﻿438 
  MR 
  W. 
  J. 
  M. 
  RANKINE 
  ON 
  THE 
  CENTRIFUGAL 
  THEORY 
  OF 
  ELASTICITY, 
  

  

  (y) 
  will 
  have, 
  in 
  those 
  two 
  operations, 
  equal 
  arithmetical 
  values, 
  of 
  opposite 
  signs. 
  

   Each 
  of 
  the 
  quantities 
  y 
  consists 
  partly 
  of 
  heat 
  and 
  partly 
  of 
  expansive 
  power, 
  

   the 
  proportion 
  depending 
  on 
  the 
  mode 
  of 
  intermediate 
  variation 
  of 
  the 
  volume 
  

   and 
  temperature, 
  which 
  is 
  arbitrary. 
  If 
  the 
  mode 
  of 
  variation 
  be 
  different 
  in 
  the 
  

   two 
  operations, 
  the 
  effect 
  of 
  the 
  double 
  operation 
  will 
  be 
  to 
  transform 
  a 
  portion 
  

   of 
  heat 
  into 
  expansive 
  power, 
  or 
  vice 
  versa. 
  

  

  Let 
  (a) 
  denote 
  the 
  first 
  operation 
  : 
  (b) 
  the 
  reverse 
  of 
  the 
  second. 
  Then 
  

  

  The 
  terms 
  of 
  y 
  which 
  involve 
  functions 
  of 
  t 
  only, 
  or 
  of 
  V 
  only, 
  are 
  not 
  affected 
  

   by 
  the 
  mode 
  of 
  intermediate 
  variation 
  of 
  those 
  quantities. 
  The 
  term 
  on 
  which 
  the 
  

   mutual 
  conversion 
  of 
  heat 
  and 
  expansive 
  power 
  depends, 
  is 
  therefore 
  

  

  J{r 
  -K) 
  d 
  ^dV 
  (6) 
  =f(r 
  -K) 
  d 
  / 
  r 
  dV 
  (a) 
  

   Hence, 
  

  

  y|| 
  dY 
  («) 
  -f*ft 
  dV(b) 
  =f 
  P 
  dV{a) 
  -fpdV 
  

  

  (*) 
  

  

  which 
  last 
  quantity 
  is 
  the 
  amount 
  of 
  the 
  heat 
  transformed 
  into 
  expansive 
  power, 
  

   or 
  the 
  total 
  latent 
  heat 
  of 
  expansion 
  in 
  the 
  double 
  operation. 
  

  

  Let 
  

  

  Then 
  because 
  

  

  r*p 
  d 
  v= 
  f-±~ 
  .^rfv=F 
  

  

  J 
  dr 
  J 
  r—K 
  d\ 
  

  

  d 
  ^dV=(r-K)dF 
  

  

  we 
  have 
  

  

  V 
  V 
  T 
  1 
  Th 
  1 
  

  

  fp 
  l 
  dV 
  (fl)-AW 
  (b) 
  = 
  f(T-K) 
  d¥ 
  (a)-f{r-K) 
  d¥(b) 
  

  

  J 
  y 
  J 
  y 
  J 
  *o 
  J 
  ?o 
  

  

  = 
  f(r 
  a 
  ~T 
  b 
  )dF=f 
  T^Zln^LdV 
  .... 
  (28.) 
  

  

  F 
  u 
  V 
  T 
  a 
  — 
  K 
  dV 
  

  

  In 
  which 
  r 
  a 
  and 
  t 
  6 
  are 
  the 
  pair 
  of 
  absolute 
  temperatures, 
  in 
  the 
  two 
  operations 
  

   respectively, 
  corresponding 
  to 
  equal 
  values 
  o/F. 
  

  

  This 
  equation 
  gives 
  a 
  relation 
  between 
  the 
  heat 
  transformed 
  into 
  expansive 
  

   power 
  by 
  a 
  given 
  pair 
  of 
  operations 
  on 
  a 
  body, 
  the 
  latent 
  heat 
  of 
  expansion 
  in 
  

   the 
  first 
  operation, 
  and 
  the 
  mode 
  of 
  variation 
  of 
  temperature 
  in 
  the 
  two 
  opera- 
  

   tions. 
  It 
  shews 
  that 
  the 
  proportion 
  of 
  the 
  original 
  latent 
  heat 
  of 
  expansion 
  finally 
  

   transformed 
  into 
  expansive 
  power, 
  is 
  a 
  function 
  of 
  the 
  temperatures 
  alone, 
  and 
  

   is 
  therefore 
  independent 
  of 
  the 
  nature 
  of 
  the 
  body 
  employed. 
  

  

  Equation 
  (28) 
  includes 
  Carnot's 
  law 
  as 
  a 
  particular 
  case. 
  Let 
  the 
  limits 
  of 
  

  

  