﻿DYNAMICAL 
  THEORY 
  OF 
  HEAT. 
  273 
  

  

  and 
  consequently, 
  by 
  (6), 
  

  

  i 
  -L/tV4 
  (8). 
  

  

  T 
  fxdt, 
  for 
  the 
  duty, 
  indicated 
  

  

  by 
  Carnot's 
  theory,* 
  we 
  may 
  expand 
  the 
  exponential 
  in 
  the 
  preceding 
  equation, 
  

   by 
  the 
  usual 
  series. 
  We 
  thus 
  find 
  

  

  W 
  = 
  ( 
  » 
  - 
  TT2 
  + 
  IX3- 
  te 
  ) 
  • 
  H/?" 
  " 
  

  

  where 
  

  

  1 
  fS 
  ** 
  

  

  (9)- 
  

  

  This 
  shews 
  that 
  the 
  work 
  really 
  produced, 
  which 
  always 
  falls 
  short 
  of 
  the 
  duty 
  

   indicated 
  by 
  Carnot's 
  theory, 
  approaches 
  more 
  and 
  more 
  nearly 
  to 
  it 
  as 
  the 
  

   range 
  is 
  diminished, 
  and 
  ultimately, 
  when 
  the 
  range 
  is 
  infinitely 
  small, 
  is 
  the 
  

   same 
  as 
  if 
  Carnot's 
  theory 
  required 
  no 
  modification, 
  which 
  agrees 
  with 
  the 
  

   conclusion 
  stated 
  above 
  in 
  § 
  22. 
  

  

  27. 
  Again, 
  equation 
  (8) 
  shews 
  that 
  the 
  real 
  duty 
  of 
  a 
  given 
  quantity 
  of 
  heat 
  

   supplied 
  from 
  the 
  source 
  increases 
  with 
  every 
  increase 
  of 
  the 
  range 
  ; 
  but 
  that 
  

  

  fxdt,as 
  Carnot's 
  theory 
  

   J- 
  

  

  makes 
  it 
  do, 
  it 
  never 
  reaches 
  the 
  value 
  J 
  H, 
  but 
  approximates 
  to 
  this 
  limit, 
  as 
  

   I 
  fxdtis 
  increased 
  without 
  limit. 
  Hence 
  Carnot's 
  remarkf 
  regarding 
  the 
  prac- 
  

   tical 
  advantage 
  that 
  may 
  be 
  anticipated 
  from 
  the 
  use 
  of 
  the 
  air-engine, 
  or 
  from 
  

   any 
  method 
  by 
  which 
  the 
  range 
  of 
  temperatures 
  may 
  be 
  increased, 
  loses 
  only 
  a 
  

   part 
  of 
  its 
  importance, 
  while 
  a 
  much 
  more 
  satisfactory 
  view 
  than 
  his 
  of 
  the 
  

   practical 
  problem 
  is 
  afforded. 
  Thus 
  we 
  see 
  that, 
  although 
  the 
  full 
  equivalent 
  of 
  

   mechanical 
  effect 
  cannot 
  be 
  obtained 
  even 
  by 
  means 
  of 
  a 
  perfect 
  engine, 
  yet 
  when 
  

   the 
  actual 
  source 
  of 
  heat 
  is 
  at 
  a 
  high 
  enough 
  temperature 
  above 
  the 
  surrounding 
  

   objects, 
  we 
  may 
  get 
  more 
  and 
  more 
  nearly 
  the 
  whole 
  of 
  the 
  admitted 
  heat 
  con- 
  

   verted 
  into 
  mechanical 
  effect, 
  by 
  simply 
  increasing 
  the 
  effective 
  range 
  of 
  tempera- 
  

   ture 
  in 
  the 
  engine. 
  

  

  28. 
  The 
  preceding 
  investigation 
  (§ 
  25) 
  shews 
  that 
  the 
  value 
  of 
  Carnot's 
  

   function, 
  fx, 
  for 
  all 
  temperatures 
  within 
  the 
  range 
  of 
  the 
  engine, 
  and 
  the 
  absolute 
  

   value 
  of 
  Joule's 
  equivalent, 
  J, 
  are 
  enough 
  of 
  data 
  to 
  calculate 
  the 
  amount 
  of 
  

   mechanical 
  effect 
  of 
  a 
  perfect 
  engine 
  of 
  any 
  kind, 
  whether 
  a 
  steam-engine, 
  an 
  air- 
  

   engine, 
  or 
  even 
  a 
  thermo-electric 
  engine, 
  since, 
  according 
  to 
  the 
  axiom 
  stated 
  in 
  

   § 
  12, 
  and 
  the 
  demonstration 
  of 
  Prop. 
  II., 
  no 
  inanimate 
  material 
  agency 
  could 
  

  

  * 
  " 
  Account," 
  &c, 
  Equation 
  7, 
  § 
  31. 
  

   t 
  " 
  Account," 
  &c. 
  Appendix, 
  Section 
  IV. 
  

   VOL. 
  XX. 
  PART 
  II. 
  4 
  E 
  

  

  