﻿DYNAMICAL 
  THEORY 
  OF 
  HEAT. 
  285 
  

  

  Now, 
  by 
  (27), 
  we 
  have 
  

  

  (y-^ 
  =1 
  28 
  >> 
  

  

  and 
  ftr 
  _x)£ 
  + 
  .Cl-.)^ 
  + 
  4Z-0 
  . 
  . 
  (29). 
  

  

  Hence 
  m 
  = 
  ^Za 
  ( 
  30 
  ). 
  

  

  • 
  ,d\ 
  dy 
  

   N= 
  C 
  (l-*) 
  + 
  A*-L 
  ^ 
  if 
  . 
  . 
  (31). 
  

  

  55. 
  The 
  expression 
  of 
  the 
  second 
  fundamenta 
  proposition 
  in 
  this 
  case 
  becomes, 
  

   consequently, 
  

  

  H-= 
  YT^ 
  ( 
  32) 
  ' 
  

  

  which 
  agrees 
  with 
  Carnot's 
  original 
  result, 
  and 
  is 
  the 
  formula 
  that 
  has 
  been 
  used 
  

   (referred 
  to 
  above 
  in 
  § 
  31) 
  for 
  determining 
  fx 
  by 
  means 
  of 
  Regnault's 
  observa- 
  

   tions, 
  on 
  steam. 
  

  

  56. 
  To 
  express 
  the 
  conclusion 
  derivable 
  from 
  the 
  first 
  fundamental 
  proposi- 
  

   tion, 
  we 
  have, 
  by 
  differentiating 
  the 
  preceding 
  expressions 
  for 
  M 
  and 
  N 
  with 
  

   reference 
  to 
  t 
  and 
  v 
  respectively, 
  

  

  rfM__l_ 
  dL_ 
  _L_ 
  d(y-X) 
  

   ~dt~y-\' 
  dt 
  (7-A) 
  2 
  * 
  dt 
  

  

  d 
  7 
  d 
  A 
  

  

  a?N 
  / 
  , 
  T 
  d 
  t 
  dt\dx 
  

  

  —— 
  = 
  ( 
  h 
  — 
  c—L 
  ==— 
  J 
  -T- 
  

  

  dv 
  \ 
  7— 
  A 
  ' 
  dv 
  

  

  r 
  h—c 
  _ 
  L 
  1 
  

  

  h—c 
  L 
  ^ 
  d(y—X) 
  

  

  dt 
  

  

  Hence 
  equation 
  (2) 
  of 
  § 
  20 
  becomes 
  

  

  dL 
  

  

  - 
  + 
  c 
  — 
  h 
  n 
  

   dt 
  _ 
  1 
  dp 
  

  

  (33). 
  

  

  7-A 
  J 
  dt 
  ' 
  

  

  Combining 
  this 
  with 
  the 
  conclusion 
  (32) 
  derived 
  from 
  the 
  second 
  fundamental 
  

   proposition, 
  we 
  obtain 
  

  

  ^L 
  1 
  Liu 
  .... 
  

  

  -di 
  + 
  c 
  - 
  h 
  = 
  -r 
  ■ 
  • 
  • 
  w 
  

  

  The 
  former 
  of 
  these 
  equations 
  agrees 
  precisely 
  with 
  one 
  which 
  was 
  first 
  given 
  

   by 
  Clausius, 
  and 
  the 
  preceding 
  investigation 
  is 
  substantially 
  the 
  same 
  as 
  the 
  in- 
  

   vestigation 
  by 
  which 
  he 
  arrived 
  at 
  it. 
  The 
  second 
  differs 
  from 
  another 
  given 
  by 
  

   Clausius 
  only 
  in 
  not 
  implying 
  any 
  hypothesis 
  as 
  to 
  the 
  form 
  of 
  Caenot's 
  func- 
  

   tion, 
  fX. 
  

  

  VOL. 
  XX. 
  PART 
  II. 
  4 
  H 
  

  

  