﻿478 
  PROFESSOR 
  WILLIAM 
  THOMSON 
  ON 
  THE 
  

  

  will, 
  when 
  any 
  set 
  of 
  data 
  are 
  proposed, 
  make 
  it 
  manifest 
  whether 
  or 
  not 
  they 
  are 
  

   sufficient, 
  and 
  will 
  point 
  out 
  the 
  methods, 
  whether 
  of 
  summation 
  or 
  of 
  analytical 
  

   integration, 
  according 
  to 
  the 
  forms 
  in 
  which 
  the 
  data 
  are 
  furnished, 
  to 
  be 
  followed 
  

   for 
  determining 
  the 
  value 
  of 
  e 
  for 
  every 
  value 
  of 
  v. 
  Or 
  the 
  data 
  may 
  be 
  such 
  that, 
  

   while 
  the 
  thermal 
  capacities 
  would 
  be 
  derived 
  from 
  them 
  by 
  differentiation, 
  values 
  

   of 
  e 
  may 
  be 
  obtained 
  from 
  them 
  without 
  integration. 
  Thus, 
  if 
  the 
  fluid 
  mass 
  consist 
  

   of 
  water 
  and 
  vapour 
  of 
  water 
  at 
  the 
  temperature 
  t, 
  weighing 
  in 
  all 
  one 
  pound, 
  and 
  

   occupying 
  the 
  volume 
  v, 
  m 
  and 
  if 
  we 
  regard 
  the 
  zero 
  or 
  " 
  standard" 
  state 
  of 
  the 
  mass 
  

   as 
  being 
  liquid 
  water 
  at 
  the 
  temperature 
  ; 
  the 
  mechanical 
  energy 
  of 
  the 
  mass, 
  

   in 
  the 
  given 
  state, 
  will 
  be 
  the 
  mechanical 
  value 
  of 
  the 
  heat 
  required 
  to 
  raise 
  the 
  

  

  temperature 
  of 
  a 
  pound 
  of 
  water 
  from 
  0° 
  to 
  t, 
  and 
  to 
  convert 
  — 
  ^ 
  of 
  it 
  into 
  vapour, 
  

  

  diminished 
  by 
  the 
  work 
  done 
  in 
  the 
  expansion 
  from 
  the 
  volume 
  X, 
  to 
  the 
  volume 
  

   v; 
  that 
  is, 
  we 
  have 
  

  

  e=J 
  ( 
  c 
  ' 
  + 
  L 
  ^)-^- 
  X 
  ) 
  ( 
  8 
  )- 
  

  

  The 
  variables, 
  c, 
  L, 
  and 
  p 
  (which 
  depend 
  on 
  t 
  alone) 
  in 
  this 
  expression 
  have 
  been 
  

   experimentally 
  determined 
  by 
  Regnault, 
  for 
  all 
  temperatures 
  from 
  0° 
  to 
  230°, 
  and 
  

   when 
  7 
  is 
  also 
  determined, 
  by 
  experiments 
  on 
  the 
  density 
  of 
  saturated 
  steam, 
  the 
  

   elements 
  for 
  the 
  determination 
  of 
  e 
  in 
  this 
  case 
  will 
  be 
  complete. 
  The 
  expressions 
  

   investigated 
  formerly 
  for 
  M 
  and 
  N 
  in 
  this 
  case 
  (§ 
  54) 
  may 
  be 
  readily 
  obtained 
  by 
  

   means 
  of 
  (4) 
  and 
  (5 
  of 
  § 
  84), 
  by 
  the 
  differentiation 
  of 
  (8). 
  

  

  89. 
  If 
  Carnot's 
  function 
  has 
  once 
  been 
  determined 
  by 
  means 
  of 
  observations 
  

   of 
  any 
  kind, 
  whether 
  on 
  a 
  single 
  fluid, 
  or 
  on 
  different 
  fluids, 
  for 
  a 
  certain 
  range 
  of 
  

  

  dp 
  

  

  temperatures, 
  then, 
  according 
  to 
  (6) 
  of 
  § 
  85, 
  the 
  value 
  of 
  ^r— 
  for 
  any 
  substance 
  

  

  whatever, 
  is 
  known 
  for 
  all 
  temperatures 
  within 
  that 
  range. 
  It 
  follows 
  that 
  when 
  

   the 
  values 
  of 
  M 
  for 
  different 
  states 
  of 
  a 
  fluid 
  have 
  been 
  determined 
  experimentally* 
  

   the 
  law 
  of 
  pressures 
  for 
  all 
  temperatures 
  and 
  volumes 
  (with 
  an 
  arbitrary 
  function 
  

   of 
  v 
  to 
  be 
  determined 
  by 
  experiments 
  on 
  the 
  pressure 
  of 
  the 
  fluid 
  at 
  one 
  particular 
  

   temperature) 
  may 
  be 
  deduced, 
  by 
  means 
  of 
  equation 
  (6) 
  ; 
  or 
  conversely, 
  which 
  is 
  

   more 
  likely 
  to 
  be 
  the 
  case 
  for 
  any 
  particular 
  fluid, 
  if 
  the 
  law 
  of 
  pressures 
  is 
  com- 
  

   pletely 
  known, 
  M 
  may 
  be 
  deduced 
  without 
  farther 
  experimenting. 
  Hence 
  the 
  

   second 
  member 
  of 
  (4) 
  becomes 
  completely 
  known, 
  the 
  equation 
  assuming 
  the 
  fol- 
  

   lowing 
  form 
  when, 
  for 
  M, 
  its 
  value 
  according 
  to 
  (6) 
  is 
  substituted 
  : 
  — 
  

  

  * 
  The 
  same 
  notation 
  is 
  used 
  here, 
  as 
  formerly 
  in 
  § 
  54, 
  viz. 
  p 
  is 
  the 
  pressure 
  of 
  saturated 
  vapour 
  

   at 
  the 
  temperature 
  t, 
  y 
  the 
  volume, 
  and 
  L 
  the 
  latent 
  heat 
  of 
  a 
  pound 
  of 
  the 
  vapour, 
  X 
  the 
  volume 
  of 
  

   a 
  pound 
  of 
  liquid 
  water, 
  and 
  c 
  the 
  mean 
  thermal 
  capacity 
  of 
  a 
  pound 
  of 
  water 
  between 
  the 
  tempera- 
  

   tures 
  and 
  t. 
  A 
  mass 
  weighing 
  a 
  pound, 
  and 
  occupying 
  the 
  volume 
  v, 
  when 
  at 
  the 
  temperature 
  t, 
  

  

  must 
  consist 
  of 
  a 
  weight 
  of 
  vapour, 
  and 
  of 
  water. 
  

  

  