﻿AND 
  ITS 
  CONNECTION 
  WITH 
  THE 
  THEORY 
  OF 
  HEAT. 
  437 
  

  

  unity 
  of 
  weight 
  of 
  the 
  substance, 
  supposing 
  that 
  there 
  is 
  no 
  chemical, 
  electrical, 
  

   magnetic, 
  or 
  other 
  action 
  except 
  heat 
  and 
  pressure 
  ; 
  and 
  its 
  value* 
  is 
  : 
  — 
  

  

  Sv=8Q 
  + 
  8Q'-VdY 
  = 
  dr 
  . 
  { 
  ft 
  + 
  *£ 
  (1 
  - 
  *.} 
  +{T 
  - 
  K) 
  ffP 
  d 
  Y 
  } 
  

  

  + 
  8Y. 
  { 
  ( 
  T 
  _jo|f-/>-/(V) 
  j 
  . 
  . 
  . 
  (26.) 
  

  

  This 
  expression 
  is 
  obviously 
  an 
  exact 
  differential, 
  and 
  its 
  integral 
  is 
  the 
  follow- 
  

   ing 
  function 
  of 
  the 
  volume 
  and 
  temperature 
  : 
  — 
  

  

  y=U(t-k) 
  +A£ 
  (i 
  ogeT 
  + 
  ^ 
  +J\ 
  T 
  -K) 
  d 
  / 
  r 
  dV 
  -ff(V)dV 
  . 
  (27.) 
  

  

  Accordingly, 
  the 
  total 
  amount 
  of 
  power 
  which 
  must 
  be 
  exercised 
  upon 
  unity 
  of 
  

   weight 
  of 
  a 
  substance, 
  to 
  make 
  it 
  pass 
  from 
  the 
  absolute 
  temperature 
  t 
  and 
  

   volume 
  V 
  () 
  to 
  the 
  absolute 
  temperature 
  t 
  1 
  and 
  volume 
  Y 
  v 
  is 
  

  

  V(Y 
  V 
  TJ-Y(V 
  ,T 
  ) 
  

  

  This 
  quantity 
  consists 
  partly 
  of 
  expansive 
  or 
  compressive 
  power, 
  and 
  partly 
  

   of 
  heat, 
  in 
  proportions 
  depending 
  on 
  the 
  mode 
  in 
  which 
  the 
  intermediate 
  changes 
  

   of 
  temperature 
  and 
  volume 
  take 
  place 
  ; 
  but 
  the 
  total 
  amount 
  is 
  independent 
  of 
  

   these 
  changes. 
  

  

  Hence, 
  if 
  a 
  body 
  be 
  made 
  to 
  pass 
  through 
  a 
  variety 
  of 
  changes 
  of 
  temperature 
  

   and 
  volume, 
  and 
  at 
  length 
  be 
  brought 
  back 
  to 
  its 
  primitive 
  volume 
  and 
  temperature, 
  

   the 
  algebraical 
  sum 
  of 
  the 
  portions 
  of 
  power 
  applied 
  to 
  and 
  evolved 
  from 
  the 
  body, 
  

   whether 
  in 
  the 
  form 
  of 
  expansion 
  and 
  compression, 
  or 
  in 
  that 
  of 
  heat, 
  is 
  equal 
  to 
  zero. 
  

  

  This 
  is 
  one 
  form 
  of 
  the 
  law 
  proved 
  experimentally 
  by 
  Mr 
  Joule, 
  of 
  the 
  equiva- 
  

   lence 
  of 
  heat 
  and 
  mechanical 
  power. 
  In 
  my 
  original 
  paper 
  on 
  the 
  Mechanical 
  

   Action 
  of 
  Heat, 
  I 
  used 
  this 
  law 
  as 
  an 
  axiom, 
  to 
  assist 
  in 
  the 
  investigation 
  of 
  the 
  

   Equation 
  of 
  Latent 
  Heat. 
  I 
  have 
  now 
  deduced 
  it 
  from 
  the 
  hypothesis 
  on 
  which 
  

   my 
  researches 
  are 
  based 
  ; 
  — 
  not 
  in 
  order 
  to 
  prove 
  the 
  law,but 
  to 
  verify 
  the 
  correct- 
  

   ness 
  of 
  the 
  mode 
  of 
  investigation 
  which 
  I 
  have 
  followed. 
  

  

  Equations 
  (26) 
  and 
  (27), 
  like 
  equation 
  (23), 
  are 
  made 
  applicable 
  to 
  unity 
  of 
  

  

  weight 
  of 
  a 
  mixture, 
  by 
  putting 
  2 
  n 
  k 
  for 
  ft, 
  and 
  2 
  n 
  -^- 
  for 
  -^p 
  

  

  The 
  train 
  of 
  reasoning 
  in 
  this 
  article 
  is 
  the 
  converse 
  of 
  that 
  followed 
  by 
  Pro- 
  

   fessor 
  William 
  Thomson 
  of 
  Glasgow, 
  in 
  article 
  20 
  of 
  his 
  paper 
  on 
  the 
  Dynamical 
  

   Theory 
  of 
  Heat, 
  where 
  he 
  proves 
  from 
  Joule's 
  law, 
  that 
  the 
  quantity 
  correspond- 
  

   ing 
  to 
  d 
  y 
  is 
  an 
  exact 
  differential. 
  

  

  (11.) 
  Mutual 
  Conversion 
  of 
  Heat 
  and 
  Expansive 
  Power. 
  Carnot's 
  Law 
  of 
  the 
  

   Action 
  of 
  Expansive 
  Machines. 
  — 
  If 
  a 
  body 
  be 
  made 
  to 
  pass 
  from 
  the 
  volume 
  V 
  

   and 
  absolute 
  temperature 
  t 
  to 
  the 
  volume 
  V 
  1 
  and 
  absolute 
  temperature 
  r 
  v 
  and 
  be 
  

   then 
  brought 
  back 
  to 
  the 
  original 
  volume 
  and 
  temperature, 
  the 
  total 
  power 
  exerted 
  

  

  VOL. 
  XX. 
  PART 
  III. 
  6 
  B 
  

  

  