﻿570 
  MR 
  W. 
  J. 
  M. 
  RANKINE 
  ON 
  THE 
  

  

  (II.) 
  Let 
  the 
  body 
  further 
  expand 
  without 
  receiving 
  or 
  emitting 
  heat, 
  till 
  the 
  

   quantity 
  of 
  heat 
  in 
  it 
  falls 
  to 
  Q 
  2 
  ; 
  the 
  heat-potential 
  varying 
  according 
  to 
  Equation 
  

   77, 
  and 
  becoming 
  at 
  length 
  F 
  c 
  . 
  The 
  heat 
  converted 
  into 
  expansive 
  power 
  in 
  this 
  

   operation 
  is 
  

  

  Qt-Q 
  2 
  

  

  (III.) 
  Let 
  the 
  body 
  be 
  compressed, 
  at 
  the 
  constant 
  heat 
  Q 
  2 
  , 
  till 
  the 
  heat-poten- 
  

   tial 
  becomes 
  F 
  D 
  ; 
  a 
  quantity 
  differing 
  from 
  the 
  initial 
  heat-potential 
  F 
  A 
  by 
  as 
  much 
  

   as 
  F 
  c 
  differs 
  from 
  F 
  B 
  . 
  In 
  this 
  operation 
  the 
  following 
  amount 
  of 
  power 
  is 
  recon- 
  

   verted 
  into 
  heat, 
  and 
  given 
  out 
  by 
  conduction 
  : 
  — 
  

  

  H 
  2 
  = 
  Q 
  2 
  (F 
  C 
  -F 
  D 
  ) 
  

  

  (IV.) 
  Let 
  the 
  body 
  be 
  further 
  compressed, 
  till 
  the 
  heat-potential 
  returns 
  to 
  F 
  A 
  , 
  

   its 
  original 
  value. 
  Then, 
  by 
  the 
  power 
  expended 
  in 
  this 
  compression 
  alone, 
  with- 
  

   out 
  the 
  aid 
  of 
  conduction, 
  the 
  total 
  heat 
  of 
  the 
  body 
  will 
  be 
  restored 
  to 
  its 
  original 
  

   amount, 
  exactly 
  reversing 
  the 
  operation 
  II. 
  

  

  At 
  the 
  end 
  of 
  this 
  cycle 
  of 
  operations, 
  the 
  following 
  quantity 
  of 
  heat 
  will 
  have 
  

   been 
  converted 
  into 
  mechanical 
  power: 
  — 
  

  

  H^H^Q, 
  (F 
  B 
  -F 
  A 
  )-Q 
  2 
  (F 
  C 
  -F 
  D 
  ) 
  

   but 
  it 
  is 
  obvious 
  that 
  the 
  difference 
  between 
  the 
  heat-potentials 
  is 
  the 
  same 
  in 
  

   the 
  first 
  and 
  third 
  operations 
  ; 
  therefore, 
  the 
  useful 
  effect 
  is 
  simply 
  

  

  I^-H^CQ.-Q,) 
  (F 
  B 
  -F 
  A 
  ) 
  . 
  . 
  . 
  

   while 
  the 
  whole 
  heat 
  expended 
  is, 
  \ 
  (i$\ 
  

  

  H 
  1 
  = 
  Q 
  1 
  (F 
  B 
  -F 
  A 
  ) 
  .... 
  

  

  Hence, 
  the 
  ratio 
  of 
  the 
  heat 
  converted 
  into 
  mechanical 
  effect, 
  in 
  an 
  expan- 
  

   sive 
  machine 
  working 
  to 
  the 
  greatest 
  advantage, 
  to 
  the 
  whole 
  heat 
  expended, 
  is 
  the 
  

   same 
  with 
  that 
  which 
  the 
  difference 
  between 
  the 
  quantities 
  of 
  heat 
  possessed 
  by 
  the 
  

   expansive 
  body 
  during 
  the 
  operations 
  of 
  receiving 
  and 
  emitting 
  heat, 
  respectively, 
  

   bears 
  to 
  the 
  quantity 
  of 
  heat 
  possessed 
  by 
  it 
  during 
  the 
  operation 
  of 
  receiving 
  heat 
  ; 
  

   and 
  is 
  independent 
  of 
  the 
  nature 
  and 
  condition 
  of 
  the 
  body. 
  

  

  This 
  theorem 
  is 
  thus 
  expressed 
  symbolically, 
  — 
  

  

  Hi-H 
  2 
  _ 
  Effect 
  _ 
  Q 
  1 
  -Q 
  2 
  

  

  ^ 
  " 
  Heat 
  Expended 
  Q 
  x 
  ^' 
  y 
  ^ 
  

  

  (51.) 
  When 
  a 
  body 
  expands 
  without 
  meeting 
  with 
  resistance, 
  so 
  that 
  all 
  its 
  

   expansive 
  power 
  is 
  expended 
  in 
  giving 
  velocity 
  to 
  its 
  own 
  particles, 
  and 
  when 
  

   that 
  velocity 
  is 
  ultimately 
  extinguished 
  by 
  friction, 
  then 
  a 
  quantity 
  of 
  heat 
  equi- 
  

   valent 
  to 
  the 
  expansive 
  power 
  is 
  reproduced. 
  

  

  The 
  heat 
  consumed 
  is 
  expressed 
  by 
  taking 
  away 
  the 
  term 
  representing 
  the 
  

   expansive 
  power, 
  P 
  d 
  V, 
  from 
  the 
  expression 
  72, 
  the 
  remainder 
  of 
  which 
  consists 
  

   merely 
  of 
  the 
  variation 
  of 
  actual 
  heat, 
  and 
  the 
  heat 
  expended 
  in 
  overcoming 
  

   molecular 
  attraction, 
  viz. 
  : 
  — 
  

  

  