﻿DYNAMICAL 
  THEORY 
  OF 
  HEAT. 
  269 
  

  

  A 
  and 
  B 
  . 
  It 
  is 
  worthy 
  of 
  remark 
  that 
  these 
  propositions 
  are 
  rigorously 
  true, 
  

   being 
  demonstrable 
  consequences 
  of 
  the 
  fundamental 
  principle 
  of 
  the 
  dynamical 
  

   theory 
  of 
  heat, 
  which 
  have 
  been 
  discovered 
  by 
  Joule, 
  and 
  illustrated 
  and 
  verified 
  

   most 
  copiously 
  in 
  his 
  experimental 
  researches. 
  

  

  19. 
  Both 
  the 
  fundamental 
  propositions 
  may 
  be 
  applied 
  in 
  a 
  perfectly 
  rigorous 
  

   manner 
  to 
  the 
  second 
  of 
  the 
  known 
  methods 
  of 
  producing 
  mechanical 
  effect 
  from 
  

   thermal 
  agency. 
  This 
  application 
  of 
  the 
  first 
  of 
  the 
  two 
  fundamental 
  propositions 
  

   has 
  already 
  been 
  published 
  by 
  Rankine 
  and 
  Clausius 
  ; 
  and 
  that 
  of 
  the 
  second, 
  

   as 
  Clausius 
  shewed 
  in 
  his 
  published 
  paper, 
  is 
  simply 
  Carnot's 
  unmodified 
  inves- 
  

   tigation 
  of 
  the 
  relation 
  between 
  the 
  mechanical 
  effect 
  produced 
  and 
  the 
  thermal 
  

   circumstances 
  from 
  which 
  it 
  originates, 
  in 
  the 
  case 
  of 
  an 
  expansive 
  engine 
  work- 
  

   ing 
  within 
  an 
  infinitely 
  small 
  range 
  of 
  temperatures. 
  The 
  simplest 
  investigation 
  

   of 
  the 
  consequences 
  of 
  the 
  first 
  proposition 
  in 
  this 
  application, 
  which 
  has 
  occurred 
  

   to 
  me, 
  is 
  the 
  following, 
  being 
  merely 
  the 
  modification 
  of 
  an 
  analytical 
  expression 
  

   of 
  Carnot's 
  axiom 
  regarding 
  the 
  permanence 
  of 
  heat, 
  which 
  was 
  given 
  in 
  my 
  

   former 
  paper,* 
  required 
  to 
  make 
  it 
  express, 
  not 
  Carnot's 
  axiom, 
  but 
  Joule's. 
  

  

  20. 
  Let 
  us 
  suppose 
  a 
  massf 
  of 
  any 
  substance, 
  occupying 
  a 
  volume 
  v, 
  under 
  a 
  

  

  pressure 
  p 
  uniform 
  in 
  all 
  directions, 
  and 
  at 
  a 
  temperature 
  t, 
  to 
  expand 
  in 
  volume 
  

  

  to 
  v 
  + 
  d 
  v, 
  and 
  to 
  rise 
  in 
  temperature 
  to 
  t 
  + 
  d 
  t. 
  The 
  quantity 
  of 
  work 
  which 
  it 
  

  

  will 
  produce 
  will 
  be 
  

  

  p 
  d 
  v 
  ; 
  

  

  and 
  the 
  quantity 
  of 
  heat 
  which 
  must 
  be 
  added 
  to 
  it 
  to 
  make 
  its 
  temperature 
  rise 
  

   during 
  the 
  expansion 
  to 
  t 
  + 
  d 
  t 
  may 
  be 
  denoted 
  by 
  

  

  M 
  dv 
  + 
  N 
  tf*. 
  

   The 
  mechanical 
  equivalent 
  of 
  this 
  is 
  

  

  J 
  (Mdv 
  + 
  Nd<), 
  

  

  if 
  J 
  denote 
  the 
  mechanical 
  equivalent 
  of 
  a 
  unit 
  of 
  heat. 
  Hence 
  the 
  mechanical 
  

   measure 
  of 
  the 
  total 
  external 
  effect 
  produced 
  in 
  the 
  circumstances 
  is 
  

  

  (p- 
  JW)dv- 
  JNtf*. 
  

  

  The 
  total 
  external 
  effect, 
  after 
  any 
  finite 
  amount 
  of 
  expansion, 
  accompanied 
  by 
  

   any 
  continuous 
  change 
  of 
  temperature, 
  has 
  taken 
  place, 
  will 
  consequently 
  be, 
  in 
  

   mechanical 
  terms, 
  

  

  JM) 
  dv 
  - 
  JN 
  dt}; 
  

  

  fiiP- 
  

  

  where 
  we 
  must 
  suppose 
  t 
  to 
  vary 
  with 
  v, 
  so 
  as 
  to 
  be 
  the 
  actual 
  temperature 
  of 
  

   the 
  medium 
  at 
  each 
  instant, 
  and 
  the 
  integration 
  with 
  reference 
  to 
  v 
  must 
  be 
  per- 
  

   formed 
  between 
  limits 
  corresponding 
  to 
  the 
  initial 
  and 
  final 
  volumes. 
  Now 
  if, 
  at 
  

   any 
  subsequent 
  time, 
  the 
  volume 
  and 
  temperature 
  of 
  the 
  medium 
  become 
  what 
  

  

  * 
  " 
  Account 
  of 
  Carnot's 
  Theory," 
  foot-note 
  on 
  § 
  26. 
  

  

  I 
  This 
  may 
  have 
  parts 
  consisting 
  of 
  different 
  substances, 
  or 
  of 
  the 
  same 
  substance 
  in 
  different 
  

   states, 
  provided 
  the 
  temperature 
  of 
  all 
  be 
  the 
  same. 
  See 
  below 
  Part 
  III., 
  §§ 
  53-56. 
  

  

  VOL. 
  XX. 
  PART 
  II. 
  4 
  D 
  

  

  