﻿H^-Q^K-^I^V 
  

  

  HEAT 
  IN 
  EXPANSIVE 
  MACHINES. 
  207 
  

  

  equation 
  to 
  the 
  cycle 
  of 
  operations 
  undergone 
  by 
  the 
  working 
  body 
  in 
  an 
  expan- 
  

   sive 
  machine, 
  as 
  denoted 
  by 
  the 
  diagram. 
  

  

  First 
  operation. 
  The 
  body, 
  being 
  at 
  first 
  at 
  the 
  volume 
  V 
  A 
  and 
  pressure 
  P 
  A 
  , 
  

   is 
  made 
  to 
  expand, 
  by 
  the 
  communication 
  of 
  heat 
  at 
  the 
  constant 
  temperature 
  

   t 
  v 
  until 
  it 
  reaches 
  the 
  volume 
  V 
  B 
  and 
  pressure 
  P 
  B 
  , 
  AB 
  being 
  the 
  locus 
  of 
  the 
  

   pressures. 
  

  

  Here 
  8 
  t=0 
  ; 
  therefore 
  the 
  total 
  heat 
  received 
  is 
  

  

  V 
  A 
  dT 
  ' 
  (a) 
  

  

  = 
  (t 
  1 
  -/c){0(V 
  b 
  ,t 
  1 
  )-(/)(V 
  a 
  ,t 
  1 
  )} 
  J 
  

  

  Second 
  operation. 
  The 
  body, 
  being 
  prevented 
  from 
  receiving 
  or 
  emitting 
  heat, 
  

   expands 
  until 
  it 
  falls 
  to 
  the 
  temperature 
  r 
  , 
  the 
  locus 
  of 
  the 
  pressures 
  being 
  the 
  

   curve 
  BC. 
  During 
  this 
  operation 
  the 
  following 
  condition 
  must 
  be 
  fulfilled, 
  — 
  

  

  o 
  = 
  8 
  q 
  - 
  8 
  q 
  

   Which, 
  attending 
  to 
  the 
  fact 
  that 
  V 
  is 
  now 
  a 
  function 
  of 
  r, 
  and 
  transforming 
  the 
  

   integrals 
  as 
  before, 
  gives 
  the 
  equation 
  

  

  _ 
  . 
  1 
  (k 
  k 
  2 
  \ 
  N 
  / 
  d 
  dV 
  d 
  \ 
  . 
  __ 
  x 
  

  

  This 
  equation 
  shews 
  that 
  

  

  ^Vb.^-^o, 
  t 
  )=^(t 
  1s 
  t 
  ) 
  . 
  . 
  (6) 
  

  

  Third 
  operation. 
  The 
  body, 
  by 
  the 
  abstraction 
  of 
  heat, 
  is 
  made 
  to 
  contract, 
  at 
  

   the 
  constant 
  temperature 
  t 
  , 
  to 
  the 
  volume 
  V 
  D 
  and 
  pressure 
  P 
  b 
  , 
  which 
  are 
  such 
  

   as 
  to 
  satisfy 
  conditions 
  depending 
  on 
  the 
  fourth 
  operation. 
  CD 
  is 
  the 
  locus 
  of 
  

   the 
  pressures. 
  The 
  heat 
  emitted 
  is 
  evidently 
  

  

  H 
  = 
  Q'o 
  = 
  (t 
  -/<) 
  [<p 
  (V 
  c 
  , 
  t 
  o 
  )-0(V 
  d 
  , 
  t 
  )} 
  . 
  (e) 
  

  

  Fourth 
  operation. 
  The 
  body, 
  being 
  prevented 
  from 
  receiving 
  or 
  emitting 
  heat, 
  

   is 
  compressed 
  until 
  it 
  recovers 
  its 
  original 
  temperature 
  r 
  v 
  volume 
  V 
  , 
  and 
  pres- 
  

   sure 
  P 
  A 
  ; 
  the 
  locus 
  of 
  the 
  pressures 
  being 
  DA. 
  During 
  this 
  operation, 
  the 
  same 
  

   conditions 
  must 
  be 
  fulfilled 
  as 
  in 
  the 
  second 
  operation 
  ; 
  therefore 
  

   <P(Y 
  a 
  ,t 
  1 
  )-cP(Y 
  d 
  ,t 
  ) 
  = 
  ^(t 
  v 
  t 
  ) 
  . 
  ..(d) 
  

  

  ■1,, 
  being 
  the 
  same 
  function 
  as 
  in 
  Equation 
  (b). 
  

  

  By 
  comparing 
  Equations 
  (b) 
  and 
  (d), 
  we 
  obtain 
  the 
  relation 
  which 
  must 
  sub- 
  

   sist 
  between 
  the 
  four 
  volumes 
  to 
  which 
  the 
  body 
  is 
  successively 
  brought, 
  in 
  order 
  

   • 
  that 
  the 
  maximum 
  effect 
  may 
  be 
  obtained 
  from 
  the 
  heat. 
  It 
  is 
  expressed 
  by 
  the 
  

   j, 
  equation 
  

  

  <t> 
  ( 
  V 
  b> 
  T 
  i)-4> 
  (V 
  A 
  , 
  ^) 
  = 
  <P 
  (V 
  c 
  , 
  t 
  o 
  )-0 
  (V 
  D 
  , 
  t 
  ) 
  . 
  (64.) 
  

  

  From 
  this, 
  and 
  Equations 
  (a) 
  and 
  (c), 
  it 
  appears 
  that 
  

  

  H 
  = 
  t^-k 
  5 
  , 
  

  

  H, 
  T,-K 
  

  

  