﻿MECHANICAL 
  ACTION 
  OF 
  HEAT. 
  573 
  

  

  portions 
  of 
  each 
  substance, 
  of 
  equal 
  weight, 
  and 
  destitute 
  of 
  heat, 
  be 
  added 
  to 
  the 
  

   original 
  masses 
  ; 
  so 
  that 
  the 
  quantities 
  of 
  heat 
  in 
  unity 
  of 
  weight 
  may 
  be 
  dimi- 
  

   nished 
  in 
  each 
  substance, 
  but 
  may 
  continue 
  to 
  be 
  in 
  the 
  same 
  ratio. 
  Then, 
  if 
  the 
  

   equality 
  of 
  temperature 
  do 
  not 
  continue, 
  portions 
  of 
  heat 
  which 
  were 
  in 
  equilibrio 
  

   must 
  have 
  lost 
  that 
  equilibrium, 
  merely 
  by 
  being 
  transferred 
  to 
  other 
  particles 
  of 
  

   a 
  pair 
  of 
  homogeneous 
  substances, 
  which 
  is 
  absurd. 
  Therefore, 
  the 
  temperatures 
  

   continue 
  equal. 
  

  

  It 
  follows, 
  that 
  the 
  quantity 
  of 
  heat 
  in 
  unity 
  of 
  weight 
  of 
  a 
  substance 
  at 
  a 
  

   given 
  temperature, 
  may 
  be 
  expressed 
  by 
  the 
  product 
  of 
  a 
  quantity 
  depending 
  on 
  

   the 
  nature 
  of 
  the 
  substance, 
  and 
  independent 
  of 
  the 
  temperature, 
  multiplied 
  by 
  a 
  

   function 
  of 
  the 
  temperature, 
  which 
  is 
  the 
  same 
  for 
  all 
  substances. 
  

  

  Let 
  t 
  denote 
  the 
  temperature 
  of 
  a 
  body 
  according 
  to 
  the 
  scale 
  adopted 
  ; 
  k, 
  the 
  

   position, 
  on 
  the 
  same 
  scale, 
  of 
  the 
  temperature 
  corresponding 
  to 
  absolute 
  privation 
  

   of 
  heat 
  ; 
  fc, 
  a 
  quantity 
  depending 
  on 
  the 
  nature 
  of 
  the 
  substance, 
  and 
  independent 
  

   of 
  temperature. 
  Then 
  the 
  quantity 
  of 
  heat 
  in 
  unity 
  of 
  weight 
  may 
  be 
  expressed 
  

   as 
  follows: 
  — 
  

  

  Q=»(^.t-^.k) 
  (81.) 
  

  

  (54.) 
  If 
  we 
  introduce 
  this 
  notation 
  into 
  the 
  formula 
  (79) 
  which 
  expresses 
  the 
  

   proportion 
  of 
  the 
  total 
  heat 
  expended, 
  which 
  is 
  converted 
  into 
  useful 
  power 
  by 
  an 
  

   expansive 
  machine 
  working 
  to 
  the 
  best 
  advantage, 
  the 
  quantity 
  fc, 
  peculiar 
  to 
  the 
  

   substance 
  employed, 
  disappears, 
  and 
  we 
  obtain 
  Carnot's 
  Theorem, 
  as 
  modified 
  by 
  

   Messrs 
  Clausius 
  and 
  Thomson, 
  viz., 
  — 
  that 
  this 
  ratio 
  is 
  a 
  function 
  solely 
  of 
  the 
  

   temperatures 
  at 
  which 
  heat 
  is 
  received 
  and 
  emitted 
  respectively, 
  and 
  is 
  indepen- 
  

   dent 
  of 
  the 
  nature 
  of 
  the 
  substance 
  ; 
  or 
  symbolically, 
  

  

  Effect 
  = 
  Q 
  1 
  -Q 
  2 
  _ 
  ^- 
  t 
  x-^-t 
  2 
  (82.) 
  

  

  Heat 
  Expended 
  Q 
  x 
  4* 
  • 
  T 
  i 
  — 
  ^ 
  • 
  K 
  

  

  (55.) 
  Let 
  us 
  now 
  apply 
  the 
  same 
  notation 
  to 
  the 
  formula 
  (67) 
  for 
  the 
  latent 
  

   heat 
  of 
  a 
  small 
  expansion 
  d 
  V 
  at 
  constant 
  heat, 
  viz 
  : 
  — 
  

  

  dV 
  

  

  Q~'dY 
  

   dQ 
  

  

  we 
  have 
  evidently 
  

  

  d]L 
  = 
  J_ 
  ££_ 
  _JL 
  ^ 
  

  

  d 
  Q 
  ~ 
  d_Q 
  ' 
  d 
  t 
  ~ 
  U 
  4-' 
  . 
  t 
  ' 
  d 
  t 
  

  

  d 
  r 
  

   and 
  consequently, 
  the 
  heat 
  which 
  disappears 
  by 
  the 
  expansion 
  d 
  V 
  is 
  

  

  Qll-<*V= 
  *- 
  T 
  -*-" 
  4^.dV 
  (83.) 
  

  

  ^ 
  d 
  Q 
  4 
  . 
  t 
  d 
  t 
  

  

  from 
  which 
  formula 
  the 
  specific 
  quantity 
  It 
  has 
  disappeared. 
  

   Now, 
  in 
  the 
  notation 
  of 
  Professor 
  Thomson 
  we 
  have 
  

  

  4 
  . 
  T— 
  4 
  . 
  K 
  J 
  

   VOL. 
  XX. 
  PART 
  IV. 
  7 
  Q 
  

  

  