﻿MECHANICAL 
  ACTION 
  OF 
  HEAT. 
  163 
  

  

  Fourthly, 
  Let 
  the 
  body 
  be 
  compressed, 
  without 
  change 
  of 
  temperature, 
  to 
  its 
  

   original 
  volume 
  V 
  ; 
  then 
  the 
  heat 
  given 
  out 
  is 
  

  

  + 
  ChTM.\V~dv) 
  

  

  while 
  the 
  power 
  absorbed 
  in 
  compression 
  is 
  

  

  -8V 
  . 
  P 
  

  

  The 
  body 
  being 
  now 
  restored 
  in 
  all 
  respects 
  to 
  its 
  primitive 
  state, 
  the 
  sum 
  of 
  

   the 
  two 
  portions 
  of 
  power 
  connected 
  with 
  change 
  of 
  volume, 
  must, 
  in 
  virtue 
  of 
  

   the 
  principle 
  of 
  vis 
  viva, 
  be 
  equal 
  to 
  the 
  sum 
  of 
  the 
  four 
  quantities 
  of 
  heat 
  with 
  

   their 
  signs 
  reversed. 
  Those 
  additions 
  being 
  made, 
  and 
  the 
  sums 
  divided 
  by 
  the 
  

   common 
  factor 
  8 
  V 
  8 
  t, 
  the 
  following 
  equation 
  is 
  obtained 
  : 
  — 
  

  

  dV 
  

  

  W!_^ 
  ( 
  Q) 
  

  

  dr 
  Cn 
  

  

  The 
  integral 
  of 
  this 
  partial 
  differential 
  equation 
  is 
  

  

  Now 
  <p 
  . 
  t 
  being 
  the 
  same 
  for 
  all 
  densities, 
  is 
  the 
  value 
  of 
  U 
  for 
  the 
  perfectly 
  

   gaseous 
  state, 
  or 
  - 
  ; 
  for 
  in 
  that 
  state, 
  the 
  integral 
  = 
  0. 
  

  

  The 
  values 
  of 
  the 
  partial 
  differential 
  coefficients 
  are 
  accordingly- 
  

  

  . 
  . 
  . 
  (11.) 
  

  

  dV 
  1 
  „ 
  „tfP 
  

   d 
  V 
  V 
  dr 
  

  

  dV 
  K 
  n 
  „ 
  r 
  .„ 
  d 
  2 
  F 
  

  

  ~- 
  GnM 
  fdY 
  

  

  dr 
  t 
  2 
  J 
  dr 
  1 
  

  

  and 
  they 
  can, 
  therefore, 
  be 
  determined 
  in 
  all 
  cases 
  in 
  which 
  the 
  quantity 
  

   K=Cn/j,b, 
  and 
  the 
  law 
  of 
  variation 
  of 
  the 
  total 
  elasticity 
  with 
  the 
  volume 
  and 
  

   temperature 
  are 
  known, 
  so 
  as 
  to 
  complete 
  the 
  data 
  required 
  in 
  order 
  to 
  apply 
  

   equation 
  6 
  of 
  this 
  section 
  to 
  the 
  calculation 
  of 
  the 
  mechanical 
  value 
  of 
  the 
  varia- 
  

   tions 
  of 
  heat 
  due 
  to 
  changes 
  of 
  volume 
  and 
  molecular 
  arrangement. 
  

  

  The 
  total 
  elasticity 
  of 
  an 
  imperfect 
  gas, 
  according 
  to 
  Equations 
  VI. 
  and 
  XTI. 
  

   of 
  the 
  introduction, 
  being 
  

  

  p 
  =,osir( 
  1 
  -*-(**})' 
  + 
  / 
  CD) 
  

  

  its 
  first 
  and 
  second 
  partial 
  differential 
  coefficients 
  with 
  respect 
  to 
  the 
  tempera- 
  

   ture 
  are, 
  — 
  

  

  = 
  L_(2-* 
  +r 
  J-\ 
  ¥ 
  (l) 
  T 
  -\ 
  

  

  C«MV\ 
  dr* 
  dr*) 
  \ 
  'k) 
  

  

  dr 
  ~C«MV 
  

   d 
  2 
  ~P 
  

  

  dr 
  2 
  

   VOL. 
  XX. 
  PAET 
  I. 
  2 
  X 
  

  

  

  