so  slow  that  terms  depending  upon  the  squares  of 
the  velocity  of  the  liquid  can  be  neglected  in  comparison  with  those  re- 
tained. Now,  if  V  is  the  velocity,  p  the  density  of  the  liquid,  the  forces  on 
the  liquid  depending  upon  the  squares  of  the  velocity,  are  proportional  to 
the  gradient  of  the  kinetic  energy  per  unit  volume — i.e.,  to  the  gradient  of 
^pV-;  the  forces  due  to  viscosity  are  proportional  to  the  gradient  of  the 
viscous  stress.  If  a  is  the  radius  of  the  sphere,  the  distance  from  the 
sphere  at  which  the  velocity  may  be  neglected  is  pio[)ortional  to  a,  hence 
the  velocity  gradient  is  of  the  order  (V/a),  and  the  viscous  stress  T)Yla. 
Hence,  if  we  can  reject  the  effects  depending  on  the  squares  of  the 
velocity  in  comparison  with  the  effects  of  viscosity,  pV^  must  be  small 
compared  with  rfVja,  or  pVa  must  be  small  compared  with  rj.  Hence,  if 
the  preceding  solution  holds,  we  see,  by  substituting  for  V  the  value  of 

the  limiting  velocity,  that   -g—^ — ^—^  must  be  small.     Lord   Rayleigh  * 

'J  jj- 

*  Lord  Kajleigh,  Phil.  Ma<j.,  \b]  36,  p.  354. 


VISCOSITY  OF  LIQUIDS.  go.'J 

has  pointed  out  how  much  this  restricts  the  application  of  Stokes*  result; 
thus,  for  example,  in  the  case  of  drops  of  water  falling  through  air,  the 
theory  does  not  apply  if  the  drops  are  moi-e  tlum  about  one-tenth  of  a 
millimeti"e  in  radius.  When  the  velocity  of  the  falling  body  exceeds  a 
certain  critical  value  the  motion  of  the  surrounding  fluid  becomes 
turbulent,  just  as  when  the  velocity  of  a  fluid  through  a  capillary  tube 
^exceeds  a  certain  value  the  flow  ceases  to  be  regidar  (see  p.  212).  When 
this  turbulent  stage  is  reached  the  resistance  beconiesjjroportional  to  the 
8quare~of  tKe  velocity.  Mr.  Allen,*  who  has  recently  investigated  the 
resistance  experienced  by  bodies  falling  through  fluids,  finds  that  this  can 
be  divided  roughly  into  three  cases — (a)  where  the  velocity  is  very  small, 
when  the  preceding  theory  holds,  and  tlie  resistance  is  proportional  to  the 
velocity ;  (b)  a  stage  where  the  velocity  is  great  enough  to  make  the  forces 
tiepending  on  the  square  of  the  velocity  comparable  with  those  depending 
on  viscosity ;  in  this  stage  the  resistance  is  proportional  to  the  velocity 
raised  to  the  power  of  3/2  ;  (c)  a  stage  where  the  velocity  is  so  great  that 
the  motion  of  the  fluid  becomes  turbulent ;  in  this  stage  he  finds  the 
resistance  to  be  projDortional  to  the  square  of  the  velocity.  When  the 
resistance  is  proportional  to  the  squai-e  of  the  velocity  the  method  of 
dimensions  shows  that  it  does  not  for  a  given  velocity  depend  upon  the 
viscosity  of  the  liquid.  For,  suppose  the  resistance  is  proportional  to 
a'^p^ifY",  this  expression  must  be  of  the  dimensions  of  a  force — i.e.,  1  in 
mass,  1  in  length,  and  —  2  in  time ;  hence  we  have 

r>^'^4^  L^         ?=^+^    ,  y  -^^^' 

■  ^^'  P^  l=x-^-z  +  n 

x  =  n,  y  =  n-l,z  =  2-n, 

and  the  resistance  is  prgportjonal  to  {^(i(i/rj)"{7]'/p);  thus,  if  w  =  2  the 
resistance  is  proportional  to  Y^aFp,  and~is  independent  of  viscosity.  The 
energy  of  the  body  is  spent  in  producing  turbulent  motion  in  the  liquid 
and  not  in  overcoming  the  viscous  resistance. 

A  great  deal  of  attention  has  been  given  to  the  resistance  of  bodies 
moving  with  high  speeds,  such  as  bullets.  It  is  doubtful,  however,  if  the 
viscosity  of  the  fluid  tlu'ough  which  the  bullet  moves  has  any  efiect  upon 
the  resistance  ;  we  shall  not,  therefore,  enter  into  this  subject,  except  to 
say  that  the  most  recent  researches,  those  by  Zahm,  seem  to  indicate  that 
for  velocities  less  than  about  30000  cm. /sec.  the  resistance  may  be  repre- 
sented by  uv'  +  iy^,  where  a  and  h  are  constants. 

*  Allen,  Phil.  Mag.,  Sept.  and  Nov.  1900. 


7--.  _// 


INDEX 


Acceleration  due  to  gravity,  7-24 
Air,  deviations  from  Boyle's  law  as  to,  126 
Airy,  hydrostatic  theory  of  earth's  crust, 
23 

Dolcoath  experiment,  35 

Harton  pit  experiment,  35 
Amagat,  minimum  value  of  pv.,  126,  127 
Angle  of  shear,  66 

Arc,  correction  for  pendulum  swing,  10 
Atmolysis,  202 

Baily'S  Cavendish  experiment,  39 
Bailie  and  Cornu's  experiment,  39 
Bars,  bending  of,  85-102 

vibration  of,  94 
Barymeter,  von  Sterneck's,  26 
Bending  of  rods  or  bars,  85-102 
Cernouilli's  correction  for  arc  of  swing 

of  pendulum,  10 
Boiling-point,  depression  of,  in  solutions, 

191 
Borda's  pendulum  experiments,  10 
Bouguer's  pendulum  experiments,  10 

experiments    on    determination    of 
density  of  earth,  32 

rule  and  exceptions,  22-3 
Boyle's  law,  125 

at  low  pressures,  128 

deviations  of  various  gases  from,  126 
Boys's  Cavendish  experiment,  40 
Braun's  Cavendish  experiment,  41 
Breaking-point  of  stretched  wires,  55 
Bubbles     and    drops,    measurement    of 
surface  tension  by,  156,  161 

Camphor,  movements  of  on  surface  of 

water,  169 
Capillarity,  135-181 

Laplace's  theory  of,  173-181 
Capillary  tubes,  rise  of  fluids  in,  140 
Carbonic  acid,  deviation  of,  from  Boyle's 

law,  126 
Carlini's  pendulum  experiment,  35 
Cassini's  and  Borda's  pendulum  experi- 
ment, 10 
Cavendish  experiment,  36 
by  other  observers,  39 
see  Earth,  determination  of  density  of 
Clairaut's  theorem,  22 


Collision,  109 

duration  of,  on  impact,  112 

of  drops,  172 

see  also  Impact 
Colloids,  186 

Compressibility  of  liquids,  see  Liquids 
Computed  times  of  pendulums,  15 
Contamination  of  films,  170 
Critical  velocity  in  viscous  fluids,  222 
Crystalloids,  186 

Defforges'  pendulum,  19 

Degree  of  latitude,  measurement  of  a, 

21 
Diaphragm,  diffusion  through,  186,  200 
Differential  gravity  balance,  26 
Diffusion  of  gases,  see  Gases 

of  liquids,  see  Liquids 

of  metals,  204 
Dilatation  under  strain,  64 
Dissociation  of  electrolytes,  194 

Earth,  determination  of  density  of,  31 
by  Airy,  35 
Baily,  39 
Buuguer,  32 
Boys,  41 
Braun,  41 
Carlini,  35 
Cavendish,  36 
Cornu  and  Bailie,  39 
von  Jolly,  42 
Maskelyne,  33 
Mendenhall,  35 
Poynting,  43 

Richarz  and  Krigar-llenzel,  42 
von  Sterneck,  36 
Wilsing,  41 
Effusion,  thermal,  202 
Elastic  after-effect,  55 
curve,  95 
fatigue,  57 
limit,  53,  69 
Elasticity,  53 

modulus  of,  69,  102 
see  also  Young's  Modulus 
Electrolytes,  dissociation  of,  194 
Ellipticity  of  earth,  23,  24 
Elongation  under  strain,  64 


226 


INDEX 


Equilibrium  of  liquids  in  contact,  139 
Equivalent  simple  pendulum,  13 

Fatigue,  elastic,  57 

Faye's  rule,  23 

Films,  contamination  of,  170 

cooline;  cflects,  on  stretching,  163 
stability  of  cylindrical,  147 

Flexure,  99 

Floating  bodies,  forces  acting  on,  153 

Fluid  motion,  effect  of,  on  pendulums,  14 
surfaces,  disruption  of,  174 

Formulae  for  pendulum  motion,  13-24 

Freezing-point,    depression   of    in  solu- 
tions, 193 

Galileo's  observations  respecting  pen- 
dulums, 8 
Gaseous  pressures  and  volumes,  124 
Gases,  diffusion  of,  196 

kinetic  theory  as  applied  to  the,  198 

obstruction  to,  offered  by  perforated 
diaphragms,  200 

through  porous  bodies,  201 
Gases,  passage  of,  through  india-rubber, 
203 

through  liquids,  203 

through  red-hot  metals,  204 
Gases,  viscosity  of,  210,  218 

influence  of  temperature  upon,  221 
Gravitation,  constant,  29 

Newton's  law,  28 

qualities  of,  45-52 

see  also  Earth,  density  of 
Gravity,  acceleration  of,  7 

history  of  research,  as  to,  7 

Clairaut's  theorem,  22 

Newton's  theory  of,  20 

Richer's  observations  on,  20 

Swedish   and  Peruvian  expeditions 
of  investigation,  21 
Gravity  balance,  Threlfall  and  Pollock's, 

27 
Gravity  meters,  differential,  26 

Half-seconds  pendulum,  von  Sterneck, 

24 
Hodgkinson's   table   of  values   of  e   on 

impact,  114 
Homogeneous  strain,  62 
Hooke's  law,  69 
Hydrogen,    deviations    of,  from  Boyle's 

law,  126 
Hydrostatic  theory,  23 
Huygens'  pendulum  clock,  9 
theory  of  pendulums,  9 

Indian  survey,   experiments  on  pendu- 
lums,'23 
Impact,  109 

duration  of  collision  on,  112 

kinetic  energy  of,  110 
Invariable  pendulum,  28 


Jaeger's  method  of  determining  mean 

surface-tension,  162 
JoUv,  von,  experiments  on  gravitation 

42 

Kater's  convertible  pendulum,  12 
and  Sabine's  experiments,  23 
Kelvin's    table   of    thermal   effects    ac- 
companying strain,  134 
Kinetic  theory  of  gases,  218 

explanation    of    diffusion    by    the, 
198 

Laplace's  theory  of  capillarity,  173 
Latitude,  determination  of  length  of  1' 

of,  21 
Liquids,  capillarity  of,  135 

compressibility  of,  116,  122 
diffusion  of,  183 

determination    of   co-efEcient  ol 

184 
through  membranes,  186 
in  contact,  139 
films,  stability  of,  147 
flow  of  viscous,  through  cylindrical 

capillary  tubes,  207 
potential  enerey  of,  due  to  surface 

tension,  137 
rise  of,  in  capillary  tubes,  140 
surface-tension  of,  137 

relation  between  curvature  and 

pressure  of  surface,  142 
methods  of  measuring,  155 
by  bubbles  and  drops,  156, 161 
by  ripples,  157 
temperature,  effects  on,  163 
table  of  compressibility  of  various, 

122 
tensile  strength  of,  122 
vapour-pressure  over  curved  surface 

of,  166 
viscosity  of,  205 
Loaded  pillar,  stability  of,  97 

wires,  anomalous  effects  in,  58 
Lubrication,  217 

Mass,  3 

constancy  of,  5 

definition  of,  4 

unit  of,  5 
Maxwell's  law  of  gaseous  viscosity,  218 
Mean  free  path,  218,  220 
Mendenhall's  gravitation  experiment,  35 
Mercury,  compressibility  of,  121 
Metals,  diffusion  of,  through  metals,  204 

elastic  properties  of,  53,  57 

viscosity  of,  57 
Michell,  I^lev.  J.,  36 
Microstructure  of  metals   under  stress 

58 
Modulus  of  elasticity,  69,  102 
Young's,  70,  73,  74,  76 

of  rigidity,  7 


INDEX 


227 


Newton's  theory  of  gravitation,  28 

theory  of  gravity,  20 
Nitrogen,  deviation  of,  from  Boyle's  law, 

126 
Normal  stress,  68 

Oil,  effect  of,  on  waves,  171 
Osmosis,  186 
Osmotic  pressure,  188 

Pendulums,  Bessel's  experiments,  13 

Borda  and  Cassini's,  10 

clock,  9 

Defforges,  19 

formulas  for,  13-24 

Half -seconds  pendulum,  24 

Huygens'  theory  of,  9 

Indian  survey  experiments,  23 

invariable,  23 

Kater's  convertible,  12 

Newton's  use  of,  9 

Papers  on  the  theory  of,  7 

Kepsold's,  18 

von  Sterneck's,  24 

U.S.  survey,  20 

variation  in  length  of  seconds,  2 

yielding  of  support  of,  18 
Permanent  set,  53 
Picard's  pendulum  experiments,  9 
Piezometer  (the),  119 
Poiseuille's  law,  209 
Poisson's  ratio,  73,  87,  120 
Poynting's  gravitation  experiments,  43 
Pressure,  effect    of,   on    viscosity,   217, 
219 
on  volume,  124 

variations  from  Boyle's  law  at  low, 
128 

QUAETZ  thread  gravity  balance,  Threl- 

fall's,  27 

Reich's  Cavendish  experiment,  39 
Repsold's  pendulum,  18 
Resolution  of  strain,  65 
Reversible  pendulum,  theory  of,  13 
Reversible  thermal  effects  accompanying 

strain,  131 
Richer,  observations  on  gravity,  20 
Rigidity,  co-efficient  of,  83 

modulus  of,  70 
Ripples,  measurement  of  surface-tension 

by,  157 
Rods,  stresses  and  strains  of,  71,  73,  79, 

83,  85-102 

Sabine's  pendulum,  23 
Salt  solutions,  viscosity  of,  217 
Schiehallion  experiment,  32 
Shear,  65 

angle  of,  66 
Soap-bubbles,  143 


Solutions,  depression  of  boiling-point  of, 
191 
of  freezing-point  of,  193 
vapour  pressure  of,  190 
Spiral  springs,  101-108 
energy  of,  104-108 
Stability  of  cylindrical  films,  147 

of  loaded  pillar,  97 
Sterneck,  von,  Barjmeter,  26 
half-seconds  pendulum,  24 
pendulum  experiments,  36 
Strain,  62 

anomalous  effects  of  alternating,  on 

wire,  58 
alteration   of   micro-structure   con- 
sequent on,  59 
axes  of,  64 
homogeneous,  62 

resolution  of  a,  65 
in  relation  to  work,  70 
thermal  effects  accompanying,  131 
Stresses,  68 

on  bars,  71 
Stretched  film,  144 

cooling  due  to  stretching,  163 
Stretched   wire,    anomalous    effects    on 

loading,  58 
Surface-tension,  137 

effects  between  two  liquids,  179 

in  thick  films,  178 
forces  between  2  plates,  due  to,  152 
Surface-tension,     Jaeger's     method    of 
measuring,  162 
oscillations    of    a     spherical    drop 

under,  160 
of  thin  films,  164 
measurement  of  by  detachment  of 

a  plate,  161 
Ripple  method,  157 
Wilhelmy's  method,  161 
Swedish  and    Peruvian   expeditions   to 
determine  length  of  1°  of  lati- 
tude, 21 

Table  of  moduli  of  elasticity,  102 

thermal  effects  of  strain,  131 
Tangential  stress,  68 
Temperature,  co-efficient  of  viscosity,  216 

effects  of,  on  surface-tension,  163 
on  breaking  stress  of  wires,  61 
Tensile  strength  of  liquids,  123 
Terminal  velocity  in  viscous  fluids,  222 
Thermal  effects  of  strain,  131 

Kelvin's  table  of,  134 
Thermal  effusion,  202 
Thickness     of    films,    influence  of,    on 

surface-tension,  178 
Thin  films,  surface-tension  of,  114 
Threlfall  and  Pollock's  gravity  balance, 

27 
Torsion,  78 

in  cylindrical  tubes,  78 

in  solid  rods,  79 


228 


INDEX 


U.S.  Survey  pendulums,  20 

VArouR,  diffusion  of,  197 
Vapour  pressure,  of  solutions,  190 

on  curved  surfaces,  166 
Vibration  of  bars,  95 
Viscosity,  60 

temperature  co-efficient  of,  216 

determination    of   co  -  efficient 
212 
by  oscillating  disc,  214 

effects  of  pressure  upon,  217 

gaseous,  effect  of  temperature 
221 

of  gases,  218 

of  liquids,  205 

of  metals,  57 

of  mixtures,  221 

of  salt  solutions.  21 7 


of, 


on, 


Viscous  fluids,  resistance  of,  to  motion  of 
solids,  221 
velocity  in,  222 
Volume  and  pressure  of  gases,  124 

Water,  compressibility  of,  121 
Waves,  calming  of,  by  oil,  171 
Weight,  1 

standards  of,  5 
Wilhelmy's  method   of  measuring   sur- 
face-tension, 162 
Wilsing's  gravitation  experiments,  41 
Work  in  relation  to  strain,  70 

Yield  point,  53 
Young's  modulus,  70,  73 
determination  uf,  74 
by  flexure,  99 
by  optical  measurement,  76 


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ADAMS  (W.  P.).  Motor  Car  Mechauism,  -    21 

AITKEN  (T.),  Road  Making,        ...    63 

ALFOKD  (C.  J.),  Miniui;  Law.    -        -        -    42 

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BILES  (Prof.),  Works  by.      -        -        -       19,31 

BJdKLlNG  (P.  R.),  Peat,      -        -        -        -    59 

BLACKMOi{E  (E  ).  Mercantile  Marine,     -    33 

BLOUXT  cfc  BLOXAM,  Obemistrv        -        -    .05 

BLYTH  I A    Wynter),  Foods  and  Poisons,    56 

BOHLE  A  ROBERTSON.  Transformers,    -    29 

BORCHERS  (Dr.).  Electric  Smelting,         -    52 

BROUGH  (B.  H.).  Mine  Surveying,    -        -    41 

BROWNe  (W.  R.),  Works  Dy,     -  -    26 

BRUCE  (Robt.),  Food  Supply,      -        -        -     45 

BUCK  (R   C. ),  Algebra  &  Trigonometry,-    35 

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BU  I'LER,  Carburettors.        ...       - 

BUTTERFIELD.  (W.  J.  A.),  Gas, 

CAIN  &  THORPE.  Synthetic  Dvestuffs, 

CARTER  (H.  R.).  Long  Fibre  Spinning,  - 

CASTELL-EVANS.  Chemical  Tables 

CHATLEY  (H).  Works  bv.  - 

CLAYWORKERS  HANDBOOK. 

COLE  (Prof.  Q.  A.  J.).  Works  by, 

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H00'4HWINKEL  (G.),  Signalling  in  Mines, 
HUGHES  (H.  W  ).  Coal  Mining, 
HUGH    S-GIBB(Mrs.).  Works  by,      - 
HURST  (Chas  ■).  Works  by, 
HURST  (G.  H.).  Works  by.  -       -        . 

INGLE  (H.),  .\gricultural  Chemistry,  -  .57 
.TAGGER  (J.  E.),  Mechanical  Drawing,  -  24 
JAMIESON  (Prof.).  Manuals,  -  -  -  27 
JOHNSON  (J.  C.  P.),  Getting  Gold,  -        -    44 

J  ■'■rgENSEN(A.),  Micro-Organisms,  -  -  5" 

JUDE  (A.).  Steam  Turbine.  -  -  -  i9 
JULIAN  (H.  F.)  and  SMART  (E), 

Cyaniding  Gold  and  Silver  Ores,      -  -  44 

KASSNER  (T.),  Gold  Seeking,    .        -  -  44 

KERR  (G    L.).  Works  by,    -        -        -  -  40 

KERSHAW  (G.  B.).  Sewage  Disposal,  -  60 

KNECHT   &  RAWSON,  Dyeing,  -        -  6 


I'AGK 

LAF.\R,  Technical  Mycologv        -        -        -  58 

LAMBERT,  Glue,  Gelatine,  Ac.  -        -        -  65 

L\RARD  (C,  E.),  Eng.  Calculations,          -  25 

LAW  (E.  F).  Alloys,     -----  49 


42 
61 
il 
19 
35 


LAWN  (J.  G.).  Mine  Accounts, 
LEEDS  (F.  H.),  Acetylene,  - 
LIECKFELD    (G.),  Oil  Motors,  - 
LIVERSIDGE,  Engine-Room  Practice, 
MACKENZIE  (T.).  Mechanics.   - 
MACLEOD  (W.  A.)  and  WALKER  (C), 

Metallurgi  ai  Chemistry,    -        -        -        -     51 
M-LAREN,  (R.  S. ).  Elem.  Meeh.  Eng.,      25, 
McMillan  (W.  G),  Electro-Metallurgy,- 

&  BORCHERS    Electric  Smelting,      - 

McWILLIAM,  Foundry  Practice, 
MaSTIN  (J.)  ScieuiiUc  Romances 
MES8UM  (S.).  Hydrographic  Surveying,  - 
MIDDLETON  (R.  E.)    Water  Supplv, 
MILL  (Dr.  R    H.)   New  Lands,   -      "- 
MILLAR  (W.  J  ).  Latitude  &  Longitude,  - 
MITCHELL  (C,  A.),  Flesh  Foods,      - 

&  HEPWORTH  (T.  C),  Inks,     - 

MORGAN  (J.  J.),  Works  by.        -        -      52, 
MUNRO  &  JAMIESON'S  Elect.  Pkt-bk..  - 
MUNRO  (R.  D.),  Works  by,         -        -        . 
NAYLOR  (W.).  Trad(>s-  Waste,  - 
NICHOLSON  (W.).  Smoke  Abatement,     - 
NORMANDY  (F.),  Sea  Water  Distillation, 


52 
53 
53 
76 
16 
61 
45 
36 
56 
63 

30 
23 
60 
61 
19 

NORTH  (S.).  Oil  Fuel, 54 

OPPENHEIMER  (C),  Works  by.  -  -  58 
OSMOND  &  STEAD.  Micro.  Anal.  Metals,  52 
PARK  (J  ).  Works  by,  -  -  -  -  41,  44 
PEARCE  (W.  J.),  PaintiniT.  -        -        -     64 

PETTIGREW  (W.  F),  Loco.  Eng.,  -  -  20 
PHILLIPS  &  BAUERMAN,  Metallurgy,  -  51 
PHIPSON  (Dr.  T.  L.),  Earth's  Atmosphere,  46 
POSCHL(V.),  Chemistry  of  Colloids,-  -  65 
POYNTING  (J.  H.),  Mean  Density,      -        -    72 

<fc  THOMSON,  Physics,        -       -        -    73 

PRAEGER  (R.  L.),  Open  Air  Botany,  -  71 
PUPPE  (J.),  Rolling  Mills,  -  -  -  -  23 
RANKINES  Works,  -----  28 
RAWSON,  GARDNER,  &  L.\YCOCK, 

Dictionary  of  Dyestuffs,  -  -  -  -  66 
REDGRAVE  (G.  R.).  Cements.  -  -  -  60 
REDWOOD  (Sir  Boverton),  Petroleum,     -    54 

&  THOMSON.  Handbook,  Petroleum,      54 

REED(Sir  E  J.),  StabiUty  of  Ships,  -  -  31 
REID  (Geo.,  M.D.),  Sanitation,  -  -  -  62 
RICHMOND  (H.  D.),  Works  by,  -  -  57 
RIEMER  (J.).  Shaft  Sinking.  -  -  -  43 
ROBERTS- AUSTEN,  Metallurgy,  -  -  48 
ROBINSON  (Prof.),  Hydraulics,  -  -  -  17 
ROSE  (T.  L.),  Gold.  Metallurgy  of,  -  -  48 
ROTHWELL.  (0,  F.  S.),  Textile  Printing,  67 
SCHWARTZ  (Dr.  von).  Fire  Risks,  -  -  61 
SEATON  (A.  E.),  Works  by,  -  -  -  18 
SEXTON  (Prof.).  Works  by,  -  -  51,  69 
SHEL  ION-BEY,  Mechanics  Guide,  -  -  '26 
SHOWELL  (P  G.),  Navigation  Deflnitions,  37 
SINDALL,  Paper  Technology.  -  -  -  59 
SMITH  (C.  A.),  Suction  Gas  Plants,  -  -  32 
SMITH  (J.  W.),  Dustless  Road.s,-  -    63 

SMITH  (Prof.  R.  H.),  Works  by,  -  -  26 
SMITH  (W.),  Shipmaster's  Medical  Help,  36 
SOLOMON  (H.  G.).  Electricity  Meters,  -  29 
SOLON,  Ceramic  Literature,  -  -  -  59 
SORSBIE  (  .),  Geology  for  Engineers,  -  46 
S  I'EWART  (R.  W.).  Elementary  Physics,  -  72 
STRICKLAND  (F,),  Petrol  Motors,  -  -  21 
SUPLEE  (H.  H.).  Mech.  Pocket-Book,      -    26 

The  Gas  Turbine. 22 

SYKES  (Dr.  W.  J.),  Brewing,  -  -  -  58 
TAYLOR  (F.  N.),  Civil  Eng.  Practice,  -  17 
TRAILL  (T.  W.),  Boilers,  .  -  -  -  19 
TROTMAN  (S.  R.),  Leather,  -  -  -  65 
TUNZELMANN  (G.).  Electrical  Theory,  -  30 
TURNER  (Thos.),  Works  by.  -  50,  51,  53 
WALTON  (T),  Works  bv,  -  -  -  32,36 
WANG  (C.  Y.).  Antimony,  -  -  -  -  50 
WEBER  (Dr.  C.  O.),  India  Rubber  -  -  65 
WELLS  (S,  H.),  Engineering  Drawing,  -  24 
WIQLEY  (T.  B.).  Jewellers  Art.  -  -  53 
WILLOUGHBY  (Dr.  E.  F.).  Milk,  -  -  57 
WOOD  (Francis),  Sanitary  Engineering,  -  62 
WORDINGH.\M,  Electrical  Stations,  -  29 
WRIGHT  (Dr.  A.).  Oils  and  Fats,  -  -  65 
YULE  (G.  U.),  Theory  "f  Statistics,  -  -  74 
ZERR  &  RUBENCAMP,  Colour  Manuf.,    -    64 


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BOILERS,  MARINE  AND  LAND: 

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A  Handbook  of  Rules,  Formulu«,  Tables,  &o.,  relative  to  Materials, 

Scantlings,  and  Pressures,  Safety  Valves,  Springs, 

Fittings  and  Mountings,  &o. 

FOR  THE  USE  OF  ENGINEERS,  SURVEYORS,  BOILER-MAKERS, 

AND  STEAM  USERS. 

By  T.   W.    TRAILL,    M.  Inst.  0.  E.,    F.  E.  R  N., 

Late  Engineer  Surveyor-in-Chief  to  the  Board  of  Trade. 
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LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAND 


20  CHARLES  ORIFFIN  <fc  CO.'S  PUBLICATIONS. 

Third  Edition,  Revised.     Pp.  i-xv  +  356.     With  Frontispiece,   8  Plates 
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A      MANUAL     OF 

LOCOMOTIVE    ENGINEERING: 

A  Practical  Text-Book  for  the  Use  of  Engine  Builders. 

Designers  and  Draughtsmen,  Railway 

Engineers,  and  Students. 

By  WILLIAM   FRANK  PETTIGREW,  M.Inst.C.E. 

Contends,  —  Historical  Introduction,  1763-1863.  —  Modem  Locomotives:  Simple. — 
Modern  Locomotives:  Compound.  —  Primary  Consideration  in  Locomotive  Design.— 
Cylinders,  Steam  Chests,  and  StufQng  Boxes. — Pistons,  Piston  Rods,  Crossbeads,  and 
Slide  Bars. — Connecting  and  Coupling  Bods. — Wbeels  and  Axles,  Axle  Boxes,  Hornblocks, 
and  Bearing  Springs.— Balancing.— Valve  Gear. — Slide  Valves  and  Valve  Gear  Details.— 
Framing,  Bogies  and  Axle  Trucks,  Radial  Axle  Boxes.— Boilers.— Smokebox.  Blast  Pipe, 
Firebox  Fittings.— Boiler  Mountings. — Tenders.  -  Railway  Brakes.— Lubrication.— Con- 
sumption of  Fuel,  Evaporation  and  Engine  Efficiency.- Repairs,  Running,  Inspection, 
anl  Renewals. — Three  Appendices  — Index. 

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LOCOMOTIVE  COMPOUNDING  AND  SUPERHEATING. 

By    J.    F.    GAIRNS. 

Contents.— Introductory.— Compounding  and  Superheating  for  Locomotives.— A 
Classification  of  Compound  Systems  for  Locomotives. — The  Hiotory  and  Development  of 
the  Compound  Locomotive. —Two-Cylinder  Xon-Automatic  Systems.  —  Two-Cylinder 
Automatic  Systems. — Other  Two-Cylindei-  Systems.— Three-Cylinder  Systems. — Four- 
Cylinder  Tandem  Systems.— Four-Cylinder  Two-Crank  Systems  (other  than  Tandem).— 
Four-Cylinder  Balanced  Systems. — Kour-Cylincier  Divided  and  Balanced  Systems. — 
Articulated  Compound  Engines. — Triple-Expansion  Locomotives. — Compound  Rack 
Locomotives. — Concluding  Remarks  Concerning  Compound  Locomotives. — The  Use  of 
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L  I  O  H  T        R  AI  IL.  laZ^  A  Y  S 

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Arrangement  of  Engines.— Ignition. —Carburettors. —Cylinders,  Pistons,  Valves,  &c.— 
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Universal  Joints.— Axles.  —  Springs.  —  Radius  Rods.  —  Brakes.  —  \^^^eels.  —  Frames.  — 
Steering  Gear.  —  Radiator.  —  Steps,  Mudguards,  Bonnets,  &c.  —  Lubrication.  —  Ball 
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—  'hip  Boat,  and  Air-ship  Engines.— Vehicles,  &c.,  Driven  by  Internal  Combustion 
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I  CHARLES  GRIFFIN  &  CO..  IIMITED.  EXbTER  STREET.  STRANB. 


22  CHARLES  GRIFFIN  d:   CO.'S  PUBLICATIONS. 

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A    TEXT-BOOK    ON 

GAS,  OIL,  AND  AIR   ENGINES. 

By  BRYAN  DONKIN,  M.Inst.C.E.,  M.Inst.Mech.F.. 
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by  Prof.  BuRSTAi.i.. 

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Br  AND 

LEONARD  ARCHBUTT,  F.I.C.,  F.C.S.,  R.  M.  DEELEY,  M.LMech.E  ,  F.G.S. 

Chemist  to  the  Mid.  lly.  Co.  Chief  Loco.  Svii>er.,  Mid.  Ry.  Co. 

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their  Sources,  Preparation,  and  Properties.— VI.  Physical  Properties  and  Methods  oJ 
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of  Lubricants. — VIII.  The  Systematic  Testing  of  Lubricants  by  Physical  and  Chemical 
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of  Bearings.— XI.  The  Lubrication  of  Machinery.— Index. 

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THE  FORCE    OF   THE  WIND. 

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KITCHEN     BOILER    EXPLOSIONS:    Why 

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2 


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&c.— Repeaters. —  -Submarine  Telegraphy. —  Wireless  Telegraphy. —Index.  — List  of 
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THE       IDESXOIN^ 

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By  JOHN   HARVARD   BILES,   M.Inst.N.A., 

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32  CHARLES  QRIFFIN  di  GO.'S  PUBLICATIONS. 

WORKS     BY     THOMAS     WALTON, 

NAVAL     ARCHITECT. 


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"Simple    language     .      .      .      clear    and    easily    followed    illustrations."  —  Times 
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these  subjects. 

KNOW    YOUR    OWN    SHIR 

By  THOMAS  WALTON,  Naval  Architect. 
Specially  arranged  to  suit  the  requirements  of  Ships'  Officers,  Shipowners. 
Superintendents,  Draughtsmen,  Engineers,  and  Others, 
Contents.  —  Disphicement  and  I)e;id weight.  —  Moments.  —  Buoyancy.  —  Strain.  — 
Structure.  —  Stability.  —  Itolling.  —  Ballasting.  —  Loading.— Shifting  Cargoes.— Effect  of 
Admission  of  Water  into  Ship.— Trim  Tonnage.— Freeboard  (Load-line).— Calculations.— 
Set  of  Calculations  from  Actual  Drawings.  — Indkx. 

"  The  work  13  of  the  liifrhest  vaUie,  and  all  who  sio  down  to  the  sea  in  ships  should  make  them- 
aelvea  acnuaiiited  with  it." -Shippinri  TVorUl  (on  tlif  new  editioiil. 

LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAND. 


NAUTICAL    WORKS.  33 


GRIFFIN'S   NAUTICAL   SERIES. 

Introductory   Volume.     Pp.  i-xix  +  248.     Price  Ss.  6d. 

THE 

British  Mercantile  Marine. 

By  EDWARD    BLACKMORE, 

MASTER  MARINER;    ASSOCIATE  OF  THE  INSTITUTION  OF  NAVAL  ARCHITECTS; 

MEMBER  OF  THE  INSTITUTION  OF  ENGINEERS  AND  SHIPBUILDERS 

IN  SCOTLAND;   EDITOR  OF  GRIFFIN'S   "  NAUTJCAL  SERIES." 

General  Contents. — Historical  :  From  Early  Times  to  1486— ProgresB 
ander  Hem^r  VIII.— To  Death  of  Mary— During  Elizabeth's  Eeign— Up  to 
the  Reign  of  William  III.— The  18th  and  19th  Centuries— Institution  ot 
Examinations  —  Rise  and  Progress  of  Steam  Propulsion  —  Development  of 
Free  Trade- Shipping  Legislation,  1862  to  1875— "  Locksley  Hall"  Case- 
Shipmasters'  Societies— Loading  of  Ships — Shipping  Legislation,  1884  to  1894 — 
Statistics  of  Shipping.  The  Personnel  :  Shipowners— Officers— Mariners- 
Duties  and  Present  Position.  Education  :  A  Seaman's  Education :  what  it 
Bhould  be— Present  Means  of  Education— Hints,  Discipline  and  Duty— 
Postscript— The  Serious  Decrease  in  the  Number  of  British  Seamen,  a  Matter 
demandiiig  the  Attention  of  the  Nation. 

"  Interesting  and  Instbuctive  .  ,  .  may  be  read  with  profit  and  enjotkbnt."- 
Blatgow  Herald. 

"  Evert  brakch  of  the  subject  is  dealt  with  in  a  way  which  shows  that  the  writer 
'knows  the  ropes'  familiarly." — Scotsman. 

"  This  admirable  book  .  .  .  teems  with  useful  information— Should  be  in  the 
luuids  of  every  Sailor." — Western  Morning  Neus. 


Fifth  Edition,  Thoroughly  Revised.  Pf.  i-xvi  +  250.  With 
Frontispiece,  24-  Plates  {3  Coloured),  and  63  Illustrations 
in  the  Text  and  new  Chapter  on  Clouds.     Price  6s. 

ELEMENTARY     SEAMANSHIP, 

BY 

D.  WILSON-BARKER,  Master  Mariner;  F.R.S.E.,  F.R.G.S.,&o.,  &c. 

younger  brother  of  the  trinity  house. 

General  Contents.— The  Building  of  a  Ship;  Parts  of  Hull,  Masts, 
&c,  -Ropes,  Knots,  Splicing,  &c.  —  Gear,  Lead  and  Log,  &c,  —  Rigging, 
Anchors  — Sailmaking  — The  Sails,  &c.— Handling  of  Boats  under  Sail  — 
Signals  and  Signalling— Rule  of  the  Road— Keeping  and  Relieving  Watch- 
Points  of  Etiquette— Glossary  of  Sea  Terms  and  Phrases— Index. 

•^*  The  volume  contains  the  new  rules  of  the  eoad. 

"  This  ADMiBABtE  manual,  by  Oapt.  Wilson-Barker  of  the  'Worcester,'  seems  to  ub 
PERFECTLT  DEsiGKED,  and  holds  its  place  excellently  in  '  Griffin's  Nautical  Series.'  .  .  . 
Although  intended  for  those  who  are  to  become  Officers  of  the  Merchant  Navy,  it  will  be 
found  useful  by  all  yachtsmen."— .4 (Aen««7n. 


LONDON  :  CHARLES  GRIFFIN  &  CO.,  LIMITED.  EXETER  STREET.  STRAND. 


34  CHARLES  QRIFFIN  d:  CO.'S  PUBLICATIONS. 

GRIFFIN'S   NAUTICAL  SERIES. 

Second  Edition,  Revised.     Pp.  i-xii  + 156.      With  61 
Illustrations.     Price  Ss.   6d. 

NAVIGATION^: 

By  DAVID  WILSON-BARKER,  R.N.R.,  F.R.S.E.,  <fec.,  «feo.» 

AND 

WILLIAM   ALLINGHAM, 

riKST-OLASa  HONOURS,   NAVIGATION,  SCIENCE  AND  ART  DEPARTMENT. 

TKIlitb  IRumerous  illustrations  anD  JEjamination  (Sluesttons, 

General  Contents. — Definitions — Latitude  and  Longitude — Instruments 
of  Navigation — Correction  of  Courses— Plane  Sailing — Traverse  Sailing — Day's 
Work  —  Parallel  Sailing  —  Middle  Latitude  Sailing  —  Mercator's  Chart— 
Mercator  Sailing — Current  Sailing — Position  by  Bearings — Great  Circle  Sailing 
— The  Tides — Questions — Appendix :  Compass  Error — Numerous  Useful  Hints, 
&c. — Index. 

"  Pbkciselt  the  kind  of  work  required  for  the  New  Certiflcatea  of  competency  in  grade» 
from  Second  Mate  to  extra  Master.  .  .  .  Candidates  will  find  it  invaluable. "—Sundw 
Advertiser. 

"A  CAPITAL  LITTLE  BOOK  .  .  .  Specially  adapted  to  the  New  Examinations.  The 
Authors  are  Capt.  Wilson-Barker  (Captain-Superintendent  of  the  Nautical  College,  H.M.S. 
'  Worcester.'  who  has  had  great  experience  in  the  highest  problems  of  Navigation),  and 
Mr.  Allinoham,  a  well-known  writer  on  the  Science  of  Navigation  and  Nautical  Astronomy.'' 
—  Shipping  World. 


Handsome  Cloth,     Pp.   i-xvi  +  lS2.      With  10  PlaltH  and  34  oihiv 
Ili'ustration^.     Price  7s.  6d. 

MARINE    METEOROLOGY, 

FOR  OFFICERS  OF  THE  MERCHANT  NAVY. 
By   WILLIAM   ALLINGHAM, 

Joint  Author  of  "Navigation,  Theoretical  and  Practical." 

With  numerous  Plates,  Maps,  Diagrams,  and  Illustrations,  and  a  facsimile 
Reproduction  of  a  Page  from  an  actual  Meteorological  Log-Book. 

SUMMARY   OF    CONTENTS. 

Introductory.— Instruments  Used  at  Sea  for  Meteorological  Purposes. —Meteoro- 
logical Log-Books.— Atmospheric  Pressure. — Air  Temperatures.— Sea  Temperatures.— 
Winds.— Wind  Force  Scales.— History  of  the  Law  of  Storms.— Hurricanes,  Seasons,  and 
Storm  Tracks. — Solution  of  the  Cyclone  Problem. — Ocean  Currents. — Icebergs. — Syn- 
chronous Charts.— Dew,  Mists,  Fogs,  and  Haze.— Clouds.— Rain,  Snow,  and  Hall.— 
Mirage,  Rainbows,  Coronas,  Halos,  and  Meteors.— Lightning,  Corposants,  and  Auroras.— 
Qukstions.-Appenwx.— Index. 

"  Quite  the  best  publication,  and  certainly  the  most  inteeestino,  on  this  subject  ever 
presented  to  Nautical  mea." Shipping  Gazette. 


iONOON:  CHARLES  GRIFFIN  &  CO..  LIMITED.  EXETER  STREET,  STRAND 


NAD  TIC  AL   WORKS.  35 


GRIFFIN'S    NAUTICAL    SERIES. 

Third  Edition,  Revised.     Pp.  i-xii  +  175.     With  54  Illustrations. 

Price  38.  6d. 

PRACTICAL    MECHANICS: 

Applied  to  the  Requirements  of  the  Sailor. 

By  THOS.  MACKENZIE,  Master  Mariner,  F.R.A.S. 

General  Contents. — Eesolution  and  Composition  of  Forces— Work  done 
by  Machines  and  Living  Agents — The  Mechanical  Powers :  The  Levej 
Derricks  as  Bent  Levers — The  VV  heel  and  Axle  :  Windlass  ;  Ship's  Capstan  . 
Crab  Winch— Tackles  :  the  "Old  Man"— The  Inclined  Plane;  the  8crew— 
The  Centre  of  Gravity  of  a  Ship  and  Cargo  —  Eelative  Strength  of  Rope  : 
Steel  Wire,  Manilla,  Henap,  Coir — Derricks  and  Shears—  Calculation  of  the 
Cross-breaking  Strain  of  Fir  Spar— Centre  of  Effort  of  Sails— Hydrostatics : 
the  Diving-bell ;  Stability  of  Floating  Bodies  ;  the  Ship's  Pump,  &c. 

"  Well  •worth  the  money  .   .   .   will  be  found  exceedingly  helpful,"— 
Shipping  World. 

"  No  Ships'  Officers'  bookcase    will   henceforth  be  complete  without 
Captain  Mackenzie's  'Practical  Mechanics.'    Notwith.standing  my  many 

f 'ears'  experience  at  sea,  it  has  told  me  how  much  more  there  is  to  acquire."— 
Letter  to  the  Publishers  from  a  Master  Mariner). 


WORKS  BY  RICHARD  C.  BUCK, 

of  the  Thames  Nautical  Training  College,  H.M.S   '  Worcester.' 

Third  Edition,  Revised  and  Corrected.      Pp.   i-viii  +  113. 
With  38  Illustrations.     Price  2s.  6d. 

A  MANUAL  OF  TRIGONOMETRY: 

With  Diagrams,  Examples,  and  Exercises. 

*,*  Mr.  Buck's  Text-Book  has  been  specially  prepared  with  a  vie-w 
to  the  Examinations  of  the  Board  of  Trade,  in  which  Trigonometry 
is  an  obligatory  subject. 

"This  KMINENTLT  PRACTICAL  and  RELIABLE  voLCME ' — Schoolma.iter 


Second  Edition,  Revised.      Pp.  i-viii-t-158.      Price  3s.  6d. 

A  MANUAL  OF  ALGEBRA. 

Designed  to  meet   the  Requirements  of  Sailors  and  others. 

%*  These  elementary  works  on  algebra  and  trioonometrt  are  written  specially  for 
those  who  will  have  little  opportunity  of  consulting  a  Teacher.  They  are  books  for  "  seli  - 
BTELP, "  All  but  the  simplest  explanations  have,  therefore,  been  avoided,  and  answers  to 
the  Exercises  are  given.  Any  person  may  readily,  by  carefui  study,  become  master  of  their 
contents,  and  thus  lay  the  foundation  for  a  further  mathematical  course,  if  desired.  It  is 
hoped  that  to  the  younger  OfQcers  of  our  Mercantile  Marine  they  will  be  found  decidedlj 
iervioeable.  The  Examples  and  Exercises  are  taken  from  the  Examination  Papers  set  for 
the  Oadets  of  the  "  Worcester, '' 

"Clearly  arranged,   and   well   got  up.     .  .A   Hrst-rate   Elementary  Algebra."— 

Nautical  Magazine. 

LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAND. 


36  CHARLES  GRIFFIN  <fe  CO:S  PUBLICATIONS. 

GRIFFIN'S  NAUTICAL    SERIES. 

Second  Edition,  Revised.     With  Diagrams.     Price  2s. 

LATITUDE  AND  LONGITUDE :  How  to  Find  ta. 

By    W.    J.    MILLAR,    C.E., 

Late  Secretary  to  the  Inst,  of  Engineers  and  Shipbuilders  in  Scotland. 

"  Concisely  and  clearly  written  .  .  .  cannot  but  prove  an  acquisition 
to  those  studying  Navigation." — Marine  Engineer. 

"  Young  Seamen  will  find  it  handy  and  useful,  simple  and  clear." — Thf 
Engineer. 


Second  Edition,  Revised.    In  Crown  8vo.    Pp.  i-xvii  +  230.    Price  48.  6d. 

THE  LEGAL  DUTIES  OF  SHIPMASTERS. 

By  benedict  WM.  GINSBURG,  M.A.,  LL.D.  (Cantab.), 

Of  the  Inner  Temple  and  Northern  Circuit ;  Barrister-at-Law. 

General  Contents. — Qualification  for  Position  of  Shipmaster. — Coi  tract  with  Ship- 
owner.— Duty  in  respect  of  the  Crew  :  Engagement  ;  Apprentices ;  Discipline ;  Pro- 
visions, Accommodation,  and  Medical  Comforts  ;  Paymem  of  Wages  and  Discharge. — 
Passengers. — Financial  Responsibilities.— Cargo.— Casualty. — Duty  to  certain  Public 
Authorities. — Pilots,  Signals,  Flags,  and  Light  Dues. — Arrival  at  the  Port  of  Discharge. 
— Appendices  on  Legal  Matters  B.O.T.  Certificates,  Dietary  Scales,  Stowage  of  Grain 
Cargoes,  Load  Line  Regula /ions,  Life-saving  Appliances,  Carriage  of  Cattle. — Index. 

"  No  intetlleent  Master  should  fail  to  add  this  to  his  list  of  necessary  books.    A  few  lines 
Of  it  may  save  a  lawyers  fee,  bkside.s  endless  worry." —Liverpool  Journal  ot  Commerce. 


FIRST   AID    AT   SEA. 


Third  Edition,   Revised.     Pp.  i-xviii-f  349.     With  82  Illustrations  and 
the   atest  Regulations  on  the  Carriage  of  Medical  Stores.     6s. 

A  MEDICAL  AND  SURGICAL  HELP 

Fop  Shipmasters  and  Officers  in  the  Merchant  Navy. 
By      WM.       JOHNSON      SMITH,     F.R.O.S., 

Principal  Medical  Oflicer,  Seamen's  Hospital,  Greenwich. 

*,*  The  attention  of  all  interested  in  our  Merchant  Navy  is  requested  to  this  exceedingly 
osefal  and  valuable  worii.  It  is  needless  to  say  that  it  is  the  outcome  of  many  years 
PEAOTICAL  experience  amongst  Seamen. 

"  SolND,  JUDICIOUS,  BBALLT  HELPFUL  " — The  LanCti. 


Eleventh  Edition,  Revised  and  Enlarged.     Price  7s.  6d. 

KNOW    YOUR    OWN    SHIP. 

By   THOMAS    WALTON,  Naval  Architect. 

Specially  arranged  to  suit  the  requirements  of  Ships'  Officers,  Shipowners, 

Superintendents,  Draughtsmen,  Engineers,  and  Others. 

For  Contents  and  further  particulars  of  this  work,  and  other  works 

by  the  same  author,  see  p.  32. 

LONDON:  CHARLES  GRIFFIN  A  CO..  LIMITED.  EXETER  STREET,  STRAMO 


NAUTIGAL   WORKS.  37 


OTHER    WORKS    OF    INTEREST    TO    SAILORS. 


In   Pocket   Size.      With   368   Pages.      3s.    6d.    net. 

ENGLISH-SPANISH  AND  SPANISH-ENGLISH 
SEA  TERMS  AND  PHRASES. 

By  Fleet-Paymastkr  GRAHAM-HEWLETT. 

"  Most  complete    .    .    .    useful        .    .    we  cau  heartily  recommend  it."— Steamsftip. 


lu  Crown  Svo.     Handsome  Cloth.     Many  Diagrams.     23.  6d.  net. 

DEFINITIONS  IN  NAVIGATION  &  NAUTICAL 

ASTRONOMY. 

By    P     GROVES-SHOWELL, 

Head  of  the  Navigation  Department,  L.C.C.  School,  Poplar. 

Contents. — Definitions.— General.—  Navigation. — Nautical  Astronomy. — 
Time.  —  Miscellaneous.  —  Notes.  Measurements.  —  Mariner's  Compass.  — 
Chronometer.  —  Azimuth  Compass.  —  Pelorus.  —  Sextant.  —  Vernier.  —  Arti- 
ficial Horizon. — Sounding  Machine. — Log. — Station  Pointer. — Barometer. — 
Thermometer. — Hygrometer. — Hydrometer.  Miscellaneous.— Star  Nomen- 
clature.— Greek  Alphabet. — Planetary  Symbols. — Weights  and  Measures. — 
Areas. — Volumes.— Useful  Notes. — Index. 

"Mr.  Groves-Showell  writes  with  a  full  knowledge  of  his  subject,  aud  with  admirable 
clearness.''— S/i(p6iti!der. 


Attention  is  also  drawn  to  the  following: — 

HYDROGRAPHIC    SURVEYING.       By    Commander    S. 

Messum,  R.N.  {See  page  16. 

THEODOLITE  SURVEYING.     By  Professor  James  Park. 

[See  page  41. 

THE  FORCE  OF  THE  WIND.     By  Herbert  Chatley,  B.Sc. 

[See  page  23. 

THE  EARTH'S  ATMOSPHERE.     By  Dr.  T.  L.  Phipson. 

[See  page  46. 

WIRELESS  TELEGRAPHY.     By  Gustave  Eichhorn,  Ph.D. 

[See  page  29. 
iONDON :  CHARLES  GRIFFIN  &  CO.,  LIMITED.  EXETER  STREET,  STRANB. 


MINING    WORKS, 

Pages   39-45. 


MINING   WORKS.  39 


■^A^OI^K:s    sir 
SIR   CLEMENT   LE   NEVE   FOSTER,  D.Sc,  F.R.S. 


Sixth  Edition.     With  Frontispiece  and  712  Illustrations.     Price  28s.  net. 

ORE  &  STONE   MINING. 

By  Sir  C.  LE  NEVE  FOSTER,  D.Sc,  F.R.S., 

LATE  PROFBSSOR   OH    MINING.   ROYAL    COLLEGE    OF  SCIENCE. 

Revised,  and  brought  up-to-date 
By   Prof.    S.    H.    COX,   Assoc.R.S.M., 

PROFESSOR  OF  MINING.   ROVAL  COLLEGE  dV  SCIR.VCE. 

GENERAL  CONTENTS. 

INTRODUCTION.  Mode  of  Oecuppence  of  Minepals.— Ppospeetlng.— Boping. 
—Speaking  Gpound.— Suppopting  Excavations.— Exploitation.— Haulage  or 
Transpopt.— Hoisting  op  Winding.  —  Dpainage.  —  Ventilation.  —  Lighting.— 
Descent  and  Ascent.— Dpessing—Ppineiples  of  Employment  of  Mining  Labour. 
—  Legislation  affecting  Mines  and  Quarpies.  —  Condition  of  the  Miner.— 
Aeeidents.— Index 

"  We  have  seldom  had  the  pleasure  to  review  a  work  so  thorough  and  complete  as 
the  present  one.     Both  in  manner  and  in  matter  it  is  FAR  superior  to  anything  on 

ITS  SPECIAL  SUBJECT  HITHERTO  PUBLISHED  IN  ENGLAND."— ^<Aena*!(?n. 

"  Not  only  is  this  worli  the  aclinowledged  text-book  on  metal  mining  in  Great  Britain 
and  the  Colonies,  but  that  it  is  so  regarded  in  the  United  .States  of  America  is  evidenced 
by  the  fact  that  it  is  the  book  on  that  subject  recommended  to  the  students  in  most  of 
the  mining  schools  of  that  country." — The  Times. 


Second  Edition,  Revised.      In  Crown  8vo.      Handsome  Cloth. 
With  nearly  300  Illustrations.     Price  7.s.  6d.  net. 

THE  ELEMENTS  OF  MINING  AND  QUARRYING. 

An  Introductory  Text-Booh  for  Mining  Students. 
By  Sir  C.  LE  NEVE  FOSTER,  D.Sc,  F.R.S., 

Late  Professor  of  iJiuing  at  the  Royal  College  of  Science. 
Revised  by  Prof.  S.   H.  Cox,  A.R.S.M.,  &c. 

General  Contents.  —  Introduction.  —  Occurrence  of  Minerals.  —  Pro- 
specting.— Boring. — Breaking  Ground. — Supporting  Excavations. — Exploita- 
tion.— Haulage  or  Transport.— Hoisting  or  Winding. —Drainage. — Ventilation. 
— Lighting. — Descent  and  Ascent. — Dressing,  &c. — Index. 

"  A  remarkably  clear  survey  of  the  whole  field  of  mining  operations." — Engineer. 

"Rarely  does  it  fall  to  the  lot  of  a  reviewer  to  have  to  accord  such  unqualified  praise  as 
this  book  deserves.  .  .  .  The  profession  generally  have  every  reason  to  be  grateful  to 
Sir  C.  Le  Neve  Foster  for  having  enriched  educational  literature  with  so  admirable  an 
elementary  Text-book." — Mining  Journal. 


In  Large  Crown  8vo.     Fully  Illustrated.     6s.   net. 

THE  INVESTIGATION  OF  MINE  AIR: 

An  Account  by  Several  Authors  of  the  Nature,  Significance,  and  Practical 

Methods  of  Measurement  of  the  Impurities  met  with  in  the 

Air  of  Collieries  and  Metalliferous   Mines. 

EDITED    BY 

Sir  CLEMENT  LE  NEVE  FOSTER,  D.Sc,  F.R.S., 
And  J.  S.  HALDANE,  M.D.,  F.R  S. 

We  know  of  nothing  essential  that  has  been  omitted.     The  book  is  liberally  supplied 
with  illustrations  of  apparatus." — Colliery  Guardian. 

LONDON :  CHARLES  GRIFFIN  &  CO..  LIMITED,  EXETER  STREET,  STRANa 


40  CHARLES  ORIFFIN  dk  CO.'S  PUBLICATIONS. 

WORKS    ON    COAL-IVIINING. 

Fifth  Edition,  Revised  and  Greatly  Enlarged.     With  4  Plates  and 
690  Illustrations.     Price  24s.  net. 

A  TEXT-BOOK   OF   COAL-MINING : 

FOR   THE    USE  OF  COLLIERY   MANAGERS   AND   OTHERS 
ENGAGED   IN   COAL-MINING. 

By    HERBERT    WILLIAM    HUGHES,    F.G.S., 

Assoc.  Royal  School  of  Mines,  General  Manager  of  Sandvvell  Park  Colliery. 

General  Contents.  —  Geology.  —  Search  for  Coal.  —  Breaking  Ground. — 
Sinking. — Preliminary  Operations.  —  Methods  of  Working. — Haulage. — 
Winding.  —Pumping. — Ventilation.  —  Lighting. — Works  at  Surface. — Pre- 
paration of  Coal  for  Market. — Index. 

"Quite  THE  BEST  BOOK  of  its  kind  .  .  .  as  practical  in  aim  as  a  book  cm  be  .  .  .  The 
illustrations  are  i.xai.hE'S'c."— Athenaeum. 

"We  cordially  recommend  the  work." — Colliery  Guardian. 

"Will  soon  come  to  be  regarded  as  the  standard  work  of  its  ^ud."  -  Birminaham  Daily  Oazette. 


Fourth  Edition,  Thoroughly  Prevised  and  Greatly  Enlarged.     Re-set 
throughout      Large  Crown  8vo.     Handsome  Cloth.      12s.  6d. 

PRACTICAL   COAL-MINING: 

A.     MANUAL      FOR     MANAGERS,      UNDER-MANAGERS, 
COLLIERY     ENGINEERS,     AND     OTHERS. 

With  Worked-out  Problems  on  Haulage,  Pumping,  Ventilation,  &c. 

By   GEORGE    L.    KERR,    M.E.,    M.Inst.M.E. 

Contents. — The  Sources  and  Nature  of  Coal. — The  Search  for  Coal. — 
Sinking.— Explosives. — ^Mechanical  Wedges. — Rock  Drills  and  Coal-cutting 
Machines. — Coal-cutting  by  Machinery. — Transmission  of  Power. — Modes  of 
Working. — Timbers.— Roadways.  —Winding  Coal. — Haulage.  Pumping. — 
Ventilation. — Safety  Lamps. — Surface  Arrangements,  Surveying,  Levelling, 
&c. 

"An  ESSENTIALLY  PRACTICAL  WORK,  and  Can  be  confidently  recommended.     No  department 
of  Coal-Mining  has  been  overlooked." — Engineers  Qazettt. 


Second  Edition,  Revised.     In  Crown  8vo.     Handsome  Cloth. 
With  200  Illustrations.     3s.  6d. 

ELEMENTARY   COAL-MINING; 

FOR  THE  USE  OF  STUDENTS,  MINERS,  AND  OTHERS 
PREPARING  FOR  EXAMINATIONS. 

By  GEORGE  L.  KERR,  M.E.,  M.Inst.M.E. 

Contents. — Sources  and  Nature  of  Coal. — Exploration  and  Boring  for 
Coal. — Breaking  Ground. — Explosives,  Blasting,  &c. — Sinking  and  Fitting 
of  Shafts. — Modes  of  Working. —Timbering  Roadways. — Winding  and 
Drawing. — Haulage. — Pumping  and  Drainage. — Ventilation. — Cleaning  and 
Sorting  Coal. — Surveying,  &c. 

"An  abundance  of  information  conveyed  in  a  popular  and  attractive  form.    .    .    .     Will  be 
of  great  use  to  all  who  are  in  any  way  interested  in  coal  mm'ms." —Scottish  Critic. 

LONDON:  CHARLES  GRIFFIN  &  CO..  LIMITED.  EXETER  STREET.  STRAIJS? 


MINING    WORKS.  41 


Fourteenth  Edition,   Revised.      With  Numerous  Diagrams. 

Cloth,  7s.  6d. 

A    TREATISE     ON     MINE-SURVEYING: 

For  the  use  of  Managers  of  Mines  and  Collieries,  Students 
at  the  Royal  School  of  Mines,  dc. 

By    BENNETT    H.    R  ROUGH.    Assoc.R.S.M.,    F.G.S. 

Rf.vimuj  kv  L.   H.  C()()KE, 

Instructor  of  Mine-Surveying,  Royal  School  of  Mines. 

Contents. — General  Explanations.  —  Measurement  of  Distances.  —  Mmers 
Dial. — Variation  of  the  Magnetic  Needle. — Surveying  with  the  Magnetic  Needle 
in  the  Presence  of  Iron. — Surveying  with  the  Fixed  Needle. — The  German  Dial. — 
The  Theodolite. — Traversing  Underground.  —  Surface  Surveys  with  the  Theodo- 
lite.— Plotting  the  Survey. — Calculation  of  Areas. — Levelling. — Connection  of  the 
Underground  and  Surface  Surveys. — Measuring  Distances  by  Telescope.— Setting- 
out. —  Mine-Surveying  Problems.  —  Mine  Plans. — Application  of  the  Magnetic 
Needle  in  Mining. — Photographic  Surveys. — Appendices. — Index. 

"Its  CLEARNESS  of  STYLE.  LUCIDITY  of  DESCRIPTION,  and  FULNESS  of  DETAIL  have  long  ago  won 
for  it  a  place  unique  in  the  literature  of  this  branch  of  mining  engineering,  and  the  present  edition  fully 
maintains  the  high  standard  of  its  predecessors.  To  the  student,  and  to  the  mining  engineer  alike.  ITS 
VALUE  is  inestimable.     The  illustrations  are  excellent." — T/te  Mimm:  yoitrnitL 


In  Crown  8vo.     Handsome  Cloth.     Fully  Illustrated.     6s.   net. 
A    HANDBOOK     ON 

THEODOLITE  SURVEYING  AND  LEVELLING. 

For  the  use  of  Students  in  Land  and  Mine  Surveying. 
By   Professor    JAMES    PARK,    F.G.S. 

Contents. — The  Scope  and  Object  of  Surveying. — Land  Surveying. — 
The  TheodoKte. — Chains  and  Steel  Bands.— Obstacles  to  AUignment. — 
Meridian  and  Bearings. — -The  Theodolite  Traverse. — Co-ordinates  of  a 
Station. — Calculation  of  Omitted  or  Connecting  Line  in  a  Traverse. — 
Calculation  of  Areas. — Subdivision  of  Land. — Triangulation. — Determina- 
tion of  True  Meridian,  Latitude  and  Time. —Levelling. — Kailway  Curves. 
— Mine  Surveying. — Surveying  Boreholes. — Index. 

"A  book  which  should  prove  as  useful  to  the  professional  surveyor  as  to  the 
student." — Sahtre.  

Second  Edition,  Revised.   Crown  8vo.    Handsome  Cloth.    Illustrated.    6s. 

MINING     GEOLOGY. 

A   TEXT-BOOK   FOR   MINING    STUDENTS  AND    MINERS. 

By  PROF.  JAMES  PARK,  F.G.S.,  M.Inst.M.M., 

Professor  of  Miniug  and  Director  of  the  Ota^o  I'niver.=lty  School  of  Mines  ;  late  Director 

Thames  School  of  Mines,  and  Geological  Surveyor  and  Mining  Geologist  to  the 

Government  or  New  Zealand 

General  Contents.— Introduction. -Classification  of  Mineral  Deposits.— Ore  Veins, 

their  Filling,  Age,  and  Structure.— The  Dynamics  of  Lodes  and  Beds.-Ore  Deposits 

Genetically  Considered— Ores  and  Minerals  Considered  Economically.— Mine  Sampling 

and  Ore  Valuation.— The  Examination  and  Valuation  of  Mines.— Index. 

"  A  work  which  should  find  a  place  in  the  library  of  every  mining  engineer."- 
Mining  World. . 

LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAND. 


42  CHARLES  GRIFFIN  Jk  CO.'S  PUBLICATIONS. 


WORKS   FOR   MINERS   AND   STUDENTS. 

Third  Edition.     In  Crown  8vo.     Handsome  Cloth.      With  30  New 
Illustrations.     7s.  Qd.  net. 

ELECTRICAL   PRACTICE    IN    COLLIERIES. 

By  Prof.  D.  BUilNS,  M.E.,  M.Inst.M.E., 

Professor  of  .Mining  and  Geology  to  the  Glasgow  antl  West  of  Scotland  Technical  College. 

Units  of  Measurement,  Conductors,  &c.— The  Theory  of  the  Dynamo.— The 
Dynamo,  Details  of  Construction  and  Working. — Motors. — Lighting  Installa- 
tions in  Collieries.  —  Pumping  by  Electricity.  —  Electrical  Haulage.  —  Coal 
Cuttmg.  —  Miscellaneous  Applications  of  Electricity  in  Mines.  —  Coal  Mines 
Regulation  Act  (Electricity). — Indkx. 

"A  clear  and  concise  introductiou  to  electrical  practice  in  collieries  "—3fiju'>ii7 
Journal.  ^ 

In  Crown  Svo,  Handsome  Cloth.     8s.  (id.  net. 

MINING  LAW  OF  THE  BRITISH  EMPIRE. 

By  CHARLES  J.  ALFORD,  F.G.S.,  M.Inst.M.M. 

Contents.— The  Principles  of  Mining  Law.— The  Mining  Law  of  Great 
Britam.— British  India.— Ceylon.— Burma.— The  Malay  Peninsula— British 
^orth  Borneo.— Egypt.— Cyprus.  -The  Dominion  of  Canada. —British 
(xuiana.— ihe  Gold  Coast  Colony  and  Ashanti.— Cape  of  Good  Hope — 
Natal.  —  Orange  River  Colony.  —  Transvaal  Colony.  —  Rhodesia  —  The 
Commonwealth  of  Australia.— Xew  Zealand,  &c.— Index. 

"  Cannot  fail  to  be  useful    ...    we  cordially  recommend  the  hook."-~irining  World. 

Fifth  Impression.     In  Large  Svo.     Price  lOs.  6d. 

MINE  ACCOUNTS  AND  MINING  BOOK-KEEPING. 

Fop  Students,  Managers,  Secretaries,  and  others. 

With   Examples   taken  from   Actual   Practice  of   Leading   Companies. 

By  JAMES  GUNSON  LAWN,  A.R.S.M.,  A.M.Inst.C.E.,  F.G.S., 

Head  of  the  Mining  Department,  Camborne  School  of  Mines. 
Edited  by  Sir  C.  LE  NEVE  FOSTER,  D.Sc,  F.R.S, 
"It  seems  impossible  to  suggest  how  Mr.  Lawn's  book  could  be  made  more  complete  or 
more  valuable,  careful,  and  exhaustive." — Accountants'  Magazine. 


Third  Edition.     In  Pocket  Size,  Strongly  Bound  in  Leather,  3s.  6d. 
Provided  with  Detachable  Blank  Pages  for  MS. 

THE    MINING    ENGINEERS'    REPORT    BOOK 

AND  DIRECTORS'  AND  SHAREHOLDERS'  GUIDE  TO  MINING  REPORTS. 

By   EDWIN   R.    FIELD,  M.Inst.M.M. 
Witfi  Notes  on  tiie  Valuation  of  Property,  and  Tabulating  Reports, 

Useful  Tables,  and  Examples  of  Calculations,  dc. 
"An  ADMIRABLY  Compiled  book  which  Mining  Engineers  and  Managers  will  find 
EXTREMELY  VSV^FUL."— Mining  Journal. 


In  Crown  Svo.     Handsome  Cloth,     Illustrated.     10s.  6d.  net. 
A  DICTIONARY  OF 

SPANISH-ENGLISH  AND  SPANISH-AMERICAN  MINING, 

METALLURGICAL,   AND  ALLIED  TERMS. 

To  which  some  Portuguese  and  Portuguese-American  i  Brazilian)  Terms  are  added. 
By    EDWARD    HALSE,    A.R.S.M., 

Mem.  Inst.  Ming,  and  Metall.,  of  the  Enp.  lust,  of  Miiis;.  and  Metall.  Engrs.,  &c.,  &c. 
"  Will  be  found  of  the  gre.-vtest  service  to  the  mining  profession."— JlJinnif?  Journal. 

LONDON:  CHARLES  GRIFFIN  &  CO..  LIMITED.  EXETEB  STREET,  STRAND, 


MINING   WORKS.  43 


■WORSS   OIT   SINEING,   EXPLOSIVES,   &c. 

In  Medium  8vo,  Handsome  Cloth.     With  IS  Figures  in  the  Text, 
and  19  Folding  Plates.     10s.  6d.  net. 

SHAFT-SINKING  IN  DIFFICULT  CASES. 

By     J.    RIEMER, 

Translated   by   J.    W.    BROUGH,   A.M.Inst. C.E. 

Contents.— Shaft  Sinking  by  Hand.— Shaft  Sinking  by  Boring.— The 
Freezing  Method.— The  Sinking  Drum  Process. —Bibliography.— Index. 

"The  translator  deserves  the  thanks  of  the  mining  commanity  for  placing  this 
valuable  work  before  them.  .  .  .  The  work  is  one  which  every  mining  engineer 
should  include  in  his  library."— ^/imn;;  World. 


Second  Edition,  Revised.      In  Large  Svo,  with  Numerous  Illustrations 
and  Folding  Plates.      10s.  6d. 

BLASTING :  AND  THE  USE  OF  EXPLOSIVES. 

By  OSOAR  GUTTMANN,  M.Inst.O.E.,  F.I.C,  F.C.S. 

Contents.  —  Historical  Sketch.  —  Blasting  Materials.  —  Qualities  and 
Handling  of  Explosives. — The  Choice  of  Blasting  Materials. — Preparation 
of  Blasts. — Chamber  Mines. — Charging  of  Boreholes. — Determination  of 
Charge. — Blasting  in  Boreholes.— Firing. — Results  of  Working. — Various 
Blasting  Operations. — Index. 

"  Should  prove  a  vade-mecum  to  Mining  Engineers  and  all  engaged  in  practical  work." 
— Iron  and  Coal  Trades  Review. 


In  Medium  Svo,  Cloth.      With  many  Illustrations  in  the   Text. 
Four  Full  Page  Plates  and  Four  I'olding  Tables.     6s.  net. 

NEW     METHODS     OF 

TESTING    EXPLOSIVES. 

By    0.    E.    BICHEL. 

Translated  and  Edited  by  ALEX.  LARSEN,  M.Inst.C.E, 

Contents.  —  Introductory.  —  Historical.  —  Testing  Stations.  —  Power 
Gauges.  —  Products  of  Combustion.  —  Rate  of  Detonation.  —  Length  and 
Duration  of  Flame. — After-Flame  Ratio. — Transmission  of  Explosion. — 
Conclusions. — Efficiency. 

"Its  pages  bristle  with  suggestions  and  actual  experimental  results  to  an  extent 
seldom  found  in  a  volume  of  five  times  its  &iz&."— Arms  and  Explosives. 


In  Crown  Svo.     Handsome  Cloth.     Fully  Illustrated. 
A   MANUAL    ON 

ELECTRICAL    SIGNALLING    IN     MINES. 

By  GERALD  J.  HOOGHWINKEL,  M.Inst.E.E.,  M.LMin.E. 

Contents.— Section  I. — Electric  Haulage  Signals — (a)  Acoustic  Signals.— (6)  Optical 
Signals. — (c)  Acoustic  Optical  Signals.— Current  Supply.— Batteries.— Accumulators.— 
Main  Supply. — Design  and  Construction  of  Signalling  Installations. — Maintenance  of 
Signalling  Installations.  Section  II.— Electric  Shaft  Signals — («)  Acoustic  Signals.— (&) 
Electro-Mechanical  Signals.— (c)  Optical  Acoustic  Signals.— Signals  for  Winding  Minerals. 
—For  Winding  Men.— Signalling  from  the  Cage.  —  Emergency  Signals. —  Bells.— iline 
Telegraphs.— Mine  Telephones.    Section  III.— Special  Applications  in  Mines. 

CONDON:  CHARLES  GRIFFIN  &  CO..  LIMITED.  EXETER  STREET.  STRANtt 


44  GHARLES  GRIFFIN  dk  CO.'S  PUBLICATIONS. 

Second  Edition,  Uevised  Throughout.     In  Medium  8t'o.      With 

Nj(7nerous  Hlates,  Mapx,  and  IlluHtration-s.     '2ls.  net. 

CYANIDING  GOLD  &  SILVER  ORES. 

A    Practical   Treatise   on   the   Cyanide    Process  ;    its    Application, 

Methods  of  Working,   Design  and  Construction  of 

Plant,  and  Costs. 

By    H.    FORBES    JULIAN, 

Mining  and  Metalhirgical  Engineer  :  Specialist  in  Gold  :  Late  Technical  Adviser  Of  the 
Deutsche  Gold  und  Silber  Scheide  Anstalt,  Frankfort-on-Maine. 

And  EDGAR  SMART,  A.M.I.O.E., 

Civil  and  Metallurgical  Engineer. 

"  A  handsome  volume  of  400  pages  which  will  be  a  valuable  book  of  reference  for  all 
associated  with  the  process." — Mining  Journal. 

"The  authors  are  to  be  congratulated  upon  the  production  of  what  should  prove  to  be 
a  standard  w oik."— Page's  Magazine. 


In  Large  Crown  Svo.      With  13  Plates  and  many  Illustrations  in  the  Text. 
Handsome  Cloth.      Is.  6d.  net. 

THE  CYANIDE  PROCESS  OF  GOLD  EXTRACTION. 

A  Text-Book  for  the  Use  of  Metallurgists  and  Students  at 
Schools  of  Mines,  &o. 

By    JAMES    PARK,    F.G.S.,  M.Inst.M.M., 

Professor  of  Mining  and  Director  of  the  Otago  University  School  of  Mines  ;  late  Director 

Thames  School  of  Mines,  and  Geological  Surveyor  and  Mining  Geologist 

to  the  Government  of  New  Zealand. 

Fourth  English  Edition.     Thoroughly  Revised  and  Greatly  Enlarged. 
With   additional   details   concerning  the  Siemens-Halske  and  other 
recent  processes. 
"  Deserves  to  be  ranked  as  amongst  the  BEST  OF  EXISTING  TREATISES.  "—J/tmn^J^owrnoi. 


Third  Edition,  Revised.      With  Plates  and  Illustrations.     Cloth,  3s.  6d. 

GETTING     GOL  D: 

A    GOLD-MININa    HANDBOOK     FOR    PRACTICAL    MEN. 

By  J.    0.    F.    JOHNSON,    F.G.S.,    A.I.M.E., 

Life  Member  Australasian  Mine-Managers'  Association. 
General  Contents. — Introductorj' :  Prospecting  (Alluvial  and  General) — 
Lode  or  Reef  Prospecting — Genesiology  of  Gold — Auriferous  Lodes — Drifts — 
Gold  Extraction — Lixiviation — Calcination — Motor  Power  and  its  Transmission 
—  Company  Formation  —  Mining  Appliances  and  Methods  —  Australasian 
Mining  Regulations. 

"  Practical  from  beginning  to  end    .    .    .    deals  thoroughly  with  the  Prospecting, 
Sinking,  Crushing,  and  Extraction  of  gold."— Brif.  Australasian. 


In  Croivn  Svo.     Illustrated.     Fancy  Cloth  Boards.     4s.  6d. 

GOLD  SEEKING  IN  SOUTH  AFRICA 

A  Handbook  of  Hints  for  intending  Explorers,  Prospectors. 

and  Settlers. 
By     THEO     KASSNER, 

Mine  Manager,  Author  of  the  Geological  Sketch  Map  of  the  De  Kaap  Gold  Fields. 

With   a    Chapter  on    the   Agricultural   Prospects   of  South   Africa. 

"As  fascin.itinsr  as  anvthius  ever  penned  by  .lulea  Verne."— J/rican  Commerce. 


HONDON :  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET.  STRANtt 


NEW  LAND  SERIES.  45 

GRIFFIN'S  "NEW  LAND"  SERIES. 

Practical  Hand-Boohs  for  the    Use  oj  Prospectors,  Explorers, 

Settlers,  Colonists,  and  all  Interested  in  the  opening 

up  and  Development  of  New  Lands. 

Edited  by  GRENVILLE  A.  J.  COLE,  M.R.I.A.,  F.G.S., 

Professor  of  Geology  in  the  Royal  College  of  Science  for  Ireland,  and  Examiner  in 

the  University  of  London. 


In  Grown  8w.     Handsome  Cloth.     5s. 
With  Numei  mis  Maps  Specially  Draion  and  Executed  for  this   Work. 

NEW     LANDS: 

THEIR    RESOURCES     AND     PROSPECTIVE 

ADVANTAGES. 

By  HUGH  ROBERT  MILL,  D.Sc,  LL.D.,  F.R.S.E., 

"A  want  admirably  supplied.    .    .     .    Has  the  advantage  of  being  written  by  a  pro- 
eased  Geographer." — Geographical  Journal. 


With  many  Engravings  and  Photographs.     Handsome  Cloth,  4b.  6d. 

FOOD      STJI>I»LY. 

By    ROBERT    BRUOE, 

Agricultural  Superintendent  to  the  Royal  Dublin  Society. 

With  Appendix  on  Preserved  Foods  by  C.  A.  Mitchell,  B.A.,  F.I.C. 

"The  work  is  one  which  will  appeal  to  those  intending  to  become  farmers  at  home 
or  in  the  Colonies,  and  who  desire  to  obtain  a  general  idea  of  the  true  principlei 
farming  in  all  its  branches." — Journal  of  the  Royal  Colonial  Inst. 


Fifth  Edition.       With  Illustrations.      Handsome  Cloth,  5s. 

PROSPECTING  FOR  MINERALS. 

A   Practical  Handbook  for   Prospectors,    Explorers,    Settlers,    and  alt 
interested  in  the  Opening  up  and  Deuelooment  of  New  Lands. 

By  S.  HERBERT  COX,  Assoc.R.S.M.,  M.Inst.M.M.,  F.G.S., 

Professor  of  Mining  at  the  Royal  School  of  Mines. 

General  Contents. — Introduction  and  Hints  on  Geology— The  Determina- 
tion of  Minerals  :  Use  of  the  Blow-pipe,  &c.  — Eock- forming  Minerals  and  Non- 
Metallic  Minerals  of  Commercial  Value :  Rock  Salt,  Borax,  Marbles,  Litho- 
graphic Stone,  Quartz  and  Opal,  &c. ,  &c.  — Precious  Stones  and  Gems — Stratified 
Deposits:  Coal  and  Ores— Mineral  Veins  and  Lodes — Irregular  Deposits — 
Dynamics  of  Lodes  :  Faults,  &c. — Alluvial  Deposits — Noble  Metals  :  Gold, 
Platinum,  SUver,  &c.— Lead— Mercury— Copper— Tin— Zinc— Iron— Nickel, 
&c. — Sulphur,  Antimony,  Arsenic,  &c.— Combustible  Minerals— Petroleum- 
General  Hints  on  Prospecting — Glossary — Index. 

"This   ADMIRABLE    LITTLE    WORK      .      .      .     written   with    SCIENTIFIC   ACCURACY    In    a 

CLEAR  and  LUCID  style.    ...    An  IMPORTANT  ADDITION  to  technical  literature    .    .    . 
— Mining  Journal. 


lONDONi'  CHARikES  GRIFflN  &  CO.,  LIMITED,  EXETER  STREET.  STRAND 

3 


46  CHARLES  ORIFFIN  cfe  GO:s  PUBLICATIONS. 


Demy  8vo,  Handsome  cloth,  34s. 

Stratigraphical  Geology  &  Paleontology, 

ox    THE    BASIS    OF  PHILLIPS. 
By  R  O  B  E  R  T    E  T  H  E  R  I  D  G  E,    F.  R.  S., 

OP  THB  NATURAL  HIST.   DEPARTMENT,   BRITISH  MUSEUM,   LATE  PALyHONTOLOGIST  TO  THE 

GEOLOGICAL  SURVEY  OF  GREAT  BRITAIN,  PAST  PRESIDENT  OF  THE 

GEOLOGICAL  SOCIETY.   ETC. 

maitb  ^ap,  flumerous  ^Tables,  ano  ^birt^'Sli  plates. 

"  If  Prof.  Shelby's  volume  was  remarkable  for  its  originality  and  the  breadth  of  its  views, 
Mr.  Et  hkridgb  fully  justifies  the  assertion  made  in  his  preface  that  his  book  differs  in  con- 
struction and  detail  from  any  known  manual.  .  .  .  Must  take  HIGH  RANK  AMONG  WORKS 
OF  RHFERBN'CK." — Atketueum. 


In  Crowu  8vo.     Fully  Illustrated.     Cloth. 

GEOiL.OG^x'    FOR    e:n^gin^£:e:i^s. 

By    major    SORSBIE,    R.E. 

Contents. — Dynamical  and  Structural  Geology. —Rocks  and  Minerals.— Historical 
•Geology.— Geological  Observation.— Practical  Geology.— Coast  Erosion.— Index. 


Sixth  Edition,  Thoroughly  Revised.      With  Illustrations.    Cloth.     los.  6d. 

PRACTICAL     GEOLOGY; 

WITH   A    SECTION  ON   PALEONTOLOGY. 

By   professor  GRENVILLE   COLE,   M.R.I.A.,  F.G.S. 
O-ENERAL    CONTENTS.— 

Part    I.— Sampling  oi  Earth's  Crust.  I  Part  III.— Examination  of  Rocks. 
Part   II.— Examination  of  Minerals.     I  Part  IV.— Examination  of  Fossils. 

"That  the  work  deserves  its  title,  that  it  is  full  of  'Aids,' and  in  the  highest  degree 
•  PRACTICAL,'  will  be  the  verdict  of  all  who  use  it." — Nature. 


OI»EN^-ii.m     STUDIES     I1»I     GEOr-OG^^' 

An  Introduction  to  Ceologry  Out-of-doors. 

By   professor   GRENVILLE   COLE,    M.R.I. A.,    F.G.S. 

For  details,  see  Griffin's  "Open  Air"  Series,  p.  71. 


In   Crozvit  ^vo.     Handsome  Cloth.      2s.  6d. 

RESEARCHES  on  the  PAST  and  PRESENT  HISTORY  of 

THE     EARTH'S    ATIYIOSPHERE. 

Including  the  latest  Discoveries  and  their  Practical  Applications. 

By  dr.  THOMAS  LAMB  PHIPSON. 

Part  I. — The  Earth's  Atmosphere  in  Remote  Geological  Periods. 
Part  II. — The  Atmosphere  of  our  present  period.     Appendices;  Index, 

"The  book  should  prove  of  interest  to  general  readers,  as  well  as  to  meteorologists  and 
other  students  of  science." — Nature. 

lONDON:  CHARLES  GRIFFIN  1^  CO.,  LIMITED.  EXETER  STREET.  STRAIiC 


METALLURGICAL   WORKS.  47 

Griffin  s  glrfalkxrgixal  Strus. 


STANDARD   WORKS   OF   REFERENCE 

FOR 

Metallurgists,   Mine-Owneps,   Assayers,    Manufacturers, 

and  all  interested  in  the  development  of 

the  Metallurgical  Industries. 

EDITED    BY 

Sir  W.  ROBERTS-AUSTEN,  K.C.B.,  D.C.L.,  F.R.S. 

In  Large  %vo.  Handsome  Cloth.      With  Illustrations. 


INTBODUCTION    to    the    STUDY    of    METALLURGY. 

By  the  Editor.     Sixth  Edition.     (Seep.  48.) 

QOLD  (The  Metallurgy  of).  By  Thos.  Kirke  Rose, 
D.Sc,  Assoc.  R.S.M.,  F.C.S.,  Chemist  and  Assayer  of  the  Royal 
Mint.     Fifth  Edition.     21s.     (Seep.  48.) 

LEAD  (The  Metallurgy  of).  By  H.  F.  Collins,  Assoc. 
K.S.M.,  M.Inst.M.M.     Second  Edition.     (See  p.  49.) 

SILVER  (The  Metallurgy  of).  By  H.  F.  Collins,  A.R.S.M., 
M.Inst.M.M.     Second  Edition.     (See  p.  49.) 

IRON  (The  Metallurgy  of)-  By  T.  Turner,  A.R.S.M., 
F.I.C,  F.C.S.     Third  Edition,  Revised.     16s.  net.     (See  p.  50.) 

STEEL     (The     Metallurgy    of).       By   F.    W.    Harbord, 

Assoc. R.S.M.,   F.I.C,   with  a  Section  on  Mechanical  Treatment  by 
J.    W.   Hall,  A.M. Inst. C.E.      Fourth  Edition.     (See  p.  50.) 

ALLOYS.  By  Edward  F.  Law,  Assoc.R.S.M.  With  Frontis- 
piece in  Colours,  and  Fine  Series  of  Micro-photographs.  12s.  6d.  net. 
(See  p.  49). 

ANTIMONY.  By  C.  Y.  Wang,  M.A.,  B.Sc.  Fully  Illustrated. 
I2S.  6d.  net.     (See  p.  50). 


Will  be  Published  at  Short  Intervals. 

METALLURGICAL  MACHINERY  :  the  Application  of 
Engineering  to  Metallurgical  Problems.  By  Henry  Charles  Jenkins, 
Wh.Sc,  Assoc.R.S.M. 

<30PPER  (The  Metallurgy  of).  By  Thos.  C.  Cloud,  Assoc. 
R.S.M. 

*»*  Other  Volumes  in  Preparation. 

LONDON:  CHARLES  GRIFFIN  &  CO..  LIMITED,  EXETER  STREET.  STRAND. 


48  CHARLES  QRIFFIN  ds  CO.'S  PUBLICATIONS. 


GRIFFIN'S     METALLURGICAL     SERIES. 


Sixth  Edition,  thoroughly  Revised  and  considerably  Enlarged.     Large 

8vo,  with  numerous  Illustrations  and  Micro- Photographic 

Plates  of  different  varieties  of  Steel.      ISs.   net. 

An  Introduction  to  the  Study  of 

BY 

Sir  W.  ROBERTS-AUSTEN,  K.C.B.,  D.C.L.,  F.R.S.,  A.R.S.M., 

Late   Chemist  and  Assayer  of  the  Royal   Mint,  and  Professor  of  Metallurgy 
in  the  Royal  College  of  Science. 

Revised   throughout    by   F.    W.    HARBORD,    A.  R.  S.  M.,    F.  I.  C. 

General  Contents. — The  Relation  of  Metallurgy  to  Chemistry. — Physical  Properties 
of  Metals. — Alloys.  The  Thermal  Treatment  of  Metals. — Fuel  and  Thermal  Measurements. 
— Materials  and  Products  of  Metallurgical  Processes. — Furnaces. — Means  of  Supplying  Air 
to  Furnaces. — Thermo- Chemistry. — Typical  Metallurgical  Processes. — The  Micro-Structure 
of  Metals  and  Alloys. — Economic  Considerations. 

*'  No  English  text-book  at  all  approaches  this  in  the  completeness  with 
which  the  most  modern  views  on  the  subject  are  dealt  with.  Professor  Austen's 
volume  will  be  invaluable,  not  only  to  the  student,  but  also  to  those  whose 
knowledge  of  the  art  is  far  advanced." — Chemical  News. 


Fifth  Edition,  Revised,  Considerably  Enlarged,  and  in  part  Re-written. 
With  Frontispiece  and  numerous  Illustrations.     21s. 

THE  METALLURGY  OF  GOLD. 

BY 

T.  KIRKE  ROSE,  D.Sc.Lond.,  Assoc.R.S.M.. 

Chemist  and  Assayer  of  the  Royal  Mint. 

General  Contents.— The  Properties  of  Gold  and  its  Alloys.— Chemistry  of  the 
Compounds  of  Gold.— Mode  of  Occurrence  and  Distribution  of  Gold.— Shallow  Placer 
Depoaits.—Deep  Placer  Deposits.— Quartz  Crushing  in  the  Stamp  Battery.— Amalgam- 
ation in  the  Stamp  Battery.— Other  Forms  of  Crushing  and  Amalgamating  Machinery. 
—Concentration  in  Gold  Mills.— Dry  Crushing.— Re-grinding.— Roasting.— Chlorination: 
The  Plattner  Process,  The  Barrel  Process,  The  Vat-Solution  Process. — The  Cyanide 
Process. — Chemistry  of  the  Cyanide  Process.— Refining  and  Parting  of  Gold  Bullion. 
—Assay  of  Gold  Ores.— Assay  of  Gold  Bullion.— Statistics  of  Gold  Production.— Biblio- 
-raphy.— Index. 

"  A  coMPBKHKNSivB  PRACTICAL  TREATISE  On  this  important  subject."— 2V(e  Times. 

•'The  MOST  COMPLETE  description  ot  the  chlorination  process  which  has  yet  been  pub- 
lished."—J/ininj'  Journal. 

"Adapted  for  all  who  are  interested  in  the  Gold  Mining  Industry,  being  free  from  tech- 
nicalities as  far  as  possible,  but  is  more  particularly  of  value  to  those  engaged  in  the 
Industry." — Cape  Times. 

LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAND, 


METALLUROIGAL   WORKS.  49 


GRIFFIN'S    METALIjURGICAL    SERIES. 

Edited  by  SIR  W.  ROBERTS-AUSTEN,  K.C.B.,  F.R.S.,  D.C.L. 
In  Large  Svo.     Handsome  Cloth.      With  Illustrations. 

Second  Edition,  Revised  Throughout  and  Enlarged.     Ilhistrated. 

THE    METALLURGY   OF    LEAD. 

By  H.  F.  COLLINS,  Assoc.R.S.M.,  M.Inst.M.M. 

A  Complete  and  Exhaustive  Treatise  on  the  Manufacture  of  Lead, 
with  Sections  on  Smelting  and  Desilveriaation,  and  Chapters  on  the 
Assay  and  Analysis  of  the  Materials  involved. 

"A  THOROUGHLT  SOUND  and  useful  digest.     May   with   every    confidence  be 
recommended." — Mining  Journal. 


Second  Edition,  Revised  Throughout  and  Enlarged.     Illustrated. 

THE    METALLURGY    OF   SILVER. 

By  H.  F.  COLLINS,  Assoc.R.S.M.,  M.Inst.M.M. 

Comprising  Details  regarding  the  Sources  and  Treatment  of  Silver 
Ores,  together  with  Descriptions  of  Plant,  Machinery,  and  Processes  of 
Manufacture,  Refining  of  Bullion,  Cost  of  Working,  &c. 

"  The  author  has  focussed  A  large  amount  of  valuable  information  into  a 
convenient  form.  .  .  .  The  author  has  evidently  considerable  practical  experience, 
and  describes  the  various  processes  clearly  and  well." — Aiiniiig  Journal. 


Frontispiece  in  Colours,  and  Beautiful  Series  of  Photo-micrographs. 

12s.  6d,   net. 

JL  L  L  O  "^  S 

AND    THEIR    INDUSTRIAL     APPLICATIONS. 
By     EDWARD     F.    LAW,    A.R.S.M. 

Contents.  —  Introduction.  —  Properties  of  Alloys.  —  Methods  of  Investigation. — 
Constitution. — Influence  of  Temperature  on  Properties. — Corrosion  of  Alloys. — Copper 
Alloys,  Brass,  Bronzes. — Special  Brasses  and  Bronzes.— German  Silver  and  Miscellaneous 
Copper  Alloys. — White  Metal  Alloys. —  Anti-Friction  Alloys.  —  Aluminium  Alloys. — 
Silver  and  Gold  Alloys.— Iron  Alloys. — Miscellaneous  Alloys  (.\malgams,  &c.).— Index. 

"Concise  and    practical    ...    a  valuable  amount  of   information    that  will    be 
appreciated  by  student  and  manufacturer  alike." — Foundry  Trades'  Joi>,rnal. 

LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAND. 


so  CHARLES  GRIFFIN  d:  CO.'S  FUBLIGATIONS. 

GRIFFIN'S    METALLURGICAL    SERIES. 


Fourth  Edition,  Thoroughly  Revised.      With  Numerous  Illustrations. 
Large  8vo.     Handsome  Cloth. 

With  Additional  Chapter  on   The  Electric  Smelting  of  Steel. 

THE  METALLURGY  OF  STEEL. 

By   F.    W.    HARBORD,    Assoc.R.S.M.,    F.I.C. 

With  37  Plates,  280  Illustrations  in    the  Text,  and  nearly   100  Micro- 
Sections  of  Steel,  and  a  Section  on 
THE    MECHANICAL    TREATMENT    OF    STEEL. 

By    J.    W.    HALL,    A. M.Inst. C.E. 

Abbiugkd  Contents. — The  Plant,  Machinery,  Methods  and  Chemistry  of  the  Bessemer 
and  of  toe  Open  Hearth  Processes  (AcilI  and  Basic). — The  Mechanical  Treatment  of  Steel 
comprising  Mill  Practice,  Plant  and  Machinery.  — The  Influence  of  Metalloids,  Ueat 
Treatment,  Special  Steels,  Microstructure.  Testing,  and  Specifications. 

The  Engineer  s&ys,  at  the  concluaion  of  a  review  of  tliis  book  :— "We  cannoj,  conclude  without 
earnestly  recommending  all  who  may  be  interested  as  makers  or  users  of  steeC  which  practically 
means  the  whole  of  the  engineering  profession,  to  make  themselves  acquainted  with  it  as  speedily 
as  possible,  and  this  may  be  the  more  easily  done  as  the  published  price,  considering  the  size 
of  tne  book,  is  extremely  moderate." 


Third   Edition,  Revised  and  Enlarged.      With  many  new  Plates. 

16s.  net. 

THE  METALLURGY  OF  IRON. 

By    THOMAS    TURNER.    Assoc.R.S.M.,    F.I.C, 

Professor  of  Metallurgy  in  the  University  of  Birmingham. 
General  Contents.— Early  History  of  Iron.— Modern  History  of  Iron.— The  Age  of  Steel. 
—Chief  Iron  Ores.— Preparation  of  Iron  Ores.— The  Blast  Furnace,— The  Air  used  in  the 
Blast  Furnace.  —  Reactions  of  the  Blast  Furnace. —The  Gaseous  Products  of  the  Blast 
Furnace —The  Fuel  used  in  the  Blast  Furnace.  — Slags  and  Fuxes  of  Iron  Smelting,— 
Properties  of  Cast  Iron,  —  Foundry  Practice,  —  Wrought  Iron.  — Indirect  Production  of 
Wrought  Iron.— The  Puddling  Process.— Further  Treatment  of  Wrought  Iron.  — Corrosion 
of  Iron  and  Steel. 

"  A  THOROUGHLY  USEFUL  BOOK,  which   brings  the  subject  up  to  date.     Of 

OREAT  VALUE  to  those  engaged  in  the  iron  industry." — Mining  Journal. 


For  Professor  Turner's  Lectures  on  L-on- Founding,  see  page  53 


In  Large  Svo.      Handsome  Cloth.      Fully  Illustrated.      123.  6d.  net. 

JINTIIMONY: 

Its  History,  Chemistry,  Mineralogy,  Geology,  Metallurgy,  Uses  and 

Preparation,  Analysis.  Production  and  Valuation. 

By  C.  Y.  WANG,  M.A.,  B.Sc, 

Mem.  Am.  Inst.  Mining  Eng. ;  Mem.  Iron  and  Steel  Institute;  Jliuing  Engineer  to 

the  Chung  Lou  General  Mining  Company;   Geologist  for  the 

Hunan  Province,  China,  &v..,  &c. 

"A  book  which  stands  alone,  inasmuch  as  there  is  not,  to  our  knowledge,  any  other 
complete  treatise  on  antimony  among  all  the  English  text-books."— /run  and  Coal 
Trades'  Review. 

LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAKtt 


METALLURGICAL   WORKS.  51 

Eleventh  Edition.     With  Tables  and  Illustrations.     Cloth,   los.  6d. 

A    TEXT-BOOK    OF    ASSAYING: 

For  the  use  of  Students,  Mine  Managers,  Assayers,  dc. 
By  J.  J.  BERINGER,  F.I.C.,  F.C.S., 

Public  Analyst  for,  and  Lecturer  to  the  Mining  Association  of,  Cornwall. 

And  C.  BERINGER,  F.C.S., 

Late  Chief  Assayer  to  the  Rio  Tinto  Copper  Company,  London, 

Gbnhral  Contents. —  Part  \.  —  Introductory;  Manipulation:  Sampling; 
Drying  ;  Calculation  of  Results — Laboratory-books  and  Reports.  Methods  :  Dry  Gravi- 
metric ;  Wet  Gravimetric— Volumetric  Assays:  Titrometric,  Colorimetric,  Gasometric — 
Weighing    and    Measuring — Reagents — Formulae,  Equations,  &c. — Specific  Gravity. 

Part  II. — Metals:  Detection  and  Assay  of  Silver,  Gold,  Platinum,  Mercury,  Copper, 
Lead,  Thallium,  Bismuth,  Antimony,  Iron,  Nickel,  Cobalt,  Zinc,  Cadmium,  Tin,  Tungsten, 
Titanium,  Manganese,  Chromium,  &c. — Earths,  Alkalies. 

Part  III. — Non-Metals  :  Oxygen  and  Oxides;  The  Halogens — Sulphur  and  Sul- 
phates— Arsenic,  Phosphorus,  Nitrogen — Silicon,  Carbon,  Boron — Useful  Tables. 

"A  RBALLY  MERITORIOUS  WORK,  that  may  be  safely  depended  upon  either  for  systematic 
instruction  or  for  reference." — Nature. 


In  Crown  8vo.     Handsome  Cloth.     Fully  Illustrated.     3s.  net. 
AN    INTRODUCTION    TO 

PRACTICAL      METALLURGY. 

By    prof.    THOMAS    TURNER,    A.R.S.M.,    F.I.C. 

"It  is  an  excellent  and  handy  book  for  its  purpose,  and  will  have  a  far  wider  range 
of  usefulness  than  for  class  work  alone." — Practical  Engineir. 


Fourth  Edition,   Revised.      Witli  iSunifrous  Illustrations.     6s. 
A      TEXT-BOOK      OF 

ELEMENTARY    METALLURGY. 

Including  the  Author's  Practical  Laboratory  Course. 
By    a.    HUMBOLDT    SEXTON,    F.I.C,    F.C.S. 

"  Just  the  kind  of  work  for  Students  co.m.vencing  the  study  of  Metallurgy,  or  for  Engineebino 
Students."— Practical  Enriinea-. 


Large  Svo.     Cloth.     With  Illustrations.      12s.  6d.  net. 

METALLURGICAL  ANALYSIS  &  ASSAYING: 

A  THREE  YEARS'  COURSE  FOR  STUDENTS  OF  SCHOOLS  OF  MINES. 
By  W.  a.  MACLEOD,  B.A.,  B.Sc,  And  CHAS.  WALKER,  F.C.S. 

"The  publication  of  this  volume  tends  to  prove  that  the  teaching  of  metallurgical 
analysis  and  assaying  in  Australia  rests  in  competent  hands." — Mature. 


Third  Edition.     With  Folding  Plates  and  Many  Illustrations.     36s. 

ELEMENTS    OF    METALLURGY. 

A  Practical  Treatise  on  the  Art  of  Extracting  Metals  from  their  Ores. 

By  J.  ARTHUR  PHILLIPS,  M.Inst.C.E.,  F.G.S.,  &c..  And 

H.  BAUERxMAN,  V.P.G.S. 

LOMDOH:  CHARLES  GRIFFIN  &  CO..  LIMITED,  EXETER  STREET.  STRAND 


52  CHARLES  GRIFFIN  dk  CO.'S  PUBLICATIONS. 

In  Crown  8vo.     Handsome  cloth.     With  102  Illustrations.     6s.  net. 
A      HANDBOOK      ON 

IVTETALLIC       ALLOYS 

Their  Structure  and  Constitution. 
By    gilbert    H.    GULLIVER,    B.Sc,    F.R.S.E. 

"A  valuable  addition  to  existing  literature  on  alloys." — Foundrtj  Trades'  Journal. 


In   Crown   8vo.       Beautifully   Illustrated   with  nearly   100 
Microphotographs  ot  Steel,  &c.      7s.  6d.  net. 

Microscopic  Anaiysis  of  Metais. 

By  FLORIS  OSMOND  &  J.  E.  STEAD,  F.R.S.,  F.I.C. 

"  There  has  been  no  work  previously  published  in  English  calculated  to  be  so  useful  to 
the  student  in  metallographic  research." — Iron  and  Steel  trades'  Journal. 


In  Large  8vo.     Handsome  Cloth.     Price  4s. 

QUANTITATIVE  METALLURGICAL  ANALYSIS. 

TABLES    FOR    LABORATORY    USE. 

ON  THE  PRINCIPLE  OF  ■'GROUP"  SEPARATIONS. 

By  J.   JAMES    MORGAN,   F.O.S.,   M.S.C.I. 

"The  Author  may  be  cosgratulated  on  the  way  hie  work  has  been  carried  out.' 
The  Engineer. 


Third  Edition,  Revised,   Enlarged,  and  Re-written.     I2s.  Gd.  net. 
A    TREATISE    ON 

ELECTRO-METALLURGY: 

Rmbracing  the  Application  of  Electrolysis  to  the  Plating,  Depositing, 
Smelting,  and  Refining  of  various  Metals,  and  to  the  Repro- 
duction of  Printing  Surfaces  and  Art-Work,  &c. 

By  W.  O.   M-^MILLAX,  F.I.C,  and  W.  R.  COOPER.  M.A.,  B.Sc. 

"  Excellent,    .     .    .    one  of  the  BEST  and  most  comi'LETK  manuals  hitherto  published 
on  Electro-Metallurgy.'' — Electrical  Review  (on  the  Second  Edition). 


Second    Edition,  Thoroughly  Prevised    and    Enlarged.      In    large   8vo. 
With  Numerous  Illustrations  and  Three  Folding-Plates.    21s.  net. 

ELECTRIC  SIELTIlir&  &  REEinif&: 

A  Practical  Manual  o*^  the  Extraction  and  Treatment 
of  Metals  by  Electrical  Methods. 

Being  the  "  Elektro-Metallurgie"  of  Dr.  W.  BORCHERS. 

Translated  from  the  Latest  German  Edition  by  WALTER  G.  MCMILLAN, 

F.I.C.  F.G.S. 
"  Comprehensive  and  authoritative    .    .    .    not  only  full  of  valuable  infor- 
mation, but  gives  evidence  of  a  THOROUGH   INSIGHT   into   the   technical   VALUE  and 
possibilities  of  all  the  methods  discussed."— r/ie  Electrician. 

LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED   EXETER  STREET,  STRAND. 


METALLURGICAL   WORKS.  53 

In  Medium  Svo,     Haiulsome  Cloth.     Fully  Illustrated.     158.  net. 

GENERAL    FOUNDRY    PRACTICE: 

A  Practical  Handbook  for  Iron,  Steel  and  Brass  Founders, 

Metallurgists,  and  Students  of  Metallurgy. 

By  a.  M^WILLIAM,  A.R.S.M.,  and  PERCY  LONGMUIR. 

"  The  student  of  foundry  work    .     .     .    needs  no  otlier  text-book.    .     .    .    The  l)Ook 
contains  a  tremendous  amount  of  information,  and  is  well  written." — Engineering  Times. 


Second  Editton.     Extra  Grown  8yo.      With  many  Illustration.-i. 

3.S.  6rf.  net. 

LECTURES  ON  IRON-FOUNDING. 

By  THOMAS  TURNER,  M.Sc,  A.R.S.M.,  F.I.C., 

Professor  of  Metallurgy  in  the  University  of  Birmingliam. 

Contents. — Varieties  of  Iron  and  Steel. — Application  of  Cast  Iron. — History. — Pro- 
duction.—  Iron  Ores. — Composition. — The  Blast  Furnace. — Materials.  —  Reactions. — 
Grading  Pig  Iron. — Carbon,  Silicon,  Sulphur,  Phosphorus,  Manganese,  Aluminium, 
Arsenic,  Copper,  and  Titanium.— The  Foundry. — General  Arrangement. — Re-melting 
Cast  Iron.  —  The  Cupola. — Fuel  Used.  ^  Changes  due  to  Re-melting.  —  Moulds  and 
Moulding. — Foundry  Ladles. — Pouring  and  Pouring  Temperature. — Common  Troubles. — 
Influence  of  Shape  and  Size  on  Strength  of  Castings. — Tests. 

"  Ironfounders  will  find  mucli  information  in  the  book." — Iron  Trade  Circular 
Ryland's).  

In  Crown  Svo.     Handsome  Cloth,      is.  6d.  net. 

BLAST    FURNACE     PRACTICE. 

By  J.  JAMES  MORGAN,  F.C.S.,  M.  S.C.I., 
Author  of  "Quantitative  Metallurgical  Tables,"  &c. 
Contents. — Requirements. — Iron  Ores. — Fuel. — Flux. — Slags.— Quantity  of  Slag. — 
Burden,  Charge,  Round. — The  Blast. — Ore  -Mixing. — Amount  and  Composition  of  the 
Iron. — Calculation  of  Flux. —  Heating  the  Blast;  Stoves  —  Drying,  Changing,  and 
Cleaning. — Drying  the  Furnace.— Filling  the  Furnace,  and  Lighting.— Charging  the 
Furnace.— Descent  of  the  Charge.— Flushing.— Tapping.— Hard  Tapping  Hole.— Running 
down  the  Beds  (Casting). — .Judging  the  Temperature. — Controlling  the  Temperature. — 
Temperature  and  the  Reduction  of  Silicon.— The  Pig  Iron.— Fuel  Consumption.— 
Economy  in  Fuel  Consumption. — Tuyeres:  Leaky,  Changing. — Cooler  or  "Jumbo." — 
Obstructions  —Pillaring.— Breakouts. — Hot  Spots.— Scaffolds  (Hanging). — Slips. — Damp- 
ing Down. — Blowing  Out. — Index. 

"Excellent.    ...     It  is  essentially  suitable  for  the  practical  man,  and  to  the 
beginner  particularly  it  should  prove  valuable." — Iron  and  Coal  Trades  Record. 


Second  Edition,  Revised.      In  Larue  Grown  Svo.      With  Numerous 
Illustrations.     7a.  6d.  net. 

THE  ART  OF  THE  GOLDSMITH  AND  JEWELLER. 

A  Manual  on  the  Manipulation  of  Gold  and  the 
Manufacture  of  Personal  Ornaments. 

By    THOS.    B.    WIGLEY, 

Headmaster  of  the  Jewellers  and  Silversmiths'  Assoc.  Tech.  School,  Birmingham. 

Assisted  by  J.  H.  STANSBIE,  B.Sc.  (Lond.),  F.I.O., 

Lecturer  at  the  Birmingham  Municipal  Technical  School. 

LONDON:  CHARLES  GRIFFIN  &  CO..  LIMITED.  EXETER  STREET.  STRAND, 


54  CHARLES  ORIFFIN  d:  CO.'S  PUBLICATIONS. 

Second  Edition,  Revised,  Enlarged,  and  Re-net  Throtighout  on  Larger  Page.. 
With  Valuable  Bibliography,  iVejc  Maps,  lUustrdtio.i^.  <t-c.     45a'.  net. 

IN      rr  vtf- o      "WO  L  xj  ivi  E  s. 

A     TREATISE     ON 

:E^ETE;OLETjnvn. 

By     sir     BOVERTON     REDWOOD, 

D.Sc.  F.U.S.E..  Assou.IssT.C.E  ,  F.I.C. 

Contents.— Skction  I.:  Historical  Account  of  tlie  Petroleum  Imiustiy.— SECTION  II.: 
Geological  and  Geographical  Distribution  of  Petioleiuii  anil  Natural  Gas. — Section  III.: 
The  Chemical  and  Physical  Properties  of  Petroleum  and  Natural  Gas.  -Section  IV.: 
The  Origin  of  Petroleum  and  Natural  Gas.— Section  V.:  The  Production  of  Petroleum, 
Natural  Gas,  and  Ozokerite.— Section  VI.:  The  Refiiung  of  Petroleum— Section  VII.: 
The  Shale  Oil  and  Allied  Industries.— SECTION  VIII.:  The  Transport,  Storage,  and  Dis- 
tribution of  Petroleum.— Section  IX. :  The  Testing  of  Crude  Petroleum,  Petroleum  and 
Shale  Oil  Products,  Ozokerite,  and  Asphalt.— Section  X.  :  The  Uses  of  Petroleum  and 
its  Products.— Section  XI.  :  Statutory,  Municipal,  and  (jther  Kegulations  relating  to 
the  Testing,  Storage,  Transport,  and  Use  of  Petroleum  and  its  Products.— APPENDICES. 
—Bibliography.— Index. 

"It  is  indisiiutably  the  most  comprehensive  and  cumplete  treatise  on  petroleum,  anil  this 
statement  is  true,  no  matter  on  what  lirancli  of  the  inilusti'y  a  test  ol  its  merits  is  made.  It  is 
the  only  book  in  existence  which  gives  tlie  oil  man  a  clear  and  reliable  outline  of  the  growth  and 
present-day  condition  of  the  entire  petroleum  world.  .  .  .  Tliere  is  a  wonderfully  complete 
collection  of  plates  and  illustrations.  '— Petroleum  IForW. 


Second  Edition,  Revised.      With  Illuttrations.      Price  Ss.  6d.  net. 

A     HANDBOOK     ON      PETROLEUM. 

FOR  INSPECTORS  UNDER  THE  PETROLEUM  ACTS, 

And  for  those  engaged  in  the  Storage,  Transport,  Distribution,  and  Industrial 

Use  of  Petroleum  and  its  Products,  and  of  Calcium  Carbide.     With 

suggestions  on  the  Construction  and  Use  of  Mineral  Oil  Lamps. 

By    captain    J.    H.    THOMSON, 

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56  CHARLES  GRIFFIN  Jc  CO.'S  PUBLIC ATlONfi. 


Sixth  Edition,  Thoroughly  Revised.     Fiiliy  Illustrated,     'ils. 

FOODS :  THEIR  COMPOSITION  AND  ANALYSIS. 

By  a.  WYNTER   BLYTH,  M.R.C.S.,  F.I.C.,  F.C.S., 

liarrister-at-Law,  Public  Analyst  for  the  County  of  Devon,  and 
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And  M.   WYNTER   BLYTH,  B.A.,  B.Sc,  F.C.S. 

General  Contents.— History  of  Adulteratiou.— Legislation.— Apparatus.— "  Ash. "— 
Sugar.— Confectionery.— Honey.— Treacle.— Jams  and  Preserved  Fruits.  — Starches. — 
Wheates-Flour.— Bread.— Oats.— Barley.  —  Kye.  —  Pace.  —  Maize.  —  Millet.— Potatoes.— 
Peas.— Lentils.— Beans  —  Milk.— Cream.—  Butter.— Oleo-Margarine.— Cheese.  — Lard.— 
Tea.— Coffee.— Cocoa  and  Chocolate. —Alcohol.  — Brandy.  — Rum.  — Whisky.— Gin.— 
Arrack.—  Liqueurs.—  Absinthe.— Yeast.  —  Beer.  —  Wine.  —  Vinegar.  —  Lemon  and  Lime 
.luiee.— Mustard.— Pepper.— Sweet  and  Bitter  Almonds.— Annatto.—Oilve  Oil.— Water 
Analysis.— Appendix  :  Adulteration  Acts,  &c. 

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General  Contents.— I.— Historical  Introduction.  II.— Classification— Statistics- 
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Ill-— Poisonous  Gases.  IV.— Acids  and  Alkalies.  V.— More  or  less  Volatile  Poisonous 
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Inorganic  Poisons.  Appendix:  Treatment,  by  Antidotes  or  otherwise,  of  Cases  of 
Poisoning. 

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A  Practical  Handbook  for  Medical  Men,  Analysts,  Inspectors  and  others. 
By   C.   AINSWORTH    MITCHELL,    B.A.,    F.LC,  F.C.S. 

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CHEMISTRY  AND  TEGHNOLOOY.  57 

Second  Edition.      With  Numerous  Tables,  Fully  Illustrated. 

DAIRY    CHEMISTRY 

FOR    DAIRY    MANAGERS,    CHEMISTS,   AND    ANALYSTS 

A  Practical  Handbook  for  Dairy  Chemists  and  others 
having  Control  of  Dairies. 

By   H.    droop    RICHMOND,    F.I.C., 

CHEMIST  TO  THE  AYLESBURY   DAIRY  COMPANY. 

Contents. — I.  Introductory. — The  Constituents  of  Milk.  II.  The  Analysis  of 
Milk.  III.  Normal  Milk  :  its  Adulterations  and  Alterations,  and  their  Detection. 
IV  The  Chemical  Control  of  the  Dairy.  V,  Biological  and  Sanitary  Matters. 
VI.  Butter.  VII.  Other  Milk  Products.  VIII.  The  Milk  ot  Mammals  other 
than  the  Cow. — Appendices. — Tables. — Index. 

"...     In  our  opinion  the  book  is  the  best  contribution  on  the  subject  that 
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By    EDWARD    F.    WILLOUGHBY, 

M.D.  (Lond.),  D.P.H.  (Lend,  and  Canib.). 

"We  cordially  reconimend  it  to  everyone  who  has  anything  at  all  to  do  with  milk." 
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DAIRY    ANALYSIS. 

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Analyst  to  the  Aylesbury  Dairy  Co.,  Ltd. 
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ELEMENTARY  AGRICULTURAL  CHEMISTRY. 

By  HERBERT  INGLE,  B.Sc,  E.LC,  F.C.S. 

Contents.— Introduction.  — The  Atmosphere. —  The  Soil.  — Natural  Waters.  — The 
Plant.— Manures.— Crops. —  The  Animal. —  Foods  and  Feeding.- The  Dairy.- Miscel- 
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Fifth  Edition,  Revised  Throughout.     With  101  Illustrations. 

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MICRO-ORGANISMS    AND    FERMENTATION. 

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TRAKSL.A.TED   BY    SAMUEL    H.    DA  VIES,   M.Sc. 

Contents.— Microscopical  and  Physiological  Investigations.- Investigation  of  Fer- 
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Index^ 

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PRINCIPLES  AND  PRACTICE  OF  BREWING. 

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Editor  of  the  Journal  of  the  Institute  of  Brewing. 

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Translated   from   the   German   by 

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By  REGINALD  E.  MIDDLETON,  M.Inst.C.E.,  M.Inst.Mkch.E.,  F.S.I. 

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VoLumE    I.      Fourth    Edition.      Price  7s.    6d.    net. 
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THE    CHEMISTRY     OF 

A  Hand-Book  on  the  Production,  Purification,  and  Testing  of  Illuminating 
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By  W.  J.  ATKINSON  BUTTERFIELD,  M.A.,  F.I.C.,  F.O.S., 

Formerly  Head  Chemist,  Gas  Works,  Beckton,  London,  E. 
"  The  BEST  WOKK  of  its  kind  which  we  have  ever  had  the  pleasure  of  re- 
viewing."— Journal  of  Gas  Lighting, 


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ACETYLENE: 

THE    PRINCIPLES    OF    ITS    GENERATION    AND    USE. 
By  F.    H.    LEEDS,    F.I.C.,  F.C.S., 

And    W.    J.   ATKINSON    BUTTERFIELD,    M.A.,    F.LC,    F.C.S., 

Consulting  Chemist,  Author  of  "The  Chemistry  of  Gas  Manufacture." 

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FIRE    AND    EXPLOSION    RISKS: 

A  Handbook  of  the  Detection,  Inuestigation,  and  Prevention  of  Fires  and  Explosions. 

By    Dr.    VON    SCHWAKTZ. 

Translated  from  the  Revised  German  Edition  by  C.  T.  C.  SALTER. 

Abbridged  General  Contents.— Fires  and  Explosions  of  a  General  Character  — 
Dangers  arising  from  Sources  of  Light  and  Heat. — Dangerous  Gases.— Risks  Attending 
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SlVIOIi:^:       ABATEIVIENT. 

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Health,  Engineers,  and  Others. 

By    WILLIAM    NICHOLSON, 

Chief  Smoke  Inspector  to  the  Sheffield  Corporation. 
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Medical  Journal. 

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A. 


62  CHARLES  O BIFFIN  ic   CO.'S  FUBLICATIONB, 

Fii-TF.EXTii  Edition,  Thoroughly  Revised.     Price  6s. 

PRACTICAL  SANITATION: 

A  HAND-BOOK  FOR  SANITARY  INSPECTORS  AND   OTHERS 
INTERESTED   IN  SANITATION. 

By    GEORGE    REID,    M.D.,    D.P.H., 

Ftliaw,  Mem.  Council.,  and  Examiner,  Sanitary  Institute  of  Great  Britain, 
and  Medical  Officer  to  the  Staffordshire  County  Council. 

TRIlltb  an  appenMi  on  Sanitatg  Xaw. 

By     HERBERT     M  A  N  L  E  Y,     M.A.,     M.B.,     D.P.H., 

Barrisier-at-Law. 

General  Con'TENPS. —  Introduction.  —  Water  Supply:  Drinking  Water,  i'ollution  of 
Water. — Ventilation  and  Warming. — Principles  of  Sewage  Removal. — Details  of  Drainage  ; 
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Characteristics  of  Good  Meat;  Meat,  Milk,  Fish,  &c.,  unfit  for  Human  Food. — Appendix: 
Sanitary  Law  ;  Model  Bye-Laws,  vc. 

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LESSONS    ON   SANITATION. 

By  JOHN  WM.  HARRISON,  M.R.San.L, 

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Contents. — Water  Supply. — Ventilation. — Drainage. — Sanitary  Building  Construction. — 
Infectious  Diseases. — Food  Inspection. — Duties  of  an  Inspector  of  Nuisances  and  Common 
Lodging-Houses. — Infectious  Diseases  Acts. — Factory  and  Workshop  Acts. — Housing  of 
the  Working-Classes  Act. — Shop  Hours  Acts. — Sale  of  Food  and  Drugs  Acts.  The  Mar- 
garine Acts. — .Sale  of  Horseflesh,  Sic,  Rivers  Pollution. — Canal  Boats  Act. — Diseases  of 
Animals. — Dairies,  Cowsheds  and  Milkshops  Order. — Model  Bye-Laws. — Miscellaneous. — 
Index. 

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SANITARY    ENGINEERING: 

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For  Sanitary  Authorities,  Engrineers,  Inspectors,  Architects, 
Contractors,  and  Students. 

By   FRANCIS   WOOD,  A.M.Inst.CE.,  F.G.S., 

Borough  Engineer  and  Surveyor,  Fulham ;  late  Borough  Engineer,  Bacup,  Lanes. 
GENERAL     CONTENTS. 

Introduction. — Hydraulics.— Velocity  of  Water  in  Pipes.  — Earth  Pressures  and  Retaining 
Walls. — Powers. — House  Drainage. —Land  Drainage. — Sewers. — Separate  System. — Sewgge 
Pumping. — Sewer  Ventilation. — Drainage  Areas. —  Sewers,  Manholes,  &c.— Trade  Refuse. — 
Sewage  Disposal  Works.  --  Bacterial  Treatment.  —  Sludge  Disposal.  —  Construction  and 
Cleansing  of  Sewers. — Refuse  Disposal. — Chimneys  and  Foundations. 

'  The  volume  bristles  with  information  which  will  be  i;reedily  read  by  those  in  need  of  assistance.  The 
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"A  VERITABLE  POCKET  COMPENDIUM  of  Sanitary  Engineering.  ...  A  work  which  may,  In 
many  respects,  be  considered  as  COMPLETE  .  .  COMMENDABLV  CAUTIOUS  .  .  .  INTERESTING 
.     .     .     SUGGESTIVE."— PK^/if  Health  Emineer 

lONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED.  EXETER  STREET,  STRAND. 


GHEMISTRY  AND  TECHNOLOGY.  63 

Second    Edition.      In    Large    8yo.     Handmme    Cloth.      Beautifully 
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ROAD    MAKING   AND    MAINTENANCE: 

A  Practical  Treatise  for  Engineers,   Surveyors,   and  Others. 

With  an  Historical  Sketch  of  Ancient  and  Modern  Practice. 

By  THOS.  AITKEN,  Assoc.M.Inst.O.E., 

Member  of  tba  Association  of  Municipal  and  County  Engineers;  Member  of  tbe  Sanitary 
Inst.;  Surveyor  to  the  County  Council  of  Fife.  Cupar  Division. 

WITH  NUMEROUS  PLATES,  DIAGRAMS,  AND  ILLUSTRATIONS. 
Contents. — Historical  Sketch. — Resistance  of  Traction. — Laying  out 
New  Roads. — Earthworks,  Drainage,  and  Retaining  Walls.  —  Road 
Materials,  or  Metal. — Quarrying.— Stone  Breaking  and  Haulage. — Road- 
Rolling  and  Scarifying. — The  Construction  of  New,  and  the  Maintenance 
of  existing  Roads. — Carriage  Ways  and  Foot  Ways. 

"The  Literary  style  is  BXCELLBNT.  .  .  .  A  compbbhensive  and  excellent  Modern  Book,  an 
rp-TO-DATE  work.  .  .  .  Should  be  on  the  reference  shelf  of  every  Municipal  and  County 
Enfcineer  or  Surveyor  iu  the  United  Kingdom,  and  of  every  Colonial  Engineer."— TAe  Surveyor. 


In  Handsome  Cloth.      Fully  Illustrated.      10s.  6cZ.  net. 

I>XJSTIL.C:SS     ]ROJLI>S. 
TAR      MACADAM. 

By    J.    WALKER     SMITH, 

City  Engineer,  Edinburgh. 

Contents.— Necessity  for  Imiiroved  and  Standard  Koad  Construction.— 'J'ar.— Standardisation 
of  Matrix.— Aggregate  for  Macadam  — Difierent  Modes  of  Preparing  and  Laying.— Mechanical 
Mixing.- Effects  of  Wear,  Density,  Porosity,  Distribution  of  Weight. — Scavenging :  Watering  and 
Maintenance.  —  Camber  :  Gradient,  Noiselessness,  Hygienic  Advantages.  —  Rolling.  —  Tractive 
Effort.— Statistics— Tar  Spraying  on  Ordinary  Macadam  Surfaces.— Appendices.— Index. 

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who  has  had  uniciue  opportunities  for  studying  the  subject,  who  has  made  himself 
master  of  all  its  details,  and  who  is  not  afraid  to  express  his  opinions  thereon.  It  is, 
therefore,  practical  in  the  best  sense  of  the  term." — County  and  Municipal  Record. 


In  Demy  Svo.     Handsome   Cloth.      With   Many  Tables.     6s.  net. 
THE    THEORY    AND    PRACTICE    OF 

ENAMELLING  ON  IRON  AND  STEEL. 

By     JULES     GRUNWALD, 

Technical  Chemist  and  Works'  Manager. 

Contents. — Introduction. — The  Eaw  Materials. — The  Mixing,  Dissolving,  and  Appli- 
cation of  Enamel. — Heating  and  Pickling  Goods  in  the  Ptough.— Correct  laying  on. — 
Baking  Enamelled  Ware. — Decoration  of  Enamelled  Objects. — Photo-Ceramics  in  their 
Application  to  Enamels. — General  and  Statistical  Chapter. — The  History  of  Enamels 
and  their  Uses, — Index. 


In  Large  8vo.     Handsome  Cloth.    With  Plates  and  Illustrations.     7s.  6d.  net. 

THE    MANUFACTURE    OF   INK. 

A  Handbook  of  the  Production  and  Properties  of  Printing,   Writing,  and  Copying   Inks. 
By  C.  a.   MITCHELL,  B.A.,  F.I.C.,  F.C.S.,  &  T.   C.   HEPWORTH. 

"Thoroughly  well  arranged    .    .     .    and  of  a  genuinely  practical  order." — British  Printer. 

lONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAND. 


64  CHARLKS  ORIFFIN  db  CO.'S  PUBLICATIONS. 

In  Large  8vo.     Handsome  Cloth.     Profusely  Illustrated.     303.  net. 

A    TREATISE    ON 

COLOUR    MANUFACTURE. 

A  Guide  to  the  Preparation,   Examination,  and  Application  of  all  the 
Pigment  Colours  in  Practical  Use. 

By  GEORGE  ZERR  and   Dr.    R.    RUBENCAMP. 

Authorised  English  Translation  by  Dr.  C.  MAYER,  of  Burgdorf. 

'  This  comprehensive  guide    .    .    .    useful  and  interesting." — Oil  and  Colour  Trades 
Journal. 


Fourth  Edition,   Revised  and  Enlarged.     With  Illustrations.     I2s.  6d. 

PAINTERS'  COLOURS,  OILS,  AND  VARNISHES: 

A    PRACTICAL    lYEANUAIi. 

By  GEORGE    H.    HURST,   F.C.S. 

General  Contents. — Introductory — The  Composition,  Manufacture, 
Assay,  and  Analysis  of  Pigments,  White,  Red,  Yellow  and  Orange,  Green, 
Blue,  Brown,  and  Black— Lakes — Colour  and  Paint  Machinery— Paint  Vehicles 
(Oils,  Turpentine,  &c.,  &c.>~Driers — Varnishes. 

"  A  THOROUGHLY  PRACTICAL  boolc.  .  .  .  Satislactorily  treats  of  the  manufacture  of 
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THE  PAINTER'S  LABORATORY  GUIDE. 

A  Student's  Handbook  of  Paints,  Colours,  and  Vapnishes. 

By   GEORGE   H.    HURST,  F.C.S. 

Abstract  of  Contents. — -Preparation  of  Pigment  Colours.  —  Chemical  Principles 
Involved. — Oils  and  Varnishes. — Properties  of  Oils  and  Varnishes. — Tests  and  Experiments. 
— Plants,  Methods,  and  Machinery  of  the  Paint  and  Varnish  Manufactures. 

"This  excellent  handbook,  .  .  .  the  model  of  what  a  handbook  shoidd  be." — Oils, 
Colours,  and  D'  ysalteries. 


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tions and  Plates  (some  in  Colours),  including  Original  Designs.     12s.  6d. 

Paintings  and  Decorating: 

A    Complete   Practical   Manual  for  House 
Painters  and  Decorators. 

By    WALTER    JOHN    PEARCE, 

LECTURKR   AT   THK   MANCHESTER  TECHNICAL   SCHOOL   FOR   HOCSE-HAINTINO   AND  DKCOEATINQ. 
"A    THOROUGHLY    USEFUL    BOOK      .       .      .       GOOD,    SOUND,    PRACTICAL  INFOR- 
MATION in  a  CLEAR  and  concise  form." — Plumber  and  Decorator. 

"A   THOROUGHLY   GOOD    AND    RELIABLE   TEXT-BOOK.       .       .       .       So    FULL   and 

COMPLETE  that  it  would  be  difficult  to  imagine  how  anything  further  could  be 
added  about  the  Painter's  craft." — Builders'  Journal. 


LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED.  EXETER  STREET,  STRAND. 


CHEMISTRY  AND    TECHNOLOGY.  65 

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THE   CHEMISTRY  OF  INDIA   RUBBER. 

Treatise  on  the  Nature  of  India  Rubber,  its  Chemical  and  Physical  Examina- 
tion, and  t»ie  Determination  and  Valuation  of  India  Rubber  Substitutes. 

Including  the  Outlines  of  a  Theory  on    Vulcanisation. 
By   carl    otto    WEBER,    Ph.D. 

"  Replete  with  scientific  and  also  with  technical  interest.     .     .    .    The  section  on  physical 
properties  is  a  complete  resume  of  every  thing  known  on  the  subject." — India-rubber  J  ourtial. 


In  Handsome  Cloth.     Fully  Illustrated.     los.   6d.  net. 
THE     lYIANUFACTURE    OF    RUBBER    GOODS. 

By  ADOLF  HEIL  and  Dr.  W.  ESCH. 
Translated  by  EDWARD  W.  LEWIS,  A.C.G.I.,  F.C.S., 

Chemist  to  Messrs.  J.  G.  Ingram  &  Son,  London. 

General    Contents. — Raw    Material    and    its    Preparation. — Vulcanisation. — Rubber 
Mixings. — Manufacture   of  ^oft   Rubber   Goods. — Manufacture   of   Hard   Rubber   Goods. 
— Regeneration  of  Waste  Rubber. — Specific  Gravity  of  Rubber  Goods.— Index. 
"  Can  be  recommended  as  a  very  practical  and  useful  work." — Nature. 


In  Large  Crown  8vo.     Fully  Illustrated.     5s.  net. 

G  L  XJ  E,      GELATIKTE, 

AND    THEIR   ALLIED    PRODUCTS, 
A  Practical  Handbook  for  the  Manufacturer,  Agriculturist,  and  Student  of  Technology. 

By    THOMAS     LAMBERT, 

Analytical  and  Technical  Chemist. 

Contents. — Historical. — Glue. — Gelatine. — Size  and  Isinglass. — Treatment  ot  Efflu- 
ents produced  in  Glue  and  Gelatine  Making. — Liquid  and  other  Glues,  Cements,  &c. — Uses 
of  Glue  and  Gelatine. — Residual  Products. — Analysis  of  Raw  and  Finished  Products. — 
Appendix.— Index. 

"A  sufficient  account  of  modern  methods  of  working,  chiefly  from  a  practical  standpoint. 
A  book    .     .     .     of  real  value." — Chemical  Neivs. 


In  Large  8vo.      Handsome  Cloth.     Fully  Illustrated.      15s.  net. 

A  Practical  Manual  on  the  Analysis  of  Materials  and  Finished  Products. 
By  S.   R.  TROTMAN,  M.A.,   F.I.C., 

Public  Analyst  for  the  City  of  Nottingham,  Member  of  the  International  Association 

of  Leather  Trades'  Chemists. 

Synopsis  of  Contents. — Standard  Solutions. — Acids,  Alkalies,  &c. — Water. — Depilation 

and  Deliming.— Fleshings,  &c.— Glue.— Spent  Liquors.— Mineral  and  Vegetable  Tanning 

Agents. — Oils. — Soaps. — Varnishes. — Skin. — Leather.  —  Dyestufifs. —  Degreasing  Agents. — 

Effluents. — Glossary. — Index. 

■' Mr.  Trotman  has  admirably  succeeded  in  his  aim.     .     .     .     Practically  every  section  ot 
the  leather  trade  chemistry  is  gone  into." — Leather  Trades'  Revievi. 


In  Crown  8vo.     Handsome  Cloth. 

THE   CHEMISTRY  OF  THE  COLLOIDS. 

Translated  from  the  German  of  Dr.  VICTOR  POSCHL 
By  Dr.   H.  H.   HODGSON. 


LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET.  STRAND. 


I 

66  OHARLES  ORIFFIN  dk    GO.'S  PUBLICATIONS. 


Second  Edition,    Thoroughly  Revised    I'kroiirihout.     In   Tivo  Large 
Volumes.     Handsome  Cloth.     45*'. 

A    MANUAL    OF    DYEING: 

FOR    THE    USE   OF  PRACTICAL    DYERS,    MANUFACTURERS,   STUDENTS, 
AND  ALL  INTERESTED   IN    THE   ART  OF  DYEING. 

BY 

E.  KNECHT,  Ph.D.,  F.I.C.,  CHR.  RAWSON,  F.I.C.,  F.C.S., 

He»d  of  the  ChemiBtry  aud  Dyeing  Department  of  Late  Head  of  the  Chemistry  and  Dyeing  Department 

the  Technical  School,  Manchester;  Editor  of  "The  of  the    Technical  College,    Bradford  ;    Member 

Journal  of  the  Society  of  Dyers  and  Colourists ;  "  Council  of  the  Society  of  Uyera  and  ColonriBt*;; 

And  RICHARD  LOEWENTHAL,  Ph.D. 

General  Contents. — Chemical  Technology  of  the  Textile  Fabrics — 
Water — Washing  and  Bleaching  —  Acids,  Alkalies,  Mordants — Natural 
Colouring  Matters — Artificial  Organic  Colouring  Matters— Mineral  Colours 
— Machinery  used  in  Dyeing — Tinctorial  Properties  of  Colouring  Matters — 
Analysis  and  Valuation  of  Materials  used  in  Dyeing,  &c.,  &c. 

"  This  authoritative  and  exhaustive  work  .  .  .  the  most  complete  we  have  yet  aeen 
on  the  subject." — Textile  Manufacturer. 


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THE    SYNTHETIC    DYESTUFFS, 

AND 

THE  INTERMEDIATE  PRODUCTS  FROM  WHICH  THEY  ARE  DERIVED. 

By  JOHN  CANNELL  CAIN,  D.Sc.  (Manchester  and  Tubingen), 

Technical  Chemist, 

And  JOCELYN  FIELD  THORPE,  Ph.D.  (Heidelberg),  F.R.S., 

Lecturer  on  Colouring  Matters  in  the  Victoria  University  of  Manchester. 

Part  I.  Theoretical.     Part  II.  Practical.     Part  III.  Analytical. 

"  We  have  no  hesitation  in  describing  this  treatise  as  one  of  the  most  valuable  books 

that  has  appeared.    .    .     .     Will  give  an  impetus  to  the  study  of  Organic  Chemistry 

generally." — Chemical  Trade  Journal. 

Companion    Volume  to  Knccld  d:  Rawson's  "Dyeing."     In  Large  Svo. 

Handsome  Cloth,  Library  Style.     16s.  net. 

A   DICTIONARY   OF 

DYES,    MORDANTS,    &    OTHER    COMPOUNDS 

USED  IN   DYEING  AND   CALICO   PRINTING. 

With  Formulce,  Properties,  and  Applications  of  the  various  substances  described 

and  concise  directions  for  their  Commercial  Valuation, 

and  for  the  Detection  of  Adulterants. 

By  CHRISTOPHER  RAWSON,  F.I.C.,  F.C.S., 

Consulting  Chemist  to  the  Behar  Indigo  Planters'  Association  ;    Co-Author  of  "  A  Manual 

of  Dyeing;" 

WALTER  M.  GARDNER,  F.C.S., 

Head  of  the  Department  of  Chemistry  and  Dyeing,  Bradford  Municipal  Technical  College  : 
Editor  of  the  "  Journ.  Soc.  Dyers  and  Colourists  ;  " 

iND  W.  F.  LAYCOCK,  Ph.D.,  F.C  S., 

Analytical  and  Consulting  Chemist. 
"  Turn  to  the  book  as  one  may  on  any  subject,  or  any  substance  in  connection  with  the 
trade,  and  a  reference  is  sure  to  be  found     The  authors  have  apparently  left  nothing  out  " 
— Textile  Mercury. 

L0ND3N:  CHARLES  GRIFFIN  &  CO..  LIMITEO.  EXETER  STREET.  STRAND. 


THE   TEXTILE  INDUSTRIES.  67 

In  Crown  8vo.      Cloth.      With  Numerous  Illustrations.     6s.  net. 

THE    COTTON    WEAVERS'    HANDBOOK. 

A  Practical  Guide  to  the  Construction  and  Costing  of  Cotton  Fabrics. 
with  Studies  in  Design. 

By     henry     B.    HEYLIN, 

Of  tlie  Royal  Technical  Institute,  Salford. 

Contents  —Weaving— Designing.  —  Weaves. — The  Power  Loom  and  its  Accessories.— 
^ualy8ing  or  Dissection  of  the  Cloth.— Cloth  Quoting  or  CostiDg.— Division  cf  the  Cotton 
Industry  in  Englaml.— Yarns  for  Manufacturing  Purposes. — The  Selection  of  W  arp  Yarns 
and  Yarn  Testing. — Read  and  Heald  Counting  and  Calculations. — Important  Labour 
Units  in  a  Weaving  Mill.— Miscellany. —  Illustrations  of  Machinery. — Examination  Ques- 
tions.— Index. 

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THE    SPINNING   AND   TWISTING   OF   LONG 
VEGETABLE   FIBRES 

(FLAX,     HEWIP,    JUTE,     TOW,    &    RAIVIIE). 

A  Practical  Manual  of  the  most  Modern  Methods  as  applied  to  the  Hachling,  Carding 
Preparing,  Spinning,  and  Twisting  of  the  Long  Vegetable  Fibres  of  Commerce. 

By  HERBERT  R.  CARTER,  Belfast  and  Lille. 

General  Contents.— Long  Vegetable  Fibres  of  Commerce.— Rise  and  Growth  of 
the  Spinning  Industry.— Raw  Fibre  Markets.— Purchasing  Raw  Material.— Storing  and 
Preliminary  Operations. — Hackling. — Sorting. — Preparing. — Tow  Carding  and  Mixing. — 
Tow  Combing.— Gill  Spinning.— The  Roving  Frame.— Dry  and  Demi-sec  Spinning.— Wet 
Spinning.— Spinning  Waste.— Yarn  Reeling. — Manufacture  of  Threads,  Twines,  and 
Cords. — Rope  Making. — The  Alechanioal  Department. — Modern  Mill  Construction. - 
Steam  and  Water  Power. — Power  Transmission. 

"  Meets  the  requirements  of  the  Mill  Manager  or  Advanced  Student  In  a  manner 
perhaps  more  than  satisfactory.  .  .  .  We  must  highly  commend  the  work  as  repre- 
senting up-to-date  practice." — Nature. 


In  Large  Svo,  Handsome  Cloth,  with  Numerous  Illustrations.      9s.  net. 

TEXTILE  FIBRES  OF  COMMERCE. 

A   HANDBOOK    OF 

The  Oeeuppence,  Distribution,  Preparation,  and  Industrial  Uses  of  the 
Animal,  Vegetable,  and  Mineral  Products  used  in  Spinning  and  Weaving. 

By    WILLIAM    I.    HANNAN, 

Lecturer  on  Botany  at  the  Ashton  Municipal  Technical  School,  Lecturer  on  Cottoii 
Spinning  at  the  Chorley  Science  and  Art  School.  &c. 
"  Obefdl  Infoemation.    .    .    .    Admirable  iLLDSTKATiONS.    .    .    ." — lexiile  Uecorder. 


In  Large  Svo,  with  Illustrations  and  Printed  Patterns.     Price  21s. 

TEXTILE      I>I^INTING: 

A    PRACTICAL    MANUAL. 
Including  the  Processes  Used  in  the  Printing  ot 
COTTON,    TAT-OOLLEN,    SILK,    and    HALF- 
SILK   FABRICS. 
By   C.    F.    SEYMOUR   ROTHWELL,   F.C.S., 

^(rn.  Soc.  of  Chem.  Ind. ;  late    Lecturer  at  the  Uunic.  Teeh.  School    Manchester. 
"  By  far  the  best  and  most  practical  book  on  textile  printing  which  has  yet  been 
brought  out,  and  will  long  remain   the  standard  work  on  the  subject." — Textile  Mercury. 

LONDON:  CHARLES  GRIFFIN  &  CO..  LIMITED.  EXETER  STREEL  STRAND. 


6S  CHARLES  ORIFFIN  di  CO.'S  PUBLICATIONS. 

Large  8vo.     Handsome  Cloth.     128.  6d. 

BLEACHING  &  CALICO-PRINTING. 

A    Short    Manual   for    Students   and 
Practical   Men. 


By     GEORGE     DUERR, 

aching,  Dyeing,  and  Printing  Department  at  the  Ace 
Technical  Schools  ;   Chemist  and  Colourist  at  the  Irwell  Print  \<^ork8. 


Director  of  the  Dleachlng,  Dyeing,  and  Printing  Department  at  the  Accrington  and  Bacup 
,„._^_  =  „,  o_,. ^,       ■  .        ,  „  ,       , "  ■   ■  Wo  • 


AS.SISTED    BY   WILLIAM    TURNBULL 

(of  Tiirnbull  &  Stockdale,  Limited). 

With  Illustrations  and  upwards  of  One  Hundred  Dyed  and  Printed  Patterns 
designed  specially  to  show  various  Stages  of  the  Processes  described. 

GENERAL  CONTENTS. -Cotton,  Composition  of;  Bleaching,  New 
Processes  ;  Printing,  Hand-Block  ;  Flat-Press  Work  ;  Machine  Printing — 
Mordants— Styles  of  Calico-Printing  :  The  Dyed  or  Madder  Style,  Resist 
Padded  Style,  Discharge  and  Extract  Style,  Chromed  or  Raised  Colours, 
Insoluble  Colours,  &c.  —  Thickeners  —  Natural  Organic  Colouring  Matters 
— Tannin  Matters — Oils,  Soaps,  Solvents — Organic  Acids  —  Salts  —  Mineral 
Colours — Coal  Tar  Colours — Dyeing — Water,  Softening  of —Theory  of  Coloura 
— Weights  and  Measures,  &c. 

"  When  a  rbadt  way  out  of  a  difficulty  is  wanted,  it  is  in  books  iikb  this  that  it  is  found."— 
Textile  Recorder. 

"Mr.  Durbr's  woEK  will  be  found  MOST  USEFUL.    .    .    .    The  information  given  is  of  orba? 
JAtt;B.    .    .    .    The  Recipes  are  THOEOUGHLY  PRACTICAL,"— TfWiZe  Jl/apiz/actMrw. 


Second  Edition,  with  New  Appendix.     In  Handsome  Cloth. 
With  76  Illustrations.      5s.  net. 

DYEING    AND    CLEANING. 

By    frank    J.     FARRELL,    M.Sc,    &c. 

General  Contents. — Technology  of  the  Textile  Fibres.  —  Dry 
Cleaning.  —  Wet  Cleaning.  —  Dyeing. — Dry  Dj-eing.  —  Special  Methods, 
Cleaning  and  Dyeing  Sliin  Pv-ugs,  Feathers,  and  Hats. — Finishing. — 
Appendices. — Index. 

"  A  timely  and  valuable  contribution  .  .  .  well  got  up  in  every  way." — Dyer  and 
Calico  Printer. 

LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET.  STRAND. 


INTRODUCTORY    WORKS.  69 

Third  Edition,  Revised,  Enlarged,  and  Re-issued.     Price  6s.  net. 
A   SHORT    MANUAL    OF 

INORGANIC     CHEMISTRY. 

By    a.    DUPRE,   Ph.D.,  F.R.S., 
And    WILSON    HAKE,  Ph.D.,  F.I.O.,  F.C.S., 

Of  the  Westminster  Hospital  Medical  School. 

"An  bxamplk  of  the  advantages  of  the  Systematic  Treatment  of  a  Science 
over  the  fragmentary  style  so  generally  followed.  Bv  A  long  way  the  best  of  the  small 
Manuals  for  Students." — Analyst. 


In  Handsome  Cloth.    With  nearly  50  Illustrations.    3s.  6d.  net. 

THE    ELEMENTS    OF    CHEMICAL    ENGINEERING. 

By  J.  GROSSMANN,  M.A.,  Ph.D.,  F.I.C. 

WITH  A  PREFACE   BY 

Sir    WILLIAM    RAMSAY,    K.C.B.,    F.R.S. 

Contents. — The  Beaker  and  its  Technical  Equivalents. — Distilling  Flasks,  Liebig's 
Condensers. — Fractionating  Tubes  and  their  Technical  Equivalents. — The  Air-Bath  and 
its  Technical  Equivalents. — The  Blowpipe  and  Crucible  and  their  Technical  Equivalents. 
— The  Steam  Boiler  and  other  Sources  of  Power. — General  Remarks  on  the  Application 
of  Heat  in  Chemical  Engineering. — The  Funnel  and  its  Technical  Equivalents. — The 
Mortar  and  its  Technical  Equivalents. — Measuring  Instruments  and  their  Technical 
Equivalents. — Materials  Used  in  Chemical  Engineering  and  their  Mode  of  Application. — 
Technical  Research  and  the  Designing  of  Plant. — Conclusion. — Chemicals  and  Materials. 
—Index. 

"Excellent.  .  .  .  Every  student  of  chemistry  attending  a  technical  course  should 
obtain  a  cooy.  ' — Chemical  News. 


LABORATORY  HANDBOOKS   BY  A.   HUMBOLDT   SEXTON, 

Profesaor  of  Metallurgy  in  the  QIasgow  and  West  of  Scotland  Technical  College. 


OUTLINES    OF    QUANTITATIVE    ANALYSIS. 

FOR  THE  USB  OF  STUDENTS. 

With  Illustrations.     Fifth  Edition.     Crown  8vo,  Cloth,  38. 

'*  A  OOBCPAOT  lABORATORY  GUiDK  for  beginners  was  wanted,  and  the  want  has 
be«n  WBLL  8UPPLIKD.     ...     A  good  and  nsefnl  book." — Lancet, 


OUTLINES   OF    QUALITATIVE   ANALYSIS. 

FOR   THE   USE  OF  STUDENTS. 

With  Illustrations,    Fourth  Edition,  Revised.    Crown  8vo,  Cloth,  Ss.  6d. 

"  The  work  of  a  thoronghly  practical  chemist." — British  Medical  Journal. 
"  Compiled  with  great  care,  and  will  supply  a  want." — Journal  of  Education. 


ELEMENTARY    METALLURGY: 

Including    the   Author's    Practical   Laboratory    Course. 


[Seep.  51. 


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70  CHARLES  GRIFFIN  d-  CO.'S  PUBLICATIONS. 


Third  Edition,  Revised  and  Enlarged.    Large  Crown  8vo,  with  numerous 

Illustrations.     3s.  6d. 

THE    FLOWERING    PLANT, 

WITH  A  SUPPLEMENTARY  CHAPTER  ON  FERNS  AND  MOSSES. 
As    Illustrating    the    First    Principles    of    Botany. 

By  J.  R.  AINSWORTH   DAVIS,  M.A.,  F.Z.S., 

Prof,  of  Bioloev,  University  College,  Aberystwyth  ;  Examiner  in  Zoology, 
University  of  Aberdeen. 

"  It  would  be  hard  to  find  a  Text-book  which  would  better  guide  the  student  to  an  accurate 
knowledge  of  modem  discoveries  in  Botany.    .  .    The  scientific  accuracv  of  statement, 

and  the  concise  exposition  of  first  principles  make  it  valuable  for  educational  purposes.     In 
the  chapter  on  the  Physiology  of  Flowers,  an  admirable  resume,  drawn  from  Darwin,  Hermann 
Miiller,  Kemer,  and  Lubbock,  of  what  is  known  of  the  Fertilization  of  Flowers,  is  given. "- 
Journal  0/  Botany. 


POPULAR    WORKS    ON   BOTANY   BY    MRS.    HUGHES-GIBB. 

With  Illustrations.    Crown  8vo.    Cloth.    2a.  6d 

HOW   PLANTS   LIVE   AND   V/ORK : 

A  Simple  IntPOduction  to  Real  Life  in  the  Plant-world,  Based  on  Lessons! 
originally  given  to  Country  Children. 

By    ELEANOR     HUGHES-GIBB. 

*.*  The  attention  of  all  interested  in  the  Scientific  Training  of  the  Young  is  requested  to  this 
DBLiSHTPDLLT  PBESH  and  CHARMING  LITTLE  BOOK.  It  ought  to  be  In  the  hands  of  every  Mother 
and  Teacher  throughout  the  land. 

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jiCcnttATE.the  first  lessons  in  plant-life  are  set  before  'A."— Natural  Science. 

"In  every  way  well  calculated  to  make  the  study  of  Botany  atteactivb  to  the  young."— 
Scotsman. 


Witii    Illustrations.      Crown    8vo.      Gilt     2s.    Od. 

THE    MAKING    OF    A    DAISY; 

*' WHEAT    OUT    OF    LILIES;" 

And  other  Studies   from   the   Plant  World. 

A  Popular  Introduction  to  Botany. 
By    ELEANOR     HUGHES-GIBB, 

Author  of  Hmv  Plants  Live  and  Work. 

"A  BRIGHT  little  introduction  to  the  study  of  Flowers." — Journa'  o'  botany. 
"The  book  will  afford  real  assistance  to  those  who  can  derive  pleasure  from  the  itudy  of 
Nature  i'l  the  open.  .     .     The  literary  style  is  comm-^n  .aoie  '  —  Knoichiige 

tONOOi«-,  CHARLES  GRIFFIN  &  CO.,  liMifEO.  EXETER  STREET,  STRANO. 


GNIFFIX'S  ''OFEX-AIL'"  SEL'IES.  71 

"  boys   COULD    NOT   HAVE    A    MORE    ALLURING    INTRODUCTION    tO   scientific   pursulU 

thau   these   charming-looking  volumes."- -Letter  to    the    Publishers  from   the  Head- 
master of  one  of  our  great  Public  Schools. 

Secdnd  Edition',   llcvised.     Handsome  Clotli.     ()s.  net. 

OPEfl-AIH  STUDIES  I]i  B0Tfl|4Y: 

SKETCHES    OF    BRITISH    WILD    FLOWERS 
IN  THEIR  HOMES. 

By  R.    LLOYD    PRAEGER,   B.A.,    M.R.LA 

Illustrated  by  Drawings  from  Nature  by  S.  Rosamond  Praeger, 
and  Photographs  by  R.  Welch. 
General  Contents. — A  Daisy-Starred  Pasture — Under  the  Hawthornt 
— By  the  River — Along  the  Shingle — A  Fragrant  Hedgerow — A  Connemara 
Bog — Where  the  Samphire  grows — A  Flowery  Meadow — Among  the  Com 
(a  Study  in  Weeds) — In  the  Home  of  the  Alpines — A  City  Rubbish-Heap — 
Glossary. 

"A  FRESH  AND  STIMULATING  book    .     .    .    should  take  a  high  place    .    .     .    The 
Illustrations  are  drawn  with  much  skill."— 2'Ae  Times. 

"Beautifully  illustrated.     .     .     .     One  of  the  most  accurate  as  well  aa 
IBTERKSTING  books  of  the  kind  we  have  seen." — Athenceum. 

"Redolent  with  the  scent  of  woodland  and  meadow." — The  Standard. 


With   12   Full-Page   Ulustrationa  from   Photographs.     Cloth. 
Second  Edition,  Reuised.      8a.    6d, 

OPEN-fllH  STUDIES  Iji  GEOLOGY: 

An  Intpoduetion  to  Geology  Out-of-doors. 

By    GRENYILLE    A.   J.    COLE,    F.G.S.,    M.R.I.A., 

Professor  of  Geology  in  the  Royal  College  of  Science  for  Ireland, 
and  Examiner  in  the  University  of  London. 

General  Contents. — The  Materials  of  the  Earth — A  Mountain  Hollow 

— Down  the  Valley — Along  the  Shore — Across  the  Plains — Dead  Volcanoe« 

—A  Granite  Highland— The  Annals  of  the  Earth— The  Surrey  Hills— The 

Isolds  of  the  Mountains. 

"The  FABCiNATiNQ  'OPBN-AiR   STUDIES'  of   Prof.   Oole  give  the  subject  a  olow  op 
ANIMATION    .    .    .    cannot  fail  to  arouse  keen  interest  in  geoXogj. " —Oeological  Magatine. 
"  A  CHABiuiKQ  BOOK,  beautifully  illustrated."  —jl<A«raa«ini. 


Beautifully  Illustrated.      With  a  Frontispiece  in  Colours,  and  Numerous 
Specially  Drawn  Plates  by  Charles  Whymper.     7s.  6d. 

OPEH-fllH  STUDIES  Ifi  BIRD-LIFE: 

SKETCHES  OF  BRITISH  BIRDS  IN  THEIR  HAUNTS. 
By    CHARLES    DIXON. 

The  Spacious  Air.— The  Open  Fields  and  Downs.— In  the  Hedgerows.— On 
Open  Heath  and  Moor.— On  the  Mountains.— Amongst  the  Evergreens.— 
Copse  and  Woodland.— By  Stream  and  Pool.— The  Sandy  Wastes  and  Mud- 
flats.— Sea-laved  Rocks. — Birds  of  the  Cities. — Index. 

"Enriched  with  excellent  illustrations.  A  welcome  addition  to  all  librariei." — West- 
minster Revieiu. 

kmm\i  I  CHARLES  GRIFFIN  A  CO.,  LIMITED  EXETER  8TREEL  STRAiii. 


72  CHARLES  ORIFFIN  «fc  CO.'S  PUBLICATIONS. 

AN  ELEMENTARY  TEXT -BOOK  OF  PHYSICS. 

By  R.  WALLACE  STEWART,  D.Sc.  (Lond.) 

In  Four  Volumes.      Crown  8uo.      Cloth.      Each  Fully  Illustrated. 

Sold  Separately. 


Vol.   I.     Uniform  with  Vols.  II.  to  IV.  and  B'ully  Illustrated. 

GENERAL    I>H:YSICS. 

Now  IN  THE  Press.     Will  be  readj'  very  short!}'. 


Vol.  II.     JcsT  Published.     Profusely  Illustrated.     2s.  6d.  net. 

SOUND. 

Contents. — Simple  Harmonic  Vibration. — Production  of  Sound. — 
Wave  Motion. — Propagation  of  Sound. — Characteristics  of  Sound. — Re- 
flection and  Refraction  of  Sound. — Velocity  of  Sound  in  Air  and  Water. — 
Transverse  Vibration  of  Strings. — Longitudinal  Vibration  of  Rods  and 
Columns  of  Air. — Index. 

"  Should  supply  the  much-felt  need  (jf  an  elementary  tieatnient  of  this  snl)ject  .  .  . 
distinctly  good."— yafure. 


Vol.  III.     .Just  Published.     Willi  \42  Illustrations,     .^s.  6d.  net. 

LIGHT. 

Contents.  —  Introductory.  —  Rectilinear  Propagation  of  Light.  — 
Photometry.  —  Reflection  at  Plane  Surfaces. — Reflection  at  Spherical 
Surfaces. — Refraction.  —  Refraction  through  Lenses. — Dispersion. — Index. 

"  This  elementary  treatise  resembles  Part  II.  (.'^o?<«d)  in  its  attractiveness  .  .  .  the 
treatment  is  good  .  .  .  excellent  <liagrams  .  .  .  very  clear." — Jowni.  of  Inst,  of 
Teachers  in  Technical  Institutes. 


Vol.  IV.     Just  Published.     With  84  Illustrations.     3s.  6d.  net. 

MEAT. 

Contents.  —  Introductorj'.  —  Thermometr}-.  —  Expansion  of  Solids.  — 
Expansion  of  Liquids. — Expansion  of  Gases.  —  Calorimetry.  —  Specific 
Heat. — Liquefaction  and  Solidification. — Vaporisation  and  Condensation. 
— Conduction  of  Heat. — Convection.- — Mechanical  Equivalent  of  Heat. — 
Radiation.  — Indfx. 

In  Large  Svo.     With  Bibliography,  Illustrations  in  the  Text,  and 

Seven  Plates.     12s.  6d. 

THE    MEAN    DENSITY    OF   THE    EARTH. 

An  Essay  to  which  the  Adams  Prize  was  Adjudged  in  1893  in  the  University  of  Cambridge. 

By  J.  H.  POYNTING,  Sc.I).,  F.R.S. 

"Cannot  fail  to  be  of  great  and  general  interest."— .'I f/icrtajon. 


LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAND. 


TEXT -BOOK.    OF  PHYSICS.  73 

In  Five  Volumes.     Large  8vo.     Sold  Separately. 

A  TEXT-BOOK  OF  PHYSICS. 

By    J.     H.    POYNTING,     Sc.D.,     F.R.S., 

Professor  of  Physics,   I'.iniiinghain   I'liiversity, 

And    Sir    J.    J.    THOMSON,    M.A.,    F.R.S., 

Professor  of  Experimental  Physics  in  the  T'nivcrsity  of  Cambridge. 


Introductory  Volume.     Fifth  Edition,  Revised.     Fully  Illustrated. 

10s.  6d. 

I>FtOF>ERTIES    OF    IVTJLTTER. 

Contents.  —  Gravitation.  —  The  Acceleration  of  Gravity.  —  Elasticity. —  Stresses  and 
Strains. — Torsion.— Bending  of  Kods.— Spiral  Springs —Collision. —  Compressibility  of 
Liquids. — Pressures  and  Volumes  of  Gases. — Thermal  Effects  Accompanying  Strain. — 
Oapillarity.  — Surlace  Tension. — Laplace's  Theory  of  Capiharity. — Diffusion  of  Liquids  — 
Diffusion  of  Gases.  — Viscosity  of  Liquids. — Index. 

"  We  regard  this  book  as  quite  indispensable  not  merely  to  teachers  but  to  iihysicists  of  every 
grade  above  the  lowest."—  University  Correspondent. 


Volume  II.     Fifth  Edition.     Fully  Illustrated.     Price  Ss.  6d. 

SOUND. 

OowTKNTS. — The  Nature  of  Sound  aud  its  chief  Characteristics. — The  Velocity  of  Sound 
IL  Air  and  other  Media. — Reflpcion  and  Refraction  of  Sound. — Frequency  and  Pitch  of 
Notes.  — Besonance  and  Forced  Oscillations. — Analysis  of  Vibrations. — The  Transvers* 
Vibrations  of  Stretched  Strings  or  Wires  —Pipes  and  other  Air  Cavities.— Rods. — Plates. 
.—Membranes. — Vibrations  maintained  by  Heat. — Sensitive  Flames  and  Jets. — Musical 
Sand.— The  Superposition  of  Waves.-  Indi  x. 

"The work    .    .     .     maybe  recommended  to  anyone  desirous  of  possessing  an  east 
DP-TO-DATE  Standard  Treatise  on  Acoustics." — Literature. 


Volume  III.     Third  Edition,  Revised.     Fully  Illustrated.     Price  153. 

HEAT. 

Contents. — Temperature.  — Expansion  of  Solids  — Liquids.  — Gases.  —  Circulation 
and  Convection. — Quantity  of  Heat;  Specific  Heat. — Conductivity. — Forms  of  Energy; 
Conservation;  Mechanical  Equivalent  of  Heat.— The  Kinetic  Theory  —Change  of  State; 
Liq aid,  Vapour.  —  Critical  Points.  —  Solids  and  Liquids. — Atmospheric  Conditions. — 
Radiation. — Theory  of  Exchanges. — Radiation  and  Temperature. — Thermodynamics. — 
Isothermal  and  Adiabatic  Changes. — Thermodynamics  of  Changes  of  State,  and  Solu- 
tious.— Thermodynamics  of  Radiation.— Index. 

"Well  up-to-date,  and  extremely  clear  and  exact  throughout.  ...  As  clear  as 
it  would  be  possible  to  make  such  a  text-book  " — Nature. 

Remaining  Volumes  in  Preparation — 

LIGHT;    MAGNETISM  AND  ELECTRICITY. 

LONDON:  CHARLES  GRIFFIN  &  CO.,  LIMITED,  EXETER  STREET,  STRAND. 


74 


CHARLES  GRIFFIN  cfc  CO.'S  PUBLICATIONS. 


In  Crown  8vo.       With  Diagrams.      Cloth. 
AN   INTRODUCTION   TO 

THE  THEORY  OF  STATISTICS. 

By  G.    UDNY    yule, 
Honorary  Secretary  of  the  Royal  Statistical  Society  of  London,  &e. 

Contents.— Introduction.  Part  I.— The  Theory  of  Attributes.— Notation  and 
Terminology.  —  Consistenee.  —  Assoriatiou.  —  Partial  Association.  —  Manifold  Classitit-a- 
tion.  Part  II.— The  Theory  of  Variables.— Freiiuency  Distribution.— Averages.— 
Measures  of  Dispersion,  &c. — Ccjrrelation. — Do.,  Practical  Applications  and  Methods. — 
Miscellaneous  Theorems  Involving  the  use  of  the  Coi-relation  Coefficient.  — Partial 
Correlation.  Part  III.— Theory  of  Sampling.— Simple  Sampling  of  Attrilnites.— 
Effects  (jf  Removing  the  Limitations  of  Simple  Sampling.— The  Binomial  Distribution 
and  the  Normal  Curve. — Normal  Correlation. — The  Simpler  Cases  of  Sampling  for 
Variables. — Percentiles  and  Mean. 


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THE     OFFICIAL     YEAR-BOOK 

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SCIENTIFIC  AND  LEARNED  SOCIETIES  OF  GREAT  BRITAIN 

AND  IRELAND. 

COMPILED  FROM  OFFICIAL  SOURCES. 

Oomprising  (together  ivith  other  Official  Information)  LISTS  of  the 
PAPERS  read  during  the  Session  1908-1909  before  all  the  LEADING 
SOCIETIES  throughout  the  Kingdom  engaged  in  the  following  Depart- 
ments of  Research : — 


$  I.  Science  Generally:  i.e.,  Societies  occupy- 
ing theniselves  with  several  Branches  of 
Science,  or  with  Science  and  Literature 
jointly. 

J  2.  Mathematics  and  Physics. 

§  3.  Chemistry  and  Photography. 

§  4.   Geology,  Geography,  and  Mineralogy. 

I  5.  Biology,  including  Microscopy  and  An- 
thropology. 


5   8. 

5  9 
5 10 

5  II. 
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Economic  Science  and  Statistics. 
Mechanical   Science,  Engineering,  and 

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Agriculture  and  Horticulture. 
Law. 

Literature 
Psychology. 
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