﻿162 
  MR 
  W. 
  J. 
  M. 
  RANKINE 
  ON 
  THE 
  

  

  Suppose 
  a 
  portion 
  of 
  any 
  substance, 
  of 
  the 
  weight 
  unity, 
  to 
  pass 
  through 
  a 
  

   variety 
  of 
  changes 
  of 
  temperature 
  and 
  volume, 
  and 
  at 
  length 
  to 
  be 
  brought 
  back 
  

   to 
  its 
  primitive 
  volume 
  and 
  temperature. 
  Then 
  the 
  absolute 
  quantity 
  of 
  heat 
  in 
  

   the 
  substance, 
  and 
  the 
  molecular 
  arrangement 
  and 
  distribution, 
  being 
  the 
  same 
  

   as 
  at 
  first, 
  the 
  effect 
  of 
  their 
  changes 
  is 
  eliminated 
  ; 
  and 
  the 
  algebraical 
  sum 
  of 
  the 
  

   vis 
  viva 
  expended 
  and 
  produced, 
  whether 
  in 
  the 
  shape 
  of 
  expansion 
  and 
  compression, 
  

   or 
  in 
  that 
  of 
  heat, 
  must 
  be 
  equal 
  to 
  zero 
  : 
  — 
  that 
  is 
  to 
  say, 
  if, 
  on 
  the 
  whole, 
  any 
  

   mechanical 
  power 
  has 
  appeared, 
  and 
  been 
  given 
  out 
  from 
  the 
  body, 
  in 
  the 
  form 
  

   of 
  expansion, 
  an 
  equal 
  amount 
  must 
  have 
  been 
  communicated 
  to 
  the 
  bodj', 
  and 
  

   must 
  have 
  disappeared 
  in 
  the 
  form 
  of 
  heat 
  ; 
  and 
  if 
  any 
  mechanical 
  power 
  has 
  

   appeared 
  and 
  been 
  given 
  out 
  from 
  the 
  body 
  in 
  the 
  form 
  of 
  heat, 
  an 
  equal 
  

   amount 
  must 
  have 
  been 
  communicated 
  to 
  the 
  body, 
  and 
  must 
  have 
  disappeared 
  

   in 
  the 
  form 
  of 
  compression. 
  This 
  principle 
  expressed 
  symbolically 
  is 
  

  

  An+AQ' 
  = 
  (8.) 
  

  

  Where 
  n, 
  when 
  positive, 
  represents 
  expansive 
  power 
  given 
  out, 
  when 
  negative, 
  

   compressive 
  power 
  absorbed 
  ; 
  and 
  Q,' 
  represents, 
  when 
  positive, 
  heat 
  given 
  out. 
  

   when 
  negative, 
  heat 
  absorbed. 
  

  

  To 
  take 
  the 
  simplest 
  case 
  possible, 
  let 
  the 
  changes 
  of 
  temperature 
  and 
  of 
  

   volume 
  be 
  supposed 
  to 
  be 
  indefinitely 
  small, 
  and 
  to 
  occur 
  during 
  distinct 
  intervals 
  

   of 
  time, 
  so 
  that 
  t 
  and 
  V 
  are 
  independent 
  variables. 
  Let 
  the 
  initial 
  absolute 
  tem- 
  

   perature 
  be 
  t, 
  the 
  initial 
  volume 
  V, 
  and 
  the 
  initial 
  total 
  elasticity 
  P 
  ; 
  and 
  let 
  the 
  

   substance 
  go 
  through 
  the 
  following 
  four 
  changes. 
  

  

  First, 
  Let 
  its 
  temperature 
  be 
  raised 
  from 
  t 
  to 
  t 
  + 
  St, 
  the 
  volume 
  remaining 
  

   unchanged. 
  Then 
  the 
  quantity 
  of 
  heat 
  absorbed 
  is 
  

  

  d 
  Q 
  t 
  — 
  k 
  d 
  U> 
  

  

  $■ 
  (d 
  Q 
  t 
  — 
  k 
  riU\ 
  

   T 
  \dr~GnM. 
  ~dr~) 
  

  

  and 
  there 
  is 
  no 
  expansion 
  nor 
  compression. 
  

  

  Secondly, 
  Let 
  the 
  body 
  expand, 
  without 
  change 
  of 
  temperature, 
  from 
  the 
  

   volume 
  V 
  to 
  the 
  volume 
  V 
  + 
  8 
  V. 
  Then 
  the 
  quantity 
  of 
  heat 
  absorbed 
  is 
  

  

  T 
  + 
  8 
  T 
  — 
  K 
  

  

  s. 
  t+Ot—k/1 
  d 
  /TT 
  dU 
  j 
  A 
  

  

  ■ 
  8Y 
  --^ 
  f 
  rw{v-dY^ 
  v+ 
  dV 
  dT) 
  ) 
  

  

  while 
  the 
  power 
  given 
  out 
  by 
  expansion 
  is 
  

  

  dr 
  

  

  Thirdly, 
  Let 
  the 
  temperature 
  fall 
  from 
  t 
  + 
  £t 
  to 
  its 
  original 
  value 
  r, 
  the 
  

   volume 
  V 
  + 
  8 
  V 
  continuing 
  unchanged 
  ; 
  then 
  the 
  heat 
  given 
  out 
  is 
  

  

  «, 
  Id 
  Q 
  t 
  — 
  k 
  d 
  T 
  <ZUft„ 
  N 
  \ 
  

   + 
  8T 
  W-(^Mdr( 
  V 
  + 
  dV 
  8Y 
  n 
  

  

  and 
  there 
  is 
  no 
  expansion 
  nor 
  compression. 
  

  

  