﻿DYNAMICAL 
  THEORY 
  OF 
  HEAT. 
  477 
  

  

  ?=JN 
  (5). 
  

  

  dt 
  K 
  J 
  

  

  Lastly, 
  if 
  we 
  denote, 
  as 
  formerly, 
  Carnot's 
  function 
  of 
  the 
  temperature 
  t, 
  by 
  //, 
  we 
  

   have 
  ( 
  § 
  21) 
  

  

  ^ 
  M 
  <«)• 
  

  

  86. 
  The 
  use 
  that 
  may 
  be 
  made 
  of 
  these 
  formulae 
  in 
  investigations 
  of 
  the 
  physical 
  

   properties 
  of 
  any 
  particular 
  fluid 
  must 
  depend 
  on 
  the 
  extent 
  and 
  accuracy 
  of 
  the 
  

   general 
  data 
  belonging 
  to 
  the 
  theory 
  of 
  the 
  mechanical 
  action 
  of 
  heat, 
  that 
  are 
  

   available. 
  Thus, 
  if 
  nothing 
  be 
  known 
  by 
  experiment 
  regarding 
  the 
  values 
  of 
  J 
  

   and 
  /*, 
  we 
  may, 
  in 
  the 
  first 
  place, 
  use 
  equations 
  (4) 
  and 
  (5), 
  or 
  the 
  following 
  de- 
  

   duced 
  from 
  them 
  ( 
  § 
  20) 
  by 
  eliminating 
  e, 
  

  

  dp_ 
  T 
  (d^JL 
  rfN 
  

   dl~ 
  

  

  \dt 
  dv) 
  ^ 
  ; 
  ' 
  

  

  and 
  equation 
  (6), 
  as 
  tests 
  of 
  the 
  accuracy 
  of 
  experimental 
  researches 
  on 
  the 
  pres- 
  

   sure 
  and 
  thermal 
  capacities 
  of 
  a 
  fluid, 
  on 
  account 
  of 
  the 
  knowledge 
  we 
  have 
  from 
  

   theory, 
  that 
  J 
  is 
  certainly 
  an 
  absolute 
  constant, 
  and 
  that, 
  in 
  all 
  probability 
  if 
  not 
  

   with 
  absolute 
  certainty, 
  we 
  may 
  regard 
  jj. 
  as 
  independent 
  of 
  v, 
  and 
  as 
  the 
  same 
  for 
  all 
  

   fluids 
  at 
  the 
  same 
  temperature 
  ; 
  and, 
  with 
  experimental 
  data 
  of 
  sufficient 
  extent, 
  

   we 
  may 
  use 
  these 
  equations 
  as 
  means 
  of 
  actually 
  determining 
  the 
  values 
  of 
  J 
  and 
  

   /*. 
  No 
  other 
  way 
  than 
  this 
  has 
  yet 
  been 
  attempted 
  for 
  determining 
  \x 
  ; 
  and, 
  if 
  we 
  

   except 
  a 
  conceivable 
  but 
  certainly 
  not 
  at 
  present 
  practicable 
  mode 
  of 
  determining 
  

   this 
  element 
  by 
  experiments 
  on 
  thermo-electric 
  currents, 
  no 
  other 
  way 
  is 
  yet 
  

   known. 
  Carnot's 
  original 
  determination 
  of 
  fx, 
  was 
  effected 
  by 
  means 
  of 
  an 
  expres- 
  

   sion 
  equivalent 
  to 
  that 
  of 
  equation 
  (6) 
  applied 
  to 
  the 
  case 
  of 
  a 
  mass 
  of 
  air 
  ; 
  and 
  

   the 
  determinations 
  by 
  Clapeyron, 
  and 
  those 
  shewn 
  in 
  Table 
  I. 
  of 
  my 
  Account 
  of 
  

   Carnot's 
  Theory, 
  were 
  calculated 
  by 
  the 
  formula 
  which 
  is 
  obtained 
  when 
  the 
  

   same 
  equation 
  is 
  applied 
  to 
  the 
  case 
  of 
  a 
  fluid 
  mass, 
  partly 
  liquid 
  and 
  partly 
  in 
  

   the 
  state 
  of 
  saturated 
  vapour 
  (§ 
  55). 
  

  

  87. 
  As 
  yet 
  experiments 
  have 
  not 
  been 
  made 
  on 
  the 
  pressure 
  and 
  thermal 
  ca- 
  

   pacities 
  of 
  fluids 
  to 
  a 
  sufficient 
  extent 
  to 
  supply 
  data 
  for 
  the 
  evaluation, 
  even 
  in 
  

   the 
  roughest 
  manner, 
  of 
  the 
  expression 
  given 
  for 
  J 
  by 
  equation 
  (7) 
  ; 
  and 
  it 
  may 
  be 
  

   doubted 
  whether 
  such 
  data 
  can 
  even 
  be 
  had 
  with 
  accuracy 
  enough 
  to 
  give 
  as 
  exact 
  

   a 
  determination 
  of 
  this 
  important 
  element 
  as 
  may 
  be 
  effected 
  by 
  direct 
  experi- 
  

   ments 
  on 
  the 
  generation 
  of 
  heat 
  by 
  means 
  of 
  friction. 
  At 
  present 
  we 
  may 
  regard 
  

   J 
  as 
  known, 
  probably 
  within 
  gi 
  g 
  of 
  its 
  own 
  amount, 
  by 
  experiments 
  of 
  this 
  kind. 
  

  

  88. 
  The 
  value 
  of 
  J 
  being 
  known, 
  equations 
  (4) 
  and 
  (5) 
  may 
  be 
  used 
  for 
  deter- 
  

   mining 
  the 
  mechanical 
  energy 
  of 
  a 
  particular 
  fluid 
  mass 
  in 
  different 
  states, 
  from 
  

   special 
  experimental 
  data 
  regarding 
  its 
  pressure 
  and 
  thermal 
  capacities, 
  but 
  not 
  

   necessarily 
  comprehending 
  the 
  values 
  of 
  each 
  of 
  these 
  elements 
  for 
  all 
  states 
  of 
  

   the 
  fluid. 
  The 
  theory 
  of 
  the 
  integration 
  of 
  functions 
  of 
  two 
  independent 
  variables 
  

  

  