﻿226 
  Mr. 
  S. 
  H. 
  Burbury 
  on 
  Irreversible 
  

  

  Corollary 
  — 
  If 
  u\ 
  v' 
  lie 
  between 
  u 
  and 
  v, 
  u'log 
  m' 
  + 
  i/ 
  logv' 
  

   < 
  u 
  log 
  u 
  + 
  v 
  log 
  ?'. 
  

  

  ^4 
  System 
  of 
  Exchanges. 
  

  

  2. 
  Let 
  w, 
  v! 
  be 
  two 
  essentially 
  positive 
  quantities, 
  for 
  example 
  

   the 
  energies 
  of 
  two 
  material 
  systems. 
  In 
  time 
  dt 
  let 
  u 
  emit, 
  

   and 
  u! 
  receive, 
  the 
  energy 
  kudt, 
  and 
  let 
  v! 
  emit, 
  and 
  u 
  receive,, 
  

   the 
  energy 
  k'u 
  r 
  dt, 
  k 
  and 
  k' 
  being 
  constants. 
  

  

  Then 
  du 
  _ 
  , 
  ^' 
  _ 
  , 
  y 
  , 
  

  

  — 
  = 
  — 
  — 
  It 
  U 
  — 
  till. 
  ~~, 
  — 
  ' 
  fill 
  lij 
  tl 
  . 
  

  

  dt 
  ' 
  dt 
  ' 
  

  

  and 
  (a) 
  let 
  S 
  = 
  u(\og(qu) 
  — 
  1) 
  + 
  w'(log 
  {q'u') 
  — 
  1) 
  

  

  = 
  (*V-A 
  u 
  )log^,. 
  

  

  If 
  then, 
  k, 
  k' 
  being 
  given 
  by 
  the 
  physical 
  conditions, 
  q, 
  q' 
  be- 
  

   so 
  chosen 
  that 
  q/q 
  l 
  ssk/k 
  1 
  , 
  -r 
  is 
  necessarily 
  negative. 
  

   S 
  may 
  be 
  put 
  in 
  other 
  forms. 
  For 
  instance, 
  

   (b) 
  let 
  S 
  = 
  qu 
  log 
  (qu) 
  — 
  (1 
  + 
  qu) 
  log 
  ( 
  1 
  + 
  qu) 
  

  

  + 
  q'u' 
  log 
  (y 
  V) 
  - 
  ( 
  1 
  4- 
  yV) 
  log 
  (1 
  + 
  q'u') 
  

  

  which 
  is 
  again 
  negative 
  if 
  q/q 
  f 
  =k/k'. 
  

  

  The 
  Physical 
  Application 
  of 
  this 
  Theorem. 
  

  

  3. 
  A 
  material 
  system 
  is 
  divided 
  into 
  n 
  parts, 
  a 
  Y 
  a 
  2 
  . 
  . 
  . 
  a 
  n 
  , 
  and' 
  

   we 
  assume 
  that 
  the 
  energy, 
  U, 
  which 
  tho 
  system 
  possesses 
  

   can 
  be 
  localized, 
  so 
  that 
  at 
  any 
  instant 
  a 
  1 
  . 
  . 
  . 
  a 
  n 
  possess 
  

   respectively 
  the 
  energies 
  Ui 
  , 
  . 
  . 
  u 
  n 
  , 
  which 
  vary 
  with 
  the 
  time 
  

   subject 
  to 
  the 
  condition 
  that 
  u 
  r 
  + 
  . 
  . 
  . 
  +w 
  w 
  — 
  2w=U. 
  

  

  For 
  the 
  law 
  of 
  this 
  time 
  variation, 
  let 
  us 
  assume 
  that 
  

   every 
  a 
  emits 
  in 
  time 
  dt 
  a 
  quantity 
  of 
  energy 
  proportional 
  to 
  

   the 
  corresponding 
  u, 
  that 
  is 
  a 
  x 
  emits 
  k^dt, 
  &c. 
  

  

  By 
  conservation 
  of 
  energy, 
  what 
  a 
  x 
  emits 
  is 
  received 
  by 
  

   some 
  one 
  or 
  more 
  of 
  the 
  other 
  parts 
  a 
  2 
  . 
  . 
  . 
  a 
  n 
  . 
  Let 
  us 
  then 
  

   assume 
  that 
  k 
  l2 
  u 
  1 
  dt 
  is 
  received 
  by 
  a 
  2 
  

   ktfUidt 
  „ 
  a 
  3 
  , 
  

  

  and 
  so 
  on, 
  where 
  k 
  12 
  -\-k 
  u 
  -\- 
  . 
  . 
  . 
  ■i-k 
  hl 
  = 
  k 
  l 
  . 
  

  

  In 
  the 
  same 
  way 
  let 
  a 
  { 
  receive 
  from 
  a 
  2 
  in 
  time 
  dt 
  the- 
  

   energy 
  k 
  2l 
  u 
  2 
  dt, 
  

  

  from 
  a 
  6 
  k 
  n 
  7/g 
  dt, 
  &c. 
  

  

  