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  PROFESSOR 
  WILLIAM 
  THOMSON 
  ON 
  THE 
  

  

  they 
  were 
  at 
  the 
  beginning, 
  however 
  arbitrarily 
  they 
  may 
  have 
  been 
  made 
  to 
  

  

  vary 
  in 
  the 
  period, 
  the 
  total 
  external 
  effect 
  must, 
  according 
  to 
  Prop. 
  I., 
  amount 
  

  

  to 
  nothing 
  ; 
  and 
  hence 
  

  

  (p-JM)dv- 
  JNdt 
  

  

  must 
  be 
  the 
  differential 
  of 
  a 
  function 
  of 
  two 
  independent 
  variables, 
  or 
  we 
  must 
  have 
  

  

  d(p-JM) 
  rf(-JN) 
  

  

  dt 
  ~ 
  dv 
  K 
  h 
  

  

  this 
  being 
  merely 
  the 
  analytical 
  expression 
  of 
  the 
  condition, 
  that 
  the 
  preceding- 
  

   integral 
  may 
  vanish 
  in 
  every 
  case 
  in 
  which 
  the 
  initial 
  and 
  final 
  values 
  of 
  v 
  and 
  

   t 
  are 
  the 
  same, 
  respectively. 
  Observing 
  that 
  J 
  is 
  an 
  absolute 
  constant, 
  we 
  may 
  

   put 
  the 
  result 
  into 
  the 
  form 
  

  

  dp 
  (dM 
  tfN\ 
  

  

  Tt 
  = 
  3 
  \dt~~dv) 
  w 
  " 
  

  

  This 
  equation 
  expresses, 
  in 
  a 
  perfectly 
  comprehensive 
  manner, 
  the 
  application 
  of 
  

   the 
  first 
  fundamental 
  proposition 
  to 
  the 
  thermal 
  and 
  mechanical 
  circumstances 
  

   of 
  any 
  substance 
  whatever, 
  under 
  uniform 
  pressure 
  in 
  all 
  directions, 
  when 
  sub- 
  

   jected 
  to 
  any 
  possible 
  variations 
  of 
  temperature, 
  volume, 
  and 
  pressure. 
  

  

  21. 
  The 
  corresponding 
  application 
  of 
  the 
  second 
  fundamental 
  proposition 
  is 
  

   completely 
  expressed 
  by 
  the 
  equation 
  

  

  5? 
  = 
  " 
  M 
  • 
  ■ 
  ■ 
  ' 
  • 
  • 
  ■ 
  < 
  3 
  )- 
  

  

  where 
  fj. 
  denotes 
  what 
  is 
  called 
  " 
  Carnots 
  function," 
  a 
  quantity 
  which 
  has 
  an 
  

   absolute 
  value, 
  the 
  same 
  for 
  all 
  substances 
  for 
  any 
  given 
  temperature, 
  but 
  which 
  

   may 
  vary 
  with 
  the 
  temperature 
  in 
  a 
  manner 
  that 
  can 
  only 
  be 
  determined 
  by 
  

   experiment. 
  To 
  prove 
  this 
  proposition, 
  it 
  may 
  be 
  remarked 
  in 
  the 
  first 
  place 
  

   that 
  Prop. 
  II. 
  could 
  not 
  be 
  true 
  for 
  every 
  case 
  in 
  which 
  the 
  temperature 
  of 
  the 
  

   refrigeration 
  differs 
  infinitely 
  little 
  from 
  that 
  of 
  the 
  source, 
  without 
  being 
  true 
  

   universally. 
  Now, 
  if 
  a 
  substance 
  be 
  allowed 
  first 
  to 
  expand 
  from 
  v 
  to 
  v 
  + 
  dv, 
  its 
  

   temperature 
  being 
  kept 
  constantly 
  t 
  ; 
  if, 
  secondly, 
  it 
  be 
  allowed 
  to 
  expand 
  farther, 
  

   without 
  either 
  emitting 
  or 
  absorbing 
  heat 
  till 
  its 
  temperature 
  goes 
  down 
  through 
  

   an 
  infinitely 
  small 
  range, 
  to 
  t 
  — 
  r 
  ; 
  if, 
  thirdly, 
  it 
  be 
  compressed 
  at 
  the 
  constant 
  

   temperature 
  t 
  — 
  t, 
  so 
  much 
  (actually 
  by 
  an 
  amount 
  differing 
  from 
  d 
  v 
  by 
  only 
  an 
  

   infinitely 
  small 
  quantity 
  of 
  the 
  second 
  order), 
  that 
  when, 
  fourthly, 
  the 
  volume 
  

   is 
  further 
  diminished 
  to 
  v 
  without 
  the 
  medium's 
  being 
  allowed 
  to 
  either 
  emit 
  or 
  

   absorb 
  heat, 
  its 
  temperature 
  may 
  be 
  exactly 
  t 
  ; 
  it 
  may 
  be 
  considered 
  as 
  consti- 
  

   tuting 
  a 
  thermo-dynamic 
  engine 
  which 
  fulfils 
  Carnot's 
  condition 
  of 
  complete 
  

   reversibility. 
  Hence, 
  by 
  Prop 
  II., 
  it 
  must 
  produce 
  the 
  same 
  amount 
  of 
  work 
  for 
  

   the 
  same 
  quantity 
  of 
  heat 
  absorbed 
  in 
  the 
  first 
  operation, 
  as 
  any 
  other 
  substance 
  

  

  similarly 
  operated 
  upon 
  through 
  the 
  same 
  range 
  of 
  temperatures. 
  But 
  -tj 
  t. 
  dv 
  

  

  is 
  obviously 
  the 
  whole 
  work 
  done 
  in 
  the 
  complete 
  cycle, 
  and 
  (by 
  the 
  definition 
  of 
  

  

  