﻿DYNAMICAL 
  THEORY 
  OF 
  HEAT. 
  479 
  

  

  de 
  J 
  dp 
  

  

  Tv 
  = 
  JxTt~ 
  p 
  < 
  8 
  > 
  

  

  The 
  integration 
  of 
  this 
  equation 
  with 
  reference 
  to 
  v, 
  leads 
  to 
  an 
  expression 
  for 
  e, 
  

   involving 
  an 
  arbitrary 
  function 
  of 
  t, 
  for 
  the 
  determination 
  of 
  which 
  more 
  data 
  

   from 
  experiment 
  are 
  required. 
  It 
  would, 
  for 
  instance, 
  be 
  sufficient 
  for 
  this 
  pur- 
  

   pose, 
  to 
  have 
  the 
  mechanical 
  energy 
  of 
  the 
  fluid 
  for 
  all 
  temperatures 
  when 
  con- 
  

   tained 
  in 
  a 
  constant 
  volume 
  ; 
  or, 
  what 
  amounts 
  to 
  the 
  same 
  (it 
  being 
  now 
  supposed 
  

   that 
  J 
  is 
  known), 
  to 
  have 
  the 
  thermal 
  capacity 
  of 
  the 
  fluid 
  in 
  constant 
  volume, 
  for 
  

   a 
  particular 
  volume 
  and 
  all 
  temperatures. 
  Hence, 
  we 
  conclude, 
  that 
  when 
  the 
  

   elements 
  J 
  and 
  p. 
  belonging 
  to 
  the 
  general 
  theory 
  of 
  the 
  mechanical 
  action 
  of 
  heat 
  

   are 
  known, 
  the 
  mechanical 
  energy 
  of 
  a 
  particular 
  fluid 
  may 
  be 
  investigated 
  with- 
  

   out 
  experiment, 
  from 
  determinations 
  of 
  its 
  pressure 
  for 
  all 
  temperatures 
  and 
  

   volumes, 
  and 
  its 
  thermal 
  capacity 
  for 
  any 
  particular 
  constant 
  volume 
  and 
  all 
  tem- 
  

   peratures. 
  

  

  90. 
  For 
  example, 
  let 
  the 
  fluid 
  be 
  atmospheric 
  air, 
  or 
  any 
  other 
  subject 
  to 
  the 
  

   " 
  gaseous" 
  laws. 
  Then 
  if 
  % 
  be 
  the 
  volume 
  of 
  a 
  unit 
  of 
  weight 
  of 
  the 
  fluid, 
  and 
  

   the 
  temperature, 
  in 
  the 
  standard 
  state 
  from 
  which 
  the 
  mechanical 
  energy 
  in 
  any 
  

   other 
  state 
  is 
  reckoned, 
  and 
  if 
  p 
  Q 
  denote 
  the 
  corresponding 
  pressure, 
  we 
  have 
  

  

  r 
  v 
  v 
  'at 
  v 
  

  

  and 
  f 
  v 
  ^Tt- 
  p 
  ) 
  dv 
  = 
  p 
  » 
  v 
  » 
  {^r-a+ 
  E 
  ')}iog^ 
  

  

  Hence, 
  if 
  we 
  denote 
  by 
  N 
  the 
  value 
  of 
  N 
  when 
  v=v 
  , 
  whatever 
  be 
  the 
  tempera- 
  

   ture, 
  we 
  have, 
  as 
  the 
  general 
  expression 
  for 
  the 
  mechanical 
  energy 
  of 
  a 
  unit 
  weight 
  

   of 
  a 
  fluid 
  subject 
  to 
  the 
  gaseous 
  laws, 
  

  

  e=P 
  v 
  ^-(l 
  + 
  Et)]\og^- 
  + 
  jf^ 
  dt 
  . 
  . 
  . 
  . 
  (9). 
  

  

  91. 
  Let 
  us 
  now 
  suppose 
  the 
  mechanical 
  energy 
  of 
  a 
  particular 
  fluid 
  mass 
  in 
  

   various 
  states 
  to 
  have 
  been 
  determined 
  in 
  any 
  way, 
  and 
  let 
  us 
  find 
  what 
  results 
  

   regarding 
  its 
  pressure 
  and 
  thermal 
  capacities 
  may 
  be 
  deduced. 
  In 
  the 
  first 
  place, 
  

   by 
  integrating 
  equation 
  (8), 
  considered 
  as 
  a 
  differential 
  equation 
  with 
  reference 
  to 
  

   t, 
  for 
  p, 
  we 
  find 
  

  

  p=e 
  J 
  fx 
  — 
  e 
  dt+^(v)e 
  • 
  • 
  • 
  ( 
  10 
  )> 
  

  

  where 
  ^ 
  {v) 
  denotes 
  a 
  constant 
  with 
  reference 
  to 
  t, 
  which 
  may 
  vary 
  with 
  v, 
  and 
  

   cannot 
  be 
  determined 
  without 
  experiment. 
  Again, 
  we 
  have, 
  from 
  (5), 
  (4), 
  and 
  (1), 
  

  

  •vr 
  1 
  de 
  

   Jdl 
  

  

  dp 
  

   1 
  de 
  1 
  /de 
  \ 
  dt 
  

  

  „ 
  1 
  de 
  1 
  (de 
  \ 
  

   K= 
  Jdl 
  + 
  j\T 
  v 
  + 
  p 
  ) 
  

  

  dp 
  

   d 
  v 
  

   VOL. 
  XX. 
  PART 
  III. 
  6 
  N 
  

  

  (11). 
  

  

  