﻿MECHANICAL 
  ACTION 
  OF 
  HEAT. 
  569 
  

  

  (50.) 
  In 
  order 
  to 
  investigate 
  the 
  laws 
  according 
  to 
  which 
  heat 
  is 
  converted 
  

   into 
  mechanical 
  power, 
  in 
  a 
  machine 
  working 
  by 
  the 
  expansion 
  of 
  an 
  elastic 
  body, 
  

   it 
  will 
  be 
  convenient 
  to 
  use 
  a 
  function 
  

  

  F 
  = 
  /"|^dV(Q= 
  const.) 
  

  

  of 
  such 
  a 
  nature 
  that 
  the 
  difference 
  between 
  two 
  of 
  its 
  values, 
  corresponding 
  to 
  dif- 
  

   ferent 
  volumes 
  of 
  the 
  body 
  at 
  the 
  same 
  total 
  heat, 
  represents 
  the 
  ratio 
  of 
  the 
  heat 
  

   converted 
  into 
  power 
  by 
  expansion 
  between 
  those 
  volumes, 
  to 
  the 
  given 
  constant 
  

   total 
  heat. 
  I 
  shall 
  call 
  this 
  function 
  a 
  heat-potential. 
  

  

  Introducing 
  this 
  function 
  into 
  Equation 
  72, 
  we 
  find, 
  for 
  the 
  total 
  heat 
  con- 
  

   sumed 
  by 
  a 
  body 
  during 
  the 
  increase 
  of 
  total 
  heat 
  d 
  Q, 
  and 
  the 
  expansion 
  d 
  V, 
  

  

  dQ 
  + 
  d.S 
  + 
  ~PdV 
  = 
  (l+<p'.(Q)\ 
  dQ 
  + 
  Qd.F 
  .... 
  (76.) 
  

  

  (obsemiigthatd.F=||dQ 
  + 
  ^dV=(/|^dV.)dQ 
  + 
  ||dV.) 
  

  

  Let 
  us 
  now 
  suppose 
  that 
  the 
  body 
  changes 
  its 
  volume 
  without 
  either 
  losing 
  

   or 
  gaining 
  heat 
  by 
  conduction. 
  This 
  condition 
  is 
  expressed 
  by 
  the 
  equation 
  

  

  = 
  (l 
  + 
  (j)' 
  .Q)dQ 
  + 
  Qd.F 
  

   from 
  which 
  we 
  deduce 
  the 
  following, 
  

  

  -d.F=^M.dq 
  .... 
  (77.) 
  

  

  which 
  expresses 
  the 
  following 
  theorem 
  : 
  — 
  

  

  When 
  the 
  quantity 
  of 
  heat 
  in 
  a 
  body 
  is 
  varied, 
  by 
  variation 
  of 
  volume 
  only, 
  the 
  

   variation 
  of 
  the 
  heat-potential 
  depends 
  on 
  the 
  heat 
  only, 
  and 
  is 
  independent 
  of 
  the 
  

   volume. 
  

  

  In 
  order 
  that 
  a 
  machine 
  working 
  by 
  the 
  expansive 
  power 
  of 
  heat 
  may 
  produce 
  

   its 
  greatest 
  effect, 
  all 
  the 
  heat 
  communicated 
  from 
  external 
  bodies 
  should 
  be 
  em- 
  

   ployed 
  in 
  producing 
  expansive 
  power, 
  and 
  none 
  in 
  producing 
  variations 
  of 
  the 
  

   quantity 
  of 
  heat 
  in 
  the 
  body 
  ; 
  for 
  heat 
  employed 
  for 
  the 
  latter 
  purpose 
  would 
  be 
  

   wasted, 
  so 
  far 
  as 
  the 
  production 
  of 
  visible 
  motion 
  is 
  concerned. 
  To 
  effect 
  this, 
  

   the 
  body 
  must 
  receive 
  heat 
  by 
  conduction, 
  and 
  convert 
  it 
  into 
  expansive 
  power, 
  

   while 
  containing 
  a 
  certain 
  constant 
  quantity 
  of 
  heat 
  Q, 
  ; 
  give 
  out 
  by 
  conduction 
  

   heat 
  produced 
  by 
  compression, 
  while 
  containing 
  a 
  smaller 
  constant 
  quantity 
  of 
  

   heat 
  Q, 
  ; 
  and 
  change 
  between 
  those 
  two 
  quantities 
  of 
  thermometric 
  heat 
  by 
  

   means 
  of 
  changes 
  of 
  volume 
  only, 
  without 
  conduction. 
  For 
  this 
  purpose 
  a 
  cycle 
  

   of 
  operations 
  must 
  be 
  performed 
  similar 
  to 
  that 
  described 
  by 
  Carnot 
  ; 
  as 
  fol- 
  

   lows: 
  — 
  

  

  (I.) 
  Let 
  F 
  A 
  be 
  the 
  initial 
  value 
  of 
  the 
  heat-potential 
  ; 
  let 
  the 
  body 
  expand 
  at 
  

   the 
  constant 
  heat 
  Q, 
  i 
  , 
  till 
  the 
  heat-potential 
  becomes 
  F 
  B 
  . 
  Then 
  the 
  heat 
  received 
  

   and 
  converted 
  into 
  expansive 
  power 
  is 
  

  

  B. 
  1 
  = 
  q 
  i 
  (F 
  B 
  -F 
  A 
  ) 
  

  

  VOL. 
  XX. 
  PART 
  IV. 
  7 
  P 
  

  

  