﻿290 
  PROFESSOR 
  WILLIAM 
  THOMSON 
  ON 
  THE 
  

  

  § 
  21 
  of 
  my 
  former 
  paper, 
  derived 
  from 
  Clausius's 
  extension 
  of 
  Carnot's 
  theory, 
  

   we 
  have 
  

  

  M=-. 
  d 
  / 
  f 
  («)* 
  

  

  fx 
  dt 
  K 
  J 
  

  

  where 
  ll 
  denotes 
  Carnot's 
  function, 
  the 
  same 
  for 
  all 
  substances 
  at 
  the 
  same 
  tem- 
  

   perature. 
  

  

  Now 
  let 
  the 
  substance 
  expand 
  from 
  any 
  volume 
  V 
  to 
  V, 
  and, 
  being 
  kept 
  

   constantly 
  at 
  the 
  temperature 
  t, 
  let 
  it 
  absorb 
  a 
  quantity, 
  H, 
  of 
  heat. 
  Then 
  

  

  r 
  Y 
  ' 
  1 
  d 
  r 
  Y 
  ' 
  

   K 
  =f 
  Y 
  Mdv 
  =Uj 
  Y 
  t> 
  d 
  °- 
  ' 
  • 
  » 
  

  

  But, 
  if 
  W 
  denote 
  the 
  mechanical 
  work 
  which 
  the 
  substance 
  does 
  in 
  expanding, 
  

   we 
  have 
  

  

  W=/ 
  pdv 
  (c), 
  

  

  Jy 
  

  

  and 
  therefore 
  

  

  H 
  = 
  il£ 
  (<). 
  

  

  fx 
  dt 
  K 
  ' 
  

  

  This 
  formula, 
  established 
  without 
  any 
  assumption 
  admitting 
  of 
  doubt, 
  expresses 
  

   the 
  relation 
  between 
  the 
  heat 
  developed 
  by 
  the 
  compression 
  of 
  any 
  substance 
  

   whatever, 
  and 
  the 
  mechanical 
  work 
  which 
  is 
  required 
  to 
  effect 
  the 
  compression 
  ; 
  

   as 
  far 
  as 
  it 
  can 
  be 
  determined 
  without 
  hypothesis, 
  by 
  purely 
  theoretical 
  con- 
  

   siderations. 
  

  

  4. 
  The 
  preceding 
  formula 
  leads 
  to 
  that 
  which 
  I 
  formerly 
  gave 
  for 
  the 
  case 
  of 
  

   fluids 
  subject 
  to 
  the 
  gaseous 
  laws 
  ; 
  since 
  for 
  such 
  we 
  have 
  

  

  pv=p 
  v 
  (l 
  + 
  Et) 
  (1), 
  

  

  from 
  which 
  we 
  deduce, 
  by 
  (c), 
  

  

  W=/, 
  t, 
  o 
  (l 
  + 
  E01og^ 
  • 
  • 
  • 
  (2), 
  

  

  dW 
  V 
  E 
  

  

  and 
  ■d7 
  =Ep 
  ° 
  p 
  °' 
  log 
  V 
  = 
  TT^V 
  W 
  " 
  ' 
  * 
  (3); 
  

  

  and 
  therefore, 
  by 
  (d), 
  

  

  which 
  agrees 
  with 
  equation 
  (11) 
  of 
  § 
  49 
  of 
  the 
  former 
  paper. 
  

  

  5. 
  Hence 
  we 
  conclude 
  that 
  the 
  heat 
  evolved 
  by 
  any 
  fluid 
  fulfilling 
  the 
  gaseous 
  

   laws, 
  is 
  proportional 
  to 
  the 
  work 
  spent 
  in 
  compressing 
  it, 
  at 
  any 
  given 
  constant 
  

   temperature 
  ; 
  but 
  that 
  the 
  quantity 
  of 
  work 
  required 
  to 
  produce 
  a 
  unit 
  of 
  heat 
  

  

  * 
  Throughout 
  this 
  paper 
  formulas, 
  which 
  involve 
  no 
  hypothesis 
  whatever, 
  are 
  marked 
  with 
  

   italic 
  letters 
  ; 
  formulas 
  which 
  involve 
  Boyle's 
  and 
  Dalton's 
  laws 
  are 
  marked 
  with 
  Arabic 
  numerals 
  ; 
  

   and 
  formulas 
  involving, 
  besides, 
  Mayer's 
  hypothesis, 
  are 
  marked 
  with 
  Roman 
  numerals. 
  

  

  

  