﻿206 
  MR 
  W. 
  J. 
  M. 
  RANKINE 
  ON 
  THE 
  ECONOMY 
  OF 
  

  

  the 
  method 
  by 
  which 
  he 
  proves 
  Carnot's 
  law, 
  I 
  have 
  received 
  from 
  him 
  a 
  state- 
  

   ment 
  of 
  some 
  of 
  his 
  more 
  important 
  results. 
  

  

  (42.) 
  I 
  have 
  now 
  come 
  to 
  the 
  conclusions, 
  — 
  First 
  : 
  That 
  Carnot's 
  Law 
  is 
  not 
  

   an 
  independent 
  principle 
  in 
  the 
  theory 
  of 
  heat 
  ; 
  but 
  is 
  deducible, 
  as 
  a 
  consequence, 
  

   from 
  the 
  equations 
  of 
  the 
  mutual 
  conversion 
  of 
  heat 
  and 
  expansive 
  power, 
  as 
  given 
  

   in 
  the 
  First 
  Section 
  of 
  this 
  paper. 
  

  

  Secondly 
  : 
  That 
  the 
  function 
  of 
  the 
  temperatures 
  of 
  reception 
  and 
  emission, 
  

   which 
  expresses 
  the 
  maximum 
  ratio 
  of 
  the 
  heat 
  converted 
  into 
  power 
  to 
  the 
  total 
  heat 
  

   received 
  by 
  the 
  working 
  body, 
  is 
  the 
  ratio 
  of 
  the 
  difference 
  of 
  those 
  temperatures, 
  to 
  

   the 
  absolute 
  temperature 
  of 
  reception 
  diminished 
  by 
  the 
  constant, 
  which 
  I 
  have 
  

   called 
  Kz=Cnfjib, 
  and 
  which 
  must, 
  as 
  I 
  have 
  shewn 
  in 
  the 
  Introduction, 
  be 
  the 
  

   same 
  for 
  all 
  substances, 
  in 
  order 
  that 
  molecular 
  equilibrium 
  may 
  be 
  possible. 
  

  

  (43.) 
  Let 
  abscissae, 
  parallel 
  to 
  OX 
  in 
  the 
  diagram, 
  Plate 
  VIII. 
  fig. 
  2, 
  denote 
  

   the 
  volumes 
  successively 
  assumed 
  by 
  the 
  working 
  body, 
  and 
  ordinates, 
  parallel 
  

   to 
  OY, 
  the 
  corresponding 
  pressures. 
  Let 
  t 
  x 
  be 
  the 
  constant 
  absolute 
  temperature 
  

   at 
  which 
  the 
  reception 
  of 
  heat 
  by 
  the 
  body 
  takes 
  place 
  : 
  t 
  , 
  the 
  constant 
  absolute 
  

   temperature 
  at 
  which 
  the 
  emission 
  of 
  heat 
  takes 
  place. 
  Let 
  AB 
  be 
  a 
  curve 
  such 
  

   that 
  its 
  ordinates 
  denote 
  the 
  pressures, 
  at 
  the 
  temperature 
  of 
  reception 
  t 
  v 
  cor- 
  

   responding 
  to 
  the 
  volumes 
  denoted 
  by 
  abscissae. 
  Let 
  DC 
  be 
  a 
  similar 
  curve 
  for 
  

   the 
  temperature 
  of 
  emission 
  t 
  . 
  Let 
  AD 
  and 
  BC 
  be 
  two 
  curves, 
  expressing 
  by 
  

   their 
  co-ordinates 
  how 
  the 
  pressure 
  and 
  volume 
  must 
  vary, 
  in 
  order 
  that 
  the 
  

   body 
  may 
  change 
  its 
  temperature, 
  without 
  receiving 
  or 
  emitting 
  heat 
  ; 
  the 
  former 
  

   corresponding 
  to 
  the 
  most 
  condensed 
  and 
  the 
  latter 
  to 
  the 
  most 
  expanded 
  state 
  

   of 
  the 
  body, 
  during 
  the 
  working 
  of 
  the 
  machine. 
  

  

  The 
  quantity 
  of 
  heat 
  received 
  or 
  emitted 
  during 
  an 
  operation 
  on 
  the 
  body 
  

   involving 
  indefinitely 
  small 
  variations 
  of 
  volume 
  and 
  temperature, 
  is 
  expressed 
  

   by 
  adding 
  to 
  Equation 
  (6.) 
  of 
  Section 
  Fourth 
  the 
  heat 
  due 
  to 
  change 
  of 
  tempera- 
  

   ture 
  only, 
  in 
  virtue 
  of 
  the 
  real 
  specific 
  heat. 
  We 
  thus 
  obtain 
  the 
  differential 
  

   equation 
  

  

  ^-^=- 
  c 
  ^ 
  v 
  (t-^H-"} 
  

  

  -ft 
  St 
  

   In 
  which 
  the 
  negative 
  sign 
  denotes 
  absorption, 
  and 
  the 
  positive 
  emission. 
  

  

  If 
  we 
  now 
  put 
  for 
  jy, 
  -^, 
  their 
  values 
  according 
  to 
  Equation 
  (11.), 
  we 
  find 
  

  

  -Kc^-S)^-*)^/^}^ 
  • 
  W 
  

  

  The 
  first 
  term 
  represents 
  the 
  variation 
  of 
  heat 
  due 
  to 
  variation 
  of 
  volume 
  

   only 
  ; 
  the 
  second, 
  that 
  due 
  to 
  variation 
  of 
  temperature. 
  Let 
  us 
  now 
  apply 
  this 
  

  

  