﻿480 
  PROFESSOR 
  WILLIAM 
  THOMSON 
  ON 
  THE 
  

  

  From 
  the 
  first 
  of 
  these 
  equations 
  we 
  infer 
  that 
  with 
  a 
  complete 
  knowledge 
  of 
  

   the 
  mechanical 
  energy 
  of 
  a 
  particular 
  fluid, 
  we 
  have 
  enough 
  of 
  data 
  for 
  determining 
  

   for 
  every 
  state, 
  its 
  thermal 
  capacity 
  in 
  constant 
  volume. 
  From 
  equation 
  (9) 
  we 
  

   infer, 
  that 
  with, 
  besides, 
  a 
  knowledge 
  of 
  the 
  pressure 
  for 
  all 
  volumes 
  and 
  a 
  parti- 
  

   cular 
  temperature, 
  or 
  for 
  all 
  volumes 
  and 
  a 
  particular 
  series 
  of 
  temperatures, 
  we 
  

   have 
  enough 
  to 
  determine 
  completely 
  the 
  pressure, 
  and 
  consequently 
  also, 
  accord- 
  

   ing 
  to 
  equation 
  (11), 
  to 
  determine 
  the 
  two 
  thermal 
  capacities, 
  for 
  all 
  states 
  of 
  the 
  

   fluid. 
  

  

  92. 
  For 
  example, 
  let 
  these 
  equations 
  be 
  applied 
  to 
  the 
  case 
  of 
  a 
  fluid 
  subject 
  

   the 
  g; 
  

   we 
  find 
  

  

  to 
  the 
  gaseous 
  laws. 
  If 
  we 
  use 
  for 
  ^ 
  its 
  value 
  derived 
  from 
  (9), 
  in 
  equation 
  (10), 
  

  

  P= 
  P 
  -^(l+Et) 
  +x 
  (v)e 
  1 
  " 
  (12)) 
  

  

  where 
  % 
  ( 
  v 
  )-> 
  denoting 
  an 
  arbitrary 
  function 
  of 
  v, 
  is 
  used 
  instead 
  of 
  4- 
  (v)-^^' 
  

  

  We 
  conclude 
  that 
  the 
  same 
  expression 
  for 
  the 
  mechanical 
  energy 
  holds 
  for 
  any 
  

   fluid 
  whose 
  pressure 
  is 
  expressed 
  by 
  this 
  equation, 
  as 
  for 
  one 
  subject 
  to 
  the 
  gase- 
  

  

  d 
  6 
  d 
  6 
  

  

  ous 
  laws. 
  Again, 
  by 
  using 
  for 
  -r-> 
  and 
  -r-> 
  their 
  values 
  derived 
  from 
  (9), 
  in 
  equa- 
  

   tion 
  (11), 
  we 
  have 
  

  

  N 
  = 
  N 
  0+J 
  /, 
  o! 
  , 
  logi 
  UL- 
  L 
  .... 
  (13 
  ), 
  

  

  , 
  d 
  f— 
  -(1 
  + 
  E<) 
  ) 
  _, 
  

   K 
  = 
  N 
  + 
  j 
  M 
  log- 
  -^—j- 
  t 
  +jS35rfi) 
  . 
  (14). 
  

  

  The 
  first 
  of 
  these 
  equations 
  shews 
  that, 
  unless 
  Mayer's 
  hypothesis 
  be 
  true, 
  there 
  

   is 
  a 
  difference 
  in 
  the 
  thermal 
  capacities 
  in 
  constant 
  volume, 
  of 
  the 
  same 
  gas 
  at 
  the 
  

   same 
  temperatures 
  for 
  different 
  densities, 
  proportional 
  in 
  amount 
  to 
  the 
  difference 
  

   of 
  the 
  logarithms 
  of 
  the 
  densities. 
  The 
  second 
  compared 
  with 
  the 
  first, 
  leads 
  to 
  

   an 
  expression 
  for 
  the 
  difference 
  between 
  the 
  thermal 
  capacities 
  of 
  a 
  gas 
  in 
  constant 
  

   volume, 
  and 
  under 
  constant 
  pressure, 
  agreeing 
  with 
  results 
  arrived 
  at 
  formerly. 
  

   (Account 
  of 
  Carnot's 
  Theory, 
  Appendix 
  hi., 
  and 
  Dyn. 
  Th. 
  of 
  Heat, 
  § 
  48.) 
  

  

  93. 
  It 
  may 
  be, 
  that 
  more 
  or 
  less 
  information, 
  regarding 
  explicitly 
  the 
  pressure 
  

   and 
  thermal 
  capacities 
  of 
  the 
  fluid, 
  may 
  have 
  been 
  had 
  as 
  the 
  data 
  for 
  determining 
  

   the 
  mechanical 
  energy 
  ; 
  but 
  these 
  converse 
  deductions 
  are 
  still 
  interesting, 
  as 
  shew- 
  

   ing 
  how 
  much 
  information 
  regarding 
  its 
  physical 
  properties, 
  is 
  comprehended 
  in 
  

   a 
  knowledge 
  of 
  the 
  mechanical 
  energy 
  of 
  a 
  fluid 
  mass, 
  and 
  how 
  useful 
  a 
  table 
  of 
  

   the 
  values 
  of 
  this 
  function 
  for 
  different 
  temperatures 
  and 
  volumes, 
  or 
  an 
  Empirical 
  

   Function 
  of 
  two 
  variables 
  expressing 
  it, 
  would 
  be, 
  whatever 
  be 
  the 
  experimental 
  

  

  