﻿DYNAMICAL 
  THEORY 
  OF 
  HEAT. 
  271 
  

  

  M, 
  in 
  § 
  20) 
  M 
  d 
  v 
  is 
  the 
  quantity 
  of 
  heat 
  absorbed 
  in 
  the 
  first 
  operation. 
  Hence 
  

   the 
  value 
  of 
  

  

  ^ 
  or 
  AL 
  r 
  

  

  M 
  dv 
  M 
  

  

  must 
  be 
  the 
  same 
  for 
  all 
  substances, 
  with 
  the 
  same 
  values 
  of 
  t 
  and 
  t 
  ; 
  or, 
  since 
  

   t 
  is 
  not 
  involved 
  except 
  as 
  a 
  factor, 
  we 
  must 
  have 
  

  

  dp 
  

   d 
  t 
  

  

  where 
  /x 
  depends 
  only 
  on 
  t 
  ; 
  from 
  which 
  we 
  conclude 
  the 
  proposition 
  which 
  was 
  

   to 
  be 
  proved. 
  

  

  dp 
  

  

  22. 
  The 
  very 
  remarkable 
  theorem 
  that 
  — 
  1 
  must 
  be 
  the 
  same 
  for 
  all 
  sub- 
  

  

  J 
  M 
  

  

  stances 
  at 
  the 
  same 
  temperature, 
  was 
  first 
  given 
  (although 
  not 
  in 
  precisely 
  the 
  

   same 
  terms) 
  by 
  Carnot, 
  and 
  demonstrated 
  by 
  him, 
  according 
  to 
  the 
  principles 
  

   he 
  adopted. 
  We 
  have 
  now 
  seen 
  that 
  its 
  truth 
  may 
  be 
  satisfactorily 
  established 
  

   without 
  adopting 
  the 
  false 
  part 
  of 
  his 
  principles. 
  Hence 
  all 
  Carnot's 
  conclusions, 
  

   and 
  all 
  conclusions 
  derived 
  by 
  others 
  from 
  his 
  theory, 
  which 
  depend 
  merely 
  on 
  

   equation 
  (3), 
  require 
  no 
  modification 
  when 
  the 
  dynamical 
  theory 
  is 
  adopted. 
  

   Thus, 
  all 
  the 
  conclusions 
  contained 
  in 
  Sections 
  I., 
  II., 
  and 
  III. 
  of 
  the 
  Appendix 
  to 
  

   my 
  Account 
  of 
  Carnot's 
  Theory, 
  and 
  in 
  the 
  paper 
  immediately 
  following 
  it 
  in 
  the 
  

   Transactions, 
  entitled 
  " 
  Theoretical 
  Considerations 
  on 
  the 
  Effect 
  of 
  Pressure 
  in 
  

   Lowering 
  the 
  Freezing 
  Point 
  of 
  Water," 
  by 
  my 
  elder 
  brother, 
  still 
  hold. 
  Also, 
  

   we 
  see 
  that 
  Carnot's 
  expression 
  for 
  the 
  mechanical 
  effect 
  derivable 
  from 
  a 
  given 
  

   quantity 
  of 
  heat 
  by 
  means 
  of 
  a 
  perfect 
  engine 
  in 
  which 
  the 
  range 
  of 
  temperatures 
  

   is 
  infinitely 
  small, 
  expresses 
  truly 
  the 
  greatest 
  effect 
  which 
  can 
  possibly 
  be 
  

   obtained 
  in 
  the 
  circumstances 
  ; 
  although 
  it 
  is 
  in 
  reality 
  only 
  an 
  infinitely 
  small 
  

   fraction 
  of 
  the 
  whole 
  mechanical 
  equivalent 
  of 
  the 
  heat 
  supplied 
  ; 
  the 
  remainder 
  

   being 
  irrecoverably 
  lost 
  to 
  man, 
  and 
  therefore 
  " 
  wasted," 
  although 
  not 
  anni- 
  

   hilated. 
  

  

  23. 
  On 
  the 
  other 
  hand, 
  the 
  expression 
  for 
  the 
  mechanical 
  effect 
  obtainable 
  

   from 
  a 
  given 
  quantity 
  of 
  heat 
  entering 
  an 
  engine 
  from 
  a 
  " 
  source 
  " 
  at 
  a 
  given 
  

   temperature, 
  when 
  the 
  range 
  down 
  to 
  the 
  temperature 
  of 
  the 
  cold 
  part 
  of 
  the 
  

   engine 
  or 
  the 
  " 
  refrigerator 
  " 
  is 
  finite, 
  will 
  differ 
  most 
  materially 
  from 
  that 
  of 
  

   Carnot 
  ; 
  since, 
  a 
  finite 
  quantity 
  of 
  mechanical 
  effect 
  being 
  now 
  obtained 
  from 
  a 
  

   finite 
  quantity 
  of 
  heat 
  entering 
  the 
  engine, 
  a 
  finite 
  fraction 
  of 
  this 
  quantity 
  must 
  

   be 
  converted 
  from 
  heat 
  into 
  mechanical 
  effect. 
  The 
  investigation 
  of 
  this 
  expres- 
  

   sion, 
  with 
  numerical 
  determinations 
  founded 
  on 
  the 
  numbers 
  deduced 
  from 
  

   Regnault's 
  observations 
  on 
  steam, 
  which 
  are 
  shewn 
  in 
  Tables 
  I. 
  and 
  II. 
  of 
  

  

  