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TRANSACTIONS 


CAMBRIDGE 


PHILOSOPHICAL    SOCIETY. 


ESTABLISHED   November  15,  1819. 


VOLUME   THE   FIFTH. 


CAMBRIDGE: 

PRINl'ED  BY  JOHN  SMITH,  PRtNTER  TO  THE  UNIVERSITY: 

AND    SOLD    BY 

JOHN    WILLIAM    PARKER,    WEST    STRAND,    LONDON; 
J.  &  J.J.   DEIGHTON;    AND  T.STEVENSON,  CAMBRIDGE. 


M.DCCC.XXXV. 


CONTENTS   OF   THE  FIFTH  VOLUME. 


Part  I. 


PAGE 


N°.  I.  Mathematical  Investigations  concerning  the  Laws  of  the  Equilibrium  of  Fluids 
analogous  to  the  Electric  Fluid,  with  other  similar  Researches:  by  George 
Green,  Esq.  Communicated  by  Sir  Edward  Ffrench  Bromhead,  Bart,  M.A. 
F.R.S.L.  &   E I 

II.     On    Elimination    between    an    Indefinite    Number   of   Unknown    quantities :    by    the 

Rev.  R.  Murphy 65 

III.  On   the    General    Equation   of  Surfaces  of  the    Second    Degree :    by    Augustus 

De  Morgan,  Esq 77 

IV.  On  a  Monstrosity  of  the  Common  Mignionette :  by  the  Rev.  Professor  Henslow.  . .     95 


Part  II. 

V.  On  the  Calculation  of  Newton's  Experiments  on  Diffraction:  by  Professor  Airv...    101 

VI.  Second    Memoir    on    the    Inverse    Method    of    Definite    Integrals:    by    the    Rev. 

R.  MUBPHY 113 

VII.  On  the  Nature  of  the  Truth  of  the  Laws  of  Motion  :  by  the  Rev.  W.  Whewkll 149 

VIII.    Researches  in  the  Theory  of  the  Motion  of  Fluids :  by  the  Rev.  James  Challis US 

IX.  Theory   of   Residuo- Capillary    Attraction;    being   an    Explanation   of  the    Pheno- 

mena  of  Endosmose   and   Exosmose  on   Mechanical   Principles:    by   the    Rev. 

J.   PovfEH 205 

X.  On  Aerial  Vibrations  in  Cylindrical  Tubes:   by  William  Hopkins,  M.A 231 

XI.     On  the  Latitude  of  Cambridge  Observatory:   by  Professor  Airy 271 


IV  CONTENTS. 


Paet  III. 


PAAB 


N°    XII.      On  the  Diffraction  of  an  Object-glass  with  Circular  Aperture:    bt    Professor 

Airy  283 

XIII.  On  the  Equilibrium  of  the  Arch:  by  the  Rev.  Hbnry   Moseley    293 

XIV.  Third    Memoir   on   the    Inverse   Method  of  Definite    Integrals:    by  the  Rev. 

R.  MuHPHY 315 

XV.  On   the  Determination  of  the  Exterior  and  Interior  Attractions   of  Ellipsoids 

of  Variable  Densities :   by  George  Green,  Esq 395 

XVI.  On   the    Position  of  the  Axes  of  Optical  Elasticity  in   Crystals    belonging   to 

the   Oblique- Prismatic  System:    by  W.  H.  Miller,  Esq 431 


ADVERTISEMENT. 

The  Society  as  a  hody  is  not  to  he  considered  responsible  for  any 
Jucts  and   opinions  advanced  in  the  several  Papers,    which   must  rest 
entirely  on   the  credit  of  their   respective  Authors. 


The  Society  takes  this  opportunity  of  expressing  its  grateful 
acknowledgments  to  the  Syndics  of  the  University  Press,  for  their 
liberality  in  taking  upon  themselves  the  expense  of  printing  this 
Part  of  its   Transactions. 


TRANSACTIONS 


CAMBRIDGE 


PHILOSOPHICAL    SOCIETY. 


Vol.  V.    Part  I. 


CAMBRIDGE: 

PRINTED  BY  JOHN  SMITH.  PRINTER  TO  THE  UNIATERSITY; 

AND   SOLD   EV 

JOHN  WILLIAM   PARKER,   445  WEST   STRAND,  LONDON; 

J.  &  J.  J.  DEIGHTON,  AND  T.  STEVENSON, 
CAMBRIDGE. 


M.DCCC.XXXIII. 


d-? 


^AL  rtV 


I.  Mathematical  Investigations  concerning  the  Laws  of  the  Equilibrium 
of  Fluids  analagous  to  the  Electric  Fluid,  with  other  similar  Researches. 
By  George  Green,  Esq.  Communicated  hy  Sir  Edward  Ffrench 
Bromhead,  Bart.  M.A.  F.K.S.L.  and  E. 


[Read    Nov.  12,    1832.] 


Amongst  the  various  subjects  which  have  at  different  times  occupied 
the  attention  of  Mathematicians,  there  are  probably  few  more  interesting 
in  themselves,  or  which  offer  greater  difficulties  in  their  investigation, 
than  those  in  which  it  is  required  to  determine  mathematically  the 
laws  of  the  equilibrium  or  motion  of  a  system  composed  of  an  infinite 
number  of  free  particles  all  acting  upon  each  other  mutually,  and  ac- 
cording to  some  given  law.  When  we  conceive,  moreover,  the  law  of 
the  mutual  action  of  the  particles  to  be  such  that  the  forces  which 
emanate  from  them  may  become  insensible  at  sensible  distances,  the 
researches  to  which  the  consideration  of  these  forces  lead  will  be  greatly 
simplified  by  the  limitation  thus  introduced,  and  may  be  regarded  as 
forming  a  class  distinct  from  the  rest.  Indeed  they  then  for  the  most 
part  terminate  in  the  resolution  of  equations  between  the  values  of 
certain  functions  at  any  point  taken  at  will  in  the  interior  of  the  sys- 
tem, and  the  values  of  the  partial  differentials  of  these  functions  at  the 
same  point.  When  on  the  contrary  the  forces  in  question  continue 
sensible  at  every  finite  distance,  the  researches  dependent  upon  them 
become  far  more  complicated,  and  often  require  all  the  resources  of 
the  modern  analysis  for  their  successful  prosecution.  It  would  be  easy 
so  to  exhibit  the  theories  of  the  equilibrium  and  motion  of  ordinary 
fluids,  as  to  offer  instances  of  researches  appertaining  to  the  former 
class,  whilst  the  mathematical  investigations  to  which  the  theories  of 
Electricity  and  Magnetism  have  given  rise  may  be  considered  as  in- 
teresting examples  of  such  as  belong  to  the  latter  class. 

Vol.  V.    Pakt  I.  A 


2  Mr  green,  ON  THE  LAWS  OF  THE  EQUILIBRIUM  OF  FLUIDS. 

It  is  not  my  chief  design  in  this  paper  to  determine  mathematically 
the  density  of  the  electric  fluid  in  bodies  under  given  circumstances, 
having  elsewhere*  given  some  general  methods  by  which  this  may  be 
effected,  and  applied  these  methods  to  a  variety  of  cases  not  before 
submitted  to  calculation.  My  present  object  will  be  to  determine  the 
laws  of  the  equilibrium  of  an  hypothetical  fluid  analagous  to  the  electric 
fluid,  but  of  which  the  law  of  the  repulsion  of  the  particles,  instead  of 
being  inversely  as  the  square  of  the  distance,  shall  be  inversely  as  any 
power  n  of  the  distance ;  and  I  shall  have  more  particularly  in  view 
the  determination  of  the  density  of  this  fluid  in  the  interior  of  con- 
ducting spheres  when  in  equilibrium,  and  acted  upon  by  any  exterior 
bodies  whatever,  though  since  the  general  method  by  which  this  is 
effected    will    be    equally    applicable    to    circular    plates   and   ellipsoids. 

1  shall  present  a  sketch  of  these  applications  also. 

It  is  well  known  that  in  enquiries  of  a  nature  similar  to  the  one 
about  to  engage  our  attention,  it  is  always  advantageous  to  avoid  the 
direct  consideration  of  the  various  forces  acting  upon  any  particle  p  of 
the  fluid  in  the  system,  by  introducing  a  particular  function  V  of  the 
co-ordinates  of  this  particle,  from  the  differentials  of  which  the  values 
of  all  these  forces  may  be  immediately  deduced  f.  We  have,  therefore, 
in  the  present  paper  endeavoured,  in  the  first  place,  to  find  the  value 
of  V,  where  the  density  of  the  fluid  in  the  interior  of  a  sphere  is  given 
by  means  of  a  very  simple  consideration,  which  in  a  great  measure 
obviates  the  difficulties  usually  attendant  on  researches  of  this  kind, 
have  been  able  to  determine  the  value  F^,  where  p,  the  density  of  the 
fluid  in  any  element  dv  of  the  sphere's  volume,  is  equal  to  the  product 
of  two  factors,  one  of  which  is  a  very  simple  function  containing  an 
arbitrary  exponent  fi,  and  the  remaining  one  J"  is  equal  to  any  rational 

*  Essay  on  the  Application  of  Mathematical  Analysis  to  the  Theories  of  Electricity  and 
Magnetism. 

t  This  function  in  the  present  case  will  be  obtained  by  taking  the  sum  of  all  the  molecules 
of  a  fluid  acting  upon  p,  divided  by  the  (n  — 1)*  power  of  their  respective  distances  from^; 
and  indeed  the  function  which  Laplace  has  represented  by  F  in  the  third  book  of  the 
Mecanique  Celeste,   is  only  a  particular  value  of  our  more  general  one  produced  by  writing 

2  in  the  place  of  the  general  exponent  n. 


Mb  green,   on   THE   LAWS   OF    THE    EQUILIBRIUM    OF    FLUIDS.        S 

and  entire  function  whatever  of  the  rectangular  co-ordinates  of  the  element 
dv,  and  afterwards  by  a  proper  determination  of  the  exponent  /3,  have 
reduced  the  resulting  quantity  ^  to  a  rational  and  entire  function  of 
the  rectangular  co-ordinates  of  the  particle  p,  of  the  same  degree  as 
the  function  f.  This  being  done,  it  is  easy  to  perceive  that  the  reso- 
lution of  the  inverse  problem  may  readily  be  effected,  because  the 
coefficients  of  the  required  factor  f  will  then  be  determined  from  the 
given  coefficients  of  the  rational  and  entire  function  V,  by  means  of 
linear  algebraic  equations. 

The  method  alluded  to  in  what  precedes,  and  which  is  exposed  in 
the  two  first  articles  of  the  following  paper,  will  enable  us  to  assign 
generally  the  value  of  the  induced  density  p  for  any  ellipsoid,  what- 
ever its  axes  may  be,  provided  the  inducing  forces  are  given  explicitly 
in  functions  of  the  co-ordinates  of  p ;  but  when  by  supposing  these  axes 
equal  we  reduce  the  ellipsoid  to  a  sphere,  it  is  natural  to  expect  that 
as  the  form  of  the  solid  has  become  more  simple,  a  corresponding  degree 
of  simplicity  will  be  introduced  into  the  results ;  and  accordingly,  as 
will  be  seen  in  the  fourth  and  fifth  articles,  the  complete  solutions  both 
of  the  direct  and  inverse  problems,  considered  under  their  most  general 
point  of  view,  are  such  that  the  required  quantities  are  there  always 
expressed  by  simple  and  explicit  functions  of  the  known  ones,  inde- 
pendent of  the  resolution  of  any  equations  whatever. 

The  first  five  articles  of  the  present  paper  being  entirely  analytical, 
serve  to  exhibit  the  relations  which  exist  between  the  density  p  of  our 
hypothetical  fluid,  and  its  dependent  function  V;  but  in  the  following 
ones  our  principal  object  has  been  to  point  out  some  particular  appli- 
cations of  these  general  relations. 

In  the  seventh  article,  for  example,  the  law  of  the  density  of  our 
fluid  when  in  equilibrium  in  the  interior  of  a  conductory  sphere,  has 
been  investigated,  and  the  analytical  value  of  p  there  found  admits  of 
the  following  simple  enunciation. 

The  density  p  of  free  fluid  at  any  point  p  within  a  conducting  sphere 
A,  of  which  O  is  the  centre,  is  always  proportional  to  the  {n  -  4)"'  power 
of  the  radius  of  the  circle  formed  by  the  intersection  of  a  plane  per- 
pendicular to  the  ray  Op  with  the  surface  of  the  sphere  itself,  provided 

A  2 


4         Mr  green,    on   THE   LAWS   OF    THE   EQUILIBRIUM   OF   FLUIDS. 

n  is  greater  than  2.  When  on  the  contrary  n  is  less  than  2,  this  law 
requires  a  certain  modification ;  the  nature  of  which  has  been  fully 
investigated  in  the  article  just  named,  and  the  one  immediately  fol- 
lowing. 

It  has  before  been  remarked,  that  the  generality  of  our  analysis  will 
enable  us  to  assign  the  density  of  the  free  fluid  which  would  be  induced 
in  a  sphere  by  the  action  of  exterior  forces,  supposing  these  forces  are 
given  explicitly  in  functions  of  the  rectangular  co-ordinates  of  the  point 
of  space  to  which  they  belong.  But,  as  in  the  particular  case  in  which 
our  formulae  admit  of  an  application  to  natural  phenomena,  the  forces  in 
question  arise  from  electric  fluid  diffused  in  the  inducing  bodies,  we 
have  in  the  ninth  article  considered  more  especially  the  case  of  a  con- 
ducting sphere  acted  upon  by  the  fluid  contained  in  any  exterior  bodies 
whatever,  and  have  ultimately  been  able  to  exhibit  the  value  of  the 
induced  density  under  a  very  simple  form,  whatever  the  given  density 
of  the  fluid  in  these  bodies  may  be. 

The  tenth  and  last  article  contains  an  application  of  the  general 
method  to  circular  planes,  from  which  results,  analagous  to  those  formed 
for  spheres  in  some  of  the  preceding  ones  are  deduced;  and  towards 
the  latter  part,  a  very  simple  formula  is  given,  which  serves  to  express 
the  value  of  the  density  of  the  free  fluid  in  an  infinitely  thin  plate, 
supposing  it  acted  upon  by  other  fluid,  distributed  according  to  any 
given  law  in  its  own  plane.  Now  it  is  clear,  that  if  to  the  general  ex- 
ponent 11  we  assign  the  particular  value  2,  all  our  results  will  become 
applicable  to  electrical  phenomena.  In  this  way  the  density  of  the 
electric  fluid  on  an  infinitely  thin  circular  plate,  when  under  the  in- 
fluence of  any  electrified  bodies  whatever,  situated  in  its  own  plane, 
will  become  known.  The  analytical  expression  which  serves  to  repre- 
sent the  value  of  this  density,  is  remai-kable  for  its  simplicity ;  and  by 
suppressing  the  term  due  to  the  exterior  bodies,  immediately  gives  the 
density  of  the  electric  fluid  on  a  circular  conducting  plate,  when  quite 
free  from  all  extraneous  action.  Fortunately,  the  manner  in  which 
the  electric  fluid  distributes  itself  in  the  latter  case,  has  long  since 
been  determined  experimentally  by  Coulomb.  We  have  thus  had  the 
advantage   of  comparing   our   theoretical    results    with    those   of  a   very 


Mit  GREEN,   ON   THE   LAWS   OF   THE   EQUILIBRIUM  OF   FLUIDS.        5 

accurate  observer,  and  the  differences  between  them  are  not  greater 
than  may  be  supposed  due  to  the  unavoidable  errors  of  experiment, 
and  to  that  which  would  necessarily  be  produced  by  employing  plates 
of  a  finite  thickness,  whilst  the  theory  supposes  this  thickness  infinitely 
small.  ]\Ioreover,  the  errors  are  all  of  the  same  kind  with  regard  to 
sign,  as  would  arise  from  the  latter  cause. 

1.  If  we  conceive  a  fluid  analogous  to  the  electric  fluid,  but  of 
which  the  law  of  the  repulsion  of  the  particles  instead  of  being  in- 
versely as  the  square  of  the  distance  is  inversely  as  some  power  n  of 
the  distance,  and  suppose  p  to  represent  the  density  of  this  fluid,  so 
that  dv  being  an  element  of  the  volume  of  a  body  A  through  which 
it  is  diffiised,  pdv  may  represent  the  quantity  contained  in  this  element, 
and  if  afterwards  we  write  g  for  the  distance  between  dv  and  any 
particle  />    under   consideration,   and  these  form   the  quantity 

the  integral  extending  over  the  whole  volume  of  A,  it  is  well  known 
that  the  force  with  which  a  particle  p  of  this  fluid  situate  in  any 
point  of  space  is  impelled  in  the  direction  of  any  line  q  and  tending 
to  increase  this  line  will  always  be  represented  by 


(1). 


I^\: 


1-n   \dq)  ' 

?^,  being    regarded  as   a  function   of    three   rectangular   co-ordinates   of 

p,   one   of   which    co-ordinates   coincides    with    the    line  q,    and    (—7-) 

being   the   partial   differential   of    V,   relative   to   this   last  co-ordinate. 

In  order  now  to  make  known  the  principal  artifices  on  which  the 
success  of  our  general  method  for  determining  the  function  V  mainly 
depends,   it  will  be  convenient  to  begin  with  a  very  simple  example. 

Let  us  therefore  suppose  that  the  body  ^  is  a  sphere,  whose  centre, 
is  at   the   origin  O  of  the   co-ordinates,    the   radius  being  1 ;   and  p  is 
such  a  function  of  x',  y',  %,  that  where  we  substitute  for  x',  y',  »'  their 
values  in  polar  co-ordinates 


6        Me  green,   on   THE   LAWS   OF    THE   EQUILIBRIUM   OF   FLUIDS. 

X  =  r'  cos  0',     y'  =  /  sin  9'  cos  tst',     %'  =  r'  sin  Q'  sin  tr', 
it  shall  reduce  itself  to  the  form 

P  =  (l-/y./(0; 

f  being    the   characteristic   of    any   rational   and   entire   function    what- 
ever: which  is  in  fact  equivalent  to  supposing 

p  =  (1  -  /'  -  y"  -  %'f.f{x"  +  y"  +  z'^). 

Now,  when  as  in  the  present  case,  p  can  be  expanded  in  a  series 
of  the  entire  powers  of  the  quantities  x,  y',  %',  and  of  the  various 
products  of  these  powers,  the  function  V  will  always  admit  of  a  similar 
expansion  in  the  entire  powers  and  products  of  the  quantities  x,  y,  %, 
provided  the  point  p  continues  within  the  body  A*,  and  as  moreover 
V  evidently  depends  on  the  distance  Op  —  r  and  is  independent  of  6 
and  -sr,  the  two  other  polar  co-ordinates  of  p,  it  is  easy  to  see  that  the 
quantity   V  when  we  substitute  for  x,  y,  z  these  values 

x  =  r  cos  9,    y  =  r  sin  9  cos  w,    z  =  r  sin  9  sin  tst 

will   become    a    function    of    r,    only    containing    none    but    the    even 
powers  of  this  variable. 

But  since  we  have 

dv  =  r"dr  d9'  d-ur  sin  0',     and  />  =  (1  -  ry.f{r'% 

the  value  of  V  becomes 

V=  f-^,  =  jr'^dr'd9'd-w'  sin  9'  (1  -  r''ff{r")  .g'"", 
J  g" 

the  integrals  being  taken  from  tst'  =  0   to   tr'  =  2  tt,    from    9'  =  0  to  9'  =  w, 
and  from  r'  =  0  to  r'  =  l. 

*  The  truth  of  this  assertion  will  become  tolerably  clear,  if  we  recollect  that  V  may  be 
regarded  as  the  sum  of  every  element  pdv  of  the  body's  mass  divided  by  the  (n—l)""  power 
of  the  distance  of  each  element  from  the  point  p,  supposing  the  density  of  the  body  A  to  be 
expressed  by  p,  a  continuous  function  of  x, y,  z.  For  then  the  quantity  V  is  represented 
by  a  continuous  function,  so  long  as  p  remains  within  A ;  but  there  is  in  general  a  violation 
of  the  law  of  continuity  whenever  the  point  p  passes  from  the  interior  to  the  exterior  space. 
This  truth,  however,  as  enunciated  in  the  text,  is  demonstrable,  but  since  the  present  paper 
is  a  long  one,  I  have  suppressed  the  demonstrations  to  save  room. 


Mb  green,    on   THE   LAWS   OF    THE    EQUILIBRIUM    OF   FLUIDS.         7 

Now  V  may  be  considered  as  composed  of  two  parts,  one  V  due 
to  the  sphere  B  whose  centre  is  at  the  origin  O,  and  surface  passes 
through  the  point  p,  and  another  V"  due  to  the  shell  S  exterior  to  B. 
In   order  to  obtain   the   first   part,   we   must   expand   the   quantity  g^~" 

t 

T 

in  an  ascending  series  of  the  powers  of  — .     In  this  way  we  get 


^1  -« _  ^1  ^2rr  {cos 9  cosff  -\-  sin  9  sin 9'  cos  (^'  -  -sr)]  +  r'^] 


l-n 
2 


=  r' " " , 


If  then  we  substitute  this  series  for  g^'"  in  the  value  of  F",  and 
after  having  expanded  the  quantity  (1  —  r'^f  ,  we  effect  the  integrations 
relative  to  r,  0',  and  w',  we  shall  have  a  result  of  the  form 

r'  =  r*-''  [A-i-Br+Cf^  +  Sic.] 

seeing   that  in  obtaining  the  part  of   V  before  represented   by    V,  the 
integral  relative  to  r'  ought  to  be  taken  from  r  =0  to  r'  =  r  only. 

To  obtain  the  value  of  F",  we  must  expand  the  quantity  g^-"  in 
an  ascending  series  of  the  powers  of  — ,  and  we  shall  thus  have 


l-n 
2 


g^-''={r^  —  2rr'  [cos  6  cos  0'  +  sin  9  sin  6'  cos  (tst - -nr')]  +  r"') 

the  coefficients  Qo,  Qi,  Q2,  &c.  being  the  same  as  before. 

The  expansion  here  given  being  substituted  in    P",  there  will  arise 
a  series  of  the  form 

of  which  the  general  term   T,  is 

T,=  fd9'd^'  sin  ff  QJr-'dr  ^^^(l-ry.f{ry, 

the  integrals  being  taken  from  r'  =  r  to  r'  =  l,   from  0'  =  O  to  & —  it,  and 
from  •z«r'  =  0  to  'ar'  =  27r.     This   will   be   evident   by   recollecting   that  the 


8        Mr  green,   ON   THE   LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS. 

triple  integral  by  which  the  value  of  V"  is  expressed,  is  the  same  as 
the  one  before  given  for  V,  except  that  the  integration  relative  to  r, 
instead  of  extending  from  /=0  to  r'=l,  ought  only  to  extend  from 
r  =r  to  r=  1. 

But   the   general   term   in   the   function  J'{r'^)   being  represented   by 
Atr'^\  the  part  of  T^  dependent  on  this  term  will  evidently  be 

(2) Atr'fde'dw'  sin  9'.QJr'''+^-^-''dr' {l-r'y-, 

the  limits  of  the  integrals  being  the  same  as  before. 

We   thus  see   that   the  value  of    T,  and  consequently  of  F'"  would 
immediately   be   obtained,    provided    we   had   the   value   of   the  general 

integral 

flr^dril-ry, 

which  being  expanded  and  integrated  becomes 

^    1   +Mzl).   1   _&c. 


b  +  l       l'b  +  3  1.2     'b  +  5 

^+1        Q    fj,+3       /3(/3-l)    r'+» 

+  T  •  i — ;r  —  ~    7  r.     •  ~i =  +  <^c- 


6  +  1       16  +  3  1.2     '6  +  5 

but  since  the  first  line  of  this  expression  is  the  well  known  expansion  of 


(f) 


r  lf\  r  li 


or 


nT 


m' 


when  n  =  2.p  —  h  +  \   and  5'  =  2(/3  +  l)  we  have  ultimately, 


By  means  of  the  result  here  obtained,  we  shall  readily  find  the 
value  of  the  expression  (2)  which  will  evidently  contain  one  term  multi- 
plied by  r'  and  an  infinite  number  of  others,  in  all  of  which  the  quantity 
r  is  affected  with  the  exponent  n.  But  as  in  the  case  under  considera- 
tion,  n   may    represent  any  number  whatever,   fractionary  or  irrational, 


Mr  green,   on  THE  LAWS  OF  THE    EQUILIBRIUM  OF  FLUIDS.         9 

it  is  clear  that  none  of  the  terms  last  mentioned  can  enter  into  V, 
seeing  that  it  ought  to  contain  the  even  powers  of  r  only,  thence  the 
terms  of  this  kind  entering  into  V",  must  necessarily  be  destroyed  by 
corresponding  ones  in  V.  By  rejecting  them,  therefore,  the  formula  (2) 
will  become 

m —^ ^-^ Air'.fde'd-Br'  smd'Qs. 

But  as  V  ought  to  contain  the  even  powers  of  r  only,  those  terms 
in  which  the  exponent  s  is  an  odd  number,  will  vanish  of  themselves 
after  all  the  integrations  have  been  effected,  and  consequently  the  only 
terms  which  can  appear  in   V,  are  of  the  form 

r(#+2-y-|)  r(/3  +  i) 

(4) ^ Atr'^fde'dTir'  sin  ff  Q,r, 


2r(^  +  /3  +  3-*'-|) 


where,  since  s  is  an  even  number,  we  have  written  2  s'  in  the  place  of 
s,  and  as  Qu-  is  always  a  rational  and  entire  function  of  cos  9',  sin  0' 
cos  w',  and  sin  9'  sin  -sr',  the  remaining  integrations  may  immediately  be 
effected. 

Having  thus  the  part  of  T'a,-  due  to  any  term  Atr'''*  of  the  function 
y(r'*)  we  have  immediately  the  value  of  T'.^'  and  consequently  of   F'", 
since 

r"= u'+  t:+  t:+  t:+  T:+kc.  -, 

U'  representing  the  sum  of  all  the  terms  in  F"  which  have  been  rejected 
on  account  of  their  form,  and  T,'  T,'  T,'  the  value  of  T,  Ty  T„  &c. 
obtained  by  employing  the  truncated  formula  (2)  in  the  place  of  the 
complete  one  (2). 

But  -v=v'+  V" = r'+  u^Ti^  r;+  t:  +  7v+  &c. 

or  by  transposition, 

r-T:-T;-Ti-Ti-hc.=r-YU, 

and   as   in   this   equation,   the   function   on   the  left   side   contains  none 
but  the  even   powers  of  the  indeterminate   quantity   r,   whilst   that  on 
Vol.  V.    Part  I.  B 


lO      Mr  green,  on  THE  LAWS  OF  THE  EQUILIBRIUM  OF   FLUIDS, 

the  right  does  not  contain  any  of  the  even  powers  of  r,  it  is  clear  that 
each  of  its  sides  ought  to  be  equated  separately  to  zero.  In  thi&  way 
the  left  side  gives 

(5) r=T:+T,'+T:+T:+kc. 

Hitherto  the  value  of  the  exponent  /3  has  remained  quite  arbitrary, 
but  the  known  properties  of  the  function  r  will  enable  us  so  to 
determine  /3,  that  the  series  just  given  shall  contain  a  finite  number 
of  terms  only.  We  shall  thus  greatly  simplify  the  value  of  F)  and 
reduce  it  in  fact  to  a  rational  and  entire  function  of  r*. 

For  this  purpose,  we  may  remark  that 

r(0)=«,     r(-l)=oo,     r  {  —  2)  =  CO,  in  infinitum. 

If  therefore   we   make  —  -  +  /3  =    any    whole    number    positive    or 

negative,  the  denominator  of  the  function  (4)  will  become  infinite,  and 
consequently   the   function   itself   will   vanish    when   s    is  so  great   that 

1-  /3  +  i  +  3  -  *'   is   equal   to   zero   or   any   negative    number,    and    as 

tit 

the  value  of  t  never  exceeds  a  certain  number,  seeing  that  f{i^^)  is 
a  rational  and  entire  function,  it  is  clear  that  the  series  (4)  will  termi- 
fMrte  of  itself,  and  V  become  a  rational  and  entire  function  of  r*. 

(2)  The  method  that  has  been  employed  in  the  preceding  article 
where  the  function  by  which  the  density  is  expressed  is  of  the  particular 
form 

may  by  means  of  a  very  slight  modification,  be  applied  to  the  far  more 
general  value 

P  =  (1  -  ryf{^,  i,  a')  =  (1  -  x" - y" - --'ff{x,  y',  z) 

tvhere  f  is  the  characteristic  of  any  rational  and  entire  function  what- 
ever :  and  the  same  value  of  /3  which  reduces  V  to  a,  rational  and  entire 
function  of  r^  in  the  first  case,  reduces  it  in  the  second  to  a  similar 
function  of  x,  y,  %  and  the  rectangular  co-ordinates  of  p. 


Ma  GREEN,   ON  THE  LAWS  OF   THE  EQUILIBRIUM  OF  FLUIDS.      H 

To  prove  this,  we  may  remark  that  the  con-esponding  value  V  will 

beeoMie 

F  =  fr"dr'de'd^'  sin  6'  (1  -  ryf{x',  y',  «')^'-"; 

tJie  integral  being  conceived  to  comprehend  the  whole  volume  of  the 
sphere. 

Let  now  the  function  y  be  divided  into  two  parts,  so  that 

fi^,  y,  %')  =/  ix',  y',  z')  +f,  ix',  y',  ^') ; 

/i  containing  all  the  terms  of  the  function  J]  in  which  the  sum  of  the 
exponents  of  af,  y,  %'  is  an  odd  number ;  and  ^  the  remaining  terms,  or 
those  where  the  same  sum  is  an  even  number.     In  this  way  we  get 

the  functions  F'l  and  V^a  corresponding  to^  andj^,  being 

V,  =  fr"dr'de'dvr'  sin'0'  {l~ryf,  {x',  y\z')g'-', 

V^  =  lr"dr'd&d-^  sin  &  (1  -  ryf,  {x',  y%  a')  g^-\ 

"We  will  in  the  first  place  endeavour  to  determine  the  value  J^j;  and 
for  this  purpose,  by  writing  for  x,  y,  %'  their  values  before  given  in 
r',  ff,  w',  we  get 

f,{x',y,%')^rW')\ 

the   coefficients    of  the   various  powers   of  r'^  in  ^{r'^)   being   evidently 

rational    and  entire  functions    of    cos  0„   sin  &  cos  w',    and    sin  0  sin  w. 

Thus 

V,  =  jr^dr'dffdTs'  sin 6' (1  - ry  />/.(/') ^'-"; 

this  integral,  like  the  foregoing,  comprehending  the  whole  volume  of 
the  sphere. 

Now  as  the  density  corresponding  to  the  function  Fi  is  - 

p,=.{l-af^-y'^-^^ff,{x',^,%% 

it  is  clear  that  it  may  be  expanded  in  an  ascending  series  of  the  entire 
powers  of  x',  y,  »',  and  the  various  products  of  these  powers  consequently, 
as  was  before  remarked  (Art.  1.),  Fl  admits  of  an  analagous  expansion 
in  entire  powers  and  products  of  x,  y,  ■%.     Moreover,  as  the  density  /i, 

B  2 


12      Mr  green,  ON  THE   LAWS  OF   THE   EQUILIBRIUM  OF  FLUIDS. 

retains  the  same  numerical  value,  and  merely  changes  its  sign  when 
we  pass  from  the  element  dv  to  a  point  diametrically  opposite,  where 
the  co-ordinates  x,  y ,  %  are  replaced  by  -  x ,  -  y' ,  —%  \  it  is  easy  to 
see  that  the  function  V-^,  depending  vxpon  /s,,  possesses  a  similar  property, 
and  merely  changes  its  sign  when  x,  y,  %,  the  co-ordinates  of  p,  are 
changed  into  -  x,  —y,  —  as.  Hence  the  nature  of  the  function  Vi  is 
such  that  it  can  contain  none  but  the  odd  powers  of  r,  when  we  sub- 
stitute for  the  rectangular  co-ordinates  x,  y,  %,  their  values  in  the  polar 
co-ordinates  r,  6,  ■zs. 

Having  premised  these  remarks,  let  us  now  suppose  Vx  is  divided 
into  two  parts,  one  V^  due  to  the  sphere  B  which  passes  through  the 
particle  p,  and  the  other  V"  due  to  the  exterior  shell  aS*.  Then  it  is 
evident  by  proceeding,  as  in  the  case  where  p  =  (1  -  r"^Yf{i%  that  Vi 
will  be  of  the  form 

the  coefficients  A,  B,  C,  &;c.  being  quantities  independent  of  the  variable  r. 

In  like  manner  we  have  also 

F/'  =  fr'^dr'ae'dsr'  sin  ff  {\-ry  .r'>\,{r'')g^-''; 

the  integrals  being  taken  from  r' =  r  to  r  =  l,  from  6' =  0  to  0'  =  7r,  and 
from  Gr'  =  0  to  'z<r'  =  2  7r. 

By  substituting  now  the  second  expansion  of  g^"  before  used  (Art.  1.), 
the  last  expression  will  become 

r,"  =  t;  +  Ti  +  r.  +  ^3  +  &c. 

of  which  series  the  general  term  is 

T,  =  fd9'dw'  sin  ff  Q,  fr"-dr' (1  - ry  ^ x/. {r"). 

Moreover,  the  general  term  of  the  function  \l^  {r'-)  being  represented  by 
Air'^\  the  portion  of  1\  due  to  this  term,  will  be 

(a) r  fdffdw'  smO'  Q,Atjr''-''^''-Ulr'  {l-ry-, 

•the  limits  of  the  integrals  being  the  same  as  before. 


Mr  green,   on  THE  LAWS   OF   THE   EQUILIBRIUM   OF  FLUIDS.      IS 

If  now  we  effect  the  integrations  relative  to  r'  by  means  of  the  for- 
mula (3),  Art.  1,  and  reject  as  before  those  powers  of  the  variable  /•, 
in  which  it  is  affected,  with  the  exponent  w,  since  these  ought  not  enter 
into  the  function   Fi,  the  last  formula  will  become 


(^^"F^V(3+i) 


(a) ;r-7rrj ^-rr: r'fde'diir' sin  6' Q,A„ 

2  r  C^  +  ^/J-w  +  g^-jyx        •'  ^' 

and  as  F,  ought  to  contain  none  but  the  odd  powers  of  r,  we  may  make 
*  =  2*'  +  l,  and  disregard  all  those  terms  in  which  s  is  an  even  number, 
since  they  will  necessarily  vanish  after  all  the  operations  have  been 
effected.     Thus  the  only  remaining  terms  will  be  of  the  form 

t^''^' fde'dsr' sin  9' Q,,.^,  A,; 


2.T  ' 


) 


where,  as  At  and  02/+ 1  are  both  rational  and  entire  functions  of  cos  0', 
sin  ff  cos  •ht',  sin  ff  sin  -sr',  the  remaining  integrations  from  6'  =  0  to  9'  =  tt, 
and  Tsr'  =  0  to  tjt'  =  2  tt,  may  easily  be  effected  in  the  ordinary  way. 

If  now  we  follow  the  process  employed  in  the  preceding  article,  and 
suppose  To',  Ti,  T2,  &c.  are  what  T^,  Ti,  71,  &c.  become  when  we  use 
the  truncated  formula  («')  instead  of  the  complete  one  (a),  we  shall 
readily  get 

F,  =  t:  +  t:  +  t:  +  r/  +  &c. 

In  like  manner,  from  the  value  of  V^  before  given,  we  get 

r,"  =  fr'dr'd&dsr'  sin &{1- ry(p{r")g' -" ; 

the  integrals  being  taken  from  r'  =  r  to  r  =  l,  from  9'  =  0  to  9'  =  ^,  and 
from  -ar  =  0  to  tsr  =  2  tt. 

Expanding  now  g^'"  as  before,  we  have 

r;'=  t;-„+t7,+  z7.+  j7, +  &c. 

where 

U.  =  fdffd-sr'  sin  ■sr'Qjy'-'^dr'il-ry  ^  0  (/*), 


14      Mr  green,   ON  THE   LAWS  OF  THE  EQUILIBinUM   OF  FLUIDS. 

acnd  the  part  of  U,  due  to  the  general  term  i?(/-''^'  in  0  (/*),  will  be 

(J) r'fd&diir'  sm  9' Q^Bt/lr'^-"^^'-' dr' {l-ry-, 

which,  by  employing  the  formula  (3')   Art.  1.,   and   rejecting  the  inad- 
missible terms,  gives  for  truncated  formula 


[  2  j 


By  continuing  to  follow  exactly  the  same  process  as  was  before 
employed  in  finding  the  value  of  Fl,  we  shall  see  that  *  must  always 
be  an  even  number,  say  2  s';  and  thu«  the  expression  immediately  {Br- 
eeding will  become 

,,  l4!-n  +  2t-2s 


^  (6-n  +  23  +  2t-2s' 

2r  I 


; — r'^'  fdO'dw  sin  d'^2.-  B,. 


2  J 

Moreover,  the  value  erf  V^  will  be 

r,  =  u: + u: + u: + u:  +  &c. ; 

U^,  Ui,  Ui,  U3,  &c.  being  what  Uo,  Ui,  Ui,  &c.  become  when  we  use 
the  formula  {b')  instead  of  the  complete  one  (h). 

The  value  of  V  answering  to  the  density 

p  =  p,  +  p,  =  (l-ry. /{:>/,  y',z'), 

by  adding  together  the  two  parts  into  which  it  was  originally  divided, 
therefore,  becomes 

r  =  r,+r,=  t:  +  t:  +  t: + t/  +  &c. 

+  £/■„' +t4'+C7;'+t7e'  +  &c. 

When  /3  is  taken  arbitrarily,  the  two  series  -entering  into  V  extend 
in  infinitum,  but  by  supposing  as  before.  Art.  1., 

—  n       n 


Ma  GREEN,  ON   THE  LAWS   OF  THE  EQUILIBRIUM  OF  FLUIDS.       15 

w  representing  any  whole  number,  positive  or  negative,  it  is  clear  from 
the  form  of  the  quantities  entering  into  JLs'+i  and  U2/,  and  from  the 
known  properties  of  the  function  F,  that  both  these  series  wiU  terminate 
of  themselves,  and  the  value  of  F'  be  expressed  in  a  finite  form ;  which, 
by  what  has  preceded,  must  necessarily  reduce  itself  to  a  rational  and 
entire  function  of  the  rectangular  co-ordinates  x,  y,  ss.  It  seems  needless, 
after  what  has  before  been  advanced,  (Art.  1.)  to  offer  any  proof  of  this: 
we  will,  therefore,  only  remark  that  if  7  represents  the  degree  of  the 
function  f{x',  y\  &'),  the  highest  degree  to  which  V  can  ascend  will  be 

7  +  2  a>  +  4. 

In  what  immediately  precedes,  w  may  represent  any  whole  number 
whatever,  positive  or  negative ;  but  if  we  make  w=  —2,  and  consequently, 

^  = ^  the  degree  of  the  function  J^  is  the  same  as  that  of  the  factor 

A^\  y',  ^), 

comprised  in  p.  This  factor  then  being  supposed  the  most  general  of 
its  kind,  contains  as  many  arbitrary  constant  quantities  as  there  are 
terms  in  the  resulting  function  V.  If,  therefore,  the  form  of  the  rational 
and  entire  function  V  be  taken  at  will,  the  arbitrary  quantities  contained 
in  fkpd,  y,  %')  will  in  case  w  =  —  2  always  enable  us  to  assign  the  corres- 
ponding value  of  p,  and  the  resulting  value  of  J'{a;',  y,  %')  will  be  a  rational 
and  entire  fimction  of  the  same  degree  as  T-^.  Therefore,  in  the  case 
now  under  consideration,  we  shall  not  only  be  able  to  determine  the 
value  of  F'  when  p  is  given,  but  shall  also  have  the  means  of  solving 
the  inverse  problem,  or  of  determining  p  when  V  is  given ;  and  this 
determination  will  depend  upon  the  resolution  of  a  certain  number  of 
algebraical  equations,  all  of  the  first  degree. 

3.  The  object  of  the  preceding  sketch  has  not  been  to  point  out 
the  most  convenient  way  of  finding  the  value  of  the  function  ^,  but 
merely  to  make  known  the  spirit  of  the  method ;  and  to  show  on  what 
its  success  depends.  Moreover,  when  presented  in  this  simple  form, 
it  has  the  advantage  of  being,  with  a  very  slight  modification,  as  ap- 
plicable  to   any   ellipsoid  whatever   as  to  the   sphere   itself.     But   when 


16       Mr  green,    on   THE   LAWS   OF   THE   EQUILIBRIUM    OF    FLUIDS. 

spheres  only  are  to  be  considered,  the  resulting  formula?,  as  we  shall 
afterwards  show,  will  be  much  more  simple  if  we  expand  the  density  p 
in  a  series  of  functions  similar  to  those  used  by  Laplace  {Mec.  Cel. 
Liv.  iii.) :  it  will  however  be  advantageous  previously  to  demonstrate 
a  general  property  of  functions  of  this  kind,  which  will  not  only  serve 
to  simplify  the  determination  of  F,  but  also  admit  of  various  other 
applications  of  dcr. 

Suppose,  therefore,  J^'''  is  a  function  of  9  and  trr,  of  the  form  con- 
sidered by  Laplace  {Mec.  Cel.  Liv.  iii.),  r,  9,  -zs-  being  the  polar  co-ordi- 
nates referred  to  the  axes  JT,  Y,  Z,  fixed  in  space,  so  that 

ar  =  r  cos  0,     y  =  r  %\w9  cos  Tsr,     x  =  /•  sin  0  sin  vr ; 

then,  if  we  conceive  three  other  fixed  axes  Xi,  Y^,  Z,,  having  the  same 
origin  but  different  directions,  P'^'^  will  become  a  function  of  0,  and  •zjti, 
and  may  therefore  be  expanded  in  a  series  of  the  form 

(6) r  <^>  =  r/"'  +  F.*'>  +  F/^'  +  F/^'  +  &c.     . 

Suppose  now  we  take  any  other  point  p  and  mark  its  various  co-ordinates 
with  an  accent,  in  order  to  distinguish  them  from  those  of  p ;  then,  if 
we  designate  the  distance  pp   by   {p, p),  we  shall  have 

^  -  =  f  r'  -  2rr'  [cos  9  cos  ff  +  sin  9  sin  &  cos  {tn-  -  •sr')]  +  r'^\  "* 


as  has  been  shewn  by  Laplace  in  the  third  book  of  the  Mec.  Cel.,  where 
the  nature  of  the  different  functions  here  employed  is  completely  ex- 
plained. 

In  like  manner,  if  the  same  quantity  is  expressed  in  the  polar  co- 
ordinates belonging  to  the  new  system  of  axes  X-,,  F„  Z,,  we  have, 
5ince  the  quantities  r  and  r'  are  evidently  the  same  for  both  systems, 

{^p, p)      r  \^  r  r  IT  I 

^nd  it  is  also  evident   from  the  form  of  the  radical  quantity  of  which 


Mr  green,   on    THE   LAWS   OF   THE  EQUILIBRIUM   OF  FLUIDS.       17 

the  series  just  given  are  expansions,  that  whatever  number  i  may  re- 
present,  Qi***  will  be  immediately   deduced  from  Q*'>  by  changing   9,  sr, 

9',  -sr'   into   0„  "sr,,  9/,  ^r,'.     But    since   the   quantity   -    is   indeterminate, 

and  may  be  taken  at  will,  we  get,  by  equating  the  two  values  of  .        , 

.       f 
and  comparing  the  like  powers  of  the  indeterminate  quantity  -, 

If  now  we  multiply  the  equation  (6)  by  the  element  of  a  spherical 
surface  whose  radius  is  unity,  and  then  by  Q<*'  =  Q/*>,  we  shall  have, 
by  integrating  and  extending  the  integration  over  the  whole  of  this 
spherical  surface, 

fdf.dwQ"^  r®  =  fdfx,  d-ar,  Q/**  {  F/"'  +  Y^  +  F*^'  +  &c. } . 

Which  equation,  by  the  known  properties  of  the  functions  Q**'  and  Y^^\ 
reduces  itself  to 

when  h  and  i  represent  different  whole  numbers.  But  by  means  of  a 
formula  given  by  Laplace  {Mec.  Cel.  Liv.  iii.  No.  17.)  we  may  imme- 
diately effect  the  integration  here  indicated,  and  there  wiU  thus  result 

"-2^  +  1-^^     ' 

F;'<*>  being  what  Fi''"  becomes  by  changing  9^,  tsti  into  0,',  •ar/,  and  as 
the  values  of  these  last  co-ordinates,  which  belong  to  p,  may  be  taken 
arbitrarily  like  the  first,  we  shall  have  generally  F,**',  except  when 
h  =  i.  Hence,  the  expansion  (6)  reduces  itself  to  a  single  term,  and 
becomes 

F®  =  F®. 

We  thus  see  that  the  function  F<''  continues  of  the  same  form  even 
when  referred  to  any  other  system  of  axes  X„  F„  Z„  having  the  same 
origin  O  with  the  first. 

This  being  established,  let  us  conceive  a  spherical  surface  whose  center 
is  at  the  origin   O  of  the  co-ordinates  and  radius  r',  covered  with  fluid. 
Vol.  V.    Part  I.  C 


18      Mr  green,   ON   THE   LAWS   OF    THE   EQUILIBRIUM    OF   FLUIDS. 

of  which  the  density  p  =  P''*'' ;  then,  if  d<r'  represent  any  element  of 
this  surface,  and  we  afterwards  form  the  quantity 

the  integral  extending  over  the  whole  spherical  surface,  g  being  the 
distance  p,  da  and  y\f  the  characteristic  of  any  function  whatever.  I 
say,  the  resulting  value  of  V  will  be  of  the  form 

V=  Y^B; 

R  being  a  function  of  r,  the  distance  Op  only  and  K<''  what  Y'^^  becomes 
by  changing  9',  w,  the  polar  co-ordinates,  into  9,  tit,  the  like  co-ordinates 
of  the  point  p. 

To  justify  this  assertion,  let  there  be  taken  three  new  axes  JT,,  I^„  Z„ 
so  that  the  point  p  may  be  upon  the  axis  Xx ;  then,  the  new  polar 
co-ordinates  of  da'  may  be  written  r',  ff,  tjt',  those  of  p  being  r,  0,  •sr. 
and  consequently,  the  distance  will  become 

g  =  ^{r"  -  2  rr'  cos  9,'  +  r^) ; 

and  as  da^'  =  r'^d9i'd'sri  sin  9,',  we  immediately  obtain 

r  =  fY'^'Vde.d-sr,  sin  9,  f  (/•--  2rr'  cos  d,' +  O 

=  r'^SZd9;  sin  0/  ^{r'-^rr'  cos  0/  -f  r'^)f^Zd-ur(  Y' <". 

Let  us  here  consider  more  particularly  the  nature  of  the  integral 

In  the  preceding  part  of  the  present  article,  it  has  been  shown  that 
the  value  of  Y'^'^,  when  expressed  in  the  new  co-ordinates,  will  be  of 
the  form  P'/*'' ;  but  aU  functions  of  this  form  (Vide  Mec.  Cel.  Liv.  iii.) 
may  be  expanded  in  a  finite  series  containing  2 « + 1  terms,  of  which 
the  first  is  independent  of  the  angle  "sr,',  and  each  of  the  others  has 
for  a  factor  a  sine  or  cosine  of  some  entire  multiple  of  this  same  angle. 
Hence,  the  integration  relative  to  ro-/  will  cause  all  the  last  mentioned 
terms  to  vanish,  and  we  shall  only  have  to  attend  to  the  first  here. 
But  this  term  is  known  to  be  of  the  form 

,  /    ,.       i.i  —  \       ,.   „      i.i—l.i-2.i  —  S      ,,•  ,     «     N 


mh  green,  on  the  laws  of  the  equilibrium  of  fluids.    19 
and  consequently,  there  will  result 

where  ni  =  cos  9^  and  ^  is  a  quantity  independent  of  6/  and  tr/,  but 
which  may  contain  the  co-ordinates  9,  -ar,  that  serve  to  define  the 
position  of  the  axis  JCi  passing  through  the  point  p. 

It  now  only  remains  to  find  the  value  of  the  quantity  k,  which  may 
be  done  by  making  0i'  =  O,  for  then  the  line  r  coincides  with  the  axis 
JTi,  and  K*''  during  the  integration  remains  constantly  equal  to  Y^\ 
the  value  of  the  density  at  this  axis.     Thus  we  have 

^     ^rin     ^     7  [-.        ii—l         i.i  —  l.i—2.i  —  3      „     \ 
V        2.2?— 1        2.4.2«— 1.2^  — 3  I 

or,  by  summing  the  series  within  the  parenthesis,  and  supplying  the 
common  factor  2  7r,  - 

•jr(i)  _  ^-^-^ ^       J, 

1.3.5 2«-l    ' 

and,  by  substituting  the  value  of  k,  draAvn  from  this  equation  in  the 
value  of  the  required,  integral  given  above,  we  ultimately  obtain 

If  now,  for  abridgement,  we  make 

^^>  =  ^'    -  2:27:11^'       +2.4.2i-1.2i-3^'       -^^- 

we  shall  obtain,  by  substituting  the  value  of  the  integral  just  found  in 
that  of  V  before  given, 

r=  r(^27rr'%i44^^^-^^^^^/_}r?^/(^H(^-2rr'M/  +  r'^); 

which  proves  the  truth  of  our  assertion. 

From  what  has  been  advanced  in  the  preceding  article,  it  is  likewise 
very  easy  to  see  that  if  the  density  of  the  fluid  within  a  sphere  of 
any  radius  be  every  where  represented  by 

c  2 


20      Mr  green,   ON   THE   LAWS   OF   THE   EQUILIBRIUM   OF   FLUIDS. 

<p  being  the  characteristic  of  any  function  whatever;  and  we  afterwards 
form  the  quantity 

where  dv  represents  an  element  of  the  sphere's  volume,  and  g  the  dis- 
tance between  dv  and  any  particle  p  under  consideration,  the  resulting 
value  of  V  wiU  always  be  of  the  form 

V^^  being  what  I^'*"  becomes  by  changing  9^,  nr ,  the  polar  co-ordinates 
of  the  element  dv  into  Q,  w,  the  co-ordinates  of  the  point  p;  and  R 
being  a  function  of  r,  the  remaining  co-ordinate  of  p,  only. 

4.  Having  thus  demonstrated  a  very  general  property  of  functions 
of  the  form  P"*'',  let  us  now  endeavour  to  determine  the  value  of  F" 
for  a  sphere  whose  radius  is  unity,  and  containing  fluid  of  which  the 
density  is  every  where  represented  by 

p  =  {l-x''-y"-zyf{x',y',z'); 

on',  y ,  z'  being  the  rectangular  co-ordinates  of  dv,  an  element  of  the 
sphere's  volume,  and  Jl  the  characteristic  of  any  rational  and  entire 
function  whatever. 

For  this  purpose  we  will  substitute  in  the  place  of  the  co-ordinates 
x',  y ,  z'  their  values 

x  =  r  cos  &\    y  =  r  sin  &  cos  w'.    z  =  r'  sin  ff  sin  -bt'  ; 

and  afterwards  expand  the  function/(a;',  y',  s)  by  Laplace's  simple  method, 
{Mec.  Cel.  Liv.  iii.  No.  16.).     Thus, 

(7) /{x,  y,  z)  =/<«>+/'" +/'<^>  +  &c +/'«; 

s  being  the  degree  of  the  function  /{x,  y',  z'). 

It  is  likewise  easy  to  perceive  that  any  term  /'■'''  of  this  expansion 
may  be  again  developed  thus, 

/'(•■)  =/;(•■>/*  +/'«/-=  +^'<V^+^  +  &c.; 

and  as  every  coefficient  of  the  last  developement  is  of  the  form  [/'", 
(Mec.  Cel.  Liv.  iii.),  it  is  easy  to  see  that  the  general  term  y'''V'+^'  may 
always  be  reduced  to  a  rational  and  entire  function  of  the  original 
co-ordinates   x,  y',  »'. 


Mb  GREEN,   ON   THE   LAWS.  OF  THE -EQUILIBRIUM  OF  FLUIDS.      21 
If  now  we  can  obtain  the  part  of  ^  due  to  the  term 

we  shall  immediately  have  the  value  of  V  by  summing  all  the  parts 
corresponding  to  the  various  values  of  which  i  and  t  are  susceptible. 
But  from  what  has  before  been  proved  (Art.  3.),  the  part  of  V  now 
under  consideration  must  necessarily  be  of  the  form  F"*'^;  representing, 
therefore,  this  part  by  F"/'',  we  shall  readily  get 

r/"  =/J/'+^'+^</r'  (1  -  ryjtl-nr'de'  sin  ^/'<'^^'-". 

Moreover  from  what  has  been  shown  in  the  same  article,  it  is  easy 
to  see  that  we  have  generally 

fV'^'^clu'de'  sin  e'^ig"-)  =  Stt  F«  ^f'f  •:'^'~^  /-i'c?Mi'  (i)  yl^{r'-2rr',x,'  +  r") ; 

\(/  being  the  characteristic  of  any  function  whatever,  and  P'^''  what  P''"* 
becomes  by  substituting  9,  w  the  polar  co-ordinates  of  p  in  the  place 
of  6',  TB-',  the  analogous  co-ordinates  of  the  element  dv.  If  therefore 
in  the   expression  immediately  preceding,  we  make 

F'«=/'«  and  fig^)  =^'.-  =  (^^)^, 

and  substitute  the  value  of  the  integral  thus  obtained  for  its  equal  in 
Vf-'^  there  will  arise 

where  yj®  is  deduced  from  Jl'^^  by  changing  9",  %r'  into  6,  w,  and  (i),  for 
abridgement,  is  written  in  the  place  of  the  function 

,;         i.i-1      ,;_;  ,   i.i-l.i-2.i-3     ,^_^      . 
'"~2.2e-l'"      ^2.4.2i-1.2«-3'"  *'''• 

As  the  integral  relative  to  n\  which  enters  into  the  expression  on 
the  right  side  of  the  equation  (8)  is  a  definite  one,  and  depends  therefore 
on  the  two  extreme  valvies  of  fj.\  only,  it  is  evident  that  in  the  deter- 
mination of  this  integral,  it  is  altogether  useless  to  retain  the  accents 


22      Mr  green,  ON  THE  LAWS  OF   THE   EQUILIBRIUM  OF   FLUIDS. 

by   which   n\  is  affected.      But   by   omitting   these   superfluous   accents, 
we  shall  have  to  calculate  the  value  of  the  quantity 


I-n 
2     , 


fj.dfi.  (^) .  (r'  -  2  rr'/m  +  r") 
where 

,.  .       i.i  —  1      .  i.i-i,i-2.i-3    ;   ,      - 

^'^  =  ''-2:2^:1''       +2.4.2i-1.2i-3-^      -^^- 

The  method  which  first  presents  itself  for  determining  the  value  of 

l-n 

the  integral  in  question,  is  to  expand  the  quantity  {r^  —  2rr'/u.  +  r'^)  ^  by 
means  of  the  Binomial  Theorem,  to  replace  the  various  powers  of  m  by 
their  values  in  functions  similar  to  (i)  and  afterwards  to  effect  the  in- 
tegrations by  the  formulee  contained  in  the  third  Book  of  the  Mec.  Cel. 
For  this  purpose  we  have  the  general  equation 

.-s  i     ...  ,     i.i  —  1     ,.     „.    ,   i.i—l.i  —  2.i  —  3,.      ,, 

^^^ '^  =^^)+ 2:271:1  (^-^^-^2X2I33:27::5(*-*) 

i.i-l.i-2.i-3.i-4!.i-5    . 
2.i.6.2i-5.2i-7.2i-9   ^'     '")  +  ^^' 

To  remove  all  doubt  of  the  correctness  of  this  equation,  we  may 
multiply  each  of  its  sides  by  (i,  and  reduce  the  products  on  the  right 
by  means  of  the  relation 

which  it  is  very  easy  to  prove  exists  between  functions  of  the  form  (?). 
In  this  way  it  will  be  seen  that  if  the  equation  (9)  holds  good  for  any 
power  fx'  it  will  do  so  likewise  for  the  following  power  ^'+^  and  as  it 
is  evidently  correct  when  i='l,  it  is  therefore  necessarily  so,  whatever 
whole  number  i  may  represent. 

Now  by  means  of  the  Binomial  Theorem,  we  obtain  when  r^r' 


=  2, 


r"'-K(r'-2rr'^  +  r"y  =  (i-2m  J  +  ^,) 
y>n—l.n  +  l.n-\-3 n  +  2s  —  3 


l—n 
3 


2s 


If  now  we  conceive  the  quantity   (2ju- rj   to  be  expanded  by 


Mr  green,   on   THE  LAWS  OF   THE  EQUILIBRIUM   OF  FLUIDS.      23 


the  same  theorem,  it  is  easy  to  perceive  that  the  term  having  f— | 
for  factor  is 


i  +  2t' 


7i-l.n-\-l.n  +  3 «  +  2^•  +  4^'-3    ,^,„  ,^„,  /r\ 

2.4.6    2e  +  4r  '^        W) 


i  +  iV 


n-l.n  +  1 n  +  2i  +  4>t'-5     ,^,,_,  ,+,„_,  fn'^"''!!  i+^t-1 


2    .    4     2?  +  4#'-2     '"  """    '  \r') 

2.4     2«  +  4ir-4     ^   '^^  VJ  /•''■  1.2 


—  &c &c &c 

/^\  i  +  2(' 

and  therefore  the  coefficient  of  I  — I        in  the  expansion  of  the  function 

will  be  expressed  by 

v"-^-^  +  l ^  +  2^'+4^'-2^- 3         ,,,,.,,  .. 

2.4    2«  +  4^'-2*        ^""^^  '^     ^ 

«  +  2#'-*.^  +  2^'-«-l ^■  +  2^-2*  +  l 


Hence  the  portion  of  this  coefficient  containing  the  function  (i),  when 
the  various  powers  of  /i  have  been  replaced  by  their  values  in  functions 
of  this  kind  agreeably  to  the  preceding  observation  will  be  found,  by 
means  of  the  equation  (9),  to  be 

.  .X  „  n  —  l.n  +  1 n  +  2i+4<t'~2s  —  3 

^^^        2    .    4     2?  +  4^'-2* 

^^  +  2/-2^.^•  +  2^'-2,y-l i  +  1 

"^  2.4 2^-2*x2^■  +  2if'-2*  +  1.2^  +  2f-2#-1...2^■  +  3 

i  +  2t'-s.i+2t'-s-l i  +  2^-2#+l 


.2*+^''-=»(-l)»x- 


1.2.3 


n-l.n  +  1. n  +  S n  +  2i+it' -2s-3     ,,,,.,.,.. 

■^^2.4.6    2«  +  4^-2«  ^     ^ 


24      Mr  GREEN,   ON   THE   LAWS   OF   THE   EQUILIBRIUM   OF  FLUIDS. 

^^  +  l.^^  +  2.^^  +  3.^'  +  4 i  +  ^t'  -s 

^  1.2.3 *x2.4.6 2^-2*  X  2? +  2^'- 2* +  1 2e  +  3 

_    .  (-1V.W-1.W  +  1.W  +  3 w  + 2? +  4^' -2.9 -3 

-2'-W-2g^ 2?x2.4 2*x2.4 2#'-2*  x  2^■  +  2^'-2*+1...2^■  +  3 

3.5.7 2«  +  l  ...      n-\.n  +  \.n  +  S ?«  +  2m2^'-3 


f^(0 


X 


1.2.3 i      ^^  3.5.7     2«  +  2r  +  l 

r  - 1 V  ^  +  ^^'  +  ^^-1 w  +  2?  +  4/-2.y-3 

^      "^        ^  2.4.6      2^-2* 

2t  +  2^'-2.y  +  3 2? +  2/'  +  1 

^  2.4.6        2* 

where  all  the  finite  integrals  may  evidently  be  extended  from  *  =  0 
to  *  =  00 ,  and  it  is  clear  that  the  last  of  these  integrals  is  equal  to  the 
coefficient  of  a^  in  the  product 

(,       w  +  2i  +  2#'-l          »  +  2i  +  2/-l.w  +  2i  +  2/'  +  l    „      „       .     .    ., 
{1+ x  + ^-^ oi?  +  hc.tninf.\ 

,,       2?  +  2^  +  l       ,   2i  +  2r  +  1.2?  +  2#'-l    „     .       .     .  ^, 
X  {1 ~ x+ ^-^ af-kc.  intnf.] 

If  now  we  write  in  the  place  of  the  series  their  known  values,  the 
preceding  product  will  become 

n  +  2i+2f-l  2i+2t'  +  l  i-n 


(l-or)        "         x(l-;r)     '      ={\-x)\ 

and  consequently  the  value  of  the  required  coefficient  of  af^  is 

«  — 2.W.W  +  2 w  +  2/'  — 4 

2    .4.    6     2^' 

This  quantity  being  substituted  in  the  place  of  the  last  of  the  finite 
integrals  gives  for  the  value  of  that  portion  of  the  coefficient  of 


which  contains  the  function  (i)  the  expression 

3.5.7 2?  +  l       n-l.n  +  1 w+2?'+2^'-3       n-2.n m+2^ -4  , ., 

1.2.8 i       ^3.5     2e  +  2^  +  l     ^        2.4 2t'       ^*^' 


Mb  green,   on  THE   LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS.      25 

By  multiplying  the  last  expression  by  (  —  )       ,   and  taking  the  sum 
of  all  the  resulting  values  which  arise  when   we  make  successively 

^  =  0,  1,  2,  3,  4,  5,  6,  &c.  in  infinitum, 
we  shall  obtain  the  value  of  the  term  I^<'>  contained  in  the  expression 

,  l-n 

(l  -  2m ^  +  ^,) ~  =  Y^'^  +  rc>  +  F(=>  +  F^^'  +  &c. 
Hence, 

1.2 i       ^^^         3.5     2i  +  2f+l 

n-2.n n  +  ^t'-i  /rV*^*' 

"^        2.4 2t'  [?)       ' 

the  finite  integral  extending  from  t'  =  0  to  t'  =  oc. 

But  by  the  known    properties    of  functions  bf  this  kind,   we  have 
by  substituting  for  F'"'  its  value 

/_\d^  (i)  (l  -  2m  p  +  ^^~=/-\d^  (i) .  F« 

3.5.7 2«  +  l   .,    ,.,„     ^n-l.n  +  l n  +  2i  +  2t'-3 

=  1.2.3 i      /^^(O^x^     3    .    5     2e  +  2^'  +  l 

n-2.n «  +  2/-4/rV*"' 


2.4 2t' 


(p) 


^1.2.3 i  n-l.n  +  1 n  +  2i  +  2t'-3 

~     1.3.5 2?-l         3     .    5     2i  +  2t'  +  2 

71  — 2. n n  +  2t'  —  ^!  fr' 


t'      ~  \r'}       ' 


2.4 2t 

since  by  what  Laplace  has  shown  {Mec.  Cel.  Liv.  iii.  No.  17.) 

■^^^  W^=  2m  I1.3.5 2e-lj  • 


Vol.  V.    Paet  I.  D 


26     Mr  green,  on  THE   LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS. 

If  now  we  restore  to  n.  the  accents  with  which  it  was  originally 
affected,  and  multiply  the  resulting  quantity  by  r'""\  we  shall  have  when 
r<r'' 


(10)    /Ad^\(i)  if-^rr't^,  +  0  ■'  =/»-y_;</M'i  ii)  (l  -  UJ-.  +  ^-.) 

_        ,j_„  1.2.3 i  w-l.w  +  1 ?^  +  2^^  +  2^-3 

~  '1.3.5 2«-l  3    .    5     2e  +  2#'+l 


;«  — 2.W «  +  2#'-4  /r 


2.4 2jf' 


9 


j  +  21' 


and  in  order  to  deduce  the  value  of  the  same  integral  when  /•'  /.  r,  we 
shall  only  have  to  change  r  into  /,  and  reciprocally,  in  the  formula 
just  given. 

We   may   now  readily   obtain    the   value    of   Vi^    by   means   of  the 
formula  (8).     For  the  density  corresponding  thereto  being 

:/;«/•'+=' (l-r"7,      ■ 

it  follows  from  what  has  been  observed  in  the  former  part  of  the 
present  article,  that  ^'®r'^^'  may  always  be  reduced  to  a  rational  and 
entire  function  of  icf,  y,  %'  the  rectangular  co-ordinates  of  the  element 
dv,  and  therefore  the  density  in  question  will  admit  of  being  expanded 
in  a  series  of  the  entire  powers  of  x,  y',  %'  and  the  various  products  of 
these  powers.  Hence  (Art.  1.)  F/''  admits  of  a  similar  expansion  in 
entire  powers,  he.  of  x,  y,  z  the  rectangular  co-ordinates  of  the  point  p, 
and  by  following  the  methods  before  exposed  Art.  1  and  2,  we  readily 
get 

^t  ^-^J     J  r'  ui  y>.     ,   )  .^     3.5     2t  +  2t'  +  l 

n-2.n.n  +  2 »  +  2#'  — 4  /r\'+"' 

X 


2    .4.    6     2t' 


-4  /r^y^-"" 

"      V'j      ' 


and  thence  we  have  ultimately, 

(,ii;     rt       Airj,    ^33    2i  +  2t' +  1  2.4 2/' 


Mr  green,   on   THE  LAWS  OF   THE  EQUILIBRIUM   OF   FLUIDS.      27 

rr-%^"-'')r(3+i)  r{/3  +  i)r(i^) 


,2r^*' 


=2Trf,^\ ^^ - — -r' 

/2t-2t'  +  2fi  +  6-n\  ■^'  /6  +  2/3-w\ 

4-n.6-n 2t~2t'+2-n  n-2.n n  +  2t'-4! 

6  +  2fi-n 2t-2t'+2l3  +  4>-n  ^        2.4 W 

n—1  .n  +  1 n  +  2i  +  2t'-3  ^ 

"^      3    .    5     2i  +  2t'  +  l     ' 

the  finite  integrals  being  taken  from  t'  =0  to  t'=cD  and  r  being  the 
characteristic  of  the  well  known  function  Gamma,  which  is  introduced 
when  we  effect  the  integrations  relative  to  r'  by  means  of  the  formula 
(3),  Art.  1. 

Having  thus  F"/"  or  the  part  of  F  corresponding  to  the  term  j^''*', 
in  J'(x',  y,  as')  we  immediately  deduce  the  complete  value  of  V  by  giving 
to  i  and  t  the  various  values  of  which  these  numbers  are  susceptible, 
and  taking  the  sum  of  all  the  parts  corresponding  to  the  different  terms 
hi  the  expansion  of  the  function  fix',  y',  &'). 

Athough  in  the  present  Article  we  have  hitherto  supposed  J"  to  be 
the  characteristic  of  a  rational  and  entire  function,  the  same  process  will 
evidently  be  applicable,  provided  y"(a;',  y,  z')  can  be  expanded  in  an 
infinite  series  of  the  entire  powers  of  x',  y,  z'  and  the  various  products 
of  these  powers.     In  the  latter  case  we  have  as  before,  the  development 

fix',  y,  z')  =/'<»>  +/'<•>  +  /'®  +/'<^)  +  &c. 

of  which  any  term,  as  for  example  f'^''>  may  be  farther  expanded  as 
follows, 

/'«  =/;«r"  +  /'«r"+^ +/'«/•"+*+ &c. 

and  as  we  have  already  determined  F"/*'  or  the  part  of  V  corresponding 
toyt'''V'+^'',  we  immediately  deduce  as  before  the  required  value  of  V, 
the  only  difference  is,  that  the  numbers  i  and  t,  instead  of  being  as 
in  the  former  case  confined  within  certain  limits,  may  here  become  in- 
definitely great. 

D  2 


28      Mr  green,  on  THE   LAWS  OF   THE  EQUILIBRIUM  OF   FLUIDS. 

In  the  foregoing  expression  (11)  /3  may  be  taken  at  will,  but  if  we 

qq ^ 

assign   to   it   such   a  value   that  -~ —    may   be   a   whole    number,   the 

series  contained  therein  will  terminate  of  itself,  and  consequently  the 
value  of  Vt^^  will  be  exhibited  in  a  finite  form,  capable  by  what  has 
been  shown  at  the  beginning  of  the  present  Article  of  being  converted 
into  a  rational  and  entire  function  of  x,  y,  %,  the  rectangular  co-ordinates 
of  p.  It  is  moreover  evident,  that  the  complete  value  of  V  being  com- 
posed of  a  finite  number  of  terms  of  the  form  Vt-'^  will  possess  the  same 
property,  provided  the  function  fix,  y ,  %)  is  rational  and  entire,  which 
agrees  with  what  has  been  already  proved  in  the  second  Article,  by  a 
very  different  method. 

(5)     We  have  before  remarked,  (Art.  2.)    that  in  the  particular  case 

where  /3  =  — — — ,  the  arbitrary  constants  contained  in  y(a;',  y' ,  %)  are  just 

sufficient  in  number  to  enable  us  to  determine  this  function,  so  as  to 
make  the  resulting  value  of  V  equal  to  any  given  rational  and  entire 
function  of  x,  y,  z,  the  rectangular  co-ordinates  of  p,  and  have  proved 
that  the  corresponding  functions  V  and  J"  will  be  of  the  same  degree. 
But  when  this  degree  is  considerable,  the  method  there  proposed  becomes 
impracticable,  seeing  that  it  requires  the  resolution  of  a  system  of 

^  +  1  .^  +  2,.s  +  3 
1.2.3 

linear  equations  containing  as  many  unknown  quantities ;  s  being  the 
degree  of  the  functions  in  question.  But  by  the  aid  of  what  has  been 
shown  in  the  preceding  Article,  it  will  be  very  easy  to  determine  for 
this  particular  value  of  /3  the  function  J'{x',  y,  %')  and  consequently  the 
density  p  when  F'  is  given,  and  we  shall  thus  be  able  to  exhibit  the 
complete  solution  of  the  inverse  problem  by  means  of  very  simple 
formulae. 

For  this  purpose,  let  us  suppose  agreeably  to  the  preceding  remarks, 
that  p  the  density  of  the  fluid  in  the  element  dv  is  of  the  form 

p  =  {l-r^)-^/{x',y',z); 


Mr  green,   on   THE   LAWS   OF   THE   EQUILIBRIUM  OF   FLUIDS.      29 

f  being  the  characteristic  of  a  rational  and  entire  function  of  the  same 
degree  as  V,  and  which  we  will  here  endeavour  so  to  determine,  that 
the  value  of  V  thence  resulting,  may  be  equal  to  any  given  rational 
and  entire  function  of  x,  y,  %  of  the  degree  s. 

Then  by   Laplace's  simple  method   {Mec.  Cel.  Liv.  iii.  No.  16.)   we 
may  always  expand  F"  in  a  series  of  the  form 

r=  r<«>  +  r(»  +  r®  +  &c +  r«. 

In  like   manner  as    has   before   been   remarked,    we   shall    have   the 
analogous  expansion 

f{x',y',  ,')=/''«' +/'<'>+/'^=>+/'<'>+  &c +/'«, 

of  which  any  termy*''  for  example,  may  be  farther  developed  as  follows, 

/'«  =^'('V'  +y;'''V"+'^  +//»/'+*  +  &c. = r"  (/'«  +y;'('V'^  +/'»;.'^ + &c.) 

y",  yj'<'>,  j^''*^,  &c.  being  quantities  independent  of  /  and  all  of  the  form 
K'"'  {Mec.  Cel.  Liv.  iii.)     Moreover  F/"  the  part  of  F'  due  to  the  general 

term  Jl'^'^r''+^*  of  the  last  series,  will  be  obtained  by  writing  for  (i 

in  the  equation  (11),  and  afterwards  substituting  for 

(n  —  2\  _  f4i-n 


r(!t^)r(l^)  us  value- 


n-2 
sin 


In  this  way  we  get 

27r;/;'V  ±-„,e-n 2t-2f  +  2-n 

'  .     fn-2    \  2.4     2t~2t' 

sm  (-^.j 

n-2.n w  +  2#'-4       w  — 1  .  w  +  1 n  +  2i  +  2f  —  3^ 

^        2.4 2?  ^       3    .     5     ......    2i  +  2t'  +  l     ' 

yj<'^  being  what  J]''-^  becomes  by  changing  6',  -ar'  into  6,  sr,  and  the  finite 
integral  being  taken  from  t'  =  0  to  t'=<x  . 

Let  us  now  for  a  moment  assume 


^(0= 


n-2.n w  +  2#'  — 4      «-l.«  +  l n  +  2i  +  2t'-3 


X 


2.4 2t'  3.5     2i  +  2t'  +  l     ' 


30      Mr  green,  ON  THE  LAWS   OF  THE   EQUILIBRIUM  OF  FLUIDS. 

then  the  expression  immediately  preceding  may  be  written 

■^J7\r        4<-n.6-n Q,t-2t'  +  2 

(n-2    \  2.4     2t-2t' 


dn  [-^  .) 


sin 

and  by  giving  to  t  the  various  values  0,  1,  2,  3,  &c.  of  which  it  is  sus- 
ceptible, and  taking  the  sum  of  all  the  resulting  values  of  F/''  the  quantity 
thus  obtained  will  be  equal  to  V^^  or  that  part  of  V  which  is  of  the 
form   Y^\    Thus  we  get 

27r^  r' 


■fr(i), 


sm 


.«/)(0)./„« 


+  &c &c &c. 


since  aU  the  terms  in  the  preceding  value  of  Vi-^  in  which  t'>t  vanish 
of  themselves  in  consequence  of  the  factor 


/2^-2^  +4-w\ 

=  0  (when  t  >  t). 


2     .     4     2i?-2^  „,.     .,  .  _  „  /4-w 


(-'■«)  rC-?) 


But    F"^'^  as  deduced  from  the  given  value  of  V  may  be  expanded  in 
a  series  of  the  form 

r«=?''.  {r„®-i-  r;(v^+  r,»./^+  v^'>t^+kc.\ 

and  if  in  order  to  simplify  the  remaining  operations,  we  make  generally 

__„  27r^  n-2.n M+2/-4      n-\.n  +  l n  +  2i+2t-3  ^„., 

'         .     [n-2    \  2.4 2t  3.5    2«  +  2^+l        ' 

sm  [—.] 


Mr  green,  on  THE  LAWS  OF  THE  EQUILIBRIUM  OF  FLUIDS.     31 

27r' 


X<J>{t).W, 


.     /n-2    \ 
sm  (— .) 

the  equation  immediately  preceding  will  become 

^(0= ^I-:£ X  {(j)  (0) .  f7o"'  +  0(l) .  t7;(W(^  (2)  C7,».  r'  +  &cc.\ 

[n  —  2    \ 

which  compared  with  the  foregoing  value  of  F^'\  will  give  by  suppressing 

the  factor '— ,   common  to  both,   and  equating  separately   the 

sm(-^.) 

coefficients  of  the  different  powers  of  the  indeterminate  quantity  r  the 
following  system  of  equations 

&c=...&c &c &c. 

for  determining  the  unknown  functions  fo'-^,  /<*',  f/\  &c.  by  means  of  the 
known  ones  f7"o*'\  Ui'-'\  ZJg*",  &c.  In  fact  the  last  equation  of  the  system 
gives  U^^=fP,  and  then  by  ascending  successively  from  the  bottom  to 
the  top  equation,  we  shall  get  the  values  of  fs^\  /,%,  f}%,  &c.  with 
very  little  trouble.  It  will  however  be  simpler  still  to  remark,  that  the 
general  type  of  all  our  equations  is 

where  the  symbols  of  operation  have  been  separated  from  those  of 
quantity  and  e  employed  in  its  usual  acceptation,  so  that 


32      Mr  green,   ON  THE   LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS. 
But  it  is  evident  we  may  satisfy  the  last  equation  by  making 

/«=(l-e)''^C7-«. 

Expanding  now  and  replacing  e?7„''';  e^UJ^^,  &c.    by   these   values    UJIi, 
U^%,  &c.  we  get 

from  which  we  may  immediately  deduce  ^'®  and  thence  successively 

/'«  =  r"  (/„'«  +/'«  r'^  +/;«  r"  +//«  r'»  +  &c.) 
fW,  !/,  85')  =/'<"' +/'«+/'<^'  +  &c +/'« 

and >  =  (1  - X'"- - y" - z'"-y^.f(x',  y ,  %), 

Application  of  the  general  Methods  exposed  in  the  preceding  Articles 
to  Spherical  conducting  Sodies. 

(6)  In  order  to  explain  the  phenomena  which  electrified  bodies 
present.  Philosophers  have  found  it  advantageous  either  to  adopt  the 
hypothesis  of  two  fluids,  the  vitreous  and  resinous  of  Dufay  for 
example,  or  to  suppose  with  jEpinus  and  others,  that  the  particles  of 
matter  when  deprived  of  their  natural  quantity  of  electric  fluid,  possess 
a  mutual  repulsive  force.  It  is  easy  to  perceive  that  the  mathematical 
laws  of  equilibrium  deducible  from  these  two  hypotheses,  ought  not  to 
differ  when  the  quantity  of  fluid  or  fluids  (according  to  the  hypothesis 
we  choose  to  adopt)  which  bodies  in  their  natural  state  are  supposed 
to  contain,  is  so  great,  that  a  complete  decomposition  shall  never  be 
effected  by  any  forces  to  which  they  may  be  exposed,  but  that  in 
every  part  of  them  a  farther  decomposition  shall  always  be  possible  by 
the  application  of  still  greater  forces.  In  fact  the  mathematical  theory 
of  electricity  merely  consists  in  determining  p*  the  analytical  value  of 

*  It  may  not  be  Improper  to  remark  that  p  is  always  supposed  to  represent  the  density 
of  the  free  fluid,  or  that  which  manifests  itself  by  its  repulsive  force;  and  therefore,  when 
the  hypothesis  of  two  fluids  is  employed,  the  measure  of  the  excess  of  the  quantity  of  either 

fluid 


Mr  green,   on   THE  LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS.      33 

the  fluid's  density,  so  that  the  whole  of  the  electrical  actions  exerted 
upon  any  point  p,  situated  at  will  in  the  interior  of  the  conducting 
bodies  may  exactly  destroy  each  other,  and  consequently  p  have  no 
tendency  to  move  in  any  direction.  For  the  electric  fluid  itself,  the 
exponent  n  is  equal  to  2,  and  the  resulting  value  of  p  is  always  such 
as  not  to  require  that  a  complete  decomposition  should  take  place  in 
the  body  under  consideration,  but  there  are  certain  values  of  n  for  which 
the  resulting  values  of  p  will  render  fpdv  greater  than  any  assignable 
quantity ;  for  some  portions  of  the  body  it  is  therefore  evident  that 
how  great  soever  the  quantity  of  the  fluid  or  fluids  may  be,  which 
in  a  natural  state  this  body  is  supposed  to  possess,  it  will  then  become 
impossible  strictly  to  realize  the  analytical  value  of  p,  and  therefore  some 
modification  at  least  will  be  rendered  necessary,  by  the  limit  fixed  to 
the  quantity  of  fluid  or  fluids  originally  contained  in  the  body,  and 
as  Dufay's  hypothesis  appears  the  more  natural  of  the  two,  we  shall 
keep  this  principally  in  view,  when  in  what  follows  it  may  become 
requisite  to  introduce  either. 

7.  The  foregoing  general  observations  being  premised,  we  will  proceed 
in  the  present  article  to  determine  mathematically  the  law  of  the  density 
p,  when  the  equilibrium  has  established  itself  in  the  interior  of  a  con- 
ducting sphere  A,  supposing  it  free  from  the  actions  of  exterior  bodies, 
and  that  the  particles  of  fluid  contained  therein  repel  each  other  with 
forces  which  vary  inversely  as  the  w""  power  of  the  distance.  For  this 
purpose  it  may  be  remarked,  that  the  formula  (1),  Art.  1,  immediately 
gives  the  values  of  the  forces  acting  on  any  particle  p,  in  virtue  of 
the  repulsion  exerted  by  the  whole  of  the  fluid  contained  in  A.  In 
this  way  we  get 

1      dV 
-  _    .-jr  =  the  force  directed  parallel  to  the  axis  X, 

1      dV 

-■  _    .  y—  =  the  force  directed  parallel  to  the  axis   Y, 

fluid  which  we  choose  to  consider  as  positive  over  that  of  the  fluid  of  opposite  name  in  any 
element  dv  of  the  volume  of  the  body  is  expressed  by  pdv,  whereas  on  the  other  hypothesis 
pdv  serves  to  measure  the  excess  of  the  quantity  of  fluid  in  the  element  dv  over  what  it 
would  possess  in  a  natural  state. 

Vol.  V.    Paet  I.  E 


34        Mr  green,  ON  THE  LAWS  OF  THE  EQUILIBRIUM  OF  FLUIDS. 

1      dV 

.-^-  =  the  force  directed  parallel  to  the  axis  Z. 


1  — »    d% 

But   since,   in   consequence   of  the  equilibrium,   each   of  these  forces  is 
equal  to  zero,  we  shall  have 

„      dV J     .  dV  ,     ,  dV  .         .-, 
0  =  -5—  dx  +  -7—  dy  +  -J-  d%  =  dV\ 
dx  dy  d% 

and  therefore,  by  integration, 

F  =  const. 

Having  thus  the  value  of  V  at  the  point  p,   whose  co-ordinates  are 
X,  y,  %,  we  immediately  deduce,   by  the  method  explained  in  the  fifth 

article, 

/w-2    \ 


sm 


P  = 


2' 


^.(l-r'*) 


seeing  that  in  the  present  case  the  general  expansion  of  K  there  given 
reduces  itself  to 

If  moreover  Q  serve  to  designate  the  total  quantity  of  free  fluid  in 
the  sphere,  we  shall  have,  by  substituting  for 

sin  f  TT  j  its  value 


rl^)r[^y 


\       2  /     rrz-ij  >i.Ji/i  '2\""S~  ^"^ 


sm 


Q  =  /pe/«;  = ^5 i  F/liW^dril-r") 


See  Legendre.     Exer.  de  Cal.  Int.  Quatrieme  Partie. 

In   the   preceding  values,   as   in   the   article  cited,  the  radius  of  the 
sphere   is   taken   for   the   unit   of    space ;    but    the    same   formula    may 

readily  be  adapted  to  any  other  unit  by  writing  —  in  the  place  of  r', 

and  recollecting  that  the  quantities  p,   V,  and  Q,  are  of  the  dimensions 
0,  4  — «,  and  3  respectively,  with  regard  to  space;  a  being  the  number 


Mi  GREEN.   ON   THE  LAWS   OF   THE   EQUILIBRIUM   OF  FLUIDS.      35 

which  represents  the  radius  of  the  sphere  when  we  employ  the  new 
unit.     In  this  way  we  obtain 

P  =  — V4 — -  r{a'-r")~,     and  Q  = lii — ; .  V. 

Hence,  when  Q,  the  quantity  of  redundant  flviid  originally  introduced 
into  the  sphere  is  given,  the  values  of  V  and  of  the  density  p  are  like- 
wise given.     In  fact,  by  writing  in  the  preceding  equation  for 

ry,    and    sin(^7r), 

their  values,  we  thence  immediately  deduce 

and  F= ^         '       ■         '  a'-.Q. 

\/7r 

The  foregoing  formulae  present  no  difficulties  where  «  >  2,  but  when 
H  <  2,  the  value  of  p,  if  extended  to  the  surface  of  the  sphere  Jl,  would 
require  an  infinite  quantity  of  fluid  of  one  name  to  have  been  origi- 
nally introduced  into  its  interior,  and  therefore,  agreeably  to  a  preceding 
observation,  could  not  be  strictly  realized.  In  order  then  to  determine 
the  modification  which  in  this  case  ought  to  be  introduced,  let  us  in 
the  first  place  make  n>2,  and  conceive  an  inner  sphere  S,  whose 
radius  is  a  —  Sa,  in  which  the  density  of  the  fluid  is  still  defined  by 
the  first  of  the  equations  (12);  then,  supposing  afterwards  the  rest  of 
the  fluid  in  the  exterior  shell  to  be  considered  on  ^'s  surface,  the  portion 
so  condensed,  if  we  neglect  quantities  of  the  order  Sa,  compared  with  those 
retained,  will  be 

--     r  f^±i) 

2*  V    2    /    ^, 


'  V2/ 

E  2 


(1) 


QJa 


2 


36      Mr  green,   ON   THE  LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS. 

and  since,  in  the  transfer  of  the  fluid  to  ^'s  surface,  its  particles  move 
over  spaces  of  the  order  ^a  only,  the  alteration  which  will  thence  be 
produced  in  V  will  evidently  be  of  the  order 

n  — 2  n 

and  consequently  the  value  of  V  will  become 

k  being  a  quantity  which  remains  finite  when  ^a  vanishes. 

In  establishing  the  preceding  results,  ti  has  been  supposed  greater 
than  2,  but  p  the  density  of  the  fluid  within  S  and  the  quantity  of  it 
condensed  on  ^'s  surface  being  still  determined  by  the  same  formulae, 
the  foregoing  value  of  V  ought  to  hold  good  in  virtue  of  the  generality 
of  analysis  whatever  n  may  be,  and  therefore  when  w  is  a  positive  quantity 
and  hi  is  exceedingly  small,  we  shall  have  without  sensible  errors 


v;^m^m«'--* ' 


Conceiving  now  P'  to  represent  the  density  of  the  fluid  condensed 
on  A's  surface,  47ra^P' will  be  the  total  quantity  thereon  contained,  which 
being  equated  to  the  value  before  given,  there  results 


-y/TT 


(I) 


and  hence  we  immediately  deduce 

fn  +  V 


n  —  4       -p 

2~ 


m 


Moreover  as   Q  represents  the  total  quantity  of  redundant  fluid  in  the 
entire  sphere  A,  the  quantity  contained  in  B  is 


Mh  green,  on  the  laws  of  the  equilibrium  of  fluids.    37 


-)„=. 


If  now  when  w  is  supposed  less  than  2,  we  adopt  an  hypothesis 
similar  to  Dufay's,  and  conceive  that  the  quantities  of  fluid  of  opposite 
denominations  in  the  interior  of  A  are  exceedingly  great  when  this 
body  is  in  a  natural  state,  then  after  having  introduced  the  quantity  Q 
of  redundant  fluid,  we  may  always  by  means  of  the  expression  just 
given,  determine  the  value  of  Sa,  so  that  the  whole  of  the  fluid  of 
contrary  name  to  Q,  may  be  contained  in  the  inner  sphere  S,  the 
density  in  every  part  of  it  being  determined  by  the  first  of  the  equa- 
tions (12).  If  afterwards  the  whole  of  the  fluid  of  the  same  name  as 
Q  is  condensed  upon  A's  svirface,  the  value  of  V  in  the  interior  of  S 
as  before   determined   will   evidently   be   constant,   provided   we   neglect 

n 

indefinitely  small  quantities  of  the  order  ht'\  Hence  all  the  fluid  con- 
tained in  J3  will  be  in  equilibrium,  and  as  the  shell  included  between 
the  two  concentric  spheres,  A  and  S  is  entirely  void  of  fluid,  it  follows 
that  the  whole  system  must  be  in  equilibrium. 

From  what  has  preceded,  we  see  that  the  first  of  the  formulae  (12) 
which  served  to  give  the  density  p  within  the  sphere  A  when  n  is 
greater  than  2,  is  still  sensibly  correct  when  n  represents  any  positive 
quantity  less  than  2,  provided  we  do  not  extend  it  to  the  immediate 
vicinity  of  A's  surface.  But  as  the  foregoing  solution  is  only  approxi- 
mative, and  supposes  the  quantities  of  the  two  fluids  which  originally 
neutralized  each  other  to  be  exceedingly  great,  we  shall  in  the  follow- 
ing article  endeavour  to  exhibit  a  rigorous  solution  of  the  problem, 
in  case  w  <  2,  which  will  be  totally  independent  of  this  supposition. 

8.  Let  us  here  in  the  first  place  conceive  a  spherical  surface  whose 
radius  is  a,  covered  with  fluid  of  the  uniform  density  P',  and  suppose 
it  is  required  to  determine  the  value  of  the  density  p  in  the  interior 
of  a  concentric  conducting  sphere,  the  radius  of  which  is  taken  for 
the  unit  of  space,  so  that  the  fluid  therein  contained,  may  be  in  equi- 
librium in  virtue  of  the  joint  action  of  that  contained  in  the  sphere 
itself,  and  on  the  exterior  spherical  surface.  "• 


38       Mr  green,   ON   THE  LAWS   OF   THE    EQUILIBRIUM   OF    FLUIDS. 

If  now  V  represents  the  value  of  V  due  to  the  exterior  surface, 
it  is  clear  from  what  Laplace  has  shown,  {Mec.  Cel.  Liv.  ii.  No.  12.)  that 

^  =  !y^  =  (3^:^  K«+^r"-(«-'-M; 

rfo-  being  an  element  of  this  surface,  and  g  being  the  distance  of  this 
element  from  the  point  p  to  which    V  is  supposed  to  belong. 

If  afterwards  we  conceive  that  the  function  V  is  due  to  the  fluid 
within  the  sphere  itself,  it  is  easy  to  prove  as  in  the  last  article,  that 
in  consequence  of  the  equilibrium  we  must  have 

V  +V=  const. 

But  V  and  consequently  V  is  of  the  form  F^"',  therefore  by  employ- 
ing the  method  before  explained,  (Art.  4.)  we  get 

/(ar',  y',  %)  =/'(»'  =/„(">  +/(">. r''  +//>. r''  +  &c.  =  B,  +  B,r''  +  B, r"  +  &c. ; 

where,    as    in    the    present    case,  ^''°>,  yi'<°',  ^''%    &c.    are    all    constant 
quantities,  they  have  for  the  sake  of  simplicity  been  replaced  by 

J?o,  jBi,  B.^,  &c. 

Hitherto  the  exponent  /3  has  remained  quite  arbitrary,  but  by  making 

/3=  — -— ^  the  formula  (11)  Art.  4.  will  become  when  «  =  0, 


ir(o)_o     7?       <2y      V    2    ;  ^  ,,,4-».6-w 2t-9.t'  +  ^-n 

^'    -2'^^'- YW) '       4     .    6     2^-2^+2 

n-2.n-l «  +  2^-3 

"^     2.3.4   2^  +  1 

{/i-2)Tr'Bt   ^     ,„  4-W.6-M 2^-2/'  +  2-«      w-2.«-l «  +  2/-3 

2.?^    . ^ -r^ ^w    .    n     ^ 


.      (71-2    \  •     4.6    2^-2^  +  2  2    .    3     2^'  +  l    ' 

sm(— .) 

Giving   now   to   /  the   successive  values   0,   1,  2,  3,  &c.   and  taking 
the  sum  of  the  functions  thence  resulting,  there  arises 

r=  F<°'  =  Fo<">  +  rr  +  T^-P  +  ^3<°*  +  &c. = s.  r/"> 

(«-2)7r'^      ^P^  ,,,  4-W.6-W 2t-2f  +  2-n      n-2.n-l «+2^'-3 

"""T^ri"^        '  4.6.8 2^-2^'  +  2    ""     2    .    3    2/'  +  l  ' 

sm  (^.) 

where  the  sign  S  is  referred  to  the  variable  t  and  2  to  ^. 


Mn  GREEN,   ON   THE   LAWS   OF   THE   EQUILIBRIUM   OF   FLUIDS.      39 

Again,   by   substituting  for   V  and   V   their  values  in  the  equation 
V^  +  V=  const,  and  expanding  the  function  V  we  obtain 

,/^*'    ti-2.n-l.n n  +  2f-3 


const.  =47rP'«'-".2 


«^""      2    .     3     .4 2^  +  1 


(w-2)7r'      „    „    ,,/4-».6-?? 2^-2^'+2-w      n-2.n-l ti+2t'+3 

"^         (n-2    X  '^^^'''       4    .    6    2f-2f  +  2    ^    2.3.4 2^  +  1 


sm 


..) 


2 

which   by   equating  separately  the  coefficients  of  the  various  powers  of 
the  indeterminate  quantity  r,  and  reducing,  gives  generally 

(n-2 


2P'«'-"-"'.  sin  /• 


TT 


2        I        „2  — «.4  — « 2s  — n 


-^     2    .    4>     2s     ^''^'-' 


Then  by  assigning  to  t'  its  successive  values  1,  2,  3,  &c.  there  results  for 
the  determhiation  of  the  quantities  B^,Bi,  JB^,  &c.  the  following  system 
of  equations, 

2P'       „     .     (n-2    \        p  ^  2-n  „       2-n.4<-n  „   ,  » 

2P'  .     fn-2    \        „      „       2  — «  „       2  — ;«.4  — ra  ^      . 

=^  «'-".sin     — r-TT     .«-^  =  P,-f  — r-  P.  +  — ^— r— P3  +  &C. 

TT  V    2        /  2  2.4 

2P'    ,        .     (n-2    \       ...      „       2-M„,2-ra.4-/< 

a'-",  sm  ( -^— t]  •  '^     =  P.'  +  —^  -^^  +  ~~;> 4 —  P4  +  &C. 

&c &c &c &c 

But  it  is  evident  from  the  form  of  these  equations,  that  we. may  satisfy 
the  whole  system  by  making 

B,  =  B^.a\  B,  =  B.a-\   B,  =  B,,(r\  B,  =  B^.a-M>ic. 
provided  we  determine  Po  by 

2P'  .     /n-2    \        „  ,,       2-n       ,      2  —  n.4>-n    ...      .     , 

=  P„(l-a-)' 
„       2P    .     (n-2    \     .  ,     ,,^ 


n-2 
,-2\    2 


n  —  2 
Hence  as  in   the  present  case,  ^  =  —^ ,    we  immediately  deduce  the 

successive  values 

2P    .     fn-2 


40     Mr  GREEN,  ON  THE  LAWS  OF  THE  EQUILIBRIUM  OF   FLUIDS. 

/(or',  y\  «')=/'(")  =  ^„  +  P,r'^  +  ^.r'^  +  &c.  =  if„  (l  -  ^']", 
and  p  =  (1  -  O  ^  ./(x',  y',  %')  =  ^  sin  {—^  ^)  .  («=  - 1)"... 

(«^  -  /'')-Ml  -  0~- 

In  the   value  of  p  just  exhibited,  the  radius  of  the  sphere  is  taken 
as  the  unit  of  space,  but   the  same  formula  may  easily  be  adapted  to 

any  other  unit  by  writing  j  and  y-  in  the  place  of  a  and  /  respectively, 

and  recollecting  at  the  same  time  that  in  consequence  of  the  equation 

•dv.p    .     rdaP' 


const, 


=  r^r^j!^  +  ji 


s       -^  g 

before   given,  ^ ,    is   a   quantity   of  the  dimension  —  1    with   regard  to 

space:  h  being  the  number  which  represents  the  radius  of  the  sphere 
when  we  employ  the  new  unit.  Hence  we  obtain  for  a  sphere  whose 
radius  is  bg,  acted  upon  by  an  exterior  concentric  spherical  surface 
of  which  the  radius  is  a, 

2P'a.sin  {—-■"]  2-n  ^ 

(/3) p  = -^ ff-b')  '    {a'-r")-'  {b'-r")  ^  ; 

7r 

P'  being  the  density  of  the  fluid  on  the  exterior  surface. 

If  now  we  conceive  a  conducting  sphere  A  whose  radius  is  a,  and 
determine  P'  so  that  all  the  fluid  of  one  kind,  viz.  that  which  is  re- 
dundant in  this  sphere,  may  be  condensed  on  its  surface,  and  afterwards 
find  b  the  radius  of  the  interior  sphere  S  from  the  condition  that  it 
shall  just  contain  all  the  fluid  of  the  opposite  kind,  it  is  evident  that 
each  of  the  fluids  will  be  in  equilibrium  within  A,  and  therefore  the 
problem  originally  proposed  is  thus  accurately  solved.  The  reason  for 
supposing  all  the  fluid  of  one  name  to  be  completely  abstracted  from 
S,  is  that  our  formulas  may  represent  the  state  of  permanent  equilibrium, 
for  the  tendency  of  the  forces  acting  within  the  void  shell  included 
between  the  surfaces  A  and  B,  is  to  abstract  continually  the  fluid  of 
the  same  name  as  that  on  ^'s  surface  from  the  sphere  S. 


Mr  green,   on  THE   LAWS   OF   THE   EQUILIBRIUM   OF   FLUIDS.      41 

To  prove  the  truth  of  what  has  just  been  asserted,  we  will  begin 
with  determining  the  repulsion  exerted  by  the  inner  sphere  itself,  on 
any  point  p  exterior  to  it,  and  situate  at  the  distance  r  from  its  centre 
O.  But  by  what  Laplace  has  shown  {Mec.  Cel.  Liv.  ii.  No.  12.)  the 
repulsion  on  an  exterior  point  p,  arising  from  a  spherical  shell  of  which 
the  radius  is  r',  thickness  dr    and  center  is  at  O  will  be  measured  by 

I'Kr'dr'p      d_  (r  +  r)^-"  -  (r-/)^-" 

1  —  ti.S  —  n'dr'  r  ' 

the   general  term   of  which   when   expanded   in    an  ascending  series  of 

r' 
the  powers  of  —  is, 

+  ^"-      2.3.4.5     27Tl ^-.r-prfr, 

and  the  part  of  the  required  repulsion  due  thereto  will,  by  substituting 
for  p  its  value  before  found,  become 
%F   .     (n-2   \     ,o    ,,,^ -2  +  wx«.»  +  l M+2*-3xw  +  l*-l 


Si'    .     fn-2   \     ,  ,    ,„,: 


2.3    .    4     2«  +  l 

,'2\   -1  »-2 

X 


It   now  remains  to  find  the  value  of  the  definite  integral  herein  con- 

tained.     But  when    11 -\        is    expanded,    and   the  integrations   are 

effected,  by  known  formulae,  we  obtain 

(14)      M  1  -  ^ I     (*' -  O  '  r''^^'dr'=/o' 2^  -jj  {b'-r")  '  .r"'+'dr' 
^^  "a''        r^,      ,3        X     "^  'f       n      3\ 


,  2^  +  3      y  2^  +  3.2^  +  5        ¥ 

*        25  +  3  +  w  a'      2*  +  3  +  ».2«  +  5  +  wa' "*■     ^'^ 


^,^,  ^  (2)  ^  r  "^  2)       {^s+l-^n){l-xy''    r^ 


_,        ,        ^,      ,.„  .  _ „  ,        ,  „     'dx 

,  1J2S  +  1 

2°  •       /        «       3> 


r  (.+  -  +  -)  (1-.^)- 


Vol.  V.    Pakt  I. 


42      Mr  green,   ON   THE   LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS. 

r  f^^l  r  (-\  — ' 

\2J       \2)        1.3.5     2s  +  lb''+'+"      (l-af)'      fx'^^'dx 


1+w     ^      ^•'^+'+"      ^o' 


2 


J  {\-x) 


Avhere  after   the   integrations   have   been   effected,  x   ought  to   be  made 
equal    to    - . 

The  value  of  the  integral  last  found  being  substituted  in  the  expres- 
sion immediately  preceding,  and  the  finite  integral  taken  relative  to  s 
from  *  =  0  to  *  =  X    gives  for  the  repulsion  of  the  inner  sphere. 


a  ^  ll+n\       f2  —  n 

¥~J       V'~2 


) 


-2.7i.n  +  2 n  +  2s-4>  (by"(l-x')''     raf'^'dx 


''^"         2.4.6       2s         [r)  ic^'+'-^"    J„ 


(1-x^y 


.2\2 


-47rv/,rP^a^/-"     »  n-2.n.n  +  2 w  +  2^-4  (a^  rj..^„  /,  _^.^\ 

„fl+n\      l2-n<:^'        2.4.6      2s        (rj    ■'"'^^'^      '  ^*     ^'    ' 

i    )      [    2   J 


since  F  (^)  =\/ir,  sin  (  tt]  = 


^'D^C-?)' 


and  as  was  before  observed,   a;  =  - . 

a 


But  we  have  evidently  by  means  of  the  binomial  theorem, 

/    _  ff-j;-\i^  _    .  n-2.n.  w4-2 w  +  2^-4  /«^y, 

I  rW  '    ""    °         2.4.6       2*         \r]    ' 

and  therefore  the  preceding  quantity  becomes 

(15) 7^^ L  dxaf    1 '   (l-x^)"  . 


Mb  green,   on   THE  LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS.      43 

T  X* 

If  HOW  we  make  x  —  — ,  the  same  quantity  may  be  written 
(16) ,  .     , -; tx'dxiX-x^)'     1 —\'- 

Having  thus  the  vahie  of  the  repulsion  due  to  the  inner  sphere  B 
on  an  exterior  point  p,  it  remains  to  determine  that  due  to  the  fluid 
on  ^'s  surface.     But  this  last  is  represented  by 

,  2  7raP^      (L_  {a-\-rf-''-{a-rf- 

' l  —  7i.S  —  ndr  r 


{Mec.  Cel.  Liv.  ii.  No.  12.)     Now    by    expanding   this  function   there  re- 
sults 

1~  ■  r  "*"     4.5     ^  "*"  4.5.6.7 


^TrP'a'-'r. ^^ ■  U  +  "T  l^  2'-  +  """,T^":"  '  ".g^+^cl 


.     r./  i»     2-w   .^,  ra.w  +  l.?«  +  2 /i  +  2*-l,       ^.r" 

=  4.PV-V.-g-.2„  ,.5.6     ,,^3      (^-^1)^- 

The  last  of  these  expressions  may  readily  be  exhibited  vinder  a  finite 
form,  by  remarking  that 

flx"dx{l-a^)  ^    (l  -  -^)  ^  =/lx''dx(l-x')  "  S^^    g    ^^^ .-^ 

/2^  +  w  +  l\      /4-/A 

«.?f  +  2.;?  +  4 w  +  2jf-2    y^'       V        2        /      V    2    / 

~     "  2.    4    .6      2s        'a^''  ^^l2s  +  5\ 

2  r 


r  {lzl\  r  fl+^^ 

V    2    /      \    2    I    2-n   ^^n.n  +  l.n  +  2 n  +  2s-l ,      ,.r^' 


© 


3     '    "  4.    5    .    6      2*  +  3     '         'a 


Hence,  since  r  (^)  =  v^tt,  the  value  of  the  repulsion  arising  from  ^'s 
surface  becomes 

F  2 


44      Mr  green,   ON  THE  LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS. 

Now  by  adding  the  repulsion  due  to  the  inner  sphere  which  is  given 
by  the  formula  (16),  we  obtain,  (since  it  is  evidently  indifferent  what 
variable  enters  into  a  definite  integral,  provided  each  of  its  limits  re- 
main unchanged) 


f'afdxCl  -x")  ''  .{1  ,  , 

1  +  w\      /2-w\  •'^  ^  '        V  (f  ) 

\    2     j    ' 

for  the  value  of  the  total  repulsion  upon  a  particle  p  of  positive  fluid 
situate  within  the  sphere  A  and  exterior  to  S.  We  thus  see  that 
when  P'  is  positive  the  particle  p  is  always  impelled  by  a  force  whicli 
is  equal  to  zero  at  JS's  surface,  and  which  continually  increases  as  p 
recedes  farther  from  it.  Hence,  if  any  particle  of  positive  fluid  is 
separated  ever  so  little  from  2?'s  surface,  it  has  no  tendency  to  return 
there,  but  on  the  contrary,  it  is  continually  impelled  therefrom  by  a 
regularly  increasing  force ;  and  consequently,  as  was  before  observed, 
the  equilibrium  can  not  be  permanent  until  all  the  positive  fluid  has 
been  gradually  abstracted  from  B  and  carried  to  the  surface  of  A, 
Avhere  it  is  retained  by  the  non-conducting  medium  with  which  the 
sphere  A  is  conceived  to  be  surrounded. 

Let  now  q  represent  the  total  quantity  of  fluid  in  the  inner  sphere, 
then  the  repulsion  exerted  on  p  by  this  will  evidently  be 

qr-', 

when  r  is  supposed  infinite.  Making  therefore  r  infinite  in  the  expression 
(15),  and  equating  the  value  thus  obtained  to  the  one  just  given,  there 
arises 

q=  — :: tclx-afil-x')'. 


2    J       \    2 


When  the  equilibrium  has  become  permanent,  q  is  equal  to  the  total 
quantity  of  that  kind  of  fluid,  which  we  choose  to  consider  negative, 
originally  introduced  into   the  sphere  A ;    and   if  now  qi  represent   the 


Mr  green,    on   THE   LAWS  OF   THE  EQUILIBRIUM   OF  FLUIDS.      45 

total  quantity  of  fluid  of  opposite  name  contained  within  A,  we  shall 
have,  for  the  determination  of  the  two  unknown  quantities  P'  and  b, 
the  equations 

5',  =  4nra'.P', 

and    ^  =  — ,     "^""^ X"  dxx"  (1  -  af)^, 

and  hence  we  are  enabled  to  assign  accurately  the  manner  in  which  the 
two  fluids  will  distribute  themselves  in  the  interior  of  A;  q  and  «/, ,  the 
quantities  of  the  fluids  of  opposite  names  originally  introduced  into  A 
being  supposed  given. 

9.  In  the  two  foregoing  articles  we  have  determined  the  manner 
in  which  our  hypothetical  fluids  wiU  distribute  themselves  in  the  interior 
of  a  conducting  sphere  A  when  in  equilibrium  and  free  from  all  exterior 
actions,  but  the  method  employed  in  the  former  is  equally  applicable 
when  the  sphere  is  under  the  influence  of  any  exterior  forces.  In  fact, 
if  we  conceive  them  all  resolved  into  three  JT,  Y,  Z,  in  the  direction 
of  the  co-ordinates  x,  y,  «  of  a  point  j9,  and  then  make,  as  in  Art.  1, 

rpdv 

we  shall  have,  in  consequence  of  the  equilibrium, 

1     dr      „     ^         \     dV      ^     ^         \     dV      „ 

0= -J—  +  X,     0  = 5-  +  ^'     0  = 7-  +  Z, 

1  —  ndx  \  —  ndy  1  —  ndz 

which,  multiplied  by  dx,  dy  and  d%  respectively,  and  integrated,  give 
const.  =  =-^  V  +  f{Xdx  +  Ydy  +  Zdz) ; 

X   ^~   ft/ 

where  Xdx  +  Ydy  +  Zd%  is  always  an  exact  differential. 

We  thus  see  that  when  X,  Y,  Z  are  given  rational  and  entire  functions 
V  will  be  so  likewise,  and  we  may  thence  deduce  (Art.  5.) 

p  =  (1  _  ;j;'^  _  y'2  _  «'^)f^  ./{a;',  y',  x'), 

where  /  is  the  characteristic  of  a  rational  and  entire  function  of  the  Same 
degree  as  V. 


46      Mr  green,   ON   THE  LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS. 

The  preceding  method  is  directly  applicable  when  the  forces  X,  Y,  Z 
are  given  explicitly  in  functions  of  x,  y,  x.  But  instead  of  these  forces, 
we  may  conceive  the  density  of  the  fluid  in  the  exterior  bodies  as  given, 
and  thence  determine  the  state  which  its  action  will  induce  in  the  con- 
ducting sphere  A.  For  example,  we  may  in  the  first  place  suppose 
the  radius  of  A  to  be  taken  as  the  unit  of  space,  and  an  exterior  con- 
centric spherical  surface,  of  which  the  radius  is  a,  to  be  covered  with 
fluid  of  the  density  U"^'^:  ZJ"'"*  being  a  function  of  the  two  polar  co- 
ordinates 6"  and  ■zsr"  of  any  element  of  the  spherical  surface  of  the  same 
kind  as  those  considered  by  Laplace  {Mec.  Cel.  Liv.  iii.).  Then  it  is 
easy  to  perceive  by  what  has  been  proved  in  the  article  last  cited,  that 
the  value  of  the  induced  density  wiU  be  of  the  form 

p  =  [/-'Wr''  (1  -  r"'y'  .f{r") ; 

r',  &,  -ar'  being  the  polar  co-ordinates  of  the  element  dv,  and  £/'<'*  what 
Z7"<'>  becomes  by  changing  Q",  -sr"  into  9',  tst'. 

Still  continuing  to  follow  the  methods  before  explained,  (Art.  4.  and  5.) 
we  get  in  the  present  case 

f{af,  y',  «')  =  t7'<Vy(r'^)  =/«, 
and  by  expanding  y(r'^),  we  have 

/(r'^)  =  i?o  +  B,r"  +  B,i''  +  B,r"  +  &c. 
Hence,  /'"  =  B,U'^\  and 

'         .     ln-%    \-"'^'''^       2.4.6   2^-2^        ^        2.4 2^' 

sm(-^.) 

n-l.n  +  1 w  +  2?  +  2if-3 

^3.5    2«-l-2«r  +  l     ■ 

Then,  by  giving  to  t  all  the  values  1,  2,  3,  &c.  of  which  it  is  sus- 
ceptible, and  taking  the  sum  of  all  the  resulting  quantities,  we  shall 
have,  since  in  the  present  case  V  reduces  itself  to  the  single  term   V^\ 

sm  (-^.) 

n-l.n  +  1 n  +  2i  +  2t'-3  ^ 

^      S    .    5    2i  +  2f  +  l     ' 

the  sign  S  belonging  to  the  unaccented  letter  t. 


Mr  green,   on   THE   LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS.      47 

If  now  V  represents  the  function  analagous  to  V  and  due  to  the 
fluid  on  the  spherical  surface,  we  shall  obtain  by  what  has  been  proved 
(Art.  3.) 

V  =  WK  27r«^  W'^ ^'7^/-i(/M  {i)  (f-2artx-V  a'i^', 

X  •  ^  ,Ot« •  •  •      t 

{i)  representing  the  same  function  as  in  the  article  just  cited. 

Moreover,  it  is  evident  from  the  equation  (10)  Art.  4,  that 

,.  ,j    ,.,  ,^     ^  ,    ,.^       ^,1.2.3 i      ^n-l.n  +  1 n  +  2i  +  2t'-S 

/ild^{t){r^-2ar^  +  a^)^   =  2«'-"  ^  3  ^■_^2    ^        ^    ^i  +  ^TTT 

n-2.n n  +  2t'-4>   IrV^^^' 

""         2.4 2^  \a)        ' 

and  consequently, 

(.9) r'=cr<o.w-..'';';''^';;;;;'-^f;^;:^7^ 

«-2.w ?«  +  2if'-4   /r\'  +  '' 


2.4 2^'         \a) 

the  finite  integrals  extending  from  t'  =  Q  to  t'=<x). 

Substituting  now  for  F'  and  ?^'  their  values  in  the  equation  of  equi- 
librium, 

(20) const.  =  r'+  f; 

we  immediately  obtain 

const.  =  i7".47rflr'  ".2 


3    .    5    2i  +  2t'  +  l 

n-2.?i w  +  2^'-4  //•\'+''' 

2    .4 2f 


Q 


^  2^"  rm  97?  v,.<+2,'  ^-1-^^  +  1 «  +  2i  +  2^'-3 

^    .     fn-2    \  '  "3.5    2»  +  2r-l 

sm(-^.) 

n-2.fi Ti-{-2t'-4>      4-W.6-W 2^-2^'  +  2-« 

"^      2     .4 2?         ^      2     .     4    2^-2^ 

the  constant  on  the  left  side  of  this  equation  being  equal  to  zero,  except 
when  i  =  0. 


48      Mr  green,   ON   THE   LAWS   OF    THE   EQUILIBRIUM    OF   FLUIDS. 

By  equating  separately  the  coefficients  of  the  various  powers  of  the 
indeterminate  quantity  r,  we  get  the  following  system  of  equations : 

^  .     (n-2    \ 

2  sm    — — —  TT  .  AC 


2    .    4 


^  •     fn-2    \ 


,      I  o       -n       Tk  4  — w       „  4i  —  n.6  —  n      . 

TT  2  2.4 


^  .     fn-2    \ 
2sm(-^.) 


TT  2  2.4 


&c &c &c. 


But  it  is  evident  from  the  form  of  these  equations,  that  if  we  make 
generally  ^,+i  =  a'^Bt,  they  will  all  be  satisfied  provided  the  first  is,  and 
as  by  this  means  the  first  equation  becomes 

2  sm    — -—  TT  ,        A  AC 


Tr»/,       4  — «      „      4  — M.b  — «      .      .     \ 
«-»-  =  J?„  (l  +  -g-«-  +     3        ^    a-^  +  &c.) 


=  J5o(l-«-'^)  ^   =  .Bo«*-"(a'-l)  ^  . 
there  arises 

„    .      /M-2 

2sm 


(n-2    \ 

^^ La-'-'{a'-l)'  ,    B,  =  B,.a-\ 


Bo  = -— -a-'-'ia'-l)'  ,    B,  =  B,.a-\    B,  =  B,.a-\  !>ic. 

TT 

Hence 

f{r")  =  B,  +  By'  +  B,r''  +  &c.  =  ^„  (^1  +  ^  +  ^  +  &c.) 

»8  2  sm  I  — — -  TTJ  ^_„ 

=  ^„fl_  !l)-i  =  ^„a=(a^_/^)-'= L* ia'+'(a^-l)~(a^-r'^)-\ 

\        a  I  If 

and  the  required  value  of  p  becomes 

(21 ) (D  =  C7'< V;  (1  -  r'^)^/(r'0 


2sm(-^.j 


(a'  -  l)'^"a  i7'»  f^')  '(«=  -  /=)-'  (1  -  r"y\ 


Mr  green,   on    THE    LAWS   OF    THE  EQUILIBRIUM   OF  FLUIDS.       49 

But  whatever  the  density  P  on  the  inducing  spherical  surface  may 
be,  we  can  always  expand  it  in  a  series  of  the  form 

P  =  C7"<°>+  Z7"<'>+  C7"<^>+  Z7"®  +  &c.  in  inf. 

and  the  corresponding  value  of  p  by  what  precedes  will  be 


„  .     /n-Q, 
2sm 


P  =  - 


<a{a'-l)  '  .{a'-r")-H^-r") 


X  { t7'W+  [/'<')-  +  t7'<='^  +  f7'<^>^  +  &c.  in  inf.] ; 


Ijm^  U'M^  jjm^  &c.  being  what  U"^'\  U"^'\  U"^%  &c.  become  by  changing 
d",  w"  into  ff,  Ts-',  the  polar  co-ordinates  of  the  element  dv.  But,  since 
we  have  generally 

^d&'d-uy"  sin  6)"PQ«  =  fdff'd^"  sin  6"  C7"<"Q»  =  ^^  C7<", 

{Mec.  Cel.  Liv.  iii.)  the  preceding  expression  becomes 

-sm(-^.) 


p  = _> a{a'-\)  '  K-r'^)-'(l-0  '  jd&'d-sr"  sin  &'. 

2:(2e  +  l)PQ«^; 


a* 
the  integrals  being  taken  from  0"  =  O  to  0"  =  7r,  and  from  ■bt"  to  ■sr"  =  27r. 

In  order  to  find  the  value  of  the  finite  integral  entering  into  the 
preceding  formula,  let  R  represent  the  distance  between  the  two  ele- 
ments dff,  dv ;  then  by  expanding  -^  in  an  ascending  series  of  the  powers 

r' 
of  —    we  shall  obtain 
a 

—  =  ^  _  2°°Q<*>.— -, 

B      Va^  -  2ar' [cos  0' cos  0"  + sine' sine"  cos  (-ar' -•23-")+/*         °       *«" 

Mec.  Cel.  Liv.  iii.).     Hence  we  immediately  deduce 

^  =  .r«»e^,    and  .^4,^^  =  K(^.>1)«?»^. 
Vol.  V.    Part  I.  G 


50      Mr  green,   ON  THE   LAWS   OF  THE   EQUILIBRIUM   OF   FLUIDS. 
If  now  we  substitute  this  in  the  value  of  p  before  given,  and  after- 

// o  ft^  __  >«'2 

wards  write  —  and        „3    in  the  place  of  their  equivalents, 


dd"dnr"  smO",    andvV'^'^, 

clr    R 


we  shall  obtain 


.    (n-2 

p- i7^ — («^-i)^  (i-O^  /-^; 

the  integral  relative  to  da  being  extended  over  the  whole  spherical  sur- 
face. 

Lastly,  if  p^  represents  the  density  of  the  reducing  fluid  disseminated 
over  the  space  exterior  to  A,  it  is  clear  that  we  shall  get  the  corres- 
ponding value  of  p  by  changing  P  into  pida  in  the  preceding  expression, 
and  then  integrating  the  whole  relative  to  a.     Thus, 

,  =  -  !iy4  (i-..)=i^/a-«.)*-?/**£i. 

But  dada  =  dvx\  dvi  being  an  element  of  the  volume  of  the  exterior 
space,  and  therefore  we  ultimately  get 

fn  —  2_ 


4— n 


.     /n  —  2\ 


(22) p= y5 -i^-r")'^   .fp^dv,        ^        , 

where  the  last  integral  is  supposed  to  extend  over  all  the  space  exterior 
to  the  sphere  and  R,  to  represent  the  distance  between  the  two  elements 
dv  and  dv^. 

It  is  easy  to  perceive  from  what  has  before  been  shown  (Art.  7.),  that 
Ave  may  add  to  any  of  the  preceding  values  of  p,  a  term  of  the  form 

h  being  an  arbitrary  constant  quantity :  for  it  is  clear  from  the  article 
just  cited,  that  the  only  alteration  which  such  an  addition  could  produce 
would  be  to  change  the  value  of  the  constant  on  the  left  side  of  the 


Mr  GREEN,  ON  THE   LAWS  OF   THE   EQUILIBRIUM   OF  FLUIDS.      51 

general  equation  of  equilibrium ;  and  as  this  constant  is  arbitrary,  it  is 
evident  that  the  equilibrium  will  not  be  at  all  affected  by  the  change 
in  question.  Moreover,  it  may  be  observed,  that  in  general  the  additive 
term  is  necessary  to  enable  us  to  assign  the  proper  value  of  p,  when 
Q,  the  quantity  of  redundant  fluid  originally  introduced  into  the  sphere, 
is  given. 

In  the  foregoing  expressions  the  radius  of  the  sphere  has  been  taken 
as   the   unit   of  space,   but    it   is  very   easy   thence   to  deduce   formula^ 

adapted   to   any  other  unit,   by   recollecting   that   —,  -p,  j^    and  y^^, 

are  quantities  of  the  dimensions  0,  —  1,  —  1  and  S  —  n  respectively  with 
regard  to  space:  for  if  h  represents  the  sphere's  radius,  when  we  employ 

any  other  unit  we  shall  only  have  to  write,  t>  j,  -j- >  -jr-  and  j-  in  the 

place  of  r,  r,  R,  dvi  and  a,  and  afterwards  to  multiply  the  resulting 
expressions  by  such  powers  of  h,  as  will  reduce  each  of  them  to  their 
proper  dimensions. 

If  we  here  take  the  formula  (22)  of  the  present  article  as  an  example, 
there  will  result, 

•         /W-Q       ^  4-n 

(23).... p= 1-|_-I(i"-/^)^   fp,dv^-^-^, 

for  the  value  of  the  density  which  would  be  induced  in  a  sphere  A, 
whose  radius  is  b,  by  the  action  of  any  exterior  bodies  whatever. 

When  w  >  2,  the  value  of  p  or  of  the  density  of  the  free  fluid  here 
given  offers  no  difficulties,  but  if  »  <  2,  we  shaU  not  be  able  strictly  to 
realize  it,  for  reasons  before  assigned  (Art.  6.  and  7.)  If  however  n 
is  positive,  and  we  adopt  the  hypothesis  of  two  fluids,  supposing  that 
the  quantities  of  each  contained  by  bodies  in  a  natural  state  are  ex- 
ceedingly great,  we  shall  easily  perceive  by  proceeding  as  in  the  last 
of  the  articles  here  cited,  that  the  density  given  by  the  formula  (23) 
will  be  sensibly  correct  except  in  the  immediate  vicinity  of  A's  surface, 
provided  we  extend  it  to  the  surface  of  a  sphere  whose  radius  is 
h—^b  only,  and  afterwards  conceive  the  exterior  shell  entirely  deprived 
of    fluid:    the    surface  of    the   conducting   sphere  itself  having  such    a 

G  2 


52      Mr  green,   ON   THE   LAWS   OF    THE   EQUILIBRIUM    OF   FLUIDS. 

quantity  condensed  upon  it,  that  its  density  may  every  where  be  repre- 
sented by 

ftl         2       \  „-4  „_2  4-n 


Application  of  the  general  Methods  to  circular  conducting  Planes,  &f:. 

10.  Methods  in  every  way  similar  to  those  which  have  been  used 
for  a  sphere,  are  equally  applicable  to  a  circular  plane  as  we  shall  im- 
mediately proceed  to  show,  by  endeavouring  in  the  first  place  to  determine 
the  value   of   V  when   the   density   of  the  fluid   on  such  a  plane  is  of 

the  form 

p  =  {\-ry.f{x',y'): 

f  being  the  characteristic  of  a  rational  and  entire  function  of  the  degree  * ; 
x\  y'  the  rectangular  co-ordinates  of  any  element  dcr  of  the  plane's 
surface,   and  r',  &  the  corresponding  polar  co-ordinates. 

Then  we  shall  readily  obtain  the  formula 

r=   ff^  =    rrrdr'd9'{l-ry.f{x',y')  ^. 
'' g"''       ''■^  {f^-Zrr' cos  {9-9')  + r"f^' 

where  r,  9  are  the  polar  co-ordinates  of  p,  and  the  integrals  are  to  be 
taken  from  9'  =  0  to  0'  =  27r,  and  from  r'  =  0  to  /•'  =  !;  the  radius  of 
the  circular  plane  being  for  greater  simplicity  considered  as  the  unit 
of  distance. 

Since  the  function  /{x',  y')  is  rational  and  entire  of  the  degree  j, 
we  may  always  reduce  it  to  the   form 

(24)  f{x',  y')  =  A^°^  +  A^'^  cos  9'  +  A^'^  cos  20'  +  ^*''  cos  39'  + 

+  ^«  sin  9'  +  B'-'^  sin  29'  +  B^'^  sin  30'  + 
the  coefficients   A'-''\  A^'\  A^'\  &c.  B^'\  B^%  B^\  &c.   being    functions 
of  r'  only  of  a  degree  not  exceeding  *,  and  such  that 

^('•'=«'o°'  +  «<V^  +  4"V'*  +  &c.;       ^«  =  («?>  +  alV  +  4'V'V)/; 
^(»  =  ( jw  -I-  J(/)r'^  +  i(»r'^  +  &c.)  r' ;   B'^  =  {bf>  +  hfr"  +  &c.)  r'\ 


Ma  GREEN,   ON   THE   LAWS   OF   THE    EQUILIBRIUM    OF    FLUIDS.      Sfi 

We  will  now  consider  more  particularly  the  part  of  V  due  to  any 
of  the  terms  in  f  as  -<4^'^  cos  i&  for  example.  The  value  of  this  part 
will  evidently  be 

rr  /dr'dff{\  -  r"fA^^  cos  iff 

{r'-^rr'  cos  (Q  -  ff)  +  r'')~^ ' 

the  limits  of  the  integrals  being  the  same  as  before.  But  if  we  make 
6'  =  9  +  (p,  there  will  result  dff  =  d<p,  and  cos  i9'  =  cos  id  cos  e0  —  sin  iO  sin  «0, 
and  hence  the  double  integral  here  given  by  observing  that  the  term 
multiplied  sin  i<p  vanishes  when  the  integration  relative  to  (p  is  effected, 
becomes 

cos  ie/lA^'^r'dr'  (1  -  ry  f^ ^"^  ^"^  ''I' —^ ; 

°  {r''  —  2rr'  cos  (p  +  r"^)~^ 

If  now  we  write  F"/*^  for  that  portion  of  V  which  is  due  to  the  term 
«/*^r"+^*  in  the  coefficient  A^'^  we  shall  have 

r,»  =  «/'> .  cos  ieflr^^'^^'dt"  (1  -ry  /" "^"^  ^"^  "^ ^  . 

"  {r^  —  2rr'  cos  (p  +  r"^)~^ 

But  by  well  known  methods  we  readily  get 

•^'^  d(p  cos  i(}) 


L 


{r'  —  ^rr'  cos  (f>  +  r'^)  ^ 


i  ^i-.-iv»^"'    n-l.n  +  1 n+2t'-S      n-l.n  +  1 n+2i+2t'-3 

-2irr.r        i„^„,.     2    _    ^     2^       ""     2    .    4     2i  +  2t      ' 

when  r'>r,  and  when   /<r,   the   same  expression  will  still  be  correct, 
provided  we  change  r  into  r'  and  reciprocally. 

This  value  being  substituted  in  that  of  Fj*''  we  shaU  readily  have  by 
following  the  processes  before  explained,  (Art.  1.  and  2.) 

F,w  =  27ra/'V*  cos  10  2o  r '  — r 7 -—p 

P^^  .  ,.„  ,S  +  2t-2f~n\ 


»-l.»  +  l «  +  2«  +  2#'-3         "^f^  '     I     \  2 

X  — ^:: -. 7;r- — :;^-r, X 


(/3  +  i)r[^ 


2    .    4     2i  +  2/'  (2li  +  5  +  2t-2f) 

^^[  2  J 


54      Mr  green,  ON  THE  LAWS  OF  THE   EQUILIBRIUM  OF  FLUIDS. 

=  TTtt/'V  COS  i6 . —5 — . 

,    „P n-1  .n  +  1 n  +  2f  —  3      7i-l  .  ji  +  1 n  +  2i  +  2t'-3 

"  2.4     2t'         ^      2     .     i     2i  +  2f 

3-n.5-n 1  +  2^-2^'-??      ^ 

^    2(i  +  5-n 2fi  +  3  +  2t+2t'-n' 

the  sign  of  integration  2  belonging  to  the  variable  f. 

Having  thus  the  part  of  V  due  to  the  term  a,'''  cos  i9'  in  the  expansion 

of  J'iaf,  if)  it  is  clear  that  we  may  thence  deduce  the  part  due  to  the 

analogous  term  J/'^  sin  i&  by  simply  changing  «/"  cos  iQ  into  J/''  sin  iO,  and 

consequently   we   shall   have   the   total  value  of  V  itself,  by  taking  the 

sum   of  the   various   parts   due   to    all    the  different   terms   which  enter 

into  the  complete  expansion  of  y(a;',  y'). 

11 3 

If  now  we  make  iS  =  — - —  and  recollect  that 

2 

sin 


the  foregoing  expression  will  undergo  simplifications  analogous  to  those 
before  noticed  (Art.  5.)     Thus  we  shall  obtain 

TT^a/"  ,        .^^„„M-1.«  +  1 n-k2t-3 


r/"  = "-:^ r'  cos  iQ .  2  r''' 

'         .     (n 
sin  I  - 
2 


n-1    \  2.4     2t' 

sin    — - —  IT 


(- 


n-1  .n  +  1 n  +  2i  +  2f-3      3-n.  5-n 1  +  2^-2^-  w 

'^      2     .     4     2i  +  2t'         ^2.4     2if-2^ 

or  by  writing  for  abridgment 

,.  ^       n-1. n  +  1 n  +  2t'-3      n-l.n  +  1 n  +  2i  +  2f-3 

"P^^'*^'-      2     .    4     2f  ""      2     .     4     2i  +  2t'        ' 

there  will  result  this  particular  value   of  /8 

,^(.  ^«/'> .         .^   „    ,„   3-n. 5-n l  +  2t-'2f-n    ^,.     „^ 

^'    =    .     (n-1     .^^osze.^r-^.      ^^^   ^^_^^,        .<p{t;t'), 


Mr  green,   on  THE   LAWS   OF   THE  EQUILIBRIUM   OF   FLUIDS.      55 
and  afterwards  by  making 

ro  =  ro»  +  r/'>  +  r.» + r «  +  r« + &c. 

we  shall  have 

TT* 

V^'^=  ,        r*  cos  i6  into  x 

/«  —  1 
sin 


sin  (-^vr) 


«<,'>.1.0(«;O) 


+  a<'>.?^.<^(e;0) +  «?>.!. 0(«;  1) .  r= 

+^'-  2.4.6  •<^(^;o)  +  «^"-2r-^— 0(^;i)-^ 

+  «^^^.0(«;  2).r'  +  af.l.cj>{i;3).f^ 

+  &C +&C +&C +&C 


Conceiving  in  the  next  place  that  F  is  a  given  rational  and  entire 
function  of  x,  y,  the  rectangular  co-ordinates  of  p,  we  shall  have  since 
X  =  r  cos  0,   y  =  r  sin  0. 

{25)  r=  C<")  +  C('>  cos  6  +  C-'^  cos  20  +  C('>  cos  3  0  +  &c. 

+  ^«  sin  0  +  ^(''  sin  29 +  E^'^  sin  30  +  &c. 

of  which  expansion  any  coefficient  as  C^'>  for  example,  may  be  still 
farther  developed  in  the  form 

C«  = ""'-^         {di\(p{i',  0)  +  c>{K(p{i;  l).r'+4^.(p{i;  2).r'  +  ke.}. 

sm  (-^  .J 

Now  it  is  clear  that  the  term  C>  cos  iO  in  the  developement  (25) 
corresponds  to  that  part  of  F  which  we  have  designated  by  F''',  and 
hence  by  equating  these  two  forms  of  the  same  quantity,  we  get 

F»  =  Cw  cos  ie, 


56      Mr  green,   on   THE    LAWS   OF   THE   EQUILIBRIUM    OF   FLUIDS. 

which  by  substituting  for  F"*''  and  C^  their  values  before  exhibited,  and 
comparing  like  powers  of  the  indeterminate  quantity  r  gives 

/>     ,      1-^      3  —  n,.     3  —  n.5  —  n,i.     3  —  n.5  —  n.7  —  n,.,      , 

2 
&iC.— &c &c 

of  which  system  the  general  type  is 

C<'>  =  (1  -  e)~  .  «« ; 

the  symbols  of  operation  being  here  separated  from  those  of  quantity, 
and  e  being  used  in  its  ordinary  acceptation  with  reference  to  the  lower 
index  u,  so  that  we  shall  have  generally 

f.m      „(i)  _  ^  (0 

The  general  equation  between  «!''  and  cll^  being  resolved,  evidently  gives 
by  expanding  the  binomial  and  writing  in  the  place  of  eci'',  e''&i\  ^c'i\  &iC. 
their  values  c„*j\ ,  cj-i\,  Cu%,  &;c. 

(26) ««=(l-e)^c«  =  c<;>+^c„«  +  ''~^-''~^ 


2     — '  ■       2 


...       «  —  3 .  ra  —  1 .  w  + 1     (i)    ,  s 
%+      2.4.6     "--  +  ^"- 


Having  thus  the  value  of  af  we  thence  immediately  deduce  the  value 
of  ^<''  and  this  quantity  being  known,  the  first  line  of  the  expansion 
(25)  evidently  becomes  known. 

In  like  manner  when  we  suppose  that  the  quantity  J5^'>  is  expanded 
in   a  series  of  the  form 

j5:«  =  — ^TTT  ^^»"-  *^  (^' '  0)  +  ^'* "^  (^' ;  1)  •  ^'  +  ^^'* <^  («■ ;  2) .  f^ + &c. ^ 

sin 


sin(^.) 


Mr  green,   on   THE    LAWS   OF  THE    EQUHJBRIUM   OF    FLUIDS.       57 
we  shall  readily  deduce 

A«=  (1  -  ef^e^^^  +  ^e.%+  ""'^'""l^  e.%  +  &c., 

and  ii,^  being  thus  given,  B'-'^  and  consequently  the  second  line  of  the 
expansion  (25)  are  also  given. 

From  what  has  preceded,  it  is  clear  that  when   V  is  given  equal  to 
any  rational   and    entire   function   whatever   of    x   and  y,  the   value  of 
f{x',  y')  entering  into  the  expression 

p={l-r'-^)-^.f{x',y'), 
will  immediately  be  determined  by  means  of  the  most  simple  formulas. 

The  preceding  results  being  quite  independent  of  the  degree  s  of 
the  function  f(x',  y)  will  be  equally  applicable  when  s  is  infinite,  or 
wherever  this  function  can  be  expanded  in  a  series  of  the  entire  powers 
of  x,  y',  and  the  various  products  of  these  powers. 

We  will  now  endeavour  to  determine  the  manner  in  which  one  fluid 
will  distribute  itself  on  the  circular  conducting  plane  A  when  acted 
upon  by  fluid  distributed  in  any  way  in  its  own  plane. 

For  this  purpose,  let  us  in  the  first  place  conceive  a  quantity  q  of 
fluid  concentrated  in  a  point  P,  where  /•  =  «  and  6  =  0,  to  act  upon  a 
conducting  plate  whose  radius  is  unity.  Then  the  value  of  V  due  to  this 
fluid  will  evidently  be 

g V' 

((^  —  9,ar  cos  Q  +  r^)~^ 

and  consequently  the  equation  of  equilibrium  analogous  to  the  one  marked 
(20)  Art.  10.,  will  be 

(27) const.  = ^ ^+  F; 

(«'-2«rcos  e  +  r^)~ 

V  being  due  to  the  fluid  on  the  conducting  plate  only. 

If  now  we  expand  the  value  of  V  deduced  from  this  equation,  and 
Vol.  V.    Part  I.  H 


58      Mr  green,  ON  THE  LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS. 

then  compare  it  with  the  forrnulag  (25)  of  the  present  article,  we 
shall  have  generally  E^^  =  0,  and 

C"'=-2ga-^.l^(r;O)+0(e;l)^+«^(/;  2)  ^ +,^  («;  3)  J  +  &c.^ 

except  when  i  =  0,  in  which  case  we  must  take  only  half  the  quantity 
furnished  by  this  expression  in  order  to  have  the  correct  value  of  C*"'. 
Hence  whatever  u  may  be, 

2  sin  I  ^-Q—  T I 
^  -  0,       and  cf  = ^^ qa}  -"-*-^" ; 

TT 

the  particular  value  f=0  being  excepted,  for  in  this  case  we  have  agreeably 
to  the  preceding  remark 


sin 

(¥') 

-i 

a}-n-2u^ 

■w' 

and  then  the  only  remaining  exception  is  that  due  to  the  constant 
quantity  on  the  left  side  of  the  equation  (27).  But  it  will  be  more 
simple  to  avoid  considering  this  last  exception  here,  and  to  afterwards  add 
to  the  final  result  the  term  which  arises  from  the  constant  quantity  thus 
neglected. 

The  equation  (26)   of  the  present    article  gives   by   substituting  for 
d"  its  value  just  found. 

^    .     (n-l    \ 
2  sm  I—- — tt) 

«»= l^f L  qa'"-'-'".  {1  +  '^.a-' 

n  —  3.n-l        ^,  n  —  3.n-l.n-l        .,  .     , 
2.4  2.4.6  ' 

^   •     /«-! 
2  sm 


(n-1    \ 

l^i i  qa'-'-'-"-  (1  -  a-')— 


2  sin  I -—^ 


TT^ 


Mr  green,  on   THE  LAWS  OF   THE   EQUILIBRIUM   OF   FLUIDS.       59 
and  consequently, 


sin  (^.) 


.     (n-\ 

2  sin    I  —7:        /r  I  3_^  ,2  ,4 


;ii^^«--(«=-ir.''(i-5)"' 


•*    sin      1  ^  "I  B-B  „M 

q{a'-V)—{a'-r'y\-, 

the  particular  value  -4'°^  being  one  half  only  of  what  would  result  from 
making  i  =  0  in  this  general  formulje. 

But  4''  =  0  evidently  gives   £^''^=0,   and  therefore  the  expansion  of 
f{a!,  y')  before  given  becomes 

fix',  y')  =  J^'^  +  A^'^  cos  ff  +  A-'^  cos  20'  +  ^''^  cos  30'  +  &c. 

= 1-^ -g(«^-l)  ^  («=-r'0-'.{|  +  -  cos0'+  -cos20'  +  &c.| 

or  by  summing  the  series  included  between  the  braces, 

.     (n  —  1    \  3-» 

JKx,y)-  -^  ^a^-2«r'cos0'  +  r"' 


sin  (^  .) 


Q 


iJ  being  the  distance  between  P,  the  point  in  which  the  quantity  of 
fluid  q  is  concentrated,  and  that  to  which  the  density  p  is  supposed  to 
belong. 

Having  thus  the  value  of /(a;',  y')  we  thence  deduce 

(n  —  1  'N  3-, 


p  =  (1  -  /»)-/(x',  y')  =  -  — i-| i  (1  -  /')  ^  ?  — ^. 


sin  1 — ::: — TT I  „_3     (a^ —  Yy^ 


H  2 


60      Mr  green,   on   THE   LAWS   OF   THE   EQUILIBRIUM   OF   FLUIDS. 

The  value  of  p  here  given  being  expressed  in  quantities  perfectly 
independent  of  the  situation  of  the  axis  from  which  the  angle  6'  is 
measured,  is  evidently  applicable  when  the  point  P  is  not  situated  upon 
this  axis,  and  in  order  to  have  the  complete  value  oi  p,  it  will  now 
only  be  requisite  to  add  the  term  due  to  the  arbitrary  constant  quantity 
on  the  left  side  of  the  equation  (26),  and  as  it  is  clear  from  what  has  pre- 
ceded, that  the  term  in  question  is  of  the  form 

n-3 

const.  X  (1  -  /')  2  , 
we  shall  therefore  have  generally,  wherever  P  may  be  placed. 


P  =  (l-r-) 


1-3 


-     [n-l     \                   3.„ 

const.  - 

^-       7?-'       1 

The  transition  from  this  particular  case  to  the  more  general  one, 
originally  proposed  is  almost  immediate :  for  if  p  represents  the  density 
of  the  inducing  fluid  on  any  element  dai  of  the  plane  coinciding  with 
that  of  the  plate,  p^da-i  will  be  the  quantity  of  fluid  contained  in  this 
element,  and  the  density  induced  thereby  will  be  had  from  the  last 
formula,  by  changing  q  into  pidai.  If  then  we  integrate  the  expression 
thus  obtained,  and  extend  the  integral  over  all  the  fluid  acting  on  the 
plate,  we  shall  have  for  the  required  value  of  p 

p=(l-0^    .jconst. \f        ^  fp^da^"    J^      }; 

B  being  the  distance  of  the  element  dai  from  the  point  to  which  p  belongs, 
and  a  the  distance  between  da^  and  the  center  of  the  conducting  plate. 

Hitherto  the  radius  of  the  circular  plate  has  been  taken  as  the  unit 

of  distance,  but   if  we   employ  any   other   unit,  and   suppose   that  b  is 

the   measure   of    the   same  radius,   in   this   case   we   shall   only   have  to 

.^     a     r'     d(Tx        ,  R  .      ,,        ,  „  ,     , 

write  ^  >   ^ '    -^  and  -g-  m  the   place   of  a,  r,  da,  and  R  respectively, 

recollecting  that  -^  is  a  quantity  of  the  dimension  0  with  regard  to  space, 
by  so  doing  the  resulting  value  of  jo  is 


Mr  green,   on   THE  LAWS  OF  THE   EQUILIBRIUM   OF   FLUIDS.      61 


(n-1 

sin  I  -— —  TT 


(28). 


.p  =  {]r-r')'  .|const. Jp.dcr,- — ^^] 


By  supposing  w  =  2,  the  preceding  investigation  will  be  applicable 
to  the  electric  fluid,  and  the  value  of  the  density  induced  upon  an 
infinitely  thin  conducting  plate  by  the  action  of  a  quantity  of  this 
fluid,  distributed  in  any  way  at  will  in  the  plane  of  the  plate  itself 
will  be  immediately  given.  In  fact,  when  n  =  2,  the  foregoing  value  of 
p  becomes 

1  7     ,     y/a'-b'] 


^  =  7ltptHst--^/^'^<^' 


B' 


If  we   suppose   the   plate   free   from   all  extraneous  action,   we  shall 
simply  have  to  make  pi  =  0  in  the  preceding  formula;    and  thus 

,„^,  const. 

(29) p  = 


Vb'-r"' 


Biot  (Traite  cle  Physique,  Tom.  ii.  p.  277.)>  has  related  the  results  of 
some  experiments  made  by  Coulomb  on  the  distribution  of  the  electric  fluid 
when  in  equilibrium  upon  a  plate  of  copper  10  inches  in  diameter,  but 
of  which  the  thickness  is  not  specified.  If  we  conceive  this  thickness 
to  be  very  small  compared  with  the  diameter  of  the  plate,  which  was 
imdoubtedly  the  case,  the  formula  just  found  ought  to  be  applicable 
to  it,  provided  we  except  those  parts  of  the  plate  which  are  in  the 
immediate  vicinity  of  its  exterior  edge.  As  the  comparison  of  any 
results  mathematically  deduced  from  the  received  theory  of  electricity 
with  those  of  the  experiments  of  so  accurate  an  observer  as  Coulomb 
must  always  be  interesting,  we  will  here  give  a  table  of  the  values  of 
the  density  at  different  points  on  the  surface  of  the  plate,  calculated 
by  means  of  the  formula  (29),  together  with  the  corresponding  values 
found  from  experiment. 


62         Mr  green,  ON  THE  LAWS  OF  THE  EQUILIBRIUM  OF  FLUIDS. 


Distances  from  the 
Plate's  edge. 

Observed 
densities. 

Calculated 
densities. 

5  in 

4 

3 

1, 

1,001 

1,005 

1,17 

1,52 

2,07 

2,90 

1, 

1,020 
1,090 
1,250 
1,667 
2,294 
infinite. 

2 

1 

,5 

0 

We  thus  see  that  the  differences  between  the  calculated  and  observed 
densities  are  trifling;  and  moreover,  that  the  observed  are  all  something 
smaller  than  the  calculated  ones,  which  it  is  evident  ought  to  be  the 
case,  since  the  latter  have  been  determined  by  considering  the  thickness 
of  the  plate  as  infinitely  small,  and  consequently  they  will  be  somewhat 
greater  than  when  this  thickness  is  a  finite  quantity,  as  it  necessarily 
was  in  Coulomb's  experiments. 

It  has  already  been  remarked  that  the  method  given  in  the  second 
article  is  applicable  to  any  ellipsoid  whatever,  whose  axes  are  a,  h,  c. 
In  fact,  if  we  suppose  that  x,  y,  %  are  the  co-ordinates  of  a  point  p 
within  it,  and  x',  y',  z'  those  of  any  element  dv  of  its  volume,  and 
afterwards  make 

X  =  a. cos  9,      y  —  i.sin  6  cos  w,      a  =  c.sin  6  sin  ■sr, 
x'=  a. cos  ff,    y'  =  J. sin  9'  cos  w',     z'=  c.sin  9'  sin  w', 
we  shall  readily  obtain  by  substitution. 


l-n 
2     . 


■  V=abcf p. r'^dr'd&d-ar' si-n  9'. {Xr"- 2 nrr'-¥vr'^) 
the  limits  of  the  integrals  being  the  same  as  before  (Art.  2.),  and 
\  =  «^  cos  9^  +  If  sin  9^  cos  ts^  -\-  &  sin  0-  sin  Tsr^, 

ft.  =  a^  cos  9  cos  9'  +  U  sin  9  sin  9'  cos  tb-  cos  in-'  ■\-<?  sin  9  sin  9'  sin  •ar  sin  w', 
V  =  «'  COS  9"  +  ¥  sin  9'^  cos  sr'^  +  e  sin  9'^  sin  ■ar'^ 


Mr  green,    on    THE   LAWS   OF   THE    EQUILIBRIUM   OF  FLUIDS.       63 

Under  the  present  form  it  is  clear  the  determination  of  V  can  offer 
no  difficulties  after  what  has  been  shown  (Art.  2.).  I  shall  not  there- 
fore insist  upon  it  here  more  particularly,  as  it  is  my  intention  in  a 
future  paper  to  give  a  general  and  purely  analytical  method  of  finding 
the  value  of  V,  whether  p  is  situated  within  the  ellipsoid  or  not.  I 
shall  therefore  only  observe,  that  for  the  particular  value 

(30) ,  =  ^\^-ii^-%-i]'  =  ^^(^ -'") '  ' 

the  series  Uo  +  U2'  +  U/  +  &c.  (Art.  2.)  will  reduce  itself  to  the  single 
term   Uo,  and  we  shall  ultimately  get 

2sin("— .) 

which  is  evidently  a  constant  quantity.  Hence  it  follows  that  the  ex- 
pression (30)  gives  the  value  of  p  when  the  fluid  is  in  equilibrium 
within  the  ellipsoid,  and  free  from  all  extraneous  action.  Moreover, 
this  value  is  subject,  when  n  <  2,  to  modifications  similar  to  those  of 
the  analagous  value  for  the  sphere  (Art.  7.)- 


G.  GREEN. 


11.  On  Elimination  between  an  Indefinite  Number  of  Unknown  Quantities. 
By  the  Rev.  R.  Muephy,  M.  A.  Fellow  of  Cuius  College,  a?id  of 
the  Cambridge  Philosophical  Society. 


[Read    Nov.  26,    1832.] 


SECTION   I. 

INTRODUCTION. 

Fourier,  in  his  treatise,  *  Theorie  de  la  Chaleur,'  *  has  given  an 
example  of  the  determination  of  an  indefinite  number  of  unknown 
quantities,  subject  to  the  same  immber  of  conditions.  If  n  be  the 
number  of  those  quantities,  in  order  to  discover  their  law  by  this 
method,  it  will  be  necessary  to  eliminate  successively  the  first  (ni-  1) 
and  the  last  («  — »^)  unknown  quantities,  thus  determining  the  »^'^  by 
a  final  equation  containing  that  quantity  only. 

This  process  is  obviously  too  laborious,  and  the  results  too  compli- 
cated, to  be  practically  useful,  in  most  cases. 

The  same  objection  applies  to  the  elegant  method  of  Laplace,  which 
makes  the  determination  of  one  of  the  unknown  quantities,  depend 
on  the  discovery  of  all  the  (w  — 1)  arbitrary  multipliers  introduced  in 
the  process.  It  has  besides  the  disadvantage  of  not  seizing,  in  many 
cases,  the  facilities  offered  by  the  peculiar  forms  of  the  proposed  equa- 
tions. 

•  Vid.  Fourier,  p.  1 69  to  174- 
Vol.  V.    Paet  I.  I 


G6  mk  murphy,  on  elimination  between  an  indefinite 

In  the  physical  investigations,  which  conduct  to  an  indefinite  num- 
ber of  equations,  it  is  of  great  importance  to  discover  the  law  of  those 
quantities,  corresponding  to  the  law  by  which  the  given  equations  are 
connected.  The  method  which  I  here  propose  for  this  object  is  founded 
on  the  two  following  principles. 

First,  if  we  make  the  right-hand  member  of  the  a;*''  equation  dis- 
appear by  transposition,  the  left-hand  member  is  then  a  function  of  x, 

which  vanishes  when  x  is  any  number  of  the  series  1,  2,  3, w;   and 

therefore  it  must  be  of  the  form 

P.(.r-  1)  {x-2)  (x-S) (x-n). 

Secondly,  if  an  identity  exist  between  two  formulas  which  are 
partly  integer,  partly  proper  algebraic  fractions  (of  which  the  numerators 
are  of  lower  dimensions  than  the  denominators)  the  integer  and  fractional 
parts  are  separately  equal. 

To  demonstrate  this  principle,  let 

represent  such  an  identity,  where  each  symbol  denotes  an  entire  function 
of  X,  and  the  dimensions  of  P,  P'  are  respectively  lower  than  those  of 
Q,  Q';  then  we  have 

(N-N)QQ  =  PQ- PQ'. 

If  therefore  N—jV'  be  not  identically  nothing,  we  shall  have  the 
entire  function,  represented  by  the  left-hand  member,  identical  with  one 
of  lower  dimensions ;  but  this  is  impossible,  because  in  integer  formulae 
we  may  equate  like  powers  of  x,  hence  we  must  have  iV=iV'  and, 
therefore  also, 

Z!  -  ^ 

Q-  Q' 

By  means  of  this  principle,  we  shall  be  able  to  expand  a  given 
entire  function  P,  in  terms  of  other  given  functions,  whenever  such  an 
expansion  is  possible. 


NUMBER  OF   UNKNOWN  QUANTITIES.  67 


SECTION   II. 

Application  of  the  First  Principle. 

The  first  principle  alone  is  sufficient,  in  a  great  number  of  instances, 
to  resolve  the  proposed  equations ;  we  shall  illustrate  its  application  by 
selecting  three  distinct  classes  of  equations  to  be  resolved. 

First,  when  the  terms  which  compose  the  general  or  a;**"  equation  are 
proper  fractions. 

Example  : 

To  find  the  values  of  the  n  unknown  quantities  ssi,  %i,  sss, s,,  sub- 
ject to  the  n  equations  following, 

»1  «2  ^  »„  ^    _    1 

3  4  "*■  5  "^ »  +  2  2' 

«i    ,   ^  ^  ,  g«     =_  1 

4  "*"  5        6  "^ «  +  3  3* 


»  +  l       «  +  2       ra  +  3      2w  »■ 

The   general,  or  a;**"  equation,  when  its  right-hand  member  is   trans- 
posed, becomes 

-  +  — Y  — -^ \- H —  =0. 

a;      x+1       x  +  ^  x  +  n 

N 
Suppose  these  fractions  are  actually  added,    and  let  -^  represent  the 

sum;  where  D  =  x{x-\-\)(x-\-9l) {x  +  n)  and  A''  is  some  function  of  x 

of  n  dimensions.  .  ,-  •       .• 

i2 


68  Mr  murphy,   ON    ELIMINATION   BETWEEN   AN    INDEFINITE 

Hence  we  have  7^  =0,  and  therefore  iV=0,  provided  a;  is  any  num- 
ber of  the  series  1,  2,  3,....ti  and  consequently  iV  (which  is  of  ?i  dimensions) 

has  a   factor   (^  —  1)   U  — 2) (x  —  ti);    and  can   therefore   admit  of  no 

other  factor,  but  a  constant  c. 

Hence  we  have  in  general, 

^^ X      x  +  1       x  +  2      x+n      x(x+l){x  +  2)  {x  +  3)...{x+n)' 

Multiply  this  equation  by  x,  and  then  put  x  =  0,  hence  c  =  (  —  1)\ 

Multiply  the  same  by  x  +  1,  and  then  put  x  =  —I; 

,  n   n  +  1 

hence  ssi  =  -  - .  — - — . 

Similarly,  multiply  by  x  +  2,  and  put  x=  —2, 

_  n.{tt—l)    {n  +  !)(«  +  2) 
•  •  '^  ~  1  .      2      •       1      .     2      ' 

and   generally,  if  we  multiply  equation    (a)   by   x  +  m,    and   then    put 
x=  -m,  we  get 

'"'^       '  'I.       2.3     m         '        1.2     m     ' 

It  is  clear  from  this  example,  that  if  the  general  or  x^^  equation  were 

a+bx^  a'  +  h'x  ^  d'\¥x  ^ «<"'  +  i<"'x  ~  "' 

we  should  find  the  sum  of  the  fractions  composing  the  left-hand  member 
to  be 

c  .{x—\){x  —  2)         {x  —  n) 

(a  +  hx)  [a  +  h'x)  («"  +  h"x) (a"  +  i^a;) ' 

then  multiplying  by  n  +  bx  and  putting  x=  —  j,  we  should  find  e, 

-I 
multiplying  by  a'+b'x  and  putting  .r=-,,,  we  should  find  s,, 

&c &c &c 


NUMBER  OF    UNKNOWN   QUANTITIES.  60 

In  the  example  above  taken,  we  have  supposed  that  the  number 
of  equations  and  unknown  quantities  were  the  same,  but  if  we  supposed 
that  following  the  same  law  as  in  that  example,  the  number  of  equa- 
tions were  n  +  m,  then  the  numerator  N  which  was  shown  to  be  of 
n  dimensions,   ought    to   vanish   when   x   is    any  number   of  the   series 

1,  2,  3 n  +  m;  that  is,  the  equation  A^=0  has  more  roots  than  it  has 

dimensions,  which   is   impossible ;   it   is  therefore   equally   impossible   to 
satisfy  all  the  given  equations. 

On  the  other  hand,  if  the  number  of  the  given  equations  was 
only  n  —  m,  then  n  would  by  the  preceding  reasoning  have  a  factor 

{x  —  l){x  —  2) {x-7i-{tn), 

and  since  it  is  of  n  dimensions,  it  must  have  another  factor  of  m  dimen- 
sions, as  C  {x  -  a^)  {x  —  a-i) (x  — a„). 


Hence    -  -\ — ^  H "*  ^  ^- 


X      x  +  1       x  +  2  x  +  n 

_  C(x—'l){x—2) .{x~  tl  +  m){x  —  ai)  jx  —  a-^ {x~a,„)  ^ 

■~  ''"'xT{x  +  l){x  +  2) [x  +  n)  ' 

following  now  the  same  steps  as  before,  we  find 

^^      cj-iy.a.az g.  .    g^(     ly  "■('»-^) jn-m  +  i) 

«.(«  — 1) {n-m  +  1)'      ''  '   '       ai.a-i      a,„ 

c(-l)".(l+ai)(l+a.> (Ifg.)  ^  (l+aiXl+aa) (!+«„,)    w    n-m  +  l 

'~     (w-1)(m  — 2) {n  —  m  +  2)  o,     .    a^     a,„      '1'         1 

^.    .,    ,  (2  +  a,)(2+a,) {2+a^)    n.{n  +  l)    (ti-m+l)  {n-m+2) 

Similarly,   %■^=  -  ~ —  •      .,    ^      • i ^ • 


The   quantities  a„  a.^ a„  are  evidently  arbitrary,  and  each  of  the 

required  quantities  »,  z-.,  he.  x„_„,  are  here  determined  in  such  a  manner, 
as   to   contain   the   m    arbitrary   constants.      This   is   therefore   the   most, 
complete  solution  of  the  problem. 


70  Mb  murphy,   ON   ELIMINATION   BETWEEN   AN    INDEFINITE 

Another  useful  observation  may  be  made  in  this  place ;  if  the  function 
which  represents  the  a;"'  eqvxation  were  discontiiuious,  i.  e.  if  any  of  the 
equations,  for  instance  the  second,  were 

3   +  5    '  7       ~        ' 

2        2        2 

and  consequently  an   exception  to  the  general  law  expressed  by  the  x^^ 

equation,  we  should  have  then  N—0  when  x=\,S,  4 w,  also  when 

A'  =  ^,  but  not  when  a;  =  2,  hence  in  this  case, 

iV=c.  (ar-i)(x-l)  {x-S)  (x-4) {x-n); 

after  this  the  remainder  of  the  process  would  be  the  same  as  before. 

We  have  been  thus  particular  about  the  preceding  example,  as  being 
well  calculated  to  shew  the  spirit  and  advantages  of  the  present  method. 

The  next  class  of  equations,  which  may  be  solved  by  the  first  principle 
alone,  consists  of  those  in  which  the  terms  composing  the  a;*''  equation 
contain  common  factors ;  for  if  we  then  assign  to  x  such  values  as  may 
successively  cause  such  factors  to  vanish,  the  unknown  quantities  will 
be  determined. 

Example : 

To   find    the    values   of  asj,  %2,   %^ a,  subject   to   the  n   equations 

following;  viz. 

a:, +  1 ,2.S!2  +  1.2.3.S53+ +  1.2.3 «s!„=  -1, 

2s!,  +  2.3.a!2  +  2.3.4.X3+ +  2  .  3  .  4...(w  +  l)x„= -1, 

3a,  +  3.4.&,  +  3.4.5.«3  + +  3  .  4.  5...(w+2)a;„=  -1, 


n8Sl  +  ?i(w  +  l)8:2+M(w  +  l)(w  +  2)S83+ +  W  (w  +  1)  («  +  2)...2W2!„=  -1. 

If  we  transpose  the  right-hand  member  of  the  above  equations,  the 
.r""  or  general  equation  becomes 

\  +  x%,+  X  {x  +  l)%.,^  X  {x-\-\){x  +  ^) .%;  + +a;(d;+l)(a:+2)...2x.!£„  =  0. 


NUMBER   OF    UNKNOWN   QUANTITIES.  71, 

This  equation   is  evidently    of  w  dimensions  with  respect  to  x,  and  its 

roots   by   the    first    principle   are  1,    2,  3 n;    the    left-hand    member 

must   therefore  be   identical   with   the  product 

c{x-l\{x -2){x-S) (x-ti), 

whatever  value  may  be  assigned  to  x. 

i  —  lY 

Put  therefore  x  =  0.     Hence  c  =  - — ^ ' , 

1.2.3 « 

X  ^^     X.»..>........V[    ^^     -^   fif 


^-~" 2a—  ,g  ^  , 


n  .  (w  —  1) 

1\¥ 


_  _      «(w-l)(w-2) 

* ^ a:,--         1222.32 

&c &c 

and  generally,  «,„  =    ^^   ^     .     3  ^)^~  •(-!)• 

We  may  verify  this  result  by  observing,  that  if  we  substitute  this 
quantity  for  25,„  in  the  general  or  x^^  equation,  then  its  left-hand  member 
becomes 

n  .  (w-1)    X .  (x  +  l)       w.(w-l)(w-2)    ^(£+_l)_(a;  +  2)      ,  ' 
^"■"'^'*"       1.2      •~T:2  1.2     .     3      •      1.2     7    3    ^*'''- 

This  quantity  is  evidently  the  part  which  does  not  contain  h  in  the 
product, 

f,         ,      x{x  +  l)    ,„     x{x  +  l)ix  +  2)    ,3,o„l       f,       n  ,  n{n-l)    1  1 

or  in  (l-//)-'.(l-|)". 

it  is  therefore  the  coefficient  of  //"  in  the  expansion  of 

But  this  coefficient  is  manifestly  0  when  x  is  any  positive  integer,  which 
evidently  agrees  with  the  proposed  conditions. 


72  Mr  murphy,   on   elimination   BETWEEN   AN   INDEFINITE 

Another  class  of  equations  which  may  easily  be  resolved  by  the 
first  principle,  occurs  when  the  x^^  equation  is  of  n  dimensions,  and 
arranged  according  to  the  powers  of  some  function  of  .r ;  it  is  then 
merely  necessary  to  expand 

c.a:{x  —  l)(x-Q) (x  —  n) 

according   to   the   powers   of  that  function ;    and  equate  the  coefficients 
of  like  powers  in  both  cases. 

Example: 

Ki   +        SSs  +        «3  + +        SS,  =  -   1, 

2a!,  +  2-S2  +  S'xs  + +  2»ss„  =  -  1, 

Sz,  +  3-S5,  +  3^X3  + +  3'%„  =  -  1, 


w^i  +  w'asa  +  ?r%3  + +  n''z^  =  —  1, 


to  find  «„  %2 


The  general  or  «""  equation  in  this  case,  is 

1  +  x»i  +  x"%i  + +  afz^  =  0, 

the  roots  of  which  equation  are  x  =  l,  2,  3 ;/. 

Hence,  the  left-hand  member  is  identical  with  the  product 

c.{x-l){x-2){x-3) (x-tt), 

or     c(-iy{S„-xS„.,  +  x'S,_,- (-l)".x"|, 

where  S„  denotes  the  sum  of  the  quantities   1,  2,  3 n  when  taken 

in  products  m  and  m  together. 

Hence,  by  equating,  we  get 

c(-lY   S      =1-         •    r  =  \^y     .. 

-  c(-1)".aS'„_,=  «,;       .-.  x,=  -aS'_,; 
c(  —  1)" .«>„_2=  »2;       .•.  SS2=— 0,2; 
and  generally  £.,  =  S,,, 

where  S.„  denotes  the  sum  of  the  reciprocals  of  the  quantities  of  which 
a9„  represents  the  sum. 


NUMBER   OF,  UNKNOWN  QUANTITIES,  73 

SECTION   III. 

Amplication  of  the  Second  Principle. 

To  expand  a  given   function  of  x  as  P,  in  terms   of  other  given 
functions 

Qo,  Q.,  Q. Qn, 

all  being  supposed  of  n  dimensions  in  x. 

Let  P=aoQo  +  «iQi  +  «2Q2+ +«nQ», 

where  a^,  ffi,  Oa «»  are  constants  to  be  determined. 

Divide  all  the  functions  by  Qo,  and  let  the  corresponding  quotients 
be  respectively 

P',  Qo,  Q'l,  Q.....Qn, 
and  the  remainders 

p',  g^o,  q\,  q'i q\- 

Then  by  attending  to  the  second  principle,  we  have 

P'  =  «oQ'o  +  «lQ'l   +  (kQ2+ +«„Q'n, 

p'  =aoq'o  +  aig-'i  +  «25''2  + +a„q'n, 

when  we  obviously  have  Q'o=l  and  §''0=0. 

Dividing  the  last  equation  by  q'l  and  using   a   similar   notation,  we 
get  in  like  manner 

P'=«.Q".  +  «2Q"2  + ««Q"„, 

p"=  aiq"i  +  (hq"2  + anq\, 

where  Q"i  =  l  and  q'\  =  0. 

Divide  the  equation  last  obtained  by  q"i,  and  we  obtain 

P"'  =  a,Q"',+ +a„Q\, 

p"'==a»q"',  + +«„^"„, 

in  the  latter  of  which  equations  the  first  term  =  0  and  in  the  former 
it  equals  unity. 

Vol.  V.    Pakt  I.  K 


74  Mr  murphy,  ON  ELIMINATION  BETWEEN  AN  INDEFINITE 

The  systems  of  the  first  equations  thus  obtained  may  be  written  in 
an  inverse  order  thus, 

&c.  = &;c 

whence  «„,  «„_i,  a„_2,  &c.  are  successively  known. 

We    have    supposed   all   the   functions   to  be   of    n   dimensions,   for 
these  necessarily  comprise  all  of  lower  degrees. 

Example  : 
To  expand  unity  in  terms  of  the  functions 

af,    {x-^hy,    (a;  +  2A)", (a;  +  «A)°. 

Put  l=«o*"  +  «i(^ +  ^0°  + '''^(^  +  2^)"  + +  «„(a;+wA)";  dividing  by 

a;",  we  get 

0  =  ao  +  ffi      +«2      + +  «„, 

1=         ai?'i  +  Oag-'s  + +  «»<?'», 

where  we  have  o'„  =  A.  {waf"*.»i-l — ,    ^  'hm^af-^+ \. 

^  1.2 

Divide  now  by  g'l  and  we  obtain 

0  =  ai  +  2a2     + +  wa», 

1=  «2g'"2+ +  a„g^'„, 

where  in  general    g'"„  =  A^  |— ^ — ^— ^  a;""*  {m^  —  m)  +  &c.  > . 

This   process   is    easily   continued,    and   we   obtain   successively   the 

equations 

0  =  1.2a2  +  2.3a3  + (»— 1).  w«„, 

0=  1.2.  3  03+ {n- 2)  (71-1)  nan. 


and  lastly,  ^=  1.2.3 «a„ 


NUMBER  OF  UNKNOWN  QUANTITIES.  75 

From  these  equations  taken  in  the  inverse  order,  we  get 

^ 1 

"'     ~  1.2  .3 nh"' 


«„-!=  -  na, 


n> 


„     _n(n-\) 

1  .  2 
&C.= &C 

Hence  the  required  expansion  is 

To  apply  this  principle  to  equations,  we  may  observe  that  when 
the  general  or  a;"'  equation  is  cleared  of  fractions  and  its  right-hand 
member  transposed,  it  is  of  the  form 

-P+  XiXi  +^2X2  + +i8„X„  =  0, 

where    ssi,    sss a,    are    the   unknown    quantities,   and  P,  Xx,  X^.... 

known  functions  of  x. 

The  left-hand  member  must,  by  the  reasoning  of  the  preceding 
Section,  be  divisible  by  («  — l)(x— 2) {x—n). 

Let  Xi,  Xi,  &c.  when  divided  by  this  quantity  leave  the  re- 
mainders Q'l,  Q'2,  &c.   and  P,  the  remainder  P',  hence 

where  all  the  functions  are  necessarily  of  less  than  n  dimensions,  the 
application  of  the  process  above  described,  would  then  determine  the 
quantities  ssi,  asj, »„. 


R.  MURPHY. 


Caics  College, 
March  5,  1833. 


K« 


III.     On  the  General  Equation  of  Surfaces  of  the  Second  Degree. 
By  Augustus  De  Morgan,  of  Trinity  College. 


[Read  Nov.  12,  1832.] 


The  present  investigations  are  a  continuation  of  those  upon  lines 
of  the  second  degree,  published  in  Vol.  IV.  Part  I.  of  these  Transactions. 
I  have  omitted  various  algebraical  developments,  as  unnecessary,  and 
tending  to  swell  this  communication  to  a  length  more  than  proportional 
to  its  importance. 

As  the  theory  of  the  reduction  of  oblique  to  rectangular  co-ordinates 
is  a  very  necessary  part  of  what  follows,  I  proceed  first  to  give  the 
equations  which  will  be  required  under  this  head.  Let  x,  y,  %,  be 
oblique,  and  x',  if,  a'  rectangular  co-ordinates  to  the  same  point,  with 
a  common  origin.     Let  the  angles  made  by  the  first  system  be 

A  A  A         ^ 

y%  =  ?,       %x  =  t),      xy  =  ^, 

and  let  the  rectangular  and  oblique  co-ordinates  be  so  related  that 

AAA 
COS  xsd  =  a,      COS  yx'  =  /3,      cos  xyf  =  a',  &c. ; 

whence  the  following  equations: 

a/  =  ax   +  fiy  +  yx, 

y'  =  a'x  +,  /3'y  +  y'z (1), 

S8'  =a"x  +  fi"y  +  7"i8; 

l  =  a'+a"  +  a'",  COS  ?  =  /37  +  (i'y'  +  fi"y'\ 

l=l3f>+  fi'^+  fi'%  COS  t,  =ya  +  y'a!  +  y"a" (2), 

1  =  y  +  y^  +  y%  cos  ^  =  a/3  +  a'/3'  +  a")8". 


78 


Mr  DE   morgan   ON   THE   GENERAL   EQUATION   OF 


Make  the  following  abbreviations,  to  which,  for  facility  of  reference, 
are  annexed  those  which  will  afterwards  appear  in  treating  the  general 
equation  of  the  surface, 

aaf  +  bif-ir  cz"  +  2ai/z  +  2bzx  +  2cxy  +  2aa?  +  2%  +  2cs!  +/  =  0 (3), 

the  co-ordinates  of  the  center  of  which  call  X,  Y,  and  Z.  Throughout 
this  paper,  all  subscript  indices  indicate  the  dimension  of  the  quantity 
signified,  in  terms  of  the  coefficients  of  (3) : 


p  =/3'7"-/3"7', 
/  =  /3"7-/37", 
p"-=^y'  -d'y. 


y'a"-y"a'. 


^1  ff  If 

q  =  y  a—ya  , 
q"=  7«'  -7'«' 


r  =a'/3"-a"/3', 
t"  =  a"(i-a(i".. 
r"=«/3'  -a' 13. 


(4), 


a^^=  be  — a', 
b^,  =  ca-  b% 
c„  —  ab  —  (?. 


tto  =  sin'  I, 
b^=  sin'*;., 
Co  =  sin^  ^. 


.(5), 


a^=  6  +  c  — 2acos^, 
b,=  c  +  a  —  2b  cos  rj, 
Cf  =  a  +  b  — 2c  cos  ^, 

l^^  —bc-aa,    I,  =  &cos^+ccos»7-«  — acos^,    \  =  cos  v  cos  ^— cos  ^, 
m,  =  ca  —  bb,     mj=:  ccos^  +  acos^—b  -bcosrj,    7»o=  cos^cosf — cos  ^...(6), 
91^1  =■  ab—  cc,     n,=a  cos  j?  +  6  cos  ^—c~c  cos  ^,    «„  =  cos  ?  cos  tj  —  cos  ^. 

=»?„«o-ao  4-^  cos  A 

=  «o  lo-boMo-T- COSri\...(7), 

.=  /o««o-Co«o-T-cosg 

(8), 

(9), 


Fo=l+2cos^cos.jcos^-cos^^-cos'»j-cos'^' 


V,  =  aai  +  bb^-\-cCa  +  2ala  +  2bma  +  2cnf, 

Vi  =  a„  +  b^,  +  c„  +  2/,,  cos  '^■\-2m„  cos  r\-\-<iLn„  cos  ^ 


=  */;C//-/,;    ^«1 


.(10), 


Vz~abc^2(ibc  —  a<i  —  b¥  —  c&  \  =.c„a„—m,J-^b\ 

r;  =  a,,a'  +  J,,6^  +  c,,c^  +  2/,,6c  +  2»?,,ca  +  2w,,«F (11), 


=  m„n„-a„l„  -=ra 

■.nj„  -b„m„^b 

=  l„m„-c„n„  ^c 


W=-^  +/=  aX+  6  F+cZ+/ 


.(12). 


SURFACES  OF  THE   SECOND  DEGREE.  79 

From  (4),  we  find  by  inspection  that  the  following  six  quantities 
are  severally  equal: 

pa       +  qft      +  ry,  pa  +  p'a'   +  p"  a'\ 

p'a'    +  g^/3'    +  r'y',  qfi  +  ^/3'  +  q"fi" (13), 

p"a"  +  q"(i"  +  r"7",  ry  +  r'y'  +  r"y", 

and  moreover,  that  any  symmetrical  interchanges  of  accents  in  the  first 
three,  or  of  letters  in  the  second,  give  results  severally  equal  to  nothing. 
Such  are  joa'  +  g'iS'  +  ry,  p li  +  p  (i' +  p" fi" ,  &c.  Let  the  common  value 
of  the  first  six  be  T.     We  have  then 

pa  -{■  q&  +  ry  =  T, 

pa  +ql3'+ry'=0 (14), 

pa"  +  qli"  +  ry"  =  0. 

From  which,  by  obvious  multiplications  and  additions,  looking  at  equa- 
tions (2),  we  have 

p  +3'  cos  (^+r  cos  ri=Ta, 

pcos  ^+q  +r  COS  ^=T(i (15), 

p  cos  t]  +q  cos^+r  =  Ty. 

From  either  of  which  sets  we  deduce 

1^ ■¥<f  -Vi^-^^qr  cos  0  +  2rp  cos  n-\-^pq  cos^=  T^ (16), 

and  similar  relations  may  be  deduced  between  jo',  g',  r',  and  jt>",  ^",  r' ; 
T  being  the  same  throughout. 

Again,  form  the  several  quantities 

flo,    /o,   &c.    or     1  -  cos^  f,     cos  n  cos  ^-  cos  f,    &c. 

from  the  second  set  of  equations  in  (2),  and  make  the  results  homo- 
geneous and  symmetrical  from  the  first  set;  for  example,  write  for 
Oa  and  /o 

(7«  +  V«'  +  7"«")  («/3  +  a')3'  +  a"/3")  -  \c?^oi^^cl'''\  (fiy+fi'y'+(i"y"), 


80  Mr  DE  morgan    ON   THE  GENERAL  EQUATION  OF 

in  which  the  factors  equal  to  unity,  and  introduced  for  symmetry, 
have  the  brackets  [].  Develope  these  expressions,  from  which  we 
obtain  the  following  equations: 

fl„=p^+y^+jo"%  l,=  qr-^qr'  +  q"r", 

h^=q'  +  q''  +  q"\  m,=  rp^-t'p' +  r"p" (17), 

Co  =  r^  +  /"  +  r"*,  n„=pq  +p'q'  +  jo'Y'. 

These,  added  together,  the  three  last  having  been  respectively  multi- 
plied by  2  cos  I,  2  cos  rj,  2  cos  ^,  give  from  (16) 

«o  +  *o+Co  +  24  cos  f +  2»?o  cos  »?  +  2«„cos  ^=3T\ 
The  first  side  of  which,  developed  from  (5)  and  (6)  gives  3  V^*  whence 

T=y/Vo (18). 

If  the  process  by  which  (17)  w^as  obtained  from  (2)  be  repeated 
upon  (17),  that  is,  if  at,ha-lo,  Wana—a^la,  &c.  be  formed,  we  shall  have 
equations  of  a  similar  form,  substituting  instead  of  p,  p'  &;c.  such  functions 
of  them,  as  they  themselves  are  of  a,  y3,  &c.,  the  first  sides  of  the  equations 
being  from  (7),  ^o  ^^  *^^  ^^^^  three,  and  V^  cos  f,  F^  cos  ri,  Vg  cos  ^,  in 
the  last  three.  These  equations  are  such  as  would  arise  from  sub- 
stituting in  (2), 

^  ^  ,^ —  instead  of  a        y~  —  and     y-^  ^   for  a   and  a",  &c...(19), 

which  are  therefore  the  values  of  a,  a',  &c.  in  terms  of  p,  q,  &c. 

From  (1),  by  means  of  (14)  and  (18),  can  be  deduced  the  following : 

^/YgX=px'+p'y'-!t-p"^, 

VT,y  =  qaf  +  q'y'+q"fi (20), 

-v/Fo  as  =  r  x'  +  /  2^'  +  r"%', 

and  the  equations  of  the  axis  of  x',  referred  to  the  oblique  axes 
X,  y,  and  k,  are  any  two  of  the  three, 

qx-py  —  0,         ry  —  q%  =  0,         p%-rx=Q (21), 


SURFACES   OF   THE   SECOND   DEGREE.  81 

The   equations   of  the   center,   central  line,   or   central   plane,  as  the 
case   may   be,   of  the  surface   expressed  by  (3)  are 

aX+'cT+bZ+a  =  0, 

cX+bY+aZ+%  =  0 (22), 

bX+aY+cZ+c  =  0, 

and  in  the  two  following  sets  of  quantities,  it  will  be  found  that  the 
sum  of  the  products  made  by  taking  a  term  from  each  in  the  same 
horizontal  line  is  =  F^ ;  while  if  the  terms  be  taken  from  different  horizontal 
lines,  it  will  be  =  0. 


a 

c 

b. 

«« 

n„ 

^..^ 

c 

b 

a. 

«„ 

K 

K, 

b 

a 

c, 

m,, 

K 

c„. 

Thus 


aa„-\-cn„-\-bm„—Vz,        an,,  +  cb^^+bl,,  =  0,  &c. 


Hence,  if  the  three  equations  in  (22)  be  independent  of  one  another, 
the  co-ordinates  of  the  center  are 

j^^  _  a,,a  +  n,J  +  m,,c^    y^  _  n„a^bj)^l„c     ^^  _  m^^a^-l^-\-c,fi 

r- 

The  equation  of  the  surface,  referred  to  this  center,  and  to  axes 
parallel  to  the  primitive  axes,  becomes,  calling  ^  {x,  y,  %)  the  first  side 
of  equation  (3), 

aa? ^-bf +cz^  +  2ay%  +  2bzx ■{■^'cxy+^{X,  Y,  Z)  =  0 ...;:.. (24), 

and  by  multiplying  the  three  equations  in  (22)  by  X,  Y,  and  Z  respec- 
tively, and  adding,  we  get 

0(X,   F,  Z)  =  aX+bY+7z+/=JV (25). 

When   only   two  of  the  equations   (22)   are  independent,  there   is  a 
central  line.     The   conditions  of  this  case   are,  that  the  numerators  and 
Vol.  V.    Paut  I.  L 


82       Mr  DE  morgan  ON  THE  GENERAL  EQUATION  OF 

denominators  in  (23)  must  be  severally  equal  to  nothing;  but  if 
f^3  =  0,  the  equations  in  (10)  shew  that  it  is  sufficient  that  one  of 
the  numerators  should  be  equal  to  nothing;  or  that  the  conditions 
may  be  stated  thus, 

r,  =  0,        a/o^,  «  +  \/T,*  +  'v/cIc  =  0 (26). 

When  F'i  =  0,   F't  is  a  perfect  square,  (10)  and  (11),  its  root  being 

the  second  expression  in  (26).     Hence  W  appears  in  the  form  - .   From 

two  of  equations  (22),  substitute  in  (25)  values  of  any  two  co-ordinates 
of  the  center  in  terms  of  the  third;  it  will  be  found  that  the  co- 
efficient of  the  third  disappears  under  the  conditions  in  (26),  and  that 
the  resulting  value  of  W,  which  we  denote  by  W,  may  be  expressed 
in  either  of  the  following  ways: 

„^,         b(^  —  2cab  +  a¥  ,   „        cb^  —  2acb  +  b<f      ^ 
ab  —  &  bc  —  tt 

^  _  ad'-^bac  +  ca' .^^. 

ac  —  V 

When  no  two  of  the  equations  (22)  are  independent,  there  is  a 
central  plane.  The  conditions  of  this  case  are,  as  appears  from  the 
equations,  that  a„,  6,,,  c^,,  /„, «»,,,  «,,,  must  be  severally  =  0 ;  of  which  how- 
ever it  is  sufficient  that  any  three  should  exist.     We  have  moreover 


a 


a  :  c  '.  b (28). 


From  all  which  it  appears  that   W  is  now  in  the  form  -.     From 

one  of  the  equations  (22)  substitute  in  (25)  the  value  of  one  of  the 
co-ordinates  in  terms  of  the  other  two;  the  coefficients  of  the  last  two 
will  disappear,  as  before,  and  the  different  forms  of  the  value  of  W, 
which  we  call  W",  will  be 

W"^  -  I  +/=  -  J  +/=  -  7  +/• (29). 

By  substituting  W  or  W",  when  necessary,  for  W  or  <p  {X,  Y,  Z) 
in  (24)  the  equation  of  the  surface  will  be  obtained,  referred  to  any 
point  in  its  central  line  or  plane. 


SURFACES  OF   THE  SECOND  DEGREE.  83 

Let  the  equation  of  the  surface,  referred  to  the  principal  axes,  be 
Aa;"  +  A'y"  +  A"z''+W=0 (30), 

which  must  be  identical  with  (24)  when  the  values  of  x',  y',  «',  found  in 
(1)  are  substituted.     We  must  then  have 

a==Aa'     +A'a"     +A"a"\ 

h  =  Ali'   +A'(i"'  +A"(i"\ 

c  =  Ay^    +A'y"    +A"y'\ 

(31), 

a  =  Al3y  +A'fi'y'  +  A"li"y", 

h=-Aya  +A'y'a   -{■A"y"a', 

'^  =  Aafi+A'a'^  +A"a"li", 

which  equations  are  reduced  to  those  in  (2)  by  substituting  unity  for 
A,  A',  A",  a,  h,  and  c;  and  cos  f,  cos  n,  and  cos  X,  for  a,  h,  and  c.  Thus, 
whatever  equation  is  deduced  from  these,  we  immediately  find  another, 
containing  a,  /3,  &c.  in  the  same  way,  by  the  last  mentioned  substitu- 
tion. Multiplying  the  first  of  these  by  p,  the  last  by  q,  and  the  last 
but  one  by  r;  and  adding,  we  obtain  by  the  use  of  (14), 

pa  +  qc        +rb         =Aa\/Vo 

p    +qcoS(^  +  rcosr]=     ay/V^ 

from  which,  and  similar  processes,  we  obtain 


(32), 


p{A  —  a)  +  q{A  cos  ^—c)  +  r  {A  cos  t}  —  b)  =  0, 

p{A  cos^-c)  +  q(A-h)         +  r(^cosf-a)  =  0 (33), 

p{Acc^ri  —  b)  +  q{Acosl^—a)  +  r{A  —  c)  =0; 

which  agree  in  form  with  (22),  if  a,  J,  and  c  be  struck  out,  and  A  — a 
substituted  for  a,  ^cos^  — a  for  a,  &c.  But  1^3  =  0  is  the  result  of  (22), 
with  the  last  terms  erased ;  that  is,  if  in  V^  the  substitutions  just  men- 
tioned be  made  for  a,  a,  &;c.  the  result  developed  and  equated  to  zero 
wiU  give  the  equation  for  determining  A,  A',  and  A".     That  equation  is 

r,A'-  r^A'+r,A-v,=o (34). 

L  8 


84        Mb  DE  morgan  ON  THE  GENERAL  EQUATION  OF 

We  also  find  from  (33),  for  substitution  in  (21), 
-:-:-::  l^  —  l^A  +  loA'  :  m^^  —  m^A  +  m^A^  :  n^,  —  n,A  +  n^A^ (35). 

The  equation  (34)  must  have  all  its  roots  possible.  For  from  (31) 
it  appears  that  A'  and  A"  cannot  be  of  the  forms  X  +  m  V-l  and  X— m \/- 1, 
unless  a'  and  a",  /3'  and  /3",  7'  and  7"  are  of  the  same  form ;  from  which, 
since 

{K  +  xV'^){o-(p\/'^)  -  («-x\/^)(0  +  0\/^T) 

is  of  the  form  k\/  —  1,  it  will  follow  that  p,  q,  and  r  (4)  must  be  of 
this  form :  which  is  inconsistent  with  (32),  if  we  suppose  V^  positive ; 
since  it  may  be  seen  from  (31),  and  will  presently  appear  otherwise, 
that  a  is  possible  when  A  is  possible. 

We  might  find  equations  of  the  third  degree  to  determine  jh  q,  &c. 
but  it  will  be  more  convenient  to  express  them  in  terms  of  A,  &c., 
supposed  to  be  found  from  (34).  To  do  this,  let  a,,,  a^,  I,,,  I,,  &c.  (5) 
and  (6),  be  found  in  terms  of  A,  a,  he.  by  substituting  the  values  of 
a,  h,  a,  h,  he.  from  (31).     The  results,  after  reduction,  are 

a,,=A'A"jf  +A"Ap"  +AA'p"',  a,=U'+'^")f  +{A"+A)p"  +{A+A')p"% 
h,^A'A"(f  +A"Aq"  +AA'q"\  b={A'+A")q'  +{A"+A)q"  +{A+A')q"% 
c„=A'A"f^  +A"Ar"  +AA'r"',       c=U'+^"V  +U"+A)r"  +{A+A')r"\ 


..(36), 


l„=A'A"qr+A"Aqr'+AA'q"r",     1,={A'+A")qr+{A"+A)q'r'+{A+A')q"r", 

m„=A'A"rp+A"Ar'p+AAy'p",    m={A'+A")rp+{A"+A)rp'+{A+A')r"p", 

n,=A'A"pq+A"Ap'q'+AA'p"q",      n=U'+^")Pq+i^"+-^)p'q'+i^+^')P'Y' 

which    equations,    with    those   marked   (17),    give   the    following   values 
of  p'',    qr,  he. 

a,-a^A  +  a,A'  _  l„  -  l,A  +  kA' 

^~  {A-A'){A-A")'        ^        {A-A'){A-A"y 

"  —    ^ii~^t^  +hoA^                 _  m,,-m,A  +  irigA^  . 

^'~  {A-A%A-A"y        ''P~  {A-A'){A-A") ^^^' 

•i  _    C//  — g,^  +CoA^  _  n,,  -  n,A  +  n^A' 

^  ~  XA^^t^  -  ^"') '       ^^~  {A-  A)  {A  -  A")  ■ 


SURFACES  OF   THE   SECOND   DEGREE.  85 

In  which  equations  the  letter  p,  q.  A,  &c.  may  be  accented  throughout 
singly  or  doubly,  striking  off  three  accents  from  any  A  which  thus 
obtains  three  or  more. 

By  squaring  the  equations  (15),  writing  V^  for  7",  substituting 
the  values  just  obtained  for  p^,  qr,  &c.  and  then  multiplying  the  same 
equations  together  two  and  two,  and  making 

Li  =  *„Co  +  b^c,,  -  2 IX,  Z/2  =  in,,n,  +  m^n,,  -  aj,,  -  a„k, 

Ni  =  «„*„  +  «o*// -  2 n„n^,  Ni  =  l,,m,  +  lotn^^  -  c^n„  —  c„«o, 

we  get 

^_F,-L,-{F,-aK)A  +  F,A'  r,-M,-{F-br,)A+ F.A^ 

**"         V,{A-A'){A-A")         '    ^~  r,{A-A'){A-A") 

2      F-N,-{r,-cr:)A+V,A'  ,„. 
^-          K{A-A'){A-A")         ^^^^' 

„    _  FiCos^  —  Lz  —  jVi  cos^—aF'o)A+  FgCosBA^ 
^'y~  F{A-A'){A-A") 

_  Fcos  tj  — Mi- (Fj  cos  tj -bFp)  A  +  FqCos  t/A^ 
'y"-  F,{A-A'){A-A") 

^      F,  cos  t-'^2-{F,  cos  ^-c  F„)  A  +  F„  cos  ^A' 
"^~  F^{A-A'){A-A") 

in  which  the  letters  may  be  singly  or  doubly  accented  as  before,  and 
from  which  the  determination  of  the  position  of  the  principal  diameters 
is  made  to  depend  directly  upon  the  solution  of  (34). 

Let  the  surface  whose  equation  is  (3)  be  referred  to  another  origin 
and  other  axes,  and  let  the  quantities  corresponding .  to  those  already 
given  or  deduced,  which  belong  to  the  new  origin  or  axes,  be  denoted 
by  the  same  letters  and  accents  enclosed  in  brackets  [  ].  Thus  the 
angles  made  by  the  new  axes  are  [^1  [>;],  and  [^] ;   the  coefficients  of 


86  Mr  DE   morgan    ON  THE  GENERAL  EQUATION  OF 

the  new  equation  are  [a],  [«],  &c.;  the  functions  of  these  coefficients 
already  noticed  are  [«J,  [/„],  [F,],  &c.  Since  the  principal  diameters 
of  the  surface  are  the  same,  from  whatever  equation  they  are  derived, 

w      r  w'l 

that  is,  since   — 'T  ~  ~  rlT '  ^^'  *^^  roots  of  (34)  bear  to  those  of  [34] 

the  proportion  of  W^  to  [  W^ ;  whence,  \  being  an  indeterminate  quan- 
tity, since  one  coefficient  in  (3)  is  indeterminate. 


.(39), 


LjO'^'k'  M-^' 


0 


These  equations*  correspond  to  the  general  relations  (6),  (7),  and  (9), 
given  in  my  former  paper,  and  from  them  may  be  deduced  the  pro- 
perties of  systems  of  conjugate  diameters,  and  the  remarkable  property 
of  the  reciprocal  squares  of  three  semi-diameters  at  right  angles  to  one 
another. 

Let  wT',  V,  and  Z',  be  the  co-ordinates  of  the  second  origin  referred 
to  the  first,  so  that  if  the  co-ordinates  be  changed,  [y]  and  (p{JC',  Y',  Z') 
will  be  corresponding  terms  of  two  equations,  the  terms  of  which  should 
be  respectively  proportional.  Assume  X,  the  indeterminate  quantity 
above-mentioned,  so  that 

[/]  =  \4>{X',  Y',  Z') (40). 

and  multiply  together  the  first  and  last  of  (39),  recollecting  that 


W 


=  -r.-^f^     [^i  =  -[Fj^t/]. 


*  These  relations  have  been  given  by  M.  Cacchv,  for  the  case  of  rectangular  co- 
ordinates, in  his  "  Lcfons  sur  les  applications  du  Calcnl  Infinitesimal  d  la  Geometrie,"  Vol.  i. 
p.  2441.  The  equation  (34)  of  this  paper,  in  as  general  a  form,  has  also  been  given,  since 
this  was  written,  by  Mr  Lubbock,  in  the  Philosophical  Magazine. 


SURFACES  OF  THE  SECOND  DEGREE. 


87 


and  we  obtain 


[^J-X^(r-,F-.Z')[^J=X.(^-/^;). 

Substitute  from  the  last  of  (39)  for  [jfl,  and  develope  <f>{X',  V,  Z'), 

removing   the   term   which   contains    it    to   the    left   hand    side;    which 
gives 


ca= 


r,  +  F, jaX"  +  br"  +  cZ''  +  &c.  &c.) 


■(41), 


answering  to  (8)  in  my  former  paper. 

We  shall  afterwards  proceed  to  some  applications  of  these  general 
formulas,  and  now  enquire  into  the  several  varieties  of  the  equation  (3), 
and  the  criteria  for  distinguishing  between  them.  The  following  table, 
immediately  to  be  explained,  gives  a  synoptical  view  of  the  various 
caseSi  inhcrfHt i->9.R   :i;f<>1  '"\'r^hr'     5r    .-^.v: 


When  the  Equations  of  the 
Center  denote 

positive, 
negative. 

W  changes  its  sign. 

^  negative, 
positive. 

A  point 

A  Right  Line.   W 
substituted  for  W. 

A   Plane.  W"  sub- 
stituted for  W. 

Impossible. 

Single  Hyper- 
boloid. 

Impossible. 

Hyperbolic 
Cylinder. 

Impossible. 

(W=o)     Point. 

(W=  oc)  Elliptic  Paraboloid. 

(W=0)     Cone. 
(W=oc)  Hyperbolic      Para- 
boloid. 

(W'=0)     Right  line. 
(W"=oc)  Parabolic  Cylinder. 

(W'=0)     Intersecting  Planes. 
(W'=oc)  Parabolic  Cylinder. 

fW"=0  \  ^. 
^,            Plane. 

Ellipsoid. 

Double  Hyperboloid. 

Elliptic  Cylinder. 
Hyperbolic  Cylinder. 
Parallel  Planes. 

88       Mr  DE  morgan  ON  THE  GENERAL  EQUATION  OF 

Taking  the  first  line  of  this  table,  and  the  signs  of  W,  V^,  V^,  and 
V^,  (on  which,  as  will  presently  be  shewn,  the  variety  of  the  equation 
depends,)  being  such  as  to  denote  that  the  equation  is  impossible,  a 
change  of  sign  in  W  only  will  indicate  the  ellipsoid,  the  elliptic  cylinder, 
or  parallel  planes,  according  as  the  centre  is  a  point,  a  line,  or  a  plane. 
When  the  sign  changes,  if  W  be  then  =  0,  the  variety  of  the  equation 
belongs  to  a  point,  a  right  line,  or  a  plane ;  while  if  W  be  infinite, 
we  have  an  elliptic  paraboloid,  a  parabolic  cylinder,  or  a  plane.  In 
using  W,  we  mean  its  real  value,   W  or  W",  when  the  primitive  form 

of  W  becomes  -  . 

The  following  table,  from  which  the  preceding  may  be  deduced,  and 
which  I  proceed  to  establish,  gives  the  signs  of  W,  &c.,  and  also  of  V^, 
&c.,  for  the  different  cases.  When  p  alone,  or  p  and  n  occur  on  the 
same  line,  p  may  signify  either  sign,  provided  n  stands  for  the  other. 
Also  when  a  sign  is  enclosed  in  brackets,  it  is  a  necessary  consequence 
of  what  precedes  it,  and  not  an  independent  assumption.  The  num- 
bers over  the  headings  are  references  to  the  equations. 

The  last  part  of  the  table,  including  all  the  varieties  under   W=  - , 

forms  a  similar  synoptical  table  for  the  curves  of  the  second  degree. 
The  following  are  the  values  of  W,  W",  V^  and  Fi,  expressed  in  the 
notation  of  my  former  paper,  the  equation  of  the  curve  being 

ay*  +  hxy  +  ca^  +  dy->rex  +f=  0 ; 

and  the  angle  made  by  the  axes  being  Q, 

.^,  _  cd^ +  ae^  —  hde       „ 

™,„ _  _  dr-^a£_      &-^^cf 
id       ~~  4c      ' 

V,  =-  (b'-iac), 

Vi  =  a  +  c  —  h  cos  Q. 


SURFACES  OF  THE  SECOND  DEGREE. 


8d 


0) 

•,    C 

0) 

•5 

1 

§       1 

B 

d 
II 

i 

i 

.5 

1 

■4-> 

s 

II 

.O"    4) 
CO    i^C 

6 
il 

a 

CQ 

d 
11^ 

o 

11^ 

o 

II 

g 

CO 

a; 

1 

» 

^    go 

11 

-   E 

s 

S 

3,  and  the  next 
y  be  =  0, 

B 

O 

5,  and  all  which 
bstituted  for  7, 

s 
•J3 
u 

2 

CO 

<» 

-.'  S 

te 

X 

■s  s 

-^    S 

e' 

« 

s 

s 

#> 

e" 

« 

^   u- 

w 

e 

O 

e 

TS 

, 

t3 

o 

u 

8 

o 

•■6  'o 

^ 

^ 

QJ 

o 

1 

1 

p-( 

2-S 
1^' 

Si 

a 
1 

en 

s 

s" 

c8      o 

*^   'o 

i^ 

}  Straight  lin 
Intersecting 

c8 

1  = 

1 

1     S 

2    CL( 

a 

02 

1 

o 

a 
o 

(J 

1 

•1  .J 

1 

'o 

d 

1 

PU 

s^ 

a,  a 

8 

a, 

a. 

a 

a,  a 

a,  a 

« 

a, 

o   8 

^— , ' 

/— \ 

+  + 

+      1 

+    1 

+    I 

+ 

+    1 

+ 

1 

© 

+    1 

o 

O  ^ 

■ • — 

^  • 

-          ■ y • 

' 

£^ 

a, 

8 

o 

0|0 

.''^S 

" — 

O  '^ 

Rh  a 

a, 

a 

O 

a. 

§ 

a. 

a, 

8 

o    ■ 

0|C 

Vol.  V.    Paet  I. 


M 


90  Mr  DE   morgan    ON  THE  GENERAL  EQUATION  OF 

First,  with  regard  to  the  coefficients  K^,  Vi,  V^,  V3  in  equation 
(34)  it  appears  from  spherical  trigonometry,  that  V^  is  always  positive 
when  '(;,  J/,  and  ^  are  the  sides  of  a  spherical  triangle;  while  from  the 
possibility  of  the  roots,  as  well  as  from  the  quantities  themselves,  we 
infer  that  if  V3  is  finite,  Fj  and  Vi  can  never  vanish  at  the  same 
time,  while  if  ^i  =  0,  and  ^  =  0,  Fj,  must  be  negative. 

If  we   suppose    TV  finite,   and    the    order  of    signs    in    (34)   to   be 

H (--    or    +  +  +  +,    in    which    case    all    its   roots   are   of  one  sign ; 

that  is,  if  K2  be  positive,  and  Vi  and  V3  of  the  same  sign,  the  equa- 
tion (30)  shews  that  the  surface  is  impossible  or  an  ellipsoid,  according 
as  W  and  F'a  have  the  same  or  different  signs.  From  (36)  it  appears 
that  in  this  case,  a^^,  b,^,  and  c„  must  be  positive,  whence  a,  h,  and  c  have 
the  same  sign ;  which  conditions,  together  with  that  of  V^  having  the 
same  sign  as  a,  are  equivalent  to  those  given  in  the  Table  for  the 
impossible  case  or  the  ellipsoid.  If  we  examine  independently  into 
the  conditions  under  which  the  aggregate  of  the  first  six  terms  of 
(24)  always  has  the  same  sign,  we  shall  find  them  to  be  that  a^,  b„, 
and  c„  must  be  positive,  and  V3  must  have  the  common  sign  of  a,  h, 
and  c.  And  it  is  evident  that  the  first  three  terms  of  (30)  are  the  first 
six  terms  of  (24)  in  a  different  form.    It  may  be  worth   noticing,   that 

these  conditions   are   equivalent  to  supposing      ,--,    --  — ,   —7=5=  to  be 

's/ he     \/ca     y/ab 

the  cosines  of  the  sides  of  a  spherical  triangle.    When  any  other  order 

of  signs  except  the  two  already  noticed,  is  found  in  (34),  we  shall  have 

one   positive   root   only,    or    one  negative   root  only,  according  as   V3  is 

positive   or   negative ;    that    is   to   say,    one    possible   axis,    or   a   double 

hyperboloid,  when   V^  and   W  have  contrary  signs ;   and  one  impossible 

axis  or  a  single  hyperboloid,  when  they  have  the  same  signs. 

When  W—0,  V^  being  finite,  equation  (30)  represents  a  point,  or 
a  cone;  the  first  when  all  the  roots  of  (34)  have  the  same  sign,  the 
second  in  any  other  case.  When  V3  =  0,  Vi  being  finite,  or  W  infinite, 
the  center  is  at  an  infinite  distance,  and  the  equation  belongs  to  an 
elliptic  or  hyperbolic  paraboloid,  according  as  V^  is  positive  or  negative. 
Since   when    V3  =  0,  «,,,  5„,  and  c,,  have  the  same  sign,   (10),   which   is 


SURFACES  OF  THE  SECOND  DEGREE.  91 

also  the  sign  of  V^,  a,^  may  be  substituted  for  Vi.  In  this  case,  (10) 
and  (9),  V2  has  the  form 

P+ Q  +  2?  +  2\/QK" cos  ?+2\/;BP  cos  .7+2 ^/PQ  cos  ^, 

which,  when  P,  Q,  and  R  have  the  same  sign,  is  always  of  that  sign; 
and  therefore  can  only  be  =  0  when  P,  Q,  and  B  are  severally  =  0. 

When  ^"3=0,  and  F'i  =  0,  in  which  case   W  appears  in  the  form  -, 

and  its  real  value  is  W  (27),  the  simplest  criteria  of  which  are  ex- 
pressed in  (26)  the  equations  (30)  and  (34)  assume  the  forms 

Aaf'  +  A'y"+Jr'=0 (42), 

KA'-  r,A  +  r,=o (43), 

the  first  of  which,  if  V^  be  positive,  and  F",  and  W  of  the  same  sign, 
is  impossible,  and  belongs  to  an  elliptic  cylinder  if  V^  be  positive, 
and  Fi  and  W  of  different  signs.  As  before,  we  may  substitute  a„ 
for  Vi.  If  V2  or  a^i  be  negative,  (42)  belongs  to  an  hyperbolic  cylinder : 
and  if  V2  —  O,  in  which  case  a^^  =  0,  h,i  =  0,  and  c^^  =  0  and  W  is  infinite, 
we  have  a  parabolic  cylinder.  It  appears  therefore,  that  any  surface  of 
the  second  order,  which  has  three  parabolic  sections,  not  having  a 
common  line  of  intersection,  is  a  parabolic  cylinder.  The  central  line 
of  this  surface  is  at  an  infinite  distance.  When  W'  =  0  and  V  is 
positive,  equation  (42),  considered  as  of  two  dimensions,  represents 
only  the  origin,  and  therefore  belongs  to  a  straight  line,  the  axis  of 
iB'.  When  Fa  is  negative,  W  being  =0,  (42)  is  the  equation  of  two 
planes  intersecting  at  an  angle  whose  tangent  is 

2^/-AA'  2V-F,r, 

A  +  A'    '         °^  r, 

When  the  equations  of  the  center  belong  to  a  plane,  and  W  as 
well  as  W  appears  in  the  form  -,  the  real  value  of  W  is  W",  given 
in  (29)  and  the  simplest  conditions  are,  as  in  (28), 

«//  =  *//  =  C//  =  0, 

a  :  b  :  c  '.:  a  :  c  :  b. 

M  2 


92  Mr  DE  morgan    ON  THE  GENERAL  EQUATION  OF 

The  equations  (42)  and  (43)  take  the  forms 

Ax"+jr"  =  0 (44), 

F,A  -  F,  =0 (45). 

The  first  of  which  is  impossible  if  W  and  Fl  have  the  same  sign, 
that  is,  if  W"  and  a  have  the  same  sign ;  for  when  a„  =  b,^  =  c„  =  0, 
T^i  takes  the  same  form  with  respect  to  a,  b,  and  c  which  Vs  took  with 
respect  to  a„,  b^,,  and  c„  in  the  last  case.  When  a  and  W"  have 
different  signs  (44)  belongs  to  two  parallel  planes,  which  coincide  in 
one  where  W"  =  0.  That  is  (29)  the  surface  is  impossible,  two  parallel 
planes,  or  one  plane,  according  as  af—c^  is  positive,  negative,  or  nothing. 
When  W  becomes  infinite,  or  a  =  0,  in  which  case  b,  c,  a,  b,  and  c  are 
severally  =  0,  the  proposed  equation  (3)  is  in  fact  of  the  first  degree. 

Though  oblique  co-ordinates  have  hitherto  been  used,  yet  they 
might  have  been  dispensed  with  so  far  as  the  criteria  of  distinction 
between  the  different  classes  of  surfaces  are  concerned.  It  would  take 
some  space,  and  complicated  algebraical  operations,  to  prove  this  in 
all  the  individual  cases,  but  the  following  general  consideration  is  equally 
conclusive.  So  long  as  we  only  consider  those  distinctions  which  are 
implied  in  calling  the  surface  bounded  or  unbounded,  of  one  sheet  or 
of  two  sheets,  &c.  in  which  no  numerical  relations  of  lines,  &c.  appear, 
it  is  evident  that  any  equation  will  preserve  the  same  character,  how- 
ever the  axes  on  which  its  results  are  measured  are  inclined  to  one 
another.  That  is,  when  the  sign  of  a  quantity  is  alleged  to  be  a  cri- 
terion of  distinction,  it  cannot  stand  as  such,  if  by  any  alteration  of 
I,  >/,  or  ^,  consistent  with  V^  remaining  positive,  the  sign  of  that  quantity 
can  be  changed.  Again,  if  the  signs  of  two  out  of  the  three,  a,  b,  and  c 
be  changed,  as  well  as  that  of  the  third  letter  in  a,  b,  and  c,  (those  of 
a,  b,  and  c,  for  example)  it  is  evident  that  the  surface  remains  the  same 
in  form  and  magnitude,  those  parts  which  were  below  one  of  the  co- 
ordinate planes  being  transferred  above  it,  and  vice  versa.  That  is, 
it  is  impossible  that  any  aggregate  of  terms  of  an  odd  degree,  with  respect 
to  a,  b,  and  c,  b,  c,  and  a,  or  c,  a,   and  b,  can  affect  the  sign  of  any 


SURFACES   OF   THE  SECOND  DEGREE.  93 

of  the  criteria.  If  we  look  at  F'l,  V^,  Fs,  and  F^,  we  find  that  those 
terms,  and  those  terms  only,  which  are  multiplied  by  cosines  of  f,  &c., 
are  of  the  first  or  third  degree,  with  respect  to  any  of  the  three  sets 
just  mentioned. 

The  case  is  very  much  altered  when  we  consider  any  numerical 
relation,  however  simple.  For  example,  I  give  the  condition  which 
expresses  a  surface  of  revolution,  or  a  surface  two  of  whose  axes  are 
equal.  If  A  and  A'  belong  to  the  equal  axes,  a,  a,  &c.  become  in- 
determinate; hence  the  numerators  of  the  six  equations  (38),  will,  when 
equated  to  zero,  have  a  common  root.  Eliminate  F  -  FA  +  FoA"  from 
the  values  of  a*  and  (3y,  &c.  in  (38),  which  gives 

.-_         XaCOsf-Z^        MiCO^ri-D        iVgCOS^-iV^  ,^ 

AFa  =  r"=-  =  -7 T~  -  y     -      V4b), 

a  cos  ^— a  0  cos  t]  —  o  ccos^— c 

which  does  not  admit  of  any  material  simplification.  There  are  evidently 
other  ways  of  obtaining  corresponding  conditions  from  (38).  I  have 
chosen  this  because  the  corresponding  formulae  have  been  given  in  the 
case  of  rectangular  co-ordinates.     In  this  case, 

cos  ^  =  cos  >;  =  cos  ^  =  0,    and  Li  =  -  l„,  &c. 

whence, 

^  _  ^/  _  ^/ 
a       b        c 

(See  Mr  Hamilton's  Analytical  Geometry,  p.  323.) 

To  apply  the  formulae  (39)  and  (41),  let  there  be  two  planes  whose 
equations,  separately  considered,  are 

\'x+  fi'y+  i/'a  +  l  =0  J 

but  which  together  must  be  one  of  the  varieties  of  equation  (3).  Let 
new  and  rectangular  axes  be  taken,  the  intersection  of  the  planes 
being  that  of  x.     Their  equation  will  then  be 

[c]  z"  +  2[a]yss  =  0, 


94 


Mr  DE  morgan    ON  THE  GENERAL  EQUATION,  &c. 


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IV.  On  a  Monstrosity  of  the  Common  Mignionette.  By  Rev,  J.  S. 
Henslow,  M.A.  Professor  of  JBotany  in  the  University  of  Cambridge, 
and  Secretary  to  the  Cambridge  Philosophical  Society. 


[Read   May  21,   1832.] 


Having  met  with  a  very  interesting  monstrosity  of  the  common 
Mignionette  {Reseda  odorata,)  in  the  course  of  last  summer  (1831), 
I  made  several  drawings  of  the  peculiarities  which  it  exhibited.  I  beg 
to  present  the  Society  with  a  selection  from  these,  as  I  think  they  may 
both  serve  to  throw  considerable  light  upon  the  true  structure  of  the 
flowers  of  this  genus,  which  is  at  present  a  matter  of  dispute  among 
our  most  eminent  Botanists,  and  also  tend  to  illustrate  the  manner 
in  which  the  reproductive  organs  of  plants  generally,  may  be  con- 
sidered as  resulting  from  a  modification  of  the  leaf. 

It  is  well  known  to  every  Botanist,  that  Professor  Lindley  has 
proposed  a  new  and  highly  ingenious  theory,  in  which  he  considers 
the  flowers  of  a  Reseda  to  be  compounded  of  an  aggregate  of  florets, 
very  analogous  to  the  inflorescence  of  a  Euphorbia.  Mr  Brown,  on 
the  other  hand,  maintains  the  ordinary  opinion  of  each  flower  being 
simple,  and  possessed  of  calyx,  corolla,  stamens,  and  pistil.  I  shall 
not  here  enter  upon  any  examination  of  the  arguments  by  which 
these  gentlemen  have  supported  their  respective  views,  but  will  refer 
those  who  are  desirous  of  seeing  them  to  the  "  Introduction  to  the 
Natural  System  of  Botany,  by  Prof.  Lindley,"  and  to  the  "Appendix 
to  Major  Denham's  Narrative,  by  Mr  Brown,"  My  present  object  will 
be  little  more  than  to  describe  the  several  appearances  figured  in  plates 
1  and  2. 


96  PROFESSOR  HENSLOW,  ON  A  MONSTROSITY 

Fig.  1.  is  one  of  the  slightest  deviations  that  was  noticed  from  the 
ordinary  state  of  the  flower.  It  consists  in  an  elongation  of  the  pistil  (a), 
and  a  general  enlargement  of  its  parts,  indicating  a  tendency  in  them  to 
pass  into  leaves.  This  is  accompanied  by  a  slight  diminution  in  the  size 
of  the  central  disk.     The  number  of  the  sepals  was  either  six  or  seven. 

Fig.  2.  is  a  portion  of  the  ovarium  of  the  same  flower  opened,  in 
which  three  of  the  ovules  are  somewhat  distorted. 

Fig.  3.  Here  the  three  valves  of  the  ovarium  have  assumed  a  dis- 
tinctly foliaceous  character  (a);  the  same  has  happened  to  some  of  the 
stamens  {b),  and  to  the  petals  (c) ;  but  the  sepals  are  unaltered.  The 
central  disk  has  entirely  disappeared. 

Fig.  4.  This  is  a  still  closer  approximation  to  the  ordinary  state  of 
a  proliferous  flower  bud,  when  developed.  Those  parts  which  would 
have  formed  the  pistil,  if  the  flower  had  been  completed,  are  no  longer 
distinguishable,  and  only  a  few  of  the  stamens  are  to  be  seen,  disguised 
in  the  form  of  foliaceous  filaments  crowned  by  distorted  anthers  (h). 

Fig.  5.  A  slight  deviation  in  one  of  the  petals  from  the  usual 
character.  The  fleshy  unguis  is  somewhat  diminished,  and  the  fimbriae 
are  becoming  green  and  leaf-like.  These  are  aggregated  into  three 
distinct  bundles,  the  middle  one  being  composed  of  a  single  strap, 
and  the  two  outer  ones  of  five  straps  each,  blended  together  at  the 
base. 

Fig.  6.  The  line  of  demarcation  between  the  unguis  and  the  fimbriae 
has  completely  disappeared,  and  the  number  of  the  latter  is  considerably 
reduced.     The  whole  is  more  green  and  leaf-like  than  fig.  5. 

Fig.  7.  The  fimbrige  reduced  to  a  single  strap ;  the  position  of  the 
lateral  bundles  being  indicated  by  slight  projections  only.  Other  in- 
stances occurred  in  which  the  petal  appeared  as  a  single  undivided 
uniform  green  strap. 

Fig.  8.  The  two  exterior  whorls  of  a  flower,  consisting  of  seven 
regularly  formed  sepals,  and  eight  petals.  The  latter  deviate  more  or 
less  from  the  forms  represented  in  fig.  6  and  7.  The  whole  of  a  green 
tint,  and  leaf-like. 


7>^nsactM>ns  afthe  Ceimi.  I'Ail.Sor.  VoLTTTf^J 


JSa-n,l,.u-  ,1,1  f 


.TD-CSmn^Jatrfi' 


Transaetians  of  the  C'affiiJ'ful.SrK.Til.KIt.Z. 


XlkKStmnfy  X'"^ 


OF   THE   COMMON  MIGNIONETTE.  97 

Figs.  9,  10.  These  are  parts  of  one  and  the  same  flower  dissected 
to  shew  the  several  whorls  more  distinctly.  The  whole  has  assumed 
a  regular  appearance,  and  is  composed  of  seven  sepals,  alternating  with 
seven  green  strap-shaped  petals,  which  are  succeeded  by  about  twenty 
stamens  without  any  fleshy  disk ;  the  pistil  is  somewhat  metamorphosed. 
This  is  perhaps  the  most  remarkable  deviation  that  was  noticed  from 
the  ordinary  state  of  the  flower,  and  as  several  examples  of  it  occurred, 
it  is  not  likely  that  there  is  any  error  in  this  account  of  it.  It  appears 
to  lead  us  in  a  very  decided  manner  to  the  plan  on  which  the  flowers 
of  the  genus  may  be  considered  to  be  constructed,  and  to  shew  us 
that  they  are  really  simple  and  not  compound. 

Fig.  11  to  15,  represent  the  appearances  assumed  by  some  of  the 
stamens,  indicating  various  degrees  of  deviation  from  the  perfect  state 
towards  a  foliaceous  structure. 

There  were  other  circumstances,  besides  the  appearances  in  figs.  9. 
and  10,  which  may  lead  us  to  conclude  the  structvire  of  the  flowers 
of  the  genus  to  be  simple  and  not  compound.  A  compound  flower 
arises  from  the  development  of  several  buds  in  the  axillee  of  certain 
foliaceous  appendages  more  or  less  degenerated  from  the  character  of 
leaves,  and  consequently  these  buds  and  the  florets  which  they  develop 
are  always  seated  nearer  to  the  axis  than  the  foliaceous  appendages 
themselves.  If  we  suppose  a  raceme  of  the  mignionette  to  degene- 
rate into  the  condition  of  a  compound  flower,  we  must  allow  for  the 
abortion  of  the  stem  on  which  the  several  flowers  are  seated,  so  that 
these  may  become  condensed  into  a  capitulum,  each  floret  of  which 
will  be  accompanied  by  a  bractea,  more  or  less  developed,  at  its  base. 
Let  us  compare  this  supposition  with  the  diagrams  represented  in 
figs.  16,  17,  18. 

Fig.  16.  is  an  imaginary  section  of  the  flower  in  its  ordinary  state, 
(a)  the  pistil,  (b)  the  stamens  on  the  fleshy  disk,  (c)  the  petals,  {d)  the 
sepals  alternating  with  them. 

Fig.  17.  represents  the  position  of  the  several  buds  (e)  which  com- 
pose the  florets  of  the  flower  on  the  supposition  of  its  being  com- 
pound.     Here  it  will  be    noticed  that  these   buds  alternate   with   the 

Vol.  V.    Part  I.  N 


98  PROFESSOR  HENSLOW,   ON   A  MONSTROSITY 

sepals   instead    of  being  placed   in   their   axils  where   we    might   rather 
expect  to  find  them. 

Fig.  18.  represents  a  fact  which  was  observed  in  the  present  case, 
where  some  of  the  latent  buds  in  the  axils  of  the  altered  petals  were 
partially  developed.  This  development  might  perhaps  be  considered  as 
indicating  the  construction  of  a  compound  flower,  and  those  buds  which 
in  ordinary  cases  compose  the  outer  and  abortive  florets,  it  might  be 
said,  are  here  manifesting  themselves.  But  the  axes  of  these  buds  lie 
nearer  to  the  axis  of  the  whole  flower  than  the  petals  in  whose  axils 
they  are  developed;  whereas  it  appears  by  fig.  17,  that  they  ought  to 
be  further  from  it,  since  the  centres  of  the  five  outer  circles  marked  (e) 
would  represent  the  axes  of  the  several  buds,  whose  partial  develop- 
ment must  be  supposed  to  be  on  the  side  next  the  axis,  if  we  allow 
any  weight  to  the  analogy  between  the  position  of  the  abortive 
stamens  on  the  supposed  calyx,  and  the  fertile  stamens  on  the  central 
disk. 

These  figures  are  all  that  I  have  thought  it  necessary  to  give  for 
the  purpose  of  illustrating  the  structure  of  the  flower;  but  as  there 
were  several  interesting  appearances  noticed  upon  dissecting  the  pistil, 
I  have  selected  some  of  them  for  the  second  plate,  as  they  may 
possibly  serve  to  throw  some  light  upon  the  relationship  which  the 
several  parts  of  the  ovarium  bear  to  the  leaf,  and  to  support  the 
theory  of  their  being  all  of  them  merely  modifications  of  that  im- 
portant organ. 

Fig.  19.  is  a  pistil  in  which  the  three  ovules  have  become  foliaceous, 
and  the  central,  or  terminal  bud  of  the  flower-stalk  is  developing  in 
the  proliferous  form  represented  in  fig.  4. 

Fig.  20.  The  central  bud  is  not  developing ;  but  the  three  axillary 
buds  in  the  bases  of  the  transformed  valves  of  the  pistil  are  here 
assuming  the  form  of  branches  on  which  one  or  two  pair  of  leaves  are 
expanded. 

Fig.  21.  22.  unite  the  appearances  in  fig.  19  and  20,  with  the 
addition    of    a    glandular    body    seated    between     the    leaves    at    their 


OF   THE  COMMON   MIGNIONETTE.  99 

junction.  This  apparently  originates  in  the  union  of  the  two  glandular 
stipules  seated  at  the  base  of  the  leaves  of  this  genus,  and  which 
may  also  be  seen  to  accompany  the  scale-like  leaves  on  the  central 
bud  within. 

Figs.  23.  to  25.  Interior  views  of  metamorphosed  pistils,  in  which 
the  ovules  are  seen  transformed  to  leaves,  and  the  glandular  stipules 
are  all  that  remain  of  the  leaves  which  should  compose  the  central 
bud,  their  limbs  having  entirely  disappeared. 

Fig.  26.  The  appearance  of  these  stipules  on  a  leaf-bud,  develop- 
ing under  ordinary  circumstances. 

Fig.  27.     One  of  them  more  highly  magnified. 

Figs.  28.  29.  Their  appearance  on  the  small  scale-like  leaves  of  the 
central  buds  in  fig.  21,  22. 

Fig.  30.  Similar  to  fig.  23,  but  without  any  appearance  of  the 
transformed  ovules;  the  glandular  stipules  are  seen  in  the  bottom  of 
the  ovarium. 

These  glandular  bodies  assume  a  very  prominent  character  in  the 
anatomy  of  the  metamorphosed  pistils,  and  I  was  for  some  time 
puzzled  to  account  for  them,  thinking  that  they  might  represent  an 
altered  condition  of  the  ovules.  I  believe  however  that  I  have  rightly 
considered  them  as  the  only  representatives  of  the  various  leaves  which 
would  have  made  their  appearance  on  the  branch  if  the  bud  had 
developed  in  the  ordinary  way.  They  do  not  appear  to  diminish  in 
size  though  the  limb  of  the  leaf  has  disappeared. 

Fig.  31.  Four  pedicillated  semitransformed  ovules,  seated  on  a  pla- 
centa of  a  pistil  metamorphosed  similarly  to  that  in  fig.  9- 

Figs.  32.  to  35.  Other  appearances  of  a  similar  kind,  all  representing 
various  approaches  of  the  ovules  to  a  foliaceous  character.  The  little 
theca-shaped  appendages  are  hollow,  with  a  perforation  at  their  apex, 
representing  the  foramen. 


100  PROFESSOR  HENSLOW,  ON  A   MONSTROSITY,  &c. 

Fig.  36.  One  of  these  dissected,  exhibiting  a  free  clavate  cellular 
body  within,  resembling  the  columella  in  the  theca  of  a  moss,  and 
"here  probably  representing  the  nucleus  of  the  ovule. 

Fig.  37.  In  this  case  the  theca-shaped  body  was  partially  open 
exposing  the  included  nucleus. 

Fig.  38.     This  nucleus  more  highly  magnified. 

These  appearances  surely  indicate  a  development  of  the  investing 
coats  of  the  nucleus  into  leaves ;  but  how  far  these  developments 
might  be  extended,  and  whether  the  nucleus  itself  is  capable  of  being 
further  separated  into  a  series  of  investing  coats  does  not  appear  from 
these  specimens. 


J.  S.  HENSLOW. 


TRANS  A  CTIONS 


CAMBRIDGE 


PHILOSOPHICAL    SOCIETY. 


Vol.  V.     Part  II. 


CAMBRIDGE: 


PRINTED  BY  JOHN  SMITH,  PRINTER  TO  THE  UNIVERSITY : 

AND  SOLD  BY 

JOHN    WILLIAM    PARKER,    WEST    STRAND,    LONDON; 

J.  &  J.  J.   DEIGHTON,   AND  T.  STEVENSON, 

CAMBRIDGE. 


M.DCCC.XXXIV. 


On  the  Calculation  of  Newton's  Experiments  on  Diffraction.  By 
George  Biddell  Airy,  M.A.  late  Fellow  of  Trinity  College, 
and  Plumian  Professor  of  Astronomy  and  Experimental  Philosophy 
in  the  University  of  Cambridge. 


[Read  May  7,  1833.] 

Since  the  publication  of  Fresnel's  experiments  on  Diffraction,  it  has 
been  usual  to  employ  as  the  source  of  light,  in  all  experiments  of  this 
class,  the  image  of  the  Sun  formed  by  a  lens  of  short  focal  length.  On 
the  undulatory  theory,  the  effect  of  light  thus  produced  is  precisely 
the  same  as  if  the  minute  image  of  the  Sun  were  the  real  origin  of 
the  light  diverging  with  equal  intensity  through  a  solid  angle  whose 
diameter  is  many  degrees.  The  spherical  or  chromatic  aberration  of 
the  lens  produces  no  sensible  effect  in  any  of  the  common  experiments, 
in  all  which  the  angle,  made  by  rays  which  afterwards  interfere,  is  small. 
In  calculating  experiments  thus  conducted  we  proceed  therefore  with 
full  confidence  that  no  consideration  is  left  out  of  sight,  the  omission 
of  which  could  cause  sensible  error. 

Newton's  experiments  however  were  conducted  in  a  different  way. 
His  origin  of  light  was  a  hole,  from  Jg^  to  ^  of  an  inch  in  diameter, 
through  which  the  Sun's  light  was  made  to  pass.  The  effect  of  this 
light,  on  the  undulatory  theory,  is  not  the  same  as  if  the  bright  hole 
were  the  origin  of  light.  It  becomes  then  a  matter  of  some  interest 
to  examine  mathematically  what  is  the  effect  produced  by  transmitting 
the  sun-beams  directly  through  a  hole  of  sensible  size ;  and  whether  this 
effect,  in  practice,  will  differ  much  from  the  effect  produced  by  forming 
an  image  of  the  Sun  with  a  lens  of  short  focal  length. 

The   integrals  which  occur  in  this  investigation  are  of  such  a  kind 
that  their  values  cannot  be  exhibited  even  in  tables  of  numbers  (except 
Vol.  V.  Part  II.  O 


102  PROFESSOR   AIRY   ON   THE   CALCULATION   OF 

of  course  in  any  particular  case,  when  by  very  tedious  summation  nume- 
rical results  might  be  obtained).  The  only  thing  which  can  be  attempted 
is,  to  shew  that  the  integrals  are  precisely  the  same  as  those  that  occur 
in  a  very  different  instance  where  Fresnel's  method  of  experimenting 
is  adopted.  Even  thus  far  however  I  have  not  succeeded  except  in  one 
case,  namely,  where  the  hole  is  a  rectangular  parallelogram  of  any  length, 
and  where  the  diffracting  aperture  is  also  a  rectangular  parallelogram 
in  a  similar  position ;  including  in  this  general  case  the  particular  instance 
in  which  one  or  both  parallelograms  have  no  boundary  on  one  side. 

To  consider,  in  the  first  place,  a  case  similar  to  Newton's.  A  plane 
wave  is  supposed  to  enter  an  external  parallelogram  and  then  to  pass 
through  a  slit  with  sides  parallel  to  those  of  the  parallelogram ;  and  the 
intensity  of  the  light  which  falls  upon  a  screen  at  a  given  distance  is  to 
be  found.  First,  it  is  to  be  observed,  that  in  estimating  the  comparative 
intensity  of  light  in  a  direction  parallel  to  one  side  of  the  parallelograms 
(suppose  for  instance  the  shorter)  there  is  no  necessity  to  take  into  ac- 
count the  length  of  the  parallelograms  in  the  other  direction ;  as  it  will 
easily  be  seen,  upon  attempting  an  integration,  that  the  intensity  of  light 
is  expressed  by  the  product  of  two  quantities,  of  which  one  depends  only 
on  the  lengths  of  the  parallelograms  and  the  position  of  the  point  of 
the  screen  in  one  dimension,  and  the  other  depends  only  on  the  breadth 
of  the  parallelograms  and  the  position  of  the  point  of  the  screen  in  the 
other  dimension.  The  intensity  of  light  along  a  given  line  parallel 
to  one  side  of  the  parallelogram  will  therefore,  so  far  as  it  depends  on 
the  other  side,  be  affected  only  with  a  constant  multiplier.  Neglecting 
therefore  the  lengths  (by  which  term  I  designate  that  dimension  of  the 
parallelograms  which  is  perpendicular  to  the  line  on  which  the  comparative 
brightness  is  to  be  ascertained),  suppose  a  normal  to  the  front  of  the 
wave  to  be  di-awn,  and  suppose  the  limits  of  the  breadth  of  the  external 
aperture  measured  from  this  line  to  be  a,  fi,  (the  distance  of  any  point 
of  the  aperture  being  v),  and  suppose  the  limits  of  the  breadth  of  the 
slit  to  be  7,  5,  (the  distance  of  any  point  of  the  slit  being  w)'.  and 
suppose  the  distance  of  the  point  on  the  screen,  whose  illumination  we 
wish  to  ascertain,  to  be  x.  Let  the  distance  of  the  external  aperture 
from  the  slit  be  a,  and  the  distance  of  the  slit  from  the  screen  h.     Suppose 


NEWTON'S  EXPERIMENTS  ON  DIFFRACTION.  103 

the  front  of  the  wave  where  it  enters  the  external  aperture  to  be  divided 
into  a  great  number  of  small  parts  ^v ;  and  suppose  each  of  these  to 
be  the  origin  of  a  small  wave  which  diverges  from  it  as  a  center.  The 
distance  from  the  point  v  of  the  aperture  to   the  point  w  of  the  slit  is 

^{a'  +  {v-wY]=a+  —(v-wy; 

and  the  disturbance  produced  at  w  by  the  small  wave  spreading  from  the 
space  Sv  at  v  will  therefore  be  proportional  to 

^tj.sin. —  {vt- A  —  a— —-(v  —  wY]. 

Integrating  this  with  respect  to  v,  the  coefficient  of  sin  —  {vt  —  A-  a) 
will  be 

L  cos  ~{v-wy, 

and  the  coefficient  of  cos  —-  (\t—  A  —  a)  will  be 

A 

-Xsin^(«-M;)^ 
The  first  of  these  integrals  =  X  cos  ^  iv  \/  —r  —  w  V -r-J   '' 

and  putting  ^(s)  for  f.  cos  f-  »M,  this  integral  between  the  limits  v  =  a, 
v  =  l3,  will  be  proportional  to 

<h\^\/  -- —W\/  -—]  — d>\a  \/  -—  -  W\/  ^\. 


(TV        \  TT 

-  xM  ,    the   integral  -  /„  sin  -—  (v  —  ivy 

between  the  same  limits  will  be  proportional  to 


-  ^  l*^  ^  -  ■"  ^^) + H°  ^Fx  -  ="  ^i)' 


o  2 


104 


PROFESSOR   AIRY   ON   THE   CALCULATION    OF 


The  whole  displacement  at  the  point  w  will  therefore  be 


sm  — 


{yi-A-a)x{    (p(iB\/^-w\/^]-<p(a\/%-w\/-^]] 
'      I     ^V  aX  aXJ      ^\  aX  aXl  j 


2ir  f         / 

+  cos  ——  (yt  —  A  —  a)  X  <  -  \l/  [j3 


aX 


—  w 


aXJ      ^  \  aX  aXl  ] 


Suppose  now  this  displacement  to  be  the  origin  of  a  small  wave 
which  diverges  from  it  as  a  center.  The  distance  of  the  point  w  of 
the  slit  from  the  point  x  of  the  screen  is 

^{¥  +  {w-xY]=h^^{w-x)\ 

and  this  distance  must  be  added  to  ^  +  a  in  the  expressions 

sin  -^{yt-A-a)  and  cos  ~  {vt—  A-a), 
X  X 

in  order  to  find  an  expression  proportional  to  the  displacement  produced 
by  it  on  the  screen  at  the  point  x.  The  expression  must  also  be  mul- 
tiplied by  Sw,  the  breadth  of  the  small  space  from  which  the  wave 
proceeds.  Thus  we  find  for  the  whole  displacement  at  the  point  x  of 
the  screen. 


sin 


-.)..{    ^(/5V^-«,v'|;)-?.(«\/|;-».V|;)}' 


COS  J—  {w 
■{vt-A-a-b)y.     {  —  — 


+COS  -:^{vt- A -a-b)x  j  ■■ 


si„i(„-.)..{-*(^  v^-.  V„4).«(<.v/|;-.  V|)}j 


.co.i  (.-.)'x{-V.(/3  V|;-»  Vi)  .+(«  V|;-«-  X/|)  Y 


NEWTON'S  EXPERIMENTS  ON  DIFFRACTION,'  iO& 

where  the  integrals  are  to  be  taken  between  the  limits  w  =  y,  w  =  S.     The 
brightness  at  the  point  x  of  the  screen  will  then  be  proportional  to  the 

sum    of  the   squares    of    the   coefficients    of    sin---(yi—A  —  a  —  h)  an(f 
cos — (yt—  A  —  a  —  b). 

A 

To  consider  in  the  second  place  a  case  in  which  the  illumination  is 
produced  in  Fresnel's  method.  Let  the  distance  from  the  origin  of  light 
to  the  aperture  be  a',  and  from  the  aperture  to  the  screen  V.  Let  a 
line  be  drawn  from  the  origin  of  light  perpendicular  to  the  screen,  and 
let  the  limits  of  the  aperture  measured  from  this  line,  in  the  same 
direction  as  the  breadths  of  the  parallelograms  in  Newton's  case,  be  e  and  ^ 
(the  general  letter  for  the  distance  of  any  point  in  this  direction  being  p), 
and  let  the  limits  in  the  direction  perpendicular  to  this  be  rj  +  np,  6  +  nj), 
where  m  is  constant.  (It  is  readily  seen  that  this  implies  the  figure  to 
be  rhomboidal,  with  two  sides  parallel  to  the  length  of  the  parallelograms 
in  Newton's  case.)  Let  q  be  the  general  letter  for  distance  in  this  second 
direction :  also  let  of  and  y'  be  the  distances,  in  the  directions  of  p  and  q, 
of  a  point  on  the  screen  from  the  same  line.  The  distance  from  the 
origin  of  light  to  the  point  p,  q,  in  the  aperture  is 

and  the  displacement  there  will  therefore  be  proportional  to 

The  distance  from  the  point  p,  q,   in  the  aperture  to  the  point  or',  y',  on 
the  screen,  is 

and  this  must  be  added  to 

A  +  a'  +  ^  +  ^,, 

in  the  expression  for  the  displacement,  in  order  to  find  the  displacement 
produced  at  the  point  x',y',  of  the  screen  by  the  wave  diverging  from 


106  PROFESSOR  AIRY  ON   THE  CALCULATION  OF 

the  point  p,  q,  of  the  aperture.  For  the  effect  of  the  wave  spreading 
from  the  small  rectangle  whose  sides  are  ip,  ^q,  we  must  multiply  by 
^p,  Sq.     Hence  we  find  that  the  quantity  to  be  integrated  is 

where,  after  integrating  with  respect  to  q,  the  limits  of  q  must  be  ex- 
pressed in  terms  of  p  before  the  next  integration. 

Puttmg  A'  +  a'  +  b'+  ^      ^     =  B',  this   expression  becomes 

The  first  integral  is 

27r 


sm 


-COS  -  {..-2?  -  ^—  (p  _  _^)  J  /^  sm  {2  .-^r^  (?  -  ^)  } 
/tt   2(«'+ft')/        «y^'l      ,        TT  /    ./2{a'+b')       ,^/      2m:'      y 

which  between  the  limits  q  =  r}-irnp,  q  —  Q-^np,  is  proportional  to 

^1  ^    «7yx      ^  ^  i' («'  +  *')  x^  «'*'x  J 

The  quantity  proportional  to  .4  sin  j^.i^lt*}  L  _  4^)  |  will  be  ex- 
pressed in  the  same  manner,  putting  >//  in  the  place  of  0. 

The    whole   displacement   of   ether   at  the  point  x',  y',  will  therefore 
be  found  to  be 


NEWTON'S  EXPERIMENTS   ON  DIFFRACTION. 


107 


cos 


\     a'h'  V     a'  +  b'J       I    ^\    ^      a'b'X        ^  ^  b'la'  +  b')\ 


sm~(yt-B')x  f  ^ 
jp 


+  sin; 


27r,     . 
f-COS—  (v^ 
A 


-B')x\  < 

Jp 


.     TT  a'+b' f       a'x'Y     r        f . .  /2  (a'  +  b')       ,./       2a'  i 


+  COS; 


TT  a'+b'  1^     a'x'  Y     r     ,  L./2(«'  +  6')       ,./      2^^ 


where  the  integrals  are  to  be  taken  between  the  limits  p  =  €,  p  =  ^. 
The  brightness  at  the  point  x',  i/,  of  the  screen  will  then  be  propor- 
tional to  the  sum  of  the  squares  of  the  coefficients  of  sin  — (v#— ^') 

A 

and  cos -— {v  t  -  B'). 

A 

We  have  now  to  shew  that,  for  a  constant  value  of  y',  and  a  vari- 
able value  of  x',  these  expressions  may  be  made  similar  to  those  ob- 
tained in  the  first  case.  For  this  purpose  it  will  be  necessary,  first,  to 
make  the  coefficients  of  the  expressions  under  the  integral  sign  equal: 
secondly,  to  make  the  limits  of  integration  the  same. 


108  PROFESSOR  AIRY  ON  THE  CALCULATION  OF 

rr,!       /.    ,  •!      i-  •  •^        IT   a'  +  b'  a'x' 

1  he   first  consideration   gives  vis  j—  =  - .  — rrr '  x  =  -; — rt ; 

^  h\      \     a'b'  af  +  b" 

"^aX      ''^      «'6'X         y  ^  b'{a'  +  b')\'      ^  a\         ^     a'b'X     ' 

and  the  second  consideration  gives  7  =  e;  S  =  ^;  whence  5  — 7  =  ^-c.     The 
first  set  of  equations,  reduced,  are 


6  =  ^'  + 


1  1  *,OaA  ^./l  'aA 

a\/  -  =  tj'S/  J-  —  y'  \/  jn\  whence  (/3  — a)  v  -  =  0  — >?;  and  w=  —  V- 


The  purport  of  these  equations,  in  common  language,  may  be  stated 
thus : 

If  in  Newton's  method  light  pass  through  a  rectangular  hole  whose 
horizontal  breadth  is  /3  — a,  and  through  a  slit  whose  horizontal  breadth 
is  5-7,  at  the  distance  a  from  the  former,  and  fall  finally  on  a  screen 
at  the  distance  b  from  the  slit: 

And  if  in   Fresnel's   method   light   pass  through  a  rhomboidal   hole, 
with  two   vertical  sides,   at   the  distance  a'  from  the   Sun's  image;  and 
fall  on   a  screen   or  eyepiece   at  the  distance  V  from  the  hole,    so  that 
1       1__  1 
a'^  b'~  b' 

And  if  the   length  of  the  vertical  sides  of  the  rhomboid  be  \/-  x 

til 

the  horizontal  breadth  of  the  external  hole  in  the  first  case  (or  /3  — a); 
and  the  horizontal  breadth  of  the  rhomboid  be  equal  to  the  horizontal 
breadth  of  the   slit    in  the  first  case   (or  5 -7);  and  the  tangent  of  the 

angle  made  by  the  sides  of  the  rhomboid  be  \/  j,   (the  acute  angle  of 

the  rhomboid  being  on  the  side  where  x  is  negative  and  y  positive). 


NEWTON'S  EXPERIMENTS   ON   DIFFRACTION,  109 

Then  the  proportion  of  the  intensities  of  light  along  the  horizontal 
line  in  the  first  case  will  be  the  same  as  the  proportion  of  the  inten- 
sities of  light  along  a  horizontal  line  in  the  second  case:   the  distance 

x'  =  x  y.  -r  in  the  second  case  corresponding  to  the  distance  x  in  the  first 
case. 

If  in  the  first  case  the  center  of  the  hole  is  opposite  to  the  center 
of  the  slit,  the  horizontal  line  in  the  second  case  must  be  drawn  over 
the  middle  of  the  illumination  on  the  screen.  But  if  in  the  first  case 
the  center  of  the  hole  is  not  opposite  to  the  center  of  the  slit,  but 
deviates  in  the  direction  which  makes  x  positive,  then  the  horizontal 
line  in  the  second  case  must  not  be  drawn  over  the  middle  of  the 
illumination,  but  on  that  side  on  which  y'  is  negative.  In  general, 
or  when  one  side  of  either  aperture  in  the  first  case  is  wanting,  the 
equations 

may  be  used. 

When  the  inequality  of  the  sides  of  the  rhomboid  is  considerable, 
the  form  of  the  illumination  is  not  very  different  from  the  illumination 
when  the  hole  is  parallelogrammic.  The  coloured  bars  will  be  a  little 
inclined,  so  that  those  which  for  a  parallelogram  would  be  perpendi- 
cular to  its  longest  sides,  will  approach  towards  the  direction  perpendi- 
cular to  the  longer  diagonal  of  the  rhomboid.  Besides  these,  there  is 
a  faint  brush  of  light  projecting  from  each  part  which  corresponds  to 
an  obtuse  angle,  and  nearly  in  the  direction  of  a  line  bisecting  that 
angle  produced.  These  general  notions  will  assist  the  reader  in  judging 
what  ought,  theoretically,  to  be  expected  in  the  different  circumstances 
of  Newton's  experiments. 

In  Newton's  experiments  the  external  hole  was  in  fact  circular. 
What  would  be  the  effect  of  this  form  it  is  impossible  (theoretically) 
to  say:  but  judging  from  the  insignificance  of  the  effect  produced  by  a 

Vol.  V.    Tart  II.  P 


110  PROFESSOR  AIRY   ON   THE   CALCULATION   OF 

rectangular  hole,    I  am  inclined  to  think  that,   when   the  apertures  are 
centrally  opposite,  the  same  investigation  will  apply  well  to  it. 

I  may  now  without  impropriety  mention  the  circumstances  which 
induced  me  to  make  this  investigation. 

In  Newton's  Optics,  Book  iii.  Observation  6,  Newton  describes  in 
very  striking  language  the  effect  of  narrowing  a  slit  on  which  the 
sun-light  fell  after  having  passed  through  a  hole  a  quarter  of  an  inch 
in  diameter.  He  states  that  when  the  breadth  of  the  slit  was  about 
—t\\  of  an  inch,  the  illumination  on  the  screen  was  interrupted  by 
a  black  shadow  in  the  middle.  It  is  certain,  theoretically  and  prac- 
tically, that  if  the  experiment  had  been  made  in  Fresnel's  method  the 
center  would  be  the  brightest  part.  It  seemed  therefore  worth  while 
to  ascertain,  by  the  best  kind  of  investigation  that  svich  an  un- 
manageable case  admits  of,  whether  the  size  of  the  external  hole 
could  account  for  the  dark  shadow.  From  consideration  of  the  form 
of  the  illumination  in  the  second  case  above,  it  appears  certain  that 
it  could  not.  The  only  resource  (which  the  dullness  of  the  weather 
at  that  time  denied  me)  was  to  repeat  the  experiment.  This  I  have 
now  done  three  separate  times  in  the  presence  of  as  many  different 
persons :  I  have  used  both  parallelogrammic  and  circular  holes  of  dif- 
ferent sizes  (the  largest  circular  hole  being  ^inch  in  diameter)  and 
have  sometimes  diminished  the  aperture  to  as  little  as  j^  inch  (by 
estimation).  The  distances  have  been  30  inches  each,  which  appear 
to  have  been  the  distances  in  Newton's  experiments.  In  every  in- 
stance the  center  has  been  bright.  I  can  account  for  this  inaccuracy 
in  Newton's  observation  only  by  supposing  that  his  eye  was  in  such 
a  state  as  not  to  recover  from  the  sudden  impression  which  is  pro- 
duced by  rapidly  diminishing  the  central  light  on  the  screen  (which 
makes  it  for  an  instant  appear  black),  and  by  referring  to  his  candid 
avowal  in  the  Advertisement,  that  "  the  third  book  and  the  last  pro- 
"  position  of  the  second  were  put  together  out  of  scattered  papers," 
and  that  "  The  subject  of  the  third  book  I  have  also  left  imperfect, 
"  not  having  tried  all  the  experiments   which  I  intended  when   I  was 


NEWTON'S  EXPERIMENTS  ON   DIFFRACTION.  HI 

"about  these  matters,  nor  repeated  some  of  those  which  I  did  try 
"until  I  had  satisfied  myself  about  all  their  circumstances."  I  may 
add  that  Newton's  measures  of  the  distances  at  which  the  first  dark 
bar  was  formed  are  so  irreconcileable  with  those  of  his  admirer  Biot 
that,  referring  to  the  avowal  above-cited,  I  think  no  reliance  ought  to 
be  placed  on  the  accuracy  of  his  observations  of  diffraction. 

Since  writing  the  above,  I  find  that  Biot  has  repeated  the  experi- 
ment with  the  same  result  which  I  have  obtained  {Traite  cle  Physique, 
Tom.  IV.  p.  749).  He  has  not  commented  on  or  even  mentioned 
Newton's  observation. 


G.  B.  AIRY. 


Observatory, 

May  6,  1833. 


p  2 


VI.     Second    Memoir    on    the    Inverse    Method    of   Definite    Integrals. 
By  the  Rev.  R.  Muuphy,  M.A.    Fellow   of  Cuius  College,   and  of 
the  Cambridge  Philosophical  Society. 


[Read  Nov.   11,    1833-3 


INTRODUCTION. 

The  object  of  my  former  Memoir  on  the  present  subject,  pub- 
lished in  the  Fourth  Volume  of  the  Society's  Transactions,  was  to 
investigate  the  principles  by  which  we  might  revert  from  a  function 
outside  the  sign  of  definite  integration,  to  the  function  under  that 
sign,  whenever  the  latter  belonged  to  any  of  those  classes  usually 
received  in  analysis.  In  that  case  the  function  outside  the  sign  of 
integration  possessed  the  characteristic  property  of  converging  to  zero 
when  a  variable  quantity  x  was  made  to  increase  indefinitely ;  in  the 
present  Memoir  I  have  endeavoured  to  complete  this  theory,  by  the 
research  of  the  forms  and  properties  of  the  functions  under  the  sign 
of  integration,  when  the  characteristic  above  mentioned  is  not  pos- 
sessed by  the  function  resulting  from  integration :  and  as  the  subject 
increased  in  difficulty,  those  methods  of  analysis  which  possessed  greater 
simplicity  and  uniformity  have  been  most  adhered  to,  in  the  follow- 
ing investigations. 

The  fourth  Section  is  devoted  to  the  research  of  the  nature  and 
properties  of  the  function  under  the  sign  of  integration,  when  the 
integral  always  vanishes  between  the  limits  (0  and  1)  of  the  indepen- 
dent variable  which  have  been  uniformly  adopted  in  this  as  in  the 
first  Memoir.  The  class  of  functions  thus  investigated  possess  the  re- 
markable property  of  vanishing  an  indefinitely  great  number  of  times 


114  Mb   MURPHY'S   SECOND    MEMOIR   ON   THE 

in  a  finite  extent;  such  functions  correspond  to  an  extended  and 
curious  class  of  pheenomena  in  nature,  when  any  principles  of  action 
which  have  been  observed,  under  peculiar  circumstances  cease  to  produce 
the  observed  effects,  as  when  equal  charges  of  opposite  electricities 
are  communicated  to  a  body,  or  when  a  body  electrised  by  influence 
is  removed  from  the  vicinity  of  the  influencing  system ;  or  lastly,  as 
when  heat  in  its  thermometric  effects  disappears  in  the  chemical 
changes  which   bodies  undergo. 

The  properties  of  this  class  of  functions  are  of  great  use  and 
importance  in  analysis,  as  they  conduct  directly  to  the  theory  of 
reciprocal  functions.  This  term  I  have  here  employed  to  denote  such 
functions,  two  of  which  being  multiplied  together  the  integral  of  the 
product  vanishes,  except  in  one  particular  case.  That  function  which 
is  in  this  sense  reciprocal  to  another,  is  also  in  general  different  in  its 
nature.  There  are  however  many  functions  which  are  reciprocal  to 
functions  of  their  own  nature,  and  to  this  class  belong  the  only  two 
species  of  reciprocal  functions  hitherto  introduced  into  analysis ;  namely, 
the  sines  or  cosines  of  the  multiples  of  an  angle,  the  integral  of  the 
product  of  which  always  vanishes  (when  taken  between  proper  limits) 
except  in  the  particular  case  of  equimultiples;  and  secondly,  such 
functions  as  satisfy  the  well-known  partial  differential  equation  in  the 
third  book  of  the  Mecanique  Celeste;  where  the  integral  of  the  product 
also  vanishes  except  in  the  particular  case  where  the  functions  are  of 
the  same  order.  It  is  this  exception  which  renders  reciprocal  func- 
tions particularly  useful,  as  is  evident  from  the  application  of  the 
trigonometrical  functions  in  the  theory  of  heat,  and  of  Laplace's  functions 
in  investigations  relative  to  the  distribution  of  electricity.  In  the  same 
Section  I  have  shewn  generally  the  means  of  discovering  all  species 
of  reciprocal  functions,  and  given  several  examples :  as  an  instance  of 
one  of  the  most  simple  species  possessing  properties  very  analogous  to 
those  of  Laplace's  functions,  but  giving  a  simpler  integral  in  the  case 
where   that   integral  does    not   vanish,    it    is  proved    in   the    succeeding 

h 

Section  that  if  T„  be  the  coefficient  of  h"  in  - — j ,  then  when  n  and  in 
are  vmequal   ftT„T„  =  0,   but  when  n  =  vu   ftT„T„  =  l. 


INVERSE   METHOD   OF  DEFINITE   INTEGRALS.  115 

The  theory  of  reciprocal  functions  is  applied  in  the  fifth  Section 
to  the  complete  solution  of  the  question,  which  was  the  object  of  this 
and  the  preceding  JMemoir,  namely,  to  revert  from  any  function  what- 
ever to  that  under  the  sign  of  definite  integration,  those  reciprocal 
functions  being  employed  which  are  most  convenient  in  each  particular 
instance. 

The  last  application  in  this  Memoir  of  the  theory  of  reciprocal 
functions,  is  to  the  development  of  given  functions  of  x  in  descending 
powers  or  other  forms  which  vanish  when  x  is  infinitely  great;  the 
results  of  which  may  be  applied  to  the  valuation  of  functions  of 
very  great  numbers,  and  to  a  great  variety  of  physical  problems. 
These  series  have  also  the  peculiarity,  generally,  to  terminate  for  the 
functions  of  integer  numbers. 


116  Mb  MURPHY'S  SECOND  MEMOIR  ON  THE 

SECTION    IV. 

Inverse  Method  for  Definite  Integrals  which  vanish;   and  Theory  of 

Reciprocal  Functions. 

1.  When  the  equation  fif{t).t'  =  (p{x)  is  supposed  to  be  restricted 
to   particular   values   of  x,   then   whatever  may   be    the   form    of  (p  {x), 

J'{t)   may   always   be   determined ;    the   values   to   which   x  is   restricted 

we  shall  suppose  to  be  the  natural  numbers   0,  1,  2,  3 (w  — 1),   and 

the  method  here  pursued  will  also  apply  if  the  values   of  n  should  be 
different  from  those  mentioned. 

2.  *  First,  let  f,f{t).t'  =  0,  the  limits  of  t  being  always  0  and  1, 
and  let  us  seek  for  f{t)  a  rational  function  of  t  of  the  lowest  possible 
dimensions,  which  shall  satisfy  this  equation  when  x  is  any  integer  from 
0  to  n  —  1  inclusive. 

Any  value  of  f{t)  which  answers  the  proposed  conditions  may  be 
divided  by  the  absolute  term,  and  the  quotient,  it  is  evident,  will 
equally  fulfil  those  conditions;  we  may  therefore  take  the  first  or 
absolute  term  in  f{f)  to  be  unity,  and  as  the  conditions  to  be  satisfied 
are  w  in  number,  we  must  have  n  coefficients  in  f{t),  which  will  hence 
be  a  rational  function  of  the  form 

1  +  Alt  +  A,f  + +  Ant"; 

and  therefore  (j>{x)  = +  — -^  +  — ^  + + "—-^, 

p 

or  0  =  T^r  by  actual  addition, 

putting  Q  for  (a; +  !)(«  + 2), (x  +  w  +  1),  and  P  representing  a  function 

oi  X  oi  n  dimensions. 

Hence  P=0,  provided  x  be  any  number  of  the  series  0,  1,  2....(/i  — 1); 
these  are  therefore  all  the  roots  of  that  equation,  P  being  of  n  dimensions  ; 
hence  we  must  have 

P  =  c.x.{x-l){x-2) {x-n  +  \)\ 

c  representing  a  constant  quantity. 

*  I  have  resolved  this  question  in  a  different  manner  in  the  "  Treatise  on  Electricity." 


INVERSE    METHOD    OF    DEFINITE    INTEGRALS.  117 

We  have  thus 

1  Ai  A2  An c,x.{x-\)   ....(ar-w  +  l) 

«  +  l  "^  x  +  ^  "^  x  +  ^6  "^ x  +  n  +  1  ~  {x  ■\-\)  .{x  +  ^)....{x  -{-n  +  1)' 

Multiply  by  x +  1,  and  then  put  x=  —\\  hence  c  =  (  — 1)", 

by  ar  +  a, a:=  —  2;    -(4i=  —  -  .  — — , 

by  .  +  3, .=  -3;    4=-"-^-^.^^^±ii^>; 

&c &c. 

1-             j^/js     -.      «  w  +  1    ^      «.(«-l)   (?i  +  l).(w  +  2)    .„     J 
hence  /(0  =  1- j  •-]-•  ^+ ^T^-^-^ H^ ^./^-&c. 

dt"  1.2.3....W 

3.  Denoting  by  P„  the  value  of  f{t)  which  has  been  investigated 
in  the  preceding  article,  it  possesses  the  remarkable  property ;  that 
ftP„P„  =  0,  except  when  m  —  n,  and  then 

r  p  jj  __       ^      . 

•''    '"     "~2w  +  l' 

the  limits  being  always  0  and  1. 

For  when  m  and  n  are  unequal,  one  of  them  as  n  is  the  greater, 
P„  contains  then  only  powers  of  t  inferior  to  n,  the  integral  of  each 
of  which  vanishes  by  the  natvire  of  P„. 

When   m  =  n,  the  last  term  of  P„,  namely 

(w  +  l)(w  +  2)....2w  , 
1    .      2      ....n    ^     ^'' 

is  the  only  term  of  which,   when   multiplied  by  P„,   the  integral   does 

•  This  value  of  _/(<)  lias  been  shewn  in  the  "  Treatise  on  Electricity  "  to  be  the  coefficient 
of /j'  in  {1-2//. (1-2/)+/*^}-^. 

Vol.  V.    Part  II.  Q 


118  Mr  MURPHY'S  SECOND  MEMOIR  ON  THE 

not  vanish ;    and  since  in  general 

/■p..  _/_ix.       ^-(^-1)     ....{x-n  +  1) 
■''    "     "  ^     '^^  •{x+l){x  +  2)....{x  +  n  +  l)' 

it  is  evident  that  in  this  case  ftPj 


2n  +  l ' 

4.  To  illustrate  the  observation  in  Art.  1,  with  respect  to  the 
generality  of  this  method,  let  it  now  be  required,  to  find  a  rational  function 
of  t,  as  f{t),  of  the  lowest  possible  dimensions,  to  satisfy  the  equation 
fif{t).t'  =  0,  when  x  is  any  number  of  the  series 

p,   p  +  1,   p  +  2, .p  +  n-l. 

Putting  as  before /(^)  =  \  +  A,t  +  A-J'-  + +  Aj\  we  have 

/•   f(f\      ft    _  1  ,  ^'  ,  -^2  ,  -^- 

■"•^^^'  x  +  1       x-^2       x  +  S     x  +  n  +  1' 

the  sum  of  all  which  fractions  must  by  the  reasoning  of  Art.  2,  be 

c .  jx—p)  {x—p—l)....{x  —  p  —  ti  +  1)_ 

{x  +  l){x  +  2){x  +  3)....{x+p  +  x)     ' 

and  determining  c,  Ai,  Ai,  &c.  in  the  same  manner  as  in  the   Article 
referred  to,  we  have 

1.2.3....W 


c  =  (-l)». 


(j9  +  l).(jO  +  2)....(jO  +  W)' 


.  _      n    n+p+1 
"^'~~1-    p^\     ' 

.   _  n.{n-\)    (»+/?  +  l).(«+jP  +  2) 
'~       1.2       •       (^  +  l).(;j  +  2)       ' 

&c &c. 

and  therefore 

'^     '  1       ^  +  1  1.2  (jo  +  l).(jo  +  2) 

t^ d^  j        /t  _  ^  ,     n.{n-l)  ^     „     1 

~ip  +  l).{p  +  2)....(p  +  n)-dt''-y     -V      1-^+      1.2      .^-&c.|; 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  119 

or,  putting  l  —  t=^t',  we  obtain 

J^'       (jo  +  l).(jo  +  2)....(jp  +  «)"        r/^" 

5.  From  this  result  it  follows  that  if  we  put 

then  shall  ftj'{t).t''  =  0,  provided  x  is  any  number  of  the  series 

0,  1,  2 (n-l); 

Op  representing  any  constant  quantity. 

Now  OpfP  may  be  taken  for  the  general  term  of  an  arbitrary  function  J^; 
hence  the  most  general  function  which  satisfies  the  equation  ftf{t)-t''  —  0, 
is  expressed  by 

.,,.  _  d^jtH'T) 

In  fact  we  have  (supposing  the  integrals  to  commence  from  ^  =  0,) 

f,f{t) . r  =  t^f, it)  -  xt^-'f, {f)\X.{x-  1) ./s it),  &C. 

representing  by  fn  {t)  the  ri^  successive  integral  oi  fit),  and  putting  for  x 
0,  1,  2....(w  — 1)   successively,  it  follows  that 

Mt)  =  0,   f,{t)  =  0 .f„{t)  =  0,  when  ^=1; 

that  is,Jn{t)  and  its  n  differential  coefficients  vanish  when  t  =  0  and  when 
t=l;  therefore  y^ (/)  contains  a  factor  of  the  form  ^".(1  —  ^)",  and  con- 
sequently the  most  general  form  of  f{t)  is 

d"(t"t'''r) 
dt"       ■ 

6.  Hence  we  deduce  the  following  general  property:  '' If  /{t)  he 
any  function  which  satisfies  the  equation  [tf{t) .  t*  =  0,  a;  being  any  integer 

from  0  to  in  —  V)  inclusive,  then  the  equation  f{f)  =  0  will  always  have  n  real 
roots  lying  between  0  and  1." 

For  the  equation  r.^'°F=0  has  n  roots  t  =  0  and  n  roots  ^=1;  and 
therefore  f{t)  which  is  the  n^^  derived  equation  must  have  n  roots  be- 
tween 0  and  1. 

q2 


120  Mr   MURPHY'S   SECOND   MEMOIR   ON   THE 

Hence,  if  we  suppose  the  equation  J,f{t)  .^  =  0  to  hold  true  for  an 
indefinite  number  of  entire  values  of  x,  the  equation  f{t)  =  0  will  also 
have  an  indefinitely  great  number  of  roots  all  lying  between  0  and  1, 
and  the  curve,  of  which  the  ordinate  is  f{t),  and  the  abscissa  t,  would 
intersect  that  portion  of  the  axis  of  x,  of  which  the  length  is  unity 
measured  from  the  origin  in  an  indefinitely  great  number  of  points; 
thus  we  have  a  property  characteristic  of  this  class  of  functions.* 

7.  We  have  supposed  J'{t)  to  consist  of  terms  involving  the 
powers  of  t,  but  as  we  may  proceed  in  like  manner  for  any  other 
assumed  form,  we  take  the  following  as  an  example,  because  it  leads 
to  some  remarkable  results. 

To  find  a  rational  function  of  h.  1.  (f)  as  y(h.  1.  t)  of  the  lowest 
possible  dimensions,  which  may  satisfy  the  equation  ftf(h.\.t).t'  =  0, 
X  being  any  integer  from  0  to  n—1  inclusive. 

Put  /(h.  \.t)  =  \  +  A,   h.\.t+  A^  (h.  1.  ff  +  +  A„  (h.  1.  t)', 

and  observing  that   J,{h.\.{t)]"'.f  =  {-\f.     •^^■^•:;\    , 

we   get   f,f{\,.l.t).t'  =  ^^-j^^^,+~^^^^- ±  -(^^nyr.T-. 

and  actually  adding  the  fractions  in  the  right-hand  member  of  this 
equation,  the  numerator  which  is  a  function  of  n  dimensions,  ought 
to  vanish  when  x  is  any  number  of  the  series  0,  1,  2...(w-l);   that  is, 


{x  + 1)»  -A,{x  +  !)"-■  +  1 .  2^2  (a;  + 1)""'  -  1 .  2 .  3  ^3  (a;  +  1)""' 

=  C.x.{x-\){x-^) {x-n->r\). 

Let  Si  represent  the  sum  of  the  natural  numbers  1,  2,  3.,..(«-l),  n, 
Si  the  sum  of  their  products  two  by  two, 
^^3  the  sum  of  their  products  three  by  three,  &c. 

*  Vide  Art.  (4)  in  my  first  Memoir  on  the  Inverse  Method  of  Definite  Integrals. 


/ 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS.  121 

Then    by   the    theory   of    equations,    the    right-hand    member    of    this 
equation  is  equivalent  to 

c  {(x  +  1)'  -  s,{x+iy-'  +  SA^+iy-'  -  SA^+i)"-',  &c.| 

whence   c  =  l,   ^i  =  aS',,   ^2=,    „,    ^3=       J  -^,   &c.   hence   the  required 
function  is 

8.  It  has  been  proved,  that  the  function  thus  obtained  (which  we 
shall  denote  by  L„)  in  common  with  all  others  which  possess  the 
property  that  ftj'(t)  .f  =  0,  when  x  is  any  integer  from  0  to  n  —  1  in- 
clusive, is  of  the  form 

d\  {ft'"  V) 
dt"         ' 

to   verify   this   in  the  present   case,   we  must   sum   the   preceding  series 
which  is  represented  by   Z/„. 

First,  by  the  nature  of  multiplication,  we  have 

hr  +  SJi^-'  +  S.h"-"-  + +S„  =  {h  +  \){h  +  ^) {h  +  n), 

and  the  development  of  an  exponential  gives 

i+7.h.l.(^+-A^  + +  i.a.3..,:,+&c.=/-, 

the   coefficient   of  h"   in   the   product   of  both   the   latter  series  is  iden- 
tical  with  that  by  which  Z/„  is  expressed. 

But  since  that  product  =^(A  +  1)  (/«  + 2) (A  +  w) 

df 

=  ^{r(l+Ah.l.^4-^^l^^&C.)|, 

it  follows  that  the  coefficient  of  h"  is  also   expressed  by 

d"  [f  {h.\.  ty\ 
1.2.3 ndf' 


122  Mr  MURPHY'S   SECOND   MEMOIR   ON    THE 

this  quantity  is  therefore  the  sum  of  the  series  which   we   proposed   to 
find. 

Now   the   equation   h.  1.  {t)  =  0   is  satisfied   by  ^  =  1 ;    hence   h.  1.  t  is 

i'       t'^ 
of  the   form  t'.  Q,  {where  Q=  —  (1  +  -  +  —  +  &;c.)},  and  therefore  if  we 

Q" 

put   — ^ =  J^,   we   get   the  value    of   L„   to   be 

d".{t''t''''F) 
df 

which  was  the  formula  we  had  required  to  verify. 

We  may  also  observe  that  since  in  the  equation  L„  =  0,  /  must  have 
n  values  lying  between  0  and  1,  therefore  h.l.  {t),  according  to  the  powers 
of  which  L„  is  arranged,  must  have  n  real  negative  roots,  which  we 
see  confirmed  by  the  positive  signs  of  all  the  terms  which  compose  L,,. 

9.  If  we  form  the  equation 

u  (1  —  h  h.  1.  u)  =t, 
we  have  by  Lagrange's  theorem 

.      J..UW.N,     ^'      d{t\iA.tf   ^       ¥        d'{t\\.\.tf      , 
«  =  .  +  ;i.h.l.(0+^.-^^^— ^   -f-^-^3.-A^^+&c. 

from    whence   it   appears  that    Li„   is   the   coefficient   of  h"  in   the  value 
of  -^.     Similarly  if  in  Article  (12)  we  form  the  equation 

u  \\  -h.  (1  -  u)]  =/, 

du 

we  have  P„  =  the  coefficient  of  h"  in  -rj . 

10.  If  Q„  i<?  the  coefficient  of  h"  in  -j-,  supposing  u  to  he  deter- 
mined by  the  equation  u{l  —  hU)  =  t,  U  bei)ig  a  function  of  u  which 
vanishes  when  u  =  l,  and  T  the  same  function  oft,  then  shall 

j,Q„f  ^x.{x-\){x-2) {x-n  +  1)    , 

j/F'T  1.2.3 n  '^       '' 


INVERSE  METHOD   OF   DEFINITE  INTEGRALS.  123 

For  if  we  put  ti  =  0  in  the  equation  u{l  —  hU)  =  t  we  get  1  =  0, 
and  putting  u  =  1  we  have  by  supposition  U  =  0  and  therefore  t  =  I, 
hence  the  limits  of  u  are  the  same  as  the  limits  of  f. 

But  j;Q„f  =  the  coefficient  of  h"  in    f^^.f, 

^i  U/t 

and  l^^.if  =  JJ^  =  !„u''{l-hUr 

expanding   the  part  under  the  sign  of  integration,   and   taking  the  co- 
efficient of  h"  we  obtain 

hHnt  -  1.2.3 n  -(-l)^^   ■^- 

11.  If  U  he  a  rational  and  entire  Junction  of  u  which  vanishes 
when   u  =  \,    and  if  Q„  be  the   term  independent   of   u   in   the  product 

U"-  \\—  —\         ,  then  shall  Q„  be  itself  a  rational  and  entire  function  of 

t  possessing  the  property   of  ftQj'^  =  0,   x  being  any  integer  from   0   to 
n—\  inclusive. 

For  it  has  been  proved  in  my  former  Memoir  on  the  Resolution 
of  Equations*,   that   the   root  of  the  rational   equation  <f){x)  =  0  is  the 

coefficient    of  -  in  —  h.  1.  ^— ,    hence    the   value    of  u   in    the   equation 

u(\  -hU)=t,   is  the  coefficient   of  -  in    -h.l.  j(l--]  -hul,   and 

differentiating,  it  follows  that  the  value  of  -tt  is  the  term  independant 
of  u  in 


u) 


*   Camb.  Trans.    Vol.  iv.  p.  131, 


124  Mr  MURPHY'S   SECOND   MEMOIR   ON    THE 

because   the   u   under   the    logarithmic   sign   is   the   same   as  if  we   had 

placed   there,  a  or  any  arbitrary  symbol,   and  is  therefore  treated  as  a 

(jLu 
constant  in  the  differentiation;   hence  the  coefficient  of  h"  in  ~t-  is  the 

term  independant  of  m  in 


(>  -  -:) 


n  +  i  ' 


that  is,  its  value  is   Q„,  and  therefore  by  the  preceding    Article  /Q.r 
vanishes  between  the  limits  of  x,  0  and  n  —  \,  its  general  value  being 

T  being  the  same  function  of  /  that  U  is  of  u. 

By  this  theorem,  every  possible  variety  of  rational  and  entire  func- 
tions which  possess  the  above-mentioned  property  may  be  found,  as  in 
the  following 

Example: 

To  find  a  rational  function  of  t,  in  which  the  powers  of  the  variable 
are  in  arithmetical  progression,  such  that  jiQ,nt'=0  when  x  is  any  number 
of  the  series  0,  1,  2 {n  —  1). 

In   this   instance  put   U  =  1  —  u"",  m  being  any  positive  integer. 
Hence  Q„  =  term  independent  of  u  in 

/        t\  "*""*"'* 

(i-^o».(i--) 

^        «    (w  +  l)(w  +  2)...(w+m)  w.(w-l)    (M+l)(w+2)...(?i+2OT)    ^,„_. 

1'  1.2...m  ■  1.2     ■  1.2, ..2m 

in  which  if  we  take  in  particular  m  =1,  we  get  the  value  of  P„  before 
found  in  Art.  (2). 

This  formula  for  Q„  may  be  written  in  another  form  by  which  it 
will  comprise  the  case  where  /w  is  a  fraction,  thus 

n   (m+l)(m+2)...(m+n)    ,„,  w(«+l)   (2»^+l)(2w^+2)...(2w^-^w)      „     „ 

^-=^-i-       r^:::^       -^  ^"ttt-        t:2::ji         -^  "*'''• 


INVERSE    METHOD   OF    DEFINITE    INTEGRALS.  .    123 

and  it  is,  moreover,  evident  that  either  of  those  values  are  identical  with 

1.2...nde'     ' 

which  is  included  in  the  general  form  given  in  Art.  5.  viz. 

d\  {ft'"  V) 
dj"        ■ 

12.  2'o  find  a  rational  and  entire  function  of  f  of  h  dimensions, 
which  if  multiplied  hy  a  rational  and  entire  function  of  t'  of  less  than  n 
dimensions,  the  integral  of  the  product  may  vanish  between  the  limits  t  =  0 
and  t=l. 

Let  the  required  function  be  represented  by  (p,  q),„  so  that 

{p,q\^l  +  A,t^  +  A,f-f  + A,,f^, 

and  by  the  proposed  conditions  we  must  have 

lAp,  qXt-"  =  0, 

ni  being  any  integer  from  0  to  «  —  1  inclusive,  put  t^  =  T,  the  limits 
of  7'  are  the  same  as  those  of  t. 

Hence  J^ip,  ?)»  T~^~' ■  T''=  0. 

1-1 
Now  ij),  q)„  T"    ,    is    a  function  of    T  of   which    the    indices  are   in 

arithmetical  progression,  -  being   the  common  difference,  and    T'      the 

first  term ;  and  as  the  nature  of  the  question  affords  m  independant 
equations  for  the  determination  of  the  n  coefficients  Au  A-,...A„,  it 
follows  that  there  is  only  one  function  of  the  kind,  which  will  satisfy 
the  proposed  conditions,  and  by  Art.  5,  it  is  evident  that  the  function 


5  2'"'''      (1 

jAyi 

1       1      ,A/       1 
n+ 1     w+- - 

\       q        l\       q 

..)...i' 
'  1 

answers    those    conditions,   and   is    manifestly    of   the    required  form,   it 
Vol.  V.    Paet  II.  R 


126  Mr  MURPHY'S   SECOND   MEMOIR   ON   THE 

l_i 
follows  that  if  we  divide  this  function  by   T"^     ,  and  then  substitute  f 

for  T,  we  shall  obtain  the  value  of  {p,  §-)„ ;   we  have  thus, 

_         ip  +  \){p  +  l+q){p^-\  +  2q)....\p  +  1+{n-l).q]    n 

ia+g){l  +  2q)....{l  +  (n-l).g\  l' 

,   (2p  +  l)(2p  +  l+q)....{2p-i-l  +  (n-l).q}    n.(?i-l) 

l.{l+q)....{l+{n-l).q}  '      1.2       ^       .'^^• 

13.  The  functions  {p,  q\  and  (5-,  jo)„  may  be  termed  reciprocal  func- 
tions, and  possess  the  remarkable  property,  that  if  n  and  «'  are  any 
different  integers,  then  shall 

ft(p,q)n.{q,p)n'  =  0. 

For  if  n>n'  then  {q,  p)„'  is  a  rational  and  entire  function  of  t^  of 
less  than  n  dimensions,  and  therefore  by  the  preceding  Article  the 
integral  of  the  product  must  vanish ;  again  if  n'  >  n,  then  {p,  q)„  is  a 
function  of  f^  of  less  than  n'  dimensions,  and  therefore  when  multiplied 
by  (q,p)„'  the  integral  ought  to  vanish. 

To  determine  the  value  of  the  same  integral  when  71  =  n',  it  is 
evident  by  the  nature  of  the  function  {p,  q)„  that  we  need  only  attend 
to  the  last  term  in  the  expansion  of  {q,p)n,  namely 

.  {nq^l){nq  +  l+p)....{nq^-l  +  {n-\).p} 

^     "->•''  1.0.+p)....{l^{n-l).p} 


INVERSE   METHOD   OF  DEFINITE   INTEGRALS.  127 

Now  if  we  put  for  {p,  q)^  the  series  assumed  in  Art.  (12)  aiid  multi- 
plying then  by  f,  integrate  from  ^=0  to  ^  =  1,  we  have 


ar  +  1       a;+j9  +  l       a;  +  2/>  +  l      "*"      x-\rnp  +  l 
and  actually  adding  these  fractions,  the  denominator  of  the  sum  is 

{x  +  '\){x  +p  +  l)(;r  +  2jo  +  1) {x  +  np  +  1); 

and  since  the  numerator  is  of  n  dimensions  in  x,  and  vanishes  when 

x  =  0,  q,  2q....{n-  I) .  q, 
it  follows  that  the  sum  is  of  the  form 

c  .X .  (x  —  q)  (x—2q)....{x  —  {n  —  l).q] 

{x  +  l).{x+p  +  l)....{x  +  np  +  l) 

Multiply  by  ^  +  1  and  then  put  x=  —1;   hence 

^^c.{-l)\l.(q  +  l)(2q  +  l)....{{n-l).q  +  l}  ^ 
p  .  2p  .  3p....np 

whence  deducing  the  value  of  c,  and  substituting  in  the  above  integral, 
we  obtain 

^'^^'^^''•^^^~P^'''l.{q  +  l){2q  +  l)....{{n-l).q+l\ 

^^x.(x-q)(x-2q)....{x-{n-l).q} 
{x+ l){x +p +  l)....{x +  np  +  l)    ' 

hence   y;(^,  g)„  .^"^  =  (-^y .  ^  ^^^  ^| ;  ^^^^^  ^^  ^y 

nq  (nq  -  q)  [nq  —  2y) ....  \nq  -  {n  —  1) .  q] 
{nq  +  l){tiq+p  +  l)....(nq  +  np  +  l) 

from  whence  we  obtain  finally 

n"  1  .  2  .  3 . . . .?{ 

f^  ip,  q)n  {q,  p).,  =  „(^  +  ^)  +  i  •  i(^q^l)„..{{n-.l).q+l\ 

nq(nq  —  q)  {nq  —  2q)....{nq—{n  —  l).  q} 

''~'Up  +  l){2p  +  l)....{{n-l).p  +  l}      ■ 

R  2 


128  Mr  MURPHY'S   SECOND   MEMOIR   ON   THE 


that  is,  it 

_        {pqY         V  .^'  .  3' 


n{p+q)  +  l'l.{p  +  l)(q+l){2p  +  l){2q  +  l)....{{n-l).p+l\{{n-l).q  +  l\ 

,  2  _     y        f  1.2.3....W 1^ 

COR.     j,(p,p}n-  2n  +  l-\l.{p  +  l).{2p  +  l)....{{n-J).p-i.l}f- 

14.      To  find  the  reciprocal  function  to   that  denoted  by  L„  in  Art.  8, 
,     d"  {f  (h.\.  ty\ 
^'  1.2....W     df  ' 


namci 


L„  consists  of  the  powers  of  h.  1.  /,  and  possesses  the  property  of 
ftL„t'  =  0  when  x<n;  suppose  now  that  we  investigate  a  rational  function 
X„  which  shall  possess  the  property  JtK {h.l.  t)' =  0  when  x<n;  then  it 
is  evident  that  j^X„i„/  =  0  when  n  and  n  are  unequal;  and  therefore  they 
are  reciprocal  functions. 

Put  K=l  +  AJ  + A,f +....AJ'', 

Put  Ar  =  2"  +  'B„         A,  =  3''  +  'B, A„  =  {n  +  lY^\B,r, 

hence  we  must  have  when  x<n, 

1'-'  +  a"-'^,  +  3"-"^2  + («  +  i)"-^jB,  =  0. 

Now  the  left-hand  member  of  the  equation  is  the  same  as 

putting  t  =  0  after  the  differentiations. 

Hence  the  differential  coefficients  from  the  1"  to  the  w*  inclusive 
of  the  function  between  the  brackets  vanishes  when  ^=0;  that  function 
of  e'  ought  therefore  to  contain  no  power  of  t  inferior  to  the  (w  +  1)"', 
and  conversely,  a  function  of  e*  which  does  not  contain  such  a  power 
of  t,  will  fulfil  the  required  conditions. 


INVERSE  METHOD  OF  DEFINITE   INTEGRALS.  129 

Now  this  is  the  case  with  (1  — e')''''"^  which  is  also  when  expanded 
of  the  same  form  as  the  part  between  the  brackets;  hence  equating  like 
terms,  we  have 

Hence     A,= -\.T,       A,=  '^^^^  .S" ^„  +  ,  =  (- 1)". («  +  !)"; 


and  therefore 


X„  =  l-p2"^+'^^.3'7^- (-1)".  («  +  !)«. r. 


Cor.  1.     When  ?^  and  n'  are  unequal,  then  ftL„'\,„  =  0. 

But  when  Ti'=fi,  we  need  only  take  the  last  term  of  L,„  namely,  (h.  1.  /)"; 
hence 

j;x„z>„  =  j;(h.i.^)"{i-^.2"^+'i^j^^.3"^^-&c.| 

=  (-l)..,...S....„{.-f.l.^^).l-.e.} 

_  (-l)''.1.2.3....w 
~  ft  +  l 

Cor.  2.      j;x„(h.l.^)^' 

=  i  -  ly  .1 .  2 .  3....X  ll'-^-'  -  n  Q.""-'  +  ^~^ .3"-"-' - kc.\ 

=  (  -  1)"-  M  .2.3  ...x  A" .  (A"-*-'), 

h  being  put  =  1  after  the  operation  of  taking  the  «*  finite  diiFerenc<» 
on  the  supposition  that  the  increment  of  k  is  unity ;  from  whence  it 
is  easy  to  deduce 

^^■^'  =  <-')-'^--^- 

Cor.  3.     All  the  roots  of  the  equation  X„  =  0  are  real,  and  lie  between 
0  and  1. 

For  if  we  put    h.  1.  (/)  =  it,    and  X„e"=  U, 

then  ;x„  (h.  1.  ty  =  f.  Uu^  =  ti^f^  U-  xw-^f.:  U+  ^4^^  //  U,  &c. 


130  Mr  MURPHY'S   SECOND   MEMOIR   ON   THE 

and  putting  x  =  0,  1,  2,  &c.  successively,  it  follows  that  fu"U  and  its  («  — 1) 
successive  differential  coefficient  vanish  when  u  =  0  and  w  =  -  oo ,  Hence 
U=0  has  n  real  negative  roots;  and  therefore  X„  =  0  has  n  real  positive 
and  fractional  roots. 

15.  In  general  let  U,„  V„  be  any  functions  of  the  variable  t  and  the 
integer  n,  and  let  A-^.-.A,,,  ai...a„  represent  constant  quantities;  or  de- 
pending on  n  only. 

Put    T„  =  C/„  +  A,U,  +  A,U,  +  ....  +  A^U„, 
and     T:=  K  +  «i^>  +  «-.F,  +  ....  +   a,r„. 
Then  the  n  equations 

j;r„r„=o,   f,T„r,=o,  j,t„v,=q ;r„r;_,=o, 

Avill  serve  to  determine  the  constants  A^,  A.,....A„. 

In  like  manner  let  the  corresponding  constants  «i,  a2....a„  be  de- 
termined from  the  n  equations 

the  functions  T„  and  T„'  which  are  thus  determined,  are  reciprocal  func- 
tions, and  possess  the  general  property  ft  T„  TJ  =  0,  except  when  n  -  n', 
and  then 

ft  2\  T:  =  aJtT^K  =  A,,  ft  T:  C7„  ; 

this  is  the  general  principle  of  reciprocal  functions. 

Cor.     Let  f{t)  be  any  function  of  t  represented  by  the  series 
f{t)  =  c,T,  +  c.  2\  +  c,  T, .... &c. 

where  Co,  c,,  Cg,  &c.  are  constant  coefficients  to  be  determined,  then 
multiply  by  T^,  T(,  T~U  &c.  and  integrate  the  successive  products, 
and  we  get 

c,ftT,Tl  =  ff{t)T^, 

c^ftT.TI^  ftf(f).Tl, 

c.fT,T^  =  f,f{t).T.I, 

&c &c. 

by  means  of  which  equations  the  required  coefficients  are  given. 


INVERSE   METHOD  OF   DEFINITE   INTEGRALS.  131 

16.  Let  «„,  h„,  c„,  &c.  be  any  functions  of  t,  the  reciprocal  functions 
to  which  for  simple  integration  are  «„',  J„',  c'„',  &c. 

Let  a„,  &c.  be  any  function  of  another  variable  T,  and  let  a/,  &c. 
represent  the  corresponding  reciprocal  function. 

Put     S„  =  a„a^     +  Kai     +  C^a^    + 

and    S,!  =  an'uo  +  i/a/  +  c„'a.2  + 

then  S„,  Sn   are   general   forms  for  reciprocal  functions  with  respect  to 
the  double  integration  relative  both  to  t  and  T. 

For  if  we  put  m  for  n  in  the  latter  series,  and  multiply  the  series 
for  S„  and  S,„'  together,  the  integral  of  the  products  of  any  two  terms 
which  do  not  hold  the  same  place  in  either  series  when  taken  relative 
to  T  must  vanish,  since  a„,  a„'  are  reciprocal  functions. 

Hence      frSnSJ  =  a„a„'  fj.aoaa   +  b„b,„'  fraiai  +  c^cj  frO^a./  + 

Integrate  now  with  respect  to  t,  observing  that  when  m  and  n  are  un- 
equal, then 

_^ «„«,„'  =  0,     ftKbJ  =  0,     ftC„c„'  =  0,  &c. 

Hence     /_4*S'„«S',„'  =  0,  when  m  is  not  equal  to  n, 

and     ftfrSuSn   =  ftfr  {a„a^aoa^  +  Kb„'aiai'  +  CnC^'a-^a^  +...]. 

Cor.  1.  In  the  same  manner  reciprocal  functions  of  any  number 
of  independent  variables  may  be  formed. 

Cor.  2.  The  equation  S„  =  0  has  n  real  roots  or  values  of  t  lying 
between  0  and  1,  whatever  value  be  assigned  to  7',  when  a„,  b„,  c„,  kc. 
are  functions  possessing  the  property  ftaj'  =  0,  &c,  x  being  any  integer 

from  0  to  w  - 1  inclusive ;    for  then  «„  must  be  of  the  form  — —jj-„ — -  > 
by  Art.  5,  and  similarly 

,       d\{t'-t"'V')         _d\{t^t"'F") 
"~         dt"        '    ^"~         df        ' 
and  therefore 

Hence  »S',=0  must  have  n  real  roots  between  0  and  1.    (Art.  6.) 


132  Mr  MURPHYs   SECOND   MEMOIR   ON   THE 

17.  If  it  is  necessary  that  the  terms  which  compose  the  reciprocal 
functions  S,„  S,!  should  follow  a  simple  law,  it  will  be  most  convenient 
to  get  first  two  reciprocal  functions  of  t,  as  R,,,  R,',  which  may  contain 
an  arbitrary  constant  r,  and  to  put  for  «„,  J„,  c„  &c.  the  values  acquired 
by  R„  when  r  =  0,  1,  2,  &c. ;  and  similarly  for  «„',  i„',  c,,',  he.  the  cor- 
responding values  of  R\. 

Example : 

Thus,   put    R^^iiff'"^-^,    and    RJ  =  {ttf'" '^,  P„  being   the 

d"  (tt'Y 
function  so  denominated  in  Art.  3,  namely,  — ^,^ ;  then,  integrating 

by  parts,  we  have 

the  part  outside  the  sign  of  integration  vanishes  between  the  limits  of 
/,  and  repeating  the  same  operation  any  number  of  times,  the  part  out- 
side the  sign  of  integration  is  evidently  of  the  form 


dt'-"   '  dt"-'  \         dt' 

the  latter  differential  coefficient  will  vanish  between  limits  when  k  is 
any  number  from  0  to  r  inclusive,  because  it  will  always  contain  the 
factor  {tt'Y~'"^^ ;   also  when   n  and  m  are  unequal  we  may  suppose  w  to 

d'P„ 

be  the   greater,   and  since  ft'' — j-^  is  of  in  +  ;•  dimensions,   it   follows 

that  if  k>  n  +  r,  then  k  -1>  in  +  7-;  and  consequently  the  latter  dif- 
ferential coefficient  will  be  identically  zero. 

^*-i    /        d'  P 
The   only   instance  in   which    the   factor     ,  .._,  iff'     ,  .'" j    does    not 

Aanish  between  limits  is,  therefore,  where  k  lies  between  r  +  1  and  r-\-n 
inclusive,  but  then  the  first  factor  is  changed  to  ft'''P„;  and  since  k  —  r 
is  now  some  immber  from  1  to  «  inclusive,  this  factor  vanishes  between 
limits  (vid.  Art.  5.),  and  therefore  the  part  outside  the  sign  of  integration 
vanishes  in  all  cases,  and  we  thus  obtain  , 

f,R.R.„  -(-1;  j^-^^,-^.-^[tt  -^jr)' 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS.  1S8 

Put  now  h  =  r,  the  first  factor  under  the  sign  of  integration  becomes 
simply  P„,  and  the  second  factor  is  then  of  m  dimensions;  and  there- 
fore, by  the  nature  of  P„,  the  integral  vanishes;  and  therefore,  when 
n>m,  ftIl„Bm'  =  0:  and  the  same  reasoning  applies  when  m>n,  only  sub- 
stituting RJ  instead  of  R^  throughout  the  process,  hence  R„  and  R,„'  are  ' 
reciprocal  functions. 

When  m  =  n,  then  in  the  general  expression 

j;R«.'  =  (-irj;p.^(«-^); 

we  need  only  take  the  term  involving  the  highest  power  of  t  in 

dr  K^    dr  )' 
namely, 

/     ,....(« +  !)•(« +  2)...(2w)   d^  (..rd-.n 
^       '  1.2...  n     dr  \       dt'  1 

.  ,,      ,       («  +  l)   .   («  +  2)...2«        ,  .         ,  ,^  ,  ,x       , 

and  observing  that     /JP„#"  =  ( —  1)" . -, .,,    ,   — ''"    /_ — -^.  ; 

it  follows  that    ftR„R,!=- .  {n  +  r)  {n  +  r-1)  {n  +  r-2)...{n-r). 

The  reciprocal  functions  a„,  a„'  may  be  obtained  by  putting  r  =  0 
in  R„  and  RJ ',  similarly,  if  we  put  r  =  l,  we  get  b„,  b„',  &c.,  and  thence 
we  obtain  the  reciprocal  functions  relative  to  double  integration,  namely, 

dP  d^P  d^P 

S,'=:ao'{tt'Y  ^n  +  «.'(«')^^'^  +  «^'(«T^"^-f"  +  «3'(«')^-"^",  &c. 

In  the  same  manner  if  we  vary  the  constant  a  while  r  remains  constant, 
we  obtain  the  reciprocal  functions 

Vol.  V.    Part  II.  S 


184  Mh  MURPHY'S  SECOND   MEMOIR  ON   THE 

Cor.  1.     The  simplest  form  for   a„   is  the  sine  or  cosine   of  the  w'" 
multiple  of  an  arc  of  which  the  limits  are  0  and  2w7r,  as 

A„  sin  (2  nicT)  +  B„  cos  (2  wtt  T), 

where  A„,  B„  are  arbitrary  constants,  then   we   have  (putting   for  sim- 
plicity a  =  0), 

S„  =  A,P„  +  {A,  sin  ^TTT  +  B,  cos  ^-n-r)  -^ 


+  {Ai  sin  4  TTT  +  Bi  cos  4  ttt) 


dt 


■i  > 


this  is  the  most  general  form  for  all  the  reciprocal  functions  which  occur 
in  the  Mecanique  Celeste.     (Vid.  Prop,  xi.     Treatise  on  Electricity.) 

CoK.  2.     If  T„,  T,'  are  arbitrary  functions  of  t,  which  do  not  become 
infinite  when  ^=0  or  1,  then,  putting 

Rn  =  {tt'f  Tr*^,    and  R,:  =  {tt'f  T; .^ , 

the  same  reasoning  as  that  used  in  the  preceding  example  will  show 
that  R^,  R„'  are  reciprocal  functions,  and  thus  we  get  for  a^^,  aS",,'  the 
very  general  forms 

S„  =  «„  T,P„  +  «.  y.  ^  («')*  +  «^  T^  -^  m  +  «3  T,  ^  {tt'f  +  &c. 

S:  =  a„'  2;'P„  +  a/  T;  "^  {tt'f  +  a.:  T^  ^  {tt')  +  ai  Ti  ^{tt'f  +  &C. 

Cor.  3.     If  f{t,  t)  is  any  function  of  the  variables  /,  t,  which  is  ex- 
panded under  the  form 

f{t,  t)  =  a,S^  +  a,Si  +  a^S;  + 

then,  to  determine  the  coefficients  a^,  Ui,  a-i,  &c.,  multiply  successively 
by  So,  Si,  SJ....  and  integrate  from  t=0  to  t=l,  and  from  t  =  0  to 
T  =  1 :  we  thus  get 

do  ft  fr  So  So    =  ftfTSo'J'{t,  t), 

aiftfrSiSi'  =  ftf,Si/{t,  t), 
aJJ^S.,S.;  =  f,f^S./f{t,T); 
from  whence  the  required  coefficients  are  known. 


INVERSE   METHOD   OF  DEFINITE   INTEGRALS.  185 


SECTION    V. 

Inverse  Method  for  Definite  Integrals  which  are  expressed  in  positive 
powers  of  x,  or  under  any  form. 

18.  Let  <^{x)  represent  any  function  of  x,  such  that  Stf(Jt)  .f  =  (p{x) 
when  X  is  any  integer  from  0  to  n  —  1  inclusive,  then  excluding  the 
case  of  (p  {x)  =  0,  which  has  been  considered  in  the  preceding  Section, 
it  is  evident  that  by  putting 

f{t)  =  A,  +  A,t  +  A,f  + +An-,tf-\ 

the   conditions  of  the   question  give   n   equations,  which  suffice   to    de- 
termine   the    coefficients    A^,   Ai,    A^, A„.^\    if   we    represent    the 

particular    value    of  f{t)    thus    deduced    by    T„^i,   and   seek   its    most 
general  value,  we  have 

;/(0  .t^  =  <p  {x), 

.-.  f,{f(t)-T„.,}.t^  =  0. 
Hence  by  the  preceding  Section,  the  most  general  value  of  f{t)—  Tn-i  is 

dt-       ' 

and  therefore  the  most  general  value  oi  f(t)  is  found  by  adding  this 
appendage  to  its  prime  value  T„_i. 

19.  When  <p{x)  is  a  rational  and  entire  function  of  x,  of  m  di- 
mensions, we  have  by  the  proposed  conditions 

'P^'^''  x+l^  x  +  2^  x  +  3^ x  +  n' 

and  actually   adding   the   terms  which   compose  the  right-hand  member 
of  this  equation,  the  common  denominator  is 

(x  +  l){x  +  2) (x  +  n), 

s2 


136  Mr  MURPHY'S   SECOND   MEMOIR   ON    THE 

and  tlie  numerator  will  be  a  function  of  ft  —  \  dimensions,   represented 
by  v„,  so  that 

v„ 


<p{x) 


{x+  l)(ar  +  2) (a;  +  M)' 


when  X  is  any  integer  from  0  to  (^^  -  1)  inclusive;  and  if  we  multiply 
by  a;+  1  and  put  x=  —  1,  and  again  by  a;  +  2  and  put  a-=  —  2,  &c.  as  in 
the  preceding  Section,  we  get 


A, 

V  - 

1 

1 

.2.3.. 

..{n- 

•1)' 

A, 

=  -  • 

n-\ 
1      * 

1.2. 

V. 

3... 

-2 

-1)' 

4 

_(^ 

-1)(« 

-2) 

«_S 

'~           1.2  ■  1.2.3....(«  -  J)' 

&c.= &c. 

Now   the   equation 

^{x) .  {x  +  1)  (a;  +  2) {x  +  n)-  v„  =  0, 

is  of  m  +  n  dimensions,  and  is  by  hypothesis  satisfied,  when 

^•  =  0,  1,  2, («-l); 

therefore  if  u^  represent  some  function  oi  x  of  m  dimensions,  we   must 
have  the  identity 

(p{x)  .{x  +  V)  (ar  +  2) (ar +  «)-»,  =  M,.ar .  (^-l)(ar- 2) (x-n  +  \), 

hence  if  we  divide 

<f>{x){x  +  \){x^2) (x  +  w)    by   x{x-'\){x-^) {x-n^l\ 

and  retain  only  the  part  of  the  quotient  which  is  an  entire  function  of  x, 
u,  will  be  completely  determined. 

Put  now  —1,  —%,...— n  successively  for  x  in  the  preceding  identity, 
and  we  get 

t;.,  =  (-l)»+M  .2.3....».«_,, 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS.  137 


«_,  =  (- 1)"+'.  1.2.3....n/-~-.u.2, 


&c.  = &c. 


from  whence  the  values  of  A^,  Ai,  A-z,  &c.  are  known,  and  being  sub- 
stituted, give 


J  .M_2^ 


n.(«.H)(.-f2)    («-l)(.-2)  I 

^  1.2  1.2  '  J 

Example : 
Let  0(a;)  =  1,  then  «,  =  1,  and  therefore 


«.(«  +  l)(w  +  2)    (w-l)(/?-2) 
"*"  1.2  ■  1.2 


.f-&e.| 


20.  The  function  Tn-\  possesses  a  property  analogous  to  the  charac- 
teristic property  of  those  in  the  former  Section,  that  is,  the  equation 
2\_^  =  0  admits  of  n  —  m-l  roofs  between  0  and  1,  and  consequently 
vanishes  an  indefinitely  great  number  of  times  between  the  limits  /  =  0 
and  t=\  when  n  is  taken  indefinitely  great. 

For  since    r„.,  =  (- 1)-' |«M.,  -  ^^^^jtil .  ^V  «.,  ^ 
n{n+l){n+2)    {n-l)(n-2)  ] 

■^         TTa        •        1.2       •«-3^&c.j 

_      (-1)-       ^\t^(u  -VlzI  u    1 1  (»-i)-(^^-a)  „  t.  .,„^l 
=  i.2.3....(.*-i)-rfrr  \  '     1   ^  ^'^^      i:%      -"-a^-^c.JI 

=  -r2.3..U-i)-£^^'^"'"-^^"'^- 


138  Mr  MURPHY'S  SECOND  MEMOIR  ON   THE 

tlie  operation  A  being  performed  on  the  supposition  that  the  finite 
increment  of  x  is  unity,  and  x  being  put  =1  after  the  operation  A""' 
has  been  performed. 

Put   i=l  —  f,   and   therefore, 
A"-'(M_,r-')  =  A"-'M_,-^A'-'M_.(a;-l)+— —  A"-'M_,(a;- 1)  (a--2)-&c. 

and  since  m_^  is  of  m  dimensions,  the  first  term  of  this  series  which  does 
not  vanish  is 

ftn-m-\ 

-  1  ■  2.-(«-/»-  1)  •^""'"-^^'^~  ^^  (^-^)--^^  -n  +  m  +  l), 
and  therefore  the  whole  expansion  is  of  the  form 

t'"-"-'  r,  1.2.3 (w-l), 

which  being  substituted  gives 

_d'{t''t"'-'"-T} 

and  since  the  equation  t''t''""'-'^F'=0  has  at  least  2n-m—l  real  roots, 
viz.  ti  of  them  =0,  and  n  —  m—l  of  them  =  1,  it  follows  that  the  w"" 
derived  equation  T„  =  0  has  n  —  m—l  real  roots  lying  between  0  and  ] . 

COK.        Since  r„_.  =  ,.,.3.1(,_^)  •  ^  {r^-^u.J-^}, 
if  we  actually  differentiate  we  get 

^-^=1.2.3.!..(«-l)-^""'^^-^-^+^>—^^  +  "~^^"-^"'^- 

21.  Let  now  <l>{x)  be  any  function  whatever,  and  let  it  be  required, 
in  general,  to  find  J'(i),  so  that  ftj'it)  ■  f  =  ^{x),  provided  x  be  any 
integer  from  0  to  w  —  1  inclusive. 

It  has  been  shewn  in  Art.  18,  that  a  function  T„.i  of  w— 1  di- 
mensions may  always  be  found  to  satisfy  the  imposed  conditions,  and 
for  the  most  general  value  oi  f{t)  we  shall  then  have 


INVERSE  METHOD  OF  DEFINITE   INTEGRALS.  1^9 

* 

Now  7'„-i  contains  only  n  constants,  being  of  »  — 1  dimensions,  and 
therefore  if  we  denote  by  P„  the  same  quantity  as  in  the  preceding 
Section,  namely  the  coefficient  of  h"  in 

{1-  2h{l-2t)  +  h:'}-i, 

we  may  put 

T„.i  =  ttoPo  +  a^P,  +  (hP2  + +  a„_,P„_i, 

the  right-hand  member  being  of  the  same  dimensions  with  the  left,  and 
containing  the  same  number  of  constants. 

Now  by  the  properties  of  P„  we  have  j;P„P„  =  0,  when  m  and  «  are 
unequal,  and 


2«  +  l 
Hence  we  have  fiP^T„_.,=  «„ 


Hi 


Jl'*   2  -*  n-1  —   "^  • 

But  by  the  conditions  of  the  question, 

jc  being  any  integer  less  than  n. 

Hence 
j;P„7;-i  =  ^r„_i  =  0(O)  =  (f>{h)   when  h  is  put  =0, 

iP  7'„_,=j;2;_,  (1  -  2o=0(o)-20  (1)= -  A  ^^y^  .<^  (A), 


140  Mr  MURPHY'S   SECOND    MEMOIR   ON   THE 

and  generally 

i;p.r..,=;r._,{i-f.^.*.'-ti^.<?!^<|±?l.f-&c.) 

=  (-l)'"A"'.^^ '\    ^  ' ^^ '-.d){h).    When  h  is  put  =0. 

'  1  . 2 m  ^     '  ^ 

and   by  comparing  the  former   integrals  with   the   latter,   the  values  of 
ffo,  «i,  a-i,  &c.  are  known,  and  being  substituted  give 

T._,  =  P,0(/^)-3P,A^.0(A)  +  5P.A^^^±^^^±^.0(A) 

J.    •   <«   •   t7 

//  being  put  =0,  after  the  operations  are  performed. 

It  should  be  observed  here  that  the  terms  of  this  expansion  are 
perfectly  independant  of  «,  which  only  fixes  the  number  of  the  terms; 
hence  this  series  may  be  continued  to  any  number  of  terms,  and  we 
shall  always  have  ftT„.it^  =  (p{x)  provided  x  is  any  integer  less  than  that 
number,  and  consequently  if  the  series  be  continued  ad  infinitum,  the 
equation  will  be  true  for  all  integer  and  positive  values  of  x. 

Cor.     Multiply  both    sides   by  if   and   integrate   from   t  =  0  to  /f=l, 
hence   «^  (^)  =  ^  •  <^  ('')  +  ^ .  ^^^j;;^^^^^  A  ^  .  0  A 

+  ^-(x  +  l)(ar  +  2)(x  +  3)^  1.2  '?>^  +  *'C- 

when  /*  is  put  =0. 

This  series  may  be  used,  not  only  for  the  integer  and  positive 
values  of  x,  but  for  any  values  which  will  not  render  it  divergent. 
(Vid.  First  Memoir,  'On  the  Inverse  method  of  Definite  Integrals,' 
Art.  2.) 


INVERSE    METHOD   OF   DEFINITE   INTEGRALS.  141 

22.  When  0(a;)  is  given  we  may  obtain  f{t)  in  an  infinite  variety 
of  forms  by  means  of  the  theory  of  reciprocal  functions  given  in  the 
preceding   Section.     For  instance,  if  we   denote   by  S^   the   sum  of  the 

products   of  the   natural  numbers   1,   2,  3. n   when   taken   m   and  m 

together,  and  put 

i.=i+«.h.i,^.j«^.(h.M.+  ^.(h.M-+....+  r^.ch.ur 

,5!lM!!iM.     (Art.  8.    Section  IV.) 

and  \„  =  l-?.2"/+^4^^.3"f- ±{n+l)''t 

=  (-l)"A"{(A  +  l)''^*},  when  h  is  put  =0, 
then  L„  and  \„  are  reciprocal  functions.     (Sect.  iv.  Art.  14.) 

Put  therefore  y*(^)=aoZ/o  +  «iZ/i  +  a2Z/2+a3i3  +  &c. 
and   observing   that 

1.2.3....W 


ftKL„  =  {-lY.- 


w  +  1 


we  have  «„  =  (  - 1)" .  ^   ^"^     ^  .  ftf{t) .  X„. 
But  jl/{t) .  \  =  ftf{f) .  ( - 1)'.  A" .  (A  + 1)" .  t.     When  h  is  put  =  0, 

=(-i)"A"(a  +  i)»j;/(o.^ 

=  (-l)».A".(A  +  l)«.0(A),     since  ft/{t).f  =  (p{x). 
Hence  «„=  ^   ^^ — - .  A» .  (A  + 1)" . 0 (k), 

and  therefore 

f(f)  =  Lo(p{h)  +  2Li — ^     ^'  r    ^.gjr,^ —  i    2        +^-^^- — i  2  3      ' 
Vol.  V.    Part  II.  T 


I 


142  Mr  MURPHY'S   SECOND    MEMOIR   ON    THE 

which  series  when  convergent  will  satisfy  the  equation  jtf{t) .  f'  =  ^  («) 
for  all  values  of  x\  but  even  if  not  convergent,  it  will  satisfy  that 
equation  for  all  the  integer  values  of  x  from  0  to  n  —  \  inclusive, 
provided  it  be  continued  for  at  least  n  terms. 

If  we  multiply  by  f  and  integrate  as  before,  we  get 

which  series  when  convergent  may  be  used  for  any  value  of  x,  but 
only  positive  and  integer  values  when  divergent. 

23.  In  Art.  21.  when  ftf(t).t'^(p(x)  a  given  function  of  x,  we  have 
found  y(0  in  a  series  expressed  by  functions  of  t  of  the  same  nature 
as  P„,  now  P„  is  only  a  particular  value  of  the  general  function  (jo,  q)„ 
investigated  in  the  former  Section,  Art.  12.,  namely,  when  p  =  q  =  l;  we 
shall  now  express  /{t)  according  to  this  more  general  class  of  functions, 
that  is,  under  the  form 

fit)  =  «o  ip,  q)o  +  «i  (p,  q)i  +  «2  {p,  q)2  +  &c. 

Now  in  Art.  12.  above  referred  to,  we  have  found 

,        .   _         {p  +  l){p  +  l+q)....{p^l  +  (m-l).q]    m 
Kp,qh-i  l(l^q)....{i^{m-\).q}  l"^ 

(2p  +  l)(2p  +  l+g)....{2jo  +  l  +  (?»-l).g}    m.jm-l) 
■^  i.(\+q)....{\  +  {m.-l).q}  '        1.2        '^    "  *''• 

To  simplify  this  expression,  put 

77    =  (/>^  +  l)(M  +  l+9)--{p^  +  l+(^-l)-g} 

'■''  l{l+q)....{l+{m-l).q} 

Let  yj^  express  the  operation  of  changing  h  into  h  +  1  (Vid.  former 
Memoir,  Note  B.  2.),  >//^  the  repetition  of  this  operation  a  second 
time,  &c. ;    the  preceding  series  will  then  become 


INVERSE    METHOD   OF   DEFINITE    INTEGRALS.  143 

{p,  9),„ = H,r  -  f .  ^H,,f" + ^^^  >\^^H,.r 

on  the  supposition  that  we  put  h  =  0  after  the  operations  above  indicated, 
are  performed. 

Separate  in  this  expression  the  symbols  of  operation  and  of  quantity, 
and  we  shall  obtain  the  equation 

(p,q),„  =  (l-fr.H,J'': 

But  \U  —  1  or  \^  T-  x//°  indicates  that  we  must  subtract  the  original 
value  of  Hp,q,  from  the  value  it  receives  when  h  +  1  is  put  for  h, 
that  is,  it  is  the  same  as  performing  the  operation  A  of  finite  differences ; 
this  consideration  transforms  the  preceding  equation,  to  this 

(p,  q)m  =  (-iy"  A" .  Hf.qt"",   when  h  is  put  =0. 

In  like  manner  if  we  put 

„     ^  i,qh-\-\){qh  +  l+p) {qh  +  1 -\-{m-l)  .p} 

"■'  1(1+^) {\  +  {m-\).p} 

we  have  (g-,  jo)„  =  (-l)"' A'"  Jf^.pi?"',   when  A  =  0. 

Now  observing  that  by  the  nature  of  reciprocal  functions  we  have 

S*  ip,  q)m  (q,  p)n  =  0,  except  when  m  =  n, 

and  by  Art.  13.,   fi{p,  q\{q,p)^ 

_      ip,  q)'"       1.1.2.2.3.  3 .m  .  m 

~  1  +  mip+q)  '1.1.  (l+ju)(l  +  9)(l  +  2^)(l+29)...{l  +  (»w-l)  .p}  {l  +  {m-l).q}  ' 

then  since  f{t)  =  «„  (p,  q)o  +  «i  (p,  q)i  +  a,  {p,  q)^  +  &c. 

we  have    ftf{t) .  (q,  p)„ 

_  (pq)'"         1.1.2.2.  3 m  .  m 

"**"'•  l  +  m(p+q)'l.l{l+p){l+q) {l  +  im-l).p}  {l+{m-l).q}  ' 

t2 


144  Mr  MURPHY'S   SECOND   MEMOIR   ON   THE 

But  if  we  put  for  (q,  p)„  the  value  above  found,  and  observe  that  the 
operations  A  and  fi  are  with  respect  to  different  variables  h  and  t,  and 
therefore  their  order  is  transmutable,  we  have  also, 

=  {-iy  A"^  H,,p<p{qh),   by  hypothesis. 
Comparing  this  value  of  the  integral  with  that  already  found,  we  get 

'"     ^       '         {pqT  I'l"     2     ■     2     •      3       •       3       ■■■ 

l  +  {m-l).p    \  +{m-\)  .q 
"mm 

X  A"  JZ",  p  0  {qh),   when  h  =  0, 
from  whence  we  have  finally 

At)  =  {p,  q).  0  (qh)  -  ip,  q),  .  ^+f/^  •  T  '  T  "  ^  ^V.  <l>  W 
,        ,      1  +  2(0  +  0)    1    1    1+p    1+q    .,„„         ,    , 

_(«  «N     l+^Ci>+g)    1    1     l+£     l+i     1±2£    1+22    A3  W^'"     ri.r«M 
^^'^'°-         (pqf  ri-      2     •^^-       3       •—^■^■^'>f't>'^W 

+  &c &c. 

h  being  put  =0,  after  the  operation,  and  H',  H",  H',  &c.  being  the 
values  of  Hp,,  when  m  =  \,  2,  3,  &;c.  successively. 

Cor.  1.     Multiply  by  t\  and  then  integrate  from  ^  =  0  to  ^=1;   for 
Itf{t).t'  put  its  value  <p{x),  and  for  ft{p,q)mt''  its  value 

/     ,v„„,„    1-2. 3. .-^^ xix-q)...{x-{m-l).q\ 

^       'P  ■  1  (1  +  g)(l  +  2y)...{l -!-(»«- 1|  .^)*(a;+l)(a;+jt)  +  l)...(a;  +  »w^+l)' 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS.  145 

by  Art.  12 ;  and  lastly,  put  for  H,,^  its  value 

(gA  +  1) {qh  +  1.  +i)). . ..\ g^  +  l+  {m-\).p\  . 

1.(1 +;?).. .|i+(»w-iy:jo"i 

we  thus  obtain 

+  »(^-g>  L+a(i>±2) ^. (^ ^    (^  (^ 

(a:  +  l)(a;+jo  +  l)(a?+2^  +  l)  Sg''  '^  '^^       ^       /rv"/    / 

x(a;  — 5')(a;  — 2^) 

■*"  {x  +  l)(a:+jt>  +  l)(ar  +  2/>  +  l)(2  +  3ja  +  l) 

^  ^  1^.2^^^  ^'^^^'  +  ^^^^^  +-^  +  ^^^^^  +  aja  +  1)  0  {qh) 
+  &c.  when  A  is  put   =  0, 
and  where  />  and  q  are  perfectly  arbitrary. 

Cor.  2.     Put  ^  =  ^  =  0,  and  make 

where   ^(0),   0'(O),   0"(O),    and    the    values   of  ^{x)   and   its   successive 
differential  coefficients  when  a;  =  0,  and  the  above  expansion  will  become 

</>(x)  =  ^„.^  +  ^,.^^-.,  +  ^..^3  +  &c. 

If,  moreover,  we  put 

rr,  ,         «     ,      ,    ,        W.(»-l)     (h.  1.  O''  „ 

which   is   the   same   as   A„  when  we  put  f  for  ^  (a;),    then   it  is  easily 
seen  by  the  principles  of  the  first  Memoir,  that  jj  Tj'  =  -, r-r—r ,  and 

/  r  r  ,  ji     n  (a;  +  l)"+'' 

since  we  have  also  fij'it) .  /^  =  0  {x),  it  follows  that 


146  Mr    MURPHY'S   SECOND    MEMOIR   ON    THE 

24.  The  functions  which  have  been  all  along  designated  by  {p,  q)„  and 
{q,  p)„,  have  been  already  shewn  to  be  reciprocal  one  to  the  other;  putting 
p  =  q,  the  resulting  function  {p,  p)„  must  be  reciprocal  to  itself;  that  is, 
ft{p, p)„{p, p),„  =  0  when  m  and  n  are  unequal  positive  integers;  when 
p  =  l  the  function  {p,  j)),,  is  then  identical  with  that  denoted  by  P„ ,  which 
has  been  before  shewn  to  be  reciprocal  to  itself;  again,  the  function  T„  or 

n  n.(n~\)    {hA.ty  ^  n.{n-l)  .{n-^)    (h.  1.  If 

is  reciprocal  to  itself,  for  if  we  mviltiply  by  (h.  1.  ty,  and  integrate,  we  get 

j;r„(h.l.0"'  =  1.2.3...«.(-ir{l-f^-^!^   (^±i)^_&e.}. 

The  expression  between  the  brackets  is  the  term  independent  of  h  in  the 

product  (1+/^)"(1  +  t)         ,    or  the  coefficient  of  //-('"+'>  in  (1+A)"-'"-'; 

it   is   therefore   zero  when    n>m,    but   when  n  =  m   its  value  is   (  — I)'", 
and  when  n<m,  its  value  is 

,  _         (?w  +  l  -n){m  +  2-n)...m 
^~    '   '  1  .2  ...n  '■ 

Hence  fi  T„  7'„  =  0,  when  m  and  n  are  unequal,  and 

1.2. ..n 

25.  Put  h.  1.  (^)  =  T,  and  substituting  in  T„,  we  have 

1.2...W  J'„e'^  =  e"|l.2...M  +  w.2.3...Wx+^^^^^\3.4...WT^  +  &C.| 
(dw  c?"-'t"      n.(n-l)    d"-W     „     1 

_  d"{e'^r'') 
„  _,        6-^d"  (e-T") 

Hence      7;  =  -^—p^ t-^  . 

1.2. ..war" 


INVERSE   METHOD   OF   DEFINITE    INTEGRALS.  147 

From ,  this  formulae  it  appears  that  the  equation  T„  =  0  has  n  real 
values  of  t  all  negative;  and  therefore  n  corresponding  values  of  t, 
which  are  all  included  between  0  and  1. 

Moreover,  if  we  form  the  equation 

u  =  T  +  hu,       or  u 


1-h' 

it  follows  by  the  theorem  of  Lagrange,  that   T„  is  the  coefficient  of  h" 

de"  e'~* 

in  ^'^•-j->  that  is,  in  - — y,  and  putting  t  for  e%   T„  is  clearly  the  co- 

h 

efficient  of  h"  in  the  expansion  of  the  function  y  . 

Conversely,  we  may  now  prove  that  the  coefficient  of  h"  in  the  ex- 

h 

pansion  of  - — -  is  a  reciprocal  function;    for  when  h  =  0,  this  function 

A  ""■  ft 

is  reduced  to  unity,  we  may  therefore  put  generally 

=  ro+T,^+T,A^  +  &c.  where  T,  =  \. 


A 
fX-k 


1-h 

Let  h'  represent  any  other  arbitrary  quantity,  and  we  have 


1-h 


j=  T,+  T,h'+T,h"  +  &ic. 


Multiply    both   series    term    by   term    and   integrate,   the   result   in    the 
left-hand  members  is 

{i-h){i-h')^'  ~  i-hh" 

/which  expanded  becomes  1  +  hh'  +  h^h'^ +  kc.;  which  being  identical  with 
the  integral  of  the  product  of  the  right-hand  members,  will  necessarily 
require  that  the  integrals  of  those  terms  which  are  not  in  corresponding 
places  in  both  series  must  vanish,  and  the  integrals  of  the  products  of 
the  corresponding  coefficients  to  be  unity,  which  are  the  same  properties 
that  have  been  demonstrated  in  Art.  24. 


148  Mr  MURPHY'S   SECOND   MEMOIR,   &c. 

Cor.     Put  .; — r  =  ^»  and  the  series 
\—n 

t~^  =  (\~.h){  T„  +T,h+  TJi^  +  &c.  \  becomes 

^^=^+  ^■•r4T^+  ^-7;:ttv3  +  *'^- 

The  principles  which  have  been  used  in  this  Section  to  obtain  ex- 
pansions such  as  the  preceding  by  means  of  reciprocal  functions  relative 
to  simple  integration,  will  apply  with  equal  simplicity  to  reciprocal 
functions  relative  to  any  number  of  integrations. 


R.  MURPHY. 


Caius  College, 
Bee.  18,  1833. 


VII.     On    the  Nature  of  the  Truth  of  the  Laws  of  Motion.      By  the 
Rev.  W.  Whewell,  M.A.  Fellow  and  Tutor  of  Trinity  College. 


[Bead  Feb.  17,  1834.] 


1.  The  long  continuance  of  the  disputes  and  oppositions  of  opinion 
which  have  occurred  among  theoretical  writers  concerning  the  elementary 
principles  of  Mechanics,  may  have  made  such  discussions  appear  to  some 
persons  wearisome  and  unprofitable.  I  might,  however,  not  unreasonably 
plead  this  very  circumstance  as  an  apology  for  offering  a  new  view  of 
the  subject;  since  the  extent  to  which  these  discussions  have  already 
gone  shews  that  some  men  at  least  take  a  great  interest  in  them ; 
and  it  may  be  stated,  I  think,  without  fear  of  contradiction,  that 
these  controversies  have  not  terminated  in  the  general  and  undisputed 
establishment  of  any  one  of  the  antagonist  opinions. 

The  question  to  which  my  remarks  at  present  refer  is  this:  "What 
is  the  kind  and  degree  of  cogency  of  the  best  proofs  of  the  laws  of 
motion,  or  of  the  fundamental  principles  of  mechanics,  exprest  in  any 
other  way?"  Are  these  laws,  philosophically  considered,  necessary,  and 
capable  of  demonstration  by  means  of  self-evident  axioms,  like  the 
truths  of  geometry ;  or  are  they  empirical,  and  only  known  to  be  true 
by  trial  and  observation,  like  such  general  rules  as  we  obtain  in  natural 
history  ? 

It  certainly  appears,  at  first  sight,  very  difficult  to  answer  the  argu- 
ments for  either  side  of  this  alternative.  On  the  one  hand  it  is  said, 
the  laws  of  motion  cannot  be  necessarily  true,  for  if  they  were  so,  the 
denial  of  them  would  involve  a  contradiction.  But  this  it  does  not, 
for  we  can  readily  conceive  them  to  be  other  than  they  are.  We  can 
conceive  that  a  body  in  motion  should  have  a  natural  tendency  to 
move  slower  and  slower.  And  we  know  that,  historically  speaking, 
Vol.  V.    Paet  II.  U 


150       Mr  WHEWELL,  ON  THE  NATURE  OF  THE  TRUTH 

men  did  at  first  suppose  the  laws  of  motion  to  be  different  from 
what  they  are  now  proved  to  be.  This  would  have  been  impossible 
if  the  negation  of  these  laws  had  involved  a  contradiction  of  self-evi- 
dent principles,  and  consequently  had  been  not  only  false  but  incon- 
ceivable. These  laws,  therefore,  cannot  be  necessary ;  and  can  be  duly 
established  in  no  other  way  than  by  a  reference  to  experience. 

On  the  other  hand,  those  who  deduce  their  mechanical  principles 
without  any  express  reference  to  experiment,  may  urge,  on  their  side, 
that,  by  the  confession  even  of  their  adversaries,  the  laws  of  motion 
are  proved  to  be  true  beyond  the  limits  of  experience ; — that  they  are 
assumed  to  be  true  of  any  new  kind  of  motion  when  first  detected,  as 
well  as  of  those  already  examined; — and  that  it  is  inexplicable  how 
such  truths  should  be  established  empirically.  They  may  add  that  the 
consequences  of  these  laws  are  allowed  to  hold  with  the  most  complete 
and  absolute  universality;  for  instance,  the  proposition  that  "the  quan- 
tity of  motion  in  the  world  in  a  given  direction  cannot  be  either 
increased  or  diminished,"  is  conceived  to  be  rigorously  exact;  and  to 
have  a  degree  and  kind  of  certainty  beyond  and  above  all  mere  facts 
of  experience ;  what  other  kind  of  truth  than  necessary  truth  this 
can  be,  it  is  difficult  to  say.  And  if  the  conclusions  be  necessarily 
true,  the  principles  must  be  so  too. 

This  apparent  contradiction  therefore,  that  a  law  should  be  neces- 
sarily true  and  yet  the  contrary  of  it  conceivable,  is  what  I  have  now 
to  endeavour  to  explain ;  and  this  I  must  do  by  pointing  out  what 
appear  to  me  the  true  grounds  of  the  laws  of  motion. 

2.  The  science  of  Mechanics  is  concerned  about  motions  as  deter- 
mined by  their  causes,  namely,  forces ;  the  nature  and  extent  of  the 
truth  of  the  first  principles  of  this  science  must  therefore  depend  upon 
the  way  in  which  we  can  and  do  reason  concerning  causes.  In  what 
manner  we  obtain  the  conception  of  cause,  is  a  question  for  the  meta- 
physician, and  has  been  the  subject  of  much  discussion.  But  the  general 
principle  which  governs  our  mode  of  viewing  occurrences  with  reference 
to  this  conception,  so  far  as  our  present  subject  is  concerned,  does  not 
appear    to  be  disturbed    by   any  of   the    arguments   which   have   been 


OF  THE  LAWS  OF  MOTION.  151 

adduced  in  this  controversy.     This  principle  I  shall  state  in  the  form 
of  an  axiom,  as  follows. 

Axiom  I.     Every  change  is  produced  by  a  cause. 

It  will  probably  be  allowed  that  this  axiom  expresses  a  universal 
and  constant  conviction  of  the  human  mind ;  and  that  in  looking  at 
a  series  of  occurrences,  whether  for  theoretical  or  practical  purposes, 
we  inevitably  and  unconsciously  assume  the  truth  of  this  axiom.  If  a 
body  at  rest  moves,  or  a  body  in  motion  stops,  or  turns  to  the  right 
or  the  left,  we  cannot  conceive  otherwise  than  that  there  is  some 
cause  for  this  change.  And  so  far  as  we  can  found  our  mechanical 
principles  on  this  axiom,  they  will  rest  upon  as  broad  and  deep  a 
basis  as  any  truths  which  can  come  within  the  circle  of  our  know- 
ledge. 

I  shall  not  attempt  to  analyse  this  axiom  further.  Different  per- 
sons may,  according  to  their  different  views  of  such  subjects,  call  it  a 
law  of  our  nature  that  we  should  think  thus,  or  a  part  of  the  con- 
stitution of  the  human  mind,  or  a  result  of  our  power  of  seeing  the 
true  relations  of  things.  Such  variety  of  opinion  or  expression  would 
not  affect  the  fundamental  and  universal  character  of  the  conviction 
which  the  axiom  expresses;  and  would  therefore  not  interfere  with  our 
future  reasonings. 

3.  There  is  another  axiom  connected  with  this,  which  is  also  a 
governing  and  universal  principle  in  all  our  reasoning  concerning 
causes.     It  may  be  thus  stated. 

Axiom  II.     Causes  are  measured  by  their  effects. 

Every  effect,  that  is,  every  change  in  external  objects,  implies  a 
cause,  as  we  have  already  said :  and  the  existence  of  the  cause  is  known 
only  by  the  effects  it  produces.  Hence  the  intensity  or  magnitude  of 
the  cause  cannot  be  known  in  any  other  manner  than  by  these  effects: 
and,  therefore,  when  we  have  to  assign  a  measure  of  the  cause,  we 
must  take  it  from  the  effects  produced. 

In  what  manner  the  effects  are  to  be  taken  into  account,  so  as 
to   measure    the    cause    for    any    particular   purpose,    will    have  to  be 

u2 


153     mk  whewell,  on  the  nature  of  the  truth 

further  considered ;  but  the  axiom,  as  now  stated,  is  absolutely  and 
universally  true,  and  is  acted  upon  in  all  parts  of  our  knowledge  in 
which  causes  are  measured. 

4.  But  something  further  is  requisite.  We  not  only  consider  that 
all  changes  of  motion  in  a  body  have  a  cause,  but  that  this  cause  may 
reside  in  other  bodies.  Bodies  are  conceived  to  act  upon  one  another, 
and  thus  to  influence  each  other's  motions,  as  when  one  billiard  ball 
strikes  another.  But  when  this  happens,  it  is  also  supposed  that  the 
body  struck  influences  the  motion  of  the  striking  body.  This  is  inclu- 
ded in  our  notion  of  body  or  matter.  If  one  ball  could  strike  and 
affect  the  motions  of  any  number  of  others  without  having  its  own 
motion  in  any  degree  affected,  the  struck  balls  would  be  considered, 
not  as  bodies,  but  as  mere  shapes  or  appearances.  Some  reciprocal  in- 
fluence, some  resistance,  in  short  some  reaction,  is  necessarily  involved 
in  our  conception  of  action  among  bodies.  All  mechanical  action  upon 
matter  implies  a  corresponding  reaction;  and  we  might  describe  matter 
as  that  which  resists  or  reacts  when  acted  on  by  force.  Not  only 
must  there  be  a  reaction  in  such  cases,  but  this  reaction  is  defined 
and  determined  by  the  action  which  produces  it,  and  is  of  the  same 
kind  as  the  action  itself  The  action  which  one  body  exerts  upon 
another  is  a  blow,  or  a  pressure;  but  it  cannot  press  or  strike  with- 
out receiving  a  pressure  or  a  blow  in  return.  And  the  reciprocal 
pressure  or  blow  depends  upon  the  direct,  and  is  determined  altogether 
and  solely  by  that.  But  this  action  being  mutual,  and  of  the  same 
kind  on  each  body,  the  effect  on  each  body  will  be  determined  by  the 
effect  on  the  other,  according  to  the  same  rule ;  each  effect  in  turn 
being  considered  as  action  and  the  other  as  reaction.  But  this  cannot  be 
otherwise  than  by  the  equality  and  opposite  direction  of  the  action  and 
reaction.  And  since  this  reasoning  applies  in  all  cases  in  which  bodies 
influence  each  others  motions,  we  have  the  following  axiom  which  is 
universally  true,  and  is  a  fundamental  principle  with  regard  to  all  me- 
chanical relations. 

Axiom  III.     Action  is  always  accompanied  by   an   equal  and  opposite 

Reaction. 


OF   THE   LAWS  OF   MOTION.  153 

5.  I  now  proceed  to  shew  in  what  manner  the  Laws  of  Motion 
depend  upon  these  three  axioms. 

Bodies  move  in  lines  straight  or  curved,  they  move  more  or  less 
rapidly,  and  their  motions  are  variously  affected  by  other  bodies.  This 
succession  of  occurrences  suggests  the  conceptions  of  certain  properties 
or  attributes  of  the  motions  of  bodies,  as  their  direction  and  velocity, 
by  means  of  which  the  laws  of  such  occurrences  may  be  exprest. 
And  these  properties  or  attributes  are  conceived  as  belonging  to  the 
body  at  each  j)^^^^  of  its  motion,  and  as  changing  from  one  point  to 
another.  Thus  the  body,  at  each  point  of  its  path,  moves  in  a 
certain   direction,  and  with  a  certain  velocity. 

These  properties,  direction  and  velocity  for  instance,  are  subject 
to  the  rule  stated  in  the  first  axiom :  they  cannot  change  without 
some  cause ;  and  when  any  changes  in  the  motions  of  a  body  are 
seen  to  depend  on  its  position  relative  to  another  body  or  to  any  part 
of  space,  such  other  body,  or  such  other  part  of  space,  is  said  to 
exert  a  Jbrce  upon  the  moving  body.  Also  the  force  exerted  upon 
the  moving  body  is  considered  to  be  of  a  certain  value  at  each 
point  of  the  body's  motion  ;  and  though  it  may  change  from  one  point 
to  another,  its  changes  must  depend  upon  the  position  of  the  points 
only,  and  not  upon  the  velocity  and  direction  of  the  moving  body. 
For  the  force  which  acts  upon  the  body  is  conceived  as  a  property  of 
the  bodies,  or  points,  or  lines,  or  surfaces  among  which  the  moving  body 
is  placed;  the  force  at  all  points  therefore  depends  upon  the  position 
with  regard  to  the  bodies  and  spaces  of  which  the  force  is  a  property ; 
but  remains  the  same,  whatever  be  the  circumstances  of  the  body 
moved.  The  circumstances  of  the  body  moved  cannot  be  a  cause 
which  shall  change  the  force  acting  at  any  point  of  space,  although 
they  may  alter  the  effect  which  that  force  produces  upon  the  body. 
Thus,  gravity  is  the  same  force  at  the  same  point  of  space,  whether  it 
have  to  act  upon  a  body  at  rest  or  in  motion  ;  although  it  still  remains 
to  be  seen  whether  it  will  produce  the  same  effect  in  the  two  cases. 

6.  This  being  established,  we  can  now  see  of  what  nature  the 
laws   of   motion   must  be,   and    can  state  in    a  few   words   the  proofs 


J  54  Mr  WHEWELL,   ON   THE   NATURE   OF   THE   TRUTH 

of  them.  We  shall  have  a  law  of  motion  corresponding  to  each 
of  the  above  three  axioms ;  the  first  law  will  assert  that  when  no  force 
acts,  the  properties  of  the  motion  will  be  constant;  the  second  law 
will  assert  that  when  a  force  acts,  its  quantity  is  measured  by  the 
effect  produced ;  the  third  law  will  assert  that,  when  one  body  acts 
upon  another,  there  will  be  a  reaction,  equal  and  opposite  to  the 
action.  And  so  far  as  the  laws  are  announced  in  this  form,  they  will 
be  of  absolute  and  universal  truth,  and  independent  of  any  particular 
experiment  or  observation  whatever. 

But  though  these  laws  of  motion  are  necessarily  and  infallibly 
true,  they  are,  in  the  form  in  which  we  have  stated  them,  entirely 
useless  and  inapplicable.  It  is  impossible  to  deduce  from  them  any 
definite  and  positive  conclusions,  without  some  additional  knowledge  or 
assumption.  This  will  be  clear  by  stating,  as  we  can  now  do  in  a 
very  small  compass,  the  proofs  of  the  laws  of  motion  in  the  form 
in  which  they  are  employed  in  mechanical  reasonings. 

7.  First,  of  the  first  Law ; — that  a  body  not  acted  upon  by  any  force 
will  go  on  in  a  straight  line  with  an  invariable  velocity. 

The  body  will  go  on  in  a  straight  line :  for,  at  any  point  of  its 
motion,  it  has  a  certain  direction,  which  direction  will,  by  Axiom  I, 
continue  unchanged,  except  some  cause  make  it  deviate  to  one  side  or 
other  of  its  former  position.  But  any  cause  which  should  make  the 
direction  deviate  towards  any  part  of  space  would  be  a  force,  and  the 
body  is  not  acted  upon  by  any  force.  Therefore,  the  direction  cannot 
change,  and  the  body  will  go  on  in  the  same  straight  line  from  the 
first. 

The  body  will  move  with  an  invariable  velocity.  For  the  velocity 
at  any  point  will,  by  Axiom  I,  continue  unchanged,  except  some 
cause  make  it  increase  or  decrease.  And  since,  by  supposition,  the 
body  is  not  acted  upon  by  any  force,  there  can  be  no  such  cause 
depending  upon  position,  that  is,  upon  relations  of  space;  for  any 
cause  of  change  of  motion  which  has  a  reference  to  space  is  force. 

Therefore  there  can  be  no  cause  of  change  of  motion,  except 
there   be   one   depending   upon  time,   such,   for  instance,  as  would  exist 


OF   THE  LAWS  OF  MOTION.  155 

if  bodies  had  a  natural  tendency  to  move  slower  and  slower,  according 
to   a  rate   depending  on  the  time  elapsed. 

But  if  such  cause  existed,  its  effects  ought  to  be  considered  sepa- 
rately ;  and  it  would  still  be  requisite  to  assume  the  permanence  of 
the  same  velocity,  as  the  first  law  of  motion ;  and  to  obtain,  in  addi- 
tion to  this,  the  laws  of  the  retardation  depending  on  the  time. 

Whether  there  is  any.  such  cause  of  retardation  in  the  actual 
motions  of  bodies,  can  be  known  only  by  a  reference  to  experience; 
and  by  such  reference  it  appears  that  there  is  no  such  cause  of  the 
diminution  of  velocity  depending  on  time  alone;  and  therefore  that 
the  first  law  of  motion  may,  in  all  cases  in  which  bodies  are  exempt 
from  the  action  of  external  forces,  be  applied  without  any  addition  or 
correction  depending   upon   the  time  elapsed. 

It  is  not  here  necessary  to  explain  at  any  length  in  what  manner 
we  obtain  from  experience  the  knowledge  of  the  truth  just  stated,  that 
there  is  not  in  the  mere  lapse  of  time  any  cause  of  the  retardation  of 
moving  bodies.  The  proposition  is  established  by  shewing  that  in  all 
the  cases  in  which  such  a  cause  appears  to  exist,  the  cause  of  retar- 
dation resides  in  surrounding  bodies  and  not  in  time  alone,  and  is 
therefore  an  external  force.  And  as  this  can  be  shewn  in  every  in- 
stance, there  remains  only  the  negation  of  all  grovind  for  the  assump- 
tion of  such  a  cause  of  retardation.     We  therefore  reject  it  altogether. 

Thus  it  appears  that  in  proving  the  first  law  of  motion,  we  obtain 
from  our  conception  of  cause  the  conviction  that  velocity  will  be 
uniform  except  some  cause  produce  a  change  in  it ;  but  that  we  are 
compelled  to  have  recourse  to  experience  in  order  to  learn  that  time 
alone  is  not  a  cause  of  change  of  velocity. 

8.  I  now  proceed  to  the  second  Law :  —  that  when  a  force  acts 
upon  a  body  in  motion,  the  effect  is  the  same  as  that  which  the  same 
force  produces  upon   a   body  at  rest. 

This  law  requires  some  explanation.  How  is  the  effect  produced 
upon  a  moving  body  to  be  measured,  so  that  we  may  compare   it  with 


156       Mk  WHEWELL,  on  the  NATURE  OF  THE  TRUTH 

the  effect  upon  a  body  at  rest?  The  answer  to  this  is,  that  we  here 
take  for  the  measure  of  the  effect  of  the  force,  that  motion  which 
must  be  compounded  with  the  motion  existing  before  the  change,  in  or- 
der to  produce  the  motion  which  exists  after  the  change:  the  rules  for 
the  composition  of  motion  being  established  on  independent  grounds 
by  the  aid  of  definition  alone.  Thus  if  gravity  act  upon  a  body 
which  is  falling  vertically,  the  effect  of  gravity  upon  the  body  is 
measured  by  the  velocity  added  to  that  which  the  body  already  has : 
if  gravity  act  upon  a  body  which  is  moving  horizontally,  its  effect 
is  measured  by  the  distance  to  which  the  body  falls  below  the  hori- 
zontal line. 

The  effect  of  the  force  which  we  consider  in  the  second  Law  of 
motion,  is  its  effect  upon  velocity  only :  and  it  is  proper  to  mark 
this  restriction  by  an  appropriate  term :  we  shall  call  this  the  accele- 
rative  effect  of  force;  and  the  cause,  as  measured  by  this  effect,  may 
be  termed  the  accelerathe  quantity  of  the  force.* 

A  law  of  motion  which  necessarily  results  from  our  second  Axiom 
is,  that  the  accelerative  quantity  of  a  force  is  measured  by  the  acce- 
lerative  effect.  But  whether  the  accelerative  effect  depends  upon  the 
velocity  and  direction  of  the  moving  body,  cannot  be  known  indepen- 
dently of  experience.  It  is  very  conceivable,  for  instance,  that  the 
force  of  gravity  being  every  where  the  same,  shall  yet  produce,  upon 
falling  bodies,  a  smaller  accelerative  effect  in  proportion  to  the  velocity 
which  they  already  have  in  a  downward  direction.  Indeed  if  gravity 
resembled  in  its  operation  the  effect  of  any  other  mode  of  mechanical 
agency,    the    result    would    be    so.      If  a   body   moved    downwards   in 


*  The  accelerative  quantity  of  a  force  (the  quantitas  acceleratrix  vis  cujusvis  of  Newton) 
is  often  called  the  accelerating  forces  and  we  may  thus  have  to  speak  of  the  accelerating 
force  of  a  certain  force,  which  is  at  any  rate  an  awkward  phraseology.  It  would  perhaps 
have  been  fortunate  if  Newton,  or  some  other  writer  of  authority,  at  the  time  when  the 
principles  of  mechanics  were  first  clearly  developed,  had  invented  an  abstract  term  for 
this  quantity :  it  might  for  instance  have  been  called  acceleralivity.  And  the  second  law 
of  motion  would  then  have  been,  that  the  acceleralivity  of  the  same  force  is  the  same, 
whatever  be  the  motion  of  the  body  acted  on. 


OF  THE   LAWS   OF  MOTION.  157 

consequence  of  the  action  of  a  hand  pushing  it  with  a  constant  effort, 
or  of  a  spring,  or  of  a  stream  of  fluid  rushing  in  the  same  direction, 
the  accelerative  effect  of  such  agents  would  be  smaller  and  smaller 
as  the  velocity  of  the  body  propelled  was  larger  and  larger.  We  can 
learn  from  experience  alone  that  the  effects  of  the  action  of  gravity 
do   not   follow   the   same   rule. 

We  assert  that  the  accelerative  quantity  of  the  same  force  of  gra- 
vity is  the  same  whatever  be  the  motion  of  the  body  acted  on.  It 
may  be  asked  how  we  know  that  the  force  of  gravity  is  the  same 
in  cases  so  compared ;  for  instance,  when  it  acts  on  a  body  at  rest 
and  in  motion  ?  The  answer  to  this  question  we  have  given  already. 
By  the  very  process  of  considering  gravity  as  a  force,  we  consider 
it  as  an  attribute  of  something  independent  of  the  body  acted  on. 
The  amount  of  the  force  may  depend  upon  place,  and  even  time,  for 
any  thing  we  know  a  priori ;  but  we  do  not  find  that  the  weight  of 
bodies  depends  on  these  circumstances,  and  therefore,  having  no  evi- 
dence of  a  difference  in  the  force  of  gravity,  we  suppose  it  the  same 
at  different  times  and  places.  And  as  to  the  rest,  since  the  force  is  a 
force  which  acts  on  the  body,  it  is  considered  as  the  same  force, 
whatever  be  the  circumstances  of  the  passive  body,  although  the  ejects 
may  vary  with  these  circumstances.  If  the  effects  are  liable  to  such 
change,  this  change  must  be  considered  separately,  and  its  laws  investi- 
gated ;  but  it  cannot  be  allowed  to  unsettle  our  assumption  of  the 
permanence  of  the  force  itself.  It  is  precisely  this  assumption  of  a 
constant  cause,  which  gives  us  a  fixed  term,  as  a  means  of  estimating 
and  expressing  by  what  conditions  the  effects  are  regulated. 

It  appears  by  observation  and  experiment,  that  the  accelerative 
quantity  of  the  same  force  is  not  affected  by  the  velocity  or  direction 
of  the  body  acted  on :  for  instance,  a  body  falling  vertically  receives, 
in  any  second  of  time,  an  accession  of  velocity  as  great  as  that  which 
it  received  in  the  first  second,  notwithstanding  the  velocity  with  which 
it  is  already  moving.  The  proof  of  this  and  similar  assertions  from 
experiment  produced,  historically  speaking,  the  establishment  of  the 
second  law  of  motion  in  the  sense  in  which  we  now  assert  it.  And 
here,  as  in  the  case  of  the  first  law,  we  may  observe  that  an  important 
Vol.  V.    Part  II.  X 


158       Mr  WHEWELL,  ON  THE  NATURE  OF  THE  TRUTH 

portion  of  the  process  of  proof  consisted  in  shewing  that  in  those  cases 
in  which  the  accelerative  effect  of  a  force  appeared  to  be  changed  by 
the  circumstances  of  the  motion  of  the  body  acted  on,  the  change  was, 
in  fact,  due  to  other  external  forces ;  so  that  all  evidence  of  a  cause 
of  change  residing  in  those  circumstances  was  entirely  negatived;  and 
thus  the  law,  that  the  accelerative  effect  of  the  same  force  is  the 
same,   appeared  to  be  absolutely  and  rigorously  true. 

9.  When  the  motions  of  bodies  are  not  affected  merely  by  forces 
like  gravity,  which  are  only  perceived  by  their  effects,  but  are  acted 
upon  by  other  bodies,  the  case  requires  other  considerations. 

It  is  in  such  cases  that  we  originally  form  the  conception  of  force; 
we  ourselves  pull  and  push,  thrust  and  throw  bodies,  with  a  view,  it 
may  be,  either  to  put  them  in  motion,  or  to  prevent  their  moving, 
or  to  alter  their  figure.  Such  operations,  and  the  terms  by  which 
they  are  described,  are  all  included  in  the  term  force,  and  in  other 
terms  of  cognate  import.  And  in  using  this  term,  we  necessarily 
assume  and  imply  the  co-existence  of  these  various  effects  of  force 
which  we  have  observed  universally  to  accompany  each  other.  Thus 
the  same  kind  of  force  which  is  the  cause  of  motion,  may  also  be 
the  cause  of  a  body  having  a  form  different  from  its  natural  form ; 
when  we  draw  a  bow,  the  same  kind  of  pull  is  needed  to  move  the 
string,  and  to  hold  it  steady  when  the  bow  is  bent.  And  a  weight 
might  be  hung  to  the  string,  so  as  to  produce  either  the  one  or 
the  other  of  these  effects.  By  an  infinite  multiplicity  of  experiments 
of  this  kind,  we  become  imbued  with  the  conviction  that  the  same 
pressure  may  be  the  cause  of  tension  and  of  motion.  Also  as  the 
cause  can  be  known  by  its  effects  only,  each  of  these  effects  may  be 
taken  as  its  measure ;  and  therefore,  so  long  as  one  of  them  is  the 
same,  since  the  cause  is  the  same,  the  other  must  be  the  same  also. 
That  is,  so  long  as  the  pressure  or  force  which  shews  itself  in 
tension  is  the  same,  the  motion  which  it  would  produce  must,  under 
the  same  circumstances,  be  the  same  also.  This  general  fact  is  not 
a  result  of  any  particular  observations,  but  of  the  general  observation 
or   suggestion  arising  unavoidably  from  universal  experience,  that  both 


OF  THE  LAWS  OP  MOTION.  159 

tension  and  motion  may  be  referred  to   force  as   their  cause,  and  have 
no  other    cause. 

We  come  therefore  to  this  principle  with  regard  to  the  actions  of 
bodies  upon  each  other,  that  so  long  as  the  tension  or  pressure  is  the 
same,  the  force,  as  shewn  by  its  effect  in  producing  motion,  must 
also   be   the   same. 

10.  This  force  or  action  of  bodies  upon  one  another,  is  that  which 
is  meant  in  the  Third  Axiom,  and  we  now  proceed  to  consider  the 
application  of  this   axiom   in  mechanics. 

Pressures  or  forces  such  as  I  have  spoken  of,  may  be  employed  in 
producing  tension  only,  and  not  motion ;  in  this  case,  each  force  prevents 
the  motion  which  would  be  produced  by  the  others,  and  the  forces 
are  said  to  balance  each  other,  or  to  be  in  equilibrium.  The  science 
which  treats  of  such  cases  is  called  Statics,  and  it  depends  entirely 
upon  the  above  third  axiom,  applied  to  pressures  producing  rest.  It 
follows  from  that  axiom,  that  pressures,  which  acting  in  opposite  di- 
rections thus  destroy  each  other's  effects,  must  be  equal,  each  measuring 
the  other.  Thus  if  a  man  supports  a  stone  in  his  hand,  the  force  or 
effort  exerted  by  the  man  upwards  is  equal  to  the  weight  or  force 
of  the  stone  downwards.  And  if  a  second  stone,  just  equal  to  the 'first, 
were  supported  at  the  same  time  in  the  same  hand,  the  force  or  effort 
must  be  twice  as  great ;  for  the  two  stones  may  be  considered  as 
one  body  of  twice  the  magnitude,  and  of  twice  the  weight;  and 
therefore  the  effort  which  supports  it  must  also  be  twice  as  great. 
And  thus  we  see  in  what  manner  statical  forces  are  to  be  measured 
in  virtue  of  this  third  axiom ;  and  no  further  principle  is  requisite  to 
enable  us  to  establish  the  whole  doctrine  of  statics. 

11.  The  third  axiom,  when  applied  to  the  actions  of  bodies  in 
motion,  gives  rise  to  the  third  law  of  motion,  which  Ave  must  now  con- 
sider. Here,  as  in  the  cases  of  the  other  axioms,  we  must  inquire 
how  we  are  to  measure  the  quantities  to  which  the  axiom  applies.  What 
is  the  measure  of  the  action  which  takes  place  when  a  body  is  put 
in  motion  by  pressure  or  force?    In  order  to  answer  this  question,   we 

X  2 


160        Mr  WHEWELL,  ON  THE  NATURE  OF  THE  TRUTH 

must  consider  what  circumstances  make  it  requisite  that  the  force 
should  be  greater  or  less.  If  we  have  to  lift  a  stone,  the  force  which 
we  exert  must  be  greater  when  the  stone  is  greater :  again,  we  must 
exert  a  greater  force  to  lift  it  quickly  than  slowly.  It  is  clear,  there- 
fore, that  that  property  of  a  force  with  which  we  are  here  concerned, 
and  which  we  may  call  the  motive  quantity  of  the  force,*  increases  both 
when  the  velocity  communicated,  and  when  the  mass  moved,  increase,  and 
depends  upon  both  these  quantities,  though  we  have  not  yet  shewn 
what  is  the  law  of  this  dependence. 

The  condition  that  a  quantity  P  shall  increase  when  each  of  two 
others  V  and  M  does  so,  may  be  satisfied  in  many  ways :  for  instance, 
by  supposing  P  proportional  to  the  sum  M+  V  (all  the  quantities  being 
expressed  in  numbers),  or  to  the  product,  MV,  or  to  MF'-,  or  in  many 
other   ways. 

When,  however,  the  quantities  ^  and  M  are  altogether  hetero- 
geneous, as  when  one  is  velocity,  and  the  other  weight,  the  first 
of  the  above  suppositions,  that  P  varies  as  M  +  V,  is  inadmissible. 
For  the  law  of  variation  of  the  formula  M+  V  depends  upon  the 
relation  of  the  units  by  which  M  and  V  respectively  are  measured; 
and  as  these  units  are  arbitrary  in  each  case,  the  result  is,  in  like 
manner,  arbitrary,   and  therefore  cannot  express  a  law  of  nature. 

« 

12.  The  supposition  that  the  motive  quantity  of  a  force  varies  as 
M^-V,  where  M  is  the  mass  moved  and  V  the  velocity,  being  thus 
inadmissible,  we  have  to  select  upon  due  grounds,  among  the  other 
formulae  MV,   MV\   M'V,   &c. 

And  in  the  first  place  I  observe  that  the  formula  must  be  propor- 
tional   to    M    simply    (excluding    M.^  &;c.)   for   both   the   forces   which 


*  The  motive  quantity  of  a  force  {vis  cujusvis  quantitas  matrix  of  Newton)  is  sometimes 
called  moving  force;  we  are  thus  led  to  speak  of  the  moving  force  of  a  force,  as  we 
have  already  observed  concerning  accelerating  force.  Hence,  as  in  that  case,  we  might 
employ  a  single  term,  as  motivity,  to  denote  this  property  of  force;  and  might  thus  speak 
of  it  and  of  its  measures  without  the  awkwardness  which  arises  from  the  usual  phrase. 


OF  THE   LAWS  OF   MOTION.  l6l 

produce  motion  and  the  masses  in  which  motion  is  produced  are  capa- 
ble of  addition  by  juxtaposition,  and  it  is  easily  seen  by  observation 
that  such  addition  does  not  modify  the  motion  of  each  mass.  If  a 
certain  pressure  upon  one  brick  (as  its  own  weight)  cause  it  to  fall 
with  a  certain  velocity,  an  equal  pressure  on  another  equal  brick  wiU 
cause  it  also  to  fall  with  the  same  velocity ;  and  these  two  bricks 
being  placed  in  contact,  may  be  considered  as  one  mass,  which  a  dou- 
ble force  will  cause  to  fall  with  still  the  same  velocity.  And  thus 
all  bodies,  whatever  be  their  magnitude,  will  fall  with  the  same  velo- 
city by  the  action  of  gravity.  Those  who  deny  this  (as  the  Aristo- 
telians did)  must  maintain,  that  by  establishing  between  two  bodies 
such  a  contact  as  makes  them  one  body,  we  modify  the  motion  which 
a  certain  pressure  will  produce  in  them.  And  when  we  find  experi- 
mentally (as  we  do  find)  that  large  bodies  and  small  ones  fall  with  the 
same  velocity,  excluding  the  effects  of  extraneous  forces,  this  result 
shews  that  there  is  not,  in  the  union  of  small  bodies  into  a  larger  one, 
any  cause  which  affects  the  motion  produced  in  the  bodies. 

It  appears,  therefore,  that  the  motive  quantity  of  force  which  puts 
a  body  in  motion  is,  cceteris  paribus,  proportional  to  the  mass  of  the 
body ;  so  that  for  a  double  mass  a  double  force  is  requisite,  in  order 
that  the  velocity  produced  may  be  the  same.  Mass  considered  with 
reference  to  this  rule,  is  called  Inertia. 

13.  The  measure  of  mass  which  is  used  in  expressing  a  law  of 
motion,  must  be  obtained  in  some  way  independent  of  motion,  other- 
wise the  law  will  have  no  meaning.  Therefore,  mass  measured  in 
order  to  be  considered  as  Inertia  must  be  measured  by  the  statical 
effects  of  bodies,  for  instance,  by  comparison  of  weights.  Thus  two 
masses  are  equal  which  each  balance  the  same  weight  in  the  same 
manner;  and  a  mass  is  double  of  one  of  them  which  produces  the 
same  effect  as  the  two.  And  we  find,  by  universal  observations,  that 
the  weight  of  a  mass  is  not  affected  by  the  figure  or  the  arrange- 
ment of  parts,  so  long  as  the  matter  continues  the  same.  Hence  it 
appears  that  the  mass  of  bodies  must  be  compared  by  comparing  their 
weights,  and  Inertia  is  proportional  to  weight  at  the  same  place. 


162       Mr  WHEWELL,  ON  THE  NATURE  OF  THE  TRUTH 

Since  all  bodies,  small  or  large,  light  or  heavy,  fall  downwards  with 
equal  velocities,  when  we  remove  or  abstract  the  effect  of  extraneous 
circumstances,  the  motive  quantity  of  the  force  of  gravity  on  equal 
bodies  is  as  their  masses ;  or  as  their  weight,  by  what  has  just  been  said. 

14.  For  the  measure  of  the  motive  quantity  of  force,  or  of  the  action 
and  reaction  of  bodies  in  motion,  we  have,  therefore,  now  to  chuse 
among  such  expressions  as  MV,  and  MV^.  And  our  choice  must  be 
regulated  by  finding  what  is  the  measure  which  will  enable  us  to 
assert,  in  all  cases  of  action  between  bodies  in  motion,  that  action  and 
reaction  are  equal  and  opposite. 

Now  the  fact  is,  that  either  of  the  above  measures  may  be  taken, 
and  each  has  been  taken  by  a  large  body  of  mathematicians.  The  former 
however  {MV)  has  obtained  the  designation  which  naturally  falls  to  the 
lot  of  such  a  measure ;  and  is  called  momentum,  or  sometimes  simply 
quantity  of  motion :  the  latter  quantity  {MV^)  is  called  vis  viva  or  liv- 
ing force. 

I  have  said  that  either  of  these  measures  may  be  taken :  the  former 
must  be  the  measure  of  action,  if  we  are  to  measure  it  by  the  effect  pro- 
duced in  a  given  time;  the  latter  is  the  measure  if  we  take  the  whole 
effect  produced.     In  either  way  the  third  law  of  motion  would  be  true. 

Thvis  if  a  ball  B,  lying  on  a  smooth  table,  be  drawn  along  by  a 
weight  A  hanging  by  a  thread  over  the  edge  of  the  table,  the  motion 
of  B  is  produced  by  the  action  of  A,  and  on  the  other  hand  the 
motion  of  A  is  diminished  by  the  reaction  of  B;  and  the  equality 
of  action  and  reaction  here  consists  in  this,  that  the  momentum  {MV) 
which  B  acquires  in  any  time  is  equal  to  that  which  A  loses :  that  is, 
so  much  is  taken  from  the  momentum  which  A  would  have  had,  if 
it  had  fallen  freely  in  the  same  time;  so  that  A  falls  more  slowly  by 
just  so  much. 

But  if  the  weight  A  fall  through  a  given  space  from  rest,  as  1  foot, 
and  then  cease  to  act,  the  eqviality  of  action  and  reaction  consists  in 
this,  that  the  vis  viva  which  B  acquires  on  the  whole,  is  equal  to  the 
vis  viva  which  A  loses ;    that  is,  the  vis  viva  of   A  thus  acting  on  B  is 


OF   THE  LAWS   OF  MOTION.  163 

smaller   by  so   much  than   it  would  have  been,   if  A   had  fallen  freely 
through  the  same  space. 

15.  In  fact,  these  two  propositions  are  necessarily  connected,  and 
one  of  them  may  be  deduced  from  the  other.  The  former  way  of 
stating  the  third  law  of  motion  appears,  however,  to  be  the  simplest  mode 
of  treating  the  subject,  and  we  may  put  the  third  law  of  motion  in 
this  form. 

In  the  direct  mutual  action   of  bodies,   the  momentum  gained  and  lost 
in  any  time  are  equal. 

This  law  depends  upon  experiment,  and  is  perhaps  best  proved  by 
some  of  its  consequences.  It  follows  from  the  law  so  stated,  that  the 
motive  quantity  of  a  force  is  proportional  to  the  momentum  generated  in 
a  given  time;  since  the  motive  quantity  of  force  is  to  be  equivalent 
to  that  action  and  reaction  which  is  understood  in  the  third  law  of 
motion.  Now,  if  the  pressure  arising  from  the  weight  of  a  body  P 
produce  motion  in  a  mass  Q,  since  the  momentum  gained  by  Q  and 
that  lost  by  P  in  any  time  are  equal,  the  momentum  of  the  whole 
at  any  time  will  be  the  same  as  if  P's  weight  had  been  employed 
in  moving  P  alone.  Therefore,  the  velocity  of  the  mass  Q  will  be 
less,  in  the  same  proportion  in  which  the  mass  or  inertia  is  greater: 
and  thus  the  accelerating  quantity  of  the  force  is  inversely  propor- 
tioned to  the  mass  moved.  This  rule  enables  us  to  find  the  accele- 
rative  quantity  of  the  force  in  various  cases,  as  for  instance,  when  bodies 
oscillate,  or  when  a  smaller  weight  moves  a  large  mass;  and  we 
can  hence  calculate  the  circumstances  of  the  motion,  which  are  found 
to  agree  with  the  consequences  of  the  above  law. 

16.  But  the  argument  may  be  reduced  to  a  simpler  form.  Our 
object  is  to  shew  that,  for  an  equal  mass,  the  velocity  produced  by  a 
force  acting  for  a  given  time  is  as  the  pressure  which  produces  the 
motion;  for  instance,  that  a  double  pressure  will  produce  a  double 
velocity.  Now  a  double  pressure  may  be  considered  as  the  union  of 
two  equal  pressures,  and  if  these  two  act  successively,  the  first  will 
communicate  to  the  body  a  certain  velocity,  and  the  second  will  com- 


164  Mr  WHEWELL,   ON   THE   NATURE  OF  THE   TRUTH 

municate  an  additional  velocity,  equal  to  the  first,  by  the  second  law 
of  motion ;  so  that  the  whole  velocity  thus  commvinicated  will  be  the 
double  of  the  first.  Therefore,  if  the  velocity  communicated  be  not 
also  the  double  of  the  first  when  the  two  pressures  act  together,  the 
difference  must  arise  from  this,  that  the  effect  of  one  force  is  modified 
by  the  simultaneous  action  of  the  other.  And  when  we  find  by  expe- 
rience (as  we  do  find)  that  there  is  no  such  difference,  but  that  the 
velocity  communicated  in  a  given  time  is  as  the  pressure  which  com- 
mimicates  it,  this  result  shews  that  there  is  nothing  in  the  circumstance 
of  a  body  being  already  acted  on  by  one  pressure,  which  modifies  the 
effect  of  an  additional  pressure  acting  along  with  the  first. 

17-  I  have  above  asserted  the  law,  of  the  direct  action  of  bodies 
only.  But  it  is  also  true  when  the  action  is  indirect,  as  when  by 
turning  a  winch  we  move  a  wheel,  the  main  mass  of  which  is  farther 
from  the  axis  than  the  handle  of  the  winch.  In  this  case  the  pres- 
sure we  exert  acts  at  a  mechanical  disadvantage  on  the  main  mass  of 
the  wheel,  and  we  may  ask  whether  this  circumstance  introduces  any 
new  law  of  motion.  And  to  this  we  may  reply,  that  we  can  conceive 
pressure  to  produce  different  effects  in  moving  bodies,  according  as  it 
is  exerted  directly  or  by  the  intervention  of  machines;  but  that  we 
find  no  reason  to  believe  that  such  a  difference  exists.  The  relations 
of  the  pressures  in  different  parts  of  a  machine  are  determined  by  con- 
sidering the  machine  at  rest.  But  if  we  suppose  it  to  be  put  in 
motion  by  such  pressures,  we  see  no  reason  to  expect  that  these  pres- 
sures should  have  a  different  relation  to  the  motions  produced  from 
what  they  would  have  done  if  they  were  direct  pressures.  And  as 
we  find  in  experiment  a  negation  of  all  evidence  of  such  a  differ- 
ence, we  reject  the  supposition  altogether.  We  assert,  therefore,  the 
third  law  of  motion  to  be  true,  whatever  be  the  mechanism  by 
the  intervention  of  which  action  and  reaction  are  opposed  to  each 
other. 

From  this  consideration  it  is  easy  to  deduce  the  following  rule, 
which  is  known  by  the  designation  of  D'Alembert's  principle,  and 
may   be  considered  as  a  fourth  law   of  motion. 


OF   THE   LAWS   OF   MOTION.  165 

WJien  any  forces  produce  motion  in  any  connected  system  of  matter, 
the  motive  quantities  of  force  gained  and  lost  by  the  different  parts 
must  balance  each  other  according  to  the  connexion  of  the  system.  ■ 

By  the  motive  quantity  of  force  gained  by  any  body,  is  here 
meant  the  quantity  by  which  that  motive  force  which  the  body's  mo- 
tion implies  (according  to  the  measures  already  established)  exceeds 
the  quantity  of  motive  force  which  acts  immediately  upon  the  body. 
It  is  the  excess  of  the  effective  above  the  impressed  force,  and  of  course 
arises  from  the  force  transmitted  from  the  other  bodies  of  the  system 
in  consequence  of  the  connexion  of  the  parts.  The  motive  quantity 
of  force  lost  is  in  like  manner  the  excess  of  the  impressed  above  the 
effective  force.  And  these  two  excesses,  in  different  parts  of  the  sys- 
tem, must  balance  each  other  according  to  the  mechanical  advantage 
or  disadvantage  at  which  they  act  for  each  part. 

This  completes  our  system  of  mechanical  principles,  and  authorizes 
us  to  extend  to  bodies  of  any  size  and  form  the  rules  which  the 
second  law  of  motion  gives  for  the  motion  of  bodies  considered  as 
points.  And  by  thus  enabling  us  to  trace  what  the  motions  of  bodies 
will  be  according  to  the  rule  asserted  in  the  third  law  of  motion, 
(namely,  that  the  motive  quantity  of  forces  is  as  the  momentum  pro- 
duced in  a  given  time,)  it  leads  us  to  verify  that  supposition  by  experi- 
ments in  which  bodies  oscillate  or  revolve  or  move  in  any  regular 
and  measurable  manner,  as  has  been  done  by  Atwood,  Smeaton,  and 
many  others. 

18.  We  have  thus  a  complete  view  of  the  nature  and  extent  of 
the  fundamental  principles  of  mechanics;  and  we  now  see  the  reason 
why  the  laws  of  motion  are  so  many  and  no  more,  in  what  way  they 
are  independent  of  experience,  and  in  what  way  they  depend  upon 
experiment.  The  form,  and  even  the  language  of  these  laws  is  of 
necessity  what  it  is;  but  the  interpretation  and  application  of  them  is 
not  possible  without  reference  to  fact.  We  may  imagine  many  rules 
according  to  which  bodies  might  move  (for  many  sets  of  rules,  dif- 
ferent from  the  existing  ones,  are,  so  far  as  we  can  see,  possible)  and 
we  should  still  have  to  assert — that  velocity  could  not  change  without 
Vol.  V.     Pakt  II.  Y 


166  Mb   WHEWELL,   ON   THE    NATURE   OF   THE   TRUTH 

a  cause, — that  change  of  action  is  proportional  to  the  force  which  pro- 
duces it, — and  that  action  and  reaction  are  equal  and  opposite.  The 
truth  of  these  assertions  is  involved  in  those  notions  of  causation  and 
matter,  which  the  very  attempt  to  know  any  thing  concerning  the  rela- 
tions of  matter  and  motion  presupposes.  But,  according  to  the  facts 
which  we  might  find,  in  such  imaginary  cases  as  I  have  spoken  of, 
we  should  settle  in  a  different  way — what  is  a  cause  of  change  of  ve- 
locity,— what  is  the  measure  of  the  force  which  changes  motion, — and 
what  is  the  measure  of  action  between  bodies.  The  law  is  necessary, 
if  there  is  to  be  a  law ;  the  meaning  of  its  terms  is  decided  by  what 
we  find,  and  is  therefore  regulated  by  our  special  experience. 

19.  It  may  further  illustrate  this  matter  to  point  out  that  this 
view  is  confirmed  by  the  history  of  mathematics.  The  laws  of  motion 
were  assented  to  as  soon  as  propounded;  but  were  yet  each  in  its  turn 
the  subject  of  strenuous  controversy.  The  terms  of  the  law,  the  form, 
which  is  necessarily  true,  were  recognised  and  undisputed ;  but  the 
meaning  of  the  terms,  the  substance  of  the  law,  was  loudly  contested; 
and  though  men  often  tried  to  decide  the  disputed  points  by  pure 
reasoning,  it  was  easily  seen  that  this  could  not  suffice ;  and  that  since 
it  was  a  case  where  experience  could  decide,  experience  must  be  the 
proper  test:  since  the  matter  came  within  her  jurisdiction,  her  authority 
was  single  and  supreme. 

Thus  with  regard  to  the  first  law  of  motion,  Aristotle  allowed  that 
natural  motions  continue  unchanged,  though  he  asserted  the  motions 
of  terrestrial  bodies  to  be  constrained  motions,  and  therefore,  liable  to 
diminution.  Whether  this  was  the  cause  of  their  diminution  was  a 
question  of  fact,  which  was,  by  examination  of  facts,  decided  against 
Aristotle.  In  like  manner,  in  the  first  case  of  the  second  law  of 
motion  which  came  under  consideration,  both  Galileo  and  his  oppo- 
nent agree  that  falling  bodies  are  uniformly  accelerated ;  that  is,  that 
the  force  of  gravity  accelerates  a  body  uniformly  whatever  be  the 
velocity  it  has  already ;  but  the  question  arises,  what  is  uniform  acce- 
leration ?  It  so  happened  in  this  case,  that  the  first  conjecture  of  Ga- 
lileo,  afterwards    defended   by   Casraeus,  (that  the    velocity    was   propor- 


OF   THE  LAWS  OF   MOTION.  l67 

tional  to  the  space  from  the  beginning  of  the  motion)  was  not  only 
contradictory  to  fact,  but  involved  a  self-contradiction;  and  was, 
therefore,  easily  disposed  of.  But  this  accident  did  not  supersede  the 
necessity  of  Galileo  and  his  pupils  verifying  their  assertion  by  refer- 
ence to  experiment,  since  there  were  many  suppositions  which  were 
different  from  theirs,  and  still  possible,  though  that  of  Casrasus  was 
not. 

The  mistake  of  Aristotle  and  his  followers,  in  maintaining  that 
large  bodies  fall  more  quickly  than  small  ones,  in  exact  proportion 
to  their  weight,  arose  from  perceiving  half  of  the  third  law  of  motion, 
that  the  velocity  increases  with  the  force  which  produces  it ;  and  from 
overlooking  the  remaining  half,  that  a  greater  force  is  required  for  the 
same  velocity,  according  as  the  mass  is  larger.  The  ancients  never 
attained  to  any  conception  of  the  force  which  moves  and  the  body 
which  is  moved,  as  distinct  elements  to  be  considered  when  we  en- 
quire into  the  subject  of  motion,  and  therefore  could  not  even  propose 
to  themselves  in  a  clear  manner  the  questions  which  the  third  law  of 
motion  answered. 

But,  when,  in  more  modern  times,  this  distinction  was  brought  into 
view,  the  progress  of  opinion  in  this  case  was  nearly  the  same  as  with 
regard  to  the  other  laws. 

It  was  allowed  at  once,  and  by  all,  that  action  and  reaction  are 
equal ;  but  the  controversy  concerning  the  sense  in  which  this  law  is  to 
be  interpreted,  was  one  of  the  longest  and  fiercest  in  the  history  of  ma- 
thematics, and  the  din  of  the  war  has  hardly  yet  died  away.  The 
disputes  concerning  the  measure  of  the  force  of  bodies  in  motion, 
or  the  vis  viva,  were  in  fact  a  dispute  which  of  two  measures  of  action 
that  I  have  mentioned  above  should  be  taken ;  the  effect  in  a  given 
time,  or  the  whole  effect :  in  the  one  case  the  momentum  {MV)  in  the 
other  the  vis  viva,  {MV'^)  was  the  proper  measure. 

20.  It  may  be  observed  that  the  word  momentum,  which  one  party 
appropriated  to  their  views,  was  employed  to  designate  the  motive 
quantity   of  force,    or   the   action    of  bodies    in    motion,    before   it   was 

Y2 


168       Mr  WHEWELL,  ON  THE  NATURE  OF  THE  TRUTH 

determined  what  the  true  measure  of  such  action  was.  Thus  Galileo, 
in  his  "Discorso  intorno  alle  cose  che  stanno  in  su  I'Acqua,'"  says,  that 
momentum  "is  the  force,  efficacy,  or  virtue  with  which  the  motion 
moves  and  the  body  moved  resists;  depending  not  on  weight  only, 
but  on  the  velocity,  inclination,  and  any  other  cause  of  such  virtue." 

The  adoption  of  the  phrase  vis  viva  is  another  instance  of  the  extent 
to  which  men  are  tenacious  of  those  terms  which  carry  along  with  their 
use  a  reference  to  the  fundamental  laws  of  our  thought  on  such  matters. 
The  party  which  used  this  phrase  maintained  that  the  mass  multiplied 
into  the  square  of  the  velocity  was  the  proper  measure  of  the  force 
of  bodies  in  motion;  but  finding  the  term  moving  force  appropriated 
by  their  opponents,  they  still  took  the  same  term  force,  with  the 
peculiar  distinction  of  its  being  living  force,  in  opposition  to  dead 
force  or  pressure,  which  they  allowed  to  be  rightly  measured  by  the 
momentum  generated  in  a  given  time.  The  same  tendency  to  adopt, 
in  a  limited  and  technical  sense,  the  words  of  most  general  and  fun- 
damental vise  in  the  subject,  has  led  some  writers  (Newton  for  instance,) 
to  employ  the  term  motion  or  quantity  of  motion  as  synonymous  with 
momentum,  or  the  product  of  the  numbers  which  express  the  mass 
and  the  velocity.  And  this  use  being  established,  the  quantities  of 
motion  gained  and  lost  are  always  equal  and  opposite;  and,  therefore 
the  quantity  which  exists  in  any  given  direction  cannot  be  increased 
or  diminished  by  any  mutual  action  of  bodies.  Thus  we  are  led  to  the 
assertion  which  has  already  been  noticed,  that  the  quantity  of  motion 
in  the  world  is  always  the  same.  And  we  now  see  how  far  the 
necessary  truth  of  this  proposition  can  be  asserted.  The  proposition  is 
necessarily  true  according  to  our  notions  of  material  causation ;  but  the 
measure  of  "quantity  of  motion,"  which  is  a  condition  of  its  truth,  is 
inevitably  obtained  from  experience. 

21.  It  is  not  surprising  that  there  should  have  been  a  good  deal 
of  confusion  and  difference  of  opinion  on  these  matters :  for  it  appears 
that  there  is,  in  the  intellectual  constitution  and  facvdties  of  man,  a 
source  of  self-delusion  in  svich  reasonings.  The  actual  rules  of  the 
motion  and  mutual  action   of  bodies  are,   and   must    be,  obtained  from 


OF  THE   LAWS   OF  MOTION. 


169 


observation  of  the  external  world :  but  there  is  a  constant  wish  and 
propensity  to  express  these  rules  in  such  terms  as  shall  make  them 
appear  self-evident,  because  identical  with  the  universal  and  necessary 
rules  of  causation.  And  this  propensity  is  essential  to  the  progress  of 
our  knowledge ;  and  in  the  success  of  this  effort  consists,  in  a  great 
measure,  the  advance  of  the  science  to  its  highest  point  of  simplicity 
and  generality. 

22.  The  nature  of  the  truth  which  belongs  to  the  laws  of  motion 
will  perhaps  appear  still  more  clearly,  if  we  state,  in  the  following- 
tabular  form,  the  analysis  of  each  law  into  the  part  which  is  necessary, 
and  the  part  which  is  empirical. 


First 
Law. 


Second 
Law. 


Third 
Law. 


Necessary. 

Velocity  does  not  change 
without  a  cause. 


The  accelerating  quantity 
of  a  force  is  measured  by  the 
acceleration  produced. 


Reaction  is  equal  and  op- 
posite to  action. 


Empirical. 

The  time  for  which  a  body  has  al- 
ready been  in  motion  is  not  a  cause  of 
change  of  velocity. 

The  velocity  and  direction  of  the  mo- 
tion which  a  body  already  possesses  are 
not,  either  of  them,  causes  which 
change  the  acceleration  produced. 

The  connexion  of  the  parts  of  a  body, 
or  of  a  system  of  bodies,  and  the  action 
to  which  the  body  or  system  is  already 
subject,  are  not,  either  of  them,  causes 
which  change  the  effects  of  any  ad- 
ditional action. 


Of  course,  it  will  be  understood  that,  when  we  assert  that  the  con- 
nexion of  the  parts  of  a  system  does  not  change  the  effect  of  any 
action  upon  it,  we  mean  that  this  connexion  does  not  introduce  any 
new  cause  of  change,  but  leaves  the  effect  to  be  determined  by  the 
previously  established  rules  of  equilibrium  and  motion.  The  connexion 
will  modify  the  application  of  such  rules ;  but  it  introduces  no  ad- 
ditional rule:  and  the  same  observation  applies  to  all  the  above  stated 
empirical  propositions. 


170       Mr  WHEWELL,  ON  THE  NATURE  OF  THE  TRUTH 

This  being  understood,  it  will  be  observed  that  the  part  of  each  law 
which  is  here  stated  as  empirical,  consists,  in  each  case,  of  a  negation 
of  the  supposition  that  the  condition  of  the  moving  body  with  respect 
to  motion  and  action,  is  a  cause  of  any  change  in  the  circumstances  of 
its  motion;  and  from  this  it  follows  that  these  circumstances  are  de- 
termined entirely  by  the  forces  extraneous  to  the  body  itself. 

23.  This  mode  of  considering  the  question  shews  us  in  what 
manner  the  laws  of  motion  may  be  said  to  be  proved  by  their  sim- 
plicity, which  is  sometimes  urged  as  a  proof.  They  undoubtedly  have 
this  distinction  of  the  greatest  possible  simplicity,  for  they  consist  in 
the  negation  of  all  causes  of  change,  except  those  which  are  essential 
to  our  conception  of  such  causation.  We  may  conceive  the  motions 
of  bodies,  and  the  effect  of  forces  upon  them,  to  be  regulated  by  the 
lapse  of  time,  by  the  motion  which  the  bodies  have,  by  the  forces 
previously  acting ;  but  though  we  may  imagine  this  as  possible,  we  do 
not  find  that  it  is  so  in  reality.  If  it  were,  we  should  have  to  con- 
sider the  effect  of  these  conditions  of  the  body  acted  on,  and  to  com- 
bine this  effect  with  that  of  the  acting  forces ;  and  thus  the  motion 
would  be  determined  by  more  numerous  conditions  and  more  complex 
rules  than  those  which  are  found  to  be  the  laws  of  nature.  The  laws 
which,  in  reality,  govern  motion  are  the  fewest  and  simplest  possible, 
because  all  are  excluded,  except  those  which  the  very  nature  of  laws 
of  motion  necessarily  implies.  The  prerogative  of  simplicity  is  possessed 
by  the  actual  laws  of  the  universe,  in  the  highest  perfection  which  is 
imaginable  or  possible.  Instead  of  having  to  take  into  account  all  the 
circumstances  of  the  moving  bodies,  we  find  that  we  have  only  to 
reject  all  these  circumstances.  Instead  of  having  to  combine  empirical 
with  necessary  laws,  we  learn  empirically  that  the  necessary  laws  are 
entirely  sufficient. 

24.  Since  all  that  we  learn  from  experience  is,  that  she  has  no- 
thing to  teach  us  concerning  the  laws  of  motion,  it  is  very  natural 
that  some  persons  shovdd  imagine  that  experience  is  not  necessary  to 
their  proof.  And  accordingly  many  writers  have  undertaken  to  esta- 
blish  all    the   fundamental  principles    of  mechanics   by   reasoning  alone. 


OF   THE   LAWS   OF   MOTION.  171 

This  has  been  done  in  two  ways: — sometimes  by  attending  only  to  the 
necessary  part  of  each  law  (as  the  parts  are  stated  in  the  last  para- 
graph but  one)  and  by  overlooking  the  necessity  of  the  empirical 
supplement  and  limitation  to  it; — at  other  times  by  asserting  the  part 
which  I  have  stated  as  empirical  to  be  self-evident,  no  less  than  the 
other  part.  The  former  way  of  proceeding  may  be  found  in  many 
English  writers  on  the  subject;  the  latter  appears  to  direct  the  reason- 
ings of  many  eminent  French  mathematicians.  Some  (as  Laplace)  have 
allowed  the  empirical  nature  of  two  out  of  the  three  laws  ;  others,  as 
M.  Poisson,  have  considered  the  first  as  alone  empirical ;  and  others,  as 
D'Alembert,  have  assumed  the  self-evidence  of  all  the  three  indepen- 
dently of  any  reference  whatever  to  observation. 

25.  The  parts  of  the  laws  which  I  have  stated  as  empirical, 
appear  to  me  to  be  clearly  of  a  different  nature,  as  to  the  cogency 
of  their  truth,  from  the  parts  which  are  necessary ;  and  this  difference 
is,  I  think,  established  by  the  fact  that  these  propositions  were  de- 
nied, contested,  and  modified,  before  they  were  finally  established.  If 
these  truths  could  not  be  denied  without  a  self-contradiction,  it  is 
difficult  to  understand  how  they  could  be  (as  they  were)  long  and 
obstinately  controverted  by  mathematicians  and  others  fully  sensible  to 
the  cogency  of  necessary  truth. 

I  will  not  however  go  so  far  as  to  assert  that  there  may  not  be 
some  point  of  view  in  which  that  which  I  have  called  the  empirical 
part  of  these  laws,  (which,  as  we  have  seen,  contains  negatives  only,) 
may  be  properly  said  to  be  self-evident.  But  however  this  may  be, 
I  think  it  can  hardly  be  denied  that  there  is  a  difference  of  a  fun- 
damental kind  in  the  nature  of  these  truths, — which  we  can,  in  our 
imagination  at  least,  contradict  and  replace  by  others,  and  which,  his- 
torically speaking,  have  been  established  by  experiment; — and  those 
other  truths,  which  have  been  assented  to  from  the  first,  and  by  all, 
and  which  we  cannot  deny  without  a  contradiction  in  terms,  or  reject 
without   putting  an   end  to  all  use  of  our  reason  on  this  subject. 

26.  On  the  other  hand,  if  any  one  should  be  disposed  to  maintain 
that,  inasmuch    as    the   laws    are  interpreted   by    the   aid  of  experience 


172  Mh  WHEWELL,   on   the    NATURE   OF   THE    TRUTH,  &c. 

only,  they  must  be  considered  as  entirely  empirical  laws,  I  should  not 
assert  this  to  be  placing  the  science  of  mechanics  on  a  wrong  basis. 
But  at  the  same  time  I  would  observe,  that  the  form  of  these  laws  is 
not  empirical,  and  would  be  the  same  if  the  results  of  experience 
should  differ  from  the  actual  results.  The  laws  may  be  considered  as 
a  formula  derived  from  a  priori  reasonings,  where  experience  assigns 
the  value  of  the  terms  which  enter  into  the  formula. 

Finally,  it  may  be  observed,  that  if  any  one  can  convince  himself 
that  matter  is  either  necessarily  and  by  its  own  nature  determined  to 
move  slower  and  slower,  or  necessarily  and  by  its  own  nature  deter- 
mined to  move  uniformly,  he  must  adopt  the  latter  opinion,  not  only 
of  the  truth,  but  of  the  necessity  of  the  truth  of  the  first  law  of 
motion,  since  the  former  branch  of  the  alternative  is  certainly  false :  and 
similar  assertions  may  be  made  with  regard  to  the  other  laws  of  motion. 

27.  This  enquiry  into  the  nature  of  the  laws  of  motion,  will,  I 
hope,  possess  some  interest  for  those  who  attach  any  importance  to  the 
logic  and  philosophy  of  science.  The  discussion  may  be  said  to  be 
rather  metaphysical  than  mechanical ;  but  the  views  which  I  have  en- 
deavoured to  present,  appear  to  explain  the  occurrence  and  result  of 
the  principal  controversies  which  the  history  of  this  science  exhibits ; 
and,  if  they  are  well  founded,'  ought  to  govern  the  way  in  which  the 
principles  of  the  science  are  treated  of,  whether  the  treatise  be  intended 
for  the  mathematical  student  or  the  philosopher. 


173 


VIII.  Researches  in  the  Theory  of  the  Motion  of  Fluids.  By  the  Rev. 
James  Challis,  late  Fellow  of  Trinity  College,  Cambridge, 
and  Fellow  of  the  Cambridge  Philosophical  Society. 


[VieaA  March  3,  1834-3 


1.  The  subjects  treated  of  in  this  communication  are  of  a  miscel- 
laneous character,  referring  to  several  points  of  the  theory  of  fluid 
motion,  respecting  which  the  author  conceived  he  had  something  new 
to  advance.  In  illustration  of  the  principles  he  has  attempted  to  establish, 
solutions  are  given  of  two  problems  of  considerable  interest: — the 
resistance  to  the  motion  of  a  ball-pendulum ;  and,  the  resistance  to  the 
motion  of  a  body  partly  immersed  in  water  and  drawn  along  at  the 
surface  in  the  horizontal  direction.  The  principal  object  in  the  solution 
of  the  latter  problem  is  to  account  for  the  rising  of  the  body  in  the 
vertical  direction  on  increasing  the  velocity  of  draught,  which  in  some 
recent  experiments  on  canal  navigation  has  been  observed  to  take  place. 
In  the  course  of  the  paper  I  have  had  occasion  to  refer  several  times 
to  a  previous  communication*  to  this  Society  respecting  fluid  motion, 
for  the  purpose  of  giving  to  the  views  there  advanced  some  corrections 
and  confirmations  which  have  been  suggested  by  more  mature  considera- 
tion. For  the  sake  of  distinctness  the  subjects  of  the  present  essay 
are  divided  into  sections. 


•  Camb.  Phil.  Trans.  Vol.  HI.  Part  in. 
Vol.  V.    Part  II.  >     Z 


174  Mb  CHALLIS's  RESEARCHES  IN  THE  THEORY 


SECTION    I. 

On  the  Integral  of  the  Equation    -j-^  +  -~  =  Q. 

2.  This  equation  is  applicable  to  all  problems  respecting  the  motion 
of  incompressible  fluids,  which  require  for  their  solutions  the  consideration 
of  motion  in  one  plane  only.  Mathematicians  have  obtained  integrals 
of  it  suited  to  the  particular  questions  they  were  discussing ;  for  instance, 
in  solving  the  problem  of  waves  propagated  in  a  canal  of  uniform 
width,  M.  Poisson  has  given  a  value  of  (p,  which,  while  it  satisfies  the 
equation  in  question,  is  exclusively  applicable  to  that  problem.  But 
it  is  well  known  that  by  the  common  method  of  finding  the  integrals 
of  linear  partial  differential  equations  of  the  second  order  between 
three  variables,  a  value  of  cp  may  be  found  prior  to  any  consideration 
of  the  circumstances  under  which  the  fluid  was  put  in  motion.  There- 
fore any  inferences  respecting  the  nature  of  the  motion,  which  may  be 
drawn  from  this  integral,  must  be  equally  applicable  to  all  problems  of 
this  class.  To  obtain  such  inferences  is  the  object  of  the  following 
reasoning. 

3.  The  integral  I  speak  of  is. 

To  ascertain  its  general  signification,  I  propose  to  determine  the  forms 
of  the  functions  F  and  jf,  independently  of  any  hypothesis  respecting 
the  mode  in  which  the  fluid  was  put  in  motion.  The  quantity  (j)  is 
subject  to  the  condition  {d(p)  =  udx-^vdy,  where  u  and  v  are  the 
velocities   at  the  poiiit   xy    in   the  directions   of  the   axes   of  x   and   y 

respectively.     Hence  ^^=«,    -j^=v,    and 

u  =  F'  {x  +  y^/'^\)+f  {x-y^/'^), 
v=V^lF'ix  +  y\/^^)-\/'^fix-yV^l). 


OF  THE  MOTION  OF  FLUIDS.  '  175 

First,  it  may  be  observed  that  u  and  v  are  not  both  possible  for  any 
values  of  x  and  y,  unless  the  functions  F'  andy  be  the  same.  Again, 
as  the  form  of  F'  we  are  seeking  for  is  to  be  independent  of  all  that 
is  arbitrary,  it  will  remain  the  same  whatever  direction  we  arbitrarily 
assign  to  the  axes  of  co-ordinates.  Let  therefore  the  axis  of  y  pass 
through  the  point  to  which  the  velocities  u,  v,  belong.     Then 

y  =  0,    u  =  2F'{x),    v  =  0. 

If  now  the  axes  be  supposed  to  take  any  other  position,  the  origin 

remaining   the  same,  u   will  be  equal   to      /  ^       ^  F'  {^/x^  +  y^). 

Hence 

F'{x  +  y^^)  +  F'(x-yV-l)=-^^y^^=^.F'(./^FTr), 

a  functional  equation  for  determining  the  form  of  F'.     Let 

x  +  yy^  -  l=m,   and  x  —  y^/—  1  =n; 
then 

2x  =  m  +  n,    and    "s/ x^ -k-y^^s/ mn. 
Therefore, 

c 

It  is  easily  seen  that  if  F\y/mn)  =— f=,  the  equation  is  satisfied. 
Hence      ^ 

^  = ^-7=+ ^==  =  A^      and^--^^.    ^-^ 

dx       x  +  yV^l^  a;-yV-l       x'^  +  y^'  dy~x^  +  y'-' 

2C  -^    - 


and  consequently  the   velocity  at  xy,   or   \/ u'^  +  v^  = —-r=^      ^-    ,.,      ,, 

These  results  shew  that  the  velocity  is  directed  to  or  from  the  origin 
of  co-ordinates,  and  varies  inversely  as  the  distance  from  it.  But  we 
must  observe  that  this  limitation  as  to  the  point  to  which  the  velocity 
is  directed,  is   owing  to  the  particular  forms,  x+y'\/~^,  x-ys/  —  \, 

z2 


176  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

of  the  quantities  which  the  function  F'  involves.  For  the  equation 
^j%  +  -T-?  =  0,   is  also   satisfied   by   the  following, 

((>  =  F  {{a^-  X  cos  d  -  y  m\  Q)  +  {^  +  xsm9  +y  cosO)  ^/^^} 

+y {("  +  a;  cos  0-y  sin  0)  -  (/3  +  a;  sin  0  f  y  cos  9)  V~^\  '■> 

and  this  analytical  circumstance  has  its  interpretation  in  reference  to 
the  motion  of  the  fluid.  By  supposing  the  function  f  to  be  the  same 
as  F,  and  giving  to  F'  the  same  form  as  before,  we  shall  find, 

'  d^  _  2  C(.r  +  g  cos  g  +  /3  sin  9) 

dx  ~  (a  +  xcos9  —  i/  sin  9y  +  (13  + x  sin  9 +  y  cos  9)- 

d^ 2C(y  +  /3cosg-asin  9) 


dy       {a -\- X  cos  9  —  y  sin  9'f  +  (/3  +  a;  sin  0  +  y  cos  9Y 

/d^y     /d^Y 4C^ 

\dx )         \dy)    ~  {a  +  x  cos9-y  sin9y  +  (^  +  x  sm9  +  y  cos9y 
Or,    if   a  cos  9-\-fisin9=  —a,    and    /3  cos  9  — a  sin  9=  —b, 

d(f>  2C(x-a) 

dx  (x  —  ay  +  {y  —  bf ' 

^     or    V-       ^^(y-*) 
dy  ^x-ay  +  iy-by 

Vu'  +  v^  = 


y/ix-af  +  iy-by' 

This  shews  that  the  velocity  is  directed  to  the  point  whose  co-ordinates 
are  a,  b,  and  varies  inversely  as  the  distance  from  it.  And  as  we  have 
arrived  at  this  result  without  considering  any  circumstances  under  which 
the  fluid  was  caused  to  move,  the  inference  to  be  drawn  is,  that  such 
is  the  general  character  of  the  motion.  Nothing  forbids  our  considering 
C,  a,  and  b,  functions  depending  on  the  time  and  the  given  conditions 
of  motion  in  any  proposed  problem.  Also  if  at  a  given  instant,  a  line 
commencing  at  any  point,  be  drawn  continually  in  the  direction  of  the 
motions  of  the  particles  through  which  it  passes,  C,  a,  and  b,  may  be 


OF  THE  MOTION  OF  FLUIDS.  177 

supposed  to  vary  in  any  manner  along  this  line.  The  foregoing- 
reasoning  only  proves  that  in  passing  at  a  given  instant  from  one 
point  to  another  indefinitely  near  along  the  line,  these  quantities  may 
be  considered  constant. 

4.  The  nature  of  the  integral  we  have  been  discussing  will  perhaps 
be  understood  by  comparing  it  to  the  general  integral  of  a  common 
differential  equation,  which  has  a  particular  solution.  The  latter,  we 
know,  is  that  which  gives  the  answer  to  a  proposed  problem,  and  the 
general  integral  is  used  (though  not  necessarily)  to  obtain  this  solution. 
So,  I  conceive,  the  integral  above  is  useful  for  arriving  at  the  particular 
functions  of  x,  y,  and  t,  which  give  the  velocity  and  direction  of  the 
velocity    at    any    point    and   instant    in    any   proposed    question.      The 

integral    of   -—^  +  -r^  =  0 ,    which    M.M.    Poisson    and    Cauchy    have 

obtained  for  the  solution  of  the  problem  of  waves,  may  be  called  the 
particular  solution  of  the  equation,  for  that  particular  problem ;  and  I 
think  it  probable  that  the  same  might  have  been  obtained  by  employing 
what  I  would  call  the  general  integral,  though  I  am  not  prepared  to 
state  exactly  the  process. 

5.  The  following  considerations  are  added  in  confirmation  of  the 
foregoing  reasoning.  In  whatever  manner  the  fluid  is  put  in  motion, 
we  may  conceive  a  line,  commencing  at  any  point,  to  be  continually 
drawn  in  a  direction  perpendicular  to  the  directions  of  the  motions  at 
a  given  instant  of  the  particles  through  which  it  passes.  This  line 
may  be  of  any  arbitrary  and  irregular  shape,  not  defined  by  a  single 
equation  between  x  and  y.  But  it  must  be  composed  of  parts  either 
finite  or  indefinitely  small,  which  obey  the  law  of  continuity.  Con- 
sequently the  motion,  being  at  all  the  points  of  the  line  in  the  directions 
of  the  normals,  must  tend  to  or  from  the  centres  of  curvature,  and 
vary,  in  at  least  elementary  portions  of  the  fluid,  inversely  as  the 
distances  from  those  centres.  An  unlimited  number  of  such  lines 
may  be  drawn  through  the  whole  extent  of  the  fluid  mass  in 
motion. 


178  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

6.     If  we  put   (f}  =  (pi  +  (p2  +  (f>3  +  Sac.   we  shall  have 

d^       d^_  (d^       d^\         Id^       dy,\         id^       d-^<pA   ,   .      _  „ 
daf  "*"  df  ~  \dx'   ^   dfj  '^   \dx'   "^   df)  ■*■   \dx'    ^  df)'^^-~^- 

Hence  if  there  be  any  number  of  functions  which  severally  satisfy  the 
given  equation,  the  sum  of  these  will  satisfy  it.  But  from  what  has 
been  proved  above,  if 


d^i                 Ci(x  —  a,) 

d(p,  _         C:(y-/3,) 

dx  ~  (ar-aO'  +  (y-/3,)" 

dy       (x-a,y  +  (y-l3,r' 

a  02                   Cs  {x  —  aj) 

d(j>,              Q{y-fi,) 

dx  ~  ix-a,r  +  oj-(i,y' 

dy        (^x-a,f  +  iy-fi,Y' 

&c.  =  &c. 

&c.  =  &c. 

<p\,  02.  03,  &c.  will  severally  satisfy  it;  therefore  0i  +  02  1- 0,  +  &c. 
will  also.     And  we  have, 

dx        dx        dx 

v=^  +^  +^  +&,c 
dy        dy        dy 

These  equations  prove  that  the  velocity  at  any  point  may  be  the  re- 
sultant of  several  velocities  produced  by  different  causes;  and  that  any 
given  cause  will  have  the  same  effect  in  producing  velocity  at  a  given 
point,  whether  or  not  other  causes  may  be  operating  to  produce 
velocities  at  the  same  point. 

7.  We  may  here  also  determine  the  manner  in  which  the  motion 
of  the  fluid  is  affected,  when  the  rectilinear  transmission  of  an  impulse 
tending  from  any  centre  is  interrupted  by  a  plane  surface.  For  suppose 
two  impulses  tending  from  two  centres  to  be  of  equal  magnitude  and  in 
every  respect  alike.  Then  if  the  straight  line  joining  these  centres  be 
bisected  at  right  angles  by  a  plane,  there  will  be  no  motion  of  the  par- 
ticles contiguous  to  the  plane  in  a  direction  perpendicular  to  it,  because 
the  resultant  of  the  velocities  from  the  two  causes  must  lie  wholly  in 


7)U>.1I-    OF  THE  MOTION  OF  FLUIDS.  179 

the  plane.  Hence  as  the  division  of  fluids*  may  be  effected  without  the 
application  of  force,  nothing  will  be  altered  if  we  suppose  the  plane  to 
become  rigid  and  to  intercept  the  communication  of  the  fluid  on  one  side 
with  that  on  the  other.  The  motion  on  each  side  will  then  be  reflected, 
and  the  angle  of  incidence  will  be  equal  to  the  angle  of  reflection.    - 

8.  I  propose  now  to  adduce  an  application  of  the  proposition 
above  demonstrated  (Art.  3.)  respecting  the  general  law  of  fluid  motion, 
which  may  serve  to  shew  its  utility.  Suppose  water  in  a  cylindrical 
vessel  (for  instance,  a  glass  tumbler,)  to  be  caused  to  revolve  with  con- 
siderable rapidity  about  the  axis  of  the  cylinder.  There  is  no  practical 
difficulty  in  making  the  fluid  revolve  so  that  every  particle  shall  de- 
scribe approximately  a  horizontal  circle  about  the  axis.  Then,  the  fluid 
being  left  to  itself  after  the  disturbance,  each  particle  may  be  considered 
to  move  as  it  does,  by  reason  of  a  centripetal  force  tending  to  the 
axis  in  a  horizontal  plane.  This  force  must  be  owing  to  the  action 
of  the  cylindrical  surface  on  the  fluid  particles  in  contact  with  it, 
deflecting  them  continually  from  a  rectilinear  course.  If  V  be  the 
velocity  of  the  particles  in  contact  with  the  surface,  and  a  the  radius 

V- 
of  the   cylinder,    the   force   tending    to    the   axis   is    — ,    the   effect   of 

friction  being  neglected.     The  deflections  which  this  force  is  continually 

producing  in  the  directions  of  radii,  are  transmitted  through  the  fluid, 

and  as  they  tend  to  a  centre,   will  vary,   according  to  the   proposition 

above  proved,  inversely  as  the  distance  from  the  centre.f      Hence  the 

V^      a  V^ 

centripetal    force   at   the   distance    r    is    —  x  -,    or    — .      This   shews 
^  a        r  r 

that   at   any  distance   r  the   velocity    is   still    V.     Experience   seems   to 

confirm  this  result.     For  if  light  substances  be  strewed  on  the  surface 

of  the  water,  those  nearer  the  centre  always   perform  their  revolutions 

*  The  introduction  of  this  consideration  here  is  merely  reverting  to  a  principle  -which 
Professor  Airy  (very  properly,  I  think,)  has  proposed  to  make  the  basis  of  the  mathematical 
treatment  of  fluids.  Without  referring  to  a  principle  of  this  nature,  I  do  not  see  that 
problems  of  reflection  can  be  satisfactorily  solved. 

+  The  total  motion  is  compounded  of  these  deflections  and  rectilinear  motions  along 
tangents  to  the  circles,  which  by  Art.  6.  may  be  considered  separately.  '''  '    •"'-'■'■ 


180  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

in  less  time  than  those  more  remote.  This  is  particularly  observable 
in  two  of  the  floating  particles  which  are  near  each  other,  and  at  nearly 
equal  distances  from  the  centre.  That  which  is  less  distant  overtakes 
the  other,  as  it  ought  to  do,  supposing  it  to  describe  a  less  circle  with 
equal  velocity.  At  the  centre  a  kind  of  eddy  is  formed,  the  more 
observable  as  the  motion  at  every  point  of  the  surface  is  more  nearly 
in  concentric  circles.  When  the  revolving  motion  takes  place  in  a 
conical  tunnel  from  which  the  water  is  issuing,  the  appearance  at  the 
axis  is  very  remarkable,  a  hollow  space  like  a  sack,  being  formed  a 
considerable  way  down  the  axis.  What  has  been  here  said  may  serve 
to  explain  in  some  measure  the  manner  in  which  eddies  in  any  case 
are  produced. 


SECTION    II. 


On   the  Integration  of  the  Equation    -^  +  -r^  +  -r^  =  0. 

9.  M.  Poisson  has  expressed  the  general  integral  of  this  equation 
by  means  of  definite  integrals  ;  {Memoires  de  rAcademie  des  Sciences, 
Ann.   1818),   and  this,   I  believe,   admits  of  a  discussion  similar  to  that 

applied  above  (Art.  3.)  to  the  integral  of  -^  +  -~  =  0.     But  perhaps 

the  following  reasoning,  analogous  to  what  was  indicated  in  Art.  5., 
may  be  considered  sufficient.  In  whatever  manner  the  fluid  is  put  in 
motion,  we  may  conceive  a  surface  to  be  described,  which  shall  be 
every  where  perpendicular  to  the  directions  of  the  motions  at  a  given 
instant  of  the  particles  through  which  it  passes.  This  surface  may  be 
of  an  arbitrary  and  irregular  shape,  not  necessarily  defined  by  a  single 
equation  between  x,  y,  and  %.  But  it  must  be  composed  of  parts  either 
finite  or  indefinitely  small,  which  are  continuous,  and  consequently  have 
radii  of  curvature  subject  to  the  same  conditions  as  those  of  regular 
curve  surfaces.  Hence  the  normals  to  all  the  points  of  any  element 
of  the  surface  will  pass  through  two  focal  lines,  situated  at  the  centres 
and   in    the    planes  of  greatest    and  least    curvature,    and   cutting    the 


OF  THE  MOTION  OF  FLUIDS.  181 

directions  of  the  normals  at  right  angles.  The  motion,  being  in  the 
normals,  will  be  directed  to  the  focal  lines.  If  we  describe  another 
surface  indefinitely  near  the  first,  and  cutting  in  like  manner  the  direc- 
tions of  the  motion  at  right  angles,  all  the  points  of  any  fluid  element 
intercepted  between  two  opposite  elements  of  the  surfaces,  will  at  a 
given  instant  ultimately  have  their  motion  directed  to  the  same  focal 
lines :  but  this  cannot  be  said  in  general  of  more  than  an  elementary 
portion.  If  we  suppose  the  form  of  the  superficial  element  to  be  a 
rectangle,  the  normals  through  all  the  points  of  its  sides,  will  inclose 
a  wedge-shaped  mass,  the  transverse  section  of  which  at  any  point,  it 
is  easy  to  shew,  will  vary  as  the  product  of  the  distances  of  that  point 
from  the  focal  lines.  Hence  the  velocity  in  passing  at  a  given  instant 
from  the  first  to  the  second  of  the  surfaces  above-mentioned  wiU  vary 
inversely  as  this  product.  Let  therefore  r  and  r  +  l  he  the  distances 
of  the   point  whose   velocity   is    V,   from  the  focal  lines   to   which  the 

C 

motion   is   directed.     Then    V=     . j-,  in  which  expression   C,  /,   and 

the  positions  of  the  focal  lines  are  constant  at  a  given  instant,  when 
r  varies  through  a  space  which  may  either  be  finite  or  indefinitely  small. 
Let  a,  /3,  7,  be  the  co-ordinates  of  the  middle  of  that  focal  line  which 
is  distant  by  r  from  the  point  in  question.     The  velocity  (m)  in  x  will 

then   be    V. ;    the  velocity    {v)   in    y,    V. ;    and    the    velocity 

{w\  in  »,  V. ^.     Hence 

'  r 

udx  +  vdy  +  wd%=  Vi — ~dx  +  ^  ,     dy  H -d%\ . 

Now  since  r-  =  {x  —  af  +  {y  —  fif-\-{%~yY,  if  we  make  r  vary  with 
X,  y,  and  %,  while  a,  )3,  7,  remain  constant  according  to  what  has  just 
been  said,  we  shall  have  rdr  —  {x  —  a)dx  +  {y-fi)dy  +  {%  —  y)dti.  Hence 
tfdx  +  vdy +  wdz=F^dr;  and  as  F"  is  a  function  of  r  and  /,  the  right 
side  of  the  equation  is  a  complete  differential  of  a  function  of 
X,  y,  %,  and  t,  with  respect  to  the  three  first  variables,  t  being  con- 
stant. Therefore  also  the  left  side  is  the  same.  Let  the  function  be  <p. 
Vol.  V.     Part  II.  A  a 


182  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

Then 

dr  '      dx         '     dy  '      d% 

We   proceed   to  shew  next   that   the   equation 

d^d)       d^d)       d^d>       „  du       dv       dw 

zr^  +   j^  +  -j^  =  0,    or    J-  +  T-  +  ^-  =  0, 
daf       df        d%^  dx      dy       dx 

is  satisfied  by  the  kind  of  motion  we  have  been  describing. 

10.  Let  P  (Fig.  1.)  be  the  point  whose  motion  we  are  considering; 
Or,  Nq,  the  focal  lines  to  which  the  motion  of  the  element  at  P  is 
directed.  Let  PNO  be  the  straight  line  which  passes  through  P  and 
the  focal  lines,  cutting  them  in  N  and  O.  Suppose  O  to  be  the 
origin  of  a  system  of  axes,  of  which  ONP  is  the  axis  of  x,  Oy  coinciding 
with  the  focal  line  Or  the  axis  of  y,  and  0%  perpendicular  to  the  plane 
yOx,  the  axis  of  %.  The  co-ordinates  of  P  referred  to  another  system 
of  rectangular  axes  AX,  AY,  AZ,  are  X,  Y,  Z:  p  is  a  point 
indefinitely  near  to  P,  Pp  is  parallel  to  AZ,  and  the  co-ordinates  of 
p  are  X,  Y,  Z+SZ:  pqr  is  the  straight  line  which  passes  through  p 
and  the  focal  lines  cutting  them  in  q  and  r.  Now  let  the  equations 
of  Pp  referred  to  the  system  Ox,  Oy,  0%,  be  x  =  a%  +  a,  y  —  b%  +  fi, 
and   the  equations   of  pqr,   x  =  dz-\-a,    y  =  b'z  +  li'.     Then 

„  l+aa'  +  bb' 

cos  ^  Ppq  =  „ , —  . 

Let  ON=l,  NP=r.  Hence  because  Pp  passes  through  P  whose 
co-ordinates  referred  to  the  axes  Ox,  Oy,  Oz,  are  I  +  r,  0,  0,  it  follows 
that  l  +  r  =  a,  and  /3  =  0.  Thus  the  equations  of  Pp  become  x  =  az  +  l  +  r, 
y  =  bz.  Again,  because  the  line  ^^gr  passes  through  r,  whose  co-ordinates 
are  x  —  0,  z  =  0,  we  have  a'  =  0 ;  and  because  it  passes  through  q,  whose 
co-ordinates    are   y  =  0,    x  =  l,    we    have   l=a'z,    and    0  =  i's:  +  /3'.     Hence 

a:  =  -  =  -  -n,    and    consequently    ft'  = r.      Thus  the  equations  of 

pqr    become    x  —  a'%^    y^V% y .      Also    because    Pp    and    pqr    pass 


OF  THE  MOTION  OF  FLUIDS.  183 

through   the   same  point  p,   a;  =  «'x  =  a«  +  /+r,   and  therefore   ^  =  -7 . 

And  y  =  hz  =  h'% ^i  therefore  z  =  -rm — tx  •     Hence  -; =  —rrr, — 7t> 

''  a'  a{b'-h)  a -a     a{b-b) 

which    gives    h'  =    — ; j^.      From  p  draw  ps   perpendicular   on    Ox, 

and  let  P.?  =  5.     Then  ^  =  x-{r  +  l).     Bnt  x  =  a'z  =  t!^±Il,     Therefore 

'  a  —a 

^  =  — 7 '- .      Hence     it     will    be     found     that     a'  =  — — » ,     and 

a  —a  6 

V  =  — J — -.     This  latter  quantity,  if  we  neglect  powers  of  S  above 

the   first,   is   equal   to (l  H — rj A.     Therefore  by  substitution 

„                                     d  r       \        r{l  +  r)J 

cos  /  Jr«o  =  — -.  .   ,        ,  ,. 

a'(^r  +  l)+(l+d'  +  b'—]s 
=  (neglecting  ^,  &c.)  V        ^   / 

Here      /         „  ===  is   the  cosine   of  the  angle  pPs.     Hence  if  ^  =  the 
V  1  +  a^  +  o^ 

velocity  at  P  in   Ox,  and  w  the  part  resolved  in  the  direction  parallel 

Va 

to  AZ,  w  —  — / 2~^i  •     ^^^  ^  ~  ^^  resolved  portion  of  the  velocity 

at  p  in  the  same  direction.      Now  the  velocity  at  p  is  ultimately  the 

same    as    that    at    s,    and    is     therefore     equal    to     V .  -, A^ \ — r-  , 

according    to    the   law   of    variation   from    P   to    s  determined .  by   the 

AA2 


w 


184  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

considerations  with  which  we  commenced  this  investigation.    Neglecting 

powers  of  I  above  the  first,  this  quantity  becomes    V  \\ ^  J  . 

Consequently 

a 

But   S  =  SZcospPs  =  SZ  X     ,       'g.Jp'     Hence 

w'-w  _  „ /    \-d         1  ¥-a^       1 

~IZ  U +«'  +  *'■ /  +  r  "^  l+«2  +  fr^'r 

If  now  a,  ft,  7,  be  the  angles  which  the  axis  AZ  makes  with 
Ox,    Oy,   0%,   respectively,   we  have 

Hence   passing   from   differences   to   differentials, 

-7-  =  (COS^'V  —  COS'o)^ +   (cOS^/3-COS*a)  - (1). 

d%  '  '  l  +  r  '  r  ' 

So  if  d,  /3',  7',  be  the  corresponding  angles  for  the  axis  of  Y,  and 
a",  /3",  7",  for  that  of  ^,  v  the  velocity  in  F",  and  u  that  in  ^,  we  shall 
have  by  a  like  process, 

^  =  (COSV  -  COs'a')  ,—  +   (coS^/3'  -  COS^a')  .......  (2) , 

^  =  (cos^y-  COs'a")  y^  +   (cos'/3"-COS^a")  ^ (3). 


OF  THE  MOTION  OF  FLUIDS.  185 

But  as  a,  a,  a",  are  the  angles  which  Ox  makes  with  three  rectangular 

axes, 

cos"  a  +  cos"  a  +  COS"  a"  =  1, 

so   cos-/3  +  cos^/3'  +  cos^/3"  =  l, 

and   cos'^7 +  cos'^7' +  cos^7"  =  l. 

Therefore   by   adding   the   equations   (1),    (2),    (3), 

du        dv       dw  _ 
7lX^  dY^dZ~ 

11.  The  general  conclusion  from  all  that  precedes  is,  that  the  law 
of  the  variation  of  the  velocity  from  any  point  to  another  indefinitely 
near  in  the  direction  of  the  motion,  at  a  given  instant,  may  be  expressed 

C 

by    -^ — J-,   the  quantities   C,  r,   and  I,  being  such  as  we  have  stated 

C 

in    Art.    9-      If  1=0,   we   have- as    a   particular   case,    V=-^.      In   my 

former  paper  on  the  motion   of  fluids,   I   assumed,   as   it   now   appears, 

C 

incorrectly,  that  —   represents  the   general   law  of  the   variation   of  the 

velocity.  None,  however,  of  the  results  in  that  paper  are  affected  by 
the   assumption.      For    instance,   the   expression   for 

as  it  only  requires  that  (p  should  be  a  function  of  r  and  /,  will  remain 
the   same.      This    expression    may   also   be   briefly    obtained   thus.      We 

have   seen   that    -~-  =  V.     Now   as  r  is   ultimately  in  tlie   direction  in 

which  the  velocity  V  takes  place,  if  a  line  commencing  at  a  given 
point  be  drawn  constantly  in  the  direction  of  the  motion  at  a  given 
instant  of  the  points  through  which  it  passes,  dr  may  be  considered  the 
increment  of  this  line.      Hence  if  we   call   its   length  s  reckoned  from 

the    fixed    point,    -j^  = -^  =  F.      Then    integrating,    c}>  =  jVds -^/{t); 


1«6  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

and    differentiating    under    the    sign    /,     ^  =  f  -r-^^  +J''(^)-      Hence 
substituting  for  -^  in   the  known   expression  for  the  pressure  {p), 

p  =  f(Xdx  +  Ydy  +  Zdz)  -  f^ds  -  ^  -fit). 

If   f^  be  always  the  same  in  quantity  and  direction  at  the  same  point, 

dr  V^ 

-^  =  0 :    so    that,   p  =  j{Xdx  +  Ydy  +  Zd%)  -  -—  -f{t). 

This  equation  may  thus  be  considered  to  be  strictly  deduced  from  the 
general   equations   of  fluid   motion. 

Considerations  analogous  to  those  applied  (Arts.  6  and  7)  to  motion 
in  a  plane,  might  here  be  introduced  to  shew  that  the  motion  at  any 
point,  when  due  to  several  causes,  is  the  resultant  of  the  motions  which 
would  be  produced  by  the  causes  acting  separately ;  and  also  to  determine 
the  same  law  of  reflection  at  a  plane  surface. 

12.  The  following  simple  instance  of  fluid  motion  may  serve  to 
illustrate  some  points  of  the  preceding  theory.  BCD  (Fig.  2.)  is  a 
conical  vessel  with  its  axis  vertical.  A  mass  of  fluid  DBhd  is  made 
to  descend  so  that  its  lower  surface  hd  is  bounded  by  a  horizontal 
plane  to  which  any  arbitrary  velocity  is  given.  The  upper  surface  is 
also  supposed  to  be  plane  and  to  be  kept  horizontal  by  the  force  of 
gravity.  It  is  required  to  find  the  consequent  velocity  and  pressure 
at  every  point  of  the  fluid. 

It  is  evident  that  the  motion  will  be  in  vertical  planes  passing  through 
the  axis,  and  will  be,  the  same  in  all  such  planes.  Take  therefore  two 
planes  making  an  indefinitely  small  angle  with  each  other,  and  let 
AB,  AE,  be  their  intersections  with  the  upper  surface,  ab,  ae,  with 
the  lower.  Let  PQSB  be  an  element  of  the  upper  surface,  P  and  B 
being  equidistant  from  A,  as  also  Q  and  S.  If  now  at  any  instant 
lines  commencing  at  the  four  points  P,  Q,  B,  S,  be  continually  drawn 


OF  THE  MOTION  OF  FLUIDS.  187 

in  the  direction  of  the  motion  at  the  points  through  which  they  pass, 
these  lines  must  be  rectiUnear,  because  there  is  no  curvilinear  motion 
at  the  boundaries  of  the  fluid,  and  therefore  no  cause  to  impress  a 
curvilinear  motion  on  the  parts  interior.  The  straight  lines  commencing 
at  P  and  R  will  intersect  ah  and  ae  at  p  and  r,  points  equidistant 
from  a,  and  those  commencing  at  Q  and  S  will  intersect  the  same 
lines  at  q  and  s  also  equidistant  from  a.  Now  from  the  law  of 
the  variation  of  the  velocity  above  found,  at  every  point  of  the  cunei- 
form element  Ps,  the  velocity  will  be  inversely  proportional  to  its 
transverse  section.  Let  therefore  V  =^  the  vertical  velocity  with  which 
(lb  is  made  to  descend,  and  v  the  vertical  velocity  with  which  the 
surface  DB  descends.  Let  AB  =  a,  AQ  =  x,  PQ  =  X,  ah  =  h,  aq  =  x', 
pq  =  'S.',  and  the  angle  BAE  =  e.  Then  the  element  PQSR^^xeX, 
and  pqsr  =  x'e\'.  These  elements  are  proportional  to  the  transverse 
sections   at  P  and  p ;    and   the   vertical    velocities     V,    v,   are    to    each 

other   as   the   velocities   at  p  and  P  in  Pp.      Hence  —  =  -; — ,  =  -V-,  • 

-'  ■*  V       X  e\        xX 

F  •     Wence  ^,  -  ^ 

because  the   motion  is  along  the  slant  surface.     Therefore  in  this  case 

X       a 

r-,  ="  T.      Suppose   X  to  be  given,  and  let   Xi    be   the   consequent   value 

X         o 

of  x'.     Then  —  =  -r,   and  -. =  y .     If  now  x  be  taken  =  «  —  X,  from 

X,       o  b-Xi       o 

what  has  been  just  shewn,  x'  will  =  6  — X, .     Hence  4 —  =  j^,    and 

\0  —  Xi)  X2       o 

consequently     —  =  t-      Therefore    X2  =  Xi;    and   so    on.      From    this    it 

X2        o 

follows  that  if  AB  and  ab  be  divided  into  the  same  number  of 
indefinitely  small  equal  parts,  the  straight  lines  joining  the  corresponding 
points  of  division  will  give  the  directions  of  the  motion,  which  is 
consequently  every  where  directed  to  the  vertex  of  the  cone.    Hence  the 

velocity  af^  any  point  W  whose  distance  Cp  W  from  C  is  p,  varies  as  —  . 

P' 
Let  CA  =h,  Ca  =  k,    z  AC W=  9  ;  then  the  velocity  at  p=V sec  9,  and 

the    velocity    at    W=  V  sec  9 . ^ — ;     this    resolved    in     the    vertical 


But  —  also  =  jj, .     Hence  ^j^,  =  71  •      If    we    take    x  =  a,    x'  must  =  h. 


188  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

f^k'sec'6                        VTf 
direction   gives  j— — ,   which  =  — —  =  velocity  at  Z.     Hence  the 

vertical  velocity  is  the  same  at  all  points  of  any  horizontal  plane,  and 
the  fluid  will  consequently  descend  in  parallel  slices.*  Let  us  now 
determine  the  pressure  at  any  point  on  the  particular  supposition  that 
V  is   uniform.      Then   if 

Vk"&eee  ^,  ,     ..       ,    „r   dw       Vsec'd     ^,dk  2F"Asec'0 

w  =  :: the  velocity  at    W,  -7-  =  ; —  x  2«-7--  = y, . 

p'  ■'at  p^  at  p- 

And 

Idt^'  ^-Jdt'^P-J 7 = -p +  ^ 

» 

Hence 

„                2r'kse&9       r'k'sec'e 
p  =  C-g.  + . __. 

And   as  when   z  =  h,  p  —  0,   and   p  cos  Q  =  h,   it   follows   that 

The  above  solution  I  do  not  consider  to  be  of  any  value,  except  as 
illustrating  the  process  to  be  followed  in  determining  mathematically 
the  way  in  which  the  interior  of  a  mass  of  fluid  is  affected  as  to 
velocity  and  pressure,  in  consequence  of  given  conditions  at  the 
boundaries.  This  part  of  the  theory  of  fluid  motion  is  very 
defective. 


*  I  obtained  this  result  in  the  number  of  the  Phil.  Mag.  and  Annals  of  Philosophy 
for  .Jan.  1831,  but  omitted  to  shew  that  it  is  entirely  dependent  on  the  arbitrary  condition 
that  the  inferior  -surface  of  the  fluid  is  bounded  by  a  horizontal  plane.  Qji  any  other 
supposition  the  problem  would  be  one  of  much  greater  difficulty.  This  omission  has  not 
without  reason  caused  a  misapprehension  as  to  the  application  of  the  solution,  on  the  part  of 
Berzelius  in  a  notice  taken  of  it  in  his  Annual  Review.     {Jahres-Bericht,  1833.) 


OF  THE  MOTION  OF  FLUIDS.  189 


SECTION    III. 


Application  of  the  Principles  of  the  foregoing  Section  to  an  instance  of 
the  Resistance  of  an  Incompressible  Fluid  to  a  Body  hounded  by  a 
Spherical  Surface  moving  in  it. 

13.  Let  a  solid  sphere,  partially  immersed  in  water,  being  of  less 
specific  gravity  than  the  fluid,  be  drawn  along  in  a  horizontal  direction 
with  a  given  uniform  velocity ;  it  is  required  to  find  the  height  of 
its  centre  above  the  horizontal  surface  of  the  water. 

We  shall  suppose  for  the  sake  of  simplicity,  that  the  fluid  is 
unlimited  in  extent  both  in  the  vertical  and  horizontal  directions,  and 
that  the  surface  of  the  sphere  is  so  smooth  that  it  impresses  no  velocity 
on  the  water  in  contact  with  it  in  the  direction  of  a  tangent  plane. 
Let  CDJBJE  (Fig.  3.)  be  the  sphere,  O  its  centre,  ADE  the  intersection 
of  the  horizontal  surface  of  the  fluid  by  a  vertical  plane  through  the 
centre  of  the  ball;  OQ  a  line  through  the  centre  parallel  to  ADE. 
This  will  be  the  direction  of  the  motion  of  O,  since  the  velocity  is 
supposed  to  have  become  uniform,  and  ON  to  be  constant.  Let  A, 
a  fixed  point  in  ADE,  be  the  origin  of  co-ordinates,  AN=a,  NO  =  'y, 

at  any  instant.     Then  the  velocity  {V)  of  O  =  -r-.     Draw  OB  vertical; 

let  P  be  any  point  of  the  surface  immersed;  through  P  draw  the 
spherical  arcs  PQ,  PB,  and  let  the  angle  QOP=6,  and  the  angle 
PQB  =  to.  The  velocity  impressed  by  the  sphere  on  the  fluid  at  P 
is  F'cos  9,  as  none  is  impressed  in  the  direction  of  a  tangent  plane. 
This  velocity  is  directed  to  the  point  O,  because  in  the  case  of  a 
spherical    surface    /  =  0.      Hence    if    «    =    the    radius    of    the    sphere, 

C 

FcosO  =  — .     (Art.  11.)     The  velocity  at   every  point   of  the  line   OP 

produced,    wiU   at    a    given   instant    be    in    the   direction    of    this  line, 
Vol.  V.    Part  II.  B  b 


190  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

because  when  the  fluid  is  of  unlimited  extent,  there  is  no  cause*  to 
produce  motion  at  any  point  of  the  line,  but  the  impression  made  at  P, 
which  is  transmitted  instantaneously,  varying  at  different  distances 
from  O  according  to  the  law  of  the  inverse  square.     Hence  if  ^  be  a 

point  in   OP  produced,  and   OR  =  r,  the  velocity  at  R  —  —,  = ^ — . 

Let  ADE  be  the  axis  of  x,  a  vertical  through  A  the  axis  of  z  reckoned 
positive  downwards,  and  a  line  through  A  perpendicular  to  the  plane 
of  these  two  the  axis  of  y.     Then  if  the  co-ordinates  of  R  be  x,  y,  %, 

we  shall  have  r- =  (:r  —  a)" -I- y^  +  (s  +  7)' ;    and    cos0= .     Therefore 

the   velocity    {v)  at  R, 

•A-A  I'? 

VoH^X-a) 


Ka;-ay^  +  y'  +  (»  +  7)-}5" 


And 


Hence 


dv        dv    da      re        rr        :i  4.      ..k 

~j-  =  7-  •  77  >    (lor    ^  and  7  are  constant), 

_  F«^(3cos-e-l)    (la 
r"  '  dt 

rV(3cos*^-l) 


/^rf,=/(o-g5:(3cos'e-i). 


Therefore,  gravity  being  the  only  force  acting  on  the  fluid,  the  pressure 
ip)   at   R, 


*  This  cannot  be  said  of  the  parts  of  the  fluid  adjacent  to  the  radii  produced  which  pass 
through  the  circle  in  which  the  surface  of  the  water  meets  the  surface  of  the  sphere,  because 
the  water  outside  of  the  conical  surface  formed  by  these  radii  must  be  put  in  motion  by  that 
within  by  reason  of  the  difference  of  pressure  occasioned  by  the  motion.  On  account  of  the 
difficulty  of  estimating  this  effect,  it  is  left  out  of  consideration  in  our  solution,  which  can 
therefore  be  only  considered  approximate. 


OF  THE  MOTION  OF  FLUIDS.  191 

=  ^«  +  -27^(3008-'^-  1)  -  -g^-cos'e  -f{t). 

When  r  is  indefinitely  great   this  equation   becomes  p=g^—f{t)',    and 

as  for  this  value  of  r  the  velocity  =  0,  p  must  =  g% ;   therefore  /{t)  =  0. 

If  now  we  put  r  =  a,  and  i8  =  ss,,  the  co-ordinate  of  P,  we   obtain   the 

?^*  cos  20 
pressure  (/>,)  at  P,  =  gz,  -\ .      The  portion  of  this  resolved  in 

the  vertical  direction  is  jo,  x  cos  i  FOB.  But  from  the  spherical 
triangle  PQB,  cos  /.  FOB  =  cos  w  sin  9.  Therefore  the  vertical  pressure 
is  p,  cos  w  sin  6.  The  element  of  the  surface  at  F  =  ad9  x  a  sin  ddw. 
Hence  the   whole   vertical  pressure  = //jOia'sin''^  cos  wt/^c^w 

=ga^ff%i  sin^OeoswdOdw  +  ———  ff sin^ 9 cos29 cos wd$dw. 

M 

The  first  term  is  plainly  the  weight  of  water  displaced,  and  is  there- 
fore  equal  to  — -(2«'  — 3«'7 +  7*),  the  specific  gravity  of  the  water 
being  1.  The  integrations  with  respect  to  u>  must  be  taken  from 
a,=  —  cos"^ — jT—r  to    -f  cos"^ — ^-;r ,    and    the    integrations    with    respect 

to  6  from  sin"'—  to  the  supplement  of  that  arc.     Between  these  limits 


of    w,    fcoswda)  =  2\/i T       ;    and    between    the    limits    of    9, 


a'^sm''9 


2fsm'9cos29d9  x/i  _    , '^^      =_  J  fi  _:>:!')  . 
•^  ^  «*sin''0  2  V        «V 

Therefore  if   JV  =  the  weight  of  the  sphere,  which  is  the  same  as  the 
whole   vertical  pressure,  and   w  =  the  weight   of  fluid   displaced, 


IV  =w- 


4 

B  B  2 


i^-i)- 


192  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

This  result  shews  that  the  weight  of  fluid  displaced  is  greater  than 
the  weight  of  the  sphere,  and  consequently  that  the  centre  O  is  lower 
than  it  would  be  in  a  state  of  rest. 

Suppose  a  portion  of  the  sphere  to  be  cut  off"  by  a  horizontal 
section  at  the  distance  of  b  from  the  centre ;  and  let  7  become  7',  the 
centre  being  still  above  the  surface  of  the  water.  Then  if  we  suppose 
the  motion  to  be  always  in  the  direction  of  the  radii*,  and  the  horizontal 
bottom  to  have  no  effect  in  impressing  motion,  the  equation  for  this 
case  will  be. 


W=w- 


TrF'a' 


ttTV 
=  w : — 


The  difference  between    W  and  w  is  here  less  than  before  on   account 

of    both   the   factors    —;  and    1  —  -yr ;    for    -?-  is    greater   than   - .     This 

a*  b*  b  ^  a 

seems  to  shew  that  curved  bottoins  tend  to  depress  the  vessel  when  it 

begins  to  move,  and  consequently  to  increase  the  resistance. 

As  the  term  —-— ff sm^6eos26 cos uidOdu)  is  positive  from  0=:sin"'  — 
to  0  =  45°,  and  from   0  =  135°  to  6  =  the  supplement  of  sin"'  —  ,  let  us 

Cv 

integrate  for  the  portion  of  the  surface  corresponding  to  these  limits, 
or  what  amounts  to  the  same,  take  the  double  of  the  integral  between 
the  first  limits,  those  of  w  remaining  the  same  as  before.  In  order  to 
abstract  from  the  consideration  of  the  portion  of  the  surface  not  taken 
into  account  in  this  integration,  we  may  suppose  the  portions  for 
which  we  integrate  to  be  connected  by  a  cylindrical  surface,  the  radius 
of  which  =  a  sin  45°.  The  length  of  this  cylindrical  part  may  be  any 
we  please :    the   vertical  pressure   against  it   will  be   only  equal  to  the 

*  This  again  cannot  be  true  in  the  direction  of  the  radii  which  pass  through  the  lower 
circular  boundary  of  the  surface. 


OF  THE  MOTION  OF  FLUIDS.  193 

weight  of  fluid  displaced.  Also  the  shape  of  the  floating  body  above 
the  part  immersed  is  of  no  importance  to  the  problem.  The  form  of 
the  whole  body  may  be  such  as  is  described  in  Fig.  4,  ABCDEF 
being  a  half  cylinder  of  which  the  axis  is  GH,  and  ALC,  FKD,  the 
extreme  portions  of  the  body,  bounded  by  spherical  surfaces  which  have 
their   centres  at  M  and   N.      Now  in  general   ^  jjsin^d  co^^O  co&  wdQdw, 

commencing  at   0  =  sin~'— ,   and   ending  at   any   other   value   of  9,   will 

be  found  to  be 


cosefssin^e  +  l-^')  V  sin^e-^  -\(\-—i 


And  if  we  put  cos  9  =  — ?= ,   we  shall  have 


COS0 


a- 


V 

W  =  w  + 


As  the  second  term  is  necessarily  positive,  the  floating  body  will  be 
higher  than  it  would  be  in  a  state  of  rest,  and  consequently  the 
surface  against  which  the  resistance  acts  becomes  less  by  an  increase 
of  velocity. 

To  obtain  a  numerical  result  respecting  the  rise  of  the  body 
corresponding  to  a  given  velocity,  we  will  suppose  for  the  sake  of 
simplicity  of  calculation  that  when  the  vessel  is  at  rest,  the  centres 
of  the  spherical  ends  and  consequently  the  axis  of  the  cylindrical  part, 
are  in  the  plane  of  the  horizontal  surface  of  the  water.  This  circum- 
stance may  be  produced  by  loading  the  upper  part  of  the  body 
.  without  altering  its  specific  gravity.  Let  /  =  the  length  of  the  axis 
of  the  cylindrical  portion.     Then   the  area  of  the  horizontal  section  of 

the   vessel   at   the   level   of    the   water   surface    is    ID  H ■— — ,   its 

4  2 

breadth  being  D.     Now    W—w  must  be  equal  to  the  difference  of  the 


19*  Mr  CHALLIS's  RESEARCHES  IN  THE  THEORY 

quantities   of  fluid   displaced   in   the   states  of  rest  and  motion,  and   is 
therefore    equal    to    yg  \ID+'^ —  j  ,    7    being    small.      Therefore 

neglecting  powers  of  —  above   the   first, 

Let  ^  =  3.     It  will  then  be  found  that  F'  =  696**  x  7.     And  if  7  =  one 

inch,  or  ^,  this  equation  gives   ^=519   miles  per   hour;    consequently 
if  ^=10*4  miles  per  hour,  7  =  4   inches. 

2 

In   general,   neglecting   "—,   &c. 


TV-w== 


r'a' 


sin  e cos  e  (2  sin'0  +  ^ )  ~ |)  ' 


also    W  —  w  =  yg  llD  +  ^---iAO  -  sin  9  cos6)\   nearly  ; 

therefore,  as  I>  =  Zasm9,  it  will  be  found   that 

F-    sin2  0(2sin'0  +  l)-0  ,    .  ^        / 

y  = -r-  •'-4 '--TT, — •    r.n     r,ny   w?   bemg  put   for   -y\. 

'      4!g     4!msm'9-sm26  +  29  "    *  D 

If  9  be  assumed  equal  to  15°,  and  711  =  3,  this  equation  gives    ^"=7-35 
miles  per  hour  when  7  =  4  inches. 

These  results,  which  probably  are  but  very  rough  approximations 
to  matters  of  fact,  may  yet  suffice  to  shew  that  when  vessels  and  boats 
of  the  usual  forms  sail  in  the  open  sea,  they  may  be  expected  to  rise 
in  some  degree  upon  an  increase  of  their  velocity,  and  so  much  the 
more  as  they  are  less  adapted  to  cleave  the  water.  Our  theory  shews 
that  the  rise  is  the  same  for  bodies  of  the  same  shape  and  proportions, 
moving  with  the  same  velocity,  whatever  be  their  absolute  magnitudes; 
also  that  this  effect  is  equally  due   to  the  pressures  on    the   front   and 


OF  THE  MOTION  OF  FLUIDS.  195 

stern  of  the  vessel.  The  theory,  in  fact,  determines  these  pressures  to 
be  in  every  respect  alike,  so  that  if  we  proceeded  to  investigate  the 
total  pressure  in  the  horizontal  direction,  we  should  find  it  to  be 
nothing,  when  the  motion  is  uniform.  This  may  serve  to  shew  that, 
if  friction  be  left  out  of  consideration,  a  front  ill-adapted  to  cleave  the 
water,  is  not  unfavorable  to  speedy  motion,  if  the  stern  be  of  the  same 
shape;  and  that  the  resistance  to  the  motion  of  vessels  in  the  open 
sea  is  principally  owing  to  the  friction  of  the  water  against  their 
surface.  This  cause  operates  to  produce  unequal  actions  on  the  front 
and  stern,  making  the  directions  of  the  motions  of  the  particles  in 
contact  with  the  surface  of  the  former,  less  inclined  to  the  horizon 
than  they  would  be  in  the  case  of  no  friction,  and  of  those  in  contact 
with  the  surface  of  the  latter  more  inclined.  To  counteract  this  inequality 
probably  the  stern  should  be  less  curved  than  the  front. 


SECTION    IV. 

General  Propositions  respecting  the  Motion  of  Compressible  Fluids. 

14.  The  considerations  applied  at  the  beginning  of  Section  II.  to 
incompressible  fluids,  are  equally  applicable  to  compressible.  I  shall 
therefore  assume  that  in  a  mass  of  fluid  in  which  the  density  varies 
as  the  pressure,  the  directions  of  the  motion  at  all  the  points  of  any 
element  pass  at  a  given  instant  through  two  focal  lines.  Let  p  be 
the  density  at  a  point  distant  by  r  and  r  -vl  from  the  focal  lines,  and 
V  the  velocity :  p  and  V  the  same  for  a  point  indefinitely  near  the 
former.  Also  let  the  transverse  section  of  a  cuneiform  element  aclk 
(Fig.  5.)  which  is  bounded  by  four  pli.nes  passing  through  the  focal 
lines  kl,  mn,  be  at  the  first  point  efgh,  and  at  the  other,  abed.  The 
pressure  and  consequently  the  density  will  be  the  same  at  all  points 
of  the  section  eg;  as  also  the  velocity;  at  least  our  reasoning  does  not 
apply  to  cases  in  which  this  condition  is  not  fulfilled.  The  same  may 
be  said  of  the  section  ac  and  of  all  sections  intermediate  to  ac  and  eg. 


196  Mr  CHALLIS's  RESEARCHES   IN   THE   THEORY 

Let  now  the  area  of  eg  =  m,  and  that  of  ac  =  m'.  Then  if  the  motion 
which  exists  at  a  given  instant,  be  supposed  to  be  continued  uniform 
for  the  small  time  t,  the  quantity  of  fluid  which  passes  the  section  eg 
in  that  time,  is  mpF^T,  and  that  which  passes  ac  is  m'p'Vr.  Hence 
the  increment  of  matter  between  the  two  sections  is  —  {m' p'V'T  —  mpVT), 
whether  the  velocity  tend  from  or  to  the  focal  lines,  being  considered 
negative  in  the  latter  case.     The  increment  of  density  {Ip)  of  the  element 

in  the  time  t,  is  consequently  —  ^^ — - — r—, — —. — —  ■      But  —  =  — ^^ =^  . 

^         •'  m{r'-r)  m        r{r  +  l) 

Hence 

pT'r'ir'  +  D-prnr  +  l)  _^^^^^^Sp_^ 

And  passing  from  differences   to  differentials, 
^^^^^'dt  ~  dr 


or 


dp  dV       ,.dp  ,^  /i         1    \ 


As    before    udx  +  vdy  +  wd%  =  V dx  +  V^^ — —  dy  +  V d%  =  Vdr, 

"  f*  T  T 

if  a,  /3,  7,  be  the  co-ordinates  of  the  middle  point  of  the  focal  line  hi. 
Now  as  we  have  supposed  that  in  passing  from  one  point  to  another 
of  tlie  element  acge,  the  change  of  velocity  at  a  given  instant  depends 
only  on  the  change  of  r,  we  may  consider  V  a  function  of  r  and  t, 
and  Vdr  a  differential  of  a  fimction  of  r  and  t.  Then  udx  ^  vdy 
+  wdfi  =  d(l),  a  complete  differential  of  a  function  of  x,  y,  and  as;    and 

-~  =  V.      But  in  this   case  we  have  the  known  equation, 

a'  Nap. log.  p^fiXdx  +  Vdy  +  Zd%)  -  ^  -  ~ (B.) 

Therefore  considering  X,    V,  Z,  to  be  independent  of  the  time, 

d'dp  _      d^(p  dV         d'(ji      d(p     d'<p 

pdt  ~       df  dt  ~d¥  ~  'dr  '  drdt ' 


OF   THE   MOTION  OF   FLUIDS.  197 

But   from   (A), 

pdt  pdr  '  dr  dr'  dr  \r       r  +  l)  ' 

And  differentiating   (B)   with  respect  to   space  only, 

^1^  =  Xdx+  Vdy  +  Zdx-d.^  -  VdV. 
p  at 

If  the  variation  be  from  one  point  to  another  in  the  direction  of  the 

motion,    dx  = dr,    dy  =  - — —  dr,    dz  = dr.     Hence, 

r  ^  r  r 

a\dp  ^  X  ^-°  ,    Y  y~^  +  Z  ^^^  _-^      d(p    d'(f> 
pdr  '     r  '     r  '     r         drat      dr  '  di^  ' 

Substituting  this   value   of   — ^  in   the   foregoing   equation,    and   then 

equating  the  two  values  of       ,'] ,  we  shall  obtain, 

/  d£\d^_Q^     d^       d^t   .   ,.d^(l   ,   J_\ 

\     ~  dt^j  dr'         dr  '  drat       df  "*"      dr\r  "^  r  +  l) 


+  ^  (x^^  +  ry^  +  Z'-^)  =0 (C.) 

dr  \         r  r  r    I 


This  is  an  equation  of  general  application.  If,  as  a  particular  case, 
I,  a,  /3,  7,  each  =  0,  we  shall  have  the  equation  I  obtained  in  my 
former  paper  (Art.  4.)  by  assuming  ^  to  be  a  function  of  v^a^  +  y^  +  s!^ 
and  t  in  the  equation  {n)  of  the  Mecanique  Analytique  (Part  II. 
Sect.  XII.  Art.  8.) 

It  may  be  proved  as  in  Art.  11,  that  -^  =  /~77  ^*'  ^^  ^'^^  incom- 
pressible  fluids,  and  that   the   equation   applicable   to   steady  motion  is, 

a'  h.  1.  p  =  fiXdx  +  Ydy  +  Zd%)  -  ^  +  fit) . 
Vol.  V.     Part  II.  Cc 


198  Mr   CHALLIS's   RESEARCHES   IN   THE    THEORY 

15.     If  r  be   indefinitely   great   in   equation   (C),    the   motion   is   in 

parallel    lines,     and    putting    r  =  c  +  s,      j=j-      Let    -^  =  w,    and 

suppose  no  force  to  act ;    the  equation  for  this  case  becomes 

d''(p         2w        (Pep  1         d'(p  _ 

~d?  ~  '^^'  ■  dsdt  "^  o^^T^  '  dF~^' 

This  equation  combined  with  a*  N.  1.  p  =  —  -^  —  — ,  gives  as  a  particular 

integral,   u]  =  al:iA.  p  =/"{«- {a +  w)t\.     By  varying  a  little  the  mode  of 

_  ^(  as. 


integrating,    I    found    also    w  —  a^A.  p  =/( ■ atj,     {Camb.   Phil. 

Trans.  Vol.  III.  Part  III.  p.  399),  and  endeavoured  to  shew  the  way 
in  which  each  integral  ought  to  be  applied.  But  this  enquiry  was 
unnecessary ;  for  the  integral  may  present  itself  under  an  unlimited 
number  of  different  forms.      The  equations 

w  =  a^.\.p=f{.^-{a  +  io)t  +  ^{w)],    or    «,  =  «N.].  jo=/(^^^^%i^l , 

will    equally    satisfy    the    differential    equations,    being,    in    fact,    only 

different  forms  of  the  first-mentioned  integral.     The  principle  according 

to  which  it  now  appears  to  me,  an   integral  of  this  nature  should   be 

employed,   is    to   apply    it    immediately  only   to    the   parts   of  the   fluid 

immediately    acted    upon    by    the    arbitrary    disturbance,    in    order    to 

determine  the  law  according  to  which  the  initial  velocity  is  transmitted 

to   the   contiguous   parts ;    then    to   determine   the    law   of  transmission 

from    these   to    the    next;    and    so    on    in    succession.     In    the   present 

instance   by   making  *   and   t  vary   so  that  w  and   p   remain   the  same, 

ds 
we  shall  find  a  +  w  for  -j~  the  velocity  of  transmission,  under  whatever 

form  the  integral  may  appear.  The  second  term  m  of  this  quantity 
is  due  to  the  transmission  of  velocity  through  space  by  the  motion  of 
the  particles  themselves ;  the  other  a  is  the  velocity  of  propagation 
along  the  particles.  In  this  example,  as  the  velocity  and  density  are 
propagated  uniformly  and  undiminished,  it  is  easy  to  determine  at  any 


OF  THE   MOTION  OF  FLUIDS.  199 

instant  the  velocity  and  density  at  any  given  point,  which  result  from 
a  given  disturbance.  In  other  cases  in  which  the  velocity  of  propaga- 
tion is  variable,  the  determination  would  be  more  difficult,  but  must 
probably  be  arrived  at  by  the  same  principle  of  reasoning.  Variable 
propagation  is  analogous  to  variable  motion,  as  uniform  propagation  to 
uniform  motion,  and  would  seem  to  require  integration  to  determine 
the  time  at  which  the  effect  of  a  given  disturbance  is  felt  at  a  given 
place. 

16.  If  in  the  equation  (C),  a  be  an  indefinitely  great  quantity, 
the  terms  which  do  not  contain  a^  as  a  factor  may  be  neglected  in  com- 
'parison  of  those  which  do,  and  the  equation  will  become 


dr^  ^  dr\r  ^  r  +  l)        ' 


which  by  integration  gives  -^  =  — j-,  the  same  as  for  incompressible 

fluids.  This  result  was  to  be  expected,  because  a,  as  is  well  known, 
is  the  velocity  of  propagation  in  the  compressible  fluid,  and  when  this 
becomes  infinite,  the  propagation  is  instantaneous,  and  the  fluid  there- 
fore incompressible. 

If  /  be  indefinitely  great,   it   will   be  found  in  the   same  way  that 

-r~  —  — ,  and  the  motion  is  such  as  was  considered   Art.  3. 
dr       r 

Let   now  -^  be  very   small  compared  to  «,  and   X,    V,  Z,  and  / 
each  =  0.     The   equation   (C)   reduces  itself  to 

"-11?^^"^  dr        df-^'    '''''•     dr'     -~dF~' 

a  particular  integral  of  which  is   r^=^'P{r  —  at).     This  gives 

d^  ^  F\r-at)  _  F{r-at) 

dr  ~         r  r^       ^  "' 

CC2 


200  Mr  CHALLIS's   RESEARCHES  IN   THE    THEORY 

At  the  same  time,  because   a^'N.l.p=  —  -^  nearly,   we  shall  have 


, T  ,          F'{r-at)  ,^, 

«.N.l.p  =  ^ -(2.) 


The  equations  (1)  and  (2),  involving  but  one  arbitrary  function,  can 
apply  only  to  a  single  disturbance,  which  takes  place  in  a  direction 
tending  from  a  centre,  as  I  have  elsewhere  shewn*.  It  is  important 
to  observe  that  when  r  is  very  small,  the  term  of  equation  (1)  which 
involves  r"-  in  the  denominator  may  be  much  greater  than  that  in- 
volving r.  In  fact,  if  we  expand  the  fxmctions,  supposing  r  to  be 
very   small. 


&c. 


_  F{-at)  _  F'{-at)  _  F"{-at) 

When  therefore  the  disturbance  is  made  by  a  sphere  of  very  small 
radius  r,  the  motion  is  transmitted  from  its  surface  to  other  parts  of 
the  fluid,  nearly  as  if  the  fluid  were  incompressible. 


SECTION    V. 


Application   of  the  Principles  of  the  foregoing  Section  to  determine   the 
Resistance  of  the  Air  to  the  Motion  of  a   Sail-Pendulum. 

17.  For  the  sake  of  simplicity,  I  will  suppose  gravity  not  to  act. 
The  ball  being  spherical  and  perfectly  smooth,  the  direction  of  the 
motion  of  the  aerial  particles  in  contact  with  its  surface  tends  at  every 

*  Camh.  Phil.  Trans.  Vol.  HI.  p.  402. 


OF  THE   MOTION   OF   FLUIDS.  201 

instant  from  its   centre.      Therefore   /  =  0.      Also   if  the   radius   of  the 

ball  be  supposed  very  small,   the  equation  -f-  =  ^-t^>   obtained   at   the 

end  of  the  preceding  Article,  will  be  approximately  applicable  to  the 
motion  of  the  fluid  in  contact  with  the  ball.  Hence  the  velocity  which 
is  impressed  at  any  point  of  the  spherical  surface  may  be  considered 
to  be  transmitted  instantaneously  in  the  direction  of  the  radius  through 
that  point,  and  to  decrease  according  to  the  law  of  the  inverse  square 
of  the  distance.  The  problem,  with  the  limitations  above  made  is 
solved  in  the  same  manner  for  air  as  for  water. 

Let  now  the  origin  of  co-ordinates  be  A,  (Fig.  6.),  the  position 
of  the  centre  of  the  ball  when  it  hangs  at  rest.  I^et  its  centre  perform 
oscillations  of  very  small  extent  in  nAN,  which  we  will  consider  to 
be  rectilinear.  Suppose  N  to  be  the  position  of  the  centre  at  the 
time  t  reckoned  from  a  given  epoch,  and  call  AN,  a.  Take  P  any 
point  of  the  surface,  join  NP  and  produce  it  to  R,  and  let  NPR  make 
an   angle   Q  with   ANQ,    and   the    plane    RNQ    an   angle   /3    with   the 

plane   SAQ.     The   velocity   of  the   centre  =  ^;    and    the   velocity   of 

da 

the  air  at  P  —  -rrCosO.     Hence  if  NP=h,   and   NR  =  r,  the  velocity 

at   ^  = -„ —  .  -^ .     Now   if  AN  be  the  axis   of  x,  AS  of  a,   and   a 

r-        at 

line  through   A   perpendicular   to  the  plane  SAN,   the   axis  of  y,  and 

the  co-ordinates  of  R  be  x,  y,  %,  then  r^  =  {x  —  aY  +  y^  +  %^.     Consequently 

the  velocity  (^)    at  R=, r^ ^ — 2  •  ;77-     Hence  differentiating   V^ 

With  respect  to  the  time  only, 

dr  _  d'a    b^cos9       2b^cose{x-a)    d^      h^    da    d.cosO 
dt  ~  W-'      r'      "^  7  •  dt^  '^  r"'  df      dt      ' 

^                    ^      x  —  a      d.cosO           1     da       cos^6    da  sin^O    da 

But   as  cos9  = ,    r: —  = --77  + 


Therefore 


dt .  r  '  dt  r     '  dt  r     '  dt 


dV_d^    ¥cos9       ^b'cos'e    da'       b'sin'0    d^ 
dt  ~  df      r^      ^        r'        '  dt'  r'      '  dt^ 


202  Mr  CHALLIS's   RESEARCHES   IN   THE   THEORY 

Hence 

J   dt  df  •       r  2?  •  df  ' 


Substituting   in   equation    (B), 

j\-.  -  d^a     FC0S9  b'^    ir,        2/,        -am  da         b*  COS^  9     da  „.^^ 

«  ^-^-P  =  df  ■  —r-  +  ap  (2cos=0-sm=0)  ^  -  -^^  .  ^  -M. 
When  /•  =  infinity,  /o  =  1 :    therefore  f{t^  =  0.     Hence   when   /•  =  A, 

„,T.  ,  «?^a  ,  .         COS  20     <:?a^ 

Where  p  =  1,  let  j9  =  n  =  a^     Hence  when  (O  =  1  +  o-,  p  =  e'  (1  4-  a-)  =  n  +  aV. 
But  because  a-  is  very  small,  «^N.  l.jo  =  «V  very  nearly.     Therefore, 

„      d^a    ,        -       cos  20    rfa^ 

^  =  n+^.*cos0  +  -^.^. 

The  total  pressure  resolved  in  the  direction  NA  is  ffp¥ eos6sm9d9dfi, 
from   /3  =  0   to   /3  =  27r,   and   from   0  =  0   to   0  =  7r.     It   will   consequently 

be  found  to  be  equal  to  — —  .  -^  :   and  if  A  =  the  ratio  of  the  specific 

gravity  of  the   ball   to   that  of  air,   the  accelerative  force  produced   by 

1      d'a 

this  pressure  is   —  .  -7-7 .      But   the  accelerative  force  of  gravity  in  the 

same  direction,  if  SA  =  A,  is  ^  ( 1  ~  t"  )  »  taking  account  of  the  weight 
of  air  displaced.     Hence 


_  cP_a  _g^(-._}\       j_    d^ 


d'a  ^ 

or 


«^__^ ±^_§^(l_l]    nearly 


1  +  K 


OF    THE   MOTION   OF   FLUIDS.  203 

Therefore  if   L  be   the  length   of   the  seconds  pendulum  in  vacuum, 

2s    * 


I  in   air,  /  =  Z«  ( 1  —  —  j 


The  correction  of  the  length  of  the  pendulum  is  thus  determined 
to  be  double  of  what  it  would  be  if  the  motion  of  the  air  were  not 
considered.  It  is  to  be  observed  that  these  calculations  apply  strictly 
only  to  the  case  of  a  very  small  ball.     The  experiments  of  M.  Bessel 

give  1"956  for   the   coefficient  of   — .     Those  of  Mr  Baily,  which  were 

made  most  nearly  under  the  circumstances  which  the  theory  supposes, 
give  1"864.  The  effects  of  friction  and  of  the  suspending  wire,  would 
tend  to  make  the  coefficient  rather  greater  than  less  than  2.  I  am 
therefore  unable  to  account  for  the  difference  between  the  experimental 
and  theoretical  determinations,  which  it  appears  by  Mr  Baily's  experi- 
ments, is  greater  as  the  radius  of  the  ball  is  greater,  excepting  perhaps 
the  confined  space  of  the  apparatus  may  have  had  some  effect  on  the 
experimental  results. 

It  would  not  be  difficult  to  shew  from  the  nature  of  the  analytical 
expressions,  that  if  the  confined  space  in  which  the  balls  vibrate  were 
taken  into  account  in  the  theory,  the  same  results  would  be  obtained 
for  two  balls  of  different  diameters,  vibrating  in  different  spaces,  if  the 
linear  dimensions  of  the  spaces  were  in  the  proportion  of  the  diameters, 
their  forms  being  alike.  If  this  could  be  verified  experimentally,  it 
would  shew  that  the  difference  of  the  values  of  the  numerical  coefficient 
which  Mr  Baily  calls  n,  for  balls  of  different  diameters,  as  well  as  its 
deviation  from  the  theoretical  value  2,  is  very  probably  owing  to  the 
confined  space  of  the  vacuum  apparatus.  It  would  at  any  rate  be  de- 
sirable to  ascertain  by  experiment  whether  the  same  ball  gives  the  same 
value  of  n,  when  it  oscillates  in  apparatus  of  different  dimensions. 

Papworth  St  Everard, 
March  S,  1834. 

*  This  result  I  obtained  in  the  London  and  Edinburgh  Philosophical  Magazine  (September, 
1833),  by  assuming  the  principle  of  the  conservation  of  vis  viva,  without  employing  equa- 
tion (B). 


205 


IX.  Theory  of  Residuo-Capillarij  Attraction;  being  an  Explanation  of 
the  Phenomena  of  Endosmose  and  Exosmose  on  Mechanical 
Principles.  By  the  Rev.  J.  Power,  M.A.  Fellow  and  Tutor 
of  Trinity  Hall,  and  late  Fellow  of  Clare  Hall,  Cambridge. 


[Read  March  17,  1834-3 


1.  The  curious  and  elegant  law,  according  to  which  an  interchange 
takes  place  between  two  fluids  separated  from  each  other  by  a  thin 
membrane,  one  of  the  fluids  generally  (but  not  universally)  the  lighter 
of  the  two,  being  transmitted  in  greater  abundance,  was  discovered  a 
few  years  ago  by  Dutrochet.* 

His  experiments  tended  to  show  that  the  unknown  force  which 
operated  this  effect,  whether  measured  by  the  fluid  transmitted  in  a 
given  time,  or  by  the  pressure  required  to  stop  the  process,  was,  for 
the  same  substances,  proportional  to  the  difference  of  densities  of  the 
mixtures  on  each  side  of  the  membrane. 

The  vast  importance  of  this  law  in  animal  and  vegetable  physiology, 
renders  it  highly  desirable  that  its  theory  should  be  investigated  on 
mechanical  principles,  and  such  is  the  object  of  the  present  enquiry. 

2.  The  opinion  which  would  attribute  this  phenomenon  to  the 
existence  of  electrical  currents,  is  now  pretty  nearly  abandoned,  even  by 
Dutrochet  himself,  with  whom  it  originated,  and  who  maintained  it  with 
great   zeal,   until    the   publication   of  his   later   researches,   in    which   he 

*  L'Agent   immedial    du    Mouvement    Vital,    (Paris,    1826),    and  Nouvelles    Recherches    sur 
I'Endosmose  et  VExosmose  (Paris,  1828). 

Vol.  V.    Part  II.  Dd 


206  Mr  POWER'S  THEORY  OF 

confesses  himself  compelled  to  resign  it,  though  he  does  so  with 
manifest  reluctance.  That  electricity,  artificially  excited,  is  capable  of 
accelerating  the  process,  is  indeed  sufficiently  established  by  the  experi- 
ments of  Dutrochet;  but  it  is  equally  certain  that  this  agent  is  by  no 
means  essential  to  the  operation,  since,  in  the  natural  process,  the  most 
delicate  galvanometer  gives  no  indication  of  its  existence. 

3.  To  me  it  appears  unquestionable,  that  the  phenomenon  results 
from  the  corpuscular  attractions,  which  the  particles  constituting  the 
membrane  and  the  fluids,  exert  upon  each  other :  that  electricity, 
by  heightening  or  modifying  these  attractions,  should  produce  a  sensible 
effect  upon  the  operation,  is  nothing  more  than  its  ordinary  chemical 
agency  would  lead  us  to  expect. 

4.  By  corpuscular  attractions  are  meant  the  forces  which  the 
ultimate  atoms  of  different  materials,  whether  simple  or  compound, 
exert  upon  each  other.  These  forces  are  enormously  great  (though  not 
infinite)  when  the  particles  are  in  immediate  contact,  but  diminish  with 
extreme  rapidity,  as  the  particles  separate,  becoming  insensible  at  a 
sensible  distance.  The  effects  of  corpuscular  attraction  are  different, 
according  as  it  is  exerted  between  particle  and  particle,  or  between 
mass  and  mass.  In  the  former  case  it  gives  rise  to  the  phenomena  of 
chemical  affinity ;  and  in  the  latter,  to  those  of  cohesion,  adhesion,  and 
capillary  attraction,  which  may  be  regarded  in  general,  as  the  mutual 
attraction  of  contiguous  masses,  being  the  combined  effect  of  the 
corpuscular  attractions  of  their  integrant  particles.  It  is  under  this 
point  of  view  that  La  Place  has  considered  the  subject  of  capillary 
attraction,  and  his  theory  will  be  of  the  greatest  use  in  the  present 
investigation. 

5.  Although  no  pores  can  be  detected  in  the  membranous  partition 
by  the  help  of  the  most  powerful  microscope,  yet  the  fact  that  the 
fluids  are  transmitted,  is  a  certain  proof  that  such  pores  exist.  They 
must  indeed  be  extremely  minute,  and  it  will  be  seen  that  it  is  on 
this  very  minuteness  that  the  energy  of  the  sustaining  force  depends. 
These  pores  must  be  regarded  as  communicating  with  the  opposite  fluids 


RESIDUO-CAPILLARY  ATTRACTION.  207 

at    their    two    extremities,    while     the    fluids    meet    and    mix    in    the 
interior. 

6.  Dutrochet  argues  that  capillary  attraction  cannot  be  the  cause 
of  endosmose,  because  it  can  only  raise  a  fluid  to  a  small  height  in  a 
capillary  tube,  and  is  utterly  incapable  of  drawing  it  beyond  the  limits 
of  the  tube. 

In  stating  these  objections,  he  perhaps  does  not  consider  that  the 
height  at  which  a  fluid  may  be  sustained  in  a  capillary  tube  is  inversely 
as  its  diameter,  and  consequently  in  a  tube  of  so  extremely  small  a 
diameter  as  those  of  which  it  is  necessary  to  suppose  the  membrane  to 
consist,  that  height  might  be  almost  indefinitely  great.  It  is  true  that 
in  the  case  of  a  single  fluid,  this  effect  would  require  for  its  production 
that  the  tubes  themselves  should  be  coextensive  with  the  fluid  raised ; 
but  this  is  no  longer  necessary  when  the  two  ends  of  the  tube  are 
immersed  in  different  fluids.  The  reason  why  a  homogeneous  fluid 
cannot  be  drawn  beyond  the  limits  of  the  tube,  is,  that,  were  it  to 
be  so,  the  tube,  acting  equally  at  its  two  ends,  would  produce  no 
effect  whatever  upon  the  fluid.  But  the  circumstances  are  very  different 
when  the  extremities  communicate  with  different  fluids.  In  that  case  the 
full  residual  effect,  consisting  of  the  difference  of  effects,  which  the  same 
tube  indefinitely  extended,  is  capable  of  impressing  separately  upon  the 
two  fluids,  might  be  produced  by  an  extremely  small  length  of  tube, 
not  exceeding  a  small  multiple  of  the  sphere  of  attraction  of  the  par- 
ticles of  the  tube,  and  there  is  no  doubt  that  the  thickness  of  the 
finest  membrane  is  a  considerable  multiple  of  this  magnitude.  In  fact, 
if  we  cut  off"  from  the  ends  of  the  tube  a  distance  greater  than  the 
tube's  sphere  of  sensible  attraction,  it  is  plain  that  the  fluids  which 
occupy  the  intermediate  part,  in  whatever  way  they  may  communicate 
there,  will  suffer  no  effective  attraction  from  the  tube,  since  every 
elementary  portion  will  be  drawn  by  it  equally  in  both  directions.  The 
only  effective  attractions  will  therefore  be  those  exerted  by  an  insensible 
portion  at  each  extremity ;  we  may  therefore  imagine  these  two  por- 
tions to  be  brought  together  as  near  as  we  please  without  any  diminution 
of  effect. 


D  D  2 


208  Mr  POWER'S  THEORY  OF 

7.  In  order  to  form  some  sort  of  estimate  of  the  forces  which  may 
be  expected  to  result  from  residual  attractions  of  this  kind,  let  us 
suppose  the  fluids  to  be  water  and  alcohol,  and  the  tube  to  be  of  glass. 
Now  Gay  Lussac  found  by  experiments  of  great  accuracy,  that  in  a 
tvibe  of  glass  whose  diameter  was  1.29441  millimetres,  water  would 
stand  at  the  height  of  23"'.3791,  and  alcohol  of  specific  gravity  O.8I96 
(that  of  water  being  1)  at  the  height  of  g^^'.SgSOS.  This  column  of 
alcohol  would  be  equivalent  to  7™.7176  of  water;  the  difference  of 
effects  would  therefore  be  measured  by  a  column  of  water  of  I5"'\66l5. 
Suppose  now  the  diameter  of  the  tube  to  be  diminished  a  thousand 
times,  or  to  become  0'"'.001294,  the  column  of  water  which  measures 
the  difference  of  effects  would  be  1566l™'.5:  or,  since  the  French 
millimetre  =  .0393708  of  an  English  inch,  a  glass  tube  of  diameter 
0'".0000507,  or  about  the  twenty-thousandth  of  an  inch,  would  produce 
a  residual  effect,  with  water  and  alcohol,  measured  by  616.6  inches  or 
51"  4'"  of  water,  which  is  equivalent  to  the  pressure  of  nearly  two 
atmospheres.  When  it  is  considered  that  a  platina  wire  of  one  three- 
thousandth  of  an  inch  in  diameter  may  be  seen  by  the  naked  eye,  it  is 
probable  that  the  magnitude  we  have  assigned  to  the  capillary  tube 
is  considerably  greater  than  the  diameter  of  the  membranous  pores, 
which  evade  the  powers  of  the  strongest  microscope.  From  this  ex- 
ample I  think  the  conclusion  may  be  fairly  drawn,  that,  so  far  at  least 
as  the  magnitude  of  the  force  is  concerned,  we  need  be  under  no 
apprehension  but  that  the  residual  capillary  forces  are  sufficient  to 
account  for  the  sustaining  force  of  endosmose.  How  far  they  will 
account  for  the  law  of  its  variation  will  be  seen  hereafter. 

8.  An  attempt  to  explain  the  phenomenon  by  the  principles  of 
capillary  attraction  has  been  already  made  by  a  distinguished  mathema- 
tician, Mons.  Poisson.  He  first  abstracts  from  the  pressure  of  the 
adjacent  fluids,  by  supposing  their  altitudes  above  the  membrane  to  be 
inversely  as  their  densities.  The  fluid  in  the  tube  being  now  equally 
pressed  on  both  sides,  he  supposes  that  that  liquid,  for  which  the  tube 
has  the  stronger  attraction  is  drawn  by  this  attraction  to  the  opposite 
end,  thus  filling  the  whole  tube.  The  fluid  within  the  tube,  he  now 
argues,   will  be  urged  by  two   forces :    1st,  the  attraction  of  the  liquid 


RESIDUO-CAPILLARY  ATTRACTION.  209 

to  which  it  belongs ;  2dly,  the  attraction  of  the  opposite  liquid.  If  then 
the  latter  attraction  be  superior  to  the  former,  the  fluid  which  fills  the 
tube,  he  says,  will  be  drawn  in  an  uninterrupted  stream  into  the 
opposite  vessel. 

Dutrochet  justly  objects  to  this  theory,  that  it  will  only  account  for 
a  motion  in  one  direction,  whereas  the  phenomenon  of  exosmose  requires 
a  corresponding  motion  in  the  opposite  direction. 

Professor  Henslow,  in  a  number  of  the  Foreign  Quarterly,  suggests 
as  a  modification  of  Poisson's  theory,  that  whilst  the  fluid  within  the 
tube  is  carried  in  the  direction  of  the  stronger  attraction,  the  natural 
tendency  of  the  fluids  to  mix,  may  carry  the  other  fluid  (or,  perhaps, 
a  slight  infusion  of  it)  in  the  opposite  direction,  and  thus  produce  the 
exosmose. 

I  perfectly  agree  with  Professor  Henslow  that  the  natural  process 
of  mixture  is  the  cause  of  the  exosmose,  it  being  only  necessary  to 
suppose  that  the  rapidity  with  which  this  process  extends  itself  witliin 
the  tube  is  somewhat  greater  than  the  velocity  with  which  the  whole 
mass  of  fluid  which  fills  the  tube  is  drawn  in  the  opposite  direction. 

But  the  theory  of  Poisson  is  further  objectionable  on  this  account, 
that  it  makes  the  continuation  of  the  process  solely  dependent  on  the 
action  of  the  fluids,  whereas  the  experiments  of  Dutrochet  incontestably 
demonstrate  that  it  depends  mainly  on  the  action  of  the  membrane. 
No  doubt,  the  effect  both  of  the  fluids  upon  themselves,  and  of  the 
membrane  upon  the  fluids,  ought  to  be  taken  into  consideration,  and 
this  will  be  done  in  the  following  theory. 

9.  If  a  capillary  tube  be  divided  into  two  parts  by  a  plane  perpen- 
dicular to  its  axis ;  the  attraction  of  one  of  these  parts  upon  a  fluid 
which  exactly  fills  the  other  part  is  \cH,  c  being  the  contour  of  the 
inner  surface  of  the  tube,  and  H  a  certain  definite  integral  or  constant, 
depending  solely  on  the  materials  of  which  the  tube  and  the  fluid 
consist.  The  contour  of  the  tube  may  be  of  any  shape  whatever,  curved 
or  polygonal.     (See  Mec.  Cel.    Sup.  au  X*  Liv.   pp.  14 — 21.) 


210  Mb  POWER'S  THEORY  OF 

It  is  convenient  to  give  a  name  to  the  quantity  H  \  we  will  call 
it  the  capillary  affinity  between  the  two  materials  of  which  the  tube 
and  fluid  are  composed. 

It  is  easy  to  see  that  the  quantity  H  will  remain  unchanged  if  we 
conceive  the  tube  and  the  fluid  to  exchange  their  materials;  for,  by 
the  equality  of  action  and  reaction,  the  elementary  attractions,  of  which 

cH 

——   is   the  sum,   will   be  equal  in  the  two  hypotheses.     The  tube  may 

be   regarded   either  as  solid  or  fluid,  and  this  fluid   may  be  either  the 
same  as  that  which  fills  its  interior  or  a  different  one. 

If  we  conceive  the  density  of  the  inner  fluid  to  be  diminished  in 
any  ratio,  all  the  elementary  attractions,  and  therefore  H,  will  be 
diminished  in  the  same  ratio ;  and  if,  further,  the  density  of  the  tube 
be  diminished  in  any  ratio,  H  will  be  diminished  in  the  compound 
ratio. 

10.  Next,  let  u  and  v  be  the  original  quantities  by  volume  of  two 
vmmixed  fluids.  Then,  if  no  penetration  of  dimensions  takes  place, 
u  +  v  will  be  their  volume  after  mixture.  If  we  regard  the  fluids  after 
mixture  as  coexisting,  each  with  a  diminished  density,  within  the  same 
volume  u  +  v,  calling  r,  and  pi  these  diminished  or  partial  densities, 
(r)  and  {p)  the  densities  of  the  unmixed  fluids,  we  shall  have 


,^ J    and   7— , 

{r)       u  +  v  \p)       u  +  v 

whence 


^  +-^  =1 

{r)  ^  (p) 

Again, 

ri  +  pi  =  r, 

r  being   the  total   or   ordinary  density   of  the   mixture.      The   two  last 
equations  serve   to  express  ri  and  pi  in  terms  of  /•,   (;•)  and  {p). 


RESIDUO-CAPILLARY  ATTRACTION.  211 

If  then  we  have  a  second  mixture  of  the  same  two  original  fluids, 
we  shall  have 

—  +  -^  =  1 

and     r-i  +  p-i  =  p , 

where  rj  and  p-i  are  the  two  partial  densities,  and  p  the  total  density  of 
this  second  mixture.  These  equations  serve  in  like  manner  to  express 
r-i  and  p-i  in   terms  of  p,   {r)  and   {p). 

11.  Let  us  now  endeavour  to  express  the  mutual  capillary  affinities 
which  exist  between  the  two  mixtures  just  mentioned,  and  a  third 
material  (as  that  of  a  membrane  or  tube),  in  terms  of  the  densities 
of  these  mixtures  and  the  mutual  capillary  affinities  between  this  same 
material  and  the  unmixed  fluids. 

Let  the  former  affinities  be  denoted  by  H,  K,  L,  M,  N,  namely, 
H  between  the  tube  and  the  first  mixture, 
K  between  the  tube  and  the  second  mixture, 
L    between  the  first  mixture  and  the  second, 
M   between  the  first  mixture  and  its  like, 
N  between  the  second  mixture  and  its  like; 

and  let   the  latter  affinities   be   denoted    by  {H),  {K),  (L),    {M),   (A^), 

namely, 

{H)    between  the  tube  and  the  fluid  of  density  (r), 
(K)    between  the  tube  and  the  fluid  of  density  (p), 
{L)    between  the  fluids  of  densities  (r)  and  {p), 
{M)    between  the  fluid  of  density  (/•)  and  its  like, 
(iV)    between  the  fluid  of  density  (p)  and  its  like. 

The  attraction  ^cH  of  No.  (9)  will  be  the  sum  of  two  partial 
attractions,  namely,  that  of  the  tube  upon  two  coexistent  cylinders  of 
the  opposite  fluids,  whose  densities  are  those  of  the  original  unmixed 
fluids  diminished  in  the  ratios  r^  :  (r)  and  p^  :  (p).  Hence  by  the  latter 
part  of  that  No., 

ic^=ic(^)^  +  ic(^).^; 


212  Mr  POWER'S  THEORY  OF 

whence 

By  similar  reasoning,  superposing  all  the  different  attractions,  each 
diminished  in  the  ratio  of  the  densities  of  the  attracting  and  attracted 
materials,  we  shall  have 

.■.i^=.(/.,^.g+(M,.^H-W^.. 

By  combining  each  of  the  last  five  equations  with  the  four  equations 
of  No.  10,  and  eliminating  r,,  ^2,  /a,,  p^,  we  shall  obtain  H,  K,  L,  M, 
N,  in  terms  of  the  actual  densities  r,  p,  the  original  affinities  (H), 
(K),  (L),  (M),  (N),  and  the  original  densities  (r)  and  (p). 

12.  Let  us  now  proceed  to  apply  the  principles  of  the  three  last 
numbers  to  explain  the  experiments  of  Dutrochet.  And  first  let  us 
consider  those  which  relate  to  the  statical  force  of  endosmose.  In 
these  experiments  the  process  was  allowed  to  continue  until  the  fluid 
raised,  or  rather  the  mercurial  column  which  was  hydrostatically  sub- 
stituted for  it,  attained  its  maximum  altitude ;  at  this  moment  the 
densities  of  the  two  liquids  were  experimentally  determined ;  and 
instituting  different  experiments  with  different  mixtures  of  the  same 
substances,  Dutrochet  found  that  the  maximum  altitudes  were  propor- 
tional to  the  corresponding  differences  of  densities. 

The  substances  employed  in  his  experiments  were  saccharine  or 
gummy  solutions   on   the  one  hand,  and  water   on   the   other,   and  the 


RESIDUO-CAPILLARY   ATTRACTION. 


213 


water  was  found  to   be  transmitted   in    greater  abundance.     Common 
treacle  is  a  very  convenient  substance  for  experiments. 

Let  us  suppose  then  that  the  lower  part  of  the  endosmometer  is 
filled  with  treacle,  and  having  a  thin  membrane  tied  over  its  mouth, 
is  immersed  in  water ;  and  let  us  suppose  that  the  fluid  is  allowed 
to  ascend  until  the  operation  ceases. 

At  this  moment  we  may  regard  the  capillary  pore  which  traverses 
the  membrane,  as  communicating  at  its  two  extremities  with  fluid  in 
the  same  state  of  mixture  as  the  fluid  in  the  contiguous  vessels, 
there  being   a   gradual  transition  from  one  end  to  the  other. 


Let  C1C1C2C2,  be  a  portion  of 
the  membrane,  AiAiAsA^  one  of 
its  capillary  pores,  with  its  axis  at 
right  angles  to  the  plane  of  the 
membrane,  communicating  originally 
with  the  water  at  A^A^,  and  with 
the  treacle  at  A^A^,  but  when  the 
fluid  has  reached  its  maximum  al- 
titude, communicating  with  the  ^ 
first  mixture  of  No.  (10)  at  AiA^, 
and  with  the  second  mixture  of 
that  No.  at  A2A2. 


^, 

^ 

c^ 

^. 

A^ 

c^ 

\ 

^. 

S* 

' 

1 

-B/ 

B, 

\ 

1 

<i 

Ay 

9 

Imagine  the  geometrical  figure  of  the  tube,  (not  its  material)  to 
be  produced  both  ways  to  D,  and  D^,  and  cut  off"  from  each  end 
of  the  tube  a  distance  AiBi,  AiB^,  equal  to  the  tube's  sphere  of 
sensible  attraction. 

Since  A^Bi,  and  A^Bi,  are  insensible,  we  may  regard  the  fluids  in 
AiA^BiBi,  and  A^A-^BiB^,  as  in  the  same  state  of  mixture  as  the 
fluids  in  the  contiguous  vessels. 

Vol.  V.    Part  II.  E  e 


214  Mr  POWER'S   THEORY   OF         >! 

Let  us  now  estimate  all  the  forces  which  tend  to  move  the  central 
column  DyD^D-iDi  in  direction  of  its  axis. 

It  is  plain  that,  in  whatever  manner  the  fluids  may  communicate 
in  the  interior  of  the  tube,  the  tube  can  produce  no  effect  upon 
ByBiBiBi,  since  every  elementary  portion  of  this  part  of  the  fluid 
will  be  drawn  in  both  directions  as  by  an  infinitely  extended  tube. 

We  may  also  neglect,  as  producing  equal  and  opposite  forces  in 
both  directions,  the  attraction  between  the  tube  A^B^  and  the  fluid 
AiAiBiBi;  between  the  tube  A^Bi,  and  the  fluid  A^AzBiB,;  be- 
tween the  fluid  tube  dA^Di,  and  AiA^D^D^ ;  between  C2A2D,,  and 
AzAzDiDs,  between  the  membrane  and  C^A^D^-,  between  the  mem- 
brane and  CiA-iD-i. 

Lastly,  we  may  neglect  all  the  mutual  actions  of  the  particles 
composing  the  central  column  DyD^DiDi,  their  tendency  being  only 
to  mix  the  opposite  fluids,  and  not  to  move  the  column  as  a 
mass. 

Of    the    remaining     attractions     we    shall     have    at    one    end    the 

attraction    of    the    tube    B^Bt,     upon     B-^B^A^A^,     (  =  \  cH)  +  the 

attraction     of    the    tube     A^B^,     upon    D^D^A^A^,     {=  ^  cH)  —  the 

attraction    of    the    fluid    tube    C^A^D^,    upon    A^A^B^B^,    {=\cM); 

c 
constituting    the   capillary   force    -    {2H  —  M).       This   will   be   opposed 

by  a   similar   force  —  {^K—N)   exerted  at   the  other  end   of  the  tube. 
The   residual  sustaining   force  is    therefore 

I  .{2H-2K-M+N'). 

It  now  only  remains  to  express  this  force  in  terms  of  the  actual 
densities  r  and  p,  and  the  initial  constants 

{r),   ip),   {H),   (K),   (L),   {M),   {N). 


RESIDUO-CAPILLARY  ATTRACTION.  215 

13.     For  this  purpose  let 


^-=...     and   ^=..; 


therefore  by  No.  (10). 

making  these  substitutions  in  the  equations  of  No.  11.,  we  have 

K  =  s,{H)  +  il-s,){K). 

L  =s,.{l-s,){L)  +  s,.{l-s,){L)  +  s,s,{M)  +  {l-s,){l-s,){N'). 

M=  2s,  (1  -*0  {L)  +  s,'  {M)  +  (1  -*,)'  (^)- 

N  =  2s,  (1  -  s,)  {L)  +  si  (M)  +  (1  -  s,y  (N). 

Hence  2H-2K-M+  N=A  (H)  +  BiK)  +  C {L)  +  D{M)  +  E{N-), 

where  A  =  2.  {s,  —  S2). 

B  =  2.{l-s,)-2.{l-s,) 

=  -2{si-s,). 
C=-2s,.{l-s,)  +  2s,  (1  - s,) 

=  -2{s,-s,)  +  2{8{'-si). 
D=-{s,'~si). 
E=-{l-s,Y  +  {l-s,Y 

=  2{s,-s,)-{s,'-'Si). 

EE2 


216  Mr  POWER'S   THEORY  OF 

The  residual  force  is  therefore 


I  {s,-s,){2{H)-^{K)-2{L)  +  2{N)} 


2 


+  l-{s,'-s.'){2{L)-{M)-{N)}. 


Again,  r  =  r^+p,  =  {r).^^+{p).■^ 

=  ('•)  *!  +  (/»)  (1-*.); 
(p)-r 


••.   *, 


=={r)s,  +  {p).{l-s,); 

■•  '~ip)-irr 

_    p  —  r 
ip)  -  {r) 

„  .  „_?ie)zik±rl. 


i_^ 


{ip)-ir)r       {{p)-ir)r' 
The  expression  for  the  residual  force  is,  therefore, 

p^  -r 


i-fvFW*'<'^>-''^>-<'^>'' 


RESIDUO-CAPILLARY  ATTRACTION.  217 

which  may  be  put  under  the  form 

cA{p-r)-{-cB(p'-t^)*, 
making 

^-(^)-(r)-r^)     ^^^)+(^)-(r)L^^>  {p)  +  {r)         J}' 


and^=..^{(Z)-W^H 


{ip)-ir)rv'       2    r 


The  agreement  of  theory  with  experiment,  then,  requires  that 

jM)  +  jN) 
(^) 2 

should  be  either  nothing,  or  very  small  compared  with 

14.  When  I  first  began  to  investigate  this  subject,  certain  con- 
siderations, which  it  would  be  tedious  to  detail,  led  me  to  imagine 
that  the  fluids  might  communicate  in  the  interior  of  the  tube, 
forming  a  series  of  interlacing  cylinders  one  within  another,  and  I 
found    the    forces   which    tended    to    protrude    the    cylinders   into   the 

opposite  fluids,   all  multiplied   by  (L)  —  - — '-—^ — - .     I  therefore  looked 

upon  this  expression  as  a  measure  of  the  tendency  of  the  fluids  to 
mix,  and  this  tendency  being,  as  experience  shows,  very  small  in  the 
case  of  treacle  and  water,  as  well  as  in  the  case  of  the  gummy 
solutions  and  water,  afforded  an  explanation  why  the  force  should 
be  so  nearly  proportional  to  the  difference   of  densities,   as  Dutrochet's 

*  I  have  elsewhere  erroneously  stated,  that  the  residual  force  is  c  A(p—r)  +  c  B(p—ry, 
a  mistake  which  I  am  glad  to  have  this  opportunity  of  correcting. 


218  Mr  POWER'S   THEORY  OF 

experiments  seemed  to  indicate.  But  the  preceding  theory  being 
perfectly  independent  of  the  mode  in  which  the  fluids  communicate, 
it  is  better  not  to  have  recourse  to  a  supposition,  which  is  in  the 
slightest  degree  precarious,  especially  as  I  am  now  prepared  to  show, 
that,  in  whatever  way  the  fluids  may  arrange  themselves  within  the 
tube,   the   rapidity  of  the  mixing  process  will   depend   upon    the   mag- 

nitude   of  (X)-(^);W. 

15.  In  fact,  in  whatever  manner  the  mixing  process  may  be 
effected,-  we  may  at  any  moment  imagine  the  fluid  to  be  divided 
into  an  indefinite  number  of  contiguous  strata,  of  any  arbitrary  or 
convoluted  form,  the  density  being  the  same  for  the  whole  extent  of 
any  one  stratum,  but  varying  from  one  to  another. 

If  the  surface  which  separates  two  contiguous  strata  be  a  perfect 
plane,  it  is  evident,  by  the  equality  of  action  and  re-action,  that  this 
would  be  a  position  of  momentary  equilibrium,  (abstracting  from 
gravity,  which  I  am  not  here  considering.) 

Suppose,  now,  that  this  surface  becomes 
undulated  in  an  arbitrary  way,  and  take  any 
point  A  upon  it,  and  draw  a  tangent  plane 
BAD,  including  with  the  surface  EAC,  a  kind 
of  lens  BDEC,  which,  with  La  Place,  we 
may  call  a  meniscus.  Draw  the  normal  FAG ; 
and  let  Ri,  and  R^  be  the  radii  of  greatest  and 
least  curvature  at  the  point  A. 

Now  La  Place  has  shown  that  the  attraction  of  such  a  meniscus 

upon  the   column  of   fluid  AF   is    ("»"+  p")--^>    where  H    is   the 

capillary  affinity  between  the  material  of  the  meniscus,  and  that  of 
the  fluid  in  the  sense  already  defined.  (See  Supp.  au  X*  Liv. 
page  14 — 17.) 


RESIDUO-CAPILLARY  ATTRACTION.  219 

He  has  also  shown  that  the  attraction  of  the  meniscus  is  the 
same  whichever  way  it  be  turned. 

If  the  meniscus  instead  of  consisting  of  the  left  hand  fluid,  (as 
in  the  figure),  consisted  of  the  right  hand  fluid,  the  common  boundary 
being  the  plane  BAD,  there  would  be  equilibrium,  the  column 
AF  being  attracted  by  the  right  hand  fluid,  just  as  much  as  the 
column  AG  is  by  the  left. 

Since  then  the  meniscus  consists  of  the  left  hand  fluid  instead  of 
the  right,  the  effect  of  the  disturbance  upon  the  column  AF,  tending 
to  draw  it  in  the  direction  FA,  is  the  attraction  of  the  meniscus 
upon  AF,  regarding  it  as  consisting  of  the  left  hand  fluid,  minus 
the  attraction  of  the  same  meniscus  regarding  it  as  consisting  of  the 
right,   that   is 

\R,  ^  EJ  \2        2 

supposing    the  left    hand    fluid   to   be   the   first   mixture   of    No.  (10), 
or  the  lower  fluid  of  No.  (12). 

If  then  we  estimate  the  effect  in  the  direction  AF,  it  is 

1         1\    /L      M\ 


(1_       J_\     (±  _M 


In  the  same  way,  the  effect  of  the  disturbance  upon  AG,  in  the 
direction  GA,  is  the  attraction  of  the  meniscus,  regarded  as  consisting 
of  the  left  hand  fluid,  minus  the  attraction  of  the  same  meniscus, 
regarded  as  consisting  of  the  right,  that  is 


Ui  "^  BJ  •  \2  ^ 


2  j- 


Hence  the  whole  attraction  in  the  direction  GF,  is 


{i^k){--^)- 


220  Me  POWER'S  THEORY  OF 

If  we  substitute  for  L,   M,  N,    the  expressions  at  the  commence- 
ment of  No.  (13),  we  shall  find 

^L-M-N  =  A{L)  +  B{M)  +  C{N),  where 

^  =  2*i.(l-«2)  +  2*2.(l-*,)-2*i.(l  -*i)-2  52.(l-«2) 
=  2  *,  -  2  *,«2  +  2  *2  -  2  «i*8  -  2  *,  +  2  «i'  -  2  *2  +  2  «/ 
=  2*,*- 4*1*2 +  2*/ 
=  2(*i-*,f. 

=  -(*!-  s^f. 

=  -{(i-*0-(i-*.)}' 
=  -(«i-*2)'; 

.-.  ^L-M-N={s,-s,)\{2 (L) - {M) - {N)} 
The  effect  of  the  disturbance  in  the  direction  GF,  is  therefore 

consequently  if  (i)  be  greater  than   ^ — '-^ — '- ,    or,    if  the   capillary 

affinity  of  the  opposite  fluids  exceed  an  arithmetic  mean  between 
the  capillary  affinities  of  the  two  fluids  for  fluids  of  their  own  kind, 
the    tendency   will    be    to    depart    still    farther    from    the  position   of 


RESIDUO-CAPILLARY  ATTRACTION.  221 

equilibrium,  and  the  tendency  is  the  greatest   where   the  curvature  is 
the  greatest. 

16.  Hence  it  is  easy  to  see  that  the  protruding  segments  of  each 
fluid  will  become  more  and  more  pointed  at  their  summits  of  greatest 
curvature  as  they  advance  into  the  opposite  fluids,  thus  forming 
interlacing  spiculse,  shooting  into  the  opposite  fluids,  and  at  the  same 
time  inosculating  with  each  other  by  their  lateral  protrusion,  and 
that  this  process  cannot  cease  until  the  fluids  have  divided  each 
other  into  segments  of  a  magnitude  comparable  with  that  of  the 
sphere  of  sensible  attraction. 

Beyond  this  limit  the  theory  does  not  hold.  It  is  very  possible  then, 
that  in  some  cases  a  limit  may  be  attained  where  the  mixing  fluids 
have  arrived  at  such  a  state  of  subdivision,  that  the  conditions  for 
continuing  the  subdivision  are  no  longer  satisfied ;  in  other  cases  it 
is  possible  that  the  subdivision  may  proceed  until  the  ultimate  atoms 
of  the  opposite  fluids  act  upon  each  other  by  ones,  twos,  and  threes, 
thus  effecting  a  chemical  decomposition :  nature  presents  numerous 
instances  of  both  kinds. 

17.  But  though  the  mathematical  theory  is  not  strictly  applicable 
when  the  subdivided  segments  are  of  less  magnitude  than  the  sphere 
of  sensible  attraction,  it  may  be  considered  as  an  approximation  to  the 
truth  considerably  beyond  this  limit.  For,  the  most  effective  part  of 
the  attraction  of  each  segment  being  that  exerted  by  the  particles 
in  immediate  contact  with  the  normal  column,  the  diminution  of 
the  segments  will  only  have  the  effect  of  removing  the  more  feeble 
part  of  the  attractions  which  the  theory  takes  into  the  account.  It  is 
therefore  probable  that,  even  in  cases  where  no  chemical  decomposition 
takes  place,  the  subdivision  of  the  fluids  may  be  carried  to  a  limit  far 
beyond  that  to  which  the  theory  is  strictly  applicable.  Besides,  the 
processes  of  nature  are  not  interrupted  of  a  sudden;  the  tendency 
therefore  to  farther  subdivision  cannot  be  suddenly  arrested,  but  in 
cases  where  it  is  ultimately  reduced  to  nothing,  it  must  be  so  by 
passing   through   all   degrees   of  magnitude.     This   reasoning  is    further 

Vol.  V.    Paet  II.  F  f 


222  Mr  POWER'S  THEORY  OF 

confirmed  by  those  experiments  which  demonstrate  the  almost  infinite 
subdivision  of  matter  by  repeated  dilution,  experiments  which  are 
familiar  to  every  one.  This  infinite  subdivision  is,  in  fact,  involved  in 
the  mathematical  conception  upon  which  this  theory  is  founded,  namely, 
that  in  the  state  of  mixture  the  two  fluids  may  be  regarded  as 
coexisting  within  the  same  volume,  each  with  a  diminished  density. 
This  conception  cannot  of  course  be  a  rigorous  representation  of  nature ; 
but  is  sufficiently  so  for  the  application  of  La  Place's  theory,  or,  which 
comes  to  the  same  thing,  for  the  summation  of  the  attractions  by  the 
principles  of  the  Integral  Calculus. 

18.  In  cases  of  simple  mixture,  unattended  with  a  chemical  change, 
the  ultimate  segments  of  the  opposite  fluids,  though  in  an  extreme 
state  of  subdivision,  have  a  separate  and  independent  existence,  which 
renders  it  highly  probable,  that  the  volume  of  the  mixed  fluids  should 
equal  the  sum  of  the  volumes  of  the  unmixed  fluids.  This  supposi- 
tion has  been  made  in  the  preceding  theory,  and  I  find  by  experiment 
that  in  mixtures  of  treacle  and  water  it  is  accurately  true.  The  same, 
I  believe,  is  true  in  all  cases  of  simple  mixture,  where  no  chemical 
result  takes  place,  such  as  the  precipitation  of  solids,  or  the  disengage- 
ment of  heat  or  other  volatile  constituents.  To  liquids  whose  union 
is  accompanied  by  such  phenomena  the  present  theory  is  inapplicable, 
not  only  on  account  of  the  penetration  of  dimensions,  with  which 
such  phenomena  are  generally  attended,  but  on  account  of  the  change 
of  affinities,  which  the  escape  of  some  of  the  constituents  must 
necessarily  produce,  including  heat,  which,  regarded  as  a  chemical 
constituent,  is  as  important  as  any. 

19.  The  addition  of  a  third  fluid  to  one  of  the  liquids,  by  altering 
the  chemical  affinities-,  must  likewise  alter  the  capillary  aflfinities,  which 
are  only  a  different  modification  of  the  same  corpuscular  attractions 
which  produce  the  former.  It  is  not  surprising  then,  that  Dutrochet 
should  have  discovered  some  substances  which  accelerated  the  process 
in  his  experiments,  and  others  which  retarded  it  or  stopped  it 
altogether. 


RESIDUO-CAPILLARY  ATTRACTION.  223 

Water  impregnated  with  sulphuretted  hydrogen  was  found  not  only 
to  stop  the  process,  but  to  destroy  the  energy  of  the  membrane  for 
subsequent  experiments  with  pure  water  and  pure  saccharine  solutions. 
No  doubt  the  sulphuretted  hydrogen  had  decomposed  the  surface  of 
the  capillary  pore,  leaving  a  coating  of  putrid  matter,  which  was  not 
possessed  of  such  capillary  properties  as  to  supply  the  place  of  the 
material  of  the  membrane.  That  this  is  the  true  explanation  is  shown 
by  the  fact,  that  when  the  membrane  was  for  a  long  time  steeped  in 
water  and  well  washed,  its  energy  was  restored :  in  fact,  the  putrid 
matter  being  washed  away,  the  membrane  presented  an  unvitiated 
surface  to  the  fluids.  Heat  and  electricity  may  be  classed  amongst 
these  chemical  agents,  as  they  operate  their  effect  precisely  in  the  same 
way,  namely,  by  changing  the  chemical  and  consequently  the  capillary 
affinities. 

20.  If  we  wish  to  compute  the  height  to  which  the  fluid  will 
rise  in  the  endosmometer,  let  ^  be  the  height  of  the  supported  column 
above  the  surface  of  the  membrane,  and  z  the  height  of  the  lower  fluid 
above  the  same,  w  the  transverse  section  of  the  tube;  the  difference 
of  the  pressures  of  the  cylindrical  columns  w^  and  wg,  having  the 
common  section  w,  is  gpco^—grioz:  this  must  be  counterbalanced  by  the 
sustaining  force  cA(p  —  r)  +  cS{p^  —  r"),  which  denotes  a  pressure  on  the 
same  scale ; 


"         o     ff       \    p    J         oi      ff       \      p      /         p 


If  a  column  of  mercury  be  hydrostatically  substituted  for  the 
ascending  fluid,  as  in  the  experiments  of  Dutrochet,  calling  Z  the 
altitude  of  the  mercury,  and  R  its  density,  we  must  have 

„      c     A     (p-r\       cB     (p^-r^\        r 

^  =  -.-  g  •  K-w)  ■"  -.^  ■K-R-)  "■  R^' 

this   of  course  being   subject   to  a  correction   when   the   cistern   of  the 
mercury  is  not  on  a  level  with  the  membrane. 

F  F2 


224  Mr  POWER'S  THEORY  OF 

21.  If  the   pore   be  circular,   let  ^  be  its  diameter,  then 

c  =  2'7r.-,    and    to  =  tt  .  —  : 
2  4t 

€  4 

•'.  -  =  -J ;. 

CO  0 

the  sustaining  force  is  therefore  inversely  proportional   to  the  diameter 
of  the  pore,  as  in  ordinary  capillary  attraction. 

Hence  we  see  how  the  membrane^s  delicacy  of  texture  contributes 
to  the  intensity  of  the  sustaining  force. 

22.  It  is  now  easy  enough  to  see  in  what  manner  the  process  is 
effected.  The  residual  force  cA{p  —  r)  +  cB{p'^  —  r"),  which  would  result 
if  the  ends  of  the  tube  communicated  with  fluid  of  the  densities 
r  and  p,  being  greater  than  the  altitudinal  pressure  upon  the  section  w, 
would  cause  the  fluid  within  the  tube  to  move  as  a  mass  into  the 
endosmometer,  thus  bringing  fluid  more  and  more  diluted  to  the 
issuing  orifice;  this  will  continue  until  the  residual  force  is  weakened 
to  such  a  degree  as  exactly  to  counterbalance  the  altitudinal  pressure. 
Contemporaneously  with  the  former  motion,  the  mixing  process  will 
transfer  the  two  fluids  in  opposite  directions,  the  current  from  the 
endosmometer  towards  the  water  producing  the  exosmose,  and  the 
opposite  current  supplying  the  deficiency  caused  by  the  exosmose,  and 
therefore  not  contributing  to  the  endosmose.  The  diluted  fluid  which 
was  carried  into  the  endosmometer  by  the  residual  force,  will  gradually 
mix  with  the  treacle  within,  whether  that  mixture  be  carried  on  near 
the  orifice  of  the  tube,  or  whether  the  diluted  fluid  be  raised  by  its 
specific  levity  higher  up  in  the  endosmometer.  The  extremely  small 
portion  of  diluted  fluid  which  has  thus  been  transmitted,  and  the 
viscosity  of  the  treacle,  render  it  most*  probable  that  it  would  not  be 


*  This  probability  amounts  nearly  to  certainty  when  we  consider  that  the  denser  fluid 
has  no  access  to  the  lower  part  of  the  transmitted  fluid.  It  is  only  when  a  lighter  body 
is  insulated,  or  partially  insulated^  in  a  denser  that  it  rises  by  its  specific  levity. 


RESIDUO-CAPILLARY  ATTRACTION.  225 

carried  up  by  its  specific  levity,  but  rather  adhere  to  the  membrane 
in  the  way  that  bubbles  of  air  adhere  to  the  sides  of  vessels  containing 
water  or  mercury.  But,  be  this  as  it  may,  the  end  of  the  tube  which 
communicates  with  the  endosmometer,  will  soon  be  surrounded  by  a 
stronger  infusion  of  the  treacle,  which  will  again  bring  the  residual 
force  into  action ;  thus  a  fresh  portion  of  the  fluid  will  be  introduced 
into  the  endosmometer,  and  the  same  process  will  be  repeated  as  before. 
For  the  sake  of  explanation,  I  have  supposed  the  residual  force  to 
produce  its  eflPect  discontinuously,  but  it  is  easy  to  see  that  the  process  will 
really  be  continuous,  the  united  actions  of  the  endosmose  and  exosmose 
always  keeping  the  orifices  of  the  tube  surrounded  by  fluid  in  such  a 
state  of  dilution  that  the  magnitude  of  the  residual  force  will  be  exactly 
sufficient  to  create  a  supply  proportioned  to  the  demand  arising  from 
the  mixing  process  which  is  continually  proceeding  within  the  endosmo- 
meter. The  residual  force  cannot  be  less  than  this,  for  if  it  were,  the 
encroachment  of  the  treacle  upon  the  issuing  orifice  would  immediately 
increase  it ;  nor  can  it  be  greater,  for  then  the  accumulation  of  the  more 
diluted  fluid  at  that  same  orifice  would  immediately  diminish  it, 

23.  The  quantity  transmitted  in  a  given  time  must  depend  more 
upon  the  rapidity  with  which  the  mixing  process  is  carried  on  within 
the  endosmometer  than  on  the  magnitude  of  the  residual  force.  This 
force  is  certainly  essential  to  the  transmission,  but  its  effect  is  no  other 
than  that  of  a  pump  which  supplies  the  fluid  from  below  as  fast  as  it  is 
wanted,  and  no  faster,  and  that  of  a  catch  or  valve  to  sustain  it  when  it 
is  once  elevated.     The  moving  force  at  the  summit  of  any  protruding 

spicula    is    by    No.    (14)    represented    by     [^  +  ~p)  ^(p  — ^)^    and    is, 

therefore,  for  spicule  of  given  shape,  as  the  square  of  the  difference 
of  densities.  It  might  appear  then,  at  first  sight,  more  probable  that 
the  quantity  of  the  lower  fluid  absorbed  by  the  fluid  in  the  endosmo- 
meter in  a  given  time,  would  be  more  nearly  as  the  square  of  the 
difference  of  densities,  than  as  the  simple  power  of  this  difference,  which 
is  the  law  the  experiments  of  Dutrochet  tend  to  establish.  But  such 
a  conclusion  would  be  very  precarious,  as  will  appear  by  the  following 
considerations. 


226  Mb  POWER'S  THEORY  OF 

24.  Let  us  imagine  two  different  experiments,  all  circumstances, 
as  regards  the  materials,  form  and  disposition  of  the  apparatus,  being 
exactly  similar,  but  the  proportions  in  which  the  substances  are  mixed 
on  each  side  the  membrane,  being  different  in  the  two  experiments. 
Let  us  suppose  also  that  the  mixing  process  takes  place  in  both  experi^ 
ments  after  exactly  the  same  type,  only  with  different  velocities,  that 
is  to  say,  that  at  certain  times,  t  and  t',  t  +  T  and  t'  +  r,  #  +  2t  and 
#'  +  2t',  &c.,  the  protruding  spiculae  from  the  lighter  fluid  exist  in 
exactly  the  same  state  in  both  experiments,  as  regards  their  number, 
shape,  size  and  situation. 

This  supposition  being  made,  the  volume  of  the  lighter  fluid  absorbed 
by  the  fluid  in  the  endosmometer  in  the  two  experiments,  will  be  equal 
in  the  intervals  t  and  t'  :  also  the  summits  of  the  spiculae  will  have 
described  the  same  paths  in  the  two  experiments  during  these  same 
corresponding  intervals.  Let  t  and  t  be  indefinitely  small,  and  let  us 
equate  the  spaces  described  by  the  summits  of  any  two  corresponding 
spiculae  between  the  epochs  t  and  #  +  r,  t'  and  t'  +  t',  and  also  between 
the  epochs  t  and  ^  +  2t,  t'  and  #'  +  2t'. 

Let  a  be  the  sphere  of  sensible  attraction,  and  imagine  a  small 
normal  column  2  a  at  the  vertex  of  each  spicula,  being  half  in  one 
fluid  and  half  in  the  other. 

The  two  spiculse  having  by  the  hypothesis  the  same  shape,  the 
moving  forces  upon  these  columns  are  as  {p  —  rf  and  [p'  —  r'f,  and  the 
masses  moved  are  as  ap  +  ar  and  ap'  +  ar',  that  is,   as  p  +  r  and  p'  +  r; 

the    accelerating;   forces   will    therefore    be    as    — and     '^ ,      /  ;    let 

^  p+r  p +r 

(p  _  rY                (p  —  r'f 
them  be  k  ^ '-  and  ^  k  .  ^ f- .     Then  if  v  and  v'  be  the  velocities 

p+r  p +r 

of  the  two  summits  at  times  t  and  t',  equating  the  corresponding  spaces, 
we   shall   have 

and 


P  +  r  "        ■  ^'     p+r' 


■^      p  +  r  ^      p  +r 


RESIDUO-CAPILLARY  ATTRACTION.  227 

These  equations  are  equivalent  to  the  following: 


VT  =  VT,    and —  = -^-—, — —. — ; 

p-irr  p  ^■r' 


whence 


v'  ~  T  ~  p'  —  r'  '         p  +  r  ' 


Let  q  be  the  volume  of  fluid  absorbed  in  the  times  t  and  t',  which 
we  have  seen  to  be  the  same  in  each  experiment;  and  let  Q  and  Q' 
be  the  quantities  absorbed  during  a  given  time  T,  T  not  being  so 
great  but  that  r,  p,  r'  and  p  may  be  considered  the  same  during  this 
interval. 

If  then  there  be  a  law  connecting  the  quantities  absorbed  in  a  given 
time  with  the  densities,  we  must  regard  this  absorption  in  each  experi- 
ment as  uniform  during  the  time  T\ 

.:  Q  :  q  y.  T  :  T, 

and  Q  '.  q  V.  T  :  r'\ 

...  Qr  =  qT=Q'T'; 


+  / 


^       T       p'  —  r'         p  +  r 

■  The  supposition  we  have  made,  as  to  the  exactitude  of  type  in  the 
two  mixing  processes,  is  particular ;  but  if  there  be  a  general  law  whicli 
is  applicable  to  all  cases,  that,  must  include  the  case  supposed,  and 
therefore  the  result  of  the  particular  case  must  coincide  with  that  of 
the  general  law.  If  then  there  be  such  a  law,  it  is  expressed  by  the 
proportion 


Q.Q  .'. 


p-r    .    p-r 


"s/p  +  r  '  \/p'  +  r' ' 


This  being   true  in   different   experiments,   must   be    true   in   different 
stages  of  the  same  experiment. 


228  Mr  POWER'S  THEORY  OF 

Now  in  the  same  experiment  p  diminishes  and  r  increases  as  the 
experiment  proceeds,  and  therefore  the  variation  of  p  +  r  is  small  com- 
pared with  that  of  p  —  r;  the  quantities  absorbed  will  therefore  be 
pretty  nearly  in  the  ratio  of  the  difference  of  densities,  as  Dutrochet 
found  them  to   be.     Whether   the   proportion 

Q:Q'  ::  -E^  :  -IzL 
y/p  +  r     Vp  +  r' 

may   be    a    more    accurate    representation   of   nature    than    the   law   of 
Dutrochet,  is  left  to  the  test  of  experiment. 

25.  It  may  perhaps  be  objected  to  the  theory  of  No.  (12),  that 
the  ordinary  theory  of  capillary  attraction  supposes  the  dimensions  of 
the  tube  to  be  incomparably  greater  than  the  sphere  of  sensible  attrac- 
tion, whereas  the  fact,  that  these  pores  are  so  small  as  to  elude 
microscopic  observation,  might  lead  us  to  apprehend  that  their  dimensions 
were  of  a  size  comparable  with  that  sphere.  The  example  which  has 
been  calculated  in  No.  (7),  does  not  seem  to  leave  any  cause  for  such 
an  apprehension.  But  supposing  this  were  the  case,  the  only  difference 
it  would  make  in  the  theory  is  this :  that,  whereas,  on  the  former 
supposition,  the  quantities  \cH,  \c K,  &c.,  denoted  the  results  of 
integrations  extending  from  nothing  to  infinity,  and  not  otherwise 
depending  on  the  form  of  the  tubes  than  by  involving  the  contour  c 
as  a  multiplier;  on  the  second  supposition,  the  limits  of  the  integra- 
tion will  depend  on  the  form  of  the  tubes  and  the  texture  of  the 
membrane :  but  these  limits  being  the  same  in  the  cases  compared,  it  is 
easy  to  see  that  the  theory  will  be  still  true  on  the  latter  hypothesis, 
provided  we  look  upon  ^c{H),  ^c{K),  &c.,  as  denoting  certain 
unknown  limited  integrals  depending  not  only  upon  the  nature  of  the 
materials,  but  also  upon  the  form  and  size  of  the  capiUary  pores.  The 
residual  force  will,  therefore,  on  this  hypothesis  also,  be  of  the  form 
a{p-r)  +  h{p'-r'). 

26.  By  the  application  of  similar  reasoning  to  the  theory  of  No.  (15), 
it  is  not  difficult  to  conclude  that  the  moving  forces  upon  the  normal 


RESIDUO-CAPILLARY  ATTRACTION.  229 

columns  at  the  summits  of  spiculae  of  given  shape  and  sixe  will  be  as 
(p  —  r)-,  even  when  the  dimensions  of  the  spiculse  are  indefinitely  less 
than  the  sphere  of  sensible  attraction.  For,  the  attraction  of  a  meniscus 
bounded  on  one  side  by  a  plane  surface,  upon  the  conterminous  normal 
column,  will  in  all  cases  be  a  definite  integral  depending  on  the  shape 
and  size  of  the  meniscus,  and  the  demonstration  of  La  Place,  by  which 
he  shows  that  the  attraction  of  such  a  meniscus  is  the  same  whichever 
way  it  be  turned,  is  perfectly  independent  of  its  size  and  the  shape  of 
its  curved  surface. 

Let  then  I  be  the  attraction  of  any  meniscus  upon  the  conterminous 
normal,  the  meniscus  consisting  of  one  mixture,  and  the  normal  of  the 
other;  m,  the  attraction  of  the  same  meniscus  when  the  meniscus  and 
column  consist  both  of  the  first  mixture;  and  n,  the  same  thing  when 
they  consist  of  the  second  mixture.  Then  reasoning  exactly  as  in 
No.  (15),  the  moving  force  upon  the  column  GF  will  be  2l—m  —  n; 
and  if  (/),   (m),   (»),  be  the  initial  values  of  I,  m,  n,  it  may  be  shown 

exactly    as    before,    that    2l—m  —  n  =  c/C_^  >.^g  •  {2(/)  -  (m)  -  (n)},    the 

theory  of  No.  (11)  being  equally  applicable  in  this  case.  Hence,  how- 
ever minute  the  spiculae  may  be,  the  moving  force  upon  the  central 
column  will,  for  spiculse  of  given  shape,  be  as  the  square  of  the  difference 
of  densities. 

This  consideration  applied  to  the  theory  of  No.  (25),  gives  it  a 
generality  which  renders  it  as  satisfactory  as  can  well  be  desired. 

J.  POWER. 

Trinity  Hall, 

Marck  29,  1834s 


Vol.  V.    Part  II.  Gg 


231 


X.  On  Aerial  Vibrations  in  Cylindrical  Tubes.  By  William 
Hopkins,  M.A.  Mathematical  Lecturer  of  St  Peter's  CoUege, 
and  FeUow  of  the  Cambridge  Philosophical  Society. 


[Read  May  20,  1833.] 


The  problem  which  has  for  its  object  the  determination  of  the 
motion  of  a  small  vibration  propagated  in  an  elastic  medium  along  a 
prismatic  tube  of  indefinite  length  (the  motion  of  every  particle  in 
each  section  of  the  tube  perpendicular  to  its  axis  being  the  same)  was 
long  since  solved  by  Euler  and  Lagrange.  The  problem,  so  nearly 
allied  to  this — to  determine  the  motion  of  an  aerial  pulsation  in  a  tube 
of  definite  length — has  not  been  so  satisfactorily  solved,  the  tube  being 
either  open  at  the  extremity  or  stopped  with  a  substance  possessing 
some  degree  of  elasticity.  In  addition  to  the  difficulties  of  the  former 
problem,  we  have  in  this  latter  one  those  still  more  formidable  difficulties 
which  exist  in  the  determination  of  the  circumstances  of  the  motion 
at  the  confines  of  two  elastic  media  in  the  closed  tube,  or  at  the 
extremity  of  the  open  one,  where  the  air  in  the  tube  communicates 
with  the  circumambient  air.  These  motions  must  no  doubt  be  deter- 
minable from  the  nature  of  the  media,  and  the  causes  producing  and 
maintaining  the  vibrations,  having  nothing  arbitrary  in  them,  except 
what  may  be  so  in  the  original  disturbance ;  but  I  am  not  aware 
of  any  progress  having  been  made  in  the  direct  solution  of  these 
questions,  which  now  forms  one  of  the  greatest  desiderata  in  the  appli- 
cation of  mathematics  to  physical  science;  and  in  our  inability  to 
determine  these  motions  at  the  extremity  of  the  tube,  either  by  theory 
or  direct  observation,  we  are  driven  to  the  necessity  of  assumptions. 
It  is   from   a  difference  in  these  assumed  conditions  that  we  have  the 

GG2 


232  Mb  HOPKINS  ON  AERIAL  VIBRATIONS 

different  solutions  which  mathematicians  have  given  of  the  problem  in 
question.  The  principle  on  which  we  ought  to  proceed  in  making  such 
assumptions  is  obvious ;  they  should  be  subjected  to  no  restrictions, 
(not  imposed  on  them  by  our  theory),  which  are  not  necessary  to  draw 
those  deductions  and  inferences  from  our  mathematical  results^  which 
admit  of  verification  by  experiment,  to  the  test  of  which  an  assumption, 
in  any  degree  arbitrary,  must  necessarily  be  subjected  before  it  can  claim 
our  confidence.  The  physical  conditions  however  on  which  the  solutions 
of  this  problem  depend,  (as  far  as  it  is  distinct  from  that  of  the  motion 
of  a  wave  along  a  uniform  tube  of  indefinite  length),  have  neither 
been  assumed  on  this  principle,  nor  subjected,  as  far  as  I  am  aware, 
to  this  experimental  test.  It  has  been  principally  with  the  view  of 
remedying  these  defects  that  I  have  prosecuted  the  researches,  an  account 
of  which  I  have  now  the  honour  of  laying  before  the  Society. 

1.  The  physical  conditions  assumed  by  Euler,  and  by  most  of  those 
who  have  since  written  on  the  subject,  are,  that  the  particles  of  air  at 
the  extremity  of  a  closed  tube  are  always  at  rest;  and  that  no  con- 
densation of  the  air  takes  place  at  the  extremity  of  an  open  one.  The 
first  condition  involves  the  supposition  of  the  perfect  rigidity  of  the 
material  with  which  the  tube  is  stopped.  This  cannot  be  accurately 
true,  but  probably  leads  to  no  error  very  appreciable  to  observation. 
The  second  condition  assumes  an  eqviality  in  the  densities  of  the  external 
air,  and  of  that  within  the  tube  immediately  at  its  open  extremity, 
during  the  whole  time  of  the  vibrating  motion,  in  the  same  manner  as 
if  the  air  were  at  rest.  This  supposition  carries  with  it  but  little 
appearance  of  being  even  very  approximately  true;  for  it  is  difficult 
to  conceive  how  a  sonorous  wave  could  thus  be  produced  and  maintained 
in  the  surrounding  air  from  the  open  extremity  of  the  tube,  and  it 
appears  perfectly  irreconcileable  with  the  fact  of  the  sudden  cessation 
of  sound  after  the  cause  producing  it  has  ceased,  M.  Poisson,  struck 
with  these  objections,  has  assumed  another  physical  condition  as  appli- 
cable to  any  tube,  whether  open  or  stopped,  viz.  that  there  exists  at  the 
extremity  of  the  tube,  during  the  whole  motion,  q  constant  relation 
between  the  velocity  of  the  particles  of  the  fluid  at  any  instant,  and 
its  condensation,  this  relation  depending  on  the  nature  of  the  substance 


IN  CYLINDRICAL  TUBES.  SSS 

with  which  the  fluid  at  the  extremity  of  the  tube  is  in  immediate 
contact.  This  condition  is  manifestly  less  restrictive  than  those  of 
Euler,  since  it  involves  no  supposition  of  the  perfect  rigidity  of  bodies, 
and  leaves  room  for  a  certain  degree  of  condensation  and  rarefaction 
of  the  fluid  at  the  extremity  of  the  open  tube,  thus  removing  the 
difficulty  above-mentioned  respecting  the  maintaining  of  aerial  pulsations 
from  the  open  end,  in  the  circumambient  air ;  while  it  enables  us  also 
to  account  in  some  measure  for  the  rapid  cessation  of  sound  with  the 
cessation  of  the  cause  producing  the  vibratory  motion  of  the  air  in 
the  tube. 

2.  The  two  authors  above-mentioned  have  written  elaborately  on 
this  subject  of  the  vibrations  of  elastic  fluids  in  tubes.  Mr  Challis 
also  in  his  paper  published  in  the  Transactions  of  this  Society,  (Vol.  III.), 
has  been  led  to  the  consideration  of  the  conditions  which  hold  at  the 
closed  or  open  extremity  of  the  tube  in  which  the  air  is  in  a  state 
of  sonorous  vibration,  though  the  determination  of  this  point  forms 
with  him  a  collateral  rather  than  a  principal  object.  He  assumes  that 
a  pulse  proceeding  along  a  cylindrical  tube  will  be  reflected  from  the 
further  extremity  if  the  tube  be  stopped,  the  intensity  of  the  reflected 
pulse  being  equal  to  that  of  the  incident  one;  and  that  if  the  extremity 
of  the  tube  be  open,  it  will  pass  into  the  circumambient  air,  sending 
back  no  reflected  wave  within  the  tube.  If  this  were  the  case,  it 
would  immediately  account  for  the  apparently  instantaneous  cessation 
of  sound  above-mentioned ;  but  there  are  other  equally  obvious 
phenomena,  for  which  this  hypothesis  appears  to  offer  no  adequate 
solution. 

3.  It  will  be  observed,  that  Euler  has  supposed  either  the  velocity 
of  the  particles  or  their  condensation  to  have,  at  the  extremity  of  the 
tube,  a  constant  value,  independently  of  the  time ;  while  M.  Poisson 
has  supposed  this  constancy  of  value  to  belong  to  the  quantity  ex- 
pressing the  relation  between  the  velocity  and  condensation.  It  does  not 
however  appear  to  me  probable  that  any  such  conditions,  independently 
of  the  time,  should  hold.  All  the  above  assumptions  are  equally 
arbitrary,  and  equally  require  to  be  put  to  the  test  of  experiment.     In 


234  Mr  HOPKINS  ON  AERIAL  VIBRATIONS 

applying  this  test,  I  find  that  the  deductions  from  the  results,  derived 
from  any  of  the  three  hypotheses  above-mentioned,  do  not  sufficiently 
accord  with  the  observed  phenomena  to  be  perfectly  satisfactory.  This 
discrepancy  is  more  particularly  observable  in  the  position  of  the  nodes 
or  points  of  minimum  vibration  in  the  open  tube.  According  to  Euler's 
hypothesis,  these  nodes  would  be  places  of  perfect  rest ;  and  they  would 

be   distant  from  the  open   end   by   an  exact  odd  multiple  of  -,  where 

\  =  length  of  a  whole  undulation.  From  the  hypothesis  of  M.  Poisson, 
their  positions  would  be  the  same  as  in  the  above  case,  but  they  would 
become  points  of  minimum  vibration,  and  not  of  perfect  rest.  Mr 
Challis's  supposition  would  lead  to  the  conclusion  that  no  nodes  existed 
in  this  case,  except  they  should  be  produced  by  some  vibration  of  the 
tube  itself,  a  cause  the  total  inadequacy  of  which  to  produce  any  appre- 
ciable effect,  must  be  immediately  recognized  by  every  one  who  has 
made  experiments  on  this  subject.  The  facts,  as  determined  by  experi- 
ment, are  very  obvious ;  and  it  appears  that  there  are  nodes,  which 
are  points  of  minimum  vibration  and  not  of  perfect  rest ;   that  they  are 

equidistant,  but  that  denoting  this  distance  by  -,  the  distance  between 

the   open    extremity   and  the   nearest  node    is   considerably  less    than   -. 

I  shall  not  in  this  place  proceed  further  with  the  detail  of  experimental 
facts ;  but  shall  first  shew  how  the  theory  of  this  subject  may  be 
generalized  by  the  assumption  of  conditions  less  restrictive  than  those 
which  have  been  made  by  the  writers  I  have  mentioned.  In  the  second 
section,  I  shall  describe  the  experiments  which  have  suggested  these 
assumptions ;  and  shall  conclude  with  some  observations  on  the  resonance 
of  tubes,  so  far,  more  particularly,  as  it  is  allied  to  the  investigations 
contained  in  this  paper. 

The  form  under  which  I  shall  consider  the  problem,  is  that  under 
which  it  presents  itself,  as  nearly  as  possible,  in  the  experiments  I  have 
to  describe. 


IN  CYLINDRICAL  TUBES.  235 


SECTION    I. 

4.  Suppose  the  tube  AB,  (fig.  I.),  open  at  A,  and  stopped  at  B, 
with  some  substance  possessing  any  degree  of  elasticity ;  and  suppose 
the  vibrations  first  produced  and  kept  up  by  a  rigid  diaphragm,  vibrating 
according  to  a  given  law  at  A,  and  perfectly  excluding  the  air  within 
the  tube  from  any  communication  with  the  external  air.  We  have 
the  usual  equations 

v=f{at-x)  +  F{at  +  x)] 

(A), 

as=f{at-x)-F{at  +  x)] 

V  denoting  the  velocity  of  a  particle  at  distance  a;  from  the  origin, 
and  s  the  condensation  at  the  same  point  at  the  time  t,  and  a  being 
the  velocity  of  propagation  of  an  aerial  pulse  along  the  tube. 

One  of  our  conditions  must  necessarily  be,  that  the  velocity  of  the 
air  within  the  tube  and  immediately  in  contact  with  the  diaphragm, 
must  constantly  have  the  same  velocity  as  the  diaphragm  itself,  con- 
strained to  move  according  to  a  given  law.  Let  this  velocity  =  <p{at). 
Then  shall  we  have 

(j){af)=/{ai)  +  F{at) (1). 

5.  To  ascertain  the  nature  of  the  second  condition,  which  must 
hold  at  B,  where  the  motion  of  the  wave  propagated  along  the  tube 
is  interrupted,  we  must  consider  the  effect  which  will  be  produced  on 
the  stop  by  the  action  of  the  air  within  the  tube.  The  vibratory  motion 
wUl  produce  alternations  of  condensation  and  rarefaction  at  the  ex- 
tremity B,  which  will  tend  to  put  the  substance  forming  the  stop  in 
vibration;  and  if  it  will  admit  of  vibrations  having  the  same  period 
as  those  of  the  air  in  the  tube,  this  effect  will  be  produced  by  the 
constant  reiteration  of  the  cause  above-mentioned.  If  the  substance  is 
not  susceptible  of  vibrations  of  this  kind,  no  appreciable  effect  will  be 
produced  upon  it. 


236  Mr  HOPKTNS  ON  AERIAL  VIBRATIONS 

The  determination  of  the  nature  of  these  vibrations,  or  of  the 
function  expressing  the  velocity  at  any  instant  of  the  extreme  section 
of  the  stop,  will  necessarily  depend  on  the  material  of  which  it  is  made; 
and  any  solution  of  the  problem  in  question,  independently  of  this 
consideration,  cannot  be  regarded  as  complete.  Still,  whatever  may  be 
the  nature  of  the  stop,  we  know  that  the  period  of  its  vibrations  must 
be  the  same  as  for  those  in  the  tube;  and  it  is  also  manifest,  that  each 
vibration  of  the  stop  must  begin  at  a  time  later  by  an  interval  at  least 

nearly  =  -,  (/=  the  length  of  the  tube),  than  the  corresponding  vibration 

in  the  diaphragm  at  A,  whence  the  original  disturbance  is  supposed  to 

proceed.      I    say   that   this   interval   is' nearly   equal   -,    because    certain 

phenomena,   of  which    I   shall    speak   hereafter,    seem   inconsistent   with 

its  being  in  particular  cases  exactly  =  -.     I  shall  therefore,  to  give  the 

ct 

assumption  all  the  generality  possible,  consider  it  as  generally  =  — f-  arbi- 

trary  quantity,  to  be  determined  in  each  particular  case  by  experiment. 
Hence  then,  if  ^  denote  the  form  of  the  function  of  the  time  expressing 
the  velocity  of  the  extreme  section  of  the  stop,  we  shall  have  the 
velocity  =  v/'l «/  —  (/  + c)},  c  being  arbitrary.  This  must  also  be  the 
velocity  of  the  extreme  section  of  the  air  at  B,  consequently  we  have 
as  a  second  condition 

•^{at-{l-^c)}=f{at-l)  +  F{at^-l) (2). 

We  have  from  (1) 

(t>{at  +  l)=f{at  +  l)  +  F{at  +  l); 

and  eliminating  F(at  +  l), 

f{at  +  l)-/{at-l)  =  <p{ai  +  l)-f{at-{l  +  c)\  ; 

or,  writing  at  + 1  (or  at, 

f(flt-ir^l)=f{at)-y\f{at-c)  +  <t>{at  +  ^l) (B). 


IN   CYLINDRICAL   TUBES.  237 

The  substance  forming  the  stop  being  known,  so  that  we  might 
regard  the  vibrations  produced  in  it  under  given  circumstances  de- 
terminable, the  relation  between  the  functions  xj^  and  J"  would  be 
known,  and  the  function  y  would  be  the  only  unknown  one  in  the 
above  functional  equation,  from  which,  any  particular  form  being 
assigned  to  (p,  that  of  y  must  be  determined.  The  arbitrary  quantity 
which  will  be  involved  in  the  solution  of  this  equation,  must  be 
determined  by  the  original  value  of  the  function  jf. 

6.  We  have  here  supposed  the  tube  to  be  stopped,  but  the 
equation  (B)  will  still  be  true  for  the  open  tube,  \|/ {«/-(/ +  c)},  de- 
noting always  the  velocity  of  the  'extreme  section  at  the  time  f. 

Equation    (2)   gives    us 

F{at  +  l)=-f{at-l)  +  y\,{at-{l+c)}, 
and  writing   at  +  x,  for  at  +  l, 

F{at  +  x)=  -f{at-{2l-x)}  +  >//  {at -{2l  +  c -x)}'. 

Hence, 

v  =  f{at-x)-f{at-{2l-x)}  +^  {at-{2l  +  c-x)\-\ 

as  =  f{at-x)+f{at-{2l-x)}-yl^{at-{2l  +  c-x)}] 

The   form  of  J"  being   determined   by  equation  J?,   these   last   equations 
will  give  the  complete  solution  of  the  problem. 

7.  Before  we  proceed  to  consider  particular  cases,  we  will  exhibit 
these  equations  (C)  under  another  form,  which  will  be  useful  in 
deducing  some  general  inferences  as  to  the  nature  of  the  motion  in 
the  tube. 

Let  T  denote  a  period  of  time,  from  the  commencement  of  the 
motion  at  A,  less  than  that  which  is  necessary  for  the  pulse  to 
travel  twice  the  length  of  the  tube ;  consequently  at  will  be  less 
than  21. 

Equation  (B)  gives  us 

/(ar  +  9.1)=/ {ar)  -  v//  («T  -  c)  +  ^  {aT  +  2l), 
Vol.  V.    Part  II.  Hh 


•(C). 


238 


Mb  HOPKINS  ON  AERIAL  VIBRATIONS 


and   for  ar,   writing  ar  —  x, 

/{(aT  +  2/)-^}=/(aT-ar)-x//{«T-(a;  +  c)}+0(«T  +  2/-;r) (3). 

Also   putting  ar  +  ^sl—x,   for  ar, 
/■{(aT  +  4/)-ar}=/(aT  +  2/-ar)-v|/{«T  +  2/-(a;  +  c)}  +0(«t  +  4/-^) 
=f{aT-x)-^{aT-{x-^c)} 

-x//{aT  +  2/-(a;  +  c)} 
+  0(aT+2/-a;) 

+  (i>{ar  +  4!l-x). 
And  similarly,   we  have 

2/^ 


/{»(.+ ^v^* 


^{«('^  +  — )-(*'  + c)}, 


■■/{aT-x)-< 


f  {"(-r +—)-(« +  c)}, 


&c. 


,  (    r        2{n-l).l-\      . 


&c. 


.^{«('^  +  -|-)-^} 


IN  CYLINDRICAL  TUBES. 


239 


In   the  same  manner, 


1. 


=f{aT-{%l-x)\-\ 


+  ( 


■>/^{«T-(2/+c-a!)}, 

(2  A 

&c. 

./.{a(x+?i^^^)-(2/  +  c-^)}. 


&c. 


2«/ 


^{«(-  +  ^)-(2/-^)}. 


Hence  we  have   at   the   time  (t+ j, 

V  =f(aT-x)-f{aT-(2l-x)} 

-{>|/[rtT-(«+c)]->|/[«T-(2/+c-a;)]} 
—  &c. 


f,  ,    r       2(n-l)J-\     ,        x>      .   ,    r      2(w-l)./n     -^,         ,J 

+  ^|/  {«  [t  +    — j  -(21  +  C-X)} 


+  &c. 


2w/^ 


+  ^{a[r  +  ^)-a^}-ct>{a[r+^)-i2l-a^)}, 


HH2 


240 


Mr  HOPKINS   ON   AERIAL   VIBRATIONS 


or, 

v=-f{aT-x)-f{aT-{^l-x)}  ,  ^ 

r^n  ^  it  / 

.  .     .         ^nl 


Similarly,   we   find 
as  =  f(fiT-x)+/{aT-{2l-x)}, 

+  2,.,  {<t>  [«  (r  +  ^)  -(.r  +  c)]  +  0  [«  (^.  4-^^)  -(2/  +  C-  .r)]}.    ^ 


\...(D)(1). 


\.. .(D)(2). 


8.  The  function  /(ar  —  x),  in  the  expression  for  v,  represents  the 
velocity  of  any  particle  produced  by  the  first  wave,  propagated 
along    the    tube    from    the    original    disturbance   at  A,    so    long    as    t 

is    less    than    - ;    and    if   this    wave   were    reflected    entirely    from    B, 


a 


the   first  line  of  the  above  expression  for  v,  would  give  us  the  velocity 
of   any  particle   within   the   sphere   of    the   reflected   wave,   the   time   t 

not  exceeding  —  . 

With  our  supposition  as   to  the   original   disturbance,   the  form  of  f 
T  less  than  —  I  will    be   immediately  known   from   that    of    (p.      The 


IN   CYLINDRICAL   TUBES.  241 

other  terms  in  the  general  vahie  of  v,  shew  how  the  general  waves 
in  which  we   have 

If 

v,  =  /,(af-x),    and   v,  =  Jl{at-(2l-x)}, 

are  formed  by  the  superposition  of  successive  waves,  as  the  time 
increases.  If  the  velocity  becomes  by  this  superposition  so  large,  that 
it  can  no  longer  be  considered  extremely  small  as  compared  with 
the  velocity  of  propagation  (a),  our  analysis  will  be  no  longer  ap- 
plicable ;  but  if  V  never  exceed  a  certain  value,  the  motion  will 
become  regular,  and  follow  the  law  which  our  investigations  indicate. 
Let  us  consider  in  what  cases  we  may  expect  these  effects  to  be 
produced. 

9.  We  have  at  present  imposed  no  restrictions  on  the  forms  of 
the  functions  denoted  by  cp,  f  and  \//,  except  that  their  greatest 
values  shall  be  small  compared  with  a.  In  order  however  that  the 
undulations  may  be  sonorous,  <p,  and  consequently  y  and  \f/,  must 
denote  periodical  functions,  so  that  the  values  of  (p  {z),  f  (2),  and  ^  (ss), 
will  recur  as  often  as  %  is  increased  by  a  certain  quantity.  We  will 
also  iinpose  an  additional  limitation  upon  them,  to  which,  in  all 
practical  cases  they  will  probably  be  subject  very  nearly,  as  will 
certainly  be  the  case  in  the  experiments  to  which  I  shall  hereafter 
more  immediately  refer.  Supposing  then  their  values  to  recur,  when 
s  becomes  %-Vm\,    {m   any  whole  number),    we   will  also  suppose  them 

to   recur    with    different    signs     when    z    becomes    x±m'  -;    {m!    being 

any  odd  number). 

10.  First  suppose  the  greatest  value  of  \//,  small  as  compared  with 
that  of  y  or  0,  as  must  be  the  case  in  a  closed  tube.  In  the  above 
expression  for  v,  it  will  be  observed  that  the  quantity  represented  by 
%  increases  as  we  proceed  from  one  term  to  the  next,  in  a  vertical 
line  by  2/. 


Suppose  then 


%l  =  m' .  -,     or  l  =  m'  - 

2  4 


242  Mh  HOPKINS   ON   AERIAL   VIBRATIONS 

In  this  case  it  is  manifest  that  the  consecutive  terms  taken  in  the 
order  just  mentioned  will  destroy  each  other ;  and  there  will  con- 
sequently be  no  accumulation  of  motion  in  the  tube,  and  the 
vibrations   will   go  on   uniformly.      Again,   let 

2l  =  m\,     or  /  =  2m.  -. 

4 

In  this  case  the  values  of  the  successive  terms  taken  as  before  in 
the  expression  for  v  will  be  equal,  and  with  the  same  sign.  Hence, 
if  we   take  x   of  any  value,   except   such   as  would   render 

<(>{at-x)  =  <p  {at-{9.l-x)], 

f  which   value   of    x   is    I  —  m  -\  ,    it  is  manifest   (since  the   value  of  (p 

is    greater   than   that  of  \|/),    that  the    motion   will    constantly  increase 

for    such    points,     and    will     soon  become    greater    than    is    consistent 

with  our  original  suppositions.  Such  a  vibration  then  cannot  be 
maintained.  . 

11.     Again   suppose  the   functions  (p,   f,    and  ^,    to  be   continuous, 
and   suppose 

2/=m'^+2\',     or    /  =  m'^+\', 
2  4 

X'  being   any  quantity  less   than    -;    the   consecutive  teims  of  1.(f>(%), 

tit 

will  not  then  destroy  each  other,  but  as  the  number  of  pairs  of  terms 
increases,  the  sum  will  increase  till  ^(s;  +  2r/)  becomes  negative,  it  will 
then  decrease,  after  having  thus  attained  a  maximum  value.  Maxima 
and  minima  values  will  thus  occur  alternately,  and  the  same  will  hold 
for  2. >//(»).  If  these  maxima  values  do  not  render  v  greater  than  our 
original  suppositions  allow,  the  vibrations  may  be  maintained. 

Since    these    maxima    values    are    0,    when    l  =  m'.-,    and    greatest 

when  l=m' .-,  we  conclude  that  they  will  be  intermediate  for  inter- 
mediate  values  of  I,   following   some   continuous   law.     Hence   we  infer 


IN   CYLINDRICAL   TUBES.  248 

the  possibility  of  maintaining  sonorous  vibrations  of  which  the  period 
is   - ,   in    stopped  tubes   of  which   the   length    differs   considerably  from 

?«' .  - ,    particularly   if    the    greatest   value    of    V/    should    not     be    very 

small.  If  the  supposition  we  have  made  respecting  the  continuity  of 
the  function  (p  more  particularly,  should  not  be  quite  true,  it  is  not 
likely  in  those  practical  cases  to  which  we  can  best  refer,  to  be  so 
far  wrong  as  to  render  the  above  reasoning  otherwise  than  at  least 
approximately  true. 

12.  Our  supposition  has  been  that  the  intensity  of  the  distvu-bance 
denoted  by  v//,  is  considerably  less  than  that  indicated  by  (p,  the  tube 
being  stopped  with  some  substance  having  a  certain  degree  of  elas- 
ticity ;  if  the  tube  be  open,  it  seems  probable  from  certain  pheno- 
mena, that  the  reverse  of  this  supposition  is  true. 

Assuming  this  to  be  the  case,  the  expansion  of  the  expression 
for  V  may  be  put  under  a  more  convenient  form. 

Let 

y{r  {at-{2l+  c-x)}  =2f{at-(2l- x)]  -  f,  {at- (21+  c' -x)], 

Then 

v=f(at-x)+f{at-{2l-x)}-xj.,{at-(2l  +  c'-x)} (a), 

and  equation  (3)  becomes 

/(aT  +  2l-x)=  -/(aT-x)  +  x|/,  {aT-(x  +  c')}  +<p(aT  +  2l-x) (4). 

By  proceeding  exactly  as  in  the  former  case,  we  obtain 

v  =  {-irif(aT-x)+flar-(2l-x)}} 


-fAci{'r+^)-(2l  +  c'-x)} 
+  2,^,(-l)-{<^[a(T  +  ^)  -X]  +  0[«  (t  +  ^y(2l-x)]} 


1 


ME)(1)- 


.(E)(2). 


244  Mr  HOPKINS   ON   AERIAL   VIBRATIONS 

Similarly,   we   find 
«*=(-l)"{/(ar-^)-/[«T-(2/-a;)]}  >^ 

+  ^l,,{a[T  +—j-{2l■^■c-x)} 

+  2,^,(-l)»-{<^[«  (t  +  ?^)  -.V]  -  0[«  {'r+~)  -  (2/  -  x)\\.  ^ 

Reasoning  on  the  expression  for  v,  exactly  similar  to  that  used 
above,    will    in    this    case     show    that    sonorous   vibrations    cannot    be 

maintained   if    /    be    too    nearly    equal   to    an     odd  multiple   of  - ;    but 

that    they  can   be   continued,     if    /   do   not     differ    too   much   from   an 

even  multiple  of  -  .* 

13.  If  we  examine  the  expressions  for  as  in  the  last  article,  and 
in  Art.  7,  it  will  appear  that  the  condensations  and  rarefactions  at 
the  surface  of  the  vibrating  plate  within  the  tube,  are  such  as  to 
produce  forces  opposing  more  strongly  the  motion  of  the  plate  as 
the  lengths  of  the  tubes  approximate  respectively  to  those  particular 
lengths    for   which    it  will   be   impossible    to    maintain    the  vibrations  in 

the  tube ;    and   when   the   lengths    differ  from   the  above   by  - ,    these 

condensations   and   rarefactions   are  such   as   to  promote    the   motion   of 
the  plate,  instead  of  opposing  it. 

14.  The  expanded  expression  for  v  may  be  put  also  under  another 
form,  which  it  may  be  useful  to  point  out  for  the  case  in  which 
the  intensity  of  the  disturbance  denoted  by  \//,  is  considerably  greater 
than  that  denoted  by  <^. 

*  The   quantity   c'   in    these   general    inferences    is   not   taken   into   account.      Its   value 
however  is  considerable,  as  will  be  seen  hereafter. 


IN  CYLINDRICAL   TUBES.  245 

This  is  deduced,  by  assuming 

i,,{at-(x  +  c')}=  (if  (at  -a;)+yl.'{ai-(x  +  c")}, 
or, 

x/.  {«/- (x  +  c)}  =  (2 - /3) /(«^ -x)+ir' {at -(x  +  c")}. 

Then  the  equation  (a)  (Art.  12)  becomes 

v=f{at-x)  +  {l-l3)f{at-{2l-x)}-i,'{at-{2l+c"-x)} (/3). 

We  may  observe,  that  since  the  vibration  denoted  by  \j/,  is  pro- 
duced by  that  denoted  by  Jl  it  seems  a  necessary  consequence  that 
their  periods  must  be  the  same.  Their  phases  also  are  nearly  so ; 
and  if  in  addition  we  assume  that  the  Jbrm  of  the  function  ex- 
pressing the  one  motion,  does  not  differ  very  widely  from  that  ex- 
pressing the  other,  (however  the  intensity  of  the  vibrations  may  differ) 
it  is  manifest  that  /3  may  be  so  taken  that  the  intensity  of  the 
vibrations  denoted  by  the  unknown  function  \j^'  shall  be  small  com- 
pared with  that  indicated  by  <p. 

Equation  (4)  becomes 

f{ar  +  2l-x)=-Cl-ft)f(aT-x)  +  i.'{aT~(x  +  c")}+(j>(aT  +  2l-x) (5), 

=  -hf{ar -x)^-^'  {a-r -  (^  +  c")}  +  0 («t  +  2/-  x), 
if  1-/3  =  *. 

This  gives  us 

And   the   equation   (/3)   becomes,    (when   t=T-\- j, 

v^{-hY  {f{aT-x)+hflar-{2l-x)]} 

+  S,,,(-&)-|>/.'{«[t+^^^^^)-^]-(x+0}+&^^1«[t+^^''^^^-/]-(2/+c"-;»^)}I 

-^'{a(r  +  ^)-{2l  +  c"-x)]. 

r-n  ^  W/  \  (t    ) 

Vol.  V.    Part  II.  1 1 


246  Mr  HOPKINS   ON   AERIAL   VIBRATIONS 

Since    b    is    less    than    unity,  and    n    soon  becomes    a    very  high 

number,  after  an    extremely  short  time  the  first  line   in   this  expression 

may  be    neglected,    as   may    also  all  the   terms  in   the    other   lines   in- 
volving high  powers  oi  h. 

Whence  it  follows  that  the  original  disturbance  (on  which  the 
form  of  the  function  f  will  depend),  will  cease  in  an  extremely  short 
space  of  time  to  have  any  effect  on  the  form  of  the  existing  vi- 
bration, supposing  the  vibrations  maintained  by  some  cause  distinct 
from  that  producing  the  original  disturbance. 

Also,  if  the  cause  maintaining  the  vibrations  cease,  the  vibrations 
themselves  may  cease  in  an  extremely  small  space  of  time. 

The  inferences  we  have  drawn  from  the  former  developement  (E) 
of  the  expression  for  v,  may  be  drawn  from  this  and  perhaps  with 
still  greater  facility. 

15.  If  we  suppose  >|/'  (ss)  always  =  0,  the  expression  for  v  will 
reduce  itself  to  the  same  as  that  given  by  M.  Poisson.  But  in  this 
case  it  will  be  observed  that  all  the  functions  involving  the  quantity  c" 
disappear,  which  renders  it  impossible  to  account  on  this  theory  for  the 
position  of  the  modes  or  points  of  minimum  vibration  as  determined 
by  experiment*.  For  the  purpose  of  determining  the  positions'  of 
these  points  theoretically  we  will  recur  to  the  equations  (C),  the  first 
of  which  is 

~      v  =  f{at-x)-f{at-{^l-  X)}  +^  {at  -{2l  +  c  -  X)} (6). 

If  we  neglect  ^{at-{2l+c  —  x)},  (or  suppose  the  substance  with 
which  the  tube  is  stopped  perfectly  rigid)  we  shall  have  »  =  0,  when- 
ever 

{at  — x)  -  {at —  {^l- x)}=Q,    or  mX, 

{m  being  any  whole  number),  or  when 

{l-x)  =  m.-. 

*  See  Art.  36,  Sec.  II. 


IN  CYLINDRICAL  TUBES.  247 

This  condition  is  independent  of  t,    and  consequently  at  all   points 

distant  from  the  stopped  end,  any  multiple  of  -,    the  motion   will  be 

the  same  as  at  that  extremity,    i.e.   it   will  always   equal  0,    and  there 
will  be  perfect  nodes  at  those  points. 

16.     We   may   take   the   general   case,   and   let 

f\at-{il-x)\-^  {a/-(2/  +  c-ar)}  =j(;  {at-{<il->rc,  -x)}, 
and   :.v=f{flt  —  x)  —  x\a't—{^l^-Cx  —  x)\, 

^  being  still  small.  The  forms  of  J"  and  x//  being  known,  that  of  ^ 
will  be  determined ;  its  period  will  also  be  the  same  as  that  of  J" 
and  ■^.  It  expresses  the  velocity  of  each  particle  produced  by  the 
whole  wave  actually  reflected  from  B.  The  nodes  will  in  this  case 
be   points   of  minimum  vibration,  and   not   of  perfect  rest. 

For  the  sake  of  clearness  we  will  assume  that  y(x),  and  >//(x),  are 
such  that 

and   therefore 

x(-»)=-x(x), 

that  y(»),  and  ^(z),   {and  therefore  x(*)}   admit  of  only  one  maximum 

value  between   x  =  0,    and    8;=-;    and   that   the   ratio  which  y(s!)   bears 

to  ylr  (%)  is  always  considerable,  as  by  hypothesis  it  is  when  those 
functions  have  their  maximum  values.  There  can  be  little  doubt  but 
that  these  assumptions  are  at  least  approximately  true  in  all  practical 
cases ;  and  appear  as  simple  as  any  we  can  make  (and  some  must 
be  made),  in  order  to  give  distinctness  to  our  inferences  as  to  the 
positions  of  these  points  of  minimum  vibration. 

17.  For  the  determination  of  c,  in  terms  of  c,  let  the  origin  of 
t  and  X  be  so  taken  that  y(0)  =  0,  then  making  at-  {2l  —  x)  —  0, 
we  have 

-^(-c)  =  x(-c,); 

or    =\l/{  —  c). 

112 


248  Mb  HOPKINS  ON  AERIAL  VIBRATIONS 

By  our  hypotheses,  x  (*)  must  be  always  greater  than  \//  (%) ;  and 
if  we  suppose  c  and  c^  less  than  the  least  value  of  z,  which  gives 
to  ^  (%),  or  X  (^)  its  maximum  value,  it  is  manifest  that  from  this 
last   equation,    c,  must    be   considerably    smaller  than    c,    and    must   be 

c 
affected  with   a  different  sign.      Suppose   c^  =  j^,    where  k  is   consider- 
ably  greater  than   unity.      It    follows  then   that    if    the  phase    of    the 
vibration   of  the  extreme   section   of  a  stopped   tube   be  retarded  by  a 
certain   quantity   c,    the  phase  of   the    actually   reflected  wave   will    be 

c 
accelerated  by  a  quantity  t. 

18.     Giving  then  the  proper  sign  to  c„  we  have 

v=f(at-x)-x{at-(2l-^-a;)} (7), 

and   to   determine  the   points    of  minimum   vibration,  we  may  observe 
that  this  expression  is  exactly  the  same,  as  if  the  wave  for  which 

v,  =  x{at-{2l-^-x)}, 
were  reflected  immediately  from  a  section  B'  whose  distance  from  A  =  l  —  —x. 

Suppose  a  rigid  diaphragm  at  this  section  constrained  to  move 
exactly  as  the  fluid  does  there ;  we  may  then  suppose  the  actual 
stop  B  removed,  and  the  points  of  minimum  vibration  will  remain 
the  same. 

Now  to  determine  them  in  this  case,  we  observe  that  whenever 
at  —  x  =  at—{2l  —  T  —  x)  +  m\. 


the  value  of  v  will  be  the  same  as  when 

c 


at—x  =  at—{2l—  T  -  x). 


In  the  latter  case 


IN  CYLINDRICAL  TUBES.  249 

and  in  the  former 

or    l-^  =  m\  +  ^^', 

consequently,  at  any  point  in  the  tube  whose  distance  from  B"  =  m  .-^, 

the  velocity  will  be  the  same  as  at  B'.  These  then  will  be  points  of 
minimum  vibration  in  this  hypothetical  case,  and  therefore  also,  from 
what  precedes,  in  the  actual  case. 

Making  c  =  0,  we  have  l—x  =  m.-,  which  will  give  the  positions 
of  the  nodes  when  there  is  no  retardation. 

Hence  we  have  this  general  conclusion  with  respect  to  the  stopped 
tube — that  if  there  be  a  retardation  in  the  phase  of  the  vibration  of 
the  extreme  section,  the  positions  of  the  points  of  minimum  vibration 

will  all   be  further  from   the   stopped  end  by    —j,   than   if  there   were 

no  such  retardation,  the  distances  between  these  points  respectively 
remaining  unaltered. 

19.  We  will  now  consider  the  case  of  the  open  tube,  in  which 
we  suppose  >|/(a!)  to  be  always  considerably  larger  than  J'{%).  Assume, 
as  in  Art.  (12), 

yl,{at-{2l  +  c-x)}-y{at-{^l-x)}~^,  [at -  {21  +  c' - x)} (8), 

v=f{at-x)-¥f{at-{2l-x)}-^,{at-{2l  +  c'-x)}. 

First  neglecting  the  function  >|/, ,   v  will  =  0  whenever 

f{at-x)^-/\at-{2l-x)};    ^ 
i.  e.    whenever 

at—x  =  at—{2l—x)  +  m'.-   {m'  an   odd  number), 
or    I—  x  =  m  .-, 


250  Mr  HOPKINS  ON  AERIAL  VIBRATIONS 

a    condition    independent    of   /.     Consequently,    at    every   point    whose 

distance  from  the  open  end  is  an  odd  multiple  of  -,  there  would  be 
a  perfect  node. 

20.     Put 

f{at-{2l-x)\-y},,{at-{2l  +  c'-a;)]  =x  {at- (2l+c,-x)\ (9). 

Then 

v=/{at-x)  +  x{at-{2l  +  c,-a;)} (10). 

To  find  the  relation  between  d  and  c,  we  have  from  equation    (8), 
(proceeding  as  in   Art.  7,  and  with  the  same  assumptions), 

^(_c)=-x/„(_c'), 

or    >|,i(c')= -x|/(c); 

and  since  >//(»)  is  much  larger  than  >/'i(i8),  we  shall  have  c'.  considerably 
larger  than  c,  and  affected  with  a  different  sign.     We  may  therefore  put 

ki  being  greater  than  unity. 
Again   from   equation    (9), 

-^.(-0=x'(-c.),  • 

or    x'(<=-^)=-Uc'). 

If  we    suppose   x'(«)   nearly   equal  to  v//^,(i8),    (which   probably  is  not 
far  from  the  truth),  we  shall  have 

C2=  —c'  nearly, 

Hence  in  this  case  if  the  phase  of  the  vibration  of  the  extreme  section 
be  retarded  by  a  quantity  c,  that  of  the  actually  reflected  wave  will 
be  retarded  by  kic;  and  it  will  appear  by  the  same  reasoning  as  in  the 
case  of  the   closed   tube,  that   the   distance   of  the   points  of  minimum 

vibration   from    the  open   end  will   be   m'  -r 1-,   {m'   being  any   odd 

number). 


IN  CYLINDRICAL  TUBES.  251 

21.  If  e  and  e'  be  the  distances  through  which  the  nodes  are  moved 
by  a  supposed  given  retardation  of  phase,  the  same  for  each,  at  the 
extremities  of  the  open  and  closed  tubes  respectively, 

e  =  —  kki  e  ; 

6  will  consequently  be  much  larger  than  e'. 

The    quantities    m' i-    in  the   open   tube,  and  m-  +  -^    in   the 

4        2  ^  4       2« 

closed  one,  must  be  determined  by  experiment. 

22.  I  will  recapitulate  the  principal  inferences  from  this  theory. 

I.  In  the  tube  AB,  open  at  the  extremity  B  opposite  to  that  at 
which   the    vibrations    are    produced,    there   will    be   a   series   of    nodes 

equidistant   from   each   other   by    -,   or   half    a   whole    undulation,    the 

distance  of  the  nearest  node  from  the  open  extremity  being  considerably 

less  than  -,  the  whole  system  of  nodes  being  thus  brought   nearer  to 

the  open  end  than  the  position  assigned  to  it  by  the  investigations  of 
Euler  or  of  M.  Poisson.  The  distance  of  each  node  from  the  open 
end  will  be  independent  of  the  length  of  the  tube.     (Art.  20.) 


II.     If  the  tube  be  closed  at  B,  the  nodes  will  still  be  equidistant  as 
X 
2 


before  by  - .     The  distance  from  B  of  the  node  nearest  that  extremity 


will  be  - ,  or  a  quantity  rather  greater  than  that,  if  we  suppose  a  cause 

of  displacement  of  the  whole  system  of  nodes  to  exist  in  this  case  of 
the  closed  tube,  similar  to  that  which  exists  in  the  open  one ;  the  dis- 
placement however  being  necessarily  much  smaller  in  the  former  than 
in  the  latter  case,  and  in  the  opposite  direction.     (Art.  18.) 

III.     These   nodes  are   not   places   in   which   the   air   is   perfectly  at 
rest,  but  points  of  minimum  vibration.     (See  Art.  16.) 


252  Mb  HOPKINS  ON  AERIAL  VIBRATIONS 

IV.  Sonorous  vibrations,  whatever  be  their  period,  may  be  main- 
tained in  a  tube  of  any  length,  except  that  of  which  the  length  does 
not   approximate   too   nearly   to   something   less   than    an    even   multiple 

of  J  in   the   closed  tube,  or  to  an  odd  multiple  of  -  in  the  open  one. 

(Arts.  11,  12.) 

V.  The  intensity  of  the  general  vibrations  in  the  tube  varies  with 
the  length  of  the  tube,  being  greatest  for  the  lengths  just  mentioned, 
and  least  in  the  closed  tube  when  its  length  is  rather  greater  than  an 

odd  multiple  of  -;  and  in  the  open  one,  when  it  is  something  less  than  an 
even  multiple  of  -r .     (Art.  10.) 

VI.  In  these  latter  cases  also  of  both  tubes,  the  opposition  afforded 
by  the  vibratory  motion  of  the  air  within  the  tube,  to  the  vibrating 
of  the  plate,  is  least;  and  greatest  for  the  lengths  which  approximate 
to  those  mentioned  in  (IV.),  as  those  with  which  the  vibrations  cannot 
be  maintained.     (Art  13.) 

VII.  When  the  cause  producing  the  vibrations  in  a  tube  ceases, 
the  vibrations  themselves  may  cease,  not  instantaneously,  but  in  a  period 
of  time  not  exceeding  the  small  fraction  of  a  second,  supposing  the 
tube  not  to  exceed  a  few  feet  in  length.     (Art.  14.) 

VIII.  If  we  suppose  the  original  disturbance  to  produce  an  un- 
dulation different  in  any  respect  to  those  produced  by  the  cause  which 
afterwards  maintains  the  vibratory  motion  of  the  aerial  column,  this 
original  disturbance  will  cease  to  affect  the  form  of  subsequent  undula- 
tions in  a  period  of  time  not  exceeding  the  small  fraction  of  a  second, 
depending  on  the  length  of  the  tube*.     (Art.  14.) 

*  Similar  inferences  to  the  above  may  be  drawn  equally  from  M.  Poisson's  investigations, 
except  that  the  phenomena  according  to  his  solution  would  take  place  for  lengths  of  the  open 
tube  materially  different  from  those  above-mentioned. 


IN  CYLINDRICAL  TUBES.  253 


SECTION    II. 

23.  I  WILL  now  proceed  to  describe  the  experiments  which  have 
been  made  with  a  view  of  putting  the  different  theories  on  this  subject 
to  an  experimental  test.  Sonorous  vibrations  are  usually  excited  in  a 
tube,  either  by  directing  a  stream  of  air  across  the  open  end,  as  in 
blowing  across  the  embouchure  of  the  flute;  by  means  of  a  vibrating 
tongue,  as  in  all  reed  instruments ;  or  by  placing  an  open  end  of  the 
tube  close  to  the  surface  of  a  vibrating  body.  In  the  two  first  cases  it 
seems  impossible  to  conceive  that  the  same  disturbance  can  be  com- 
municated to  each  part  of  the  extreme  section  of  the  air  in  the  tube 
where  the  original  motion  is  produced,  a  condition  which  is  always 
assumed  to  hold  at  least  approximately  in  all  our  mathematical  investi- 
gations of  the  subject.  This  irregularity  of  the  motion  will  no  doubt 
extend  to  some  distance  within  the  tube,  and  it  is  impossible  to  say 
how  it  will  affect  the  phenomena  even  in  those  parts  of  the  tube  in 
which  the  motion  may  become  more  uniform.  In  the  second  case  too 
in  particular,  a  stream  of  air  must  constantly  be  passing  through  the 
tube,  a  circumstance  not  contemplated  in  our  analysis  of  the  problem. 
This  may  or  may  not  influence  materially  the  observed  phenomena, 
but  at  all  events  the  danger  of  derangement  from  any  such  cause 
must  be  avoided,  if  we  would  render  our  experiments  decisive  tests 
of  the  truth  of  any  theory  professing  to  account  for  phenomena  of  so 
delicate  a  nature  as  those  which  are  now  the  objects  of  our  investigation. 
The  third  method,  however,  above-mentioned,  is  entirely  free  from  the 
latter  objection,  and  may  be  made  almost  entirely  so  from  the  former, 
and  is,  therefore,  that  which  I  have  adopted. 

24.  The  apparatus  is  very  simple.  Figure  I.  represents  it.  A 
plate  of  common  window  glass  is  held  firmly  in  a  horizontal  position 
by  a  pair  of  pincers  at  its  middle  point.  AB  is  a  gltiss  tube,  having 
a  short  brass  tube  closely  sliding  within  it  at  the  upper  end  B,  so 
that  the  whole  tube  AB  can  be  lengthened  or  shortened  at  pleasure. 
Within  the  tube  a  small*  brass  frame  M,  having  a  delicate  membrane 

*  Fig.  (2)  represents  this  frame  with  the  membrane  ab,  which  may  be  tuned,  or  rendered 
sensitive  in  different  degrees,  to  the  vibrations   produced   by  any  proposed   note,   either   by 
Vol.  V.    Paet  II.  K  k 


254  Mr  HOPKINS  ON  AERIAL  VIBRATIONS 

stretched  across  it,  is  suspended  by  a  fine  wire  or  thread  from  the  upper 
extremity  of  the  tube,  in  such  a  manner  that  it  can  be  heightened  or 
lowered  at  pleasure.  The  other  parts  of  the  apparatus  are  merely  such 
as  are  adapted  for  facility  and  -accuracy  of  arrangement  of  the  tube 
and  plate. 

25.  The  air  in  the  tube  is  put  in  a  state  of  sonorous  vibration 
by  means  of  the  plate,  which  is  made  to  vibrate  by  drawing  the  bow 
of  a  violin  equably  across  its  edge  in  a  direction  perpendicular  to  its 
plane ;  the  vibratory  motion  of  the  air  is  communicated  to  the  membrane 
suspended  in  the  tube,  and  the  degree  of  motion  is  indicated  by  the 
agitation  of  a  small  quantity  of  light  dry  sand  sprinkled  upon  it*. 
Suppose  the  tube  to  be  open  at  the  upper  end  B,  and  let  the  membrane 
be  drawn  up  near  that  extremity.  Tf  the  sand  indicate  a  considerable 
motion  when  the  plate  is  vibrating,  let  the  membrane  be  gradually 
lowered ;  a  position  will  thus  be  found  in  which  the  sand  has  little 
or  no  apparent  motion,  thus  indicating  the  existence  of  a  node.  On 
lowering  the  membrane  still  further,  the  sand  will  again  become  strongly 
agitated,  and  will  then  come  to  another  place  of  rest,  (or  at  least  of 
minimum  vibration),  and  so  on  till  it  reach  the  lower  end  of  the  tube. 
These  alternations  of  points  of  rest  and  motion  can  of  course  only  take 
place  when  the  tube  is  sufficiently  long  in  comparison  with  the  length 
of  an  undulation  produced  by  the  vibrating  plate,  to  admit  of  them. 
These  nodal  points  are  thus  found  to  be  at  equal  distances  from  each 
other,  the  distance  of  the  upper  one  from  the  top  of  the  tube  being  less 
than  half  that  between  the  nodes.  This  is  independent  of  the  length  of 
the  tube.  These  results  are  accordant  with  our  theory,  (Art.  22,  I.),  from 
which  it  appears   that   this   constant   distance   between   two   consecutive 

nodes  must  be  -. 
2 

If  we  call  the  distance  of  the  upper  node  from  B,  -—  C,  C  denotes 
what  I  have  termed  the  displacement  of  the  nodes. 

altering  the  tension  by  means  of  the  small  cylinder  round  which  the  end  b  of  the  membrane 
passes,  or  by  moving  the  small  bridge  cd,  and  thus  altering  the  length  of  the  vibrating  part. 

*  This  was  the  method  adopted  by  Savart  in  such  a  variety  of  caseSj  in  which  he  wished  to 
ascertain  the  intensity  of  sonorous  vibrations  in  air. 


IN  CYLINDRICAL  TUBES.  255 

26.  If  the  membrane  be  rendered  very  sensitive  by  being  exactly 
tuned  to  the  note  produced  by  the  vibrating  plate,  it  will  not  indicate 
perfect  rest  at  the  nodal  points,  shewing  them  in  fact  to  be  points  of 
minimum  vibration,  which  agrees  with  our  theory,  (Art.  22,  III.). 
With  such  a  membrane  it  will  be  difficult  to  determine  the  position  of 
these  points  with  accuracy,  and  its  sensibility  should  be  diminished, 
till  the  sand  appears  perfectly  at  rest  when  it  is  placed  exactly  at  the 
node.  If  the  membrane  be  rendered  still  less  sensitive,  it  will  appear 
at  rest  for  a  space  on  each  side  of  the  node,  the  position  of  which  will 
in  such  case,  be  determined  by  observing  those  points  immediately 
above,  and  below  the  node  at  which  the  motion  of  the  sand  is  just 
sensible.  The  middle  point  between  them  will  of  course  be  the 
node. 

27.  Now   suppose   the   length   of  the  tube   to  be  any  odd  multiple 

of  -,  and  the  membrane  to  have  such  a   degree   of  sensibility,   as  just 

to  remain  at  rest  only  when  placed  in  a  node  or  within  a  very  small 
distance  of  it.  After  it  has  been  placed  in  this  position,  let  the  brass 
tube  sliding  within  the  upper  part  of  the  glass  one  be  raised  through  a 

space  less  than  - .     While  the  whole   tube   is  thus  lengthened,   let  the 

distance  of  the  membrane  from  the  upper  end  B  remain  the  same; 
the  membrane  will  consequently  be  still  in  a  node.  The  plate  being 
now  put  in  vibration,  the  membrane  will  remain  perfectly  at  rest,  not 
only  in  this  position,  but  also  when  moved  to  one  considerably  above 
or  below  the  node,  the  new  length  of  the  tube  remaining  the  same. 
This  indicates  a  less  degree  of  motion  in  the  tube  than  in  the  former 
case,  and  we  find  that  the  intensity  of  the  vibration  in  the  open  tube 
is    least    when    its    length    is    equal    to    something    less    than    an    even 

multiple    of    -T,    or    2m.j  —  C;    and    becomes    greater   as    the    length 

approximates  to  rather  less  than  an  odd  multiple  of  -,  or  {2m'  +  1)-—C, 

m  and  m'  being  any  whole  numbers.  (Art.  22.  V.).  This  diminution  of 
motion  is  also  very   obvious    when   the  membrane   is   placed   in    those 

KK2 


Si56  Mr  HOPKINS  ON  AERIAL  VIBRATIONS 

parts  of  the  tube  where  the  motion  is  most  sensible.  In  all  cases, 
however,  the  distances  of  the  nodes  from  B  is  independent  of  the 
length  of  the  tube. 

28.  If  we  take  a  tube  closed   at  B  instead  of   the  open  one,    we 

observe  the  same  alternations  of  points  of  greatest  and  least  vibration, 

and   (the   plate   being   made   to   vibrate  in  the   same  manner  as  before) 

at    exactly    the    same  distances  from   each  other  as  in  the  closed  tube; 

but   the   distance   of  the  upper  node  from  the  closed  extremity  of  the 

X  I 

tube  is   now  observed  to  be  -,    the   same  as   the  distance  between    the 

2 

nodes.      Proceeding   as   in   the   former   case,    it   is   found   also   that   the 

strongest  vibrations  are  excited  when  the  length  of  the  tube  is  about  equal 

to  a  multiple  of  - ;    and  the  least  vibrations  when  the  length  =  an  odd 

multiple  of  - .     I  find  also  that  in  the  open  tube  stronger  vibrations  exist 
4 

in  the  nodal  points  than  for  corresponding  cases  of  the  closed  tube. 

29.  In  performing  the  above  experiments  with  reference  to  the 
intensity  of  the  vibrations  in  the  tube,  care  must  of  course  be  taken 
to  prevent  the  influence  of  any  other  cause  than  that  of  which  I  have 
spoken,  viz.  the  length  of  the  tube  with  respect  to  X.  It  has  been  as- 
sumed that  the  vibration  of  the  part  of  the  plate  immediately  in  contact 
with  the  mouth  of  the  tube  is  in  all  cases  the  same,  which  requires 
that  the  tvibe  should  always  be  placed  over  exactly  the  same  portion 
of  the  plate.  This  portion  also  should  be  included  in  one  and  the  same 
ventral  segment;  for  if  a  nodal  line  on  the  plate  pass  across  the  mouth 
of  the  tube,  the  vibrations  transmitted  from  opposite  sides  of  this  line 
will  be  in  exactly  opposite  phases,  and  will  consequently  neutralize  each 
other  in  a  degree  depending  on  the  ratio  which  the  intensity  of  one 
of  these  undulations  bears  to  the  other.  If  the  nodal  line  divides  the 
part  of  the  plate  in  contact  with  the  mouth  of  the  tube  into  two 
equal  portions,  parts  of  similar  ventral  segments,  the  interference 
will    be    so    complete    as     to    destroy    all    sensible    motion     in    the 


IN  CYLINDRICAL  TUBES.  257 

tube*.  It  is  only  however  as  regards  the  intensity  of  the  vibrations 
that  this  precaution  respecting  the  relative  position  of  the  nodal 
lines  and  mouth  of  the  tube  is  important ;  it  does  not  affect  the 
positions  of  the  nodes.  The  reason  is  obvious — it  does  not  affect  the 
value  of  X. 

30.  Again,  taking  the  tube  open  at  B,  let  the  extreme  section 
at  A  be  made  to  coincide  nearly  with  the  surface  of  the  vibrating 
plate.  If  the  plate  (the  bow  being  applied  to  it)  vibrate  freely,  let 
the  length  of  the  tube  be  gradually  increased  or  diminished.  It  will 
thus  be  found,  that  as  the  tube  approximates  to  certain  lengths,  the 
plate  vibrates  with  less  facility,  requiring  a  greater  pressure  of  the 
bow,  and  continuing  to  vibrate  audibly  for  a  shorter  time  after  its 
removal;  and  in  many  cases,  between  certain  limits  in  the  length  of 
the  tube,  it  becomes  almost  impossible  to  make  the  plate  assume  that 
state  of  vibration  which  it  assumes  freely  for  other  lengths ;  and  the 
vibration,  if  it  be  produced,  appears  to  cease  almost  instantaneously 
on  the  removal  of  the  bow,  instead  of  being  audible  for  several 
seconds,    as    it    would    be    if    the    tube    were    removed,    or    were    of    a 

different   length.      These   phenomena   recur   for   every    increase   of   —  in 

the  length  of  the  tube ;   and  if  I  be  any  length  with  which  it  becomes 
almost   impossible   to   make   the  plate   vibrate  in   the  manner   proposed, 

then  will   /  +  -   be   that  length   with  which  it   vibrates  with  the   same 

facility  as  if  the  tube  were  removed. 

*  It  is  easy  by  a  very  simple  experiment  to  give  ocular  demonstration  of  the  fact  that  the 
union  of  two  intense  sounds  may  produce  perfect  silence.  Take  a  branch  tube  ABA'  (Fig.  3.),"and 
stretch  over  the  open  end  B  a  fine  membrane  or  a  piece  of  common  writing  paper.  Place  the 
open  extremities  A,  A'  of  the  equal  and  similar  branches  CA,  CA'  over  portions  of  two  ventral 
segments  of  a  vibratory  plate  in  the  same  phase  of  vibration.  A  small  quantity  of  sand  strewed 
over  the  membrane  at  B,  will  immediately  shew  it  to  be  in  a  state  of  strong  vibration.  Let  A,  A 
be  then  carefully  placed  over  suiiilar  portions  of  similar  ventral  segments  of  the  plate,  in  opposite 
phases  of  vibration ;  the  sand  on  the  membrane  will  remain  perfectly  at  rest,  shewing  that  the 
waves  propagated  along  AC  and  A'C  in  opposite  phases  so  completely  interfere  at  Cas  to  produce 
no  undulation  along  CB.  In  other  words,  no  sound  would  in  this  case  be  transmitted  along  the 
tube  to  its  mouth  B. 


258 


Mk  HOPKINS  ON  AERIAL  VIBRATIONS 


So  far  these  phenomena  are  in  accordance  with  the  results  of 
theory,  (Art.  22,  VI.) ;  but  when  we  examine  the  length  I  just  men- 
tioned, we  find  it  entirely  at  variance  with  them.  In  fact  on 
investigating  the  circumstances  more  narrowly,  we  find  that  the  value 
of  /  depends  in  a  considerable  degree  on  the  small  distance  between 
the  vibrating  plate,  and  the  extreme  section  A  of  the  tube,  a  cir- 
cumstance which  nothing  in  our  theoretical  deductions  has  led  us 
to  anticipate.  This  will  be  seen  in  the  results  of  the  following  ex- 
periment made  with  an  open  tube. 

Diameter   of  the   tube  =  1 .  35  inches. 

Value   of- ,.=4.82   for   temperature  63°. 


Position  of  the  mouth  (^A)  of  the 
tube  (Fig.  I.) 


Value  of  the  length  / 
above  mentioned. 


Theoretical  value  of  /. 


As  close  to  the  plate  as 
possible  without  interfering 
with  its  vibrations 


About  T^  inch   from   thel 

lo  I 


vibrating  plate. 


.12.25  inches. 


*  1 1 .  46  inches. 


12.   6 


31.  This  discrepancy  however  between  the  theoretical  and  ex- 
perimental results  is  only  apparent.  It  arises  from  the  circumstance 
of  one  of  the  conditions  assumed  in  our  mathematical  investigation, 
not  being  accurately  satisfied,  namely,  the  perfect  prevention  of  all 
communication  between  the  external  air  and  that  within  the  tube  at 
the  extremity  next  the  plate.  And  this  is  easily  proved  experi- 
mentally,   by   placing    the   extremity    of    the   tube  as   near   as   possible 


*  In  this  value  of  Z  I  have  taken  account  of  the  displacement  of  the  nodes,  which  is  .59 
inches,  as  determined  by  experiment.     (See  Table,  Art.  S6.) 


IN  CYLINDRICAL  TUBES.  259 

to  the  surface  of  the  plate,  without  interfering  with  its  vibrating 
motion,  and  then  putting  round  the  edge  of  the  tube,  a  small 
quantity  of  fluid  which  by  its  adherence  to  the  tube  and  the  plate 
fills  up  the  interstice  between  them,  and  prevents  communication  with 
the  external  air.  When  this  precaution  is  taken,  the  lengths  of  the 
tube  which  correspond  to  the  above  mentioned  phenomena  exactly  agree 
with  theory;    that  is — 

The  .vibration  of  the  plate  is  unaffected  by  the  presence  of  the  open 
tube,  Avhen  its  length  is  equal  to  something  less   than  an  even  multiple 

of  — ,  or  2  m.  J—  C,  and  of  the  closed  one  when  its  length  is  equal  to 

4  4 

an    odd   multiple   of  -;    but   as    the   lengths   of  the   tubes   approximate 

respectively    to   quantities   differing    by  - ,    from   the   above    lengths    it 

becomes  almost  impossible  to  make  the  plate  assume  the  same  vi- 
bratory motion.     (Art.  22,  VI.) 

32.  It  might  at  first  appear  probable  that  the  neglect  of  this 
precaution  might  have  some  effect  on  the  position  of  the  nodes,  as 
well  as  on  the  phenomena  above  mentioned.  This  however  is  not 
the  case;  and  the  reason  will  be  obvious  if  we  recollect  that  the 
position  of  the  nodes  depends  on  the  periodicity  of  the  vibrations,  or 
the  value  of  X,  which  is  unaffected  by  the  communication  with  the 
external  air  at  A ;  whereas  the  force  opposing  the  vibration  of  the 
plate  depends  on  the  condensations  and  rarefactions  of  the  air,  at  the 
surface  of  the  plate  within  the  tube,  which  will  necessarily  be  much 
affected  by  the  communication  just  mentioned.* 

33.  If  we  take  a  closed  tube,  a  similar  discrepancy  or  accordance 
in  the  results  of  theory  and  experiment  will  be  found  under  the 
same  circumstances  as  above  described. 

*  It  does  not  appear  so  easy  to  account  for  the  phenomena  as  above  described,  when  the 
influence  of  external  air  is  not  prevented.  This,  however,  does  not  immediately  belong  to  the 
object  I  have  proposed  to  myself  in  this  paper,  which  is,  to  establish  as  accurately  as  possible  the 
identity  of  the  results  of  theory  and  of  experiment  in  those  cases  in  which  the  conditions  assumed 
in  our  mathematical  investigations  are  experimentally  satisfied. 


260  Mr  HOPKINS  ON  AERIAL  VIBRATIONS 

The  phenomena  above  mentioned,  agree  with  those  observed  by 
Mr  Willis,  and  described  in  his  paper  on  the  Vowel  sounds,  pub- 
lished in  the  Transactions  of  this  Society,  Vol.  III.  The  manner 
however  in  which  his  experiments  (having  a  different  object  from 
mine)  were  conducted,  render  them  unfit  for  the  verification  of  any 
of  our  mathematical  results  in  this  subject. 

34.  From  what  I  have  above  stated,  respecting  the  difficulty  of 
making  the  plate  vibrate  with  certain  lengths  of  the  tube,  it  is  manifest 
how  we  may  avail  ourselves  of  this  phenomenon,  in  the  determination 
of  the  value  of  X,  corresponding  to  any  particular  mode  of  vibration 
of  the  plate,  supposing  those  particular  lengths  of  the  tube  can  be 
ascertained  with  sufficient  accuracy.  Now  this  can  be  done  almost 
as  accurately  as  the  position  of  a  node  can  be  determined  by  the 
vibrating  membrane,  and  consequently  the  value  of  X  may  thus  be 
found.     For  if  A  and  4  denote  two  observed  values  of  /,   we  shall  have 

—  = ,  n  being  a  whole  number  easily  ascertained.     (See  Arts.  30,  31.) 

35.  Though  I  have  had  frequent  occasion  to  speak  of  this  displace- 
ment of  the  nodes  in  the  open  tube,  from  the  positions  assigned  to  them 
by  the  common  theory,  I  have  hitherto  said  nothing  as  to  the  ex- 
perimental determination  of  its  magnitude.  The  most  direct  way  of 
accomplishing  this,  is  to  determine  the  actual  positions  of  the  nodal 
points  by  means  of  the  vibrating  membrane ;  but  this  method  becomes 
inconvenient  when  the  diameter  of  the  tube  is  small,  as,  for  instance, 
less  than  an  inch.  Those  which  I  have  used  most  commonly  are 
from  1.3  in.  to  1.5  in.  diameter.  If  the  tube  be  larger  than  this,  it 
will  generally  be  too  large  to  admit  of  the  extreme  section  of  it 
being  placed  entirely  upon  the  same  ventral  segment  of  the  plate, 
as  is  always  desirable,  (see  Art.  29) ;  and  if  much  smaller  it  becomes 
necessary  to  make  the  surface  of  the  membrane  so  small  as  to  be 
inconvenient,  in  order  that  it  may  not  bear  too  great  a  ratio  to  the 
area  of  the  section  of  the  tube,  in  which  case  the  presence  of  the 
membrane  might  be  supposed  to  render  the  vibrations  in  the  tube 
materially  different  from  what  they  would  otherwise  be. 


IN  CYLINDRICAL  TUBES.  261 

The  best  method  therefore  of  determining  the  positions  of  the 
nodes  in  tubes  considerably  smaller  than  those  I  have  mentioned,  is 
that  by  which  the  value  of  \  is  determined,  as  described  in  the 
last  Article. 

Thus,  suppose  /  to  be  the  length  of  tube,  with  which  it  is  found 
most  difficult  to  make  the  plate  vibrate ;  then  (the  tube  being  open) 
we  shall  have 

l={2m  +  l)^-C, 

where  m  is  a  whole  number,  which  will  be  known  when  \  is  de- 
termined by  either  of  the  methods  pointed  out  above.  The  quantity 
C  evidently  shews  how  much  the  distance  between  the  open  ex- 
tremity, and  the  nearest  node  differs  from  — ,  or  it  expresses  the 
displacement. 

From   the  above  equation, 

C={2m  +  l))-l, 

4 

and  the  displacement  is  thus  determined. 

36.  The  following  table  exhibits  the  magnitude  of  this  displace- 
ment in    a  tube   of  given    diameter,    as  determined    experimentally   for 

different  values  of  - .     The  positions  of  the  nodes  were  in  these  cases 

carefully  ascertained  by  means  of  the  membrane  suspended  in  the 
tube. 


Vol.  V.    Paet  II.  L  i. 


262 


Mr  HOPKINS  ON  AERIAL  VIBRATIONS 


Diameter  of  the  tube  =  1.35.* 

Value  of  — . 
at  temp.  63". 

Computed  dist.  of  a  Node 
from  B,  (fig.  1.) 

Observed  dist.  of  the 
same  Node. 

Displacement  of 
the  Node. 

2.044 

3.994 
4.   82 

fll.24 
I  7.15 

9.98 

7.23 

10.88 
6.78 

9.51 
6.64 

.36] 

}  mean  =  .36.'i 
.37] 

.47 

.59 

The  above  values   of  -  were  determined  by  means  of  a  membrane 

2  •' 

and  a  tube  closed  at  the  upper  end,  nearly  100  inches  in  length.     The 

distance    of    a   node   from   the    closed    end    being    found  =  b,   we   must 


Or,   if    bf   be   the  observed   distance,   sub- 


have    n .  -  =  o,    or    -  =  -  , 
2  2       w 

ject    to    an    error    /3,    and   therefore   b  ±  (i    the   true   distance,   we    have 
-  =  -  +  —.     The   value   of  /3   will    probably  be  less   than   ^   inch,  and 

t^  Tt  ft 

in    the    determination,    for    example,    of   the   first    of  the   above   values 

of  -,  «   was  about  45,  so  that  that  value  of   -  may  probably  not  be 

subject  to  an  error  exceeding  .001   inch.     We   may  also  remark,  as  an 
indication  of  accuracy  in  the  numbers  10.88  and  6.78,  given  in  the  third 


*  The  measures  are  all  given  in  inches. 

t  In  the  determination  of  the  quantity  b,  the  temperature  at  the  time  of  observation  must 
be  carefully  noted,  since  the  variation  in  the  velocity  of  aerial  undulations  produced  by  a  varia- 
tion of  temperature  of  even  less  than  1°,  is  sufficient  to  make  a  very  sensible  difference  in  the 
value  of  h,  this  value  being  as  much  as  nearly  100  inches. 

Since  the  distance  of  any  proposed  node  from  the  upper  end  of  the  tube  will  be  proportional 
to  the  velocity  of  the  undulation,  it  is  manifest  that  by  observing  the  values  of  b,  corresponding 
to  different  temperatures,  we  may  estimate  directly  the  effect  of  temperature  on  the  velocitj'  of 
sound.     This  method  is  capable  of  great  accuracy. 


IN  CYLINDRICAL  TUBES. 


263 


column,   that   10.88-6.78  =  4.10   must   =2.-,  which  gives  us  -  =  2.05, 

differing  but  .006  from  the  more  accurate  value.  The  error  in  the 
two  numbers  above  mentioned,  10.88  and  6.78,  does  not  probably  exceed 
.01  or  .02,  and  cannot,  I  conceive,  exceed  .04,  and  consequently,  I  think, 
the  utmost  limit  to  the  error  in  the  corresponding  numbers  in  the 
fourth  column  cannot  exceed  .05,  and  is  probably  considerably  less.  The 
same  may  be  concluded  respecting  the  numbers  .47,  .59,  in  the  same 
column. 

The  above  results  may,  then,  be  considered  sufficiently  accurate  to 
determine  the  fact  of  the  magnitude  of  the  displacement  increasing 
with  increased  values  of  X,  though  not  sufficiently  so  to  determine  with 
certainty  the  law  of  this  corresponding  increase. 

The  displacement  does  not  depend  only  on  the  value  of  \ ;  it  depends 
also  on  the  area  of  the  mouth  of  the  tube,  as  appears  from  the  following 
table. 


Values  of  ^  • 

Displacement. 

Diameter  of  tube  =  1.35. 

Diameter  of  tube  =  .8. 

2.04,4 
3.99^ 

.23 
.4 

.08 
.1 

These  values  of  the  displacement  of  the  nodes  have  been  obtained 
by  the  method  mentioned  in  Art.  35,  as  that  best  applicable  to  small 
tubes.  The  results  in  the  second  column  of  this  table  ought  to  be 
the  same  as  the  two  first  in  the  last  column  of  the  former  table;  but 
this  method  is  liable,  I  conceive,  to  greater  error  and  uncertainty  than 
the  former,  and  to  this,  I  doubt  not,  the  discrepancy  is  due.'  All 
these  latter  results,  however,  are  probably  subject  to  an  error  of  the  same 

L  L2 


264  Mr  HOPKINS  ON  AERIAL  VIBRATIONS 

kind,  and  are  too  small  both  in  the  large  and'  small  tube.  They  can 
leave  no  doubt  of  the  fact  of  the  magnitude  of  the  displacement  being 
dependent  on  the  diameter  of  the  tube. 

It  is  important  to  observe,  that  the  values  of  X  determined  in  the 
large  tube  and  the  small  one,  from  the  consideration   that  the  distance 

between  any  two  nodes   must   equal   some  multiple  of  - ,    was   exactly 

the  same,  being  for  the  first  case  in  the  table  2.05,  very  nearly  agreeing 
with  the  accurate  value  2.044.  This  proves  that  the  distance  between 
the  nodes  is  independent  of  the  diameter  of  the  tube,  provided  the  dis- 
turbance take  place  uniformly  throughout  its  extreme  section. 

37.  I  have  before  remarked,  that  there  can  be  nothing  arbitrary 
or  indeterminate  in  the  vibratory  motion  of  the  air  at  the  extremity 
of  the  open  tube  when  the  vibrations  in  it  are  excited  according  to 
some  known  law ;  and  consequently,  if  our  theoretical  knowledge  of 
the  subject  were  complete,  we  should  undoubtedly  find  in  our  investiga- 
tions the  cause  of  the  retardation  of  phase,  of  which  I  have  spoken, 
in  the  reflected  wave  of  the  open  tube,  supposing  it  to  be  the  actual 
cause  of  that  displacement  of  the  whole  system  of  nodes  which  I  have 
established  as  an  experimental  fact.  Our  knowledge  at  present,  how- 
ever, is  totally  inadequate  to  this  purpose,  and  therefore  we  can  only 
conjecture  what  may  be  the  probable  cause  of  this  retardation  in  the 
reflected  wave;  but  at  all  events,  our  formulse,  with  the  modifications 
1  have  introduced  into  them,  do  become  perfect  representations  of  all 
those  phenomena  which  we  can  distinctly  determine  by  experiment, 
in  the  cases  to  which  our  mathematical  investigations  apply.  The  fact 
too,  of  a  retardation  of  phase  in  the  reflected  wave  may  not  be  very 
difficult  to  conceive,  or  appear  improbable,  if  we  suppose  the  undulation 
proceeding  from  the  open  end  of  the  tube  to  advance  through  a  certain 
space  before  it  assumes  that  form  in  diverging  into  free  space,  which 
it  must  ultimately  assume  when  it  sends  back  no  reflected  wave  from 
any  point  of  its  path.  Before  it  reaches  this  state,  a  partial  wave  may 
be  reflected  in  its  course  from  each  point  towards  the  tube;  and  an 
indefinite  number  of  these  reflected  waves  will  form  a  general  reflected 


IN  CYLINDRICAL  TUBES.  265 

wave,  of  which  the  period  will  be  the  same  as  that  of  each  of  its 
component  waves,  but  the  phase  of  which  will  be  retarded  as  compared 
with  that  of  a  wave  reflected  immediately  from  the  extremity  of  the 
tube.  This  is  equivalent  to  our  supposing  a  certain  space  beyond  the 
extremity  of  the  tube  as  subject  to  a  disturbance  (acting  at  consecutive 
instants  along  this  space)  such  as  to  produce  a  wave  diverging  in  all 
directions,  and  consequently  sending  a  portion  of  this  general  wave 
back  along  the  tube. 

To  give  generality  to  the  investigations  of  the  preceding  section, 
I  have  considered  the  effect  on  the  position  of  the  nodes  which  would 
be  produced  by  any  retardation  of  the  phase  of  the  wave  reflected  from 
the  stopped  end  of  a  tube.  It  appears,  however,  that  there  is  not  in 
this  case  any  displacement  of  the  nodes  appreciable  by  the  mode  of 
experimenting  I  have  described.  The  only  reason,  in  fact,  for  supposing 
any  retardation  of  phase  in  this  case,  is  founded  in  the  imperfect 
analogy  between  the  cases  of  the  open  tube  and  the  tube  closed  with 
an  elastic  substance.  The  cases  are  far  too  different,  however,  to  admit 
of  any  thing  but  vague  inferences  from  such  analogy ;  and  it  is 
manifest  that  no  reasoning  similar  to  that  above  applied  to  the  open 
tube,  can  be  applied  to  the  closed  one.  If  any  retardation  do  exist  in 
this  case,  I  can  only  conceive  it  to  arise  from  a  cause  similar  to  that 
suggested  by  Mr  Willis*,  viz.  that  time  must  be  necessary  for  the 
action  between  the  elastic  stop  and  the  air  to  produce  its  effect.  This, 
however,  appears  much  less  probable  in  this  case  than  in  that  which 
suggested  the  idea  to  Mr  Willis,  in  which  the  action  between  the  air 
and  the  vibrating  body  (a  membrane)  was  lateral  instead  of  being  direct, 
as  in  the  present  instance.  I  have  not  been  able  to  detect  any  indica- 
tion of  such  law  of  force  in  a  displacement  of  the  nodes  in  the  closed 
tube,  though  I  have  examined  the  case  with  great  care,  conceiving 
that  any  facts  bearing  directly  upon  the  nature  of  the  mutual  action 
of  two  elastic  media  at  their  common  surface  must  necessarily  be  of 
importance. 


*  Cambridge  Transactions,  Vol.  IV.  Part  III.  p.  346. 


266  Mr  HOPKINS  ON  AERIAL  VIBRATIONS 

The  experimental  deductions  in  the  preceding  part  of  this  section 
are  based  on  the  evidence  afforded  by  the  exploring  membrane,  because 
it  is  more  direct  than  any  other  evidence  which  the  phenomena  appear 
to  admit  of,  and  therefore  better  calculated  to  supply  those  decisive 
and  positive  tests  for  ascertaining  the  accuracy  or  fallacy  of  our  theoretical 
results,  which  it  is  my  object  to  supply.  We  have  seen  the  perfect 
accordance  of  these  results  with  the  general  indications  of  the  membrane, 
and  also  with  the  striking  and  well-defined  phenomenon  of  the  im- 
possibility of  making  the  plate  vibrate  in  a  certain  manner  with  tubes 
of  certain  lengths.  It  remains  for  us  to  consider  also  how  far  our 
theory  agrees  with  the  phenomena  of  resonance,  in  those  cases  in 
which  the  conditions  assumed  in  our  mathematical  investigations  are 
satisfied,  viz.  where  the  communication  between  the  external  air  and 
that  in  the  tube  at  the  surface  of  the  plate  is  prevented,  and  the 
disturbance  extends  uniformly  over  the  whole  orifice.  In  such  cases 
it  will  appear  from  the  following  enunciation,  that  the  intensity  of 
the  sound  is  proportional  to  that  of  the  aerial  vibrations,  as  indicated 
by  the  membrane,  and  by  the  difficulty  or  facility  with  which  the 
vibrations  of  the  plate  may  be  maintained.     (See  Arts.  27,  31.) 

The  resonance  of  the  open  tube  is  scarcely  perceptible  when  the  length 
of  it  does   not  differ    much  from   something  less    than    an   even   multiple 

of  -,  or  2m  •  j  -  C  ;    but  as  it  approximates  to  something  less  than  an  odd 

multiple  of  that  quantity,  or  (2m'+  1)-  —  C,  the  resonance  increases,  and 

at  length  becomes  of  painful  intensity,  increasing  till  it  is  no  longer  possible 
to  maintain  the  same  mode  of  vibration  of  the  plate.  Whether  the  length 
of  the  tube  be  gradually  increased  or  diminished  in  approximating  to 
the  above-mentioned  lengths,  the  phenomena  are  precisely  the  same. 

I  was  the  better  pleased  to  obtain  this  result,  inasmuch  as  those 
which  I  first  obtained  (when  the  precaution  of  preventing  communication 
with  the  external  air  was  not  attended  to*),  as  well  as  those  of  previous 

*  In  such  cases  the  resonance  was  always  greatest  (as  in  the  case  considered  in  the  text) 
when  the  difficulty  of  making  the  plate  vibrate  was  greatest.  The  corresponding  lengths  of  the 
tube  may  be  seen  in  Art.  30. 


IN  CYLINDRICAL  TUBES.  267 

experimenters,  appeared  either  to  contradict  theory,  or  at  least  to  be 
altogether  anomalous.  According  to  our  common  notion  on  the  subject, 
an   open  tube  gives  the  strongest   resonance  when   its   length    is   nearly 

equal   to   an   even  multiple  of  7,  instead  of  an   odd  multiple,  as  above 

stated ;  and  Savart*  has  given  this  as  the  result  of  his  own  experiments 
for  tubes  of  about  the  same  diameter  as  those  I  have  usually  employedf ; 
but  asserting  also  that  the  length  is  less  as  the  diameter  is  increased, 
and  this  too  whether  the  disturbance  extend  over  the  whole  orifice  of 
the  tube  or  not.  My  results,  however,  are  entirely  at  variance  with 
this  latter  assertion,  for  I  confidently  conclude  from  them  that  if  the 
disturbance  extend  uniformly  and  equably  over  the  orifice  of  the  tube, 
the  phenomena  will  be  independent  of  its  diameter:]:,  with  the  exception 
of  the  effect  it  may  have  on  the  displacement  of  the  nodes  |.  If,  however, 
the  disturbance  extend  but  partially  over  the  orifice,  I  see  no  reason 
to  doubt  the  accuracy  of  the  last-mentioned  results  of  M.  Savart ;  and 
this  supposition  will  also  account  for  the  apparent  discrepancy  between 
his  results  and  mine  as  respects  the  length  of  the  open  tube  (of  which 
the  diameter  does  not  much  exceed  an  inch)  producing  the  greatest 
resonance;  for  it  is  manifest  that  with  this  partial  disturbance  none 
of  that  condensation  and  rarefaction  on  the  surface  of  the  plate  can 
take  place,  which  in  my  experiments  necessarily  attends,  and  may  be 
considered  as  causing,  that  powerful  resonance  of  which  I  have  spoken. 
It  is   easily   seen,   in  fact,  that  when   the  length  of  the   tube  is  neany 

equal  to  an  odd  multiple  of   -,   the  phase  of  the  waves  reflected  from 

any  considerable  part  of  the  orifice  not  occupied  by  the  vibrating  plate, 
will  be  directly  opposite  to  that  of  the  waves  propagated  by  the  plate 
itself;  and  that  thus  a  great  part  of  the  vibration  within  the  tube  will 
be  destroyed  by  interference. 

There  is  no   difficulty,  therefore,  in  explaining  the  non-existence  of 
resonance  in  this  case.     If  the  tube,  however,  be  lengthened  or  shortened 

by    about   - ,    (still   supposing   the   disturbance   at   its  mouth    partial),   a 

*  Annates  de  Chimie,  Tom.  XXIV.  p.  56.  t  See  Art.  S6. 

t  See  Art.  36,  p.  264.  §  Art.  36. 


268  Mr  HOPKINS  ON  AERIAL  VIBRATIONS 

resonance  will  be  heard,  though  extremely  feeble  as  compared  with 
that  I  have  found  in  my  experiments.  This  is,  in  fact,  the  kind  of 
resonance  which  has  been  observed  by  all  experimenters.  It  does  not 
appear  to  me  to  admit  of  the  same  obvious  explanation  which  the 
other  admits  of ;  that  which  is  usually  received  being,  as  I  conceive, 
in  itself  insufficient,  when  subjected  to  those  restrictions  which  must 
be  imposed  upon  it  by  the  general  laws  which  govern  the  communication 
of  motion  from  one  particle  of  matter  to  another.  At  present,  however, 
it  is  not  my  object  to  enter  on  the  discussion  of  this  and  of  some 
other  points  relative  to  this  part  of  the  subject.  It  is  sufficient  for 
me  now  to  have  shewn  that  that  powerful  resonance  which  I  have 
observed  in  my  experiments  is  exactly  accordant  with  the  results  of 
our  mathematical  investigations,  when  the  conditions  assumed  in  those 
investigations  are  fully  satisfied.  I  hope  to  return  to  the  careful  examina- 
tion of  other  cases  at  a  future  period. 

I  have  already  alluded*  to  a  paper  by  Mr  Willis,  published  in  the 
Transactions  of  this  Society,  in  which  he  has  described  some  experiments 
bearing  on  this  subject,  and  affiarding  a  general  corroboration  of  some 
of  the  results  above  stated.  He  fixed  a  reed  to  a  sliding  tube,  and 
observed  the  intensity  of  the  sound,  when  the  reed  was  made  to  speak, 
produced  by  different  lengths  of  the  tube,  and  by  means  of  a  microscope 
carefully  adjusted,  he  was  able  to  observe  the  excursions  of  the  reed 
in  its  vibration,  and  to  obtain  micrometer  admeasurements  of  them. 
He   thus   found  that   when   the   length  of  the  tube  equalled   about  an 

even  multiple  of  - ,  it  gave  the  exact  note  of  the  reed  with  no  perceptible 

resonance.     As  the  tube  was  gradually  lengthened,  the  tone  was  flattened, 

and  as   the  length   approximated   to  about   an  odd   multiple  of  -,   the 

extent  of  the  reed's  excursions  was  diminished,  its  vibrations  became 
irregular  and  convulsive,  till  at  length  it  ceased  to  produce  any  musical 
tone.  When  the  tube,  however,  was  a  little  lengthened  beyond  this 
point,  the  reed  suddenly  assumed  its  original  form  of  vibration, 
producing   a   note  of  painful   intensity,    similar    to   that    which   I    have 

*  See  page  260. 


IN  CYLINDRICAL  TUBES.  269 

described  in  my  own  experiments,  although  the  extent  of  excursion  of 
the  reed  was  in  this  case  less  than  in  that  in  which  no  resonance  was 
produced. 

One   discrepancy   is   observable   between    this    experiment   and   mine, 
inasmuch   as   the   intensity   of  the   sound,   instead  of   increasing   as    the 

length  of  the  tube  approximated  to  the  odd  multiple  of  - ,  as  in  my 

experiments,  gradually  decreased*.  The  explanation,  however,  of  this 
fact,  is  easily  found  in  the  diminished  excursion  of  the  reed,  and  still 
more,  I  suspect,  in  the  irregularity  of  its  vibration,  by  which  the 
undulations  produced  by  it  are  probably  rendered  imperfectly  sonorous^. 
With  this  explanation  of  this  apparent  discrepancy,  the  general  results 
of  Mr  Willis's  experiments  afford  as  strong  a  corroboration  of  those 
Avhich  I  have  obtained,  as  the  difference  between  our  modes  of  experi- 
menting will  allow.  The  flexibility  of  the  reed,  however,  and  its 
consequent  ready  obedience  to  the  vibrations  of  the  air,  as  compared 
with  the  inflexible  obstinacy  of  a  glass  plate,  together  with  the  partial 
disturbance  produced  by  the  reed,  render  it  a  totally  unfit  agent  in 
obtaining  experimental  tests  for  our  mathematical  results,  though  it 
presents  to  us  in  its  own  motions  many  interesting  points  of  enquiry. 


Our  theory  will  also  perfectly  account  for  one  of  the  most  striking 
phenomena  observable  in  wind  instruments,  viz.  the  rapidity  with  which 
different  states  of  vibration  are  assumed  within  the  tube,  corresponding 
to  different  effective  lengths  of  it,  as  determined  by  the  opening  or 
closing   of  the  finger  holes.      We  have   seen   (Art.  22,  VII.  VIII.)   that 

*  For  a  very  clear  and  distinct  account  of  these  experiments,  I  must  refer  the  reader  to  the 
excellent  paper  from  which  the  above  is  taken.  It  will  be  observed,  however,  that  the  results 
mentioned  in  the  text  were  not  the  direct  objects  of  Mr  Willis's  investigations,  but  were  such  as 
naturally  offered  themselves  in  the  course  of  his  experiments  on  the  production  of  the  vowel 
sounds. 

t  I  think  it  very  possible  that  \heform  of  the  aerial  vibrations  may  have  more  to  do  with  our 
sense  of  the  intensity  of  sound  than  has  been  generally  supposed ;  and  perhaps  some  cases  of 
resonance  may  admit  the  most  satisfactory  explanation  on  this  hypothesis. 

Vol,  V.     Part  XL  M  m 


27tt    Mb  HOPKINS  ON  AERIAL  VIBRATIONS  IN  CYLINDRICAL  TUBES. 

according  to  theory,  if  the  cause  maintaining  the  vibratory  motion  in 
a  tube  be  suddenly  changed,  (as  in  passing  from  one  note  to  another), 
the  effect  of  the  former  mode  of  disturbance  on  the  form  of  the 
succeeding  vibration  will  become  inappreciable  in  an  exceedingly  short 
period  of  time.  Now  in  the  most  rapid  musical  passages,  the  number 
of  notes  played  in  a  second  never  probably  exceeds  ten  or  twelve, 
and  these  usually  embrace  only  the  higher  notes  of  the  scale,  for 
which  there  must  be  many  hundred  vibrations  in  a  second.  Suppose 
this  number,  however,  not  greater  than  about  two  hundred ;  any  undula- 
tion transmitted  from  the  reed  or  embouchure  would  still  be  reflected 
about  twenty  times  at  the  open  end  in  the  interval  between  two 
consecutive  notes  in  the  most  rapid  musical  passage.  Now  assuming 
unity  to  represent  the  intensity  of  a  wave  incident  at  the  open  extremity 
of  the  instrument*,  let  1  — )3  represent  that  of  the  reflected  wave, 
(1  — /3)".  will  represent  (at  least  sufficiently  approximately)  its  intensity 
after  n  reflections ;  and  consequently,  as  we  have  no  reason  to  suppose  /3 
very  small  as  compared  with  unity,  it  is  probable  that  after  five  or  six 
reflections,  the  intensity  of  this  wave  will  be  quite  inappreciable.  Hence 
the  apparently  instantaneous  cessation  of  sound  after  the  exciting  cause 
has  ceased,  and  the  most  rapid  transition  from  one  note  to  another, 
are  perfectly  accordant  with  theory. 

M.  Poisson,  in  the  Memoir  referred  to  in  the  early  part  of  this 
paper,  has  also  investigated  the  vibratory  motion  of  air  in  two  tubes 
of  different  diameters  united  together  at  one  extremity.  I  hope  to 
examine  this  case  also  experimentally.  His  results  must  necessarily  be 
erroneous,  as  far  as  they  depend  on  the  physical  condition  he  has  assumed 
to  exist  at  the  extremity  of  the  open  tube,  and  which  I  have  shewn  to 
be  inconsistent  with  observed  phenomena  in  the  uniform  tube. 

*  See  Art.  14. 

W.  HOPKINS. 

St  Peter's  College, 
i\r««/  20,  1833. 


271 


XI.  On  the  Latitude  of  Cambridge  Observatory.  By  George  Biddell 
Airy,  M.A.  late  Fellow  of  Trinity  College,  Plumian  Professor 
of  Astronomy  and  Experimental  Philosophy,  and  one  of  the 
Flce-Presidetits  of  the  Society. 


[Read  April  14,  1834.] 


The  accurate  determination  of  the  latitude,  with  an  instrument 
like  the  Mural  Circle  now  in  use  at  the  Observatory,  seems  at  first 
sight  to  be  an  easy  business.  In  practice,  however,  it  is  not  without 
difficulties.  I  do  not  here  allude  to  the  correction  for  refraction ; 
since,  though  there  may  be  a  trifling  uncertainty  in  regard  to  its  magni- 
tude, it  is  easy  to  leave ,  a  result  subject  to  that  uncertainty,  and 
admitting  of  correction  without  any  trouble  whenever  a  correction  of 
the  refraction  shall  be  established.  Nor  do  I  allude  to  the  uncertainty 
in  the  corrections  by  which,  from  a  star's  apparent  place  on  any  day 
of  observation,  its  mean  place  at  a  fixed  epoch  can  be  inferred;  since 
the  uncertainty  about  any  of  these  is  far  less  than  the  smallest  quantity 
for  which  we  could  pretend  to  answer  in  fixing  the  latitude  of  any 
place ;  and  its  effects  being  periodical,  would  in  a  comparatively  short 
series  of  observations,  produce  no  sensible  effect.  The  difficulties  to 
which  I  allude  are  instrumental:  they  are  not  periodic  in  time,  like 
the  latter;  nor  do  they  admit  of  correction  from  posterior  researches, 
like  the  former  of  the  causes  of  uncertainty  which  I  have  mentioned ; 
they  are  moreover  such  as  would  scarcely  be  suspected  to  exist,  until 
their  effects  are  discovered  from  the  discordance  of  the  results  of 
observations. 

The  Mural  Circle  is  an  instrument  which  gives  simply  the  reading 
of  that  point  of  the  graduated  limb  which  is  opposite  to  an  imaginary 
fixed  index  when  the  telescope  is  pointed  to  the  object  of  observation. 

M  M2 


272  PROFESSOR  AIRY  ON  THE  LATITUDE 

A  single  observation  therefore  gives  us  no  tangible  result.  It  is 
necessary  to  have  one  other  observation,  or  a  series  of  observations, 
by  which  the  reading  of  that  point  of  the  limb  can  be  found  which 
is  opposite  to  the  same  index  when  the  telescope  is  directed  to  some 
point  of  reference;  then  the  difference  between  this  reading  and  the 
former  is  the  angular  distance  of  the  object  observed  from  the  point 
of  reference.  It  was  intended  originally  by  the  maker  that  this  point 
of  reference  should  be  the  celestial  pole.  In  practice,  however,  it  is 
found  necessary  to  descend  one  step  nearer  to  terrestrial  things,  and  to 
adopt  for  the  point  of  reference  the  zenith ;  a  point  which,  though  not 
marked  any  more  than  the  pole  by  any  obvious  phenomena,  can  yet 
be  discovered  by  a  process  which  involves  less  of  astronomical  assump- 
tions, and  requires  a  shorter  time  for  the  complete  determination. 

The  method  of  determining  the  zenith  point  from  observations 
by  reflexion  at  the  surface  of  mercury,  has  been  introduced  into 
observatories  almost  entirely  by  the  practice  of  the  present  Astronomer 
Royal  at  the  Greenwich  Observatory.  The  use  of  two  similar  circles 
(as  at  Greenwich)  makes  the  process  one  of  little  labour,  though  requiring 
the  co-operation  of  two  observers.  The  same  celestial  objects  being 
repeatedly  observed  by  direct  vision  with  both  circles,  the  differences 
of  the  corresponding  readings  of  the  two  circles  are  found ;  and  any 
observations  made  with  one  can  be  referred  to  the  other.  Then  when 
any  bright  star  passes  the  meridian,  one  circle  is  employed  in  observing 
it  by  direct  vision,  and  the  other  at  the  same  time  is  employed  in 
observing  it  by  reflexion  at  the  surface  of  mercury ;  the  reading  of  the 
latter  circle  is  referred  to  the  former  circle;  and  then  the  reading 
which  is  a  mean  between  the  reading  for  the  direct  observation  and 
the  referred  reading  for  the  reflected  observation,  is  the  reading  that 
corresponds  to  a  horizontal  position  of  the  telescope;  and  by  adding 
or  subtracting  a  quadrant,  the  reading  which  corresponds  to  a  zenithal 
position  of  the  telescope  is  obtained. 

With  a  single  circle  this  process  cannot  be  adopted.  In  some 
instances  it  has  been  imitated  by  observing  a  star  directly  on  one 
night,  and  observing  the  same  star  by  reflexion  on  another  night.     The 


OF  CAMBRIDGE  OBSERVATORY.  273 

calculation  for  the  zenith  point  then  relies  on  our  perfect  acquaintance 
with  the  variations  of  refraction  and  other  corrections  from  one  night 
to  another ;  and  thus  a  cause  of  inaccuracy  is  introduced,  which  does 
not  exist  in  the  other  method.  In  the  Cambridge  Observatory  a  different 
method  is  regularly  employed  (for  the  idea  of  which  I  am  indebted 
to  a  suggestion  of  Mr  Sheepshanks).  When  a  star  is  to  be  observed 
by  reflexion,  the  circle  is  set  approximately  for  the  reflected  observation, 
and  the  six  microscopes  are  read;  when  the  star  has  entered  the  field, 
and  before  it  has  reached  the  center,  it  is  bisected  by  the  micrometer 
wire,  (which  in  fact  measures  its  distance  from  the  fixed  wire,  and  thus 
gives  a  correction  to  be  applied  to  the  mean  of  the  six  microscopes,) 
and  then  there  is  ample  time  to  allow  the  circle  to  be  turned  to  the 
position  in  which  the  star  can  be  observed  directly,  shortly  after  it 
has  passed  the  center  of  the  field.  Thus  a  direct  and  reflected  observa- 
tion are  obtained  at  the  same  transit.  This  method  is,  in  my  opinion, 
much  preferable  to  the  second  that  I  have  mentioned,  and  in  some 
respects  superior  to  the  first. 

Either  of  the  methods  which  applies  to  one  circle  enables  us,  as 
will  shortly  be  seen,  to  examine  severely  into  the  consistency  of  the 
results  obtained  in  different  positions  of  the  circle ;  and  this  must  be 
considered  as  a  most  valuable  property  of  this  method  of  determining 
the  zenith  point,  and  one  which  places  it  far  above  the  use  of  a  collimator 
or  any  similar   instrument. 

I  had  hoped,  on  commencing  observations  with  the  Mural  Circle 
at  the  beginning  of  the  year  1833,  to  be  able  in  a  very  short  time  to 
obtain  a  very  approximate  latitude.  I  proposed  to  observe  some  stars 
every  night  in  the  manner  above  described,  as  well  as  circumpolar  stars 
(which  might  or  might  not  be  observed  in  the  mercury):  by  the  former 
I  should  obtain  a  very  good  zenith  point;  and  then  each  observation 
of  the  latter,  above  and  below  the  pole,  would  give  me  a  value  of  the 
co-latitude. 

But  after  a  few  nights'  observations,  I  found  that  the  reading  for 
the  zenith  point,  as   determined  by  different   stars,   was  not   the   same. 


274  PROFESSOR  AIRY  ON  THE  LATITUDE 

Had  the  discordance  been  wholly  without  regularity,  this  would  have 
given  me  no  anxiety.  But  the  first  Aveek's  observations  enabled  me  to 
see  with  certainty  that  one  general  rule  could  be  laid  down :  the  reading 
for  the  zenith  point  as  determined  by  northern  stars  was  invariably 
greater  than  that  fovmd  from  southern  stars.  As  the  readings  increase 
while  the  telescope  is  turned  towards  the  south,  this  discordance  is  of 
the  same  kind  as  that  which  would  be  produced  if  the  object  end  of 
the  telescope   dropped  by  its  own  weight. 

After  much  anxious  thought  and  many  fruitless  attempts  to  explain 
this  discordance,  I  was  obliged  to  give  it  up  entirely.  The  method 
which  was  adopted  for  approximate  reduction  of  the  observations,  easily 
admitting  of  future  correction,  was  the  following.  When  in  one  night, 
or  in  several  nights  which  it  appeared  practicable  to  group  together, 
stars  had  been  observed  by  reflexion  in  different  parts  of  the  meridian, 
1  took  the  three  means  of  zenith  points  determined  by  stars  far  north, 
by  stars  far  south,  and  by  stars  near  the  zenith,  as  three  separate  results ; 
and  then  I  took  the  mean  of  these  three  for  the  zenith  point.  For  an 
approximate  co-latitude  I  used  37°.  47'.  6",83. 

At  the  beginning  of  March  the  telescope  was  moved  about  thirty 
degrees  on  the  circle;  at  the  beginning  of  August  it  was  again  moved 
thirty  degrees,  and  on  this  occasion  (as  it  appeared  that  the  circle  was 
not  exactly  balanced)  a  pound  of  lead  was  attached  to  the  eye  end  of 
the  telescope ;  at  the  beginning  of  December  it  was  again  moved 
about  thirty  degrees.  It  does  not  appear  however  that  the  fact  of  the 
discordance  has  been  affected,  but  its  law  seems  to  have  been  in  some 
degree  altered. 

A  discordance  of  the  same  kind  exists,  I  believe,  in  every  circle 
that  has  been  properly  examined.  I  am  informed  by  Mr  Henderson 
(late  Cape  Astronomer)  that  he  has  found  it  in  the  Cape  Circle.  It 
was  recognized  as  existing  in  the  Greenwich  Circles :  and,  though  the 
system  of  observing  there,  which  I  have  described,  does  not  allow  us 
to  trace  the  unmixed  faults  of  either  circle,  yet  from  the  discordance 
in  the  places  of  stars  as  determined  by  the  two  circles,  and  its  variation 


OF  CAMBRIDGE  OBSERVATORY.  275 

in   different  points   of  the   meridian,    I   am   inclined   to   think   that   the 
defect  in  one  circle  is  different  from  that  in  the  other. 

In  vain  have  I  endeavoured  to  discover  the  cause  of  this  discordance. 
I  once  thought  that  it  might  be  owing  to  the  circumstance,  that  for 
the  reflection-observation  the  circle  is  at  rest  for  some  minutes  after 
the  microscopes  are  read,  and  possibly  it  might  (though  clamped)  have 
changed  its  position.  A  series  of  observations  expressly  made,  showed, 
however,  that  there  was  no  sensible  change  either  in  a  few  minutes 
or  in  many  hours.  I  thought  that  the  surface  of  the  mercury  might 
be  sensibly  curved,  and  that  from  a  habit  of  observing  in  one  part  of 
the  trough,  an  error  might  be  produced.  A  set  of  experiments  proved, 
however,  that  there  was  not  the  least  sensible  difference  in  the  results 
found  from  observing  at  one  or  the  other  end  of  the  trough.  A  flexure 
of  the  wire  in  the  field  of  view  would  not  explain  it,  as  the  discordance 
which  that  would  produce  is  of  the  opposite  kind.  There  appeared 
to  be  no  reason  for  supposing  an  error  in  the  determination  of  the 
coincidence  of  the  micrometer  wire  with  the  fixed  wire,  in  the  value 
of  the  micrometer  screw,  or  in  the  observation  with  the  micrometer 
wire.  The  object  glass,  repeatedly  examined  by  myself  and  once  by 
Mr  Simms,  did  not  appear  to  be  loose  in  its  cell.  I  am  driven  at  last 
to  the  supposition  that  the  circle  sensibly  changes  its  figure ;  but  I 
have  no  proof  of  this,  nor  do  I  see  distinctly  how  it  should  produce 
the  discordance  in  question.  Three  sets  of  readings  of  every  10°  under 
all  the  microscopes,  have  not  assisted  me  to  discover  such  change. 
My  a  priori  opinion  is,  that  a  change  in  figure  is  hardly  possible.  The 
telescope,  it  must  be  remembered,  is  attached  at  its  ends  to  the  limb 
of  the  circle :  the  limb  is  in  one  piece  (cast  in  several  pieces  and  burnt 
together) ;  and  the  whole  arrangement  of  parts  seems  admirably  adapted 
to  prevent  any  change.  If  I  had  to  fix  on  an  astronomical  instrument 
which  appeared  less  likely  to  change  than  any  other,  I  should  certainly 
choose  the  Mural  Circle. 

To  discover  experimentally  the  law  of  discordance,  I  proceeded 
as  follows.  The  observations  being  reduced,  and  those  of  each  star 
being    digested    under    the    heads   of    D,  R,    SP.  D.,    and    SP.  R.,    I 


276  PROFESSOR  AIRY  ON  THE  LATITUDE 

selected  for  the  three  first  positions  of  the  telescope  all  the  un- 
exceptionable corresponding  observations  D  and  R.  (The  stormy 
weather  of  December  made  it  impracticable  to  observe  low  stars  by 
reflexion).  In  each  case  of  a  double  observation,  the  difference  of 
the  results  D  and  R  would  be  double  the  difference  between  the 
zenith  point  as  found  from  that  star,  and  the  zenith  point  adopted 
in  the  reductions.  The  mean  of  the  differences  of  all  the  correspond- 
ing results  D  and  R,  would  therefore  be  double  the  mean  of  all 
the  differences  between  the  zenith  points  found  from  the  particular 
star,  and  the  zenith  points  found  from  all  by  a  tolerably  uniform 
system :  and  thus  it  might  be  considered  as  double  the  difference 
between  the  zenith  point  found  without  error  of  observation  from 
that  star,  and  a  certain  imaginary  well  defined  point.  These  values 
for  all  the  stars,  and  for  each  position  of  the  telescope,  were  arranged 
in  tables  (for  which,  as  well  as  for  some  other  numerical  values,  I 
must  refer  to  the  Cambridge  Observations,  Vol.  VI.) 

The  next  step  was,  to  connect  these,  approximately  at  least,  by  a 
law.  I  soon  found  that  to  attempt  this  by  calculation  was  almost  hope- 
less. Combinations  of  constants,  sin  Z.D.,  sin  Z.D.  cos^  Z.D.,  cos2Z.D., 
were  tried  in  vain.  I  therefore  adopted  a  graphical  method  similar 
to  that  used  by  Sir  John  Herschel,  in  the  reduction  of  his  sweeps, 
and  described  by  him  in  the  Phil.  Trans.  1833.  Taking  the  line  of 
abscissae  for  zenith  distances,  and  the  ordinates  to  represent  the  mean 
of  the  differences  above  mentioned,  I  made  a  curve  to  pass  among 
the  points  so  determined,  as  well  as  I  could,  giving  to  each  point 
an  importance  depending  on  the  number  of  observations.  From  this 
curve  I  measured  off"  the  ordinates  for  every  10°  of  zenith  distances; 
half  of  this  quantity  I  considered  to  be  the  correction  to  the  ob- 
served zenith  distance,  to  be  applied  with  different  signs  to  the 
direct  and  the  reflected  observation.  The  only  respect  in  which 
theoretical  consideration  may  be  said  to  have  assisted  me  is  the 
following.  Since  the  error  in  the  relation  between  the  position  of  the 
telescope  and  the  reading  of  the  circle,  to  which  the  discordance 
must  be  due,  is  periodical  and  never  infinite,  it  may  be  expressed  by 
sines   and  cosines  of  the   Z.  D.   and  its  multiples.      Now  it  is  useless 


OF  CAMBRIDGE  OBSERVATORY.  '  '  27T 

to  take  sines  of  even  multiples,  or  cosines  of  odd  multiples,  because 
when  180°  — Z.D.  is  substituted  for  Z.  D.,  the  result  is  equal  in 
magnitude  but  opposite  in  sign ;  and  therefore  when  the  two  are 
added  together,  (as  they  are  in  finding  the  zenith  point  from  each 
star),  no  trace  of  these  terms  would  remain.  Thus  there  may  be 
sensible  flexure  in  the  circle  which  cannot  be  discovered  from  ob- 
servation by  reflexion.  The  sines  of  odd  multiples,  and  the  cosines 
of  even  ones,  (all  which  may  be  expressed  in  finite  series  of  powers 
of  sin  Z.D.),  will  produce  the  same  values  with  the  same  signs  for 
180°  — Z.D.  as  for  Z.  D.,  and  these  will  affect  the  zenith  point. 
Thus  it  appears  that  the  terms  which  aff*ect  the  zenith  point  are 
the  same  for  a  direct  observation  and  for  the  corresponding  observation 
by  reflexion,  and  it  is  this  which  justifies  us  in  applying  half  the 
discordance  to  each.  It  appears  also  that  when  Z.  D.  =  90°,  the 
function  is  maximum  or  minimum,  and  hence  the  curve  in  the 
graphical  process  above  described  must  there  be  parallel  to  the  line  of 
abscissEB. 

The  tables  of  corrections  being  thus  formed,  I  now  considered 
myself  entitled  to  apply  them  to  the  reduced  r^ults  of  all  the 
observations,  whether  there  were  corresponding  observations  of  the 
opposite  kind  or  not.  '     >»/    ^    i    - 

The  principal  steps  of  the  succeeding  process  may  be  gathered 
from  the  subjoined  table.  The  first  column  contains  the  name  of 
the  star,  its  position  with  regard  to  the  pole,  (the  lower  transit  being 
marked  by  S.P.),  and  the  method  of  observing  it  (the  letters  D  and 
R  being  always  used  for  direct  and  reflected  vision).  Here  it  is  to 
be  observed  that  a  star  above  the  pole  and  the  same  star  below  the 
pole  are  reduced  as  separate  stars,  which  is  necessary,  because  the 
observations  have  been  reduced  with  an  assumed  co-latitude,  or  an 
assumed  place  of  the  pole,  the  error  in  which  assumption  can  be 
found  only  by  comparing  the  separate  results  for  the  same  star  above 
and  below.  The  second  column  contains  the  number  of  observations. 
The  third  contains  its  mean  N.P.D.  for  Jan.  1,  1833,  as  found  from 
the  mean  of  all  the  results  in  each  position  and  mode  of  observation. 
Vol.  V.     Part  II.  Nn 


278  PROFESSOR  AIRY  ON  THE  LATITUDE 

and  reduced  with  the  assumed  co-latitude  37° .  47' .  6,"83 :  those  de- 
termined from  the  lower  transits  of  the  star  have  the  negative  sign. 
For  refraction,  Bessel's  tables  are  used.  The  fourth  column  contains 
the  seconds  only,  as  corrected  for  the  errors  above  described ;  this 
has  been  done  by  taking  the  number  of  observations  in  each  position 
of  the  telescope  on  the  circle,  and  finding  the  mean  correction, 
supposing  that  to  each  observation  the  correction  proper  to  that 
position  was  applied.  The  negative  sign  has  still  been  retained  for 
the  lower  observations.  The  fifth  column  contains  the  whole  number 
of  observations  in  each  position  of  the  star :  and  the  sixth  contains 
the  mean  N.P.  D.  for  each  position,  as  inferred  from  the  combina- 
tion of  direct  and  reflected  observations.  The  seventh  contains  the 
whole  number  of  observations  for  both  positions.  The  eighth  contains 
the  algebraic  sum  of  the  two  determinations  of  N.P.D.,  as  the  star 
is  above  or  below  the  pole.  If  the  assumed  co-latitude  were  correct, 
this  sum  would  =  0 ;  if  the  assumed  co-latitude  be  increased  by  x, 
this  sum  would  be  increased  by  ^x,  and  therefore  to  make  it  now 
=  0,  X  must  be  taken  =  —  i  x  sum  in  8th  column.  The  results,  as 
might  be  expected,  are  however  different  for  different  stars,  though 
the  difference  is  much  smaller  than  I  could  almost  have  hoped ;  the 
extreme  difference  in  the  correction  of  latitude  being  1,"3,  and  this 
being  the  difference  between  two  results  from  stars  nearly  in  the 
same  parallel  (shewing  that  it  does  not  arise  from  error  in  the  cor- 
rections above  described),  and  which  had  been  not  much  observed. 
It  now  becomes  necessary  to  determine  how  the  relative  importance 
of  these  results  shall  be  estimated.  It  would  not  be  right  to  give 
a  value  proportionate  to  the  number  of  observations,  because  part  of 
the  discordance  may  be  produced  by  errors  of  division  and  other 
causes  which,  in  the  observations  of  a  single  star,  produce  constant 
errors.  The  ninth  column  contains  the  immbers  by  which  (from  my 
estimation  of  the  comparative  influence  of  constant  and  variable  errors) 
I  suppose  the  value  of  each  result  to  be  estimated.  The  tenth  con- 
tains the  product  of  the  corresponding  numbers  in  columns  8  and  9- 
The  sum  of  the  numbers  in  column  10  being  divided  by  the  sum  of 
those  in   column   9,   gives  +  2",82  for  the  double   correction,    or  + 1",41 


OF  CAMBRIDGE  OBSERVATORY.  279 

for  the  single  correction,  of  the  co-latitude ;  and  the  co-latitude  thus 
corrected  is  37° .  47' .  8",24,  or  the  latitude  52°.  12'.  51",76.  This  result 
I  conceive  to  be  correct  within  a  small  fraction  of  a  second.  The 
number  of  circumpolar  stars  used  for  this  determination  is  10,  and 
the  whole  number  of  observations  917. 

In  describing  the  process  by  which  I  have  arrived  at  the  above 
result,  it  has  been  my  wish  to  present  to  the  Society  not  only  a 
determination  possessing  considerable  local  interest,  but  also  an  account 
of  instrumental  anomalies  which  are  of  general  scientific  importance. 
In  further  illustration  of  the  latter  point  I  will  allude  to  the  dis- 
cordances in  the  determinations  of  the  obliquity  of  the  ecliptic.  It 
is  well  known  that  most  astronomers  have  found  the  obliquity  smaller 
from  observations  at  the  winter  solstice  than  from  those  at  the 
summer  solstice.  Now  if  I  had  used  only  the  latitude  found  from 
direct  observations  of  circumpolar  stars,  and  had  applied  no  correction 
to  the  observations  of  the  Sun,  I  should  also  have  found  two  values 
for  the  obliquity  discordant  by  about  5",  the  winter  obliquity  being 
the  smaller.  With  the  corrections  above  described,  (and  which  were 
formed  entirely  from  observations  of  stars,  and  before  I  had  even 
examined  my  sun  observations)  the  two  values  of  the  obliquity 
agree  within  1".  I  might  have  altered  the  corrections  so  as  to  re- 
move part  of  this  discordance,  but  I  prefer  leaving  them  in 
the  shape  in  which  they  were  given  by  independent  considerations. 
Indeed  if  I  had  confined  myself  to  the  January  observations  for  the 
winter  solstice,  and  omitted  those  of  December  when  the  correction  is 
less  certain,  the  discordance  would  wholly  have  disappeared.  A  very 
small  alteration  of  the  constant  of  refraction  (such  as  would  not  alter 
the  latitude  much  more  than  0",1),  or  a  very  small  alteration  in  the 
law  of  refraction  (which  would  not  be  sensible  in  the  latitude)  would 
remove  this  difference.  But  I  hardly  venture  to  assume  that  obser- 
vations of  the  Sun,  near  the  winter  solstice,  can  be  relied  on  to  this 
degree  of  accuracy. 

I  will  only  add,  in  conclusion,  that  I  believe  the  method  which 
I   have   used   to   be   the   only    one   of   those   in   practice   from   which   a 


280  PROFESSOR  AIRY  ON  THE  LATITUDE  OF  CAMBRIDGE  OBSERVATORY. 

good  result  can  be  obtained.  Had  I  determined  my  zenith  points 
by  a  floating  collimator,  the  result  of  observations  on  Polaris  and 
^  U.  Minoris  would  have  given  the  latitude  more  than  a  second 
wrong,  and  the  polar  distance  of  every  southern  body  more  than 
two  seconds  wrong :  the  result  of  observations  on  the  Sun  would  have 
given  nearly  the  same  error  in  the  latitude  but  with  the  opposite 
sign.  If  a  circle  reversible  round  a  vertical  axis  had  been  used 
(as  at  Dublin,  Palermo,  &;c.)  its  errors  would  (supposing  the  mere 
circle  exactly  as  good,)  have  been  just  as  great  as  if  a  collimator 
were  employed.  The  method  adopted  above  appears  most  valuable, 
not  only  because  it  gives  numerical  conclusions  more  accurate  than 
any  other,  but  also  because  it  enables  us  to  observe  discordances  and 
to  suspect  faults  which,  though  they  confused  our  results,  might 
otherwise  have  wholly  eluded  our  discovery. 


G.   B.   AIRY 


Obskkvatouy, 

March  23,  1834. 


Tratnsajclions  of  theCsanb.rhil.Soc-.Vol  VTl.  7. 


W MOicalfi^.  UefM^^CamJirui^e-. 


jTvnja-ciimus  ofOu.  Camji.   f/vu.  Jodefy,  Vol  V  Tt  S. 


Ft^.  I 


jri^    Z. 


J^r^.  3. 


Jkfetcalfl,  Ucho^'^  Cojnbnd^e'. 


281 

Table  exhibiting  the  Calculations  j^r  correcting  the  Latitude  o/"  Cambridge 
Obseuvatory  ;  the  Observations  having  been  reduced  with  the  assumed 
Latitude   52M2' .  53",17. 


star's  Name. 


No. 

of 

Obs. 


Polaris D 

Polaris R 

Polaris  S.P D 

Polaris  S.P R 

8  Urste  Minoris D 

S  Ursa;  Minoris R 

S  Urs»  Minoris  S.P.  ..D 
2  Ursae  Minoris  S.P.  ..R 

/3  Ursae  Minoris D 

/3  Ursae  Minoris R 

/3  Ursifi  Minoris  S.P.  ..D 
/3  Ursae  Minoris  S.P.  ..R 

/3Cephei D 

/3  Cephei R 

ySCephei  S.P D 

/3  Cephei  S.P R 

2  Draconis D 

I  Draconis R 

S  Draconis  S.P D 

S  Draconis  S.P R 

a  Draconis D 

a  Draconis R 

a  Draconis  S.P D 

a  Draconis  S.P R 

a  Ursae  Majoris D 

a  Ursae  Majoris R 

a  Ursae  Majoris  S.P.  ..D 
a  Ursae  Majoris  S.P.  ..R 


Uncorrected  mean 
N.P.D. 


Corrected 

for 

Discordance. 


«  Cephei D 

a  Cephei R 

a  Cephei  S.P D 

a  Cephei  S.P R 

S  Ursae  Majoris D 

S  Ursae  Majoris R 

0  Ursae  Majoris  S.P.  ..D 
g  Ursae  Majoris  S.P.  ..R 

a  Cassiopeiae D 

a  Cassiopeiae R 

o  Cassiopeiae  S.P D 

o  Cassiopeiae  S.P R 


113 
42 

111 
58 

43 
37 
39 
23 

20 
17 

22 

2 

4 

none 

12 

7 


3 

10 

6 

14 
11 

7 


32 

32 

5 

none 

43 

35 

13 

8 

26 

26 

3 

3 

34 

15 

26 

9 


1  .  34  .  52,22 
51,00 

1  .  34  .  53,77 
55,70 


-    3 


3.24.45,21 
43,48 
24  .  46,34 
48,42 


15.    9-41,65 

42,02 

■15.    9.44,58 

47,54 

20  .  10  .  15,91 

-  20  .  10  .  16,97 

17,24 

22  .  37  .  54,40 
53,95 

-  22  .  37  .  54,99 

56,31 

24  .  49  .  24,80 
24,51 

-  24  .  49  .  26,90 

28,90 

27  .  20  .  55,75 
55,73 

-  27  •  20  .  59,30 


28, 


■28. 


7.11,43 
11,64 

7 .  12,67 
13,47 


32.    2.18,12 

18,40 

■32.    2.20,38 

23,23 

34  .  22  .  45,38 

46,36 

-  34  .  22  .  47,37 

47,00 


No. 

of 

Obs. 


51,38 
51,78 

-  54,62 

-  54,83 

44,44 
44,24 

-  47,40 

-  47,28 

41,25 
42,40 

-  45,52 
-46,18 

15,74 

-  17,74 

-  16,52 

54,34 
54,01 

-  56,14 
-55,31 

24,76 

24,47 

-28,10 

-  27,90 

55,84 

55,64 

-59,86 


11,58 

11,50 

■13,27 

•  12,93 

18,41 

18,09 

■21,16 

-  22,45 

45,73 
46,03 

-  47,93 

-  46,54 


Concluded 
N.P.D. 


155 

169 

80 
62 

37 
24 

4 
19 

6 
16 

25 
10 

64 
5 

78 
21 


51,49 

-  54,70 

44,35 

-  47,35 

41,78 

-  45,57 

15,74 

-  17,29 

54,17 

-  55,83 

24,63 
•  28,04 

55,74 
■  59,86 

11,54 
-13,13 


No.       Algebraic 

of  Sum  of 

Obs.      Determin. 


52 

18,25 

6 

-21,80 

49 

45,82 

35 

-  47,57 

324 


142 


61 


23 


22 


35 


99 


58 


84 


•3,21 


-3,00 


■3,79 


-  1,55 


Weight 

of 
Result. 


1,66 


■3,41 


69    -  4,12 


1,59 


•3,55 


-1,75 


Product. 


-  16,05 


•9,00 


-7,58 


-1,55 


1,66 


-3,41 


•4,12 


-3,18 


-3,55 


-3,50 


Vol.  V.    Part  II. 


Oo 


TRANSACTIONS 


CAMBRIDGE 


PHILOSOPHICAL    SOCIETY. 


Vol.  V.     Part  III. 


CAMBRIDGE: 

PRINTED  BY  JOHN  SMITH,  PRINTER  TO  THE  UNIVERSITY: 

AND   SOLD    BY 

JOHN    WILLIAM    PARKER,    WEST    STRAND,    LONDON; 

J.  &  J.  J.   DEIGHTON,    AND  T.   STEVENSON, 

CAMBRIDGE. 


M.DCCC.XXXV. 


XII.  On  the  Diffraction  of  an  Ohject-glass  with  Circular  Aperture.  By 
George  Biddell  Airy,  A.M.  late  Fellow  of  Trinity  College, 
and  Plumian  Professor  of  Astronomy  and  Experimental  Philosophy 
in  the   University  of  Cambridge. 


[Read   Nov.    24,    1834.] 


The  investigation  of  the  form  and  brightness  of  the  rings  or  rays 
surrounding  the  image  of  a  star  as  seen  in  a  good  telescope,  when  a 
diaphragm  bounded  by  a  reetihnear  contour  is  placed  upon  the  object- 
glass,  though  sometimes  tedious  is  never  difficult.  The  expressions 
which  it  is  necessary  to  integrate  are  always  sines  and  cosines  of  mul- 
tiples of  the  independent  variable,  and  the  only  trouble  consists  in 
taking  properly  the  limits  of  integration.  Several  cases  of  this  problem 
have  been  completely  worked  out,  and  the  result,  in  every  instance, 
has  been  entirely  in  accordance  with  observation.  These  experiments, 
I  need  scarcely  remark,  have  seldom  been  made  except  by  those  whose 
immediate  object  was  to  illustrate  the  undulatory  theory  of  light. 
There  is  however  a  case  of  a  somewhat  different  kind;  which  in 
practice  recurs  perpetually,  and  which  in  theory  requires  for  its  com- 
plete investigation  the  value  of  a  more  difficult  integral ;  I  mean  the 
usual  case  of  an  object-glass  with  a  circular  aperture.  The  desire  of 
submitting  to  mathematical  investigation  every  optical  phaenomenon  of 
frequent  occurrence  has  induced  me  to  procure  the  computation  of  the 
numerical  values  of  the  integral  that  presents  itself  in  this  inquiry : 
and  I  now  beg  leave  to  lay  before  the  Society  tlie  calculated  table, 
with  a  few  remarks   upon  its  application. 

Let  a  be  the  radius  of  the   aperture  of  the  object-glass,  f  the  focal 
length,   h  the  lateral  distance   of  a  point  (in  the  plane  which  is  normal 
Vol.   V.     Part   III.  Pp 


284  PROFESSOR    AIRY,   ON   THE    DIFFRACTION   OF 

to  the  axis  of  the  telescope)  from  the  focus.  Then,  the  lens  being 
supposed  aplanatic,  and  a  plane  wave  of  light  being  supposed  incident, 
the  immediate  effect  of  the  lens  is  to  give  to  this  wave  a  spherical 
shape,  its  centre  being  the  focus  of  the  lens.  Every  small  portion  of 
the  wave,  as  limited  by  the  form  of  the  object-glass,  must  now  be 
supposed  to  be  the  origin  of  a  little  wave,  whose  intensity  is  propor- 
tional to  the  surface  of  that  small  portion ;  and  the  phases  of  all  these 
little  waves,  at  the  time  of  leaving  the  spherical  surface  above  alluded 
to,  must  be  the  same.  If  then  Sx  x  Sy  be  the  area  of  a  very  small  part 
of  the  object-glass,  q  the  distance  of  that  part  from  the  point  defined 
by  the  distance  b,  the  displacement  of  the  ether  at  that  point,  caused 
by   this   small   wave,   will   be   represented   by 

Sx  X.  Sy  X  sin—-  {vt  —  q  —  A) ; 

A 

and  the  whole  displacement  caused  by  the  small  waves  coming  from 
every   part   of  the   spherical  wave  will   be  the  integral  of 

sin  —  (vt—q  — A) 

through  the  whole  surface  of  the  object-glass,  q  being  expressed  in 
terms  of  the  co-ordinates  of  any  point  of  the  spherical  surface. 

Now  let  X  be  measured  from  the  center  of  the  lens  in  a  direction 
parallel  to  i;  y  perpendicular  to  x  and  also  perpendicular  to  the  axis 
of  the  telescope;  and  %  from  the  focus  parallel  to  the  axis  of  the 
telescope.      Then 

q=.^{{x-  by  +  y-  +  x}  =  -y/ix'  +f+x'-2bx) 

omitting  squares  and  superior  powers  of  b.  But  x^  +  y^  +  z'  —f^^ 
since  the  wave  is  part  of  a  sphere  whose  centre  is  the  focus ;  therefore, 

q  =  VW^-^bx)=f-j,x  nearly; 
and  the  quantity  to  be  integrated  is 

sm—  \vt  -  f  -  A  +  -x). 
^  J 


AN   OBJECT-GLASS  WITH   CIRCULAR  APERTURE.  285 

The  first  integration  with  regard  to  y  is  simple,  as  y  does  not 
enter  into  the  expression,  which  is  therefore  to  be  considered  as  con- 
stant. Putting  y,  and  y^  for  the  smallest  and  greatest  values  of  y 
corresponding   to   x,  the  first  integral  is 

{yi-yx)y-^m-^{vt-f-A^r-x). 

To  this  point  of  the  investigation  the  expressions  are  general,  including 
every  form  of  contour  of  the  object-glass. 

We  must  now  substitute  the  values  of  y^  and  y^  in  terms  of  x, 
before  integrating  with  regard   to   x.      For   a   circular   aperture 


y,  -  y.  —  ^y/a^-x" 

where  the  sign  of  the  radical  is  essentially  positive.  Hence  the  dis- 
placement of  the  ether  at  the  point  defined  .by  the  distance  A  is  re- 
presented by 

2  f,  Va' - x" .  sin  — {vt-f-  A  +  ^x) 
=  2sm -^{vt-/— A)  f^\/a^-af  .cos-— .^x 

\  Ay 

+  2cos  — -  {vt  —f—  A)  X  a/«^  —  x\sm—-.^x, 

A  ^      J 

and  the  limits  of  integration  are  from  x  =  —  a  to  x  =  +  a.  Between 
these   limits  it  is  evident   that 

;-     /-: «      .     2-ir    b 

f^Va'  — x^  .sm—- .  ^x  =  0, 

^    J 

(as  every  positive  value  is  destroyed  by  an  equal  negative  value) ;  and 
the  displacement  is  therefore  represented  by 

2sin— -(«^— /— ^)  ji\/«^  — ar'.cos  ^  .^x, 
^  ■  ,  ^    ./ 

the  integral  being  taken  between  the  limits  x=  -a,    x—  -^a. 

p  p2 


286  PROFESSOR   AIRY,    ON    THE   DIFFRACTION    OF 

If  we  make  -  =  w,    — —  .  -2r  =  n,   the  expression  becomes 
2a^.sm-—{vt-f~A)J^V^-uf'Cosnw,   fromw=— 1  tow=+\, 

A 

or    4«^  sin  — -  {vt-f~  A)  j„\/l  —  tt;''.  cos  nw,   irom  w  =  0  to  w  =  l. 

A 

It  does  not  appear,  so  far  as  I  am  aware,  that  the  value  of  this 
integral  can  be  exhibited  in  a  finite  form  either  for  general  or  for 
particular  values  of  w.     The  definite  integral 


J„^/\  —  vf .  cos  nw  (from  w=-0  to  w  =  \,) 

(which  will   be  a  function  of  7i  only)  being  expressed  by  N,  it  may  be 
shewn  that  N  satisfies  the  linear  differential  equation 

n '  dn         dv?         ' 

which   may   be   depressed   to   an  equation    of  the   first   order   that   does 
not  appear  to  yield  to  any  known  methods  of  solution. 

If  we  solve  the  equation  by  assuming  a  series  proceeding  by  powers 
of  n,  or  if  we  expand  cos  nw  and  integrate  each  term  separately,  we 
arrive  (by  either  method)  at  this  expression  for  the  integral 

TT  . rf_  n^  _         _    ""  Xr      \ 

4  ""  ^         2.4"^2.4^6       ^:^\Q'.S^^^-' 

The  table  appended  to  this  paper  contains  the  values  of  the  series 
in  the  bracket,  for  every  0,2  from  w=0  to  w  =  12.  Each  value  has 
been  calculated  separately,  the  logarithms  used  in  the  calculation  have 
been  systematically  checked,  and  the  whole  process  has  been  carefully 
examined.  The  calculations  were  carried  to  one  place  further  than  the 
numbers  here  exhibited.  I  believe  that  they  will  seldom  be  found  in 
error  more  than  a  unit  of  the  last  place;  except  perhaps  in  some  of 
the  last  values,  where  the  rapid  divergence  of  the  series  for  the  first 
five  or  six  terms  made  it  difficult  to  calculate  them  accurately  by 
logarithms. 


AN   OBJECT-GLASS  WITH   CIRCULAR  APERTURE.        .  287 

In   the   use  of  tins   table  n  must  be  taken  =  — -.-^.     If  instead   of 

using  the  linear  distance  h  to  define  the  point  of  the  field  at  which 
we  wish  to  ascertain   the  illumination,  we  use  the  number  of  seconds  *, 

then  A  =  /. *.sin  1",  and  n  must  be  taken  =  —  as  sin  1".     If  \  be  taken 

for  mean  rays  =  0,000022  inch,  n  must  be  taken  =  1,3846  x  as,  a  being 
expressed  in  inches.  From  this  expression,  and  from  the  numbers  of 
the  table,  we  draw  the  following  inferences. 

1.  The  image  of  a  star  will  not  be  a  point  but  a  bright  circle 
surrounded  by  a  series  of  bright  rings.  The  angular  diameters  of  these 
(or  the  value  of  s  corresponding  to  a  given  value  of  n)  will  depend 
on  nothing  but  the  aperture  of  the  telescope,  and  will  be  inversely  as 
the  aperture. 

2.  The  intensity  of  the  light  being  expressed  (on  the  principles 
of  the  undulatory  theory)  by  the  square  of  the  coefficient  of 

sin-^ivt-f-  A), 

and  the  intensity  at  the  center  of  the  circle  being  taken  as  the  standard, 
it   appears   that   the  central   spot  has  lost  half  its   light  when  «  =  l,6l6, 

I  17 
or  s  =  — — ;    that  there  is  total  privation  of  light,  or  a  black  ring,  when 

2  76 
n  =  3,832,   or  *  =  — — ;    that  the  brightest  part  of  the  first  bright   ring 
a 

Q    WQ  -I 

corresponds  to  w  =  5,12,  or  *  =  — — ,  and  that  its  intensity  is  about  —  of 

a  Oi 

5  16 
that  at  the  center;  that  there  is  a  black  ring  when  n  =  7,14,  or  s=  -- — ; 

a 

that  the  brightest  part  of  the  second  bright  ring  corresponds  to  ra  =  8,43, 

or  *  =  — — ,   and   that  its   intensity   is  about  — r  of  that  of  the  center ; 

7  32 
that  there  is  a  black  ring  when  w  =10,17,  or  *=  — — ;   that  the  brightest 


288  PROFESSOR    AIRY,    ON   THE    DIFFRACTION   OF 

part  of  the  third  bright  ring  corresponds  to  w  =  11,63,  or  *=  — — ,  and 
that  its  intensity  is  about  ^— -  of  that  of  the  center. 

The  rapid  decrease  of  light  in  the  successive  rings  will  sufficiently 
explain  the  visibility  of  two  or  three  rings  with  a  very  bright  star 
and  the  non-visibility  of  rings  with  a  faint  star.  The  difference  of 
the  diameters  of  the  central  spots  (or  spurious  disks)  of  different  stars 
(which  has  presented  a  difficulty  to  writers  on  Optics)  is  also  fully 
explained.  Thus  the  radius  of  the  spurious  disk  of  a  faint  star,  where 
light   of   less   than    half    the   intensity    of   the   central    light    makes   no 

1  17 

impression  on  the  eye,   is   determined  by  making  /*  =  1,616,   or  s=— — : 

whereas   the  radius  of  the   spurious  disk  of  a  bright   star,   where   light 
of  —  the   intensity   of   the  central   light    is   sensible,   is    determined   by 

1  97 
making  n  =  2,73,   or  *  =  — — . 

The  general  agreement  of  these  results  with  observation  is  very 
satisfactory.  It  is  not  easy  to  obtain  measures  of  the  rings;  since 
when  a  is  made  small  enough  to  render  them  very  distinct  as  to  form 
and  separation,  the  intensity  of  their  light  (which  varies  as  a^)  is  so 
feeble  that  they  will  not  bear  sufficient  illumination  for  the  use  of 
a  micrometer.  Fraunhofer  however  obtained  measures  agreeing  pretty 
well  (as  to  proportion  of  diameters,  &c.)  with  the  results  above. 

For  verification  of  the  numbers  it  would  probably  be  best  to  use 
an  elliptic  aperture.  By  an  investigation  of  exactly  the  same  kind  as 
that  above  it  will  be  found  that  the  rings  will  then  be  ellipses  exactly 
similar  to  the  ellipse  of  the  aperture,  but  in  a  transverse  position ;  that 
the  major  axes  of  the  rings  for  the  elliptic  aperture  will  be  the  same 
as  the  diameters  of  the  rings  for  a  circular  aperture  whose  diameter 
—  minor  axis  of  ellipse  of  aperture,  but  that  the  intensity  will  be 
greater  in  the  proportion  of  the  squares  of  the  axes.  I  have  not  yet 
had  an   opportunity  of  examining  this  in  practice. 


AN   OBJECT-GLASS   WITH   CIRCULAR  APERTURE.  289 

I  shall  now  apply  the  numbers  of  the  table  to  the  solution  of 
the  following  problem.  To  find  the  diameters,  &c.  of  the  rings  when 
a  circular  patch,  whose  diameter  is  half  the  diameter  of  the  object- 
glass,  is  applied  to  its  center,  so  as  to  leave  an  annular  aperture. 

The  radius  of  the  patch  being  -,  it  is  easily  seen  that  the  dis- 
placement  (using  the  same  notation)  is 

2sm-—-(vt—J'—A)fr\/a^-x'.cos—-.^a;  (from  a;—-a  to  x=+a) 
-  2sin  ~(vt-f-A)J\/---af.  cos-^  .^x  (from  x=  --  to  x=  +^. 
Putting  -  =w,  —  =  u,  this  becomes 

4a^ .  sin  -T-{vt  —f  —  A)  /„ \/l  —  vf  .  cos  —  .—^w 

A  A       / 

-4.^.sm  yC^^-/- ^)/«vl-M'.cos— .— .M, 

the  limits  of  integration  both  for  w  and  for  u  being  0  and  1.  Omitting 
the  factor  oV,  the  intensity  will  be  expressed  by 

V(»)-i*(l)}". 

where  (p{n)   is  the  number  given  in  the  table. 

Upon   forming   the   numerical  values   we   find   that   the   black   rings 
correspond  to  values  of  w=3,15,    7,18,    10,97:   and  that  the  intensities 

of  the  bright  rings  (in  terms  of  the  intensity  of  the  center)  are  — ,   — . 

Thus  the  magnitade  of  the  central  spot  is  diminished,  and  the  bright- 
ness of  the  rings  increased,  by  covering  the  central  part  of  the  object- 
glass. 

In  like  manner,  if  the  diameter  of  the  circular  patch  =  a  ( 1  — />),  the 
intensity  of  light  would  be  proportional  to  {<p  {n)  —  {l— pf  .^{n—pn)}". 


MiO  PROFESSOR   AIRY,   ON   THE    DIFFRACTION   OF 

The  quantity  under  the  bracket,  if  p  is  very  small,  is  equal  to 

X)       ft 

2p  .<p  {fi)  + pn<p' (n)  =  - . -J— {n^<p{n)}. 

In  the  case  of  a  very  narrow  annulus  therefore  the  diameters  of  the 
black  rings  will  be  determined  by  making  ?i^(p  (»)  maximum  or 
minimum.  It  appears  then  that  there  ought  to  be  only  one  black 
ring  corresponding  to  each  black  ring  with  the  full  aperture,  and  that 
its  diameter  ought  to  be  somewhat  smaller.  This  conclusion  does  not 
agree  with  the  experiments  recorded  by  Sir  J.  Herschel,  in  the  Encyc. 
Metrop.  Article  Light,  page  488 :  but  it  is  acknowledged  there  that 
the  results  are  discordant  with  Fraunhofer's :  and  I  am  inclined  there- 
fore to  attribvite  the  phasnomena  observed  by  Sir  J.  Herschel  to  some 
other  cause. 

The  investigation  of  cases  of  diffraction  similar  to  that  discussed 
here  appears  to  me  a  matter  of  great  interest  to  those  who  are 
occupied  with  the  examination  of  theories  of  light.  The  assumption 
of  transversal  vibrations  is  not  necessary  here  as  for  the  explanation 
of  the  phasnomena  of  polarization :  and  they  therefore  offer  no  argu- 
ments for  the  support  of  that  principle.  But  they  require  absolutely 
the  supposition  of  almost  unlimited  divergence  of  the  waves  coming 
not  merely  from  a  small  aperture,  but  also  from  every  point  of  a  large 
wave :  and  the  results  to  which  they  lead  us,  shew  strikingly  how 
small  foundation  there  was  for  the  original  objection  to  the  undulatory 
theory   of    light,    viz.    that    if    waves    spread    equally    in    all    directions. 


there  could  be  no  such  thing  as  darkness. 


Obsbrvatory,  Cambridge, 

November  20,  1834. 


G.   B.   AIRY. 


AN  OBJECT-GLASS  WITH  CIRCULAR  APERTURE. 


291 


4j  r  • 

Table  of  the  values  of  0(w)  =  —  f^vl  —  vf.co&  nw  from  w  =  0  iow=\. 


11 

</>(«) 

n 

0(«) 

0,0 

+  1,0000 

6,0 

-  0,0922 

0,2 

+  ,9950 

6,2 

-  ,0751 

0,4 

+  ,9801 

6,4 

-  ,0568 

0,6 

+  ,9557 

6,6 

-  ,0379 

0,8 

+  ,9221 

6,8 

-  ,0192 

1,0 

+  ,8801 

7,0 

-  ,0013 

1,2 

+  ,8305 

7,2 

+  ,0151 

1,4 

+  ,7742 

7,4 

+  ,0296 

1,6 

+  ,7124 

7,6 

+  ,0419 

1,8 

+  ,6461 

7,8 

+  ,0516 

2,0 

+  ,5767 

8,0 

+  ,0587 

2,2 

+  ,6054 

8,2 

+  ,0629 

2,4 

+  ,4335 

8,4 

+  ,0645 

2,6 

+  ,3622 

8,6 

+  ,0634 

2,8 

+  ,2927 

8,8 

+  ,0600 

3,0 

+  ,2261 

9,0 

+  ,0545 

3,2 

+  ,1633 

9,2 

+  ,0473 

3,4 

+  ,1054 

9,4 

+  ,0387 

3,6 

+  ,0530 

9,6 

+  ,0291 

3,8 

+  ,0067 

9,8 

+  ,0190 

4,0 

-  ,0330 

10,0 

+  ,0087 

4,2 

-  ,0660 

10,2 

-  ,0013 

4,4 

-  ,0922 

10,4 

-  ,0107 

4,6 

-  ,1116 

10,6 

-  ,0191 

4,8 

-  ,1244 

10,8 

-  ,0263 

5,0 

-  ,1310 

11,0 

-  ,0321 

5,2 

-  ,1320 

11,2 

-  ,0364 

5,4 

-  ,1279 

11,4 

-  ,0390 

6,6 

-  ,1194 

11,6 

-  ,0400 

6,8 

-  ,1073 

11,8 

-  ,0394 

6,0 

-  ,0922 

12,0 

-  ,0372 

Vol.  V.    Paet  III. 


Qa 


»  J  »  0'  { 


^9 


•  JI.JL.A       4  4-  ^li 


XIII.  On  the  Equilibrium  of  the  Arch.  By  the  Rev.  Henry  Moseley, 
B.A.  of  St  John's  College;  Professor  of  Natural  Philosophy  and 
Astronomy  in  King's  College,  London. 


[Read  Dec.  9,  1833.] 

1.  Let  a  mass  acted  upon  by  forces  applied  to  any  number  of 
points  in  it  be  imagined  to  be  intersected  by  an  infinite  number  of  planes, 
dividing  it  into  exceedingly  small  laminse.  Suppose  the  direction  of  the 
resultant  of  the  forces  acting  upon  one  of  these,  having  for  its  ex- 
ternal face  a  portion  of  the  surface  of  the  body,  to  be  determined. 
Combining  this  force  with  those  acting  upon  the  different  points  of 
the  next,  contiguous  lamina;  let  their  common  resultant  be  ascertained. 
Proceed  similarly  with  the  next,  and  with  each  succeeding  lamina. 

These  lines  will  then  be  the  tangents  to  a  curved  line,  called  in 
the  following  paper  the  line  of  pressure,  whose  intersection  with  each 
lamina,  marks  the  point  where  a  single  force  might  be  applied  so  as 
to  produce  the  same  effect  with  all  those  impressed  upon  that  lamina, 
this  single  force  being  impressed  in  the  direction  of  a  tangent  to  the 
curve. 

If  any  of  these  imaginary  intersecting  planes  be  supposed  to  become 
real  sections  of  the  mass,  so  as  to  separate  it  into  distinct  parts,  the 
conditions  necessary  that  no  one  of  these  parts  may  slip  or  turn  over 
on  those  contiguous  to  it,  will  manifestly  be  determined  by  the  direc- 
tion of  the  line  of  pressure  in  reference  to  the  plane  of  the  section. 

In  general  it  will  be  observed  that  forces  applied  to  a  system  of 
variable  form  are,  when  in  equilibrium,  subject  to  the  same  conditions 
as  though  its  form  were  invariable,  together  with  certain  other  conditions, 
dependant  upon  the  nature  of  the  variation  to  which  the  form  of  the 
system  is  liable.  In  other  words  the  conditions  of  the  equilibrium  of 
a  system  of  invariable  form  are  necessary  to  the  equilibrium  of  a  system 
of  variable  form ;   but  they  are  not  sufficient.     We  shall  first  determine 

QQ2 


294  Mr  MOSELEY,   ON  THE 

the  form  and  position  of  the  line  of  pressure  on  the  hypothesis,  that 
the  form  of  the  system  is  invariable,  and  then  consider  the  modifica- 
tion  to   which   these   are  subjected   by   the  opposite  hypothesis. 

2.     Let  there  be  conceived  a  mass,  the  connexion  between  the  parts 

of  which   may  be   any    whatever,   and   the   nature  of   whose   surface  is 

determined   by   the  equation 

^  xy%  =  0. 

Let  it  be  intersected  by  an  imaginary  plane  whose  position  in  reference 
to  a  given  system  of  rectangular  co-ordinates  is  determined  by  the  arbi- 
trary constants  A,  B,  C,  and  whose  equation  is 

z  =  Ax  +  By  +  C (1). 

Let  Ml,  Mi,  Ms  represent  the  sums  of  the  forces  acting  upon  one 
of  the  parts  into  which  the  mass  is  divided  by  the  intersecting  plane, 
resolved  in  directions  parallel  to  the  axes  of  x,  y,  z,  respectively.  Also 
let  JVx,  N'z,  ^3  be  the  moments  of  these  forces  about  the  same  axes. 
Then  Mi,  Mi,  Ms;  Ni,  N^,  iVs  are  given  in  terms  of  the  arbitrary 
constants  A,  B,  C — of  the  given  forces — and  of  the  constants  involved 
in  the   given   equation  to  the   surface   of  the   mass. 

Let  the  position  of  the  intersecting  plane  be  supposed  to  be  such, 
that  the  forces  acting  upon  the  above  mentioned  portion  of  the  mass 
may  have  a  single  resultant,  an  hypothesis  which  involves  the  known 
condition 

MiN,  +  M,N,  +  M,N,  =  0 (2). 

The  equations  to  the  resultant  in  any  given  position  of  the  inter- 
secting  plane,  are 

Ml      ^Ni 

^'=M^-'Ms 

Let  the  arbitrary  constant  C  be  eliminated  from  this  equation,  and 
from  the  equation  to  the  intersecting  plane  by  means  of  equation  (2) ; 
and  let  the  plane  be  then  supposed  to  take  up  a  series  of  positions, 
the  law  of  which  is  fixed  by  its  equation,  and  of  which,  each  is  im- 
mediately adjacent  to  the  former. 


EQUILIBRIUM  OF   THE  ARCH. 


295 


Further,  let  it  be  supposed  that  the  resultant  of  the  forces  upon 
the  portion  of  the  mass,  cut  off  by  the  plane,  in  each  of  its  positions, 
intersects  with  the  resultant  similarly  taken  in  its  immediately  previovis 
position — an  hypothesis  which  introduces  a  new  condition  into  the 
question  and  establishes  a  second  relation  between  the  quantities 
M„   M„  M,;   A,   B,    C. 

That  relation  is  determined  as  follows. 


Since  x,  y,  as  are  to  be  considered  as  the  co-ordinates  of  a  point  of 
intersection  of  two  consecutive  resultants;  we  may  differentiate  the 
equations  (3)  with  respect  to  the  arbitrary  constants  A  and  B,  consi- 
dering X  and  y  as  constant.  From  this  differentiation,  the  following 
equations   are   obtained: 


0  =  ss 


0  =  » 


KS../(t).J,KSL/_i)., 


dA 


■  dAv 


dB 


.dB 


+ 


dA 


dA 


dB 


dA+-^dB{ 
dB 


■(4), 


whence,  eliminating  z 


^M^^A^-^dB 


dA 


dB 


dA  +  r^ft— aJ?  I 


dA 


dB 


dA 


d 


dA  + 


N, 
M, 


dB 


dA 


dB 


dBi 


^^^^clA+-^dB{ 


=  0...(5). 


This  last  equation  determines  the  relation  between  A  and  B  ne- 
cessary to  the  continual  intersection  of  the  consecutive  resultants ;  and 
the  elimination  of  these  quantities  between  equations  (3)  and  (4), 
produces  two  equations  in  x,  y,  %  which  are  those  to  the  locus  of 
that    intersection.      That   is,    they   are   the   equations    to   the   line    of 

PRESSURE. 


296  Mr  MOSELEY,   ON  THE 

3.  By  the  elimination  of  A,y  B  and  C  between  the  equations  (2),  (3) 
and  (5),  a  relation  is  obtained  between  the  co-ordinates  of  a  point  in 
the  direction  of  the  resultant  force,  applicable  to  every  position  of  the 
intersecting  plane.  Being  in  fact,  the  equation  to  that  developable 
surface  which  is  the  locus  of  the  resultants,  and,  which  has  for  its 
edge  of  regression,  the  line  of  pressure.  This  surface  will  be  properly 
called  the  surface  of  pressure. 

It  is  evident  that  at  that  point  where  the  line  of  pressure  even- 
tually cuts  the  surface  of  the  mass,  there  must  be  applied  a  force  equal 
to  the  resultant  of  all  the  other  forces  impressed  upon  the  system 
and  in  the  direction  of  a  tangent  to  the  line  of  pressure  at  that  point, 
or  there  must  be  applied  to  the  surface  of  the  last  lamina  cut  off 
by  the  intersecting  plane,  forces  whose  resultant  is  of  that  magnitude 
and  in  that  direction. 

4.  These  conditions  may  be  expressed  as  follows. 

Let  P'  be  the  force — or  the  resultant  of  the  forces — applied  to  the 
last  lamina,  x^,  y,,  ss,  the  co-ordinates  of  the  intersection  of  the  line  of 
pressure  with  it,   a,  fi,  y  the  inclinations  of  P'  to  the  axps  of  x,  y,  %. 
^  Also  let 

be  the  equations  to  the  line  of  pressure. 

Since  the  point  Xx,  y^,   %i   is  a  point  in  the  surface  of  the  mass, 

.-.     ^Xiy.z,  =  0. 

Also,  since  it  is  a  point  in  the  line  of  pressure, 

.-.     Xi  =  Fi%i 
y,  =  F^%x 

Since  the  direction  of  P'  is  that  of  a  tangent  to  the  line  of 
pressure, 


tan  a  = 

tan  /3  = 


d%,   ' 
d%i 


EQUILIBRIUM   OF   THE   ARCH.  297 

Also 

p  =  VM^VMr+~m, 

where  M^,   M.^,   M-j    are   supposed   to    be  taken   throughout    the   tvhole 
mass. 

Thus  there  are  six  equations  of  condition,  which  together  with  the 
equation 

cos^  a  +  cos^  /3  +  cos^  7  =  1. 
determine  the  seven  quantities  P',  Xi,  y^,   %i\    a,  /3,   7  in  terms  of  the 
forces    (other   than  P')   which    compose   the    system,   and   the   constants 
which  enter  its  equation.     These  fix  the  relations  necessary  to  the  equi- 
librium of  the  mass  considered  as  one  continued  geometrical  solid. 

Before  proceeding  to  the  discussion  of  the  additional  conditions 
requisite  to  the  equilibrium  when  the  mass  passes  from  the  invariable 
form  here  supposed,  to  a  variable  form,  it  will  be  well  to  give  an 
example  of  the  application  of  the  principles  which  have  been  already 
laid  down  to  the  actual  determination  of  the  line  of  pressure  in  a  par- 
ticular instance. 

5.  Let  then  ABCD  (fig.  1.)  represent  a  heavy  mass,  bounded  at  its 
extremities  by  parallel  planes  AB  and  CD,  and  laterally,  by  the  planes 
AC  and  BD  inclined  at  any  angle  to  one  another. 

Let  the  mass  be  imagined  to  be  intersected  by  an  infinite  number 
of  planes  parallel  to  AB,  of  which  one  is  mn,  and  to  be  supported 
by  forces  acting  at  p  and  p'  at  angles  cp  and  <f>'  with  the  horizon. 

It  is  required  under  these  circumstances  to  determine  the  form  and 
position   of  the   line   of  pressure. 

Let  the  line  P'G  bisect  AB  and  CD.  Draw  P'E  horizontal  and 
PM  vertical. 

Let      P'M=  A,            CD  =  2b,  P'p  =  k, 
AB  =  2a,          P'G  =  h,  Gp'  =  k'. 
Inclination  of  P'G  to  the  horizon  =7,  - 
AB =  /3. 


298  Vi    Mil  MOSELEY,   ON  THE 

Now, 

BP' -  Pm  _  BP  -  DG 
PP'       ~        GP'       ' 

2a  —  (mn)  _  2  (a  —  i)  _ 
'  ■      ^sec7     ~"         ^        ' 

/      V       «         2  («  -  A)  . 

.'.  (mn)  =  2a —r — -  secy  .  A; 

.-.  area  (BAnm)  =  ^  {{AB)  +  {mn)\  .  (P'P) .  sin  (PP'A) 

=  sin  {(3 +y)  secy  {2a  A- ^-^  .secy  .A"}; 

d\aYea(BAnm)}       -    /o  ,     \          ia        2(a  —  b) 
.'.  -^ V^ ^  =  8111  (/3  +  7)  sec 7  {2a ^  ,      ^sec7  .  A\. 

-  Now   each   element  of  the  area  has  its  centre  of  gravity  in   P'G ; 
,-,  moment  of  area  =  2^sin  (/3  +  7)  sec 7 ^{a^ ^— sec7^'} 

=^sin(/3  +  7)sec7{«^* ^"7    ^sec7^H. 

Now, 
iVs  =  moment  of  p  +  moment  of  area  (BAmn) 

2  (a  —  b) 
=  PffK  sin  (0  +  /3)  +  ^sin (/3  +  7)  sec7  {aA^ ^    ,    '  sec 7 A""). 

Also, 
Mx  —  pg  cos  (p,  Mz  =  0, 

M^  =pgsm(j>  -  g sin  (/3  +  7)  sec 7  {2a^ ^  sec  7 ^^}, 

iv,  =  0,  iV3  =  o. 

Calling  therefore  x  and  »  the  co-ordinates  of  any  point  in  the  re- 
sultant of  the  forces  applied  to  the  area  (ABmn),  we  have  for  the 
equation  to  that  resultant. 


EQUILIBRIUM  OF   THE  ARCH.  299 

or,  zpgcoscp  -  pgsivKp  +  xgsm{(i  +  7)sec7{2a^ -r—secy.A^} 

=pgK  sin  ((p  +  /3)  +gsm{fi  +  7)8607  {aA^ ^-— r — ^sec7  .  A^}. 

Differentiating  which  equation  with  regard  to  the  arbitrar}'  constant 
A,  we  obtain 

A  =  x, 

whence   by  elimination  and  reduction, 

^^l(^-  ^]   (seey.sm{f3  +  y)\    ^ 

^  \   pk    J    \  COS(p  j  ' 

_   /_a\   [sec  7.  sin  (^  +  7)]    ^ 
\pj  \  cos(p  /■ 

+  tan  (f) .  X 

_^     sin((/>  +  /3) 

COS0 

The  above  is  the   equation    to  the  line  of  pressure.     It  indicates   a 
point  of   contranj  flexure  corresponding   to 

ah 

X  =■ 7  cos  <h. 

a  —  h        ^ 

The  curve  is  concave  to  the  axis  of  x,  between  the  origin  and  this 
point.      It  is  afterwards  continually  convex. 

A  minimum  value  of  %  is  determined  by  the  equation 

\Jta\   f  /         ip\^   sin>cos^7) 

^      \a-AV^   ^  ^       Ui   •sinMi8  +  7)r         * 

It  will  be  observed  that  since  all  the  forces  applied  to  the  system 
may  be  supposed  to  act  in  the  same  plane,  the  two  conditions. 

First,    "  That   in   every  position  of  the  intersecting  plane,  the  forces 
shall  have  a  single  resultant,"   and  Secondly,    "That  the  consecutive  re- 
sultants shall  intersect,"  are  necessarily  satisfied. 
Vol.  V.    Paet  III.  Re 


300  Mr  MOSELEY,   ON   THE 

To  simplify  the  question,  let  the  planes  AC,  BD  which  bound  the 
mass  laterally  be  supposed  to  be  parallel,  the  figure  ABCD  assuming 
the  form  of  a  rectangle.     Fig.  6. 

This  hypothesis  will  introduce  the   following  conditions : 

«  =  *'     ^  =  J  -  7- 

Hence,  by  substitution  the  equation  to  the  line  of  pressure  becomes 

%  — .  sec  7 .  sec  0 .  a;^ 

+  tan  ^  .X 

cos  (7  -  0) 
cos  0 
Avhich  may  be  put  under  the  form 

,  p    .     ^  1 ,      P  ,     (    cos  (y  —  (j))        P   sin^  0  cos  7       -, 

^x  —  -~  sm  0  cos^r  =  -  cos  7  .cos  0  .  \k '  +  -7 ^-— ; — -  —  *|. 

*         2a        ^         "a         '  ^    ^        COS0  4«       cos  0  ' 

It   is   manifest    therefore,    that   the   line  of  pressure   is   in  this   case 

a  parabola — having   its  axis  vertical  and   at   a  distance  =  —-  sin  0  cos  7 

^  a 

from  the  origin — having  its  concavity  downwards — its  vertex  at  a  height 

_     cos  (7  —  0)       p    sin'  0  cos  7 
~         cos  0  4 «       cos  0      ' 

above  the  axis  of  x — and  having  for  its  parameter  the  quantity 
•  (^j  .  cos  0  cos  7. 

Let  us  now  seek  to  determine  what  relation  must  exist  between 
the  forces  impressed  upon  the  mass  which  we  have  hitherto  considered 
of  invariable  form,  that  the  equilibrium,  may  continue  under  the  same 
circumstances  when  its  form  and  dimensions  are  made  to  admit  of 
variation.     And  let  us  suppose 


EQUILIBRIUM   OF   THE  ARCH.  SOI 

First.  That  certain  of  the  sections,  which  we  have  imagined,  be- 
come real  sections  of  the  mass,  dividing  it  into  separate  and  distinct 
parts,  each  of  which  retains  the  properties   of  a  perfect  solid. 

Secondly.  Let  us  suppose  every  point  in  the  system  to  admit  of 
displacement,  subject,  within  certain  limits,  to  the  law  of  perfect 
elasticity. 

The  determination  of  the  conditions  of  the  equilibrium  in  these 
two  cases,  will  constitute  a  complete  theory  of  construction. 

The  discussion  contained  in  the  remainder  of  this  paper  will  be 
confined  to  the  first  case. 

6.  Let  the  mass  AB  (fig.  2.)  have  for  its  line  of  pressure  the 
line  PP'.  Now  it  is  clear,  that  if  this  line  cut  the  plane  QQ  of  any 
section  of  the  mass  in  a  point  n'  without  the  surface  of  the  mass ; 
the  tendency  of  the  opposite  resultants  of  the  forces  acting  upon  the 
two  parts  AQQ  and  SQQ',  into  which  that  section  divides  the  mass, 
will  be  to  cause  them  to  revolve  about  the  nearest  point  Q'  of  its 
intersection  with  the  surface  of  the  mass.  And,  this  tendency  being 
wholly  unopposed,  motion  will  ensue.  And  so  in  the  mass  represented 
(fig.  6.)  the  force  p  and  with  it  the  line  of  pressure  pp'  being  given, 
it  appears  that,  being  cut  transversely  as  shewn  in  the  figure,  the  mass 
cannot  be  supported  by  any  single  force  p  if  it  extend  beyond  CD': 
any  such  force  must,  to  produce  equilibrium,  be  applied  at  q;  and 
being  applied  there,  the  portion  C'C"Z)"'iy  will  be  wholly  unsupported. 
The  line  of  pressure  being  continued  cuts  the  planes  of  the  sections 
CD',  CD',  &c.,  without  the  surface  of  the  mass. 

Thus  then  it  is  a  condition  of  the  equilibrium,  that  the  line  of 
pressure  should  intersect  the  plane  of  every  section  of  the  body  within 
its  mass. 

This  condition  will  be  satisfied  if  this  line  nowhere  cut  the  surface 
of  the  mass  except  at  the  points  P  and  P.  Fig.  2.     Or  if  the  equation 

■^F^z,     F^z,  ■  «  =  0, 

RR  Z 


302  Mk  MOSELEY,   ON   THE 

found  by  eliminating  the  values  of  a;  and  y  between  the  equation  to 
the  surface  and  the  equation  to  the  line  of  pressure,  involve  only 
such  possible  values  of  z  as  correspond  to  the  points  P  and  i*,  where 
the  intersecting  plane  touches  the  surface,  or  to  points  where  the  line 
of  pressure  touches  it. 

It  is  a  further  condition  of  the  equilibrium  that  the  line  of  pressure 
should  not  cut  any  section  of  the  mass,  at  an  angle  with  the  perpen- 
dicular to  that  section  greater  than  a  certain  given  angle,  dependant 
upon  the  friction  of  the  surfaces  in  contact,  and  having  for  its  tangent 
the  coefficient  of  friction. 


The  resistance  of  surfaces  is  not  exerted  exclusively  in  the  direction 
of  the  normal,  according  to  an  hypothesis,  which  was  probably  in- 
troduced into  the  theory  of  Statics  in  order  to  simplify  the  investi- 
gations of  those  who  originated  that  science,  but  which  there  seems 
no  reason  for  retaining  any  longer.  It  is  exerted  in  an  infinity  of 
different  directions  included  within  a  certain  angle  to  the  normal,  or 
rather  within  the  surface  of  a  certain  right  cone,  having  the  normal 
for  its  axis  and  the  point  of  resistance  for  its  vertex.  Any  force, 
however  great,  applied  within  this  conical  surface  will  be  sustained 
by  the  resistance  of  the  surface  of  the  mass — and  no  force  however 
small,  without  it. 

Let  R  represent  a  single  force  on  the  resultant  of  any  number  of 
forces  applied  to  a  fixed  surface,  and  let  R'  and  R"  be  the  resolved 
parts  of  R  in  the  directions  perpendicular  and  parallel  to  the  surface. 
Also  let  p  be  the  inclination  of  R  to  the  vertical,  and  f  the  coefficient 
of  friction.  The  friction  of  the  surfaces  in  contact  is  therefore  repre- 
sented by  fR,   and  motion  will,  or   will  not,  ensue  according  as  R"  is 

K" 

greater  or  is  not  greater  than  /R'.      Or,   according  as  -p,  is  greater  or 

is   not   greater   than  f.      Or,   if  y  =  tan  (p,    according  as   tan  p   is,   or   is 
not,   greater   than   tan  (p,   or   as  p   is   greater   or  is   not   greater   than  (p. 


EQUILIBRIUM   OF  THE  ARCH.  308 

In  the  remainder  of  this  paper  the  angle  0,  or  tan-'^  will  be  called 
the  limiting  angle  of  resistance*. 

From  the  above  then  it  appears,  that  unless  the  tangent  to  the 
line  of  pressure  at  the  point  where  it  cuts  any  section  of  the  mass, 
make  with  the  perpendicular  to  the  plane  of  that  section  an  angle, 
which  is  not  greater  than  the  limiting  angle  of  resistance,  the  surfaces 
there  in  contact  will  slip  upon  one  another. 

This  condition  may  be  expressed  analytically  as  follows : 

%  =  Ax  +  By  +  C 
is  the  equation   to  the  plane  of  any  section  of  the  mass,   therefore 

x-x^  =  -  Ai^-z),     y-y,=  -B  {x-z), 

are  the  equations  to  the  perpendicular  to  that  section.  And  the  angles 
which  that  perpendicular  makes  with  the  co-ordinate  axes  have  for 
their  cosines 

-A  -B  -1 

VA^TW+\'     VA'  +  B'+l'     VA'  +  B'  +  l' 

Also  it  appears  from  the  given  equations  (3)  to  the  resultant 
force,  or  tangent  to  the  line  of  pressure,  that  this  line  makes  angles 
with   the  co-ordinate  axes  which  have  for  their  cosines  the  quantities 

M,  M,  Ms 


Hence,    therefore    if   /    be    the    inclination    of   these    lines    to    one 
another, 


*  It  is  here  supposed  that  the  coefficient  of  friction  f  is  constant  for  the  saifie  surfaces, 
whatever  be  the  force  B!  by  which  they  are  pressed  together.  This  is  usually  assumed 
to  be  the  law  of  friction.  It  is  only  however  an  approximation  to  that  law.  The  ex- 
periments of  Mr  Rennie  shew  that  f  must  be  considered  a  function  of  R'  increasing  con- 
tinually, but  very  slowly,  up  to  the  limits  of  abrasion. 


:fs:. 


304  Mr  MOSELEY,    ON   THE 

AM,  +  BM,  +  Ms 


cos  /=  — 


{{A'  +  B'  +  1)  {Mr'  +  Mi  +  Mi)}k ' 
in  which  expression  M^,  M^,  M^,  and  B,  are  known  functions  of  A. 

Now  /  must  not  exceed  the  limiting  angle  of  resistance.  Therefore 
cos  /  must  not  be  less  than  the  cosine  of  that  angle. 

On  the  whole  then  we  have  these  two  conditions  necessary  to  the 
equilibrium  of  a  mass  intersected  by  a  series  of  planes,  under  the  cir- 
cumstances supposed. 

1.  That  the  equation 

-VF,%,     F^z,    » =  0, 

shall  involve  no  possible  roots,  except  such  as  correspond  to  the  ex- 
tremities of  the  line  of  pressure,  or  to  points  where  it  touches  the 
surface  of  the  mass. 

2.  That  the  fraction 

AM,  +  BM,  +  Ms 


shall  for  all  values  of  A,  corresponding  to  real  sections  of  the  mass, 
be  not  less  than  the  cosine  of  that  arc,  whose  tangent  is  the  coefficient 
of  friction. 

The  first  of  these  conditions  being  satisfied,  the  parts  of  the  mass 
cannot  turn  upon  one  another.  The  second  being  satisfied,  they  can- 
not slip  upon  one  another. 

We  have  supposed  the  whole  of  the  forces  impressed  upon  the 
system  to  be  known  excepting  the  force  P',  which  has  been  deter- 
mined in  terms  of  the  rest.  The  force  P'  may  be  supplied  by  the 
resistance  of  a  point  in  a  fixed  surface,  in  which  case  the  amount  and 
direction  of  that  resistance  will  be  known. 


EQUILIBRIUM   OF  THE  ARCH.  805 

If,  however,  there  enter  two  or  more  resistances  of  surfaces  among 
the  forces  which  compose  the  equilibrium,  since  the  magnitudes  of 
these  and  also  their  directions  may  be  any  whatever,  within  the  limits 
imposed  by  the  friction  of  the  surfaces;  the  problem  remains,  in  so 
far  as  the  known  conditions  of  equilibrium  are  concerned,  indeterminate, 
and   recourse   must  be  had  for   its   solution  to  other  principles. 

7.  Suppose  the  mass  AJS  to  be  acted  upon  by  any  number  of  forces 
among  which  is  the  force  P  being  the  resultant  of  certain  resistances, 
supplied  by  different  points  in  a  surface  Sb,  common  to  the  inter- 
sected mass  and  to  an  immoveable  obstacle  SC. 

Now  it  is  clear  that  under  these  circumstances  we  may  vary  the 
force  P',  both  as  to  its  amount,  direction,  and  point  of  application, 
without  disturbing  the  equilibrium,  provided  only  the  form  and 
direction  of  the  line  of  pressure  continue  to  satisfy  the  conditions  im- 
posed by  the  equilibrium  of  the   system. 

These  are  manifestly,  that  it  no  where  cut  the  surface  of  the  mass, 
except  at  P"  and  within  the  space  JSb,  and  that  it  no  where  cut  a 
section  of  the  mass  or  the  common  surface  of  the  mass  and  obstacle, 
at  any  angle  with  the  perpendicular  greater  than  the  limiting  angle 
of  resistance.         " 

Thus,  varying  the  force  P',  we  may  destroy  the  equilibrium,  either, 
first,  by  causing  the  line  of  pressure  to  take  a  direction  without  the 
limits  prescribed  by  the  resistance  of  the  section  through  which  it 
passes ;  or,  secondly,  by  causing  the  point  P  to  fall  without  the  surface 
Bb,  in  which  case  no  resistance  can  be  opposed  to  the  resultant  force 
acting  in  that  point ;  or,  thirdly,  the  point  P  lying  within  the  surface 
Bb,  we  may  destroy  the  equilibrium  by  causing  the  line  of  pressure 
to  cut  the  surface  of  the  mass  somewhere  between  that  point  and  P'. 

Let  us  suppose  the  limits  of  the  variation  of  P'  within  which  the 
first  two  conditions  are  satisfied,  to  be  known ;  and  varying  it,  within 
those  limits,  let  us  consider  what  may   be  its  least  and  greatest  values 


306  Mr  MOSELEY,   ON  THE 

so  as  to  satisfy  the  third   condition ;   and   where,  and   in  what  direction 
they   must  be  applied. 

In  the  first  place  it  will  be  observed,  that  by  diminishing  the  force 
P',  its  direction  and  point  of  application  remaining  the  same,  the  line 
of  pressure  is  made  continually  to  assume  more  nearly  that  direction 
which  it   would  have,   if  P'  were   entirely   removed. 

Provided  then,  that  if  P  were  thus  removed,  the  line  of  pressure 
would  cut  the  surface,  that  is,  provided  the  force  P'  be  necessary  to 
the  equilibrium ;  it  follows  that  by  diminishing  it,  we  may  vary  the 
direction  and  curvature  of  the  line  of  pressure  until  we  at  length  make 
it  touch  some  point  or  other  in  the  surface  of  the  mass. 

And  this  is  the  limit;  for  if  the  diminution  be  carried  further,  it 
will  cut  the  surface,  and  the  equilibrium  will  be  destroyed.  It  ap- 
pears then  that  under  the  circumstances  supposed,  when  P'  acting  at 
a  given  point  and  in  a  given  direction,  is  the  least  possible,  the  line 
of  pressure  touches  the  surface  of  the  mass. 

In  the  same  manner  it  may  be  shewn,  that  when  it  is  the  greatest 
possible,  the  line  of  pressure  touches  the  surface  of  the  mass. 

Now  by  varying  the  direction  and  point  of  application  of  P',  as 
well  as  its  amount,  this  contact  may  be  made  to  take  place  in  infinite 
variety  of  different  points,  and  each  such  variety  supplies  a  new  value 
of  P',  producing  the  required  contact.  Among  these,  therefore,  it 
remains  to  seek  the  absolute  maximum  and  minimum  values  of  that 
force. 

To  express  these  conditions  analytically,  let  Xi,  y^,  z.^  represent  the 
co-ordinates  of  a  point  where  the  line  of  pressure  touches  the  surface 
of  the   body. 

Since  the  point  x^,  y^,  &  is  common  to  the  line  of  pressure  and 
to   the   surface   of  the   body, 

.-.  -^Xty^Xi  =  0,     Xi  =  F%o,     y.,  =  F^x^. 


EQUILIBRIUM  OF   THE  ARCH.  907 

Also,  since  it   touches  the  surface  in  the  point  ar^ysSSj; 

dz-i  id-^x-iyiZj^^ 

\     dXi      J 

(d'^x^.yi%i\ 
dF,%.,       V      dz,      )  ^  ^ 
d%2  ld'^X2yi%i\ 

\      dyi      I 

Eliminating  x.^,  y^,  z.^  among  these  Jive  equations  two  relations  are 
established  between  the  force  P'*,  the  co-ordinates  of  its  point  of  ap- 
plication, and  the  angles  which  fix  its  direction  (see  Art.  4) ;  by  elimi- 
nation between  which  a  further  relation  is  established  between  six  of 
these  seven  quantities,  and,   finally,   by  the  equations  of  condition 

COS^  a  +  COS^  /3  +  COS*  7=1. 

a  relation  is  obtained  between  four  of  them. 

Thus  then  we  may  obtain  the  value  of  P'  in  terms  of  three  of 
the  quantities  x^,    y^,    s, ;     a,    /3,    7. 

Its  maximum  and  minimum  values  are  then  at  once  determined  by 
the  known  conditions  of  the  maxima  and  minima  of  functions  of 
three  variables. 

8.  It  is  evident  that  the  minimum  value  of  P',  being  that  which 
just  counteracts  the  tendency  of  the  mass  to  revolve  about  the  point 
where  the  line  of  pressure  touches  its  surface,  is  also  precisely  that 
force  which  would  be  exerted  there  by  another  equal  and  similar  mass, 
acted  upon  by  equal  forces,  under  the  same  circumstances,  but  placed 
in  a  contrary  position,  so  that  its  line  of  pressure  shall  have,  at  P, 
a  common  tangent  with  the  line  of  pressure  of  the  first  mass. 

*  The   line  of  pressure  is   here  supposed  to  commence  at  P',   and   the  force  P"  to  enter 
among  the  other  forces  which  determine  its  equation. 
Vol.  V.    Part  III.  S  s 


308  Mr  MOSELEY,   ON   THE 

Two  masses,  therefore,  thus  placed  together  would  remain  in  equi- 
librium, without  the  aid  of  any  external  force,  and  by  reason  only  of 
their  mutual  pressures  and  the  resistance  of  their  abutments. 

It  is  also  evident  that  since  the  line  of  pressure  is  similarly  situated 
in  both,  they  cannot  be  thus  placed  together  so  that  their  lines  of 
pressure  may  meet  and  have  a  common  tangent  at  the  point  where 
they  meet,  unless  both  lines  of  pressure  be  perpendicular  to  the  com- 
mon  surface   at  that   point. 

This  condition  throws  two  new  equations  into  the  system,  and  de- 
termines the   value  of  P'  in  terms   of  a   single   variable. 

The  value  of  P'  is  not  in  this  case  that  which  we  have  called 
the  absolute  minimum  or  minimum  minimorum,  but  simply  the  greatest 
or  least  force,  which  applied  at  a  given  point,  in  a  given  direction  will 
support  the  system. 

If  however  instead  of  a  single  point  of  contact  we  suppose  the 
masses  to  be  in  contact  throughout  the  whole  surfaces  of  two  planes, 
it  is  evident  that  the  point  P' *  will  take  up  for  itself  that  position, 
which  we  have  supposed  to  correspond  with  the  absolute  minimum ; 
a  condition  to  which  the  form  of  the  line  of  pressure,  and  the 
position  of  its  point  of  contact  with  the  surface  of  the  mass,  will  also 
be  subjected. 

Hence  it  appears  that  two  masses,  thus  in  contact  throughout  the 
surfaces  of  two  planes,  sustain  a  less  aggregate  of  pressure,  on  their 
common  surface  of  contact,  than  two  similar  masses  in  contact  only 
by  a  single  point,  unless  that  point,  and  the  position  of  the  masses, 
be  such  as  to  correspond  to  the  minimum  minimorum. 

In  the  preceding  pages  we  have  supposed  the  form  of  the  solid 
to  be  given,  together  with  the  positions  of  the  different  sections 
made  through  it,  and  we  have  thence  deduced  the  form  of  its  line 
of  pressure  and  the  direction  of  that  line  through  its  mass. 

.  ?;*  The  point  P  is  here  the  point  of  application  of  the   resultant   of  the  resistances  on 
the  different  points  of  either  plane.  .:iorte«pi  sJi  -^..rnaHb 

..H  .Ui  T  ■ 


EQUILIBRIUM  OF   THE   ARCH.  809 

It  is   manifest  that  the  converse  of  this  operation  is   possible. 

9.  Having  given  the  form  and  position  of  the  line  of  pressure,  and 
the  positions  of  the  different  sections  to  be  made  through  the  mass,  we 
may,  for  instance,  enquire  what  form  these  conditions  impose  upon  the 
surface   which   bounds  it. 

Or  we  may  make  the  direction  of  the  line  of  pressure  and  the 
form  of  the  bounding  surface  subject  to  certain  conditions  not  abso- 
lutely  determining   either.  '"*  t'^  oxi'>»^i'«  inn  ouJ    ifion 

For  instance,  if  we  suppose  the  form  of  the  intrados  of  an  arch  to 
be  given,  and  the  direction  of  the  intersecting  plane  to  be  always  per- 
pendicular to  it,  and  if  we  suppose  the  line  of  pressure  to  intersect  this 
plane  always  at  the  same  given  angle  with  the  perpendicular  to  it, 
so  that  the  tendency  of  the  pressure  to  thrust  each  from  its  place  may 
be  the  same, — we  may  determine  what  under  these  circumstances  must 
be  the  extrados  of  the  arch.  ,''^'^",'   '"'^ 

If  this  angle  equal  constantly  the  limiting  angle  of  resistance,  the 
arch  is  in  a  state  bordering  upon  motion,  each  voussoir  being  upon 
the  point  of  slipping  downwards  or  upwards,  according  as  the  constant 
angle  is  measured  above  or  below  the  perpendicular  to  the  surface  of 
the  voussoir. 

The  systems  of  voussoirs  which  satisfy  these  two  conditions  are  the 
greatest  and  least  possible. 

If  the  constant  angle  be  zero,  the  line  of  pressure  being  every- 
where perpendicular  to  the  joints  of  the  voussoirs,  the  arch  would 
stand   even   if  there  were   no   friction   of  their  surfaces. 

It  is  then  technically  said  to  be  equilibrated.  It  is  impossible  to 
conceive  any  arrangement  of  the  parts  of  an  arch  by  which  its  stabi- 
lity can  be  more  effectually  secured*. 

10.  The  theory  stated  above  readily  explains  the  phenomena  ob- 
served in  the  settlement  and  fall  of  the  arch. 

*  The  great   arches   of  late   years   erected  by   Mr  Rennie,  in   this  country,  have  for  the 
most  part  been  so  loaded  as  very  nearly  to  satisfy  this  condition. 

ss  2 


8M)  Mr  MOSELEY,   ON   THE 

Thus  let  ABS"  (fig.  3)  represent  an  arch  having  the  joints  of  its 
voussoirs  perpendicular  to  the  intrados  as  they  are  usually  made. 

Let  RQPQR'  be  the  line  of  pressure,  touching  the  intrados  in  the 
points  Q  and  Q'.  It  is  manifest  that  this  curve  is  then  perpendicular 
to  the  joints  of  the  voussoirs  at  Q  and  Q,  and  inclined  in  respect  to 
those  above  and  below  these  points.  The  inclination  being  downwards, 
or  towards  the  intrados,  in  reference  to  the  former,  and  upwards,  or 
from   the  intrados,   in   reference  to  the  latter. 

Hence,  therefore,  it  appears  that  the  tendency  of  the  pressure  is 
to  cause  all  the  voussoirs  above  the  points  Q  and  Q'  to  slide  down- 
wards, and  those  beneath  those  points,  upwards. 

And  that  these  effects  may  be  expected  to  follow  the  striking  of 
the  centre  of  the  arch ;  the  weight  being  then  suddenly  thrown  upon 
the  voussoirs,  and  these  admitting  of  a  certain  degree  of  motion  in 
the  directions  of  the  forces  impressed   upon   them. 

Now  this  is  precisely  what  was  observed  at  the  bridge  of  Nogent, 
of  the  construction  of  which  Perronet  has  left  a  detailed  account. 

Three  straight  lines  were  drawn  upon  the  face  of  the  arch  before 
the  striking  of  the  centre,  shewn  in  the  figure  4,  by  the  polygon 
nmm'n',  mm'  being  horizontal,  and  the  other  two  mn  and  m'n'  stretch- 
ing from  the   extremities  of  mm'  towards  the  springing  of  the  arch. 

After  the  centre  had  been  struck,  the  lines  were  observed  to  have 
assumed  the  curved  forms  indicated  by  the  dotted  lines  MM',  MN', 
M'N',  indicating,  in  accordance  with  the  theory,  a  downward  motion 
in  all  the  voussoirs  above  Q  and  Q',  and  an  upward  motion  in  those 
beneath  those   points. 

These  observations  have  been  confirmed  by  numerous  others,  and 
especially  by  those  (made  also  by  Perronet)  at  the  Pont  de  Neuilly. 

The  sinking  of  the  voussoirs  at  the  crown  necessarily  tends  to  pro- 
duce a  separation  of  their  joints  at  the  intrados  in  the  neighbourhood 
of  that  point,  and  thus  to  cause  the  actual  contact  of  the  key  and 
adjacent  voussoirs  to  take  place  only  at  their  superior  edges. 


EQUILIBRIUM    OF    THE    ARCH.  311 

If  therefore  the  settlement  be  considerable,  we  may  conclude  that 
the  line  of  pressure  touches  the  extrados  at  the  crown,  and  for 
some  distance  on  either  side  of  it.  The  material  of  the  arch  may 
therefore  be  expected  to  yield  more  particularly  about  that  point  and 
the  points  Q  and  Q'  than  any  other;  a  great  proportion  of  the 
pressure  being  there  thrown  upon   the  edges  of  the  voussoirs. 

11.  If  by  reason  of  such  yielding,  or  from  any  other  alteration  in 
the  forces  impressed  upon  the  mass,  or  in  the  circumstances  of  their  ap- 
plication, the  form  of  the  line  of  pressure  be  altered,  it  may  manifestly 
be  expected  to  intersect  the  surface  of  the  mass  first  about  those  points; 
the  least  possible  alteration  of  form  being  there  sufficient  to  produce 
the  intersection.  And  this  being  the  case,  the  portion  of  the  arch  above 
Q  and  Q'  must  separate  into  two  portions,  revolving  at  those  points 
about  the  lower  portions  of  the  arch  (see  fig.  5)  and  at  A,  upon  the 
extremities    of  one   another. 

Nevertheless  this  revolution  is  manifestly  impossible  unless  the 
points  Q  and  Q  yield  outwards.  And  this  can  only  take  place  by 
the  yielding  of  the  material  at  Q  and  Q',  by  the  slipping  back  of 
the  voussoirs  there,  or  by  the  portions  of  the  arch  or  its  abutments 
beneath  those  points  revolving  outwards,  in  consequence  of  the  inter- 
section of  the  extrados  by  the  extremities  QR  and  QR'  of  the  line 
of  pressure  (fig.  3). 

The  last  is  in  point  of  fact  the  cause  which  leads,  in  the  great 
majority  of  cases,  to  the  fall  of  the  arch. 

The  extremity  R  of  the  line  of  pressure  is  made  to  cut  the 
extrados  of  the  arch,  or  the  outer  surface  of  the  pier,  by  the 
diminution  or  removal  of  some  force  which  acted  there  in  opposition 
to  the  tendency  of  the  arch  to  spread  itself,  and  which  kept  the 
direction  of  the  line  of  pressure  within  its  mass, — the  resistance  of 
a  mass  of  earth  for  instance,  or  the  opposite  thrust  of  some  other 
arch  springing  from  the  same  pier  or  abutment. 

On  the  whole,  then,  it  appears  that  in  the  commencement  of  its 
fall  the  arch  will   divide  itself  into  six  distinct  portions,  of  which  four 


312  Mr  MOSELEY,   ON   THE 

will  revolve  about  the  points  S,  S',  Q,  Q'  and  A,  as  represented  in 
the  figure  5.  Now  this  is  what  is  uniformly  observed  to  take  place 
in  the  fall  of  the  arch. 

12.  Gauthey,  having  occasion  to  "destroy  a  bridge,  caused  one  of  its 
arches  to  be  insulated  from  the  rest;  and  the  adhesion  of  the  cement 
being  sufficient  to  counteract  the  tendency  of  the  pressure  to  rupture 
the  piers,  he  caused  them  to  be  cut  across.  The  whole  then  at  once 
fell,  the  falling  portion  separating  itself  into  four  parts.  Having  con- 
structed small  arches  of  soft  stone,  and  without  cement  he  loaded  them 
until  they  fell.  Their  fall  was  always  observed  to  be  attended  with  the 
same  circumstances.  Before  the  arch  finally  yielded  the  stone  also  was 
observed  to  chip  at  the  intrados  about  the  points  Q  and  Q',  round 
which  the  upper  portions  of  it  finally  revolved. 

Some  experiments  made  by  Professor  Robinson  with  chalk  models 
were  attended  with  slightly  different  results.  Having  loaded  them  at 
the  crown  until  they  fell,  he  observed  first,  that  the  points  where 
the  material  began  to  yield  were  not  precisely  those  where  the  rupture 
finally  took  place. 

This  fact  presents  a  remarkable  confirmation  of  the  theory  expounded 
in  this  paper. 

It  is  manifest,  that  according  to  that  theory,  with  any  variation 
in  the  least  force  P',  which  would  support  the  semi-arch  if  applied 
at  its  crown,  there  will  be  a  corresponding  change  in  the  position  of 
the  point  Q. 

Now   as   the   load   upon   the   crown    is  increased,   this   least   force  P' 
is  manifestly  increased.      The  result  is  a  corresponding  variation  in  the 
•  form   of  the  line  of    pressure,    tending   to  carry   its    point   of    contact 
with  the  intrados  lower  down  upon  the  arch. 

This  is  precisely  what  Professor  Robinson  observed.  The  arch 
began  to  chip  at  a  point  about  half  way  between  the  crown  and  the 
point  where  the  rupture  finally  took  place. 


EQUILIBRIUM   OF   THE   ARCH.  313 

The  existence  of  the  points  Q  and  Q',  about  which  the  two  upper 
portions  of  the  arch  have  a  tendency  to  turn,  and  about  which  the 
material  is  first  observed  to  yield,  has  long  been  known  to  practical 
men.  The  French  engineers  have  named  these  points  the  points  of 
rupture  of  the  arch ;  and  the  determination  of  their  position  by  a 
tentative  method  forms  an  important  feature  in  the  very  unsatisfactory 
theory  which  they  have  applied  to  this    important  branch   of  Statics. 

13.  The  theory  of  the  equilibrium  of  the  groin  and  that  of  the 
dome  are  precisely  analogous  to  the  theory  of  the  arch. 

In  the  former  case  a  mass  springs  from  a  small  abutment  spread- 
ing itself  out  symmetrically  with  regard  to'  a  vertical  plane  passing 
through  the  centre  of  its  abutment.  It  is  in  fact  nothing  more  than 
an  arch,  whose  voussoirs  vary  as  well  in  breadth  as  in  depth.  The 
centres  of  gravity  of  the  different  elementary  voussoirs  of  this  mass 
lie  all  in  its  plane  of  symmetry.  Its  line  of  pressure  is  therefore  in 
that  plane,  and  its  theory  is  embraced  in  that  which  has  been  already 
laid  down. 

Four  groins  commonly  spring  from  one  abutment ;  each  opposite 
pair  being  addossed,  and  each  adjacent  pair  uniting  their  margins. 
They  thus  lend  one  another  mutual  support,  partake  in  the  properties 
of  a  dome,  and  form  a  continued  covering. 

The  groined  arch  is  of  all  arches  the  most  stable ;  and  could  ma- 
terials be  found  of  sufficient  strength  to  form  its  abutment  and  the 
parts  about  its  springing,  it  might  be  safely  built  of  any  required 
degree  of  flatness,  and  spaces  of  enormous  dimensions  might  readily 
be  covered  by  it. 

It  is  remarkable  that  modern  builders,  whilst  they  have  erected  the 
common  arch  on  a  scale  of  magnitude  nearly  approaching  perhaps  the 
limits  to  which  it  can  be  safely  carried,  have  been  remarkably  timid 
in  the  use  of  the  groin. 


H.  MOSELEY. 


King's  College,  London, 
Ocl<^er  9,   1833. 


XIV.  Third  Memoir  on  the  Inverse  Method  of  Definite  Integrals. 
By  the  Rev.  R.  Mukphy,  M.A.  F.R.S.,  Fellow  of  Cuius  College, 
and  of  the  Cambridge  Philosophical  Society. 


i;;Read  March  2,   1835.] 


INTRODUCTION. 

In  the  two  preceding  Memoirs  on  the  Inverse  Method  of  Definite 
Integrals,  the  limits  of  integration  had  been  fixed  throughout  at  0  and 
1,  but  in  the  sixth  Section,  which  is  the  first  of  the  present  Memoir, 
the  integrations  terminated  by  arbitrary  limits  are  fully  considered;  and 
when  performed  with  respect  to  any  function  of  the  independant  vari- 
able, the  proper  methods  for  discovering  reciprocal  functions  are  given, 
and  it  is  remarkable  that  the  forms  thus  obtained  for  the  trigonome- 
trical functions,  for  Laplace's  and  an  infinite  variety  of  other  reciprocal 
functions,  are  all  similar,  differing  only  by  a  constant. 

In  identities  obtained  between  the  »""  differential  coefficient  of  a 
function  not  containing  n,  and  its  expanded  value,  we  may,  generally, 
by  changing  the  sign  of  n,  obtain  a  corresponding  identity  between 
the  ra""  successive  integral  and  its  expansion,  abstracting  from  the  ap- 
pendage of  integration  which  ought  to  contain  ?«  arbitrary  constants ; 
this  property  however  extends  also  to  certain  reciprocal  functions  which 
contain  n ;  and  this  consideration  leads  in  the  same  section  to  the  com- 
plete resolution  of  Laplace's  equation  for  the  reciprocal  functions  of 
one  variable,  which  are  the  coefficients  in  the  developement  of  the  reci- 
procal of  the  distance  of  two  points;  the  w*""  coefficient  when  multiplied 
by  an  arbitrary  constant,  satisfies  that  equation,  as  is  well  known,  but 
as  the  equation  is  of  the  second  order,  another  function  multiplied  by 
■^  Vol.  V.     Part   III.  Tr 


316  Mr  MURPHY'S   THIRD    MEMOIR    ON    THE 

an  arbitrary  constant  must  be  also  represented  by  the  same  equation, 
this  function,  which  is  here  found,  is  altogether  different  in  its  form  and 
properties  from  Laplace's  coefficients. 

The  great  class  of  reciprocal  functions  above  alluded  to  possess  the 
remarkable  property,  that  their  integrals  vanish  between  any  of  their 
own  maxima  or  minima  values. 

In  this  Section  I  have  noticed  some  curious  trigonometrical  func- 
tions of  which  the  properties  are  very  elegant,  particularly  as  affording 
simple  means  of  representing  by  Definite  Integrals  the  general  differ- 
ential coefficients  of  rational  and  integral  functions ;  another  applica- 
tion of  trigonometrical  functions  is  made,  in  representing  the  sum  of 
the  divisors  of  any  given  number,  by  means  of  a  Definite  Integral. 

The  seventh  Section  is  on  Transient  Functions.  The  way  of  forming 
reciprocal  functions  by  means  of  arbitrary  coefficients,  when  the  form  of 
the  general  term  was  given,  has  been  shewn  in  the  Second  Memoir  on 
this  subject.  To  this  I  have  here  added  the  method  of  finding  the 
functions  which  shall  be  reciprocal  to  any  proposed  one,  and  applied 
the  method  to  the  cases  where  the  given  function  is  r,  (log.  t)",  and 
cos"  {t) ;  the  reciprocal  functions  which  thence  resulted  are  transient,  that 
is,  they  have  but  a  momentary  existence  between  the  limits  of  inte- 
gration ;  that  existence  is  however  sufficient  to  make  their  integrals 
finite,  and  to  endow  them  with  remarkable  properties.  They  are  capa- 
ble of  representing  the  electrical  state  of  a  body  when  an  electrical 
spark  is  infinitely  near,  and  about  to  form  a  part  of  the  system ;  they 
are  also  capable  of  representing,  under  continuous  forms,  the  state  of  a 
body  considered  as  composed  of  absolute  mathematical  centres  of  forces, 
separated  mutually  by  infinitesimal  intervals. 

The  eighth  and  last  Section  is  on  the  Resolution  of  Equations  which 
contain  Definite  Integrals;  the  first  method  for  this  purpose  is  to  de- 
compose the  integrals  into  elements,  and  then  determine  the  unknown 
functions  by  elimination.  This  tedious  process  is  useful  in  verifying 
results  otherwise  obtained,  and  in  giving  numerical  approximations  in 
the    most    difficult   cases.      Afterwards    I     have    considered    separately, 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS.  317 

Equations  to  Definite  Integrals ;  first,  when  they  contain  but  one  Defi- 
nite Integral  and  one  parameter ;  second,  when  they  contain  two  or 
more  Definite  Integrals  and  as  many  parameters;  third,  Simultaneous 
Equations ;  fourth,  Definite  Integral  Equations  of  superior  orders  and 
degrees;  besides  which,  the  nature  of  the  appendage  analogous  to  the 
arbitrary  constant  of  integration  is  discussed  in  the  same  Section. 

Throughout   the   whole    of  this   Memoir,   a   considerable   number   of 
examples,  illustrative  of  the  corresponding  theories,  are  dispersed. 


TT2 


318  Mh  MURPHY'S   THIRD    MEMOIR   ON    THE 


SECTION    VI. 

Method  of'  discovering  Reciprocal  Functions  when  the  integrations  are  per- 
formed with  respect  to  any  Junction  of  the  independant  variable. 


(l)      When  the  limits  of  integration  are  arbitrary. 

1.  The  investigations  of  reciprocal  functions  contained  in  the  Second 
Memoir  on  the  Inverse  Method  of  Definite  Integrals,  are  founded  on  the 
supposition  that  0  and  1  are  always  the  limits  of  the  independant 
variable,  but  it  is  often  of  importance  to  possess  reciprocal  functions  in 
which  the  limits  of  integration  are  different  from  those  quoted.  The 
principle  by  which  this  is  most  easily  accomplished,  is  to  suppose  the 
integrations  performed  relative  to  a  function  of  the  independant  vari- 
able, which  must  be  so  chosen,  that  when  the  values  0  and  1  are 
assigned  to  the  independant  variable,  the  corresponding  values  or  the 
function  may  be  the  proposed  limits  of  integration. 

2.  Let  Q„,  R„,  be  functions  of  a  variable  (^),  the  limits  of  which 
are  arbitrary,  as  a  and  h,  between  which  limits  f^Q^Rm  always  must 
vanish,  except  when  the  integers  m  and  n  are  equal. 

Suppose  that  a  function  of  <p,  as  t,  is  found  such  that  when  ^  =  a 
t  =  0,  and  when  (p  =  h,  t=l,  conditions  which  it  is  always  easy  to  satisfy. 

We  may  now  conversely  regard  0  as  a  function  of  t,  and  then  the 
preceding  integral  becomes  fiQ„Rm-jr,  the  limits  being  now  reduced  to 

0  and  1.     Suppose  that  -~  is  separated  into  any  two  factors,  X  and  X'; 

then  since  f,QnX  x  R,„\'  =  0,  except  when  7n  —  n,  it  follows  that  Q„X, 
R„\'  are  mutually  reciprocal,  and  may  therefore  be  found  in  an  inde- 
finite variety  of  modes  by  the  principles  explained  in  Section  iv;  and 
dividing  these  functions  respectively  by  X,  X',  and  substituting  in  the 
quotients  the  value  of  t  expressed  in  terms  of  ^,  the  required  functions 
Q„,  Rm  will  be  obtained. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  319 

If   it    be   desired    that    Q„,   R^    should    be   functions    of    the   same 
nature,    differing    only    in    the    order    expressed    by    m   and   n,   that   is 

self-reciprocal,    put    \  =  W  =  \/{-~\,   and    having    found    any   kind    of 

self-reciprocal  functions  in  which  the  limits  are  0  and  1,  as  for  ex- 
ample, the  functions  denoted  by  P,„,  P„  in  the  preceding  Memoirs,  we 
then   obtain 

3.     If  a  function   V   can   he  determined   so  that  the  quantity 

d°f(ttyV}     dt^ 
dt"         ■  d0 

may  he  of  n  dimensions  in  t,  (where  t'  =  1  —  t  as  in  the  former  Memoirs), 
this  quantity  will  he  a  self-reciprocal  function  when  the  integrations  are 
performed  relative  to  (p. 

Denote  this  quantity  by  Q„,  and  supposing  m  to  be  an  integer 
less   than   n,  it  is   necessary   to  show   that   f,pQmQn  —  0,   or   that 

d''{{ttyn 

^'^-     di"     -^' 

the  limits  of  t  being  0  and  1. 

Now  Q„  being  of  m  dimensions  in  t,  let  its  general  term  be  re- 
presented by  Oj.f,  where  it  is  evident  that  p  cannot  exceed  n  —  1, 
since  m<n;  the  part  of  the  preceding  integral  dependant  on  this  term  is 

""'^'^       dv' — • 

The  latter  integral  may  by  partial  integration  be  put  in  the  form, 

the  last  term  being 

and  therefore  the  index  of  differentiation  never  becomes  negative. 


320  Ma  MURPHY'S  THIRD    MEMOIR   ON   THE 

The  first  term,  and  'a  fortiori',  all  the  succeeding  terms  of  this 
series  vanish  between  the  limits  ^=0,   and  t=\,   or  t'  =  0,   for 

d''-'{{tt'rV}  _rrd"-'{tt'Y    ,,^      ^,dV  (f-^tty 
dt--'  ~  dt^-'       ^^  '   dt       dt"-' 

{n-l){n~2)  dT    d^-^itt'f      , 
"^  1.2  dt'  ■     dt"-'     "^       ' 

the  first   term   of  this  latter   series   contains   a  factor   tt',    the  second  a 
factor  {tt'f,  &iC.,  and  therefore  the  whole  vanishes  between  limits. 

The  following  exception  to  this  theorem  must  however  be  attended 
to;  V  must  not  he  of  the  form  {tt')".Vj,  where  r  is  equal  to,  or 
greater  than  unity,  for  the  above  reasoning  will  not  be  applicable, 
since  then 

d"-mtt'Yr\  _d'-'{{tty-^r,} 


_i 


dt"-^  dt' 

which  being  expanded  as  above,  will  not  vanish  unless  r  be  less  than 
unity. 

4i.     If  a  function   V   can  he  determined  so   that  the  quantity 

d''f(ttrvi  d0 

at"      ■  dt 

may   he    of   n   dimensions    in  t,    then    the  factor  hy   which   -^  is    here 

multiplied,  will  he  a  self-reciprocal  function  when  the  integrations  are 
performed  relative  to  cp. 

Denote  this  coefficient  by  q„,  then 

r  -r  ^0  _   /■  <^"  (tf'Y  ^        d<t> 

and    as   we  may    suppose   m<n,    the    general    term  of   qm-^,  as  a^t^ 

cannot  be  of  greater  dimensions  than  n  —  1,  and  therefore  the  part  of 
the  whole  integral  dependant  on  this  term  vanishes,  as  has  been 
shewn  in  the  preceding  article,  hence  f^qmq„  =  0,  when  m  and  n  are 
unequal. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  321 

We  must  except,  as  before,  from  the  application  of  this  theorem 
the  case  where  V  is  of  the  form  {tt')-\Vi,  and  r  greater  than,  or 
equal  to  unity. 

5.  If  (f)  be  any  of  the  transcendants  contained  in  the  indefinite 
integral  jj  (tt')",  where  m  is  between  —  1  and  +  x  exclusive,  and  if 

^"~  1.2.3...ndt"'^"^     ' 
then  shall  Qn  be  a  self-reciprocal  function  for  integrations  relative  to  <p. 

For   Q„    is   evidently   of    the   form  — ~rp: ■  ~TZ'   ^'^d  ^  is   not 

of  the  form  excepted  in  Art.  3.,  since  m  is  between  —1  and  +  oo. 
Moreover,   by  actual  differentiation   we   get 

1  .^.S-.-ndt" 
where   a,  b,  c,  &c.  are  constant  quantities. 

Hence, 

Q„  =  at"'  ^btt'"-'  +ctH"'-^  +  kc., 

which  is  of  m  dimensions  in  t,  and  therefore  all  the  conditions  re- 
quired in  Art.  3.  are  here  fulfilled;  therefore  Q„  is  a  self-reciprocal 
function  relative  to  <p. 

6.  If  (p  be  any  of  the  transcendants  expressed  by  the  indefinite 
integral  jj  (tt')"",  where  m  is  between  + 1  and  —  oo  exclusive,  and  if 


qn  = 


-       d°  (tt')°- 


1.2.3....ndt' 


n»     ■ 


then  is  qn  a  self-reciprocal  function  relative  to  (p. 

d'  (tfy  V 
For  §-„   is   here   of    the   form         i  ' — ,   and   V  does    not  belong   to 

the   excepted   cases,   moreover 

#  _  d'^.jtt'y-'" 

^''  dt  ~  \.2...ndt"-^^^^ 
is    evidently    of    n    dimensions    in    t,    therefore    all    the   conditions    of 
Art.  4.  are  here  satisfied. 


322  Mr  MURPHY'S   THIRD    MEMOIR    ON   THE 

7.  For  the  purpose  of  convenience  both  in  evaluating  and  using 
reciprocal  functions,  the  knowledge  of  the  functions  which  they  generate 
is  very  useful.  The  generating  function,  for  example,  being  the  quan- 
tity denoted  by  q^,  Art.  (6),  the  process  for  finding  in  this  case  the 
function  generated,  will  sufficiently  exhibit  the  general  principle,  and 
therefore  it  is  now  proposed  tb  sum  the  series  q^  +  q^h  +  q^k'  +  q^h^,  &c. 

Substituting  for  q„  its  value  given  in  the  preceding  article,  and 
representing  the  required  sum  by  S  we  have 

o    /. 'V       J.  ditty-'"      le    d'itt'f-'"       M     dHtfy-'"  ,  , 

But   if  we  form   the  equation,   u  =  t  +  ku  (1  —  u),   and   suppose  y'(M)   to 
be  the  derived  function  from  J'{u),  we  have  generally 

^^    r(«\-f'(A^h^it^^)-^A.    *'     d^{f'it).(ttj} 

,_f^_  d?\f{t).{tty\ 

+  17273  •  df  *'''■ 

which   is   obtained   by    differentiating   the   value   of  /(«)   given   by    La- 
grange's Theorem. 

The  preceding  series  coincide  by  supposing 

f(f)  =  {ttf)-"'  =  t-"  il-t)"", 

and  therefore  /'(«)  =  «""  (l-w)""  =      j^-L 

by  the  assumed  equation. 

(u-t)-'"    du 
Hence  5- =  -^^  .  ^^  . 

Now  the  actual  solution  of  the  assumed  quadratic  equation  gives 


u  = 


2h 


,    where  R=  {l-2h{l-2t)  +  h'}K 


,      B-l-\-h{l-2t)         .  du       1 
whence  u-t=  ^ ,  and  -^  =  ^i 


INVERSE    METHOD   OF    DEFINITE    INTEGRALS.  323 


therefore  S  =  |/2  -  1  +  A  (1  -  2/)}  -" 


R 


Knowing  thus  the  generated  function  S,  we  can  conversely  find  q„  by 
taking  the  coefficient  of  A"  in  the  quantity  S,  and  substituting  for  t 
its  value  in  terms  of  <^. 

An  exactly  similar  process  applied  to  the  function  Q„  of  Art.  (5), 
woxild  give 

as  the  function  generated, 

and  observing  that 

R'-{\-h{\-^t)\"  ^  4^h'tt', 

this  quantity  may  be  transformed  to 

a  III 

^\R^\-h{\-^t)}-"; 

so  that  Q„  is  the  coefficient  of  h"  in  the  expansion  of  this  function. 

8.  From  the  theorems  given  in  Arts.  (5)  and  (6),  we  can  determine 
reciprocal  functions  relative  to  <p,  which  quantity  may  denote  any 
transcendant  contained  in  the  formula  Jt{tt'y,  from  m—-<xi  to  »« =  +  x ; 
circular  arcs  are  amongst  these  transcendants,  namely,  when  m  =  —  ^, 
and  since  both  theorems  are  true  simultaneously,  when  m  is  between 
—  1  and  +  1,  we  shall  get  in  this  instance  the  two  species  of  circular 
self-reciprocal  functions,  namely,  the  sines  and  cosines  of  the  multiples 
of  the  simple  arc. 

I.     To  evaluate  Q„  when  «/  =  —  i- 

For  the  variable  with  respect  to  which  the  integrations  must  be 
performed,  we  have 

^  =  jXtty^  =  l^y^r^  -■=  COS-  (1-2^), 

neglecting  the  constant  which  is  unimportant. 
Vol.  V.     Part   III.  Uu 


324  Mr  MURPHYs   THIRD   MEMOIR  ON   THE 

By  Art.  (7), 

2-4 
Q„  =  coefficient  of  Jf  in  -^  {^  +  1  -  A  (1-2^)}^, 

in  which  R  represents   |1  —  2// (1  -  2/)  + /i'}*. 

Putting  for  t  its  value  in  terms  of  0,  we  obtain 

J?  =  {1-2/i  cos^  +  /i^}-i  =  (l-Ae*^^)i.(l-//e-'''^^)^^ 
and  l-/i(l-2it)  =  l-/4cos0  =  1(1 -Ae*^^)  +  ^  (1 -//e"*^^). 

Hence,   ^  +  1 -/i  (1  -  2/f)  =  |{(l-^e*^^)^  +  (1 -//e-*^^)-^''; 

therefore,  Q„  =  coefficient  of  /r  in  x  .  Jj  "  ^t-v-!!-^4' "r'^^! 

=  ^  coefficient  of  A"  in  (l-/<e*^^)-*  +  (1  —  Ae-*^'^)-* 


=  c 


c .  cos n<p] 


13    5     (2«  —  1) 
where   c  =     '     '    '"^      ^ ,   the  limits  of  ^  are  0    and  w. 


2.4.6...      2ra 

II.     To  evaluate  q„  when  y«  =  —  i. 

As  above,  we  have  (p  =  eos"'  (1  —  2t), 


and  q„  =  coefficient  of  h"  in   ^-^  .  {^-  1  +/<  (1  —  2/)}''. 

But  i? - 1  +  /.  (1  -  20  =  i  p-^;!:l'^>'  -  (i::.^-_:!:^^)H'^ . 

I       V  -1  V-l        j 


-.  q„  =  ^  coefficient  of  /«"  +  '  in 


\/-i  V-i 


c — . =  c  sin  (1  +  n)  (p, 

*  v  —  1 

,  1.3.5...(2m  +  1) 

^^^••^  ^'  =  2.4.6■.■(2;^4-2) ' 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  325 

9.  But  whatever  may  be  the  value  of  m,  the  quantities  Q„,  q„  may 
always  be  simply  expressed  in  terms  of  t  by  the  theorem  of  T^eibnitz, 
viz. 

d"(uv)_    cV'v        clu    d'"^v       7i.{n  —  l)    dHi^    f/"'^ v 

after  {yiplying  which  we  may  substitute  for  t  its  value  in  terms  of  (p. 
Thus  when  m=  —  ^ 


1.3.5....(2«-1)  n    2n-l 

2.4.6....2W  ^'    ~1-      1      " 


•      '  J.  n{n-l)    {2n-l){2n-3)   ,  ,„_, _  „      . 

_  1.3.5....{2n  -  1)      ,  2n{2n-l) 

-"  2.4.6... .2»      ^^  1.2       " 

,   2^(2?^-l)(2>^-2)(2«-3),,^,„.,  . 

"^  1.2.3.4  ^^       "^''•^ 

•^         2.4.D....2ra  ^  '  '    ^ 

and  in  the  same  way  we  have 

d''(tt'Y*i 

^"~  rr2. 3. ...«<//" 

_3.5.7....(2»  +  l)  »    2;»  +  l 

~       2.4.6....2«       ^^^  1-       3      ^'     " 

w(w-l)    (2w  +  l)(2«-l)   5,    3     „     , 

= I 3.5.7.--.(2??  +  l)        ,        ^---    .     ^  , — i  /.h2»+2) 

2V"=l"2.4.6....(2«  +  2)  ^^^    +^^^)      -(^    _V-1#0       \, 


uu2 


826  Mr  MURPHY'S  THIRD  MEMOIR  ON  THE 

and   passing   to  the  variable   cp,   since   1  — 2^=cos^;   therefore   ^  =  sin  — 

id 

and    /'  =  cos-^,    whence    #'^ +\/  — 1  #*  =  cos^  +  \/  — 1  sin^      by    substi- 
2  2  ~"  2        -^ 

tuting  which  we  obtain 

„       1.3.5....(2w-l)  , 

^'-^       2.4.6....2/.      •^"^^'^' 


1.3.5....(2«4-J)      .     ,        ,,     ^ 

'?''=2:476::::(2¥T2)-''"^''^'^-'^' 


which  values  are  the  same  with  those  in  Art.  8. 

The  numerical  coefficients  in  these  formuhe  may  be  rejected  as  having 
no  importance  in  self-reciprocal  functions ;  it  is  also  observable  that  q„ 
contains  a  different  multiple  arc  from  that  in  Q„,  the  reason  of  which 
is  that  Q„,  <7„  are  to  be  self-reciprocal  functions  for  all  entire  values 
of  n  from  0  to  +  oo,  and  then  f,j,q„q,n  =  0  except  when  7i  =  »i,  this  ex- 
ception (on  which  the  main  value  of  reciprocal  functions  depends)  would 
not  hold  universally  true  if  q„  were  of  the  form  sin(«0),  for  then  5-0  =  0, 
and  therefore  f^qo.qo=f>  contrary  to  the  principle  of  the  exception, 
but  in  the  form  above  found  this  irregularity  does  not  occur. 

10.     From  the  results  found  in  Art.  9,  it  follows  that  if  we  put 
the  real  functions  Q„,  q„  possess  a  common  property,  viz. 

except  when  m  =  n,  which   exception  does  not  apply  to  the  last  integral 
when  m  =  ?i  =  0. 

From  the  same  results  the  following  identities  are  obtained : 

,    ff'^*!T\.  1.  ■  («')^  =  cos  {n  cos-'  (1-20} 
1.3.5....(2tt-l)</^"    ^     '  '  \  Ji 

(»  +  l)2''+'rf"(«T^        •     J/     ,  -,.  w,      o*\\ 

i-3-^i-^^^-^,,  =sm  U«  +  1)  cos  '  (1 -20}. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  327 

We  shall  now  consider  whether  analogous  formulee  hold  true  for  negative 
values  of  n  the  index  of  differentiation. 

Generally   if  u  and  v  be  functions  of  t  and  fi'u  denote  the  w"'   suc- 
cessive integral  of  ti,  then 

for  if  we  take  the  w*  differential  coefficient  of  each  term  in  this  series, 
all  the  terms  resulting  mutually  destroy  each  other  except  the  first 
term  tiv. 


Putting  u-=t'-"-^,  v  =  t""-^,  and  rejecting  the  constants  of  integration 
in  the  latter,  we  have 

also  —  -  ^^L±lt'--l,      ^  -  (2«  +  l)(2«  +  3)    .,_„_!   » 

Hence  fiitf)'"-^ 
(-2)"(^0--      (.,-.    n   2«+l     ,_^_,      n{n-l)    (2^+l)(2«  +  3)      „_    „     „      , 

=  i.3.5....(2«-i)^^    "i-~T~-^      ^+-r¥-- Ts ^      ^-^^-'^ 

«r  ^327 •        dt-'^        ■  ^^^  > 

=  cos  {mcos-'  (1  —  2^)^, 

the  appendage  which  contains  all  the  arbitrary  constants  being 

{^o  +  ^it  +  ^.f+...A„_J"-'\  .  {tt')K 


328  Mr  MURPHY'S  THIRD  MEMOIR  ON  THE 

Dividing    the  last   equation    by  {tty,    and    integrating   witt  respect 
to  t,   we  get 

1.3.5...(2w-l)    d-'^^-'iffy-i       1     .     c  i/i      o.M 
7 — x; — ^  • TT — 1 =  -  Sin  \ncos-Ul-2t)}. 

Putting   »  =  ???- 1 ,   we  get 

,        ,,    1.3.b...(2m-S)    <?-"(«') -'"  +  -i        .     <,,         ,  ,,,      ^,, 

('"-1)  • {-2)'"-' • dt^ "''"  {(i-»i)  cos-'  (l-2t)}, 

thus  are  obtained  the  corresponding  formulae  for  negative  indices. 

11.  The  two  series  of  reciprocal  functions  arising  from  the  theorems 
in  Arts.  5.  and  6.,  differ  essentially,  only  in  reference  to  the  inde- 
pendant  variable  of  integration,  for  in  Art.  5.,  ni  may  be  any  quantity 
between  —1,  and  +x,  and  in  Art.  6.  any  quantity  between  +1  and 
—  00 ;  change  in  the  latter  theorem  m  into  —  m,  and  the  limits  of  w 
Avill  then  be  the  same  in  both ;  for  distinctness,  also  let  6  be  used 
instead  of  (p  in  the  value  of  §'„. 

d"  itt'Y*'" 
Hence,    Q^=  i   ^.s.'.ndf  ■^*^'^'""'     ^"'^   <t>  =  !^itfY, 

d"  (ttY'^" 

^-lALndt"^  ^"d  ^  =  ;(«')-'". 

Now  the  reciprocal  functions  of  Art.  5.,  give  the  equation 
UQnQn=0,     or   feQM,.   ^=0. 

But  ^  =(«')",     and  ^  =(«')-'" ;     therefore  ^  =  («7'». 

Hence,    feQAtt'Y  ^  QAtt'Y  =  0. 

And  since  QAtt'T  =  qn,  and  QAtt'y  =  q„;  it  follows  that  UQnQn'  is 
equivalent  to  [dq„qn;  the  only  difference  being  with  respect  to  the 
variables  (f>  and  9  employed  for  integration. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  329 

If  in  the  formulEe  of  Arts.  5.  and  6.,  we  assign  to  m  all  possible 
values  between  —1  and  4-1,  we  obtain  two  series  of  self-reciprocal 
functions,  which  when  m  =  0  become  identical  with  each  other,  and 
with  the  functions  denominated  P„  in  the  preceding  memoirs.  For 
every  other  value  of  m  between  those  limits,  there  are  two  different 
kinds  of  reciprocal  functions,  one  of  which  only  is  a  rational  and  entire 
function  of  t,  for  instance  when  m= —\,  we  have  found  the  functions 
cos  n(p  and  sin  {7i  +  l)<p,  the  former  of  which  only  is  a  rational  fvmc- 
tion  of  cos  <p. 

12.     (1.)     W/ien  m=  -i- 

To  determine  cp  in  this  case,  make  sin  9  =  ii  —  / ',  squaring  and  ob- 
serving that  t  +  f'  =  l,  we  get  sin^  6  =  1  -2  {tt')K   whence  » 

/■i  +  ?;'i  =  V^(2  -  sin- 0),     and  2'\tty  =  cose. 
Differentiate  the  assumed  equation,  and  we  get 

^os  ^  =  a  ^**'\\  •  ~7^ '     therefore  — -^  .  -r^  =  2  cos  0 .  -7 


;i  5 


2 {tty  •  cie'    ^"-'-—  ^tt')i  ■  d0~         ■ti  +  t'i 

hence,   (p  =  2E  {e)-F(d). 

The  extreme  values    of  the  amplitude  0    of  these    elliptic   functions 

being   —  -,    and  +  -;    the   limits    of    ^   are    0,    and    4:Ei  —  2JF\,    where 

El  and  Ei   denote  the  complete  functions  when  the   amplitude  extends 
from  zero  to  a  right  angle. 

The  reciprocal  functions  for  integrations  relative  to  (p,  are 

_>3.7.11...(4?i-l) 


Q,= 


4.8.12 4.W 


4n{^n-l)  4n{4>n-l)(in-i){4u-5)      „     , 

^*  sTi       *     ^^  3.4.7.8  ^      (,^^-U 


330  Mb  MURPHYs  THIRD  MEMOIR  ON  THE 

5.913...(4«  +  1) 


q. 


4.8.12 4« 


,  (,0J  {t--  (4>^  +  l)-4^,.-.,^  (4.  +  l).4^».^(4«-3)(4»-4)  ^,„.,^,    ^^^^ 


(2.)     When   m=  —  1. 


In   this   case   <?„  =  1.2.3...,,^^, 


and   <^  =  j;(«')-'=h.l.  (I). 


Hence,     ^  =  e*,      ^  =  1  +  e*  ; 


therefore 

_(«  +  l)_        ^^    ^+1  nin-X)     {n^\){n)    ^_^    , 

9»      (1 +  £</>)»+'=  »^        1-     2     -^      ^       1.2      ■        2.3        "^         '^'^■^' 

where  the  limits  of  ^  are  —  «  and  +  w . 

13.     To  express  the  functions  Q^  and  qn  z«  terms  of  t  alone. 
By  Art.  6.,  we  have 

^'~  1.2.3...ndf 

=  (w-?»)(w-OT-l)...(l-OT)  _       „_  71    n-m 

1 .2.3...W  v     ;     •  J  j^  .  ^  _^ 

«j^_l)     (/^-m)(«->»-l)  , 

^      1.2      •      (l-»»)(2-»/)      ^       ^>&c.}. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  331 

Suppose  1- 1  substituted  for  t'  in   each  term   between  the  brackets, 
then  expanding  each,  the  coefficient  of  t"  in  the  whole  will  be 

n{n-'l)...{n-r  +  l)  ,     ^y  t-.  ,  „  »-"^   ,   r(r-l)    {n~m){n-m-l)   ,  ,_  , 
1.2...r  "^     ^^   ^^^^l-m^      1.2     '     {l-m){2-m)     ^^^'^ 

«(«-!). ..(M-r+l).(-l)'        (^„-„  d't'-'"         d.f-'"    </'->. r-" 


1.2...rx{l-tn)  (2-m)...{r-m)  '^  dt'  dp     '      dt'-' 

r.(r-\)    d^ .  t"-"    d'-' .  p-'" 
■^      1.2      •     dt^     '~dF^'        ^^•^' 

when  t  is  put  equal  to  unity  after  the  differentiations. 

But    by    the    theorem    of    Leibnitz,     the    part    within    the    latter 
brackets  is  equivalent  to 

fjr  fr+n—2m 

— -^- —  =(n-2m  +  l)  (n-2m  +  2)...{n  —  2m  +  r).t"-'"", 

hence,  the  required  coefficient  of 

,_,_.,  «■(»-  l)...{n-r+l)     {n-2m+  1)  {n  —  2m  +  2)...{n  —  2m  +  r) 
~^       ''  1.2...r  ^  il-m)  (2-m)...(r-m) 

Henpe, 

«  _  (»-ffi)(n-m-l)...il-m)  .       .         ,        n    n-2m+l 
7"-  1.2.3...n  ^'~^'      ^^~T'       l-m      '^ 

n(n-l)     (n-2m-\- 1)  (n  -  2m  +  2)    ^    ,     , 
"^      1.2      •  {l-'tn)i2-m)  *  >  ^<^'h 


Again,   by  Art.  5., 

_  d"(ttT'" 

^"-  l.2...ndf-^^^' 

(n  +  vi)  (n  +  m—l)...(l+m)    j^,„      n    n  +  m     ,„_, 


1.2...n  *■  1     1+m 


t'"-'  t 


,  n.jn-X)    (n  +  m)  (n  +  m -I)  ^„.,^,  \_, 
"*■       1.2       •     (l+m)(2  +  m)      '     ^'^^.j, 


Vol.  V.    Part  III.  Xx 


3SS  Mr  MURPHY'S  THIRD  MEMOIR  ON  THE 

the  reduction  of  which  to  the  powers  of  t  is  effected  as  before,  putting 
—  m  for  m,  whence 

(w  +  ot)  (w  +  m-l)...(l +»?)  (        n    n  +  2m  +  l 
^"~  1.2...ra  ^        1*       1+m      ■ 


When  j»  =  0, 


«    o    1     **   "  +  i        M«:il)   (w  +  i)(w  +  2) 

which  is  the  same  as  the  value  of  P„,   Sect.  ii.  Art.  2. 
When  m=  —  ^ 


2 

and  t  =  sin^  ^ 


2 


{(w  +  ly-l^{(>^  +  ly-2-}  _,^.^,0 

2.3.4.5 


Q»=2.4 2n       •il-1.2'^  ''"  2  +   1.2.3.4  -^  ''"  2      *'''-^- 

14.     To    express   the    quantities  Q„,   q„   by    means   of    a   differential 
equation. 

Suppose  /{t)  is  a  function  of  #,  subject  to  the  condition 

t(l-i)  ./"it)  +  (»»+  1)  (1-20  ./'  (0  +  «  •  («  +  2/»+  1)  ./{t)  =  0, 

where   /"(^    denotes    the    second,    and  /'(^)    the   first   differential   co- 
efficient of  f{t)  relatively  to  t ;    differentiating  this  equation,  we  get 

t{l-t)  .f"{t)  +  (»« +  2)  (1  -  2^  ./"(/)  +  (»-  1)  («  +  2»?  +  2) ./'  (^)  =  0, 

^  (1  -0  ./""  {t)  +  (w  +  3)  (1  -  20  ./'"  (0  +  («  -  2)  (W  +  2»«  +  3) ./"  {t)  =  0, 

and  generally, 

/(i-0/""^'no+('»+^-i)(i-20-/""'^'"'MO+(^-^+2)(«+2/»+r-i)./"'<'-»(0=o. 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS.  333 

Put  #  =  0  in  all  these  equations   successively,  thence  we  have 

{m  +  l)./'{0)  =-n.{n  +  2m  +  l).f{0), 

(m  +  2)  ./"  (0)  =-  in~l){n  +  2m  +  2) ,/'  (0), 

m  +  3  ./'"  (0)  =  -  (ra  -  2) .  («  +  2w  +  3)  ./(O), 

&e. 

it  follows  from  this  by  Maclaurin's  Theorem,  that  the  preceding  equa- 
tion will  be  satisfied,  as  a  particular  solution,  by  taking 

^•/^v     ^/«v(i     ^  n  +  2m+l    .n.{n-\)   (w  + 2w^  + 1)  (w  +  2>»  +  2)    .,     ,     . 
./(0=/(0){l-i.-i-^^-.^  +  ^^. (i  +  ^).(2  +  «.)       ^^^-&c-}» 

and  ,/(0)    being   arbitrary   if  we   put   it  equal  to 

(1  +  m)  (2  +  w<)  (3  +  ?») (n  +  m) 

i     '.     2     ;     3      TTTT^     n      ' 

this  value   oi  f{t)  will  become  the  same  as  the  value  found  for  Q„  in 
the  preceding  article ;   hence,  replacing  \  —  thy  its  equal  t',  we  get 

(«')  ^  +  («  +  1)  (1-2^)  .  ^  +  «  .  (w  +  2m  +  1)  .  Q„  =  0. 

But   if  in   the  value  of  /{()   we  change  the  sign  of  m,  putting 

...  _  (l-m)(2-?w) {n-m) 

'^^"^~       1.2       n       ' 

then  y*(#)  becomes   equivalent  to   q„  {tt')" ;   and  if  we  put  this  for  y  (#) 
in  the  first  supposed  equation,  and  divide  the  result  by  {tf)"',  we  get 

{ti')^  +  im  +  l){l-2t).^+{n+l){n-2m).q„  =  0. 

(2)     Particular  inferences  resulting  from  the  preceding  theory. 

15.     Denoting   as   before    by   <f>   the"   indefinite    integral   fi(tt')'",  and 
putting 

xxS 


834  Mr  MURPHY'S  THIRD   MEMOIR   ON   THE 

then  assigning  to  m  all  possible  values  from  —  oc  to  +00,  the  functions 
Q».  qn  will  give  an  infinite  series  of  reciprocal  functions  relative  to  all 
the  transcendants  contained  in  ^  considered  as  the  variable  of  integra- 
tion ;  and  when  m  is  between  -  1  and  +  1,  pairs  of  reciprocal  functions 
will  be  obtained,  except  when  ?»  =  0,   when  both  coincide. 

In  this  series  are  included  the  trigonometrical  functions,  namely, 
when  m-= —\'.,  and  Laplace's  functions,  when  /«  =  0. 

In  all  the  reciprocal  functions  thus  arising,  there  exists  one  common 
property,  namely,  the  definite  integral  always  vanishes  between  the 
limits  which  make  the  functions  themselves  maxima  and  minima;  this 
remarkable  property  I  have  had  occasion  in  another  place  to  notice,  in 
the  particular  case  of  Laplace's  functions.* 

To  prove  this  generally  take  the  equations  of  the  preceding  article, 
viz. 

/#'^  +  (»»  +  l)(l-20.-^''  +  «(w  +  2»«  +  l)Q„  =  0, 

«'^  +  (?»  +  l)(l-20•-^+(«  +  l)(w-2»^)9„  =  O. 

Multiply  both  equations  by  {tiy,  and  integrate  reserving  the  con- 
stants under  the  integral  sign ;   hence, 

{tt'Y^^  ^  +  « («  +  2m  + 1)  j:  Q„  (tty  =  0, 

'      (^0'""''-^+  («  +  l)(«-2»^)/,^„  («')"*  =  0; 

and  changing  the  independant  variable  by  the  condition  7;r  =(^^')"'"»  we 
have 

(«')""+^^  +  w  («  +  2/»  +  1)  4  Q„  =  0, 

(<0*"*' ^  +  (»  + 1)  (»« -  2»w) /^  ^„  =  0. 

Electricity,  Introduction. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  335 

But  when  Q„,  q„  are  maxima  and  minima,  -^  and  -^  respectively 

vanish ;   therefore,  between  the  corresponding  limits  of  (p,  we  must  have 
U  Q»  =  0,   f^qn  =  0,  which  general  property  is  easily  verified  when 

Q„  =  «cos«0   and   q„  =  a  sin  {n  +  1)  (p. 
16.     To  find  the  complete  integral  of  the  differential  equation 
tt' ,  -^-^  +  (m  +  1)  (1  -  2t)  ^  +  n  (n  +  2m  +  1)  tr  =s  0, 
where  n  is  integer  and  m  any  constant. 

The  differential  equation  for  Q„  (Art.  15.)  is  of  the  same  form  as 
the  above  equation,  and  therefore  u=cQ„  is  a  particular  solution,  c 
being  an  arbitrary  constant. 

The  form  of  the  differential  equation  for  q„  will  become  the  same 
as  that  of  the  given  equation,  if  —  (w  +  1)  be  written  instead  of  n  in 
the  former;   hence,  another  particular  solution  is  c'q-^„+^y 

The  complete  solution  is  therefore 

u  =  cQ„  +  c'q.^„+iy 

This  solution  fails  first  when  m  =  0,  for  then  the  functions  Q„, 
5'-(«+i)  in  their  expanded  forms  become  both  identical  with  Laplace's  func- 
tion P„,  and  consequently  the  two  constants  c,  c'  merge  into  only  one, 
viz.  their  sum ;   but  if  we  put  generally 

b         ,     ,  b 

c  =  a  -\ —   and   c  = , 

m  m 


then   M=«Q„  +  &.^"~  ^-'"^' 

m 


And  putting  m  =  0,  the  latter  term  becomes  a  vanishing  fraction,  and 
therefore, 

u  =  aP,  +  ^^{Qn-  S'-(«4.i)}    when  m  =0. 


336  Mr  MURPHY'S   THIRD   MEMOIR   ON   THE  A 

The    term    by  which   b    is   here   multiplied,   is   the    coefficient  of  m  in 
Qn-q-i,„^^),  which  is  easily  found  from  the  expansions  in  Art.  13;  hence, 

n{n-\)   («  +  ])(w  +  2)  r        1  1  I  1  1     Ux,. 

The  general  solution  also  fails  when  m  is  an  integer,  for  then  some 
of  the  terms  in  the  expansion  of  Q„  or  g-.^+u  will  become  infinite,  and 
the  principle  of  vanishing  fractions  will  simply  enough  in  this  case 
also  be  applicable  in  determining  the  complete  solution ;  but  if  we  put 
for  Q„,  q„  their  differential  forms,  the  solution  will  never  fail,  for  the 
failures  arise  from  the  entrance  of  logarithms  into  the  result,  and  these 
will  actually  enter  in  the  latter  forms;  changing  our  constants,  the 
complete  solution  for  all  cases  is 

it  is  therefore   necessary  to  shew   that  the  functions  by  which  the  ar- 
bitrary constants  are  multiplied,  are  particular  solutions. 

Putting  v-itty-"^,  then  -t- =(»  +  »»)  (1-2^)  («')"+""-', 
and  -^  =(m  +  »?)  {n  +  m -  1)  (1  - 2tf  («')"+'"-^-  2  (?<  +«?)  {tt'f  *''-'. 

Hence  tt' .-^ -{n  +  m—\){l-2f) -r.-\-2{n-\-m)  .v-0, 
and  by  successive  differentiations  the  following  equations  arise: 
(«■').^-(«  +  '«-2)(l-20.^+2(2«  +  2«^-l)^=0, 

(«')f^-(»  +  ^«-3)(l-20.^+2(3«  +  3»e-3)g=0. 


d"v 
dF 


INVERSE   METHOD   OF  DEFINITE   INTEGRALS.  337 

and  the  law  of  the  successive  formation  of  these  equations  being  very 
simple,  we  have  generally 

(„.,^-(»+».-*-i)(.-.o^>.{(*+i)(»+»)-*i*±L>}.g=o. 

Put  k  =  n,  hence 

d"v 
Transpose  n(n  +  2m  +  l)-j—,  and  multiply  by  {tt')-",  hence 

from  which  it  follows  that  M  =  (it')""  .d".       '       satisfies  the  equation   of 

Art.  16. 

dv' 
Again  put  «j'  =  («')"'"*'"^^  or  tt' -j- +(n  +  m  +  l)(l-2t)v=0,  and  by 

successive  integrations  we  obtain 

tt' .  v'  +  (n  +  m)  (1  -2t)  ftv'  +  2  (n  -\-m)  ft'v  =0, 
tt' .  ftv'  +  {n  +  m-l)  (1  -2t) .  ft' v  -\-2(2n  +  2m-  1)  J^'v;  =  0, 
and  generally  * 

tt'  ft*-'v'  +  {n  +  m-k  +  l){l-2t)  ft'v'  +  2h{n  +  m)  -  ^  '^^~^^\ .  ft"-'  v==0. 

Put  k  =  n,  hence 

tt'  fr'  V  +  (m  +  1)  {1  -2t) .  ft"  v'  +  91  (n  +  2m  +  1) .  ft"*'  v'  =  0 ; 

from  which  it  appears  that  m  =  jJ" +*(»')  is  also  a  particular  solution,  and 
therefore  the  complete  solution  of  the  general  equation  is 


338  Mr  MURPHY'S  THIRD  MEMOIR  ON  THE 

Laplace's  equation  occurs  when  we  put  m  =  0,  and  therefore 

the  first  term  alone  of  which  is  the  type  of  Laplace's  functions,  the 
equation  is  therefore  more  general  than  the  functions  it  was  used  to 
designate. 

The  term  ^"+' («')-<"+"'+')  gives  n  +  1   constants   of  integration   which 

enter   as   coefficients   of  the   appendage  which  is   a  rational  function   of 

n   dimensions,   but   this   must  be  rejected,  since   the  constants   must  be 

determined  so  that  the  rational  function   of  n   dimensions   may   satisfy 

the   given  equation,    and   this   only   identifies    the   appendage   with    the 

d'  ift'Y^"' 
other  term  in  u,  viz.  aitf)"" —    ,  / — . 

17.     To  find  explicitly  the  omitted  part  of  the   complete  integral  in 
Laplace's  equation. 

The  general  equation  of  Art.  16.  becomes  in  this  instance 

and  the  complete  solution  is 

u  =  a^^^^+bfr'{tt')-^'-'\ 

the  first  term  being  Laplace's  function,  and  the  second  the  transcendant, 
it  is  required  to  find  explicitly. 

Let  a,  /3  be  any  arbitrary  quantities,  then  we  have 

dar\t-a'  fi-a)   ~  ^-a    da^\t-a)^'^da\fi-a]  da'-'\t-al 

n{n-l)d^  /    1  X   d'-^  /    1   N 

^       1.2      Ma'[(i-a)  da"''[t-a)'       ^' 


or 


INVERSE   METHOD  OF   DEFINITE   INTEGRALS.  339 


hence 

^-/ 1 I 

+  1         I 


_  \«-a)(/3-a)|     ^  1  r        1  W  + 


,   (n  +  mn  +  2)         1 L_  +  &cl 


(«  +  !)(« +  2) 


Commuting  in  this  equation  the  quantities  a  and  /3,  we  have 

(»  +  !)(» +  2)         1  1 


,.,      '^"{(f-/3)(a-/3)}     _         1        r.      ^  ^  +  11  1 


(»  +  l)U  +  2)         1  1  1 

1.2        ■  {a-(iy{t^(iy^  ^^■j 


If  both  equations  be  added  observing  that 


1  1 

+ 


(#-a)(/3-a)   ^  (/-/3)(«-/3)         {t-a){^-t)' 

the  sum  of  the  left-hand  members 

fpn  £ 


1^2'.3^...M'«?a"C?/3" 


rf".-^  ef-.   ^ 


^_1)„^  ^-«  /3-^ 


1.2.3...wrfa"'  1  .2.3...W6?/3" 
1  1 


Vol.  V.     Part   III.  Yy 


340  Mr  MURPHY'S  THIRD  MEMOIR  ON  THE 

Hence,  we  get  the  general  identity, 

1  1 


K^-«)(/3-^)l"^'        (/3-ar' 

i  1  ,^  +  11 1  (n  +  l){n  +  2)         1  _JL_&cl 

(^-ar'  "^       1      -fi-a-it-ay^  1.2  •(/?- a)^ '  (#-«)'-        '[ 

^_^ ^^±1    ^^ 1  («  +  l)(>^  +  2)  1  _J__£,c 

■^(/S-O'*""^      1      •/3-a-(/3-0"  1.2  •(/3-af-(/3-#)'-'        •] 


Put  now  a  =  0,  /3  =  1,  and  therefore  (i  -  t  =  f,  hence, 
>+'  "^      1      •/"  "^  1.2  ■^"-' 

[+F^"*"    1    V""^         172        •^-^  +  *'c.j 

in  which  identity  n  must  be  one  of  the  natural  numbers  0,  1,  2,  3,  &c. 
and  the  number  of  terms  in  each  series  must  be  limited  to  w-f  1. 

Suppose  the  (ra  + 1)*  successive  integral  of  each  term  of  this  expansion 

is   taken  after  multiplying,   for  convenience,  by   1.2.3 n,  the  result 

will  consist, 

1st,  of  a  logarithmic  part,  viz.  . 

(-,)-.h.i.w{i-f.^.^.^^.<''";'.'r''''-M 

where   the    part  between   brackets    in   the   upper   line    is    equivalent   to 
the  function  P,„  and  in  the  lower  to  (-1)".P„,  and  therefore  the  whole 

to  (-l)".P„.h.l.  ^,. 

*  This  method  is  applicable  in  every  case  to  the  decomposition  of  fractions,  the  denomi- 
nators of  which  contain  equal  factors. 


INVERSE   METHOD  OF  DEFINITE   INTEGRALS.  341 

2d,  of  a  rational  and  entire  function  />„  which  satisfies  the  equation, 


dP 

since  the  term  2  ( -  1)".  -^  is  the  result  which  arises  if  the  logarithmic 

term  ( —  1 )"  P„ .  h.  1.  ->  be  put  for  u  in  the  actual  equation. 

3d,    of  an   appendage   containing  n  +  1   arbitrary   constants,   which  as 
before  remarked  must  be  rejected  altogether. 

Differentiating  the  equation  for  p„  above  obtained,  we  get 

(«',^-  +  Mi-20.^-+>-i)(«+^)#  +  2(-ir.^--o,. 


(«-l)(l-20.^"  +  2(2«-l)^^"  +  2(-l)»^^=0, 

•  :         ■  ^"   df-^    +  ^^^>   ~dF~^' 

when  these  equations  terminate,  since  j9„  is  of  « —  1  dimensions. 

Put  ^  =  0,  in  all  these  equations  beginning  with  the  last,  observing 
that  then 

^  =  (-l)-.(«  +  l)(»  +  2)...(2«), 
^^  =  -  (  -  1)" .  «  («  +  1)  («  +  2)...(2«  -  1), 

'^=^-^)"-^^i^^-^'*''^)^''  +  ^) (2«-2),&c. 

Y  Y2 


342  Mr  MURPHY'S   THIRD   MEMOIR   ON   THE 

Hence  -^^  =  - 2.  («  +  !)(» +  2)... (2»- 1), 

^^"  =  (m  +  1)(«  +  2)...(2m-1),  &c. 
and  the  value  of  j9„  is  the  rational  function 

^  1.2...{n-l) ^^      +^'^      +A,f    ...+A„^,], 

in  which  the  coefficients  are  successively  formed  from  the  equation 

{n-m-lf.A„  +  {m  +  2){2n-m-l).A„^i 

+  2(-ir    "i^-'^)-("^  +  ^)     n{n-l)...(n-m-l)    _ 
'    '1.2...{n-m-l)'  2n{2n-l)...{2n-m) 

and  the  omitted  part  in  the  integral  of  the  proposed  equation  is 

6|p„h.l.  (I)  +  (-l)».^. 

18.  When  m  =  —^,  the  general  equation  of  Art.  16.  becomes 

and  putting  ^  =  cos"^(l  — 2#),  we  have  Q„  =  cosn<p,  §'_,„+,)  =  sin  ncp,    the 
complete  solution  is  therefore  M  =  a  cos«^  +  6  sin  «^. 

Though  the  trigonometrical  functions  were  the  first  used  in  analysis 
as  reciprocals,  for  the  purposes  of  expressing  functions  by  means  of 
definite  integrals  and  of  expanding  them,  in  the  former  instance  of 
their  application  there  remain  a  few  remarkable  cases  which  do  not 
seem  to  have  been  noticed,  with  which  we  shall  conclude  this  Section. 

19.  The  two  functions  which  possess  the  remarkable  properties  al- 
luded to,  are 

e  =  e^'<«» .  COS  {x  sin  &),  and  6'  =  e^  ""^^  sin  {x  sin  B). 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  343 

The  successive  differential  coefficients  with  respect  to  x  of  the  func- 
tions 0,  6'  follow  simple  and  elegant  laws,  thus 

do  dQ' 

=  6^'=°'*  cos  fa:  sin  0  +  0},  ^— =  e^'=»*«  sin  {a;  sin  0  +  0}, 

d'Q  d^Q' 

d^  =  6"^"°'' cos  {ar  sin  0  +  20},        ^^=  e^'^"**  sin  {«  sin  0 +  20}, 

and  generally 

d"  0  d°  0' 

•  ^-;  =  e^'°'^  cos  {x  sin  0  +  «0},        -j—  =  e^  ">"*  sin  {x  sin  0  +  «0} .     ■ 

Again,   the   successive  integrals  relative  to  x,   follow  the  same  laws, 
omitting  the  arbitrary  constants  of  integration, 

/^0  =  e^cose  cos  {a;  sin  0-0},  /,©'  =  e^<=°'»  sin  {«  sin  0-0}, 

//e  =  e^cose  cos  1^  sin  0  -  20},  f,'Q'  =  e^'^"^*  sin  {x  sin  0-20}, 

fj-Q  =  e^'^"'^  COS  far  sin0-w0},  f/O'  =  6^<=<««  sin  {a;sin0-«0}, 

for  it  will  readily  be  seen  by  actual  differentiation  that 

d"  d" 

0  =  ^-;;  {e^'="'''cos(xsin0-«0)},         0'  =  T-^  i^icose  sin  (a;  sin0- m0)}. 

Again,  changing  the  forms  of  the  proposed  fimctions,  we  get 

0  =  1  {e-'^  +  6"-'^^},         0'  =  -4==  {e"'^^  -  e"''"^'}, 

whence,  expanding  and  passing  from  the  exponential  to  trigonometrical 
functions 

a;*  of 

0  =  1  +  a;  cos  0  +  - — -  .  cos  20  +  ,    .    ^,  cos  30  +  &c. 
1.2  1.2.3 

0'  =  a;  sin  0  +  — —  .  sin  20  +  ,    ^   „  sin  30  +  &c. 

1.2  1.2.3 


344  Mr  MURPHY'S   THIRD    MEMOIR   ON    THE 


jeW  cos  wy  —  -.  j— g— g       ^    the  limits  of  0  being  0  and  ir,  these  formulee 

\      apply  for  all   integer  values   of  n,   except 

Now  e  cos  nO  ±  e'  sin  nO  =  e^^s*  cos  {x  sin  0  +  «0|. 

Hence   /ee"""^  cos  {arsin^- w0|  =7r  .  — — -— , 

l^^xcose  cos  {arsin0  +  wej  =0. 

The  particular  case  where  w  =  0  is  included  in  the  first  of  these  two 
equations. 

20.  By  the  results  thus  obtained,  we  are  enabled  to  represent  any 
rational  and  integer  function  of  a;  in  a  form  adapted  to  general  differen- 
tiation. 

By  applying  Maclaurin's  theorem,  we  first  have 

(}>(x)  =  Ao  +  A,.x  +^2- j^  +  ^3  -^    g   3  +  &c.; 

and  passing  to  definite  integrals  by  the  formulae  of  the  last  article, 

(h{x)  =  -  /ge^cose  1^^  cos  (x  sin  9)  +  A^  cos  {x  sin  6-9) 

+  ^2  cos  (.r  sin  0 -  20)  +  &c.  J 
also  if  A^u  A^i,  A_3,  &c.  represent  arbitrary  constants, 

0  =  -  /ee^'=°^*  {A-i  cos(x  sin  0  +  0)  +  ^_2  cos  (ar sin  0  +  20) 

+  ^_3  cos  (a;  sin  0  +  30)  +  &c.} 

both  of  which  integrals  must  be  added  before  <p  (x)  can  be  subjected  in 
a  complete  form  to  general  differentiation. 

We  then  obtain  the  w*''  differential  coefficient  by  adding  n9  under 
each  cosine  in  this  sum,  that  is. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  345 

'-pp-  =  i /ee'-^o'S  {Ao  cos  (x  sin  0  +  n9)  +  A,  cos  [a; sin  d  +  {n-l)e'\ 
(toe  '^ 

+  ^a  cos  [ic  sin  0  +(»  — 2)0]  +  &c.} 
+  _|ge'^cose  |^_jCos[xsine  +  (re  +  l)0]  +  ^_,cos[xsin0  +  (w  +  2)0] 

+  ^_3  cos  [a;  sin  0  +  (w  +  3)  6*]  +  &c4 . 

I.  When  n  «'*  a  positive  integer,  the  whole  of  the  second  line 
vanishes,  there  will  then  be  no  arbitrary  constants;  also,  the  first  n 
terms  of  the  upper  line  disappear. 

II.  When  n  is  a  negative'  integer,  the  first  n  terms  of  the  second 
line  remain,  and  these  contain  n  arbitrary  constants. 

III.  When  n  is  jractional,  the  whole  of  the  second  line  remains, 
giving  an  infinite  number  of  constants. 

21.  The  theory  of  numbers  as  connected  with  definite  integrals, 
afibrds   another  remarkable   application  of  reciprocal  functions. 

Let  n  be  any  integer  of  which  the  divisors  are  n,  Ji',  n" 1;  also 

let  m  be  any  intger,  and  d  an  arc  of  which  the  limits  are  0,  tt. 

Then,  generally, 

1  -2Acos»?0  +  A''  =  (l  -  A6'»e^^)(l- Ae-"*^^); 
and  hence, 
h.  1.  (1  —  2  A  cos  ra 0  +  A")  =  -  2  {  A  cos  m 0  +  ^  ^'  cos  2 »J  0  +  ^  A^  cos  3  /w  0  +  &c.  I . 

Suppose  now   that   m  is   one    of  the   numbers   n,   n',   n" 1;    this 

series  must  contain  one  term  involving  cos»0,  viz. 


—  A^cos  w0: 
n 


and  therefore. 


Tit         — 

j^cosw^h.l.  (1  —  2Acos»»0  +  A^)  =  —  TT.  — .  A"". 

But  when   m   is   not   a   divisor   of  n,  there  will  be  no  term  in   the 
expansion  found  to  contain  the  arc  n9,  and  therefore, 
^cos«0h.l.  (1 -2ACOSJW0  + A^)  =  0. 


346  Mil  MURPHY'S  THIRD    MEMOIR   ON   THE 

Put  now  for  m  successively  every  integer  from  1  to  w  inclusive,  and 
take  the  sum  of  all  the  definite  integrals  thus  resulting,  hence 

/ecos»0h.l.  {(l-2Acos0  +  A^)(l-2Acos2e  +  A^)...(l-2Acos«0  +  A*)} 

\n    ,       w'    -^,       w"    -4,  1,1 

=  -  ttX-  .h  +  -  .h"  +  — .  A"    +  ...-  .  h"). 
\n  n  n  »       J 

Now   the   quantities  -,   — ,   — ,   &c.    are    the    reciprocals   of   all   the 

Tt        ft         Ti' 

possible  divisors  of  n,  and  therefore  this  definite  integral  may   also   be 
expressed  by 

"^  -^{k  +  -,h'''  +  \h""  +  ...-h"}. 

'^         n'  n  n     ' 

For  9   in   the  preceding  equation   write   20,  the   limits  of  the  latter 

variable  will  be  0  and  - . 

2 

Also  put  h  =  1,  and  therefore, 

1  -2hcosd  +  h'  =  2{l-cos2(p)  =  4!sm^(f), 
1  -  2A  cos  20  +  ^2  =  2 (1  -  cos  40)  =  4  sin' 20, 
&c. ; 
.-.  h.l.  {{l-2heos9  +  h')  {I  -2hcos2e  +  h'')...{l  -  2hcosne  +  h')} 
=  2w  h.  1,  (2)  +  2  h.  1.  {sin  0  sin  20. ..sin  w0}. 

The  integral  of  the  constant  multiplied  by  cos2«0  vanishes,  and  therefore 
7^  h.l.  {sin  0  sin  20  sin  30. .. sin  »0}  ,  cos2»0  =  —  t|~  +  —  +— ,  +...4-ll; 
and  multiplying  both   sides   by ,  we  get  this  theorem. 

The   sum   of  all   the   divisors   of  a  given   number   n,   including   the 
number  itself  and  unity,  is  expressed  by  the  definite  integral 

4t7l 

/^h.  1.  {sin  0  sin  20 sin  30. .. sin  w0}  .  cos2w0. 


INVERSE    METHOD   OF   DEFINITE    INTEGRALS.  347 

SECTION    VII. 
On  Transient  Functions. 


22.  Let  ^  (h,  t)  be  such  that  when  h  has  a  particular  value  as- 
signed, the  whole  function  vanishes  whatever  may  be  the  value  of  t, 
except  in  one  case ;  0  (/^,  t)  under  those  circumstances,  is  a  transient 
function  having  only  a  momentary  existence. 

Thus  the  function  _  ,  (^ —0.t\l.hH^'  ^^*^"  ^'  ^^  P"^  equal  to 
unity  is  a  transient  function,  because  its  value  is  zero  in  every  case 
except  when  t  =  0,  for  then  it  becomes  t- — j-^  when  h  is  put  equal 
to  1,  that  is,  it  acquires  momentarily  an  infinite  value. 

If  the  value  of  the  function  had  been  always  zero,  its  definite  inte- 
gral relative  to  t  would  also  be  zero;  but  if  we  actually  integrate  from 
^  =  0  to  t  =  \  without  previously  assigning  a  particular  value  to  h,  the 
definite  integral 

2A  \\-h       \+h\~    ' 

thus  this  integral  is  independent  of  h,  and  therefore  remains  the  same 
when  h=\,  that  is,  for  the  transient  function. 

By  the  principles  of  the  Second  Memoir  we  can  always  form  a 
self-reciprocal  function  in  which  the  general  term  may  be  of  any  par- 
ticular kind ;  thus  if  f{t,  n)  were  the  type  of  the  general  term,  and 
if  we  put  generally, 

Fit,  n)  =  a,f{t,  0)+a,f{t,  l)  +  a^f{t,  2)+ +  a„fit,  n), 

lastly,  if  we  determine  the  coefficients  a,,  «2, a„  in  terms  of  «„  and  n, 

by  the  n  equations  (arising  from  the  definite  integrals)  following, 

f,F{i,n).fit,0)=0, 
Vol.  V.     Part  III.  Zz 


348  Mr  MURPHY'S  THIRD   MEMOIR  ON   THE 

!,F(t,  n)  ./{t,  1)  =  0, 


SF{t,n).f(t,n-l)  =  0; 
then  the  function  F(t,  n)  will  obviously  be  self-reciprocal. 

But  if  f{t,  n)  not  containing  arbitrary  coefficients,  but  being  abso- 
lutely given  as  P,  (cos^%  &c.  is  proposed  as  a  function  to  which  some 
unknown  function  is  reciprocal,  the  discovery  of  the  latter,  which  is 
effected  in  the  next  article,  is  of  a  more  difficult  nature  than  the  pro- 
cess above  mentioned;  and  in  the  particular  cases  quoted,  as  well  as  in 
many  others,  this  required  function  is  transient,  it  is  therefore  in  this 
character  that  transient  functions  are  here  introduced. 

23.  Given  f  (t,  n)  a  Junction  of  known  form  with  respect  to  the  vari- 
able t  and  the  integer  n,  it  is  required  to  find  another  Junction  of  t  and 
n,  as  ^  (t,  n),  such  that  the  definite  integral  jjf  (t,  n)  ^(t,  n')  may  always 
vanish  when  the  integers  n  and  n'  are  unequal. 

Begin  with  forming  a  self -reciprocal  function  F{t,  n),  the  general 
term  of  which  may  be  of  the  given  form  J{t,  n) ;   thus 

F{t,  n)  =  a,f{t,  0)  +  a,f{t,  l)+a,f{t,  2)+ +a„f{t,  n), 

where   the   coefficients   are   determined   in    the  manner  indicated  in   the 
preceding  article. 

Suppose  next  that  the  required  function  0  {t,  n)  is  expanded  in  an 
infinite  series  of  which  the  general  term  is  of  the  form  F  (t,  n),  thus 

<p{t,  n):=Ao.F(t,0)+A,F{t,l)  +  ...+A„F{t,n)  +  A„^,F{t,  (n  +  l)},  &c. 

Multiply  by  f(t,  0),  f(t,  1),  f{t,  2) f(t,  n  -  1)    successively,   and 

integrate  the  products  between  the  given   limits  of  t,  observing  that 

f,F{t,  1)  .fit,  0)  =  0,         f,F(t,  2)  .fit,  0)  =  0. ..J,F{t,  n)f{t,  0)  =  0, 

by  the  property  of  the  functions  F  {t,  n) ; 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  M9 

and  similarly, 

jlFit,  2)  fit,  1)  =  0,         f^Fit,  3)/{t,  l)  =  0...f,F{t,  n)  .f{t,  1)  =  0, 
&c.     &c., 
we  thus  obtain  the  following  equations ; 

j;  0  {t,  n)  .fit,  0)  =  A,  j,f{t,  0) .  F  (f,  0), 
^,<^{t,  n)  .fit,  1)  =  A,S,f{t,  1) .  F{t,  1), 


^0  (A  n)  .fit,  w  -  1)  =  A._,!,f{t,  n)  .  Fit,  n-l); 

hence  the  imposed  condition  of  reciprocity  requires  that  the  first  n  co- 
efficients Ao,  Ai...A„-i  in  the  expansion  of  0(#,  w),  may  be  each  equal 
to  zero ;   and  therefore, 

0(^,  n)=A„F{t,  n)  +  A„^,F{t,  n  +  1)  +A„+,F{t,  n  +  2),  &c.  ad  inf. 

Multiply  successively  both  sides  by  f{t,  n  +  1),  f{t,  n  +  2),  &;c.,  and 
integrate;  and  since  n  +  \,  n  +  2,  &c.  are  each  >  n,  the  definite  integrals 
must  vanish. 

Hence, 

AJ,F{t,  n)  .fit,  n  +  l)  +  A„^J,Fit,  n  +  1)  .fit,  m  +  1)  =  0, 

AJtFit,  n)  .fit,  »  +  2)  +  An^,^,Fit,  w  +  1)  .fit,  n  +  2) 

+  A„^2ftFit,n  +  2).fit,n  +  2)  =  0, 
&c.     &c., 

from  whence  the  coefficients  An+i,  ^„+2,  &c.  are  known  in  terms  of  A^ 
and  ti,  and  therefore  the  required  function  ^  it,  n)  is  known. 

24.     To  find  the  function  which  is  reciprocal  to  t°. 

First,  we  must  form  a  self-reciprocal  function,  of  which  the  general 
term  is  of  the  form  /";  this  has  been  already  effected  in  Section 
IV.,  namely, 

:,.      n    n  +  1  ^  ,  nin-\)     in  +  !)(»  +  2)     ^ 
^,,-1-  j.-y-   t+       ^^       .  j-^  .t  -&C.. 

z  z2 


350  Ma  MURPHY'S   THIRD   MEMOIR   ON   THE 

which  has  been  also  proved  to  be  the  coefficient  of  A",  in  the  ex- 
pansion of    {l  —  2h{l  —  2f)  +  h^\~^,  (Section  IV.  Art.  9),   and  to  be  equal 

cl"  (tt'Y 
*^  1 — oQ — ~Tf^'   where  t'  =  \  —  t,  (Section  iv.  Art.  2.) 

Then   representing  by  V„  the  required   function  which  is   reciprocal 
to  f,  we  have  by  the  preceding  article 

where  it  is  obvious  that  when  n'  is  less  than  w  fiVj"' =  0,  and  it  is 
only  necessary  that  the  coefficients  may  be  so  determined,  that  the 
same  equation  may  remain  true  when  n  is  greater  than  n ;  and  since 
one  of  these  coefficients  is  arbitrary,  we  may  put  ^„  =  1. 

Now  in  general,  we  have  by  Section  iv.  Art.  2. 

x{x~l)  {x  —  2)...{x-n  +  \) 


f,Pj''  =  {-iy. 


(a;  +  1)  (a:  +  2)  (a;  +  3)...(;r  +  w  +  1) ' 

hence,  i  F„  #" +^  =  (  -  1 )"  { 7 ^^ zr~r ^^-h-r^ — v^ 

■'  ^       ^   \(w +  a;  + 1)  (w  +  a;  +  2)...(2«  +  ar  +  l) 

_.  {n-¥x)  {n+x-\)...x  .  {n+x)  {n+x~\)...{x—l)         .     1 

~     "*'■  (w+ar+l)(ra+ar+2)...(2w+a;+2)^     '"'"{n+x+\){n+x+9)...{'in+x+S)~       ] 

Therefore,   when  x  is  any  integer  from  1  to  x ,  we  must  have 

A  ^  A  X{X  —  Y) 


2w  +  X  +  2  "■"     (2»  +  a;  +  2)  (2 w  +  j;  +  3) 

.        x{x-\)  (;r-2) 

~     "*"  (2»  +  x  +  2)  (2w  +  a;  +  3)  (2w  +  ar  +  4)  "^       ' 

and  putting  for  x  the  successive  integrals  1,  2,  3,  &c. 
1 


0  =  l-^„+i. 


2m +  3' 


«     ,        ^  2  .  2.1 

0=1— .4„+i.  X—-—:  -r-4„+2 


2«  +  4  "+'■  (2w  +  4)(2m  +  5)' 

^     ,         .  3  .  3.2  .  3.2.1 


2w+5  ^"""+^-  (2w+5)  (2w+6)  "+"  (2«+5)  (2«+6)  (2m+7)* 

&c.      &c. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  351 

From  whence  we  obtain 

A„^,  =  2n  +  3,     A„^,=    ^   ^    .(2n  +  5),     Jn+3=~ fgg '.(2n  +  7), 

and   to   prove  that   this  law   of  formation   is   general,    we   may  observe 
that  since 

/        iy+2^+i_         2n  +  2x+l     1  (2w  +  2a;  +  1)  (2n  +  2x)...{x  +  1) 

V~  h)  ~  1  '  h^ "^  1.2...(2w  +  ^  +  l) 

iy+^-^'    I  X  1  x.jx-l)  i,^\ 

^  \      hi  '\        2n  +  x  +  2' h       {2n  +  x  +  2){2n  +  x  +  S)' K"  ]' 

Qfi  4-  2 
and   (l-hy^'-^'Hl  +  h)  =  l  +  {2n+.3) .  h  +  . {2n  +  5).k' 

(2«  +  2)  (2w  +  3)    ,„        „.     ,3      J 
+  ^ o    9 •  ^^^  +  7).h^  +  &c. 

Multiply  both,  and  take  the  coefficient  of  ,^„^^^,  in  the  products, 
and  we  get 

{2n  +  2x  +  ])  {2n  +  2x),..{x  +  1)  ^ x  , 

\.2.3...{2n+x  +  \)  *    ~  2n  +  x  +  2'^  ' 

x{x-l)  (2w  +  2)(2w  +  5) 

■*"(2w  +  a;  +  2)  (2«  +  a;  +  3)'  1.2  '         ^^ 

=  coefficient  of  -1-  in  (-IV    (l+^Xl-^r'' 

=  (-1)'.  coefficient  of  A'  in  (1 +^)  (1-^)^'-. 

Now  the  coefficient  of  h"  in  (1 +^)  (1 -Af""',  is  evidently  the  sum 
of  the  coefficients  of  h'~\  and  of  h\  in  the  expansion  of  (1  — A)*'-'; 
that  is,  the  sum  of  the  coefficients  of  the  two  middle  terms  in  a 
binomial  raised  to  an   odd   power,  and  with    alternate  signs   of  +   and 


352  Mr  MURPHY'S  THIRD  MEMOIR  ON  THE 

— ,   hence   the   quantity   we  are  considering    must   be   zero,    and   there- 
fore 

2w  +  3  X  {2n  +  2)  (2w  +  5)    x(x—l) 

~  i       '2ra  +  a;  +  2'*'  1.2  '  (2n +  x +  2)  {2n  +  x  +  3) 

_  (2»  +  2)  (2?^ +  3)  i2n  +  7)     x{x-l)  {x-2) „ 

~  1.2.3  '  {2n  +  x  +  2){2n  +  x  +  3){2n  +  x +  4>)  "' 

which  shews  the  generality  of  the  observed  law  of  the  coefficients  ^„+i, 

Substituting  now    these  values  in   the    general   formula   for    V,„    we 
get  the  required  function  which  is  reciprocal  to  t",  namely, 

rr-        r.        /„          „v      T,            (2«  +  2)(2w  +  5)      „ 
K  =  P„+{2n  +  3).F„^,  +  ^ ^ ^  .  P,+8 

(2w  +  2)  (2w  +  3)  (2w  +  7)     „         „ 
"-  17273  •^"*^'  *'''• 

25.     The  Junction  which  is  reciprocal  to  t"  is  transient. 
For  in  general 

d".(tt'Y     _  d^itt'T 


\.2.3...ndt''      ^       ^  '  1.2.3. ..ndt'"' 


and    putting    1-t'    for    t,    and    expanding    the   binomial    (l-t')",    and 
lastly  actually  performing  the  differentiations  indicated,  we  have 

^         A)    .^n-A  11*^1.2  1.2 

and  therefore 

V-l)  ''n-JA      ii'^+i2  1.2  * 

-(2w  +  3)|i T"  •  ~T~         ~~T72~~  •         1T2  ' 

(2«+2)  (2W+5)  ,,      »+2     n+3    -     (w+2)_(^+l)     (w+3)  (w+4)    ,,2_^    > 
+ 172  t^~     11  1.2         •         1.2         ■  "^ 

-&c.  &c. 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS  353 

The  term  which  is  independent  of  t'  is 

but  in  general  we  have 

(l-A)(l+A)-(^"+^'  =  l-(2«  +  3).A  +(^"  +  ^^)(|^  +  5)    ^._^^ 

and  putting  h  =  \,  we  find  that  the  term  independent  of  if  is  zero. 
Again,  multiplying  the  last  equation  by  A"+'",  we  get 
(*"+"■  -  A"*""*')  (1  +^)-<'"+'*  =  A"+'» -  (2 w  +  3)  .  A"-^""*'  +  (^"+yv^"+^) .  ^.+".. >^  _  &c. 
Now  it  is  easily  seen  that  when  h  =  1,  we  have 

-Tj^  (A"+'"-A"+"'+')  =  (w+w)(«+»?-l)...(«-/»+l)-(«+/»  +  l)(«+»»)...(«-w+2) 
=  —  2»w  .  (w  +  >w)  (w  +  TW  —  l)....(w  — m  +  2), 


-(A"+'"-A"+'"+')=  -  (2m-l)(«+m)  («+m-l)....(«-»w+3). 


&c.  &e. 

and  therefore  when  A  is  put  =1  after  differentiation,  we  have 

Jim 

_^^  {(A"+'»-A"+'"+>)(l+A)-'"+'}  =  -2-<""+''.2>«. («+»«)  (w+»«-l)...(w-M  +  2)x 

,        ,    2w+2      2OT-1  1     (2w  +  2)(2w  +  3)        (2m  - 1)  (2?»-2)  .      . 

*        ^*      i       ■«-»^^-2"^2^■  1.2  ■(w-w  +  2)(»-»»  +  3)        *^"» 

which   series   consists   of  only    Im   terms,    and   is   equal   to   the   infinite 
series  obtained  by  differentiating  the  other  side  of  the  equation,  viz. 

w(«  — !)....(« -OT+l)x  (w  +  l)(«  +  2)....(w  +  »«) 

-  (2«+3. («+!)». ...(«-»»  +  2)  X  (w  +  2)(»  +  3)....(»  +  »«  +  l) 

+  (^^+^)(^"  +  5)  (n  +  2)(w  +  l)...■(w-»^+3)x(w+3)(w  +  4)....(w  +  »?  +  2) 

—  &c.  oe?  infinitum. 


354  Mk  MURPHY'S  THIRD  MEMOIR  ON  THE 

Now  it  is   obvious   by   putting  m  =  1,  2,  &c.    successively,   that  the 
finite   series   is   always   =0,    and   therefore   the  infinite  series   [which   is 

( —  t'Y 
the  same  as  the  coefficient  of  ^     ^     — ^  in  the  expression  for  (  — 1)°^„] 

vanishes  also,  so  that  if  V„  be  arranged  according  to  the  powers  of  t', 
it  is  0  +  0.  ^' +0^'^  +  &c,,  nevertheless  its  value  is  in  one  instance  infinite, 
namely,  when  t  =  0,  for  then  P„  =  P„+i  =  &c.  =  1,  and  therefore 

F„  =  l  +  (2.  +  8)  +  ("^  +  f^f-^^^^^"  +  ^)^fV")(^^  +  ^)  +  &c. 

=  (l+A)(]-A)-«''+^  when  h  is  put   =1. 

=       X  . 

And   if    V„   did  not   possess   this  infinite   element  ft  Vj",  from   i>  =  0 
to  t  =  1  would  vanish,  whereas  its  actual  value  is  the  same  as 

26.     To  express  the  transient  Junction  Vn  in  a  finite  form . 
Since  by  Art.  (24.)  K  =  P,  +  (2w  +  3)P„+, 

,   (2w  +  2)(2w  +  5)                 (2«  +  2)(2w  +  3)(2w+7)     «        „ 
"•  j~^2  ■      '^^  2    2   3 •  "..+3,  «c. 

therefore 

1.2.3...(2w+l)  r'„  =  l  .2.  3.. .2??  X  (2«  +  l)P„ 

+  2.3...(2w  +  l)  X  (2w  +  3)P„+,./f  +  3.4...(2w  +  2)  x  (2«  +  5)P,+s^\  &c. 
when  A  is  put  equal  to  unity. 

But  in  general, 

{1  -  2A  (1  -  20  +  K"]  -^  =  P„  +  P,h  +  P^A^  +  ...P„A"  +  P„+, A"*'  +  &c. ; 
c?^''^"{l-2^(l-2j?)+^'|-i^ 

=  1.2.3...2?«.P„  +  2.3,.,(2w4-l).P„+,  A  +  3.4...(2«  +  2).P„+2AH&c. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  355 

Multiply  by  2A"+i,  and  diiFerentiating  once  more,  we  get 

^Thr  djf^ 1 

=  1.2.3...2wx(2«  +  l)P„A'-i  +  2.3...(2w  +  l)x(2»  +  3)P„+,A"+J+&c. 

Hence,   F„  =  ^h-^^ .  j.  [jf^l  ^^  j^  I  "^ '  "  "^>  -^,f  ^ . 

c?A\  1.2.3...(2«  +  1).</A^°     j' 

when  A  is  put   =  1. 

Put  for  abridgment  the  radical   {1  —  2A  (1  -  2#) -f  A^}-J  =  ^,  then 

rf^ . {Rh")  _  2«.(2w-l)...(w  +  l)  ^^B 

rfA^       ~  1.2...«  •     •  </A'' 

.   2w(2w-l)...w   ^    ^        ,c?"+'J?       . 
^  '      '  '  \c?A"  W  +  1     1      C?>&»+1 

w(w-l)        _A^  c?»+^B  1 

(«  +  l)(«  +  2)'1.2'rfA»*'"  *''^7 

Whence    2  ~  {^-i  ^^^^|  =  2»  (2«- !)...(« +  1)  {(2«  +  1)  A«-4  ^ 

2w  +  3       n       ,^^  d'^'R      2n  +  5  h"*^ .n .{n-l)   d'^^R 
■^       1       •«  +  !  •  dh"^'   "^    1  .  2    *  (»  +  l)  (w  +  2)  ••  d¥^  ■•■  *'*'• 

c?A"+'         w  + 1      c?A"+-  J 

u           100        Tr       d'R   .    2w  +  3       «       .d^^'R 
Hence  1 .2.3.. .wF^,  = -jT- +  r — -^. .^     „  ,, 

2w  +  5        w  (w-1)  ^^      rf»^^jB 

"''  2»  +  1  ■  (»  +  1)  (ra  +  2)  ■  1 .  2  •  rfF^  "*■  *'''• 

2       ^^  c?"^^Jg  2  n  d'^'R 

2«  + 1   *  t/A"+'    "^  2w  +  l»  +  l        ■  </A"+^ 

2  n(n-l)         h^      d'^^R 

2w  +  l*(w  +  l)(«  +  2)*1.2'  <^A»^'  ■•"  *'^- 
Vol.  V.    Part  HI.  3  A 


356  Mr  MURPHY'S   THIRD   MEMOIR   ON   THE 

h  being  put  =  1,   after  the  differentiations;    this  value  of  1.2,..?iF^,  is 
expressed  in  two  finite  series,  each  containing  only  w  + 1  terms. 

If  we   actually   add   the   terms    in   this  formula,   which   contain    the 
same  powers  of  A,  we  get 


V  -        1         K:?       w  +  2    h    d"^^R  (n  +  3) 

"~  1 .  2...W  \dh"   "^  w  +  1  ■  1  '  dh"^'    "^  (w  +  1)  (« 


,  n 


h'      d'^'R 


+ 


n  +  2)'  1.2"  c?A"+' 

(«  +  4).ra(w-l)  k"       d'+^E 


^  SL        1 


{n  +  l){n  +  2){n  +  3)'l.2.3'  dh 
when  h  is  put  equal  to  unity. 

27.     -Discussion  of  the  transient  function  N ^. 

Put  «  =  0  in  the  general  expression  for  ^„  in  the  preceding  article, 

dR 

hence  V.  =  R  -^  2h  -y^- 

dh 

=  U-2A(l-20+A^}-^  +  2A(l-2jf-A)  {l-2A(l-2^)  +  A^}-i 
(1-A)(1+A)  ,         ,   . 

This  function,  as  has  been  observed  in  Section  vii.  (22),  is  in  general 
zero,  except  in  the  particular  case  when  ^  =  0,  when  its  value  is  infinite. 

If  we  imagine  a  curve  of  which  the  equation  is 

y       {l-2A(l-2a;)  +  ^^}4' 

where  h  is  less  than  unity  but  nearly  equal  to  it,  the  limiting  values  of 
y  as  A  approaches  unity,  will  give  the  geometrical  interpretation  of 
the  transient  function   V^. 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS. 


357 


Take  (Fig.  1.)  AB  =  1,  AH  =  1  -  h,  or 
BH  =  h  both  along  the  axis  of  x,  and  make 
A    the    origin,    then    putting    x  =  0,    we    have 

1  -i-  h 

y  —  jz — T7^,  which  is  very  great,  and  tends  to  be 

infinite  as  h  approaches  unity,  and  is  represented 
by  AC\  next  putting  x  =  l  —  h  =  AH,  we  get  the 

corresponding    ordinate  HE  =  (jZThf) '  (T+W' 
which  also  tends  to  infinity  ;  lastly,  putting  x  =  l 


we  have  y  = 


l-h 


=  BD,  which  tends  to  vanish 


(1  +  h)i 
in  the  ultimate  case  representing  V^. 

Now  varying  the  parameter  h  so  as  to  make 
it  approach  unity,  the  points  C  and  £1  recede 
indefinitely  from  the  axis  of  x,  and  the  point 
7>  approaches  it  indefinitely. 

Yet  the  area  DBACE  remains  constant  (for 
the   integral   between   x  =  0,   and   x  =  \    of 

{l-2^(l-2x)  +  A''}J 
relative  to  x  is  evidently  unity). 

And   the  altitude   GN  of  the  centre  of  gravity  of  this  area   is   also 
constant,  for 


ANH      H 


h!^f 


and   therefore    is    the    same    as    that    of    the   parallelogram   HF,    when 
AF=^AB,  for  the  distance  Gg  from  the  axis  of  y 

x{\-h){\^h)  \~h      AH 


{l-^h(X-2x)  +  ¥\l         2  2' 

Hence   G  tends  ultimately  to   the  point  g  in  the  axis  of  y,   which 
shews   that  the   area  DBH'E'   tends  absolutely  to  vanish,  HE'  being 

3  A  2 


358 


Mr  MURPHY'S  THIRD  MEMOIR  ON  THE 


an   ordinate    drawn    near  the   origin   at    any    small   distance   not  varying 

with  the  parameter  Ji,   and  since  -r-  has   the  same  sign    in    the   interval 

from  B  to  H',  H  or  A,  it  is  evident  that  the  portion  of  the  curve 
BE'  tends  to  coincide  with  the  axis  BH',  the  curve  therefore  which 
represents  V„  coincides  with  AB,  except  infinitely  near  the  origin 
A,  when  it  suddenly  mounts  to  an  infinite  height. 

Since  the  general  function  V„  is  reciprocal  to  t",  it  follows  that 
fi  Vaf  =  0,  except  when  w  =  0,  and  then  the  definite  integral  is  unity ; 
hence  if  f{t)  be  any  function  containing  only  the  positive  and  integer 
powers  of  t,  the  transient  function  Vo  possesses  the  remarkable  pro- 
perty expressed  by  the  equation    [tV'o  ■/{f)=J^{Q). 

Fig.  2.  Let  2a  =  AB,  equal  the 
length  of  the  axis  in  a  solid  of  revo- 
lution, the  surface  of  which  is  covered 
with  an  indefinitely  thin  stratum  of 
fluid,  let  any  abscissa  ON  measured 
from  the  centre  O  be  put  equal  to 
a  (1  -  2#),  the  limits  of  t  will  evidently 
be  O  and  1 . 

Let  the  law  of  density  or  accumulation  at  any  point  P  of  a  section 
perpendicular  to  the  axis  be  expressed  by  the  transient  function  \V^, 
X  being  constant,  and  let  the  total  action  of  the  fluid  on  any  point  Q 
in  the  axis  be  required,  the  law  of  force  being  capable  of  expansion 
according  to  the  positive  and  integer  powers  of  t. 

Put   PA''  =  y,  then    the   whole    quantity   E   of   fluid    is    manifestly 

ds 
equal  to  ^Xtt  ft  T^^y -r, ,   s  representing  the  arc  AP. 

ds 
Now  it  is  easily  seen  that  the  value  of  y-yr  at  the  point  A  where 

y  vanishes  is  iaR,  R  being  the  radius  of  curvature  at  that  point,  and 
by  the  nature  of  V^  this  quantity  is  the  value  of  the  above  integral, 
or  E  =  SXttuR. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  359 

Again,  if  we  represent  the  distance  PQ  by  r,  and  the  law  of  force 
by  y(/-)   and   put   AQ  =  k  the   initial   value  of  r,  the  total  action  is 

r  rr       ds        „,    ^       ON 

2ATJ;r,y^./(r).^, 

which  by  the  property  of  F"o  is  equal  to  S\traRf{k),  or  to  E  .f{k). 

Let  us  now  suppose  an  equal  quantity  of  fluid,  but  of  a  contrary 
nature  in  its  action,  and  therefore  represented  by  —  E  to  be  collected 
in  a  single  point  C  in  the  axis  produced  to  a  small  distance  AC- a. 

The   total   action   of  the   compound   system   on   Q    will   then   be 

E{f(,k)-f(k  +  a)}, 

which  tends  to  vanish  as  C  approaches  A. 

Lastly,  suppose  a  unit  of  fluid  when  distributed  over  the  surface 
according  to  a  law  expressed  by  0  {t),  which  depends  on  the  figure  of 
the  solid,  will  exert  no  action  on  any  point  Q  in  the  axis;  then  if 
the  law  of  distribution  of  the  fluid  be  expressed  by  X  V^  +  c  (p  {t),  the 
total  action  on  Q  including  that  of  C,  will  be  still  E  {f{k)  -  f{k  +  a)  ^ . 

From  which  it  follows  that  when  an  electrical  spark  -Eh  in- 
finitely near  to  the  vertex  of  a  conducting  solid  of  revolution  charged 
with  a  quantity  of  electricity  E',  the  distribution  of  the  latter  under 
the  influence  of  the  former  is  expressed  by   the  law 

pi 

\Va  +  c<i>{t)   where   \  —  ^ ^, 

otraH 

and  where  c  is  determined  by  the  equation* 

Having  thus  given  the  geometrical  and  physical  interpretations  of 
Vo,  it  will  not  be  necessary  to  discuss  the  transient  functions  V^,  V^, 
&c.,  of  which  the  properties  are  very  analogous. 

*    Vide   First    Memoir,   Art.  35,   the   expression   there   obtained  for   a   sphere   being   in- 
cluded in  that  obtained  above,  when  the  influencing  point  is  infinitely  near  the  sphere. 


S60  Mb  MURPHY'S  THIRD  MEMOIR  ON  THE 

28.     To  find  the  quantity  to  which  V„  is  the  generating  function. 

By  Art.  26. 

_  d^'iRh")  ^hd''*^  (Rh") 


I  .2.S.,.2ndh;"'       1  .2.3...(2»  +  l)rfA"^" 
where  R=  {l~2h(l  —  2t) +h^}~K  and  h  is   ultimately  equal  to   1. 

Forming  the  equation  u==h  +  &uK  we  have 

A^   d^f{h)h\       k^     d^f{h).m} 

hence  {l-2«(l-2^)  +  «1-^  ■^^  =  ^iZ  +  ^'     ^^   ^  1T2      <//^'      +&c.; 


dh' 

du 
dh 


^l^^'-^^^^^'  rS^F^  =  '^'  "°"^'^""'  °^  ^""  ^"  {l-2t.(l-20  +  >/'M 
In  like  manner, 

«-i{l-2«*(l-2/)  +  «n-^.^  =  2?A-^+^'|f^ 

/fe^     d\{RM)  ¥       d'jRh) 

therefore, 

_^M!!!1(^)         =  the  coefficient  of  >P«-  in  .,      ,    ^      f^^    .,,; 
1  .2...(2«  +  l)rf^''+*  {1-2m(1-20  +  «'H 

(*  +  2Am-^)^ 
consequently,  V„  =  the  coefficient  of  ¥'^*'  in  v^  _  ^^(i  _  20  +  «'}^' 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  361 

Now,  by  the  assumed  equation  we  have 

ui  —  hu~i  =  k, 

du 
and  by  differentiation  («J  +  Am"*) -jr  =  2m  ; 

but  also  (m^  +  hu~^)  =  k  +  2hu~i; 

hence,  {k  +  2hu~i)'Tr=2u; 

and  therefore,  F„  =  2  the  coefficient  of  ^"+'  in  u {l-2u(l  -  2t)  +  u"}  -^ 
from  which  it  follows  that  if  we  form  the  two  equations, 

u'=h  +  (ku')i]  .        \U'  =u'  {l-2u'  (l-2t)  +  u''}->^ 

}  putting  i 

u"  =  h  +  (ku")i]  [t7"  =  «"{l-2«"a-20  +  «"'i^^; 

then  ^^j—  =  ^0  +  J^^k  +  V^¥  +r,k'kc.  ad  inf. 

supposing  that  in  the  left-hand  member  h  is  finally  put  equal  to  unity. 
It  may  be  observed  that  the  quantities  u',  u"  are  the  two  roots  of  the 
equation  u'^  —  {2h  +  k)u'  +  h!'  =  0. 

29.     To    expand   a   given  ^function    (pit),    in   terms   of  the  transient 
function    \ ^ . 

Let  the  general  term  of  the  expansion  be  A„V„,  then  by  the  nature 
of  reciprocal  functions  we  have 

=  AJtPj",         (Art.  24.) 

M.(W-l) 1 


=  (-l)".^„ 


(w +  !)(«  + 2) (2«  +  l)' 


lience,(p(t)=Kft<l>(t)-^^rj,(p(t)t-\-^^.FJt<}>(t).f-&ic. 


362  Mr  MURPHY'S   THIRD    MEMOIR    ON   THE 

Examples  : 

/»  -  -J_    V  _       ^-3         V  J.      g-4-5         ^      . 

p_j^    2«+3     „        (2w+4)(2«+5)  (2m+5)(2«  +  6)(2w+7)    .^     , 

-rn—r„  J  .    ;'„+,+  I        ^  .    f'n+a J .    f'n+g&C. 

the   latter   series   would   also   result   by  reverting  the   series   for    V„,    in 
Art.  24. 

30.      To  find  a  function  U„  which  shall  he  reciprocal  to  (h.l.ty. 

Following  the  steps  indicated  in  Art.  23,  we  must  first  form  a  self- 
reciprocal  function  of  which  the  general  term  is  a  constant  multiplied 
by  (h.  1.  ty ;   this  has  been  already  effected  in  Sect,  v,    namely, 

and  then  the  form  of  the  required  function  will  be 

[/„  =  t; + a  2;+ ,  +  6 1;^^  +  c  7;+3 + &c. 

Multiply  by  (h.  1.  ty,  supposing  m>n,  and  observing  that 
j;r„(h.l.0"  =  1.2.3...>^.(-ir."-^'^-;)^:-f-("-'^-^^)bySect.v, 

and  ir7„(h.  1. /)•"  =  0 
by  the  nature  of  reciprocal  functions,  we  get  the  general  identity 

m{m  —  \){m  —  Q)...{m  —  n  +  l)    ,  m  —  n     ,    {m-n){m -n—\) 

® r.2.3...«  •^^""•^TT+*-  («  +  i)(«  +  2) — *'*'-^' 

but  on  the  same  supposition  that  m  is  greater  than  w,   we  also  have 
0  =  (l-l)'"-"  =  l-(m-w)+^^ 4p— ^-&c.; 


INVERSE    METHOD   OF    DEFINITE    INTEGRALS.  363 

and  by  comparing  the  corresponding  terms 

-  ^  +  1       ^_(«  +  l)(w  +  2)         _{n  +  l)(w  +  2)(w  +  3)     . 

therefore, 
rr-7--L''  +  ^    T       ,  (w +!)(«  + 2)  .  (w  +  1)(w  +  2)(m  +  3) 

Ly„  —  J  „  -I         J       .  ^  „+i  +  r — .   -I  „+2  i r — - — .  X  „+3,   Cue. 

31.  To   express   the  function    Un    which   is   reciprocal  to   (h.  I.  t)"   in 
a  finite  form,  and  also  the  function  which  Un  generates. 

l.2.3...nU„  =  1.2.3...nT„+2.3.  4...(w  +  l)  T„+,+3  .  4  .  5...(w  +  2) .  T,.+,+&cc. 

=^  { r„  +  r,A  +  7;a^+ ...  T^h'+T^^.h"^^  +  &,c.], 

h  being  put  equal  to  unity  after   the  differentiation. 
But  by   Section  v,  we  have 

h 

J— ^  =  T,+  T,h  +  T^h'  +  &c.  ad  inf. ; 

"■■(A) 

therefore,    U„  =  - — — — -jt  when  A  =  1. 

1.2.  3...ndh" 

Now  by  Taylor's  Theorem,  this  quantity  is  the  coefficient  of  k"  in 

the   expansion   of  - — j~r^   the   latter  is   therefore   the    function   which 
U„  generates. 

32.  Properties  of  Un- 

.1.  jiUn  (h.  1.  ty  =  f,T„  (h.  1.  ^)"  =  1  .  2  .  3...W,  by  Sect.  v. 

II.     Changing   the    sign   of  k  in    the   quantity    which    U„   generates, 
we  get 

Vol.  V.     Paet   III.  sB 


364  Mr  MURPHY'8   THIRD   MEMOIR    ON  THE 


pT''  =  Uo-  U,k  +  U,¥  -  U^¥  +  he. 


— f-f^§--} 


III.     Since  f  =\ +x\i.\.t+  -^ .  (h.  1.  tf  +  j-^-j  ■  (h.  1.  <)'  +  ««:■ 

by  means  therefore  of  a  single  integral,  x"   may  be  adapted  to  general 
differentiation. 

As   this   result   is    remarkable,   we    may   confirm    it    by   the   general 
rule  in  the  First  Memoir.     (Vide  Sect,  i.)     Thus, 

1 

put  0  (x)  =  :j T—  =l-\-xk  +  a^J^  +  &c. 

*±i 
/-* 

then  fit)  =  — -T  =U,+  U,k+  U^k"  &c. ; 

for  all  values  of  k,  whence  JiUnf  —  xf  as  before. 
33.     Discussion  of  the  Junction  U^. 

h 
fl-h 

By  Art.  31.    Uo  =  - — r   when   h  is  put   eqifal   to   unity.      Like   the 

transient  function  Vo,  discussed  in  Art.  27-,  the  quantity  Uo  is  always 
zero  for  values  of  t  between  0  and  1  ;  but  when  t  =  1  its  value  is 
infinite,  and  thence  its  integral  between  the  limits  0  and  1  of  /  is  finite, 
viz.  unity. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS. 


365 


To  prove  this  property,   conceive  a  curve  Y 
APC,  of  which   the  abscissa  measured  from 
A    along   AB  is   taken  equal  to  /,  and  the 

corresponding  ordinate  y  is  equal  to  f''', 
and  let  us  suppose  h  very  nearly  equal  to 
unity,  and  at  any  point  P  draw  a  tangent 
PT;  then   since 


dt 


h 


y  =  t'-\ 

therefore,  11^  is  the  limiting  value  of  the  tangent  of  the  angle  PTB. 

Take  AB  =  1  and  the  ordinate  BC  =  1,  then  it  is  evident  that 
A  and  C  are  constantly  points  of  the  curve  when  the  parameter  h 
varies   so   as  to  approach   unity. 

Again,  for  the  entire  area  APCB  the   expression  is  Ji^'~*,  from  t=0 

1  —  A 
to  t=l,   that  is,  - — Y,   which   evidently   tends   to   vanish   as   the   para- 

.«  —    fl 

meter  k  approaches  unity ;  and  as  no  part  of  the  area  is  negative,  it 
follows  that  the  curve  APC  tends  ultimately  to  coincide  with  the  two 
right  lines  AB,  BC,  and  therefore  when  T  is  sensibly  distant  from 
B  the  tangent  of  the  angle  PTB  tends  to  vanish,  but  when  indefinitely 
near  to  B  it  tends  to  infinity ;  and  therefore  Ug,  which  ultimately  re- 
presents these  tangents,  is  zero  from  A  to  indefinitely  near  to  B  where 
t  is  unity,  when  its  value  becomes  infinite. 

In  like  manner  the  remaining  functions  C/i,  U^,  &c.  may  be  dis- 
cussed with  similar  results. 

It  may  be  observed  that  for  values  of  t>l  (which  however  do  not 
enter  the  definite  integral),  the  values  of  t/'^  are  infinite. 

34.     Expansion  of  given  Junctions  in  terms  of  the  functions  Un . 
The  general  formula  for  this  purpose  is 

0(0  =  V^kW)  +  UJt<p{t).  hA.it) 


1.2 


^,<t>{f).ih.\.tf  + 


u. 


1.2.3 


.j;<^(0.(h.i. /)'  +  &c. 


3  b2 


366  Mr  MURPHY'S  THIRD   MEMOIR  ON  THE 

T  -u  -  ^±1  u    +  (^  +  i)(^+^)  r/    _  &c 

1  1.2 

which  latter  series  is  also  produced  by  reverting  to  that  which  expresses 
C7„  in  terms  of  71  in  Art.  30. 

35.     To  find  a  function  reciprocal  to  t"   when  the  limits  of  t  are  0, 
and  GO . 

Let  M„  be  the  required  function,  and  put  t  =  e"', 

then  ^lUj'^  =  0,     from    #  =  0  to   ^  =  x  ; 

«»    /V-     1  N™  ^  i>_ =    0     to     T    =    1, 


therefore    /"-  (h.  1.  t)-"  =  0,     from  t  =  0 


^»  " 


h_ 
1-* 


hence  m„  =  t  C/„  =  t  ; — ^ttt   when  h  =  \ 

1 .  ^...ndh'' 


36.     Tb  ^«c?  « function   F„   ^t>A^cA   *Aa?/  ie   reciprocal  to   cos"  ^,   ^A^ 

-  awa  -. 

2  2 


ZmeV*  o/*^  Je^?^  —  -  and  - 


Following  similar   steps  to  those   adopted   in  the  preceding  Articles 
we  shall  obtain, 

w  +  2 
in  cosines  F„  =  cos  n<p — .  cos  {n  +  2)(f> 

(«  +  l)(w  +  4)          ,     ,,,^       (w+l)(w  +  2)(«  +  6)         ,     ,-,-,     . 
+  —^-12 ^•cos(«  +  4)0 -^ ^    g    g ^cos(w  +  6)<^,  &c. 


INVERSE   METHOD   OF  DEFINITE    INTEGRALS.  367 


in  sines  Fn  =  2  sin  (p  {sin  {n  +  l)(p — .  sin  {n  +  3)(p 


+  ^^ -^ .  sm  («  +  5)  <^  -  &c.  ] 


37.     The  Junction  F„  is  transient. 

Either  of  the  preceding  values  of  F„  give  F„  =  Fn—F",  where 

F:  =   cos  {n(t>)       -  ^±i  .  cos  {n  +  2).(p  +  ^^"^^^^^^^  •  ^^^  («  +  4)  <^  +  &c. 
F„"=  cos  (w  +  2)  <^  -  ^^  .  cos  («  +  4) .  <^  +  {n  +  l){n  +  2)   ^^^  („  ^  g^  ^  ^  ^^. 
passing  from  trigonometrical  to  exponential  values, 

1  1.2' 

1  1.2" 

_  ("g.^vrr  ^  g-</>vrT\-" 
=  2cos«0, 

2F„"  =  £("+2)*^^  -  ""'"^  .  e("+4)*v^  +  ("  +  l)(^  +  2)  _  ^(„^g)^^—  _  ^^ 

1  1.2 

+  g-(n  +  2),^V:rT   _   ^jt2,e-(n  +  4)0V^  ^   (w  +  1)  (W  +  2)  ^-(„  +  4)^vri  _  j^^.^ 

=  2  COS  «^, 
hence  F„=-F';-F„"=0. 


368  Mr  MURPHY'S   THIRD   MEMOIR   ON    THE 

However,   if  n  be  even,  and  our  limits  be  —  ^  and  ^,  the  function 

becomes    suddenly    infinite   at    the   limits,    for   the   expansion   of    F„    is 
then  identical  with  that  of  (1 -!)-'"+'•. 

38.     To  express  infinite  terms  the  transient  function  Fn. 
Put 

i?'„' =  cos  ra(^  -  ^  .  ^  cos  («  +  2)  0  +  ^^i|^-^^ .  A'' cos  («  +  4) «/)  -  &c. 

F:'=  hcos  (n+2)(f>  -  ^  .  A^  cos  {n+4>).<l>  +  <"+!)  (^  +  ^)  ^^  cos  (w  +  6)0-&c. 

1  1  •  ^1 

Then  F„  is  the  limit  of  F^—F"  when  h  approaches  unity. 

Put  also  2  cos  0  =  a;  +  - , 

^  X 

hence  2-F„' 

x~^  X 


~  {l+A(a:^+a;-^) +  *'}"+' 

W  +  1  (W  +  X^    Tt  W  +  1 

COSW0+— j— .Acos(w-2)0+^— — ^^.A^cos(«-4)0...— — -.A"cosw0+A"+^cos(«+2)0 

~  fiTaFcosa^TFp^^  * 

the  number  of  terms  in  the  numerator  being  w  +  2. 

In  like  manner, 
2F„"  X 


+ 


h         (aT-'  +  Aa;)"+i       (x  +  Aa;-')"*' 

_  X  (a;  +  Aa;-')"+'  +  a;-'  (a;"'  +  Aa;)"+' 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  369 

eos{n+2)(l)+'^r—  .h  cosn(f)+- -^ .k^cosin-2)<p+ . . .  ——  .h"  cos{n—2)(p+ h"*'  cosn<p 

^  fTT2rcos2^1~Fp^'  ' 

Hence, 

costKj)  .{I  -h"*^)+h\——cos{n-2)(p-cos{n+2)<p\+hH— — ~-cos(w-4)0 — — cosw0> 
^"  {l  +  2Acos20  +  ^^}»+'  '    ^' 

when  h  is  put  =  1. 

Thus  F  =       (^-^)(^  +  ^^ 
which   is   evidently   a  transient   function,   as   its   general  value  for  A  =  1 

7r 
2 


is   zero,    except   ^   is  an   odd  multiple   of  -,   when  its   value   becomes 


infinite. 

And    in    general   F„'   and   F„"    are   equal,   when    h  is   put   equal  to 
unity,  and  therefore  F„  has  a  factor  1  —  A  in  its  numerator,  which  causes 

TT 

its  general  vanishing  state,  except  when  ^  =  „,   or  an  odd  multiple  of 

^,    when  the  denominator   becomes  (1-A)^"'*"^  and  as   the  numerator  is 

of  only  n  +  2  dimensions,  it  is  evident  F„  in  this  case  is  infinite,  when 
k=  I. 

In  general    f  ,       ~i.        r.^     1:2  =  2  tan"' .  I^^^  .  tan  0>  +  const., 

which  taken  from  0  =  0  to  ^  =  -  is  equal  to  tt,  a  quantity  independent 

of  h,    a   result   similar   to   those   already   obtained   from   other   transient 
functions. 

39.  When  the  sum  of  a  series  containing  transient  functions  is 
required,  the  following  process,  with  only  such  modifications  as  may 
simplify  particular  cases,  will  apply. 


370  Mk  MURPHY'S   THIRD   MEMOIR   ON    THE 

Let  S  =  UaVo  +  fli Fi . ss  +  (hV.i.%^  +  ...  +  «x ^^K'  +  &c. 
be  the  series  proposed. 

By  the  inverse  method,  put  a^  =  U/{t)  .  t'  from  t  =  0  to  t  =  1 . 

Then   S  =  /^/(r)  \V,  +  F.tz  +  V.t'z'  +  &c.| 

But  V^+  V^k  +  VJe^,  &;c.  is  the  function  which   V„  generates,  and  may 
be  represented  by  ^  {t,  k),  we  have  then 

S  =  X-/(t)  .  <p{t,  T%),  from  T  =  0  to  T  =  1. 


INVERSE  METHOD  OF  DEFINITE   INTEGRALS. 


S7l 


SECTION    VIII. 


On  ike  Resolution  of  Equations  involvings  Definite  Integrals. 


(l)     By  the  Decomposition  of  the  Integrals  into  Elements. 

40.  The  utility  of  the  method  of  decomposition  consists  principally 
in  the  verifications  it  offers  to  results  obtained  by  other  analytical  pro- 
cesses, the  difficulty  in  the  eliminations  which  it  requires. 


Pm-i-^    Pm+if     Pa       Put/ 


Suppose  a  cylindrical  shell  exerts  no  force  on  any  point  in  its  axis 
AB,  the  law  of  force  tending  to  each  particle  of  the  shell  being  given, 
but  the  law  of  density  of  the  shell  unknown,  then  the  application  of 
the  method  of  decomposition  is  this : 

Divide  the  shell  into  « + 1  equal  portions  by  planes  perpendicular 
to  the  axis  PiQi,   P2Q2,  &c. 

Let  the  density  throughout  each  portion  be  supposed  uniform,  and 
let  the  successive  densities  be  pa,  pi,  p->....pn- 

Let  the  total  actions  on  the  points  of  division  Qj,  Q2...Q„  be  equated 
to  zero,  which  will  give  n  equations,  and  another  will  be  obtained  by 
considering  the  mass  of  the  shell. 

From  these  n  +  1  equations,  let  po,  p^,  p^,  &cc.  be  determined  in  terms 
of  ».  .. 


Finally,  make  n  infinite. 
Vol.  V.    Part  III. 


3C 


S72  Mr  MURPHY'8  THIRD  MEMOIR  ON  THE 

41.     General  Calculus  for  the  Cylinder  with  any  law  of  force. 

Let  «o,  «i.  a^.-Mn  represent  the  total  actions  on  the  point  A  which 
would  be  exerted  by  the  successive  portions  P^Pi,  PiPz  of  the  shell, 
if  the  density  of  each  were  unity;  these  quantities  are  given,  since  the 
law  of  force  is  supposed  known. 

Then   aopo,  a^p^,  a^p^,  &c.   represent  the  actual  forces  on  A. 

Again,  the  action  of  any  portion  as  P„+4P„+5  on  any  point  Q„  of 
division  in  the  axis,  will  be  to  the  action  of  the  similarly  situated 
portion  P^P^  on  the  point  A  in  the  ratio  of  the  corresponding  densi- 
ties, and  in  this  case  would  be  atpm+i. 

By  this  consideration  the  total  actions  on  the  points  Qi,  Q2...Q„ 
are  easily  estimated,   and   equating  each  to  zero,  we  get   the  following 

system  of  n  equations,  which  serve  to  determine  the  ratios  — ,  — ,  &c.  viz. 

pa     Pa 

aopo-  Ctopi  —  aipz  —  Chps  —  asPi —  dn-lpn-l  —  (tn-iPn  =  0, 

aipo  +  aopi  —  aop-i  —  aipi  —  aipi —ctn-spn-i  —  an-ip„  =  0, 

Oipo  +  aipi  +  ttopi-  ttops—aipi —  «n-4/'n-l  —  «n-3/'«  =  0, 

a^po  +  a^pi  +  aip2  +  aopa— Oopi —ctt,-ipn-i  —  (i„^ip„  =  0. 


a„-2po  +  an-3pl  +«„_4p2  +  «n-5/03  +  «»-6P4 —  «2/0«_l  —  fl!l/0»  =  0, 

ffn-lpO  +  fin-2Pl  +  «»-3p2  +  «»-4j03  +  «n-5/04 +«l/On-l  "  aoPn  =  0. 

Comparing  the  first  equation  with  the  »*'',  the  second  with  the  (w-l)"", 
&c.   it  is   obvious  that  po  is  involved  in  the  same  manner  as   /o„,  p^   as 

Pn-l,    &c. 

Hence,        p„  =  po,        p„_i  =  pi,        p„.i  =  p^,  &c. 
Form  now  two  functions  in  the  following  manner: 
a  known  function,   M  =  «osin0 +«isin30  +  a8sin50+ .,.«„_isin(2w-l)0, 
an  unknown,  aS'„  =  jOoCOSw0+|OiCOs(w-2)0  +  ^sCOs(«-4)0+ 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  373 

the  first  series  may  be  continued  to  n  terms  or  infinity  indifferently, 
and  the  last  term  in  the  second   series  will  be  ^p^  when   n  is  even, 

2 

and  p„_x .  cos  9  when  n  is  odd. 


Suppose  now  that  the  product  9,u.Sn  is  decomposed  into  the  sines  of 
the  multiples  of  9,  and  that  all  the  multiples  higher  than  the  «'"  are 
rejected  from  this  product,  the  remaining  part  will  evidently  be, 

—  {aopo—aopi  —  aip2 — a»-i/Oo}  .sin(n  —  l)0, 

—  {aipo  +  Uopi  —  aopi —a„.2po}.sm{n  —  3)0,  • 

—  {(hpo  +  ctipi  +  aop2 —a„-3po}.sm{n  —  5)$,  &c. 

the  whole  of  which  by  the  given  equations  is  equal  to  zero. 

Hence, 

2S.u  =  A„sm{n  +  l)9  +  B„sin(n  +  3).9  +  C„sm{n  +  5).B,  &c. ; 
.-.  4  cos  0 .  S„u  =  A„  sin  (nB)  +  {A„  +  B„)  sin  in  +  2)9  +  {B„  +  C)  sin  («  +  4)0,  &c. 
and  2Sn.iU  =  A„^i  sin  {n9)+B„.i  sin  {n  +  2).9  +  C„_i  sin  («  +  4) .  0 ; 

.-.  2{2cos9.Sn-j^S,.^}  .u  =  i^A„  +  B„-A„.^\.sm{n  +  2) .  9,  &c. 

Hence  it  follows  that  if  we  put  So=po,  S^  =  po  cos  9, 

and  u  =  aoSin9  +  a^  sin 3 0  +  a^  sin 5 9  &c.  ad  inf.,  then. 

First,  Supposing  S^.^  and  S,n  known,  form  a  quantity  \„  by  dividing 
the  coefficient  of  sin(/»  +  l)0  in  2S,„u,  by  the  coefficient  of  sin(/»0) 
in  S/S'm.i  .u.  • 

Secondly,    Form  a  quantity  S^^^,  by  the  equation 
-S'„+i  =  2  COS0  . /y^  -  X^iS*™.! , 
by  which  S^,  S3 a^^  may  be  successively  formed. 

Then  it  is  obvious  that  the  product  2S„u  contains  no  multiple  of  9 
below  the  «'\  and  therefore  the  coefficients  in  S„  must  be  the  required 

quantities  po,  p^,  pi pn-j^  when   n  is  odd,  or  p^,  pi,  pt ^p^  when 

2  3 

n  is  even.  _ 

Sc2 


374  M^a  MURPHY'S  THIRD  MEMOIR  ON  THE 

42.     Applications,  when  the  law  of  force  is  the  inverse  square  of  the 
distance. 


(1)  Let  AB  be  the  axis  of  a  very  broad  cylindrical  plate,  the 
round  side  of  which  is  covered  with  a  fluid,  attractive  or  repulsive, 
and  so  distributed  as  to  exert  no  action  on  any  point  in  the  axis. 

Put  AB  =  1,   APo  =  a  the  radius  of  the  base. 

Let  ab  be  one  of  the  very  small  annuli  into  which  the  edge  is 
divided,  and  put  aPo  =  x. 

Then  it  is  easy  to  prove  that   the  action  of  the  annulus  a  5  on   the 

11 

point  A  is  proportional  to  -^ -jr,   or  ultimately  to  the   differential 

1  ■  X  . 

coefficient  of  -7—  with  respect  to  x,  that  is,  to  -t-t? ryj,  which  quan- 

Aa  {o''  +  arj» 

tity  expanded  is  proportional  to  a;  —  f  rj  +  &c. ;  and  as  b  is  very  great 

compared  with  x,  we  need  only  take  the  first  term  of  this  expansion. 

In  this  case  we  may  therefore  put  ao  =  l,  a,  =  2,  ai  =  3,  &c., 
and  therefore,  M  =  sin0  +  2sin30  +  3sin50  +  4sin70  +  &c. 

The  calculus  of  S„  as  indicated  in  the  preceding  article  will  be  as 
follows : 

So  =  pay  Si  =  |OoCOS&, 

f     ^  coefficient  of  sin2g  in  ZS^u  ^31 
\  '  "~  coefficient  of  sin0      in  2SoU  ~  2)' 

■    •  Si  =  2coseSi  -  x^So 

=  Po  Jcos20-^};  ■    • 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  375 

i     _  coefficient  of  sin30  in  2SiU  _2\ 
\  *  ~  coefficient  of  sin29  in  2«S^m  ~  3 j 

S:i=2COS0S,-\Si 

=  Po{cos39  — -cosO],  .... 

J     _  coefficient  of  sin  40  in  2S3U  _  5\ 
\  '  ~  coefficient  of  sin30  in  ZSaU  ""6/ 

*S.  =  2cos9S3  -  Xs.S^  . 

=  po|cos40  —  -cos  20  —  ->, 

f         coefficient  of  sin 50  in  2SiU  _  91 
\     '^  coefficient  of  sin4!0  in  gAysM  ~  lOj 

Si  =  2cos9S^  -  XiSi    - 

=  po  |cos  50  -  -  cos  30  — -> . 

{2  2  11 

cos60  —  ^cos40  —  ^cos20  —  ^> 

{2  2  2  1 

cos  70  —  =  COS  50  —  -  COS  30  —  -  cos  0> . 

Generally  when  n  is  an  odd  integer,  suppose 
-^^  =  cos(w-l)0-^—  {cos(m-3)0  +  cos(«  -  5)0  +  ...  +  cos20  +  i}, 

and  —  =  cos  »0 {cos  (w  —  2)  0  +  cos  («  —  4)  0  +  ...  +  cos  30  +  cos  0}. 

po  n  -  - 

The  coefficient  of  sin  w0  in  2»S'„_,«  = -.  p„, 

n  —  1   ^ 

of  sin(M+l)0  in  2S„u  = .p^; 

n 


therefore,    x„  =  <^±fi^  =  1  -  ^  +  ^ . 
n(n  +  l)  n      n  +  1 


Stfi  Mw  MURPHY'S  THIRD  MEMOIR  ON  THE 

Hence,  ^=2cos0.— -X^*^ 

pa  pa  po 

=  cosln+l)9 -{cosin-l^e  +  cos(n-S)e  +  ...+cos29+i}, 

and  by  a  repetition  of  the  same  process, 

O  Q      .    ,■>   .*,{  ,-; .    •-:.. 

— ^  =  cos(»+2)0 -{cbsw0  +  cbs(«-  2)  .  0  +  ...  +  cos30  +  cos^}. 

pa  n-TXt 

Hence  the  laws  by  which  S^-x  and  S^  are  expressed  are  uniform, 
and  therefore  we  get  for  the  required  unknown  quantities, 

2  2  2  _ 

po  =  Pa,      pi=--pa>      P8=~^Po P—i -P<>>      Pn-Po- 

The  positive  values  may  be  taken  for  the  repulsive  and  the  negative 
for  the  attractive  parts  of  the  fluid,  and  if  E  denote  the  excess  of  the 
former,  we  have 

[n       n      n  n        n]  n      [  n    }         n 

.-.  po=»^-T — >  which  gives  the  complete  solution  of  the  problem. 

Thus  the  application  of  a  process  purely  algebraical,  conducts  in  this 
instance  to  a  transient  function,  for  if  we  suppose  the  final  and  equal 
densities  po,  p^  to  be  finite,  all  the  intermediate  values  of  the  densities 
p^,  pa pn-x  become  indefinitely  small  when  n  is  made  infinite;  yet 

they  are  not  to  be  rejected,  for  if  so,  the  total  charge  would  be  4nra^, 

it 

whereas  its  actual  value  is  only  inra  ~,   an  infinitesimal  of  the  second 


pa 
as  Its  itutuiu    vaiuc   is    fJiii-y    •±'iru,    — 

order. 


.Vo 


Pm/ 


INVERSE   METHOD   OF  DEFINITE   INTEGRALS  377 

(2)  Let  AB  be  a  right  line  perpendicular  to  the  bounding  planes, 
which  terminate  a  solid  composed  of  parallel  strata  of  indefinite  extent, 
but  uniformly  dense  throughout  that  extent ;  and  let  the  law  of  den- 
sity of  the  different  strata  be  such  that  there  is  no  action  on  any 
point  Q„  within. 

Let  the  solid  be  decomposed  into  n  +  1  equal  portions  in  which  the 
densities  are  as  before  represented  by  po,  pi,  p% /o„. 

In  this  case  the  quantities  ao,  flj,  02 ci„  are  all  equal,  and  putting 

them  equal  to  unity,  we  have 

u  =  sm9  +  sin39  +  sm56  +  SiC. 

So  =  po,  Si=poCOS9,  \  =  1» 

S2  =  2cos9 .  S^  —  XiSo=poCos29,  X8  =  l, 

Ss=-2cos9.Sz  —  \2Si  =  poCOs39,  X3=l, 

and   generally,   S„=pocosn9,  and\„  =  l. 

Hence  the  solution  is  pi  =  0,  p2  =  0 p»_i  =  0,  pn  =  po' 

And  if  E  be  the  whole  mass  and  A  the  area  of  the  bounding  planes, 
which  is  supposed  very  great,  we  have 

E  =  2iA.po. 

This  result  is  analogous  to  the  well-known  fact,  that  electricity  can 
reside  only  on  the  surfaces  of  bodies,  and  affords  another  instance  of 
a  transient  function. 

The  method  of  decomposition  may  always  be  applied  to  obtain 
numerical  approximations  in  cases  which  involve  Definite  Integrals; 
for  instance,  in  the  distribution  of  electricity  on  bodies,  and  in  esti- 
mating the  forces  between  bodies  which  are  electrised. 

(2)     By  means  of  Reciprocal  Functions. 

43.     Equations  which  contain  only  one  definite  integral. 

Let  f(f,  a)  be  a  function  involving  a  variable  f,  and  an  arbitrary 
parameter  a;  F{a)  a  function  containing  a  only,  and  (p  (t)  a  function 


378  Mr  MURPHY'S   THIRD  MEMOIR  ON  THE 

containing  t  only,  the  first  and  second  of  these  functions  being  given, 
it  is  required  to  find  the  third  so  as  to  satisfy  the  definite  integral 
equation 

!,<l>{t).f{t,a)  =  F{a), 

the  limits  of  t  being  given. 

Suppose  (p  {t)  expanded  according  to  any  given  class  of  self-reciprocal 
functions  as  P„,  that  is, 

^(^)  =  CoPo  +  CiPi  +  C2P2  +  C3P3,  &c.  ad  infinitum, 
where  the  coefficients  Co,  c,,  Ca,  &c.  are  unknown. 

Let  J^{t,  a)  be  expanded  according  to  the  same  reciprocal  functions, 
f{t,  a)  =  AoPt,  +  A^P^  +  A2P2  +  A3P3,  &c.  ad  infinitum. 

Then  j?P„P„  =  0,  and  fiPnPn  =  a„  a  known  numerical  quantity  depend- 
ant on  n,  and  on  the  particular  species  of  reciprocal  functions  which 
are  employed. 

Multiply  both  series  and  integrate  between  the  given  limits  of  /, 
and  the  proposed  equation  gives  us 

F  (a)  =  Aoao.Co  +  Ai  ai.c^+  A2  as .  C2  +  ^303 .  C3,  &c.  od  infinitum. 

Now  An  being  a  known  function  of  a  and  n,  we  can  by  Art.  23. 
Sect.  VII.,  find  another  function  of  a  and  n,  as  An  such  that  fiA„A„'  =  0, 
when  m  and  n  are  unequal  integers. 

Multiply  the  equation  successively  by  Ao,  A^',  Ai,  &c.  and  take  the 
definite  integrals  relative  to  a,  hence 

jaA(s-P\a)  =  CoOojaAaAt,  ',  .'.   Co  ^  C  A  '  A    ' 

f.A,'F{a)  =  e,aJ.A,A,';       •.:  c,  =  ^^4^, 
and  generally  c„  =        r'^'j  • 
Hence  <b(f)  =  ^  /^^°'--^(«)  +  ^  fgA^Fja)  ^  P.  fa-A^'Fja)  ^  ^^ 

^    '         ao  '      faAo'Ao  a,'    faAi'Ai  aj  ■     faAa'A^ 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS.  379 

44.    Examples. 

In  the  following  examples  two  things  are  to  be  observed.  First, 
that  the  given  functions  are  supposed  to  be  continuous,  and  therefore 
the  equation  proposed  must  hold  true  for  all  values  of  the  parameter  a. 

Secondly,  In  the  final  equation  for  determining  the  unknown  coeffi- 
cients, instead  of  using  a  reciprocal  multiplier  any  means  more  simple 
may  be  occasionally  employed. 

Ex.  1.  Given  ^^(/),  cos  («^)  =  1  to  determine  <^{t\  ^he  limits  of  t 
being  0  and  tt. 

Put  (^{t)  =  Co  +  Ci  cos/  +  d  cos  (2/)  +  d  cos  (3/),  &c.  ml  infinitum, 

and  cos  {at)  =  Ao  +  A^  cos  t  +  A2  cos  (2/)  +  A^  cos  (3/),  &c., 

where   to   determine   Ao,  A^,  A,.,   &c.   we   multiply    successively    by    1, 
cos  t,  cos  2  A  &c.,  and  integrate  from  t  =  0  to  t  =  ir,  whence 

,   _  sin  {a-n)  J   _      2asin«-7r  ,  _  2«sin«7r 

J              n       ^        /     ,x„   2«  sin  air      , 
and  generally  A^  =  ( —  1)  .  — ri ^  when  n>  0. 

7r  {a  —  n  J 

Multiply   both   series   and   integrate,    and   we   get    by   the    proposed 

equation, 

[Co       a.€i     ,     «C2  aCi         .    \ 

1  =  sm  a-K  { — =-  +  ~„ — -„ z — -5  +  &c.> 

{a       a^—1       «^  — 2^       «^  — 3^  J 

Put  a  =  0,  1,  2,  3,  &c.  successively,  and  we  get 
_  1  2  3     . 

C(j  —       ,         Cj  —       ,         C2  =  —  ,   oZC. 

•TT  TT  TT 

Hence   ■tr(p{t)  =  1  +  2cos/  +  2cos2#  +  2cos3#,  &c. 

The  value  of  <t)(t)  is  therefore  the  transient  function  -  .  =^^ — ^-.'^    ^ — i-^ . 
^^  ^  TT   I  — Hh  cos  t  +  h^ 

{Vide  Art.  38.  Function  Fo),  when  h  is  put  equal  to  unity. 
Vol.  V.    Part  III.  3D 


330  Mr  MURPHY'S   THIRD   MEMOIR   ON   THE 

Ex.  2.     Given  fi(p(t)  .cos  {at)  =  cos  (a  0). 

As  before  (p{t)  =  c„  +  Cj  cos  t  +  c^  cos 2t  +  CaCosSt  +  &c. 

sin  flTT  fl       2«cos^       2«  cos  2^       2a  cos  3^       „     ] 

cos  at  =  < J \ h  &c.> 

therefore  cos  a0  =  sin(a7r)l-  -  -~-  +    f^' „ ^^j  +  &c.l 

'  [a       a^  —  1       a^  —  2'       a-  — 3^  J 

But  also  by  reciprocal  functions  we  get 

sinaTrQ       2acos0       2acos20       2acos30       „    1 
cosae  =  __  |-  _  _,__  +  --^-^^ -,__  +  &c.} 

TT                   1                  2COS0                 2cos20                 2cos30     „ 
Hence  Co  =  -  ,       c,  = ,       c,  = ,       Cs  — ,  &c. 

TT  TT  TT  TT 

therefore  7r^(#)  =  1 +2cos0  cos^  + 2  cos20  cos  2^  +  2  cos36  cos  3#  + &c. 

or  27r(pt=  1  +2COS  {9  +  t)  +  2  cos  2{e  + 1)  +  2COS 3{9  + 1)  +  Sic. 

+  l  +  2cos{9-t)  +  2cos2{9-t)  +  2  cos  3{9-t)  +  &c. 

^        (1-A)(1+^)  (1-A)(1+^) 

l-2h  cos{9  +  0  +  A'       1  -  2A  cos  (0  -  ^  +  *' 

when  A  is  put  equal  to  unity. 

Ex.  3.     Given  ft  <{> (t) :  cos  {at)  =  27'(a). 

jP(a)  must  be  such  (in  continuous  functions)  as  not  to  change  when 
—  a  is  put  for  a,  since  cos  (at)  which  is  under  the  sign  of  integration 
will  not  then  alter  its  value. 

Proceeding  as  in  the  former  examples  we  get 

ET/  ^        •  \<^o        ac,  ac,  acs         „     ] 

F{a)  =  sm«.  |-  _  -,_^  +  -,_^  _  -^-^^  +  &c.} 

Put  successively  a  =  0,  1,  2,  3,  &c.  hence 

Co  =  - .  1^(0) ,         c.  =  - .  F{1),         c,  =  -.  F{2),  &c. 

ir  IT  TT 

hence  7r(p{t)  =  F(0) +2F(1).  cos  #+ 2F(2) .  cos  2#  +  2F(3)  cos(30 +  &c. 


INVERSE   METHOD   OF   DEFINITE   INTEGRALS.  381 

Ex.  4.     Given  ft(p{t) .  {/{a  +  t)  +f{a-t)\  =  F{a), 

where   the   forms   of  the   functions  f  and   F  are   known,    and    that   of 
0  required. 

Put  (f){t)  =  Co  -1-  Ci  cos  t  +  d  cos  %t  +  c-i  cos  St  +  kc. 
f{a)  =  oo  +  a,  cos«  +  02  cos  2«+  "3  cos  3a  +  &c. 
where  a,„  a^,  a^,  &c.  are  known  numerical  quantities;  hence 
J'{a  +  t)+J^(a-t)=^2ao+2ai  cosacos  t+2a2  cos  2a  cos  2l  +  2a3  cos  3«  cos  3^-r&c. 

and    JP(«)  =  27raoCo  +  7raiCi  COS«  +  wa^d  COS2«  +  TrogCs  cos3«  +  &c. ; 

therefore  Co  =  — — - ,     c,  =  —^—  .  fa  F{a) .  cos  « ,     c^  =  -^—  L  F{a)  cos  2«,  &c. 

J         w..  1     r  t:t/    X    f  1  2C0S«C0S^         2cOS2«COS2#         „      ] 

and  7r(p{t)  =  —  f„F{a)  \—-  +  + +  &c.} 

Tr  [Zao  Oj  as  J 

the  hmits  of  all  the  integrals  being  0  and  tr. 

Ex.  5.       /■*%=J-,. 
Jta  —  t       a  —  h 

In  this  case  we  shall  employ  the  functions   V^  reciprocal  to  t". 

Put  <^{t)  =  CflFo  +  Ci  F",  At  C'^V-i  +  &c.  «c?  infinitum, 

1      1  !?;<'.  ,  •  ^   . 

and  ;:  =  — \ — ;  ^ — :  +  &c.  ««  infinitum; 

a-t       a       a'       {^  "^ 

^,       f  1  c„         1       c,         2.1       6-2         3.2.1      Cs       „ 

therefore  r  = ;r^  •  -5  +  ^    .    e  •  ^  —  ~. — ^  c   <-,  •  -7  +  &c- 

a-o      a       2.3   e^       3.4.5   a^       4.5.6.7   «' 

1        *        A=       6' 

=  -+—,+—  +  —4 ,  &c. 
a       a^       ci^       a 

TJ                   1                   2.3    ,  3.4.5    ,,  4.5.6.7    ,3    „ 

Hence  Co  =  1,     c,  = ~ .  *,     c,  =     ^    ^    .  V,     c^  =    .^    ^    ^   .  h\  &c. 

and  <^{t)  =  r,-^.br,  +  ^^.b^F.,-^^^^.b^r..  &c.   ■ 
=  r„  - 1.  r,.(4i)  +  —  •FAuy  -  1^^.  r3(4*r +  &c. 

3d2 


382  Mb  MURPHY'S  THIRD  MEMOIR  ON   THE 

Put   F{k)  =  Fo  +  F,A  +  V-it  +  kc.  ad  infinitum, 
as  found  in  Art.  28.    Sect.  vii. 

Hence  F{  -  kr')  =  K-  V.kr"  +  V.,kr'  -  &c. 

theretore  j^  ^^^-— ^   -  2  ^^»     a  ^^*+  2.4" '^'^  2.4.6-'^'^''^ 

the  limits  of  t  being  0  and  1 ; 

TT    ^^    v'l-T^  »2'  2.4  2.4.6 

Ex.  6.     j,<l>{t).f{a-t)=f{a-h). 

Denote  by  Pi,„  the  reciprocal  function  P„  when  ^  is  the  variable, 
by  Pi_ri  when  6  is  the  variable. 

Let/(«-0  =  J,P,,,  +  A,Pt,,  +  A,P,,,  +  A^Pt,^  +  &c. 
and  (j>{t)  =  CoPf.o  +  c,Pu  +  CaP,,^  +  c^.Pt.z  +  &c. 

.-.  f{a-h)  =  ^oCo  +  g  .  AxCi  +  g  .  ^sC^  +  \  .A^Ci  +  &c. 
but  changing  t  into  J  in  the  expansion  of  J'{a  —  t)  we  get 

f{a-h)  =  AoP,,o  +  ^,Pm  +  A,P,,,  +  A,P,,  +  &c. 
which  values  are  identical  when  Cc  =  Pi.,o,   c,  =  SPs,,,  c^  =  SPh.n,  &c. 
therefore  (j){t)  =  PmP.o  +  SP^P.,  +  5P„,P,.  +  7 P.^P.s  +  &c. 

45.  Ow  ^A^  appendage  necessary  to  complete  the  Solution  of'  a 
Definite-integral  Equation. 

In  the  examples  in  which  f{a,  t)  =  cosa^  given  in  the  last  article, 
the  function  F{a)  is  adapted  to  general  differentiation  relative  to  a, 
under  the  definite  integral ;  but  besides  the  prime  value  thus  obtained, 
there  must  be   an   appendage   to  represent   the   same  operation  on   zero. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  383 

which  contains  an  infinite  number  of  constants  multiplied  by  functions 
of  a,  which  may  vanish  or  not,  and  be  connected  or  unconnected  ac- 
cording both  to  the  nature  of  the  particular  operation  and  the  nature 
of  the  calculus  in  which  it  is  employed ;  this  has  been  already  shewn 
by  Mr  Peacock*,  and  in  Art.  20.  Sect.  vi.  of  this  Memoir.  The  same 
remark  applies   to   the  value   of  0  (t)  in  the   general  equation 

to  complete  it  we  must  add  ^{t)  where  ft^{t)  .f{t,  a)  =  0. 

To  obtain  \|/(/)  in  the  equation  ft<p{t)  .cos (at)  =  F {a)  above  mentioned. 

Let  us  suppose  (pi  {t),  (p-^  {t),  found  by  the  method  of  Art.  44.,  to 
satisfy  the  equations 

Jt(px  (t) .  cos  (at)  =  1  for  continuity, 
ft(p2{t) .  sin  {at)  =  1  for  discontinuity, 

differentiating  with  respect  to  a,  the  first  2n  times,  the  second  2«— 1 
times,  we  get 

f,<pi{t).f"' cos  (at)  =  0, 
ft  (p.2  it),  t"-' cos  {at)  ^  0. 

Hence, 

^{t)  =  (p,{t)  {At  +  Bf  +  Ct\  &c.}  +  0,(0  {A'f  +  Bt^  +  C't'^c.}, 
where  A,  B,  C,  &c.  A',  B',  C,  &;c.  are  absolute  constants. 

When  transient  functions  appear  in  the  appendage  or  even  in  the 
prime  solution,  they  must  not  be  neglected  (particularly  in  the  mole- 
cular investigations)  except  they  are  inadmissible  by  the  nature  of  the 
particular  question,  for  they  have  a  physical  as  well  as  a  geometrical 
meaning,  as  they  are  capable  of  expressing  in  continuous  analytical 
forms,  the  state  of  bodies  and  their  mutual  actions  when  they  are  com- 
posed of  absolute  mathematical  centres  of  forces,  all  separated  mutually  by 
infinitesimal  intervals. 


Q/   Q;  a>  04  Qf  Jr 

*  Third  Vol.  Report  of  British  Assoc,  p.  212,  &c. 


384  Mr  MURPHY'S  THIRD  MEMOIR  ON  THE 

Thus  let  the  ratio  of  the   weight  to  the  extent  of  an  element  P  of 
a   straight  rod   AB  be    expressed  by    the   transient  function 

(\-h){\+h)  I,       .       , 

- — —^ /o    -^N  ■    J.2 '  when  ^  =  1 ; 

and  where  AP=(p,  and  the  whole  length  AJB  =  ir,  and  n  is  very  great 
and  integer. 

Then  the  whole  weight  is  finite,  viz.    f  - — ^—^ — '—- — - — '—n  =  1,    vet 
"  J^l  —  2hcos2n(p  +  h^  ^ 

this  function  has  only  an  existence  when  0  =  0,  -,   — ,   — ...&c.,  and 

therefore   the  rod  is  actually  composed  of  disjoint  particles   Q,,  Qa,  Q3, 

&c.   which  are   separated   by  equal  intervals,  each  infinitesimals,   viz.  -, 

when  n  is  very  great,  and  equal  to  the  actual  number  of  particles ; 
the  action  of  such  a  system  on  another  given  one,  may  always  be 
estimated  by  using  the  transient  function  in  its  general  form,  and  lastly, 
putting  h  equal  unity. 

46.     Equations  which  contain  two  or  more  Definite  Integrals. 

Given,        jj  cp  (t)  .f(t,  a,  b)  +  f,^l.{t)  .F  {t,  a,  b)  =  E  {a,  b), 

the  forms  of  the  functions^  F,  E  being  known,  the  forms  of  0  and 
■<\f  are  required. 

Put       /(#,  a,  b)  =  ^oPo  +  A,P,  +  A^P^  +  A^Ps  +  &c.  ad  inf. 

where  A^,  Ai,  A2,  &c.  are  known  functions  of  a  and  b,  and  Po,  Pi,  &c. 
any  self-reciprocal  functions  of  t,  such  that  ftPr!^  =  a„,  which  will  be  a 
known  numerical  quantity. 

Similarly,  F  {t,  a,  b)  =  B,Po  +  B,Pi  +  B,P,  +  B,P^  +  &c.  ad  inf., 
where  B^,  Bi,  B^,  &c.  are  known  functions  of  a  and  b. 

Again,  let  (p{t)  =CoPo +  c,Pi +CaP2  +  C3P3,  &c.  ad  inf. 
where  Co,  c,,  Ca,  &c.  are  unknown  numerical  quantities, 
and  \l/{t)  =  eoPo  +  ejPj+e2P2  +  e3P3,&iC.  ad  inf.. 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  885 

where  eo,  e^,  e.,,  &c.  are  also  unknown. 
The  proposed  equation  then  becomes 

+  eoOoBo  +  eiUiBi  +  eia-iS^  +  &c.) 

Now  to  the  function  A„  there  may  be  found  a  function  A„  reciprocal  relative  to  a, 
and  to  B„  B„  b. 

Let     f„AaB„  =  U„  a  function  of  b  only, 
ftBoA,,  =  V„  a  only. 

Hence,  f^AoE  (a,  b)-CoaofaAoAo  =  eoaoUo  +  eia^Ui  + 6.^0^112  + kc.  ad  inf. 
ft  BoE  {a,  b)  -  e^a^  fiBoB^  =  c^a^  K  +  c, a,  F;  +  c^a^  F;  +  &c.  ad  inf. 

Let  t/„  be  the  function  of  b,  which  is  reciprocal  to  f7„, 
V„  of  «,    V^. 

\L k  {Ao U,E  {a,  b)  -  c,a,A,A^  t7"„)  =  e„a„ /j  t7„  U^ 
Hence,      \ \, 

\fJ,{Bo  KEia,  b)  -  e„a,B,B,K)  =  c„«„/„F„rJ 

by  which  equations  the  constants  Co,  e,,  are  immediately  determined. 

\fa  fb  (Ao  U„E  (a,  b)  -  c^a^A^A^  U„)  =  e„ a„  /j  Un  UA 

Also,         \ >; 

(/„  /,  (^0  KE{a,  b)  -  e,a,B,B,  K)  =  c„a„  f„  V„  Vj\ 

and  since  c„,  e^,  have  been  found,  the  latter  equations  determine  gene- 
rally the  coefficients  c„,  e„,  and  therefore  the  required  functions  <p{t), 
^  {t)  are  known. 

In  like  manner  by  employing  reciprocal  functions  relative  to  double 
integration,  we  may  solve  equations  containing  three  unknown  func- 
tions,   &c. 

The  problem  of  the  distribution  of  electricity  on  bodies  of  which 
the  surfaces  are  not .  continuous,  introduces  equations  of  this  nature. 


386  Mb  MURPHY'S  THIRD  MEMOIR  ON  THE 

47.     Simultaneous  Equations  to  Definite  Integrals. 

Given  l-^*^^^^  ••^^^'  *)  +-'^^(^)  •  ^^^' «)  =  ^^^H 
\!i<p{t)Mt,  a)  +f,i.(t) .  FM  a)  =  EAa)i  ' 

the  forms  of  the  functions  j^  F,  E,fi,  Fi,  Ei,  being  known,  the  forms 
of  (p  and  \{^  are  required. 

Multiply  the  second  equation  by  an  arbitrary  quantity  \,  and  adding 
to  the  first,  put 

f(t,  a)  +  X/ {t,  a)  =  A,Po  +  A,P,  +  A,P,  +  &c. 

F{t,a)  +xF^{t,a)  =  AoQo  +  A,Q,  + A,Q2  +  &c. 

(pit)  =  CoPo   +  c,P,'  +  c,P^  +  &c. 

^^(0  =  ^oQo'  +  e,Q,  +  e,Q:  +  &c. 

where  P„,  P,,  P2,  &c.\  «       ..  /?  ^      i 

^     ^     .-k     o     f  are  functions  of  t  only, 

Qo,  Qi,  Q2,  &C.J  ^ 

A^,  Ai,  Ai,\  known  functions  of  a,  X,  and  self-reciprocal  relative 
to  a, 

PI,  Qn  reciprocal  to  P„,  Q„  respectively,  hence 

(putting  /,P„P„'=;),„  j;Q„Q„'=^„)         E{a)  +  XEM 
=  c^poAo  +  c,p,Ai  +  C2P2A2  +  &c.  +  eoqoAo  +  e^q.A,  +  e^q^A-^,  &c. ; 
.-.  faAoEia)  +  X  faAoE^  (a)  =  {copo  +  eoqo)faAo\ 
faA,E{a)  +  XlA,E,{a)  =  iCiP^  +  e,q,)faA^ 

and  giving  to  X  any  two  values  in  each  of  these  equations,  the  first 
will  produce  two  equations  which  determine  Co,  eo,  the  second  will 
similarly  give  Ci,  e^,  &c.,  and  thence  the  functions  (p{t),  \l/t  are  known. 

The  same  method  is  applicable  to  any  number   (n)   of  simultaneous 
equations  involving  n  unknown  functions. 

48.     Definite-integral  Equations  of  superior  orders  and  degrees. 

Methods   similar   to   the   preceding   are    applicable   in   most   cases   of 
the  former  class  thus : 


INVERSE   METHOD  OF   DEFINITE   INTEGRALS.  387 

Given  fj,<t>{t,  T)f(t,  T,  a)  =  F{a), 
the  forms  F  and  J"  being  known  to  determine  (p. 

By   Art.  16.    Sect.  iv.  let  a  function  Q„   be  formed   which   shall  be 
self-reciprocal,  relative  to  double  integration  for  t  and  t. 

Put  ^(#,t)  =  Co Q„  +  CiQi  + C2Q2  +  &C.      1  _rrri'i 

and/(^,T,«)  =  ^„Q„+^,Q,  +  ^,Q,  +  &c.r'''*  ^^^  a»-i.^t^», 

hence  F(a)  =  aoCo^o  +  aiCi^, +0203^2 +  &c. 

Let  ^„'  be  a  function  of  a  reciprocal  to  A„, 
then  faA,'F{a)  =  c^aJaA.A^, 

faA,'F{a)  =  c,aJaA,A„ 
&c.  &c. 

whence  Co,  Ci,  &c.  being  determined,  the  function  (p{f,T)  is  known. 

Equations  of  superior  degrees  must  generally  be  converted  into  equa- 
tions of  superior  orders  to  be  easily  solved,  thus; 

Given  f,(p{t)  .fit,  a)  x  [,cp{t) .  F{t,  «)  =  >/.(«), 
the  forms  ^  F,  and  -^^f  being  given  to  find  the  function  0. 

Introduce  another  variable  t  having  the  same  limits  as  t,   then  it  is 
evident  that 

J,<p{t) .  F(t,  a)  =  /^«^(t)  .  F{t,  a) ; 

.-.  U^cp{t)  .(pi-r)  ./{t,  a) .  F{t,  a)  =  f  (a), 

and  since  y(#,  a) .  F{t,  a)  is  a  given  function  of  t,  t  and  a,  the  unknown 
function  (p{t).(p(T)  will  be  determined  as  above,  and  representing  it  by 
<p^(t,T),  let  a  be  a  root  of  the  equation  0(t)  =  1,  then  since  (p{t).(p{T) 
=  0i(#,  t),  we  get  the  required  function  (}>{t)  =  <pi{t,  a),  and  again  putting 
^  =  a  we  get  ^1  (a,  a)  =  1,  from  which  equation  a  is  known,  and  there- 
fore <p{t)  =  <pi{t,a)  is  also  known. 

49.     In  researches  on  the  subjects  of  electricity,  and  the  phaenomena 
dependent  on  the  molecular  construction  of  bodies,  the  only  data  which 
can  be  furnished  by  experience  are  the  total  actions,   and  consequently 
Vol.  V.    JPart  III.  SE 


388  Mr  MURPHY'S  THIRD   MEMOIR   ON   THE 

the  analytical  processes  of  calculation  require  the  solution  of  definite 
integral  equations:  some  of  these  have  been  resolved  by  Laplace  and 
others,  by  means  of  particular  artifices  by  which  the  unknown  functions 
were  subjected  to  differential  equations ;  but  as  no  general  method 
existed  for  this  purpose,  the  resolution  of  such  equations  has  been  ex- 
tremely limited,  and  apparently  simple  physical  problems,  such  as  the 
distribution  of  electricity  on  surfaces,  (with  the  exception  of  a  very 
few  cases)  have  consequently  defied  the  powers  of  analysis.  Besides, 
an  abundance  of  facts  connected  with  the  interior  arrangement  of  the 
molecules  of  bodies  are  of  such  a  nature,  that  mathematics  possessed 
but  little  power  of  reducing  them  to  analytical  forms,  calculated  to 
produce  any  valuable  inferences ;  these  facts  are  daily  increasing  in 
number,  and  the  analyst  is  far  behind  the  cultivator  of  Experimental 
Physics.  The  Memoirs  on  the  Inverse  Method  of  Definite  Integrals 
which  are  now  concluded,  and  which  have  been  pursued  when  the 
absence  of  ordinary  engagements  permitted,  originated  in  the  belief 
that  by  proceeding  gradually  from  the  simplest  classes  of  Definite 
Integrals  to  the  more  complex,  the  general  principles  of  an  Inverse 
Method  would  be  discoverable.  The  formation  of  all  possible  classes 
of  Reciprocal  Functions,  and  the  Transient  Functions  included  amongst 
them,  have  at  length  furnished  means  for  the  resolution  of  equations 
to  Definite  Integrals.  The  author  is  however  well  aware  that  there 
must  exist  numerous  imperfections  in  the  manner  in  which  his  design 
is  executed,  but  believing  also  that  by  those  endeavours,  however  weak, 
some  fresh  powers  have  accrued  to  analysis,  as  an  instrument  of  investi- 
gation, he  trusts  they  will  deserve  the  approbation  of  the  Society. 


R.  MURPHY. 


Caius  Colleob, 
Dec.  24,  1834. 


INVERSE   METHOD   OF  DEFINITE   INTEGRALS.  389 


Analytical  Table   of  Reference  to  the  "Memoirs  on  the  Inverse 
Method  of  Definite  Integrals." 


FIRST    MEMOIR,  Vol.  IV.  Page  353,  &c. 

PAGE 

Introduction 353 

Section  I.     Principles  relative  to  Continuous  Functions. 

Art.  1.     Method  of  reducing  the  given  limits  of  integration  to  0  and  1  in  all  cases 358 

Arts.  2,  3,  4.  In  the  general  equation  ft f{t)  .t''=  <p(x),  x  is  understood  to  lie  between 
—  1  and  +  00 ,  then  cyj  (x)  converges  to  zero  as  x  increases,  when  y(<)  is  any  of  the 
functions  usually  received  in  analysis;  consequent  division  of  the  subject 35g 

Art.  5.     Rule;   When  the  known  function  <p{x)  is  rational,  seek  the  coefficient  of  -  in 

X 

<p  (x) .  t~',  dividing  it  by  /  we  obtain  fit) 362 

Art.  6.     Examples s6S 

Arts.  7,  8.     Means  of  facilitating  the  Calculus  oi  f{t) 3Q5    . 

Art.  9.  and  Note  (A).     When  0  {x)  is  a  logarithmic  function 366,  400 

Art.  10.     When  (pix)  is  expressed  by  an  equation  to  finite  differences 367 

Art.  1 1 .     When  <f>  (x)  is  a  fraction,  the  denominator  containing  imaginary  factors 369 

Art.  12.     When  0  {x)  is  irrational 37O 

Art.  13.     Cases  when  equations  of  the  form  J',f(J,').(t'' ±t~')=if>x,  may  be  resolved  by 

the  preceding  method 37] 

Art.  14.     Extension   of  the   general   rule   to   successive   integration   with   respect  to  any 

number  of  variables , 373 

Section  II.     Principles  relative  to  Discontinuous  Functions. 

Art.  15.     Cases  of  discontinuity  in  Physical  Problems  quoted 374 

Art.  16.     To  find  a  formula  which  shall  represent  the  least  of  the  two  quantities  a,  /3.  .  375 
Art.  18.     To  find  a  formula  which  shall  represent_/(a)  or y(/3)  according  as  a  is   <  or  >  /3.  376 

Art.  19.     To  find  a  formula  which  shall  represent  r^,  or  -= — j— ,  according  as  a  is 

a  —  lip  p  —  na 

<  or  >    /3 377 

Art.  20.     To  find  a  formula  which  shall  represent  ■~^^ ,  or  ^^^, ,  according  as  a  is  <  or  >  /3  378 

Arts.  21,  22.     Method  of  representing  discontinuous  functions  of  any  number  of  breaks 380 

Arts.  23,  24.     Geometrical  Illustrations  of  the  theory  of  discontinuity 382 

3E2 


390  Mr  MURPHY'S  THIRD   MEMOIR  ON  THE 

PAGE 

Section  III.     Application  of  the  preceding  principles  to  the  Phaenomena   of  Developed 

Electricity/ 386 

Note  (A),  No.  2.     On  the  general  separation  of  the  positive  powers  of  the  variable  from 

the  negative 402 

Note  (B),  No.  1 .     On  the  apparently  improper  forms  of  (p  (x) 404 

No.  2.     Method  of  valuing  the  results  of  operative  functions 406 

SECOND   MEMOIR.     Vol.  V.    Page  113,  &c. 
Introduction 113 

Section  IV.     Inverse  Method  Jbr  Defitiite  Integrals  which  vanish,  and  theory  of  Reci- 
procal Functions. 

Arts.  1,2.     X   being  restricted  to  the  natural  numbers  0,  1,  2, (»— 1)  to  &nd  fit)  so 

tha.tf,f(t).f  =  0 116 

Art  3.     P„  denoting  the  function  y(<)  above-found,  when  m  and  n  are  unequal /,P„P„=0, 

and  when  equal /,P,P„  =  117 

Art.  4.     To  find  a  rational  function _/(<)  which  may  satisfy  the  equationy^y(<)'''=0,  x  being 

any  number  of  the  series  p,  ^+  1,. .  .p-{-n—l 118 

Art.  5.     The  general  form  of  fit),  when  x  is  from  0  to  n— 1  inclusive,  is 

d"  (ft'"  V\ 
At)  =      •  ^^^„      ^  ,  where  t' =  \  -  t 118 

Art.  6.     In  this  case  the  equation^" (/)  =  0,  has  n  real  roots  lying  between  0  and  1 1 19 

Arts.  7,  8.     To   find  a  rational  function  of  h.  1. /,    such   that /,y{h.  1.  (<)}.<'  =  0,    when 

X  is  from  0  to  n—  1  inclusive 120 

Art.  9.     Denoting  this  function  by  Z.„,   the  function  which  it  generates  is  the  value  of 

u  in  the  equation  u  {l  —  hh.\.  u)  =  t 122 

du 
Art.  10.     If    Q„    be    the    coefficient   of   *"   in    -r- ,    u   being   found  from    the   equation 

m(1  — A  £/)  =  <,    where   J7  is  a  function  of  u  vanishing  when  u  =  \,   and   T  the  same 

function  of   t,    then -^l  =  "("- V""^"~""^'^ 122 

Art.  11.     If  t/  be  a  rational  and  entire  function  of  u  vanishing  when  m=1,  and  Q,  be 
the  term  independent  of  u  in  the  product  i/"  I  1 I  ,  then  shaliy]Q„<'=  0,  when 

X  is  from  0  to  «—  1  inclusive 12S 

Art.  12.     To   find    (p,  q)„   a  rational   and  entire   function   of  P"  of  n  dimensions,   which 

multiplied  by  a  rational  and  entire  function  of  f  of  less  than  n  dimensions,  the  integral 

of  the  product  may  vanish  from  t  —  Otot=l 125 

Art.  13.     Reciprocal  Functions;    such   are   ip,q)„,    {<l>p\';    value  of  the  integral  of  the 

product  when  n=n' 126 

Art.  14.     To  find  a  function  A„  reciprocal  to  the  function  L„  found  in  Art.  8 128 


INVERSE  METHOD  OF  DEFINITE  INTEGRALS.  391 

PAGE 

Art.  15.     General  principle  for  finding  Reciprocal  Functions  to  simple  integration 130 

Art.  l6.     The  same  extended  to  integration  for  any  number  of  variables 131 

Art.  17.     Examples 132 

Section  V.     Inverse  Method  for  Junctions  which  contain  positive  ponters  of  x,  or  are 
under  any  other  form. 

Art.  18.     An  appendage  must  be  annexed  in  all  such  cases 135 

Arts.  19,  20.     When  ^j;  is  a  rational  and  entire  function  of  x ;  and  particular  example 

when  ^(.r)  =  l 136 

Art.  21.     To  find /(<)  when7(/(O-''=0W,  an<i  ^  is  from  0  to  n—\  inclusive 138 

Arts.  22,  23.     Various  modes  of  determining  y(<)  in  this  case 141 

h 

Arts.  24,  25.     The  coefficient  of  h"  in  the  expansion  of        ,   is  a  self-reciprocal  function  146 


THIRD    MEMOIR.     Vol.  V.     Page  315,  &c. 
Introduction 315 

Section  VI.     Method  of  discovering   Reciprocal  Functions,    when   the   integrations  are 

performed  with  respect  to  any  fonction  of  the  variable. 

Arts.  1,  2.     General  principle  for  varying  the  limits 318 

d'.^t'f'V)    dt 
Art.  3.     If  V  can  be  found  so  that  — "'      —  'Tl  "™^y  "'^  ''^  "  dimensions  in  t  (where 

t'  =  \  —  t)  then  this  quantity  will  be  self-reciprocal  relative  to  (p $iq 

Art.  4.     If  V  can  be  found  so  that  —   j^ ■  j?  "^^y  be  of  n  dimensions  in  t,  then 

the  factor  by  which  -j^  is  multiplied  will  be  self-reciprocal  relative  to  0 320 

Arts.  5,  6.     If  <p=f,(Jt(y  indefinite,  and  m  between  —1  and  -j-oo,  and  if 

d^  {(tt'Y'^"'\ 
Qn  =  1    2       xdf-^" '  ("')"""  *^"  *^^^  ^  ^^  self-reciprocal  relative  to  (p 321 


n  —  m 


rhfti'\ 

If  0  z=  ftittfy  indefinite,  and  m  be  between  -1-1  and  —00,  and  if  a»  =  —      ' 

^      -"^     '  >  '  1"      i.2...ndt'" 

then  shall  q,  be  self-reciprocal  relative  to  0 321 

Art.  7.     To  find  the  functions  which  Q„,  5,  generate 322 

Arts.  8,  9.     When  7«  =  —  §,  Q„,  q,  are  the  trigonometrical  reciprocals 323,  325 

Art  10.     In  the  identities  thus  obtained,   the  sign  of  n  may  be  changed  so  as  to  pass 

from  differential  coefficients  to  integrals 326 

Art.  1 1 .     The  two  series  of  reciprocal  functions  obtained  from  the  theorems  of  Arts.  5  81.  6. 

differ  only  with  respect  to  the  variable  of  integration 328 

Art.  12.     Examples  of  the  preceding  theory 329 

Art.  13.     To  express  Q„  and  q„  in  terms  of  t  alone 330 

Art.  14.     To  express  Q„  and  q„  by  means  of  differential  equations 332 


892  Mr  MURPHY'S   THIRD   MEMOIR   ON   THE 

PAGE 

Art.  15.  The  reciprocal  functions  expressed  by  the  general  formulae  for  Q,,  q,  all  possess 
a  common  property,  viz.,  their  integrals  vanish  when  taken  between  limits  which 
render  the  functions  maxima  or  minima 333 

Art.  16.     To  find  the  complete  integral  of  the  equation 

i<'^  +  (m+l)(l-2/)  -^^+n(n+2m  +  l)u  =  0 335 

Art.  1 7.  To  find  explicitly  the  omitted  part  of  the  complete  integral  in  Laplace's  equation, 
for  the  coefficients  in  the  expansion  of  the  reciprocal  of  the  distance  between  two 
points  in  a  plane 338 

Art.  18.  When  m  =  —\  the  general  equation  of  Art.  l6.  represents  the  trigonometrical 
functions 342 

Art.  19.     Remarkable  properties  of  the  functions 

G  =  g^cose  cos(a;sin0),  0'=  £■>;  ™s  e  sin  (a;  sin  0) 342 

Art.  20.     Application  to  the  general  differentiation  of  rational  and  integer  functions  oi  x . . .   344 
Art.  21.     The  sum  of  all  the  divisors  of  a  number  n,  including  itself  and  unity 

= fy^- 1-  {sin^  sin  2^ sinn^}  .  cos  2n(^ 346 

TT 

Section  VII.     On  Transient  Functions. 

Art.  22.     Nature  of  transient  functions 347 

Art.  23.     To  find  a  function  reciprocal  to  f(t,  n)  any  given   function   of  the  variable  t 

and  integer  n ' 348 

Art.  24.     To  find  a  function  V„  reciprocal  to  /" 349 

Art.  25.     The  function  V„  is  transient 352 

Art.  26.     To  express  the  transient  function  F„  in  a  finite  form 354 

Art.  27.     Discussion  of  the  transient  function  Fo ;   it  represents  the  state  of  a  body  which 

an  electric  spark  is  about  to  enter 356 

Art.  28.     To  find  the  quantity  to  which  F„  is  the  generating  function 360 

Art.  29.     To  expand  a  given  function  in  terms  of  the  functions  F„ 36l 

Art.  30.     To  find  a  function  t/„  reciprocal  to  (h.  1.  t)' 362 

Art.  31.     In  a  finite  form   U„  = H — 2_  when  A  =  l 363 

1  .2...ndh" 

Art.  32.  Properties  of  C7„  &&f,Uj'  =  x',  &c S6S 

Art.  33.  Discussion  of  the  function   Uq 364 

Art.  34.  To  expand  a  given  function  in  terras  of  the  functions  I/„ 365 

Art  35.  To  find  a  function  reciprocal  to  /"  when  the  limits  of  t  are  0  and  =0 366 

Art.  36.  To  find  a  function  F„  reciprocal  to  cos"^  between  the  limits  ^  =  0  and  <p=-ir.  366 

Art.  37.  The  function  F„  is  transient - 367 

Art.  38.  To  express  F,  in  a  finite  form 368 

Art.  39.  Means  of  summing  a  series  expressed  in  transient  functions 369 


INVERSE   METHOD   OF  DEFINITE   INTEGRALS.  393 

PAGE 

Section  VIII.     On  the  Resolution  of  Equations  which  involve  Definite  Integrals. 

Art.  40.     Method  of  decomposition  into  elements 371 

Art.  41.     Density  of  a  cylindric  shell  which  exercises  no  action  on  any  point  in  its  axis 

with  any  law  of  force 372 

Art.  42.     Examples  when  the  law  of  force  is  the  inverse  square  of  the  distance 374 

Art.  43.     Resolution  of  equations  which  contain  but  one  definite  integral  and  one  parameter  377 

Art.  44.     Examples 379 

Art.  45.     On   the   appendage  necessary  to   complete   the  solution  of  a  Definite   Integral 

Equation 382 

Transient  functions  capable  of  representing  in  a  continuous  form  the  state  of  a  body 

composed  of  mathematical  centers  of  forces  separated  by  infinitesimal  intervals 383 

Art.  46.     Equations  which  contain  two  or  more  Definite  integrals  and  as  many  parameters  384 

Art.  47.     Simultaneous  equations  to  Definite  integrals 386" 

Art.  48.     Definite  integral  equations  of  superior  orders  and  degrees 386 

Art.  49.     Conclusion 387 


ERRATA. 

PAOE 
First  Memoir.    359,  line  9.  16.  18.    dele  y  in  the  sign  ooy. 
Vol.  IV.    377,  line  8.  for  A</3  read  A-^  1. 

406,  lowest  line  and  third  from  bottom,  for  terms  read  times. 

407,  line  17.  supply  the  word,  equation. 

Second  Memoir.    Vol.  V.    134,  line  6,  after  -j^  supply  (tf)*. 

d'P 
line  7.  after        ,*  supply  (tf). 
dt 

136,  line  3,  4.  18.  for  v,  put  v.. 

Third  Memoir.    Vol,  V.    332,  lowest  line,  for  /'"(•-')(()  read  /'"<'-"(«). 
333,  line  5,  for  /(O)  read  f"(0). 
for  m+3  read  (m+3). 

337,  line  8,  put  (tf)-"  before  ~     in  the  last  term. 
346,  line  16,  for  intger  read  integer. 
357.  line  8,/or  (1-ft*)  reod  (1-A)*. 


XV.  Oil  the  Determination  of  the  Exterior  and  Interior  Attractions  of 
Ellipsoids  of  Variable  Densities.  By  George  Green,  Esq., 
Caius   College. 


[Read  May  6,   1833.] 


The  determination  of  the  attractions  of  ellipsoids,  even  on  the  hypo- 
thesis of  a  uniform  density,  has,  on  account  of  the  utility  and  difficulty 
of  the  problem,  engaged  the  attention  of  the  greatest  mathematicians. 
Its  solution,  first  attempted  by  Newton,  has  been  improved  by  the  suc- 
cessive labours  of  Maclaurin,  d'Alembert,  Lagrange,  Legendre,  Laplace, 
and  Ivory.  Before  presenting  a  new  solution  of  such  a  problem,  it 
will  naturally  be  expected  that  I  should  explain  in  some  degree  the 
nature  of  the  method  to  be  employed  for  that  end,  in  the  following 
paper;  and  this  explanation  will  be  the  more  requisite,  because,  from 
a  fear  of  encroaching  too  much  upon  the  Society's  time,  some  very 
comprehensive  analytical  theorems  have  been  in  the  first  instance  given 
in  all  their  generality. 

It  is  well  known,  that  when  the  attracted  point  p  is  situated  within 
the  ellipsoid,  the  solution  of  the  problem  is  comparatively  easy,  but 
that  from  a  breach  of  the  law  of  continuity  in  the  values  of  the 
attractions  when  p  passes  from  the  interior  of  the  ellipsoid  into  the 
exterior  space,  the  functions  by  which  these  attractions  are  given  in  the 
former  case  will  not  apply  to  the  latter.  As  however  this  violation 
of  the  law  of  continuity  may  always  be  avoided  by  simply  adding  a 
positive  quantity,  u"  for  instance,  to  that  under  the  radical  signs  in 
the  original  integrals,  it  seemed  probable  that  some  advantage  might 
thus  be  obtained,  and  the  attractions  in  both  cases,  deduced  from  one 
common  formula  which  would  only  require  the  auxiliary  variable  u  to 
become  evanescent  in  the  final  result.  The  principal  advantage  how- 
ever which  arises  from  the  introduction  of  the  new  variable  u,  depends 
Vol.  V.     Part  III.  SF 


396        Mr  green,  ON  THE  DETERMINATION  OF  THE 

on  the  property  which  a  certain  function  F'*  then  possesses  of  satisfy- 
ing a  partial  differential  equation,  whenever  the  law  of  the  attraction 
is  inversely  as  any  power  n  of  the  distance.  For  by  a  proper  applica- 
tion of  this  equation  we  may  avoid  all  the  difficulty  usually  presented 
by  the  integrations,  and  at  the  same  time  find  the  required  attrac- 
tions when  the  density  p  is  expressed  by  the  product  of  two  factors, 
one  of  which  is  a  simple  algebraic  quantity,  and  the  remaining  one 
any  rational  and  entire  function  of  the  rectangular  co-ordinates  of  the 
element  to  which  p  belongs. 

The  original  problem  being  thus  brought  completely  within  the  pale 
of  analysis,  is  no  longer  confined  as  it  were  to  the  three  dimensions  of 
space.  In  fact,  p'  may  represent  a  function  of  any  number  s,  of  in- 
dependent variables,  each  of  which  may  be  marked  with  an  accent,  in 
order  to  distinguish  this  first  system  from  another  system  of  s  analo- 
gous and  unaccented  variables,  to  be  afterwards  noticed,  and  F'  may 
represent  the  value  of  a  multiple  integral  of  s  dimensions,  of  which  every 
element  is  expressed  by  a  fraction  having  for  numerator  the  continued 
product  of  p  into  the  elements  of  all  the  accented  variables,  and  for 
denominator  a  quantity  containing  the  whole  of  these,  with  the  un- 
accented ones  also  formed  exactly  on  the  model  of  the  corresponding 
one  in  the  value  of  V  belonging  to  the  original  problem.  Supposing 
now  the  auxiliary  variable  u  is  introduced,  and  the  s  integrations  are 
effected,  then  will  the  resulting  value  of  ^  be  a  function  of  u  and  of 
the  s  unaccented  variable  to  be  determined.     But  after  the  introduction 

*  This  function  in  its  original  form  is  given  by 

-. /•  p'  dx  dy  dz 

J  {{X  -  xy  +  (/  -  yf  +  (.'  -  2)2}"-^' 

where  dx  dy  dz  represents  the  volume  of  any  element  of  the  attracting  body  of  which  p' 
is  the  density  and  x ,  y ,  z  are  the  rectangular  co-ordinates ;  x,  y,  z  being  the  co-ordinates 
of  the  attracted  point  p.  But  when  we  introduce  the  auxiliary  variable  u  which  is  to  be 
made  equal  to  zero  in  the  final  result, 

jr  _     r  p  dx  dy  dz 

J{(^a:'-xf-\.{y-yy  +  {z-zf  +  u^yr' 
■     -  .YOii 
both  integrals  being  supposed  to  extend  over  the  whole  volume  of  the  attracting  body. 


ATTRACTIONS   OF   ELLIPSOIDS   OF   VARIABLE  DENSITIES.         397 

of  w,  the  function  V  has  the  property  of  satisfying  a  partial  differen- 
tial equation  of  the  second  order,  and  by  an  application  of  the  Cal- 
culus of  Variations  it  will  be  proved  in  the  sequel  that  the  required 
value  of  V  may  always  be  obtained  by  merely  satisfying  this  equation, 
and  certain  other  simple  conditions  when  p  is  equal  to  the  product 
of  two  factors,  one  of  which  may  be  any  rational  and  entire  function 
of  the  s  accented  variables,  the  remaining  one  being  a  simple  algebraic 
function  whose  form  continues  unchanged,  whatever  that  of  the  first 
factor  may  be. 

The  chief  object  of  the  present  paper  is  to  resolve  the  problem 
in  the  more  extended  signification  which  we  have  endeavoured  to  ex- 
plain in  the  preceding  paragraph,  and,  as  is  by  no  means  unusual,  the 
simplicity  of  the  conclusions  corresponds  with  the  generality  of  the 
method  employed  in  obtaining  them.  For  when  we  introduce  other 
variables  connected  with  the  original  ones  by  the  most  simple  rela- 
tions, the  rational  and  entire  factor  in  p  still  remains  rational  and 
entire  of  the  same  degree,  and  may  vmder  its  altered  form  be  ex- 
panded in  a  series  of  a  finite  number  of  similar  quantities,  to  each  of 
which  there  corresponds  a  term  in  V,  expressed  by  the  product  of  two 
factors;  the  first  being  a  rational  and  entire  function  of  s  of  the  new 
variables  entering  into  V,  and  the  second  a  function  of  the  remaining 
new  variable  h,  whose  differential  coefficient  is  an  algebraic  quantity. 
Moreover  the  first  is  immediately  deducible  from  the  corresponding 
part  of  p    without  calculation. 

The  solution  of  the  problem  in  its  extended  signification  being  thus 
completed,  no  difficulties  can  arise  in  applying  it  to  particular  cases. 
We  have  therefore  on  the  present  occasion  given  two  applications 
only.  In  the  first,  which  relates  to  the  attractions  of  ellipsoids,  both 
the  interior  and  exterior  ones  are  comprised  in  a  common  formula 
agreeably  to  a  preceding  observation,  and  the  discontinuity  before 
noticed  falls  upon  one  of  the  independent  variables,  in  functions  of 
which  both  these  attractions  are  expressed ;  this  variable  being  con- 
stantly equal  to  zero  so  long  as  the  attracted  point  j)  remains  within 
the  ellipsoid,  but  becoming  equal  to  a  determinate  function  of  the  co- 

3f2 


398  Mr  green,  ON   THE   DETERMINATION   OF    THE 

ordinates  of  p,  when  p  is  situated  in  the  exterior  space.  Instead  too 
of  seeking  directly  the  value  of  V,  all  its  differentials  have  first  been 
deduced,  and  thence  the  value  of  V  obtained  by  integration.  This 
slight  modification  has  been  given  to  our  method,  both  because  it 
renders  the  determination  of  V  in  the  case  considered  more  easy,  and 
may  likewise  be  usefully  employed  in  the  more  general  one  before 
mentioned.  The  other  application  is  remarkable  both  on  account  of 
the  simplicity  of  the  results  to  which  it  leads,  and  of  their  analogy 
with  those  obtained  by  Laplace.  (Mdc.  C^.  Liv.  iii.  Chap.  2.)  In  fact, 
it  would  be  easy  to  shew  that  these  last  are  only  particular  cases  of 
the  more  general  ones  contained  in  the  article  now  under  notice. 

The  general  solution  of  the  partial  differential  equation  of  the  second 
order,  deducible  from  the  seventh  and  three  following  articles  of  this 
paper,  and  in  which  the  principal  variable  1^  is  a  function  of  #  +  1 
independent  variables,  is  capable  of  being  applied  with  advantage  to 
various  interesting  physico-mathematical  enquiries.  Indeed  the  law  of 
the  distribution  of  heat  in  a  body  of  ellipsoidal  figure,  and  that  of  the 
motion  of  a  non-elastic  fluid  over  a  solid  obstacle  of  similar  form, 
may  be  thence  almost  immediately  deduced;  but  the  length  of  our 
paper  entirely  precludes  any  thing  more  than  an  allusion  to  these  ap- 
plications on  the  present  occasion. 


1.  The  object  of  the  present  paper  will  be  to  exhibit  certain 
general  analytical  formulae,  from  which  may  be  deduced  as  a  very 
particular  case  the  values  of  the  attractions  exerted  by  ellipsoids  upon 
any  exterior  or  interior  point,  supposing  their  densities  to  be  represented 
by  functions  of  great  generality. 

Let  us  therefore  begin  with  considering  p  as  a  function  of  the  s 
independent  variables 

»r J ,     x<i ,     x^  ••••• o/i, 

and  let  us  afterwards  form  the  function 

dxjdx^ dxj dxl .  p .^. 

'{{x,-xiJ^{x,-xl)^^ ^(x.-xlJ^u'-S^ 


r=f- 


n-1 
2 


ATTRACTIONS  OF   ELLIPSOIDS  OF  VARIABLE  DENSITIES.         399 

the  sign  /  serving  to  indicate  *  integrations  relative  to    the    variables 

x^,  x-i,   X3', x/,  and  similar  to  the  double  and  triple  ones  employed 

in  the  solution  of  geometrical  and  mechanical  problems.  Then  it  is 
easy  to  perceive  that  the  function  V  will  satisfy  the  partial  differen- 
tial equation 

t/vr     ^,                    d^     ^     n-sdV 
"  ~  dx,^  "^  dxi  "*" ^  dx^  '^  du^^     u     du ^^' 

seeing  that  in  consequence  of  the  denominator  of  the  expression  (1), 
every  one  of  its  elements  satisfies  for  V  to  the  equation  (2). 

To  give  an  example  of  the  manner  in  w^hich  the  multiple  integral 
is   to   be   taken,   we   may   conceive    it    to    comprise   all   the   real   values 

both  positive  and  negative  of  the  variables  ar/,  x^, x,,  which  satisfy 

the  condition 

the  symbol  / ,  as  is  the  case  also  in  what  follows,  not  excluding  equality. 

2.  In  order  to  avoid  the  difficulties  usually  attendant  on  integra- 
tions like  those  of  the  formula  (1),  it  will  here  be  convenient  to  notice 
two  or  three  very  simple  properties  of  the  function  F". 

In  the  first  place,  then,  it  is  clear  that  the  denominator  of  the 
formula   (1)   may   always   be   expanded   in   an    ascending    series    of    the 

entire  powers  of  the  increments  of  the   variables  x^,  x^, x„   u,   and 

their  various  products  by  means  of  Taylor's  Theorem,  unless  we  have 
simultaneously 

and  therefore  V  may  always  be  expanded  in  a  series  of  like  form, 
unless  the  s  +  1  equations  immediately  preceding  are  all  satisfied  for 
one  at  least  of  the  elements  of  V.  It  is  thus  evident  that  the  func- 
tion V  possesses  the  property  in  question,  except  only  when  the  two 
conditions 


4!fl#  Mr  green,   on  THE   DETERMINATION  OF  THE 

%  +  %^%+ +%  z  1   and   u  =  0 .(3) 

are  satisfied  simultaneously,  considering  as  we  shall  in  what  follows 
the  limits  of  the  multiple  integral  (1)  to  be  determined  by  the  conr 
dition  (a)*. 

In  like  manner  it  is  clear  that  when 

Z^2+  Jl+  +  77-2>^ (4)» 


a?' 


the   expansion   of   V  in  powers   of  u   will   contain   none   but   the   even 
powers  of  this  variable. 

Again,  it  is  quite  evident  from  the  form  of  the  function  f^  that 
when  any  one  of  the  *  +  1  independent  variables  therein  contained  be- 
comes infinite,  this  function  will  vanish  of  itself. 

3.  The  three  foregoing  properties  of  F  combined  with  the  equa- 
tion (2)  will  furnish  some  useful  results.  In  fact,  let  us  consider  the 
quantity 

fd.,d^,...d..duu-'.[[^)\  [^)\ +  (g)\  (^^)] (5) 

where   the  multiple  integral  comprises  all  the  real   values  whether  posi- 
tive or  negative  of  x^,  x^, x,,  with  all  the  real  and  positive  values 

of  u  which  satisfy  the  condition 

/!«  2  A<  2  «  2  /|/2 

^■^^^  + -^^^^F^^ ^^^ 

*  The  necessity  of  this  first  property  does  not  explicitly  appear  in  what  follows,  but 
it  must  be  understood  in  order  to  place  the  application  of  the  method  of  integration  by 
parts,  in  Nos.  3,  4,  and  5,  beyond  the  reach  of  objection.  In  fact,  when  V  possesses  this 
property,  the  theorems  demonstrated  in  these  Nos.  are  certainly  correct:  but  they  are  not 
necessarily  so  for  every  form  of  the  function  V,  as  will  be  evident  from  what  has  been 
shewn  in  the  third  article  of  my  Essay  on  the  Application  of  Mathematical  Analysis  to 
the  Theories  of  Electricity  and  Magnetism. 


ATTRACTIONS  OF  ELLIPSOIDS  OF   VARIABLE   DENSITIES.         401 

«,,  «2, a,  and   h   being   positive   constant   quantities;    and   such   that 

we  may  have  generally 

Ur  >  dr. 

In  this  case  the  multiple  integral  (5)  wiU  have  two  extreme  limits, 
viz.  one  in  which  the  conditions 

V  ^  or  IT  71 

-\  +  -^  + +  -^  +  t;  =  1  and  u—  a.  positive  quantity (7) 

are  satisfied;  and  another  defined  by 

%  +  %+ +-,  /I  and  «  =  0. 

jVIoreover,  for  greater  distinctness,  we  shall  mark  the  quantities  be- 
longing to  the  former  with  two  accents,  and  those  belonging  to  the 
latter  with  one  only. 

Let  us  now  suppose  that  J^"  is  completely  given,  and  likewise  F,' 
or  that  portion  of  f^'  in  which  the  condition  (3)  is  satisfied ;  then  if 
we  regard  F/  or  the  rest  of  T^'  as  quite  arbitrary,  and  afterwards  endea- 
vour to  make  the  quantity  (5)  a  minimum,  we  shall  get  in  the  usual 
way,  by  applying  the  Calculus  of  Variations, 

/7F'' 
-fdx.dx, clx,u"-^^r,'~- (8) 

seeing  that  ^V"  =  0  and   SFj'  =  0,   because   the  quantities    V"  and    F,' 
are  supposed  given. 

The  first  line  of  the  expression  immediately  preceding  gives  generally 

0  =  2'+'—      —      ^Hf^  {^•\ 

'      dxr        du'  u     du ^    ' 

which  is  identical  with  the  equation  (2)  No.  1,  and  the  second  line  gives 

dV 
0  =  u'"''  ~7-^(^'  being  evanescent) (9). 


402        Mr  green,  ON  THE  DETERMINATION  OF  THE 

From  the  nature  pf  the  question  de  minimo  just  resolved,  there  can 
be  little  doubt  but  that  the  equations  (2')  and  (9)  will  suffice  for  the 
complete  determination  of  V,  where  V"  and  V-l  are  both  given.  But 
as  the  truth  of  this  will  be  of  consequence  in  what  follows,  we  will, 
before  proceeding  farther,  give  a  demonstration  of  it;  and  the  more 
wiUingly  because  it  is  simple  and  very  general. 

4.  Now  since  in  the  expression  (5)  u  is  always  positive,  every  one 
of  the  elements  of  this  expression  will  therefore  be  positive;  and  as 
moreover  V"  and  F"/  are  given,  there  must  necessarily  exist  a  function 
Fo  which  will  render  the  quantity  (5)  a  proper  minimum.  But  it 
follows,  from  the  principles  of  the  Calculus  of  Variations,  that  this 
function  Va,  whatever  it  may  be,  must  moreover  satisfy  the  equations 
(2')  and  (9).  If  then  there  exists  any  other  function  F",  which  satisfies 
the  last-named  equations,  and  the  given  values  of  V"  and  V^,  it  is  easy 
to  perceive  that  the  function 

will  do  so  likewise,  whatever  the  value  of  the  arbitrary  constant  quan- 
tity A  may  be.  Suppose  therefore  that  A  originally  equal  to  zero 
is  augmented  successively  by  the  infinitely  small  increments  SA,  then 
the  corresponding  increment  of  V  will  be 

Sr={F,-V,)SA,' 

and  the  quantity  (5)  will  remain  constantly  equal  to  its  minimum 
value,  however  great  A  may  become,  seeing  that  by  what  precedes 
the  variation  of  this  quantity  must  be  equal  to  zero  whatever  the 
variation  of  V  may  be,  provided  the  foregoing  conditions  are  all  satis- 
fied. If  then,  besides  F"o .  there  exists  another  function  F";  satisfying 
them  all,  we  might  give  to  the  partial  differentials  of  F",  any  values 
however  great,  by  augmenting  the  quantity  A  sufficiently,  and  thus 
cause  the  quantity  (5)  to  exceed  any  finite  positive  one,  contrary  to 
what  has  just  been  proved.     Hence  no  such  value  as  F,  exists. 

We  thus  see  that  when  F""  and  F"/  are  both  given,  there  is  one 
and  only  one  way  of  satisfying  simultaneously  the  partial  differential 
equation  (2),  and  the  condition  (9). 


ATTRACTIONS  OF   ELLIPSOIDS  OF  VARIABLE  DENSITIES.  403 

5.  Again,  it  is  clear  that  the  condition  (4)  is  satisfied  for  the  whole 
of  F"/;  and  it  has  before  been  observed  (No.  2.)  that  when  V  is  deter- 
mined by  the  formula  (1),  it  may  always  be  expanded  in  a  series  of 
the  form 

r  =  ^  +  J?«'  +  Cu'  +  &c. 

Hence  the  right  side  of  the  equation  (9)  is  a  quantity  of  the  order 
?/"-'+' ;  and  v!  being  evanescent,  this  equation  will  then  evidently  be 
satisfied,  provided  we  suppose,  as  we  shall  in  what  follows,  that 

n  —  s  \  \  is  positive. 

If  now  we  could  by  any  means  determine  the  values  of  V"  and 
V(  belonging  to  the  expression  (1),  the  value  of  V  would  be  had 
without  integration  by  simply  satisfying  (2')  and  (9),  as  is  evident  from 
what  precedes.  But  by  supposing  all  the  constant  quantities  a,,  «2>  «3 
a,  and  h  infinite,  it  is  clear  that  we  shall  have 

0  =  V", 

and  then  we  have  only  to  find  V^,  and  thence  deduce  the  general 
value  of  V. 

6.     For  this  purpose  let  us  consider  the  quantity 

w    ^       ^7     n-AdVdU     dVdU  ,  dVdU     dVdU\ 

jdxidx.i...dx,duvr  '{-r—-j—  +  -f— -j— +  •••  + i—n—  +  -i t-)\ (10^ 

{dxidx^      dx.dxi  dx.dx,      du  du  j '        ^     ' 

the  limits  of  the  multiple  integral  being  the  same  as  those  of  the 
expression  (5),  and  U  being  a  function  of  ;r,,  x^, x,  and  u,  satis- 
fying the  condition  0=  U"  when  «,,  a^, a,  and  h  are  infinite. 

But  the  method  of  integration  by  parts  reduces  the  quantity  (10)  to 

—  fdXidxi dx,—j — u'"-' .  V 

du 

-/..........x..»».-.r|.,«^+^.^^} (H, 

since  0  =  V"\  and  as  we  have  likewise  0  =  U",  the  same  quantity  (10) 
may  also  be  put  under  the  form 
Vol.  V.    Part  III.  SG 


404  Mk  green,   on  THE  DETERMINATION   OF   THE 

dV 
—  fdxidXi dxi—r—u'"-' .  U' 

.fdx,dx,...dx,duu''-'.u\^r'^,+^  +  '^^^ (12). 

Supposing  therefore  that  U  like  V  also  satisfies  the  equation  (2'), 
each  of  the  expressions  (11)  and  (12)  will  be  reduced  to  its  upper  line, 
and  we  shall  get  by  equating   these  two  forms  of  the  same  quantity : 

idx^  dx2...dxs-j~  u'"-'  V  =  fdxi  dXi...dxs  -y-  «'"*  U' : 
au  au 

the  quantities  bearing  an  accent  belonging,  as  was  before  explained,  to 
one  of  the  extreme  limits. 

Because  V  satisfies  the  condition  (9),  the  equation  immediately  pre- 
ceding may  be  written 

dU'  dV 

fdxidx2...dxs-j — u'"~'  V  =  fdxidxi...dx,—y^u'"-'  U,'. 
du  du 

If  now  we  give  to  the  general  function   U  the  particular  value 

u=  {{x,  -  x,"y  +  {x,  -  x,y  + +  {x,  -  xjy  +  u']^-, 

which  is  admissible,  since  it  satisfies  for  V  to  the  equation  (2),  and  gives 
U"  =  0,  the  last  formula  will  become 

dVi 

/dxidx-i dxsu'"''  —j-^ 
du 
{{x,  -  x^y  +  {x,  -  x:j  + +  (a;,  -  xlj  +  m'^}^ 

_r dxydx^ c?;g,.(l-w) «'"-'+'  V , 

\{x,  -  xlj  +  (ar,  -  xij  + +  {x,  -  x:j  +  u''\'^ 

in    which    expression    «'    must    be    regarded    as    an   evanescent   positive 
quantity. 

In  order  now  to  effect  the  integrations  indicated  in  the  second 
member  of  this  equation,  let  us  make 


ATTRACTIONS   OF   ELLIPSOIDS  OF  VARIABLE  DENSITIES.        405 

x^  —  Xi"  =  u'p  COS  6i ;  x-i—x"  =  u'p  sin  Qx  cos  0^ ;  Xi—x^'—u'p  sin 0,  sin  02  cos 03>  &c. 
until  we  arrive  at  the  two  last,  viz., 

«,_!  -x[^-^  =  u'p  sin^i  sin  ^^ sin0,_2  cos0,_i, 

X,    —    ar,"  =  «'/o  sin  ^1  sin  02 sin  0,_2  sin  0,_i; 

u'  being,  as  before,  a  vanishing  quantity. 

Then  by  the  ordinary  formulas  for  the  transformation  of  multiple 
integrals  we  get 

dxi  dx-i dx,  =  u''f/~^  sin^i'"^  sin  02*"' smO^^.^dp  d6i  dOi...dd,.i, 

and  the  second  number  of  the  equation  (13)  by  substitution  will  become 

fdp  d9i  de, d9,_,p'-'  sm9r''  sin  9,'-' sin  e,_2 .  (1  -  ») r' 

/  »+i  (1*)- 

But  since  u'  is  evanescent,  we  shall  have  p  infinite,  whenever  x^,  Xi,...x, 
differ  sensibly  from  x",  x^',,..x"\  and  as  moreover  w  — *  +  l  is  positive, 
it  is  easy  to  perceive  that  we  may  neglect  all  the  parts  of  the  last 
integral  for  which  these  differences  are  sensible.  Hence  V  may  be 
replaced  with  the  constant  value  VI  in  which  we  have  generally 

Jbf     ^^    vUf     • 

Again,  because  the  integrals  in  (14)  ought  to  be  taken  from  0,_,  =  o 
to  0r-,  =  27r,  and  afterwards  from  0,.  =  O  to  9r  =  -n-,  whatever  whole  number 
less  than  5—1  may  be  represented  by  r,  we  easily  obtain  by  means  of 
the  well  known  function  Gamma: 

» 

/sin^i'-''  sin  02'"'  sin  03'"' sin0,_2C?0,</02...c?0,.,  =  ^ZL; 

and  as  by  the  aid  of  the  same  function  we  readily  get 

r»  — *  +  l> 


f  P'~'dp    _      V2/      V       2       ) 

Wi  +  ,f-^  2r(^) 


3g2 


406        Mr  green,  ON  THE  DETERMINATION  OF  THE 

the  integral  (14)  will  in  consequence  become 

and  thus  the  equation  (13)  will  take  the  form 

dx.dx, dx,u"^-^^  ~^'^-^^~V^) 


J  {{x,-x; 


J  +  {x,  -  x;j  +....  +  (x,  -  x:j + m'^  }  -^  r  (■ 


w-l 


In  this  equation  V '\?,  supposed  to  be  such  a  function  of  x^,  x.^ x, 

and  u,  that  the  equation  (2)  and  condition  (9)  are  both  satisfied.     More- 
over V'^O,  and  Vo   is  the  particular  value  of  F'  for  which 

Let  us  now  make,  for  abridgment, 

dV 
P  =  u"-' -r-,  {when  u  =  0) (A), 

and  afterwards  change  x  into  x\   and  x"  into   x  in   the   expression   im- 
mediately preceding,  there  will  then  result 


_-    s      ffi—s  +  V 

r  dx^  dx2 ....dx,'P,' ""^^'^  I      2      /  „,     ,,^^ 

/ ^^ rr — r; f^  •••U5), 

{{x,'-x,f+{x,'-x,y  +  ...+{'>':-^sY+u"]—         r(^) 


--^(^).„ 


P'  being  what  P  becomes  by  changing  generally  Xr  into  x,',  the  unit 
attached  to  the  foot  of  P'  indicating,  as  before,  that  the  multiple 
integral  comprises  only  the  values  admitted  by  the  condition  {a),  and 
V  being  what  V  becomes  when  we  make  u  =  0. 

The  equation  just  given  supposes  u'  evanescent;  but  if  we  were  to 
replace  u  with  the  general  value  u  in  the  first  member,  and  make  a 
corresponding  change  in  the  second  by  replacing  F''  with  the  general 
value  F,  this  equation  would  still  be  correct,  and  we  should  thus  have 


ATTRACTIONS  OF   ELLIPSOIDS  OF  VARIABLE  DENSITIES.         407 

r  dx'dxi dxlP,'  \     2      J  ,^     ,  „ 

f '■ '—^ ^,  = -L-jl— ir...(l6). 

-^  \{x,'-x,y+{x,'-x,y+...  +  {x:-x.y+u'\—         r(^^) 

For  under  the  present  form  both  its  members  evidently  satisfy  the 
equation  (2),  the  condition  (9),  and  give  V"  =  0.  Moreover,  when  the 
condition  (3)  is  satisfied,  the  same  members  are  equal  in  consequence 
of  (15).  Hence  by  what  has  before  been  proved  (No.  4),  they  are 
necessarily  equal  in  general. 

By  comparing  the  equation  (16)  with  the  formula  (1),  it  will  become 
evident,  that  whenever  we  can  by  any  means  obtain  a  value  of  V  satis- 
fying the  foregoing  conditions,  we  shall  always  be  able  to  asSgn  a  value 
of  p  which  substituted  in  (1)  shall  reproduce  this  value  of  V.  In  fact, 
by  omitting   the   unit  at  the  foot  of  P",  which  only  serves  to  indicate 

the  limits  of  the  integral,  we  readily  see  that  the  required  value  of  p  is 
p'= \     P' {c). 

r^      'r.   fn~S+l\  ' 

7.  The  foregoing  results  being  obtained,  it  will  now  be  convenient 
to  introduce  other  independent  variables  in  the  place  of  the  original 
ones,  such  that  . 

^1  =  «i?i»     «a  =  02^2j x,  =  as^„     u  =  hv, 

Oj,  ttj, flj  being  functions  of  h,  one  of  the  new  independent  variables, 

determined  by 

a,'  =  «;*  +  h',    a,-  =  (h'  +  h\ a/  =  aj'  +  /^^ 

and  V  a  function  of  the  remaining  new  variables,  f,,  ^2,  ^3, ^s  satis- 
fying the  equation 

1  =  v'  +  |;^  +  e/+ +  U; 

a,',  a/,  Os', 0/  being  the  same  constant  quantities  as  in  the  equation 

(a),  No  1.     Moreover,   Oi,  a.^, a,  will   take    the  values   belonging   to 

the  extreme  limit  before  marked  with  two  accents,  by  simply  assigning 
to  h  an  infinite  value. 


408        Mr  green,  ON  THE  DETERMINATION  OF  THE 

The  easiest  way  of  transforming  the  equation  (2)  will  be  to  remark, 
that  it  is  the  general  one  which  presents  itself  when  we  apply  the 
Calculus  of  Variations  to  the  quantity  (5),  in  order  to  render  it  a 
minimum.     We  have  therefore  in  the  first  place 

and  by  the  ordinary  formula  for  the  transformation  of  multiple  integrals, 
dx.dx, dx,du=^^^^  (l-2r'  ^')  d^,dl,...dldh. 

•      But  since  1  -  2,'+'  ^^  =  v  +  ^»S,'+'  ^, 


a;  '     Ur 


the  expression  (5)  after  substitution  will  become 

fd^^d^i d^sdhui  tti  ih a.A""'!/""'"'. 


Applying  now  the  method  of  integration  by  parts  to  the  variation  of 
this  quantity,  by  reduction,  we  get  for  the  equivalent  of  (2)  the  equation 

^-  dh^^  V"  ,^  a;)  hdh^^^     ^^''^  ar'dl'      ^'     ""     ^^^a:-dlr' 
+  A^2^  X  2-^^  -  A'22-Ml  -^^  (2") 

where  the  finite  integrals  are  all  supposed  taken  from  r  =  l  to  r  =  *  +  l, 
and  from  r'  =  1  to  r'  =  *  +  1. 


The  last  equation  may  be  put  under  the  abridged  form, 

d^  .  (     ^«:^  dV 

dJi 
provided  we  have  generally 


o  =  -^+(»-s5-)^  +  vr (n. 


ATTRACTIONS  OF   ELLIPSOIDS  OF   VARIABLE   DENSITIES.        409 

coefficient  o£^-mvV=~  {1  -^^'-2.'*'  1^  ^ "'  +  ^  ^"}> 
coefficient  of  ^,  i»  V  ^  =  -  ^.  lU 

coefficient  of  -j^  in  vF=-^|-»  +  2  ^ ^i. 

Moreover,  when  we  employ  the  new  variables 

du  "  y-     ^    ^ .  ;     •  Y'  a?   d^r      dh  ]' 

and  therefore  the  condition  (9)  in  like  manner  will  become 

— -(>-^r"i^ff-^} «'»^ 

where  the   values  of  the   variables  ^1,^2, ?,  must  be  such  as  satisfy 

the  equation  i;"  =  0,  whatever  h  may  be;  and  as  n-s-\-l  is  positive,  it 
is  clear  that  this  condition  will  always  be  satisfied,  provided  the  partial 
differentials  of  V  relative  to  the  new  variables  are  all  finite. 

8.     Let   us   now   try  whether   it   is  possible  to  satisfy  the  equation 
(2'")  by  means  of  a  function  of  the  form 

r^Hct> (/?); 

H  depending  on  the  variable  h  only,  and  cp  being  a  rational  and  entire 
function  of  ^1,  f^, f,  of  the  degree  7,  and  quite  independent  of  h. 

By  substituting  this  value  of  V  in  (2'")  and  making 

^      d'H       (        ^«:^  dH  ,      „  ,,„^ 

we  readily  get 

0  =  v<^  -  '«P (18); 

where,  in  virtue  of  (17)  k  must  necessarily  be  a  function  of  h  only; 
and  as  the  required  value  of  (p,  if  it  exist,  must  be  independent  of  k, 
we  have,  by  making  h  =  0  in  the  equation  immediately  preceding, 

0  =  v'0  -  ko(p (19); 

ko  being  the  value  k,  and  v'^  that  of  v^  when  h  =  0. 


410        Mb  green,  ON  THE  DETERMINATION  OF  THE 

We  shall  demonstrate  almost  immediately  that  every  function  ^  of 
the  form  (20),  No.  9,  which  satisfies  the  equation  (19),  and  which  there- 
fore is  independent  of  h,  will  likewise  satisfy  the  equation  (18);  and 
the  corresponding  value  of  k  obtained  from  the  latter  being  substituted 
in  the  ordinary  differential  equation  (17),  we  shall  only  have  to  integrate 
this  last  in  order  to  have  a  proper  value  of  V. 

9.     To  satisfy  the  equation  (19)  let  us  assume 

<^  =  ^(e.^  ?/,  ?3^ ?/)?,.?„  &c (20); 

F  being  the  characteristic  of  a  rational  and  entire  function  of  the 
degree  2y',  and  the  most  general  of  its  kind,  and  f,,  ^„  &c.  designating 
the  variables  in  0  which  are  affected  with  odd  exponents  only;  so  that 
if  their  number  be  v  we  shall  have 

7  =  27'  +  c, 

the  remaining  variables  having  none  but  even  exponents.  Then  it  is 
easy  to  perceive,  that  after  substitution  the  second  member  of  the 
equation  (19)  will  be  precisely  of  the  same  form  as  the  assumed  value 
of  (p,  and  by  equating  separately  to  zero  the  coefficients  of  the  various 

powers    and   products   of  ^1,  |s, ^,,   we   shall   obtain   just    the    same 

number  of  linear  algebraic  equations  as  there  are  coefficients  in  <p,  and 
consequently  be  enabled  to  determine  the  ratios  of  these  coeflScients 
together  with  the  constant  quantity  ^0. 

In  fact,  by  writing  the  foregoing  value  of  (p  under  the  form 

</)  =  aS'^„„„, „„?.•"' ?."» ?»•" (20'); 

and   proceeding    as    above  described,  the   coefficient   of    ^ri  ^/"t  ^,'', 

will  give  the  general  equation 

K  +  2)(m.  +  ]) 

^..iK+2)K  +  i)   . 

"r 


ATTRACTIONS    OF   ELLIPSOIDS   OF    VARIABLE   DENSITIES.  411 

the  double  finite  integral  comprising  all  the  values  of  r  and  r,  except 
those  in  which  r  =  r ,  and  consequently  containing  when  completely 
expanded  s  (s  -  1)  terms. 

For  the  terms  of  the  highest  degree  7  and  of  which  the  number  is 
7  +  1.7  +  2 y  +  s  —  l  _   ^ 

the  last  line  of  the  expression  (21)  evidently  vanishes,  and  thus  we 
obtain  JV  distinct  linear  equations  between  the  coefficients  of  the  degree 
7  in  <p  and  ko. 

Moreover,  from  the  form  of  these  equations  it  is  evident  that  we 
may  obtain  by  elimination  one  equation  in  ko  of  the  degree  JV,  of 
which  each  of  the  iV  roots  will  give  a  distinct  value  of  the  function 
(p^'y\  having  one  arbitrary  constant  for  factor;  the  homogeneous  function 
^''1''  being  composed  of  all  the  terms  of  the  highest  degree,  7  in  (p. 
But  the  coefficients  of  (p'-^''  and  kg  being  known,  we  may  thence  easily 
deduce  all  the  remaining  coefficients  in  (j>,  by  means  of  the  formula  (21). 

Now,  since  the  A''  linear  equations  have  no  terms  except  those  of 
which  the  coefficients  of  ^'^^  are  factors,  it  follows  that  if  ^0  were  taken 
at  will,  the  resulting  values  of  all  these  coefficients  would  be  equal  to 
zero.  If  however  we  obtain  the  values  of  N' —  1  of  the  coefficients 
in  terms  of  the  remaining  one  A  from  iV-  1  of  the  equations,  by  the 
ordinary  formulas,  and  substitute  these  in  the  remaining  equation,  we 
shall  get  a  result  of  the  form 

K.A=0, 

where  jRT  is  a  function  of  ka  of  the  degree  iV.  We  shall  thus  have 
only  two  cases  to  consider :  First,  that  in  which  A  =  0,  and  consequently 
also  all  the  other  coefficients  of  0*^'  together  with  the  remaining  ones 
in  <p,  as  will  be  evident  from  the  formulae  (21).     Hence,  in  this  case 

0  =  0: 

Secondly,  that  in  which  kg  is  one  of  the  iV  roots  of  0  =  K,  as  for 
instance,  ko  in  this  case  all  the  coefficients  of  0  will  become  multiples 
of  A,  and  we  shall  have 

Vol.  V.     Part   III.  S« 


412  Mr  green,   ON   THE  DETERMINATION   OF   THE 

(j)  =  Acpr. 

01  being  a  determinate  function  of  ^,,  ^a, E<- 

We    thus   see    that   when   we    consider   functions   of  the   form    (20) 
only,  the  most  general   solution   that  the   equation 

0  =  v'^  -  *o'0 (19') 

admits  is  

or,     (p  =  0;         or,     (p  =  atp; 

a   being  a  quantity  independent  of  ^,,   ^2, ^„   and   (p   any  function 

which  satisfies  for  <p  to  the  equation  (19').     But  by  affecting  both  sides 
of  the  equation 

with  the  symbol  v,  we  get 

0  =  V  •  v'  0  -  *o' .  V  ^ ; 

and   we   shall   afterwards   prove   the  operations  indicated   by    v   and   v' 
to  be  such,   that  whatever  0  may  be, 

V  v'  0  =  V'  V  0- 

Hence,   the  last  equation  becomes 

v'  (v  ^)  -  k„'  V  (p; 

and   as    V  0   like  (p  is  of  the  form  (20),   it  follows  from    what   has  just 
been  shewn,  that 

either     0  =  v  cp,     or,  \7  (p  =  acp, 

a  being  a  quantity  independent  of  ^i,  ^2, ?«• 

The  first  is  inadmissible,  since  it  would  give  ^  =  0;  therefore  when 
(p  satisfies  (19'),  we  have 

V  0'  =  a(p,   i.e.   0  =  V  0  —  "0- 

But  since    a    is    independent    of   ^1,    ^2, Bs,   this  last   equation   is 

evidently  identical  with  (18),  since  the  equation  (18)  merely  requires  that 
K  should  be  independent  of  fi,  ^2, ^s- 


ATTRACTIONS   OF   ELLIPSOIDS  OF  VARIABLE   DENSITIES.         41^ 

Having  thus  proved  that  every  function  of  the  form  (20)  which 
satisfies  (19)  will  likewise  satisfy  (18),  it  will  be  more  simple  to  deter- 
mine the  remaining  coefficients  of  (j>  from  those  of  cp^^^  by  means  of 
tlie  last  equation,  than  to  employ  the  formula  (21)  for  that  purpose. 

Making  therefore  h  infinite  in  (18),  and  writing  ~  in  the  place 
of  K,  we  get 

where  (22)  comprises  the  — ^ — —!■  combinations  which  can  be  formed  of 

1.2 

the  s  indices  taken  in  pairs. 

If  now  we  substitute  the  value  of  0  before  given  (20'),  and  recol- 
lect that  for  the  terms  of  the  highest  degree  we  have  2»«r  =  7,  we  shall 
readily  get 

0=(7-2»«,)(7+2»?r+»-l)^™,,»,,....,+(7».+2)(»w,+l)^„^,  „^+2,...„^...(22), 

from  which  all  the  remaining  coefficients  in  0  will  readily  be  deduced, 
when  those  of  the  part  0'^'  are  known. 

10.  It  now  remains,  as  was  before  observed,  to  integrate  the  ordi- 
nary differential  equation  (17)  No.  8.  But,  by  the  known  theory  of 
linear  equations,  the  integration  of  (17)  will  always  become  more  simple 
when  we  have  a  particular  value  satisfying  it,  and  fortunately  in  the 
present   case  such  a  value  may  always  be   obtained  from   0  by  simply 

changing  f,  into  '  ,    .      In  fact  if  we  represent  the  value  thus  ob- 

tained  by  Ho  we  shall  have 

cih     ^'  </e/«v(2«:')' 

and  by  a  second  differentiation 

3H2 


414        Mr  green,  ON  THE  DETERMINATION  OF  THE 

(22)  as  before  comprising  all  the  ~ — —  combinations  of  the  *  indices 
taken  in  pairs. 

Hence,  the  quantity  on   the  right   side  of  the  equation   (17),  when 
we  make  H  =  Ho,  becomes 

+  2(22)^^.-^;^,+  («-2^)2^.--^ (23). 

d^rd^r    a,«r'2«/      V  «r  /     «^r   o, 'v/(2«;^)  ^ 

But  if  we  recollect  that  we  have  generally 

it  is  easy  to  perceive  that  in  consequence  of  the  equation  (18)  the 
quantity  (23)  will  vanish,  and  therefore  the  foregoing  value  of  Ha 
will  always  satisfy  the  equation  (17). 

Having  thus   a    particular   value    of    H,    we    immediately    get    the 
general  one   by   assuming 

H=  Hfzdh. 

In  fact,  there  thence  results 

H  =  KHj„„    '^"^^ . 

±l(,~  Ux,  Oi,  (h a, 

the  two  arbitrary  constants  which  the  general  integral  ought  to  con- 
tain being  K,  and  that  which  enters  implicitly  into  the  indefinite  in- 
tegral.  But  the  condition  0  =  V"  requires  that  H  should  vanish  when 
h  is  infinite,  and  consequently  the  particular  value  adapted  to  the 
present   investigation  is 

n  -  jr   rr  f        ^""dh 

J^  Mo'a^,  «2 «» 

11.     The  values   of  (f>  and   H  being   known,   we  may   readily   find 
the  corresponding  values  of  V  and  p.     For  we  have  immediately 


ATTRACTIONS  OF   ELLIPSOIDS  OF   VARIABLE   DENSITIES.         415 

r=i/^  =  jir^//./-gj£2^ «6), 

and  as  the  function  (p  is  rational  and  entire,  and  the  partial  differen- 
tial of  f^  relative  to  h  is  finite,  it  follows  that  all  the  partial  differ- 
entials of  F^  are  finite;  and  consequently,  by  what  precedes  (No.  7.) 
the  condition  (9')  is  satisfied  by  the  foregoing  value  of  F',  as  well  as 
the  equation  (2)  and  condition  0  =  F".  Hence  the  equations  {b)  and 
(c)  No.  6  will  give,  since 

du-    "V     ^'    ~^)     Y"    ^■d^~~dh\' 

and  h  must  be  supposed  equal  to  zero  in  these  equations 

-  r  f^^ii) 

p'  =  — ,  A.  .---V^-^^ (where  h  =  0); 

since  where  A  =  0,   a,  =  «/ ;   and  therefore, 

1  -  2/^'  ^^  =  1  -  ^r'  V  =  ^'. 

If  now  we  substitute  for  V  its  value  (26),  and  recollect  that  « —  *  + 1  is 
always  positive,  we  get 

-r(^)  ^ 


27r^r 


(^4^) 


since  it  is  clear  from  the  form  of  Ho  that  this  quantity  may  always 
be  expanded  in  a  series  of  the  entire  powers  of  A^  In  the  preceding 
expression,  (27),  H^  indicates  the  value  of  Ho  when  h  =  0,  and  (p! 
the   corresponding   value    of    0    or    that   which   would    be  obtained   by 

simply  changing  the   unaccented  letter   fi,  ^2, ^,   into    the    accented 

ones  ^1',  f/, ?/  deduced  from 

(7)        x;  =  a,'?/ ;        x.^  =  «;  ^/ ;        x/  =  «/  ^;. 


416         Mr  green,  ON  THE  DETERMINATION  OF  THE 

It  will  now  be  easy  to  obtain  the  value  of  V  corresponding  to 


without  integrating  the  formula  (1)  No  1,  where  F  is  the  character- 
istic of  any  rational  and  entire  function.  In  fact  it  is  easy  to  see  from 
what  precedes  (No.  9),  that  we  may  always  expand  JF'  in  a  finite  series 
of  the  form 

F{xl,  x-l xl)  =  bo^o  +  ii0i'  +  bo(p2  +  63^3'  +  &c. 

after  a;/,  x-J,  &c.  have  been  replaced  with  their  values  (7).  Hence,  we 
immediately  get 

p'  =  „"-«-'  .  {bo(po'  +  b,<p!  +  h(p;  +  &c.} (29). 

By  comparing  the  formulae  (26)  and  (27)  it  is  clear  that  any  term, 
as  5,0/  for  instance,  of  the  series  entering  into  p,  will  have  for  cor- 
responding term  in  the  required  value  of  V,  the  quantity 

^       ^       i^„'«/< a:.b.<pM.f-jj^^'"/^     ^  (30): 

''co   -'^O    "1  Ms (Is 

Ha  being  a  particular  value  of  H  satisfying  the  equation  (17),  and 
immediately  deducible  from  (p  by  the  method  before  explained. 

12.     AU  that  now  remains,  is  to  demonstrate  that 

V'V0  =  VV> (31), 

whatever  <p  may  be.  For  this  purpose  let  us  here  resume  the  value 
of  A0,  as  immediately  deduced  from  the  equation  (2")  No.  7,  viz. 

+  A^2^-A^2lx2i^ (32), 


P  /w-1' 


ATTRACTIONS   OF   ELLIPSOIDS   OF  VARIABLE   DENSITIES.  417 

where  for  simplicity  the  indices  at  the  foot  of  the  letters  ^  and  a  have 
been  omitted,  and  their  accents  transferred  to  the  letters  themselves. 
Moreover  all  the  finite  integrals  are  supposed  taken  from  1  to  «+l. 

By  making  A  =  0  in  the  last  expression  we  immediately  get  v'<^, 
and  if  for  a  moment,  to  prevent  ambiguity,  we  write  h,  in  the  place 
of  the  original  «'r  and  omit  the  lower  indices  as  before,  we  obtain 

V>=(l-2a2^4p+(*-«-l)2|^ (33); 

where  to  avoid  all  risk  of  confusion  r  has  been  changed  into  r" ,  and 
the  double  accent  of  this  index  transferred  to  the  letters  ^  and  h 
themselves. 

We  will  now  conceive  the  expression  (32)  to  be  written  in  the 
abridged  form 

the  order  of  the  terms  remaining  unchanged. 

If  then  we  recollect  that  the  accents  have  no  other  office  to  per- 
form than  to  keep  the  various  finite  integrations  quite  distinct,  and 
consequently  that  in  the  final  results  they  may  be  permuted  in  any 
way  at  will,  we  shall  readily  get 

V'Va^  —  VaV'0  = 

(l-Sf)(4S2   ,^„,„.-T^>-'      +22-^,  X  2^-^     I 
+  42^'x22-^.-,p^^,      +22^x2^x2^-^ 


418         Mr  green,  ON  THE  DETERMINATION  OF  THE 
^      '  [   a^b^        aa^\^^^        a^        c^V    af  (n)  J 

(1  -  2P^|-422— i^^^^—   _22-L  ^   1 

P        g     rf'0  P      (/'^ 

2(l-2f)2JL.^   +22i:x2^^^ 

-2  (1-2^2^x2—  ^   -2  2-x2^x2-^ 

all  the  finite  integrals  being  taken  from  ?•  =  1  to  r  =  s  +  1,  and  from 
/  =  1   to  r'  =  s  +  1. 

In  order  to  obtain  the  required  value 

v'v^  -  w'<p, 

it  is  clear  that  we  shall  only  have  to  add  the  first  of  the  five  preceding 
quantities  to  the  sum  of  the  four  following  ones  multiplied  by  A',  and 
to  render  this  more  easy,  we  have  appended  to  each  of  the  terms  in  the 
preceding  quantities  a  number  inclosed  in  a  small  parenthesis. 

Now   since  the  accents  may  be  permuted  at  will,  and  we  have  like- 
wise or  =  b^  +  U,  it  is  easy  to   see   that   the  terms  marked  (1),   (6)  and 

(12)  mutually  destroy  each  other.  In  like  manner,  (2),  (3),  (7)  and 
(18)  mutually  destroy  each   other;    the  same  may   evidently  be  said  of 

(13)  and  (16),  of  (15)  and  (17),  of  (9)  and  (19),  and  of  (8)  and  (14). 
Moreover  the  four  quantities  (4),  (5),  (10)  and  (11)  will  do  so  likewise, 
and  consequently,  we  have 

V'V0  -  VV>  =  0. 
Hence  the  truth  of  the  equation  (31)  is  manifest. 


ATTRACTIONS   OF   ELLIPSOIDS   OF  VARIABLE   DENSITIES.         419 

Application  of  the  preceding  General  Theory  to  the  Determination  of  the 

Attractions  of  Ellipsoids. 

13.  Suppose  it  is  required  to  determine  the  attractions  exerted  by 
an  ellipsoid  whose  semi-axes  are  a',  b',  c'  whether  the  attracted  point 
p  is  situated  within  the  ellipsoid  or  not,  the  law  of  the  attraction  being 
inversely  as  the  w"*"  power  of  the  distance.  Then  it  is  well  known 
that  the  required  attractions  may  always  be  deduced  from  the  function 

j^  _    r p'  dx'  dy'  dx 

{{x  ~x'f  +  {y-yj  +  {x-%jy^ ' 

p    being   the  density   of    the   element   dx'  dy'  d%'  of   the   ellipsoid,    and 
X,  y,  %  being  the  rectangular  co-ordinates  of  p. 

We  may  avoid  the  breach  of  the  law  of  continuity  which  takes 
place  in  the  value  of  V,  when  the  point  p  passes  from  the  interior  of 
the  ellipsoid  into  the  exterior  space,  by  adding  the  positive  quantity 
M*  to  that  inclosed  in  the  braces,  and  may  afterwards  suppose  u  eva- 
nescent in  the  final  result.     Let  us  therefore  now  consider  the  function. 


r=/ 


p'  dx'  dy'  d%' 


{{X  -  x'y  +  (y-  y'y  +  (z-  zy  +  M^p ' ' 


this  triple  integral  like  the  preceding  including  all  the  values  of  x',  tf,  »', 
admitted  by  the  condition 


,/2  ^-^ 


—    +    —    +    —    Z     1 

If  now  we  suppose  the  density  /o'  is  of  the  form 

f^'^i^-T^^-h-z^  '  /(^',y.«') (34).  . 

which  will  simplify  /  {x',  y,  »')  when  p  is  constant  and  n'  =  2,  and  then 
compare  this  value  with  the  one  immediately  deducible  from  the  general 
expression  (28)  by  supposing  for  a  moment  n'  =  n,  viz. 

Vol.  V.     Part  III.  3 1 


420        Mr  green,  ON  THE  DETERMINATION  OF  THE 

we  see  that  the  function  f  will  always  be  two  degrees  higher  than  F. 
But  since  our  formula  become  more  complicated  in  proportion  as  the 
degree  of  F  is  higher,  it  will  be  simpler  to  determine  the  differentials 
of  V,  because  for  these  differentials  the  degree  of  F  and  f  is  the  same. 
Let  us  therefore  make 

,  _      1      dV  _    r  /o'  (« —  x)  dx  dy'  d  % 

~  1       m'    fir    ~  J  >rTi  ' 

i,ia;-x'Y  +  iy-i/r  +  {z-zr  +  u''}  — 

then  this  quantity  naturally  divides  itself  into  two  parts,  such  that 

A  =xA'  +  A", 

,           ,,           /-                    p  dx  dy  d% 
where  A'  —  -^r  J '^ ;;rr\ , 

{{x -x'Y  +  {y- yj +  [%-%)' +  u^}~ 


and  A"=-f~ 


x'p  dx'  dy'  dx 


{{x-xy  +  {y-yy  +  {z-%y  +  u^~ 

By  comparing  these  with  the  general  formula  (1),  it  is  clear  that 
M  —  1  =  n'  +  1,  and  consequently  n  =  n  +  2.  In  this  way  the  expression 
(28)  gives 

which  coincides  with  (34)  by  supposing  F=f. 

The  simplest  case  of  the  present  theory  is  where  y(a;',  y',  x')  =  l,  and 
then  by  No  11,  we  have  0o'=  1  and  &„  =  1.  when  A  is  the  quantity 
required,  and  as  the  general  series  (29),  No  11,  then  reduces  itself  to 
its  first  term,  we  immediately  obtain  from  the  formula  (30),  the  value 
of  A!  following, 

*  A= , — --    (the    \  — 7 — (35), 


2 
because  in  the  present  case  H^,  =  1,  «  =  3,  and  n  =  n'  ^  2. 

Again,   the   same  general   theory  being   applied  to  the  value   of  A" 
given  above,  we  get 


ATTRACTIONS    OF  ELLIPSOIDS  OF  VARIABLE   DENSITIES.         421 

F  («',  y',  %')  =  -  x'fix,  y',  a')  =  -  ar'  (when  /  =  1), 

and  hence  by  Noll,  F{x',y',%')  —  —  a'l'.     In  this  way  the  series  (29) 
again  reduces  itself  to  a  single  term,  in  which 

and  the  particular  value  H^  corresponding  thereto,  by  omitting  the  super- 
fluous constant    /(fj2,i,'i — tk  will  be  (No  10), 

Ho  —  a. 
These  substituted  in  the  general  formula  (30)  as  before,  immediately  give 

A"      .  ^""^^UJ      ,3,,  ,„     c^'-'^dh 
A  =  -^ ; — 7-    «  o  C  Pa  /  — rr — , 

and  consequently  by  reduction  since  a^  =  x, 

A=xA'  +  A''=-^-l-^r^l  a'b'c'x  f^^ (36). 

r  (   "^  ]  " 

The  value  of  A  just  given  belongs  to  the  density 

'  _  fi  _  ^"    y!  _  ?!\^ 


Hence  we  immediately  obtain  without  calculation  the  corresponding 
values 

1      dV_      ^^^^U)     ,,,  ,     fh^-'^dh 


1       dV         *'"''  (2)     ,-,  ,     rU-'^'dh 


2^5  r 
r 


C  =  :; ,  -y—  = ;f — r"  abc%  /  — ,  ^  . 

l~2~j 


31  2 


422        Mr  green,  -ON  THE  DETERMINATION  OF  THE 

If  now  we  suppose  moreover 

__      1      d^  _       f  f>  dx  dy  d% 

the  method  before  explained  (No  11),  will  immediately  give 

Z)  =  —  , — T—  a  0  cu   /  T — , 

p/w'+  1\  J^    abc 

and  therefore  if  for  abridgment  we  make 


-'^d')  .,, 


the  total  differential  of  V  may  be  written 
rfr=i»f{2^rf^/^-^^  +  2s.rfy/^-^^  +  2.t/./^— j-^  +  2«rf./^-^^}, 

which  being  integrated  in  the  usual  way  by  first  supposing  h  constant, 
and  then  completing  the  integral  with  a  function  of  h,  to  be  after- 
wards determined  by  making  every  thing  in   F  variable,  we  get 

A  being  a  quantity  absolutely  constant,  which  is  equal  to  zero  when 
w'  >  1.  What  has  just  been  advanced  will  be  quite  clear  if  we  recollect 
that  h  may  be  regarded  as  a  function  of  x,  y,  %  and  u,  determined  by 
the  equation 

»  =  ?^'  +  4^  +  ?4^  ^- 1  -  f  +  -'  +  f'  -^  ' <''^>' 

seeing  that  a'  =  a'*-\-h\  V  =  b''  +  h\  and  &  =  c"  +  h\ 

After  what  precedes,  it  seems  needless  to  enter  into  an  examination 
of  the  values  of  V  belonging  to  other  values  of  the  density  p,  since 
it  must  be  clear  that  the  general  method  is  equally  applicable  when 


ATTRACTIONS  OF  ELLIPSOIDS   OF  VARIABLE  DENSITIES.         423 

where  f  is  the  characteristic  of  any  rational  and  entire  function. 

The  quantity  A  before  determined  when  we  make  u  =  0,  serves  to 
express  the  attraction  in  the  direction  of  the  co-ordinate  x  of  an  ellipsoid 
on  any  point  p,  situated  at  will  either  within  or  without  it.  But  by 
making  «  =  0  in  (37)  we  have 

,  _     a^  y"  z'  ^ 

a"  +  h'  "*■  b'-' +  li' '^  c"  +  h'  ^  h' ^^  ^' 

and  it  is  thence  easy  to  perceive  that  when  p  is  within  the  ellipsoid, 
h  must  constantly  remain  equal  to  zero,  and  the  equation  (38)  will  always 

be   satisfied   by  the  indeterminate   positive   quantity  — .     When   on  the 

contrary  p  is  exterior  to  it,  h  can  no  longer  remain  equal  to  zero,  but 
must  be  such  a  function  of  x,  y,  %,  as  will  satisfy  the  equation  (38),  of 
which  the  last  term  now  evidently  vanishes  in  consequence  of  the 
numerator  o'.  Thus  the  forms  of  the  quantities  A,  B,  C,  D  and  F" 
all  remain  unchanged,  and  the  discontinuity  in  each  of  them  falls  upon 
the  quantity  k. 

To  compare  the  value  of  A  here  found  with  that  obtained  by  the 
ordinary  methods,  we  shall  simply  have  to  make  n'  =  2  in  the  expression 

(36),  recollecting  that  r(l)  =  1,  and  r  (-]  =i\/7r.     In  this  way 

,  .,,,,/-  hdh  ^      ,,,  ,       r  da 

A  =  —  Aiiraoc X  \  -rrr-  =  —  4nrab  c  x  /  -7^— 
J^ctbc  J„drbc 

=  +  ^a'h'c'x  f  4?-  =  4-«'*'c'  J  .   ,  f "^ 

But  the  last  quantity  may  easily  be  put  under  the  form  of  a  definite 

integral,   by   writing  -  in   the  place  of  a  under  the  sign  of  integration, 

and  again  inverting  the  limits.     Thus  there  wiU  result 
J       47r«'J'c'    /•!  v"dv 

^  =  'n^~  J 

a         •'o 


a  +  ^«^)(i  +  ^-/-^^) 


a'        '  a- 


424  Mr  green,   ON   THE   DETERMINATION   OF    THE 

which  agrees  with  the  ordinary  formula,  since  the  mass  of  the  ellipsoid 

47r«'6'c' 


IS 


3 


and  «^  =  d^  +  h\ 


Examination  of  a  j)articular  Case  of  the  General  Theory  exposed  in  the 

former  Part  of  this  Paper. 

14.     There  is  a  particular  case  of  the  general  theory  first  considered, 

which   merits  notice,  in  consequence  of  the  simplicity  of  the  results  to 

which  it  leads.     The  case  in   question  is  that  where  we  have  generally 

Avhatever  /•  may  be 

a/  =  a. 

Then  the   equation  (19)   which    serves   to   determine  0,   becomes  by 
supposing  kn  =  k .  a"' 

0  =  il-'2r'^r')^r'^,  +  (.v-»-l)2/*'?.^-A<^ (39). 

If  now  we  employ  a  transformation  similar  to  that  used  in  obtaining 
the  formula  (14),  No  6,  by  making 

^i  =  P  cos  9i,  ^2  =  p  sin  9i  cos  On,  ^3  =  p  sin  0,  sin  9.^  cos  63,  &c. 

and  then  conceive  the  equation  (39)  deduced  from  the  condition  that 

'"^'"i- "f-d-sf =)'^ {^■*'  (f )'  - r^l 

must  be  a  minimum  (vide  No  8),  we  shall  have 

rf^,rf^2 c?f,,  =  p'-'  sin0/-^  sina/-^ sin  9^-,  dp  de,d9, «?0,_„ 


\d^rl         \dp)        p"    '  sin0,^sin0/ sine^,-,' 

and  1  -  2^,=  =  1  -  p\ 

Proceeding  now  in  the  manner  before  explained,   (No  8),  we  obtain 
for  the  equivalent  of  (39)  by  reduction 

d''(j)  ,   (  _    _  ,  V  cos  g,.  d(f> 
d'<p       s-l-np'    dcl>       1        d9;''^^^~''       Um9rd9,.  k 

dp'  ^    p{l-p')   'dp       p'    '     sm9,'sm9^' sin0^,.,         1  _^2  9-V*U). 


ATTRACTIONS  OF   ELLIPSOIDS  OF   VARIABLE  DENSITIES.         425 
But  this  equation  may  be  satisfied  by  a  function  of  the  form 

0  =  Pe,e,e3 e,_,; 

P  being  a  function  of  p  only,  and  afterwards  generally  6,  a  function 
of  dr  only.  In  fact,  if  we  substitute  this  value  of  (p  in  (40),  and  then 
divide  the  result  by  ^,  it  is  clear  that  it  will  be  satisfied  by  the  system 


e,,.,</e\_, 


—  ^<- 1 


d'Q,.^      ,    _    cos0,_2      o?e,_2      ,       X._,  ^  .^^^ 


Os-2  de^._s  '  sin  0,_2  9s_2</0s_2      sin  6^,-2 

+  2  .  -7— -p^ —  77 j7i h  -; — —^ —  =  X,- 


B._3d0\.3  '  sin  9^-3  Qs-3d9,_3       sin0^ 

&;c.  &c.  &c.  &c. 

combined  with  the  following  equation, 

d'P       s-l-np'     dP        \,  k  ^ 


P^p"  ^  /" (1  -p')  ■  P«?/'     />'     1  -p' 

where  k,  X,,  X^,  X3,  &c.  are  constant  quantities. 

In  order  to  resolve  the  system  (41),  let  us  here  consider  the  general 
type  of  the  equations  therein  contained,  viz, 

0  -  ^'Q'-   ,   (r-i\  ^"^^--  '^^^  +   (  ^'-^'         X     "i  ft 
d9\_,.       ^         >sm9,_/d9,.r        \sm9\.,      a,_,j  «,.,. 

Now  if  we  reflect  on  the  nature  of  the  results  obtained  in  a  preceding 
part  of  this  paper,  it  will  not  be  difficult  to  see  that  6,_r  is  of  the  form 

e,_.  =  (sine._,)*;j  =  (1-M^)«;>; 

where  j9  is  a  rational  and  entire  fimction  of  m  =  cos0s_r,  and  /  a  whole 
number. 

By  substituting  this  value  in  the  general  type  and  making 

\..r^i  =  -  i{i  +  r  -  2) (43) 

we  readily  obtain 

0  =  {1-M.')^:  -{2i  +  r)^^-  {X._.  +  i{i  +  r-l)}p. 


426  Mr  green,   ON  THE   DETERMINATION   OF   THE 

To  satisfy  this  equation,  let  us  assume 

Then  by  substituting  in  the  above  and  equating  separately  the  coefficients 
of  the  various  powers  of  yu,  we  have  in  the  first  place  from  the  highest 

X._,  =  -  e{e^-r—\) (44), 

and  afterwards  generally 

. e-i-9.t  .e-i-M-\     . 

'*' ~  ~    2/  +  2x2e  +  r-2#-3      " 

But  the  equation  (43)  may  evidently  be  made  to  coincide  with  (44),  by 
writing  «*'''  for  i,  and  t^''+'^  for  e,  since  then  both  will  be  comprised  in 

\,_,+,  =  -  e*--'  {e<'-*  +  r-2| (45). 

Hence  we  readily  get  for  the  general  solution  of  the  system  (41), 

"^  2.4  X  {2f<'->  +  r-3|{2«"-'  +  r-5}  "  -  &C.J ; 

where  w  =  cos  9,_r,  and  i*''*  represents  any  positive  integer  whatever,  pro- 
vided ^''■'  is  never  greater  than  ^*'■*". 

Though  we  have  thus  the  solution  of  every  equation  in  the  system 
(41),  yet  that  of  the  first  may  be  obtained  under  a  simpler  form  by 
writing  therein  for  X^.i  its  value  —  i®  deduced  from  (45).  We  shall 
then  immediately  perceive  that  it  is  satisfied  by 


cos  [  J 


In  consequence  of  the  formula  (45),  the  equation  (42)  becomes 

^-  dp'   ^    pO-p')      dp       \  /  '^T^'i^' 

which  is  satisfied  by  making  ^= —\, -(«'*'  + 2ft))  (e"*'' +  2a)  +  w  —  l),  and 


ATTRACTIONS  OF   ELLIPSOIDS  OF  VARIABLE   DENSITIES.  427 

p  =  „i")  f„2.  _  2a>x2i'"+2^  +  v?-2    o„_2 
t'     -(H  2  ^  2i''»  +  4w  +  »-3  ^ 

,    2«)  .  2a)-2  X  2i'~'  +  2ai  +  *-2  .2i'^'  +  2w  +  *-4   ,      -      „     , 
2.4  X  2«<-'  +  4to  +  «-3,2«  +  4w  +  w-5         ^  ' 

where  w  represents  any  whole  positive  number. 

Having  thus  determined  all  the  factors  of  (f>,  it  now  only  remains 
to  deduce  the  corresponding  value  of  H.  But  Ho  the  particular  value 
satisfying  the  differential  equation  in  H,  will  be  had  from  0  by  simply 
making  therein 

since  in  the  present  case  we  have  generally  «/  =  «'. 

Hence,  it  is  clear  that  the  proper  values  of  0,,  di,  9^,  &c.  to  be  here 
employed  are  all  constant,  and  consequently  the  factor 

0102  ©3 ©s-l 

entering  into  (f>  is  likewise   constant.     Neglecting  therefore  this  factor 
as  superfluous,  we  get  for  the  particular  value  of  H, 

a' 

since  ,0^  =  ?.=  +  ?/  + +  ?/  =  ^^  =  fj  , 

ga        a 

and  Pa  represents  what  P  becomes  when  p  is  clianged  into  —  . 

a'  ^ 

Substituting  this  value  of  Ho  in  the  equation  (25),  No  10,  there 
results  since  a'  =  a'^  +  h^ 

H=K.P„  f      ^'"""^^     , (46) 

a' 

K  being  an  arbitrary  constant  quantity. 

Thus  the  complete  value  of  V  for  the  particular  case  considered  in 
the  present  number  is 

Vol.  V.    Part  III.  3  K 


428        Mr  green,  ON  THE  DETERMINATION  OF  THE 

v=  pe.e,......e..,.irp«  f  "^    , (47) 

a' 

and  the  equation  (27),  No  11,  will  give  for  the  corresponding  value  of  p', 

in  -  IN 


-m 


M-4-1 


K 


where  P/,  9/,  62',  &c.  are  the  values  which  the  functions  P,  0i,  02,  &c. 
take  when  we  change  the  unaccented  variables  fi,  ^2, ^,  into  the  cor- 
responding accented  ones  ^/,  ^/, f/,  and 

p «-^  +  l-w  — ^  +  3 n  —  s  +  2a}-l 

'  ~  »  +  2i  +  2ft)-l  .7^  +  2^  + 2(0  +  1 n  +  2i  +  4<w-3' 

or  the  value  of  P  when  p  =  1 ;  where  as  well  as  in  what  follows  i 
is  written  in  the  place  of  i'''. 

The  differential  equation  which  serves  to  determine  H  when  we 
introduce  a  instead  of  h  as  independent  variable,  may  in  the  present 
case  be  written  under  the  form 

.     0  =  a=(a^-«'^)  Vr  +  «M»«'-(*- !)•«"}  ^ 
^  '   dcf  *  '  ada 

+  {?■(«  +  *- 2)  a'' -(«  +  2ft.)(«  +  2a)  +  w-l)a'}  H, 

and  the  particular  integral  here  required  is  that  which  vanishes  when 
h  is  infinite.  Moreover  it  is  easy  to  prove,  by  expanding  in  series,  that 
this  particular  integral  is 


*-l-n-2<o 


provided  we  make   the   variable   r   to  which  A"  refers,   vanish  after  all 
the  operations  have  been  effected. 

But  the  constant  k'  may  be  determined  by  comparing  the  coefficient 
of  the  highest  power  of  a  in  the  expansion  of  the  last  formula  with 
the  like  coefficient  in  that  of  the  expression  (46),  and  thus  we  have 

"  yfc'  =  Kd'^"" (-\Y «  +  2^'  +  2a)-l.w  +  2?  +  2a>  +  l ?^  +  2^^  +  4a,^.-3 

^       ^  2.4  .  6 2o)    . 


ATTRACTIONS   OF   ELLIPSOIDS  OF  VARIABLE  DENSITIES.         429 

Hence  we  readily  get  for  the  equivalent  of  (47), 

rr     vtc^  c^  ^  «  +  2«  +  2a)-l  .M  +  2e  +  2a)4-l n  +  2i  +  4im-3 

2.4.  6 2w 

xKa'^'-'''"{-l)''a'A''a"-fdaa'-"'-'-"{a^-a"')     ^ 

■  ■  GO 

In   certain  cases   the   value   of  V  just  obtained  will  be  found  more 
convenient  than  the  foregoing  one  (47).     Suppose  for  instance  we  repre- 
sent the  value  of  f^  when  h  =  0,  or  a  =  a'  by  V^.    Then  we  shall  hence 
get 
r^      i»c>  o  o         n  +  2i+2a)-l  .n  +  2i  +  2w  +  l »  +  2i  +  4ft.-3 

2.4,6 2a> 


g  —  l~n~Suo 


OD 

which  in  consequence  of  the  well  known  formula 

r(,-p)r(H±f^) 

/"'a-'da  (a'  -  a'')-"  =  -  «''-"-^?  x -J^ i  , 

by  reduction  becomes 

fl+s  —  n\^[n  +  2i  +  4!w  —  l'^ 


r(l±|z^)rp±-±i^) 


2r(a,+  i)r(^  +  ^'^+^") 

since  in  the  formula  (5),  r  ought  to  be  made  equal  to  zero  at  the  end 
of  the  process. 

By  conceiving  the  auxiliary  variable  u  to  vanish,  it  will  become  clear 
from  what  has  been  advanced  in  the  preceding  number,  that  the  values 
of  the  function  P  within  circular  planes  and  spheres,  are  only  particular 
cases  of  the  more  general  one,  (49),  which  answer  to  *  =  2  and  s  =  3 
respectively.  We  have  thus  by  combining  the  expressions  (48)  and 
(49),  the  means  of  determining  Vo  when  the  density  p  is  given,  and 
vice  versa;  and  the  present  method  of  resolving  these  problems  seems 
more  simple  if  possible  than  that  contained  in  the  articles  (4)  and  (5) 
of  my  former  paper. 

GEORGE  GREEN. 

3k2 


XVI.  On  the  Position  of  the  Axes  of  Optical  Elasticity  in  Crystals 
belonging  to  the  Ohlique-Prismatic  System.  By  W.  H.  Millek, 
A.M.  Fellow  and  Tutor  of  St  John's  College,  and  Professor  of 
Mineralogy. 


[Read  Dec.  8,  1834.] 


1.  Fresnel  has  proved  that  whatever  be  the  regular  arrangement 
of  the  medium  which  by  its  elasticity  produces  the  optical  properties 
of  a  crystal,  there  are  always  three  directions  at  right  angles  to  each 
other,  which  may  be  considered  as  axes  of  optical  elasticity.  This 
being  understood,  it  is  further  already  established,  that  crystals  belong- 
ing to  the  tesseral  system  have  three  equal  axes  of  optical  elasticity ; 
that  rhombohedral  and  pyramidal  crystals  have  two  axes  of  elasticity 
equal  to  each  other  and  perpendicular  to  the  crystallographic  axis, 
which  therefore  is  the  third  axis  of  elasticity  and  also  an  optic  axis; 
and  that  crystals  belonging  to  the  remaining  systems  have  three  unequal 
axes  of  elasticity,  and  consequently  two  optic  axes  (that  is,  axes  of 
optical  phenomena)  making  with  each  other  angles  which  are  bisected 
by  the  axes  of  greatest  and  least  elasticity. 

Sir  David  Brewster,  who  discovered  the  mutual  dependence  of  the 
forms  and  optical  properties  of  crystals,  has  determined  the  angles  be- 
tween the  optic  axes  of  a  great  number  of  biaxal  crystals;  his  obser- 
vations, however,  do  not  contain  any  data  from  which  the  positions  of 
the  axes  with  respect  to  the  faces  of  the  crystals  can  be  found. 

2.  In  the  right  prismatic  system  the  axes  of  elasticity  coincide  (as 
might  have  been  expected)  with  the  rectangular  crystallographic  axes. 
In  the  oblique  prismatic  system,  if  the  three  axes  be  XX',  YY',  ZZ', 
the  crystallographic  axis  {YY'),  which  is  perpendicular  to  the  other 
two  {XX',  ZZ'),  is  always  one  of  the  axes  of  elasticity.  This,  in 
Gypsum,  at  the  ordinary  temperature  of  the  air,  and  in  many  other 
crj'stals,  is  the  mean  axis,  or  it  is  perpendicular  to  the  optic  axes;   in 


482  PROFESSOR  MILLER,  ON  THE  POSITION  OF  THE  AXES  OF 

Borax,  Acetate  of  Soda,  Felspar,  Tartaric  Acid  and  Gypsum,  when  heated 
to  about  lOO^C,  as  was  first  observed  by  Mitscherlich,  it  is  the  greatest 
or  least  axis  of  elasticity,  and  is  therefore  in  the  same  plane  with  the 
optic  axes  and  makes  equal  angles  with  them. 

The  position  of  one  axis  of  elasticity  having  thus  an  evident  rela- 
tion to  the  crystallographic  form,  we  are  naturally  led  to  inquire  if 
any  relation  can  be  discovered  between  the  other  two  axes  of  elasti- 
city and  the  crystallographic  form.  The  only  attempts  to  discover  any 
such  relation,  with  which  I  am  acquainted,  are  those  of  M.  Soret, 
(Memoires  de  la  Socidte  de  Physique  de  Geneve,  tome  I.)  and  Pro- 
fessor Neumann  of  Konigsberg  (Poggendorff's  Annalen,  B.  xxvii.  S. 
240).  Neumann  shews,  that  in  Gypsum  the  axes  of  elasticity  and  also 
the  thermal  axes,  or  the  three  lines  in  the  crystal  which  remain  at 
right  angles  to  each  other  at  all  temperatures,  constitute  a  system  of 
rectangular  crystallographic  axes.  It  appeared  at  first  sight  not  im- 
probable that  a  similar  relation  might  be  found  to  exist  between  the 
form  and  axes  of  elasticity  of  other  oblique-prismatic  crystals.  Though 
my  observations  appear  to  disprove  the  law  which  has  thus  been  sug- 
gested, they  do  not  establish  any  other  in  the  place  of  it.  The  only 
general  fact  which  I  have  noticed  is,  that  in  many  instances,  though  not 
in  all,  one  of  the  two  axes  of  elasticity  which  are  perpendicular  to 
W,   is   also  the  axis   of  one  of  the  principal  zones  of  the  crystal. 

3.  To  find  the  angle  between  a  normal  to  any  face  {T)  of  a 
crystal,  and  the  apparent  direction  of  one  of  the  optic  axes  as  seen  in 
air  through  any  parallel  faces  of  the  crystal. 

Let  the  crystal  be  attached  to  an  index,  moveable  on  a  graduated 
circle  having  its  plane  parallel  to  the  axis  of  the  polarizing  instrument, 
or  a  table  on  which  the  position  of  the  index  may  be  marked  by  a 
line  drawn  along  its  edge  with  a  tracing  point.  Let  the  crystal  be 
placed  in  such  a  position,  that  the  apparent  direction  of  the  optic  axis 
in  air  and  a  normal  to  T  may  be  parallel  to  the  circle.  Move  the 
index  till  the  center  of  the  coloured  rings  coincides  with  a  mark  in 
the  axis  of  the  polarizing  instrument,  and  observe  the  points  in  which 
it  meets  the  circle.      Turn   the  crystal  half  round  in  the  plane  of   T, 


OPTICAL  ELASTICITY  IN  OBLIQUE-PRISMATIC  CRYSTALS.  433 

taking  care  not  to  alter  the  inclination  of  T  to  the  index,  (this  may 
be  effected  by  moving  the  crystal,  the  index  being  fixed,  tiU  the  image 
of  some  well  defined  object  seen  by  reflexion  in  T  appears  in  the 
same  direction  after  the  crystal  is  turned  as  it  did  before.)  If  the 
index  be  now  turned  till  the  center  of  the  coloured  rings  coincides 
with  the  mark,  the  angle  it  has  described  between  the  observations 
will  be  manifestly  equal  to  twice  the  angle  between  the  apparent 
direction  of  the  optic  axis  in  air  and  a  normal  to  T.  The  angle 
between  the  optic  axis  in  air  and  a  normal  to  any  other  known  face 
of  the  crystal  being  found  in  the  same  manner,  the  direction  of  the 
optic  axis  in  air  wiU  be  completely  determined. 

4.  To  find  the  optic  axes,  their  apparent  directions  in  air  being 
known. 

Let  Qlt,  Q'K  (Fig.  1.)  be  tangents  to  the  circular  and  eUiptic  sec- 
tions of  a  wave  diverging  from  O  made  by  a  plane  through  the  optic 
axes,  and  therefore  OQ,  OQ',  perpendiculars  to  QB,  will  be  the  optic 
axes;  OP  the  direction  in  which  the  optic  axis  OQ  is  seen  in  air; 
OS  a  perpendicular  to  the  faces  through  which  it  is  seen. 

The  vibrations  in  that  part  of  the  wave  which  has  a  circular  sec- 
tion are  perpendicular  to  the  plane  QOQ,  consequently  a  ray  polar- 
ized in  the  plane  QOQ  is  refracted  in  that  plane  according  to  the 
law  of  sines.  Let  m  be  the  ratio  of  the  sine  of  incidence  to  the  sine 
of  refraction  for  such  a  ray  out  of  air  into  the  crystal,  D  the  mini- 
mum division  of  the  ray  when  refracted  in  the  plane  QOQ'  through 
the  prism  formed  by  two  natural  or  artificial  planes  meeting  at  an 
angle  /  in  a  line  perpendicular  to  QOQ.  Then  ^  sin  ^  /  =  sin  ^  (Z)  + /), 
and  fM  sin  QOS  =  sin  POS.  Whence  the  direction  of  QO  is  known. 
0*0  being  found  in  the  same  manner,  the  axes  of  elasticity  O^,  Oi[, 
which  bisect  the  angles  qOQ,  QOQ,  are  also  known. 

5.  The  diagram  which  accompanies  the  description  of  each  crystal, 
is  the  representation  of  a  sphere,  to  the  surface  of  which  the  faces  of 
the  crystal  are  referred  by  means  of  perpendiculars  drawn  from  the 
center  of  the  sphere.      The  point   in   which    the  perpendicular  to  any 


434  PROFESSOR  MILLER,  ON  THE  POSITION  OF  THE  AXES  OF 

face  meets  the  surface  of  the  sphere,  will  be  called  the  pole  of  that 
face.  The  measurements  express  the  angles  between  the  perpendiculars 
to  the  faces,  or  the  supplements  to  the  angles  between  the  faces  them- 
selves. This  method  of  representing  crystalline  forms  appears  to  have 
been  first  employed  by  Neumann,  in  his  Beitrage  zur  Krystallonomie, 
and  afterwards  by  Grassmann  and  Uhde.  It  has  the  advantage  of  ex- 
hibiting all  the  faces  of  a  crystal  without  confusion  in  one  figure, 
each  zone  being  distinguished  by  a  great  circle  drawn  through  the 
poles  of  the  faces  composing  it,  and  also  of  allowing  all  the  requisite 
calculations  to  be  performed  by  spherical  trigonometry  applied  to  the 
equations 

T  cos  PX  =  T  cos  PY  =  7  cos  PZ, 
h  k  I 

or  to  formulse  deduced  therefrom,  X,  Y,  Z  being  the  points  in  which 
radii  parallel  to  the  axes  of  the  crystal  meet  the  surface  of  the  sphere, 
and  P  the  pole  of  the  face  {h;  k\  l),  which  is  parallel  to  the  plane 

h-  +  k\-^  I-  =  0. 
a         b        c 

ad,  /3/3',  ^f,  ^^'  will  be  used  to  denote  the  extremities  of  diameters 
drawn  parallel  to  the  optic  axes,  and  the  two  axes  of  elasticity  which 
are  perpendicular  to  YY' .  In  Figs.  5,  6,  7,  8  the  faces  are  denoted 
by  the  same  letters  as  in  the  treatises  of  Mohs  and  Naumann.  The  in- 
clinations of  the  faces  of  crystal  (1)  and  (2)  are  deduced  from  a  mean 
of  the  best  measurements  of  thirty  or  forty  crystals,  and  are  probably 
within  1'  of  the  truth. 

The  chemical  notation  and  atomic  weights  are  those  employed  by 
Dr  Turner,  in  the  fifth  edition  of  his  Elements  of  Chemistry. 


OPTICAL  ELASTICITY  IN  OBLIQUE-PRISMATIC  CRYSTALS.  435 


EXAMINATION    OF   VARIOUS   CRYSTALS   ACCORDING    TO    THE    METHODS 

ABOVE    EXPLAINED. 

(1).  Sulphate  of  Oxide  of  Iron  and  Ammonia.  According  to  Mit- 
scherlich  (Jahresbericht  13),  the  composition  of  this  salt,  which  belongs 
to  an  extensive  plesiomorphous  group,  is  expresssed  by  the  formula 
H'^NS  ■\-  FeS  -^  1 H.  Fig.  2.  represents  the  poles  of  its  faces.  Their 
symbols  are  A{1;  0;  0),  C(0;  0;  1),  H{0;  1  ;  1 ;)  M{1',  1;  0), 
P(l;  1;  1),    Q(-l;  1;  1),    T{2;  0;  1). 


AT      42",  14' 

CYQ 

28»,48' 

HH' 

129°,  18' 

MQ    42»,23' 

TC      64  ,34 

QYA' 

44  ,54 

CH 

25  ,21 

QC     34 ,20 

CA'      73  ,12 

MM' 

109  ,36 

QQ' 

140  ,55 

TP    35,14 

AYP  68  ,12-1- 

AM 

35,12 

CP 

44,45 

MT  52  ,46. 

PYC   38  ,35i 

PP' 

130,37 

PM 

58  ,32 

When  yellow  light  is  refracted  through  the  faces  TC  in  the  plane 
AC  A',  the  minimum  deviation  of  a  ray  polarized  in  the  plane  AC  A', 
is  41",  26'.  The  apparent  direction  of  the  optic  axis  aa  in  air,  when 
seen  through  the  faces  TT',  makes  an  angle  of  7°,10'  with  2'2"; 
and  the  optic  axes  appear  to  be  inclined  to  each  other  at  an  angle  of 
79"  when  the  crystal  is  immersed  in  oil,  of  which  the  index  of  re- 
fraction is  1,47.  From  these  data  we  find  Ta  =  4'',47',  Tfi  =  71°,2', 
r^  =  33'',8',    A^  =  9'',6'. 

Tan  T^  is  nearly  equal  to  4tan>4f.  The  value  of  A^  deduced  from 
the  equations  tan  Tf  =  4tan^^,  T^  +  ^^  =  42°,  14'  is  9°,  13'i.  This 
would  make  Q  =  82'',25'i.  Now,  46"  tan  9'^  13'^  =  tan  82'',22'| ;  therefore, 
if  we  refer  the  faces  T,  A,  C,  to  the  rectangular  axes  ff,  YV,  ^^', 
neglecting  the  difference  of  3'  in  the  value  of  C^,  their  simplest  symbols 
will  be  (1;  0;  1),  (4;  0;  -1),  (2 ;  0 ;  -23).  The  magnitude  of  the  last 
index  renders  the  hypothesis  that  ^f ,  ^^'  are  crystallographic  axes  highly 
improbable. 

(2).  The  composition  of  Tartrate  of  Ammo7iia  is  expressed,  according 
to  Dulk,  (Jahrbuch  fiir  Chemie  und  Physik,  1831.  B.  1.)  by  the  for- 
mula H^NT+^H.      The  poles  of  its  faces  are  represented  in  Fig.  3. 

Vol.  V.    Paet  III.  sL 


486  PROFESSOR  MILLER,  ON  THE  POSITION  OF  THE  AXES  OF 

A{i;  0;  0),  C(0;  0;  1),  H{0;  1;  1),  2^(1;  0;  1),  L{-1',  0;  1), 
3f(l;  1;  0),    P(l;  1;  1),    Q(-l;  1 ;  1). 

Cleavage  parallel  to  the  face  A. 

AK    520,31'  AM  55»,2'  Qd  gr^a?'  QA'  6o»,54'i 

KC      39,53  HH' 81, 4.6  QL    41  ,l6^  CP     55,34 

CL      38  CH   49  ,7  ^i*    63  ,22  PM  35  ,48^ 

L^'     49,36  PP'   94,55  PIT  28  ,12  JWQ' 34  ,53 

iIfil/'69,56  PX    42,32  /TQ  27,31i  QC    53,44-i. 

Z>  =  25'',17',  the  light  being  refracted  through  CK.  The  apparent 
angle  in  air  between  the  optic  axes  aa  and  AA',  is  4°,55'.  In  oil, 
the  index  of  refraction  of  which  is  1,741,  the  apparent  angle  between 
the  optic  axes  =42'',20'.  This  gives  ^0  =  8", 7',  A(i  =  35'',54>',  A^=W,M', 
i'^  =  33°,12. 

In  this  case  the  positions  of  some  of  the  faces  A,  K,  C,  L  must  be 
altered  half  a  degree  before  they  can  be  referred  to  the  rectangular 
axes  ff,   W,  ^^'  with  tolerably  simple  indices. 

(3).  A  solution  of  Benzoic  acid  in  alcohol,  when  suffered  to  eva- 
porate, affords  crystals  of  which  the  faces  C,  K,  I  (Fig.  4)  alone  are 
bright.  Cifr=  69",25',  C/=97'',20'  nearly.  Z)  =  64°,45',  refraction  taking 
place  through  the  faces  CK.  The  apparent  direction  of  ad  in  air 
when  seen  through  CC  makes  with  CC  an  angle  of  4°,  30'.  When 
immersed  in  oil  of  which  the  index  of  refraction  is  1,471,  the  appa- 
rent angle  between  the  optic  axes  is  75".  Hence  Ca =2'',47',  C/3=59'',50', 
C^=28»,31',    ^^=40^54'. 

tan  K^,   tan  I^,   tan  C^  are  nearly  as   the  numbers  3,  1,  5. 

The  equation  1  tan /iT^  =  tan  7^  =  |^  tan  C^  is  satisfied  by  making  Cf= 
27°,56'^,  JC  =  97'',17'.  Hence  the  faces  C,  I,  K  may  be  referred  to  the 
rectangular  axes  ^f,  YY ,  l^  without  greatly  altering  the  observed 
angles,  and  their  symbols  will  be  (-1 ;  0;  5),  (1 ;  0;  1),  (1 ;  0 ;  3)  re- 
spectively. 


OPTICAL  ELASTICITY  IN  OBLIQUE-PRISMATIC  CRYSTALS.  437 

(4).  In  Felspar  (Fig.  5.)  the  optic  axes  lie  in  the  plane  of  the 
most  perfect  cleavage,  and  make  with  a  normal  to  M,  angles  of  about 
57"  or  58°,  (58^^  according  to  Sir  David  Brewster)  which  increase  when 
the  crystal  is  heated.     Hence,  ^^'  is  the  axis  of  the  zone  PM. 

(5).  The  optic  axes  of  Pyroxene  (Fig.  6.)  seen  in  air  through  a 
slice  cut  perpendicular  to  MM  are  in  the  plane  Pr,  and  make  angles 
of  16"  with  the  axis  of  the  zone  MM.  Hence,  ^^'  is  the  axis  of  the 
zone  MM',  a,  /3  approach  ^  when  the  crystal  is  heated.  At  ordinary 
temperatures  a/3  is  probably  about  19"^.  The  best  measurements  of 
Pyroxene  shew  that  Pr,  tr  are  nearly  but  not  exactly  equal,  and 
therefore,  that  its  faces  cannot  be  referred  to  ^^',  YY',  X,'C  as  crystallo- 
graphic  axes.  In  all  the  crystals  of  Pyroxene  which  I  have  examined, 
the  rings  surrounding  ad  are  brighter  than  the  rings  surrounding  /3/3'. 

(6).  The  form  of  Borax  (Fig.  7.)  closely  resembles  that  of  Py- 
roxene ;  its  optic  axes  however  are  very  differently  situated.  It  was 
observed  by  Sir  John  Herschel  and  also  by  Professor  Nbrrenberg,  that 
the  optic  axes  for  different  colours  do  not  lie  in  the  same  plane.  This 
being  the  case,  we  cannot  expect  to  find  any  simple  connexion  between 
the  form  and  the  directions  of  the  axes  of  elasticity. 

The  mean  directions  of  the  axes  seen  in  air  through  the  faces  7'T" 
make  angles  of  aO"^,  with  a  normal  to  the  faces  TT',  and  a  perpendi- 
cular to  them  makes  an  angle  of  55°  with  MM'.  The  rings  sur- 
rounding ad,  /3/3'  are  indistinct  on  the  sides  towards  M'P  and  MP' 
respectively,  the  extremities  a,  /3  of  the  axes  being  next  to  the  eye  of 
the  observer.  This  shews  that  the  positions  of  ^f',  ^^'  vary  slightly 
with  the  colour  of  the  light  employed. 

(7).  In  Chromate  of  Oxide  of  Lead,  as  I  have  been  informed  by 
Professor  Norrenberg  of  Tiibingen,  the  axis  of  the  zone  MM  (see  the 
figure  in  Phillips  or  Naumann)  bisects  the  angle  between  the  optic 
axes,  and  is  therefore  one  of  the  axes  of  elasticity.  The  other  two 
axes  of  elasticity  are,  without  doubt,  the  lines  which  bisect  the  angles 
formed  by  normals  to  MM'. 


438 


PROFESSOR  MILLER,  ON  THE  POSITION  OF  THE  AXES,  &c. 


(8).  In  Epidote,  (Fig.  8.)  the  optic  axis  aa  seen  in  air  through  the 
faces  r,  r,  makes  with  r  r'  an  angle  of  8^  50',  /3/3'  seen  in  air  through 
the  faces  M,  M',  makes  with  MM'  an  angle  of  Sl^SO'.  The  determina- 
tion of  m  is  rendered  difficult  by  the  complete  absorption  of  the  light 
polarized  in  the  plane  MT.  Assuming  /x  =  1,7,  which  is  probably  near 
the  truth,  we  get  /•a=5'',ll',  M/3=18°,5'.  According  to  Mohs  2V=51'',41', 
TM=64>'>,30',  therefore,  T'a=46",30',  2)3  =  46'\31'.  Hence  ^^'  is  the  axis 
of  the  zone  PT.  The  near  approximation  of  the  values  of  7'a,  Tfi  to 
equality  must  be  considered  accidental,  as  the  positions  of  the  optic  axes 
are  usually  uncertain  to  the  amount  of  some  minutes. 

The  question  whether  any  proposed  lines  are  crystallographic  axes 
must  be  decided,  as  has  already  been  intimated,  by  the  simplicity  and 
symmetry  of  the  numerical  relations  which  the  expression  of  the  faces 
requires  with  reference  to  these  axes.  This  according  to  the  old  Hauyian 
views  of  the  structure  of  crystals,  is  equivalent  to  saying  that  the  pri- 
mitive form  must  be  such  that  the  other  forms  can  be  derived  from 
it  by  simple  laws  of  decrement.  Now,  we  find  that  by  assuming  the 
axes  of  elasticity  to  be  crystallographical  axes,  we  have  in  the  crystal 
(1)  a  face  (2;  0;  —23),  which  though  not  very  probable  is  not  im- 
possible, and  in  (5)  a  face  (  —  1;  0;  5);  in  (2)  the  observed  and  com- 
puted positions  of  some  of  the  faces  differ  half  a  degree. 

In  (6),  the  optical  properties  are  not  symmetrical. 

In  (4),  (5),  (7),  (8)  one  of  the  axes  of  elasticity  f^'  or  ^^'  is  the 
axis  of  a  zone. 


St  John's  Collegb, 
Dec.  8,  18S4. 


W.  H.  MILLER. 


Tr^ff/.sojcHen.'  (/^  Vu  Cam^./'hjrZ.  S&c.  Vol.S  /'i. 


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