﻿MECHANICAL 
  ACTION 
  OF 
  HEAT. 
  579 
  

  

  in 
  which, 
  if 
  we 
  substitute 
  the 
  symbol 
  of 
  real 
  specific 
  heat, 
  

  

  P 
  V 
  

  

  « 
  = 
  *r; 
  ( 
  96 
  -> 
  

  

  we 
  obtain 
  the 
  formula 
  already 
  given 
  (86) 
  for 
  the 
  relation 
  between 
  heat 
  and 
  tem- 
  

   perature.* 
  

  

  (59.) 
  The 
  introduction 
  of 
  the 
  value 
  given 
  above 
  of 
  the 
  quantity 
  of 
  heat 
  in 
  

   terms 
  of 
  temperature, 
  into 
  the 
  formula 
  67, 
  gives 
  for 
  the 
  latent 
  heat 
  of 
  a 
  small 
  

   expansion 
  d 
  V 
  at 
  constant 
  temperature 
  

  

  (r-K^-dV 
  (97.) 
  

  

  The 
  formulae 
  79 
  and 
  82, 
  for 
  the 
  proportion 
  of 
  heat 
  rendered 
  available 
  by 
  an 
  

   expansive 
  engine 
  working 
  to 
  the 
  greatest 
  advantage, 
  becomes 
  

  

  ^£ 
  < 
  98 
  -> 
  

  

  or 
  the 
  ratio 
  of 
  the 
  difference 
  between 
  the 
  temperatures 
  of 
  receiving 
  and 
  emitting 
  

   heat, 
  to 
  the 
  elevation 
  of 
  the 
  former 
  temperature 
  above 
  that 
  of 
  total 
  privation 
  of 
  heat. 
  

   This 
  is 
  the 
  law 
  already 
  arrived 
  at 
  by 
  a 
  different 
  process 
  in 
  Section 
  V. 
  of 
  this 
  

   paper. 
  

  

  When 
  the 
  same 
  substitution 
  is 
  made 
  in 
  Equation 
  80, 
  which 
  represents 
  the 
  

   total 
  energy, 
  whether 
  as 
  heat 
  or 
  as 
  compressive 
  power, 
  which 
  must 
  be 
  applied 
  to 
  

   unity 
  of 
  weight 
  of 
  a 
  substance 
  to 
  produce 
  given 
  changes 
  of 
  heat 
  and 
  volume, 
  the 
  

   following 
  result 
  is 
  obtained 
  : 
  — 
  

  

  d. 
  Y=dq 
  + 
  d. 
  S= 
  { 
  ft 
  +/(t) 
  + 
  (t-k) 
  Hpr 
  d 
  V 
  \dr 
  

  

  =d. 
  (ftr+/(T)+ 
  ((t-k)^ 
  .-l)JPdv) 
  . 
  • 
  • 
  (99.) 
  

  

  As 
  it 
  cannot 
  be 
  simplified, 
  it 
  is 
  unnecessary 
  here 
  to 
  recapitulate 
  the 
  investi- 
  

   gation, 
  which 
  leads 
  to 
  the 
  conclusion 
  that 
  the 
  functions 
  /(t) 
  and 
  /' 
  (t) 
  have 
  the 
  

   following 
  values 
  : 
  — 
  

  

  /(r) 
  = 
  ftN(/chyp. 
  log.r 
  + 
  £.) 
  ;/'(t)=*-N(£-£) 
  • 
  (99 
  A.) 
  

  

  We 
  have 
  thus 
  reproduced 
  Equation 
  26 
  of 
  the 
  paper 
  formerly 
  referred 
  to, 
  on 
  the 
  

   Centrifugal 
  Theory 
  of 
  Elasticity. 
  

  

  The 
  coefficient 
  of 
  the 
  variation 
  of 
  temperature 
  in 
  the 
  first 
  form 
  of 
  Equation 
  99 
  

   is 
  the 
  specific 
  heat 
  of 
  the 
  substance 
  at 
  constant 
  volume. 
  Denoting 
  this 
  by 
  K 
  v 
  , 
  

   the 
  formula 
  becomes 
  

  

  d. 
  Y=K 
  v 
  .dr+ 
  | 
  ( 
  T 
  -K)~-P}dY 
  . 
  . 
  . 
  (100.) 
  

  

  * 
  See 
  Appendix, 
  Note 
  A. 
  

  

  