﻿AND 
  ITS 
  CONNECTION 
  WITH 
  THE 
  THEORY 
  OF 
  HEAT. 
  439 
  

  

  variation 
  of 
  temperature 
  and 
  volume 
  be 
  made 
  indefinitely 
  small. 
  Then 
  

  

  dpd\ 
  = 
  . 
  -W? 
  d\ 
  

  

  ^ 
  t 
  — 
  k 
  d\ 
  

  

  and 
  dividing 
  by 
  d 
  t 
  d 
  V 
  

  

  6?jt) 
  _ 
  1 
  rfQ' 
  

   fl?T 
  ~~ 
  t 
  — 
  k 
  ' 
  dV 
  

  

  This 
  differential 
  equation 
  is 
  also 
  an 
  immediate 
  consequence 
  of 
  equation 
  (25.) 
  

  

  If 
  ^ 
  be 
  put 
  for 
  - 
  = 
  -, 
  and 
  J 
  M 
  for 
  -pk 
  it 
  becomes 
  identical 
  with 
  the 
  equation 
  

  

  by 
  which 
  Professor 
  William 
  Thomson 
  expresses 
  Carnot's 
  law, 
  as 
  deduced 
  

   by 
  him 
  and 
  by 
  Mr 
  Clausius 
  from 
  the 
  principle, 
  that 
  it 
  is 
  impossible 
  to 
  transfer 
  

   heat 
  from 
  a 
  colder 
  to 
  a 
  hotter 
  body, 
  without 
  expenditure 
  of 
  mechanical 
  power. 
  

  

  The 
  investigation 
  which 
  I 
  have 
  now 
  given 
  is 
  identical 
  in 
  principle 
  with 
  that 
  in 
  

   the 
  fifth 
  section 
  of 
  my 
  paper 
  on 
  the 
  Mechanical 
  Action 
  of 
  Heat 
  ; 
  but 
  the 
  result 
  is 
  

   expressed 
  in 
  a 
  more 
  comprehensive 
  form. 
  

  

  Equation 
  (28) 
  like 
  (25), 
  (26), 
  and 
  (27), 
  is 
  applicable 
  to 
  a 
  mixture, 
  composed 
  of 
  

   any 
  number 
  of 
  different 
  substances, 
  in 
  any 
  proportions, 
  provided 
  the 
  temperature, 
  

  

  /7 
  Y) 
  ct 
  Y) 
  

  

  the 
  pressure, 
  and 
  the 
  coefficients 
  -ry 
  ~ 
  2 
  ' 
  are 
  the 
  same 
  throughout 
  the 
  mass. 
  

  

  (12.) 
  Apparent 
  Specific 
  Heat. 
  — 
  The 
  general 
  value 
  of 
  apparent 
  specific 
  heat 
  of 
  

   unity 
  of 
  weight, 
  is 
  

  

  ^%+%+£-%-<»*-<> 
  [&+/&*+£■%} 
  • 
  (-) 
  

  

  agreeing 
  with 
  equation 
  13 
  of 
  my 
  previous 
  paper. 
  

  

  The 
  value 
  in 
  each 
  particular 
  case 
  depends 
  on 
  the 
  mode 
  of 
  variation 
  of 
  volume 
  

   with 
  temperature. 
  Specific 
  heat 
  at 
  constant 
  volume, 
  is 
  

  

  K^ 
  H 
  r- 
  K) 
  (AE, 
  + 
  f£* 
  a 
  v) 
  .... 
  (30.) 
  

  

  When 
  the 
  pressure 
  is 
  constant, 
  we 
  must 
  have 
  

  

  d~P 
  ?TT 
  dp 
  , 
  _ 
  

   -p^ 
  dY 
  + 
  -^- 
  dr 
  = 
  

   d\ 
  dr 
  

  

  and, 
  consequently, 
  dp 
  

  

  dV 
  _ 
  _dr 
  

   I7~~~d~P 
  

   d~Y 
  

  

  therefore 
  specific 
  heat 
  at 
  constant 
  pressure, 
  is 
  

  

  K=K 
  v+ 
  (r-K)^lL 
  (31 
  ; 
  } 
  

  

  ~ 
  dV~ 
  

   This 
  agrees 
  with 
  equation 
  (16) 
  of 
  Professor 
  Thomson's 
  paper, 
  if 
  J 
  fx 
  in 
  his 
  notation 
  

  

  — 
  T 
  — 
  K. 
  

  

  