﻿MECHANICAL 
  ACTION 
  OF 
  HEAT. 
  567 
  

  

  small 
  quantity 
  d 
  Q. 
  Then 
  the 
  mechanical 
  power 
  of 
  expansion 
  P 
  d 
  V 
  will 
  vary 
  

   by 
  the 
  indefinitely 
  small 
  quantity 
  

  

  d 
  P 
  

  

  dQ 
  x 
  -j-^dY 
  

   d 
  m 
  

  

  This 
  is 
  the 
  development 
  of 
  power 
  for 
  the 
  expansion 
  d 
  V, 
  caused 
  by 
  each 
  indefi- 
  

   nitely 
  small 
  portion 
  d 
  Q 
  of 
  the 
  total 
  heat 
  possessed 
  by 
  the 
  body 
  ; 
  and 
  conse- 
  

   quently, 
  the 
  whole 
  mechanical 
  power 
  for 
  the 
  expansion 
  d 
  V 
  due 
  to 
  the 
  whole 
  heat 
  

   possessed 
  by 
  the 
  body 
  Q, 
  is 
  expressed 
  as 
  follows 
  : 
  — 
  

  

  ^■ 
  dV 
  • 
  (WO 
  

  

  and 
  this 
  is 
  the 
  equivalent 
  of 
  the 
  heat 
  transformed 
  into 
  mechanical 
  power, 
  or 
  the 
  

   latent 
  heat 
  of 
  expansion 
  of 
  unity 
  of 
  weight, 
  for 
  the 
  small 
  increment 
  of 
  volume 
  

   d 
  V, 
  at 
  the 
  volume 
  V 
  and 
  total 
  heat 
  Q. 
  

   Now 
  a 
  part 
  only 
  of 
  this 
  power, 
  viz. 
  — 
  

  

  PdV 
  

  

  is 
  visible 
  mechanical 
  energy, 
  expended 
  in 
  producing 
  velocity 
  in 
  the 
  expanding 
  

   body 
  itself, 
  or 
  in 
  overcoming 
  the 
  resistance 
  of 
  the 
  bodies 
  which 
  enclose 
  it. 
  The 
  

   remainder 
  

  

  0»?f- 
  p 
  ) 
  ,,v 
  •■•••• 
  (68.) 
  

  

  is 
  therefore 
  expended 
  in 
  overcoming 
  molecular 
  attraction. 
  

  

  Molecular 
  attraction 
  depends 
  on 
  the 
  density 
  and 
  distribution 
  of 
  the 
  particles 
  

   of 
  the 
  body 
  ; 
  and 
  is 
  consequently 
  a 
  function 
  of 
  the 
  volume 
  and 
  total 
  heat 
  of 
  unity 
  

   of 
  weight. 
  It 
  is 
  therefore 
  possible 
  to 
  find 
  a 
  potential 
  S, 
  being 
  a 
  function 
  of 
  V 
  and 
  

   Q, 
  of 
  such 
  a 
  nature, 
  that 
  the 
  difference 
  between 
  its 
  two 
  values 
  

  

  S 
  2 
  — 
  S 
  x 
  

   corresponding 
  respectively 
  to 
  two 
  sets 
  of 
  values 
  of 
  the 
  volume 
  and 
  total 
  heat 
  

   (V 
  1? 
  Q 
  x 
  and 
  V 
  2 
  , 
  Q 
  2 
  ), 
  shall 
  represent 
  the 
  power 
  which 
  is 
  the 
  equivalent 
  of 
  the 
  heat 
  

   consumed 
  in 
  overcoming 
  molecular 
  attraction, 
  during 
  the 
  passage 
  of 
  the 
  body 
  

   from 
  the 
  volume 
  Y 
  1 
  and 
  heat 
  Q 
  x 
  to 
  the 
  volume 
  V 
  2 
  and 
  heat 
  Q 
  2 
  . 
  The 
  form 
  of 
  the 
  

   expression 
  (68) 
  shews 
  that 
  this 
  potential 
  has 
  the 
  following 
  property 
  : 
  — 
  

  

  H=4q- 
  p 
  • 
  •. 
  <w 
  

  

  The 
  integration 
  of 
  which 
  partial 
  differential 
  equation 
  gives 
  the 
  following 
  value 
  

   for 
  the 
  potential 
  of 
  molecular 
  action 
  : 
  — 
  

  

  s= 
  /( 
  Q 
  ^- 
  p 
  ) 
  dV 
  + 
  * 
  (Q) 
  (7a) 
  

  

  (p 
  (Q) 
  being 
  some 
  unknown 
  function 
  of 
  the 
  heat 
  only, 
  and 
  the 
  integral 
  being 
  taken 
  

   as 
  if 
  the 
  heat 
  Q 
  were 
  constant. 
  

  

  The 
  heat 
  which 
  disappears 
  in 
  overcoming 
  molecular 
  action, 
  during 
  a 
  small 
  

  

  