﻿Processes 
  and 
  Planck'' 
  s 
  Theory. 
  227 
  

  

  Then 
  -~ 
  — 
  k 
  2l 
  n 
  2 
  + 
  k 
  3i 
  u 
  3 
  + 
  . 
  . 
  . 
  + 
  k 
  nl 
  u 
  

  

  du 
  x 
  

  

  — 
  AigWl 
  — 
  &I3W1— 
  • 
  • 
  • 
  —kinU^ 
  

  

  Similarly 
  —7— 
  = 
  k 
  u 
  u 
  x 
  + 
  k 
  32 
  u 
  3 
  + 
  . 
  . 
  . 
  4- 
  &„ 
  2 
  w» 
  

  

  — 
  «^21 
  w 
  2 
  — 
  "23^2""" 
  • 
  * 
  — 
  ^2iJ 
  f 
  i 
  

  

  and 
  so 
  on. 
  

  

  4. 
  Let 
  us 
  now 
  assign 
  to 
  u 
  x 
  . 
  . 
  . 
  w 
  w 
  respectively 
  the 
  constant 
  

   positive 
  coefficients 
  q 
  x 
  . 
  . 
  . 
  y 
  n 
  , 
  and 
  form 
  the 
  function 
  

  

  S 
  = 
  2(ulog 
  (qu) 
  -u) 
  =tu 
  log 
  2^, 
  if 
  6 
  = 
  2-7182818. 
  

   Then 
  — 
  

  

  ^-=2logM^ 
  

  

  = 
  log 
  (yjMj) 
  (^ 
  21 
  W 
  2 
  - 
  *U«l) 
  + 
  ^g 
  fe^g) 
  (k 
  u 
  u 
  x 
  — 
  k 
  n 
  u 
  2 
  ) 
  

  

  + 
  log 
  (? 
  3 
  %) 
  (*l 
  S 
  *«l 
  — 
  ^31^3) 
  + 
  &C. 
  

  

  + 
  log 
  (^Wj) 
  (At 
  31 
  m 
  3 
  - 
  & 
  13 
  «i) 
  + 
  log 
  (? 
  2 
  w 
  2 
  ) 
  (£32% 
  — 
  *23%) 
  

   4- 
  log 
  (^ 
  3 
  %) 
  (k 
  23 
  u 
  2 
  -k 
  32 
  u 
  3 
  ) 
  + 
  &c. 
  

  

  + 
  log 
  (firi«i)(* 
  4 
  i 
  M 
  4 
  — 
  *i4«i) 
  +1 
  °g 
  (q2U 
  2 
  )(k 
  42 
  u 
  4 
  -k 
  2i 
  u 
  2 
  ) 
  

  

  + 
  Iog(q 
  3 
  u 
  d 
  )(k 
  i3 
  u 
  4 
  — 
  k 
  34 
  u 
  3 
  ) 
  + 
  &c. 
  

   + 
  <fcc. 
  

  

  =loe^^(A:2i?/2 
  — 
  A 
  12 
  Mi) 
  + 
  log-^-^ 
  (hiu 
  s 
  — 
  k 
  n 
  Ui) 
  + 
  &c. 
  

   ^q 
  2 
  u 
  2 
  K 
  *q 
  3 
  u 
  3 
  ' 
  

  

  ^3^3 
  

  

  and 
  this 
  is 
  necessarily 
  negative, 
  provided 
  that 
  constants 
  

  

  2i 
  £» 
  exist 
  such 
  that 
  k 
  12 
  /k 
  21 
  = 
  q 
  l 
  /q 
  2 
  , 
  k 
  u 
  /k 
  il 
  = 
  q 
  1 
  /q 
  3 
  , 
  &c 
  , 
  

  

  and 
  generally 
  k 
  P 
  r[k 
  rp 
  = 
  q 
  p 
  /q 
  r 
  . 
  

  

  If 
  therefore 
  the 
  exchanges 
  of 
  energy 
  between 
  the 
  parts 
  of 
  

  

  our 
  system 
  take 
  place 
  according 
  to 
  the 
  above 
  law, 
  the 
  

  

  function 
  S 
  necessarily 
  diminishes 
  with 
  the 
  time, 
  and 
  we 
  may 
  

  

  call 
  it 
  the 
  entropy 
  of 
  the 
  system. 
  Further, 
  when 
  S 
  has 
  

  

  reached 
  its 
  minimum, 
  we 
  have 
  : 
  # 
  

  

  (1) 
  By 
  conservation 
  of 
  energy 
  2 
  -7- 
  =0, 
  and 
  (2) 
  by 
  con- 
  

  

  j 
  at 
  

  

  servation 
  of 
  entropy 
  2 
  log 
  (qu) 
  — 
  =0, 
  and 
  therefore 
  log 
  qu 
  

  

  is 
  an 
  absolute 
  constant, 
  having 
  the 
  same 
  value 
  for 
  every 
  u. 
  

  

  Any 
  function 
  whose 
  time-variation 
  has 
  always 
  the 
  same 
  

   sign 
  until 
  a 
  certain 
  state 
  is 
  reached, 
  and 
  is 
  then 
  zero, 
  may 
  be 
  

   called 
  an 
  entropy 
  function. 
  It 
  is 
  not 
  necessary 
  that 
  it 
  should 
  

  

  Q2 
  

  

  