﻿476 
  PROFESSOR 
  WILLIAM 
  THOMSON 
  ON 
  THE 
  

  

  tions 
  established 
  in 
  the 
  first 
  part 
  of 
  my 
  paper 
  on 
  the 
  Dynamical 
  Theory 
  of 
  Heat, 
  

   and 
  expressing 
  relations 
  between 
  the 
  pressure 
  of 
  a 
  fluid, 
  and 
  the 
  thermal 
  capacities 
  

   and 
  mechanical 
  energy 
  of 
  a 
  given 
  mass 
  of 
  it, 
  all 
  considered 
  as 
  functions 
  of 
  the 
  

   temperature 
  and 
  volume, 
  and 
  Carnot's 
  function 
  of 
  the 
  temperature, 
  are 
  brought 
  

   forward 
  for 
  the 
  purpose 
  of 
  pointing 
  out 
  the 
  importance 
  of 
  making 
  the 
  mechanical 
  

   energy 
  of 
  a 
  fluid 
  in 
  different 
  states 
  an 
  object 
  of 
  research, 
  along 
  with 
  the 
  other 
  

   elements 
  which 
  have 
  hitherto 
  been 
  considered, 
  and 
  partially 
  investigated 
  in 
  some 
  

   cases. 
  

  

  84. 
  If 
  we 
  consider 
  the 
  circumstances 
  of 
  a 
  stated 
  quantity 
  (a 
  unit 
  of 
  matter, 
  a 
  

   pound, 
  for 
  instance) 
  of 
  a 
  fluid, 
  we 
  find 
  that 
  its 
  condition, 
  whether 
  it 
  be 
  wholly 
  in 
  

   the 
  liquid 
  state, 
  or 
  wholly 
  gaseous, 
  or 
  partly 
  liquid 
  and 
  partly 
  gaseous, 
  is 
  com- 
  

   pletely 
  defined 
  when 
  its 
  temperature, 
  and 
  the 
  volume 
  of 
  the 
  space 
  within 
  which 
  

   it 
  is 
  contained, 
  are 
  specified 
  (§§ 
  20, 
  53, 
  ....56), 
  it 
  being 
  understood, 
  of 
  course, 
  that 
  

   the 
  dimensions 
  of 
  this 
  space 
  are 
  so 
  limited, 
  that 
  no 
  sensible 
  differences 
  of 
  density 
  

   in 
  different 
  parts 
  of 
  the 
  fluid 
  are 
  produced 
  by 
  gravity. 
  We 
  shall 
  therefore 
  consider 
  

   the 
  temperature, 
  and 
  the 
  volume 
  of 
  unity 
  of 
  mass, 
  of 
  a 
  fluid 
  as 
  the 
  independent 
  

   variables 
  of 
  which 
  its 
  pressure, 
  thermal 
  capacities, 
  and 
  mechanical 
  energy, 
  are 
  

   functions. 
  The 
  volume 
  and 
  temperature 
  being 
  denoted 
  respectively 
  by 
  v 
  and 
  t, 
  

   let 
  e 
  be 
  the 
  mechanical 
  energy, 
  p 
  the 
  pressure, 
  K 
  the 
  thermal 
  capacity 
  under 
  con- 
  

   stant 
  pressure, 
  and 
  N 
  the 
  thermal 
  capacity 
  in 
  constant 
  volume 
  ; 
  and 
  let 
  M 
  be 
  

   such 
  a 
  function 
  of 
  these 
  elements, 
  that 
  

  

  dp 
  

  

  K 
  = 
  N 
  + 
  1£_M 
  (1), 
  

  

  dp 
  

  

  dv 
  

  

  or 
  (§§ 
  48, 
  20), 
  such 
  a 
  quantity 
  that 
  

  

  Mdv 
  + 
  N 
  dt 
  (2), 
  

  

  may 
  express 
  the 
  quantity 
  of 
  heat 
  that 
  must 
  be 
  added 
  to 
  the 
  fluid 
  mass, 
  to 
  elevate 
  

   its 
  temperature 
  by 
  dt, 
  when 
  its 
  volume 
  is 
  augmented 
  by 
  dv. 
  

  

  85. 
  The 
  mechanical 
  value 
  of 
  the 
  heat 
  added 
  to 
  the 
  fluid 
  in 
  any 
  operation, 
  or 
  the 
  

   quantity 
  of 
  heat 
  added 
  multiplied 
  by 
  J 
  (the 
  mechanical 
  equivalent 
  of 
  the 
  thermal 
  

   unit), 
  must 
  be 
  diminished 
  by 
  the 
  work 
  done 
  by 
  the 
  fluid 
  in 
  expanding 
  against 
  re- 
  

   sistance, 
  to 
  find 
  the 
  actual 
  increase 
  of 
  mechanical 
  energy 
  which 
  the 
  body 
  acquires. 
  

   Hence, 
  (d 
  e, 
  of 
  course, 
  denoting 
  the 
  complete 
  increment 
  of 
  e, 
  when 
  v 
  and 
  t 
  are 
  in- 
  

   creased 
  by 
  d 
  v 
  and 
  d 
  t,) 
  we 
  have 
  

  

  rfe=J 
  (M 
  dv 
  + 
  ~N 
  dt)-pdv 
  (3). 
  

  

  Hence, 
  accordiug 
  to 
  the 
  usual 
  notation 
  for 
  partial 
  differential 
  coefficients, 
  we 
  have 
  

  

  