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  PROFESSOR 
  WILLIAM 
  THOMSON 
  ON 
  THE 
  

  

  my 
  former 
  paper, 
  constitutes 
  the 
  second 
  part 
  of 
  the 
  paper 
  at 
  present 
  com- 
  

   municated. 
  

  

  Part 
  II. 
  — 
  On 
  the 
  Motive 
  Power 
  of 
  Heat 
  through 
  Finite 
  Ranges 
  

  

  of 
  Temperature. 
  

  

  24. 
  It 
  is 
  required 
  to 
  determine 
  the 
  quantity 
  of 
  work 
  which 
  a 
  perfect 
  engine, 
  

   supplied 
  from 
  a 
  source 
  at 
  any 
  temperature, 
  S, 
  and 
  parting 
  with 
  its 
  waste 
  heat 
  to 
  

   a 
  refrigerator 
  at 
  any 
  lower 
  temperature, 
  T, 
  will 
  produce 
  from 
  a 
  given 
  quantity, 
  H, 
  

   of 
  heat 
  drawn 
  from 
  the 
  source. 
  

  

  25. 
  We 
  may 
  suppose 
  the 
  engine 
  to 
  consist 
  of 
  an 
  infinite 
  number 
  of 
  perfect 
  

   engines, 
  each 
  working 
  within 
  an 
  infinitely 
  small 
  range 
  of 
  temperature, 
  and 
  

   arranged 
  in 
  a 
  series 
  of 
  which 
  the 
  source 
  of 
  the 
  first 
  is 
  the 
  given 
  source, 
  the 
  

   refrigerator 
  of 
  the 
  last 
  the 
  given 
  refrigerator, 
  and 
  the 
  refrigerator 
  of 
  each 
  inter- 
  

   mediate 
  engine 
  is 
  the 
  source 
  of 
  that 
  which 
  follows 
  it 
  in 
  the 
  series. 
  Each 
  of 
  these 
  

   engines 
  will, 
  in 
  any 
  time, 
  emit 
  just 
  as 
  much 
  less 
  heat 
  to 
  its 
  refrigerator 
  than 
  is 
  

   supplied 
  to 
  it 
  from 
  its 
  source, 
  as 
  is 
  the 
  equivalent 
  of 
  the 
  mechanical 
  work 
  w 
  r 
  hich 
  

   it 
  produces. 
  Hence, 
  if 
  t 
  and 
  t 
  + 
  d 
  t 
  denote 
  respectively 
  the 
  temperatures 
  of 
  the 
  

   refrigerator 
  and 
  source 
  of 
  one 
  of 
  the 
  intermediate 
  engines 
  ; 
  and 
  if 
  q 
  denote 
  the 
  

   quantity 
  of 
  heat 
  which 
  this 
  engine 
  discharges 
  into 
  its 
  refrigerator 
  in 
  any 
  time, 
  

   and 
  q 
  + 
  d 
  q 
  the 
  quantity 
  which 
  it 
  draws 
  from 
  its 
  source 
  in 
  the 
  same 
  time, 
  the 
  

   quantity 
  of 
  work 
  which 
  it 
  produces 
  in 
  that 
  time 
  will 
  be 
  J 
  d 
  q 
  according 
  to 
  Prop. 
  

   I., 
  and 
  it 
  will 
  also 
  be 
  q 
  /jl 
  dt 
  according 
  to 
  the 
  expression 
  of 
  Prop. 
  II., 
  investigated 
  

   in 
  § 
  21 
  ; 
  and 
  therefore 
  we 
  must 
  have 
  

  

  Jdq=.qfldt. 
  

  

  Hence, 
  supposing 
  that 
  the 
  length 
  of 
  time 
  considered 
  is 
  that 
  during 
  which 
  the 
  

   quantity, 
  H, 
  of 
  heat 
  is 
  supplied 
  from 
  the 
  first 
  source, 
  we 
  find 
  by 
  integration 
  

  

  . 
  H 
  1 
  Cs 
  . 
  

  

  l0 
  S7 
  = 
  dt 
  f* 
  dL 
  

  

  7 
  

  

  But 
  the 
  value 
  of 
  q, 
  when 
  t 
  = 
  T, 
  is 
  the 
  final 
  remainder 
  discharged 
  into 
  the 
  refrigera- 
  

   tor 
  at 
  the 
  temperature 
  T 
  ; 
  and 
  therefore, 
  if 
  this 
  be 
  denoted 
  by 
  R, 
  we 
  have 
  

  

  H 
  1 
  rs 
  

  

  from 
  w 
  T 
  hich 
  we 
  deduce 
  

  

  loo* 
  

  

  ° 
  R 
  

  

  R 
  = 
  He 
  

  

  \f*P*t 
  ■ 
  ■ 
  • 
  ' 
  (5); 
  

  

  Now, 
  the 
  whole 
  amount 
  of 
  work 
  produced 
  will 
  be 
  the 
  mechanical 
  equivalent 
  of 
  

   the 
  quantity 
  of 
  heat 
  lost; 
  and, 
  therefore, 
  if 
  this 
  be 
  denoted 
  by 
  W, 
  we 
  have 
  

  

  W 
  = 
  J(H-R), 
  (7), 
  

  

  