﻿AND 
  ITS 
  CONNECTION 
  WITH 
  THE 
  THEORY 
  OF 
  HEAT. 
  431 
  

  

  (5.) 
  Case 
  of 
  a 
  Perfect 
  Gas. 
  — 
  As 
  a 
  substance 
  is 
  rarefied, 
  it 
  gradually 
  approaches 
  

   a 
  condition 
  in 
  which 
  the 
  pressure, 
  under 
  like 
  circumstances 
  as 
  to 
  heat, 
  varies 
  pro- 
  

   portionally 
  to 
  the 
  density. 
  This 
  is 
  because 
  the 
  effect 
  of 
  the 
  molecular 
  attrac- 
  

   tions 
  and 
  repulsions 
  on 
  the 
  pressure 
  diminishes 
  with 
  the 
  density, 
  so 
  that 
  $, 
  w, 
  

   and 
  G 
  approximate 
  to 
  constant 
  quantities. 
  In 
  the 
  limiting 
  or 
  perfectly 
  gaseous 
  

   condition, 
  therefore, 
  

  

  G, 
  

  

  H 
  '=T> 
  

  

  and 
  

  

  hjj.6 
  _ 
  h 
  jj. 
  / 
  2 
  Q 
  

  

  ? 
  "MV~MV 
  

  

  (4**0 
  • 
  cuo 
  

  

  (6.) 
  Equilibrium 
  of 
  Heat 
  : 
  Nature 
  of 
  Temperature 
  and 
  Real 
  Specific 
  Heat. 
  — 
  

   When 
  the 
  atmospheres 
  of 
  atoms 
  of 
  two 
  different 
  substances 
  are 
  in 
  contact 
  at 
  their 
  

   common 
  bounding 
  surface, 
  it 
  is 
  necessary 
  to 
  a 
  permanent 
  condition, 
  that 
  the 
  pres- 
  

   sure 
  in 
  passing 
  that 
  surface 
  should 
  vary 
  continuously. 
  

  

  Let 
  (a) 
  and 
  (b) 
  be 
  taken 
  as 
  characteristics, 
  to 
  distinguish 
  the 
  specific 
  quantities 
  

   peculiar 
  to 
  the 
  two 
  media 
  respectively. 
  Let 
  d 
  m 
  denote 
  the 
  volume 
  of 
  an 
  indefinitely 
  

   thin 
  layer, 
  close 
  to 
  the 
  bounding 
  surface. 
  Then 
  the 
  following 
  equations 
  must 
  be 
  

   fulfilled, 
  to 
  ensure 
  a 
  permanent 
  condition 
  : 
  — 
  

  

  P(a)=p(b); 
  ^(<*)= 
  J^J>) 
  when// 
  = 
  p 
  .... 
  (12.) 
  

  

  By 
  making 
  the 
  proper 
  substitutions 
  in 
  equation 
  (4), 
  it 
  appears, 
  that 
  

  

  p' 
  = 
  pe 
  ii 
  °L 
  

  

  Hence 
  

  

  d 
  w 
  

  

  d 
  ^(p'=p)= 
  P 
  (e 
  d 
  -P± 
  + 
  _+\ 
  

  

  dm 
  \r 
  rj 
  f 
  y 
  dm 
  dm 
  j 
  

  

  Now^> 
  is 
  the 
  same 
  for 
  both 
  media: 
  ^ 
  . 
  ^-= 
  e 
  * 
  * 
  "* 
  l} 
  is 
  either 
  a 
  maximum 
  

   or 
  a 
  minimum, 
  so 
  that 
  its 
  differential 
  is 
  null 
  ; 
  and 
  d 
  m 
  is 
  a 
  continuous 
  function 
  

   of 
  k 
  (p, 
  so 
  that 
  -f^P- 
  («) 
  = 
  d 
  ~^~- 
  (p). 
  There 
  remains 
  only 
  the 
  function 
  of 
  heat 
  

  

  Therefore 
  the 
  condition 
  of 
  a 
  permanent 
  state 
  of 
  molecular 
  motion, 
  that 
  is 
  to 
  

   say, 
  the 
  condition 
  of 
  equilibrium 
  of 
  heat, 
  is 
  that 
  this 
  function 
  shall 
  be 
  the 
  same 
  

   for 
  the 
  two 
  substances 
  ; 
  or 
  that 
  

  

  A 
  a 
  k„ 
  h 
  h 
  k„ 
  \ 
  16 
  ') 
  

  

  