﻿AND 
  ITS 
  CONNECTION 
  WITH 
  THE 
  THEORY 
  OF 
  HEAT. 
  429 
  

  

  1 
  dp' 
  d$> 
  

  

  g 
  dx 
  d 
  

  

  3> 
  _ 
  _ 
  / 
  d 
  d 
  d\ 
  

  

  1 
  dp' 
  d 
  <b 
  on/ 
  ^ 
  n 
  d 
  ^ 
  \ 
  o_ 
  n 
  

  

  g' 
  dy 
  dy 
  ^ 
  \ 
  dx 
  dy 
  ^ 
  dz)^~ 
  

  

  1 
  dp' 
  d<& 
  on/ 
  d 
  o 
  d 
  d 
  \ 
  „ 
  

  

  (30 
  

  

  9 
  d 
  - 
  

  

  Let 
  7* 
  be 
  the 
  radius 
  of 
  curvature 
  of 
  the 
  path 
  of 
  the 
  particles 
  through 
  (x, 
  y, 
  z) 
  ; 
  

   and 
  a' 
  j3' 
  y, 
  its 
  direction-cosines 
  ; 
  then 
  the 
  above 
  equations 
  obviously 
  become 
  

  

  _ldp^_d^_ 
  2 
  a 
  L=Q 
  

   g 
  dx 
  d 
  co 
  ^ 
  r 
  

  

  Q 
  dy 
  dy 
  r 
  

  

  _i^/_«u> 
  y 
  = 
  

  

  g 
  dz 
  d 
  z 
  r 
  

  

  If 
  these 
  equations 
  are 
  integrable, 
  

  

  a' 
  ft, 
  y 
  , 
  

  

  — 
  dx+ 
  — 
  d 
  y 
  + 
  —a 
  z 
  

   r 
  r 
  r 
  

  

  must 
  be 
  an 
  exact 
  differential. 
  Let 
  - 
  be 
  its 
  primitive 
  function 
  ; 
  the 
  negative 
  

   sign 
  being 
  used, 
  because 
  a', 
  (3', 
  y' 
  must 
  be 
  generally 
  negative. 
  Then 
  the 
  integral 
  

   of 
  the 
  equations 
  (3) 
  is 
  

  

  log, 
  p 
  = 
  jfy- 
  = 
  i 
  (2 
  Q 
  - 
  #) 
  + 
  constant 
  ; 
  

   or 
  taking 
  ^ 
  to 
  denote 
  the 
  pressure 
  at 
  the 
  bounding 
  surface 
  of 
  the 
  atom 
  : 
  — 
  

  

  2 
  1 
  

  

  9 
  = 
  9i 
  e 
  k 
  ( 
  4 
  )- 
  

  

  Our 
  present 
  object 
  is 
  to 
  determine 
  the 
  superficial-atomic 
  density, 
  g 
  v 
  and 
  thence 
  

  

  the 
  pressure 
  p-h 
  9 
  V 
  in 
  terms 
  of 
  the 
  mean 
  density 
  y- 
  and 
  heat 
  Q. 
  For 
  this 
  pur- 
  

   pose 
  we 
  must 
  introduce 
  the 
  above 
  value 
  of 
  § 
  into 
  equation 
  (2), 
  giving 
  

  

  ^g.-ff^e^^^-^^-^dxdydz 
  

   whence 
  

  

  2Q 
  1 
  

  

  p 
  = 
  hp 
  l 
  = 
  hp+frf,.e~ii' 
  ( 
  *~* 
  l) 
  ~h 
  ( 
  *~* 
  l) 
  dzdydz 
  . 
  . 
  (5.) 
  

  

  Let 
  the 
  volume 
  of 
  the 
  atom 
  be 
  conceived 
  to 
  be 
  divided 
  into 
  layers, 
  in 
  each 
  of 
  

   which 
  <£ 
  has 
  a 
  constant 
  value. 
  Then 
  we 
  may 
  make 
  the 
  following 
  transforma- 
  

   tions. 
  

  

  vol. 
  xx. 
  part 
  in. 
  5 
  z 
  

  

  