﻿MECHANICAL 
  ACTION 
  OF 
  HEAT. 
  

  

  161 
  

  

  and 
  consequently 
  that 
  

  

  — 
  3 
  / 
  d 
  u 
  . 
  u8u-^(u,1>,t) 
  = 
  

   Hence, 
  making 
  

  

  9 
  r 
  xd 
  ~ 
  n 
  du 
  . 
  u 
  2 
  ^(u,d,t) 
  = 
  v 
  

  

  Jo 
  u 
  Jo 
  

  

  u 
  . 
  v? 
  -^ 
  (u, 
  J), 
  t) 
  

  

  (5.) 
  

  

  The 
  second 
  integral 
  in 
  Equation 
  (2.) 
  is 
  transformed 
  into 
  

  

  d 
  *. 
  -~ 
  d 
  

  

  + 
  

  

  l(' 
  T 
  £ 
  +an 
  ro) 
  

  

  u. 
  

  

  By 
  means 
  of 
  those 
  substitutions 
  we 
  obtain, 
  for 
  the 
  mechanical 
  value 
  of 
  the 
  

   heat 
  developed 
  in 
  unity 
  of 
  weight 
  of 
  a 
  fluid 
  by 
  any 
  indefinitely 
  small 
  change 
  of 
  

   volume 
  or 
  of 
  molecular 
  distribution 
  — 
  

  

  or 
  taking 
  V= 
  -^ 
  to 
  denote 
  the 
  volume 
  of 
  unity 
  of 
  weight 
  of 
  

   the 
  substance, 
  

  

  (6.) 
  

  

  Of 
  this 
  expression, 
  the 
  portion 
  rf^ 
  ■ 
  ~Tr 
  =: 
  ~ 
  c~M 
  ' 
  ~V~ 
  re 
  P 
  resen 
  * 
  s 
  * 
  ne 
  va_ 
  

   riation 
  of 
  heat 
  arising 
  from 
  mere 
  change 
  of 
  volume. 
  

  

  ^ 
  — 
  =^ 
  8 
  V 
  -py 
  = 
  ~ 
  — 
  ^ 
  8 
  D 
  j-jt, 
  denotes 
  the 
  variation 
  of 
  heat 
  produced 
  by 
  change 
  

   of 
  molecular 
  distribution 
  dependent 
  on 
  change 
  of 
  volume. 
  

  

  8 
  t 
  -=— 
  expresses 
  the 
  variation 
  of 
  heat 
  due 
  to 
  change 
  of 
  molecular 
  dis- 
  

  

  C 
  n 
  M 
  w 
  ' 
  d 
  t 
  

   tribution 
  dependent 
  on 
  change 
  of 
  temperature. 
  

  

  (7.) 
  The 
  function 
  U 
  is 
  one 
  depending 
  on 
  molecular 
  forces, 
  the 
  nature 
  of 
  

   which 
  is 
  as 
  yet 
  unknown. 
  The 
  only 
  case 
  in 
  which 
  it 
  can 
  be 
  calculated 
  directly 
  is 
  

   that 
  of 
  a 
  perfect 
  gas. 
  Without 
  giving 
  the 
  details 
  of 
  the 
  integration, 
  it 
  may 
  be 
  

   sufficient 
  to 
  state, 
  that 
  in 
  this 
  case 
  

  

  and 
  therefore 
  that 
  

  

  dU 
  

   d 
  t 
  

  

  T 
  

  

  .2 
  ' 
  

  

  dV 
  

   dY 
  

  

  = 
  

  

  (7-) 
  

  

  In 
  all 
  other 
  cases, 
  however, 
  the 
  value 
  of 
  this 
  function 
  can 
  be 
  determined 
  

   indirectly, 
  by 
  introducing 
  into 
  the 
  investigation 
  the 
  principle 
  of 
  the 
  conservation 
  

   of 
  vis 
  viva.' 
  

  

  