﻿Processes 
  and 
  Planck's 
  Theory. 
  235 
  

  

  Also 
  

  

  S' 
  — 
  S= 
  log 
  -, 
  — 
  j-dt 
  

  

  av 
  to 
  bv 
  dt 
  

  

  = 
  1, 
  U 
  3c 
  2 
  

  

  , 
  U3 
  C 
  WU 
  __\ 
  . 
  %a 
  . 
  

  

  Therefore 
  

  

  y 
  + 
  8'-(, 
  + 
  8)=-i^ 
  S 
  in^(f- 
  Kl 
  )lo 
  g 
  ^^ 
  

  

  which 
  is 
  necessarily 
  positive 
  owing 
  to 
  the 
  minus 
  sign 
  

   prefixed. 
  

  

  But 
  s 
  x 
  >s 
  by 
  art. 
  1. 
  

  

  Therefore, 
  a 
  fortiori, 
  V 
  + 
  S'^ 
  + 
  S; 
  or 
  the 
  entropy 
  of 
  the 
  

   resonator 
  plus 
  the 
  hypothetical 
  entropy 
  of 
  the 
  wave 
  after 
  

   incidence 
  is 
  greater 
  than 
  the 
  sum 
  of 
  the 
  actual 
  entropies 
  of 
  

   wave 
  and 
  resonator 
  before 
  incidence. 
  

  

  17. 
  But 
  also 
  Si>s 
  by 
  art. 
  1. 
  

  

  It 
  does 
  not 
  therefore 
  yet 
  follow 
  that 
  ■ 
  s' 
  + 
  S'>s 
  + 
  S, 
  which 
  

   is 
  the 
  proposition 
  we 
  wish 
  to 
  prove. 
  

  

  To 
  make 
  the 
  proof 
  complete 
  at 
  this 
  point 
  it 
  is 
  necessary 
  to 
  

   assume 
  that 
  K 
  2 
  and 
  K 
  3 
  are 
  after 
  incidence 
  the 
  principal 
  inten- 
  

   sities; 
  and 
  therefore 
  */==#'. 
  And 
  this 
  Planck 
  asserts 
  to 
  be 
  the 
  

   case 
  (p. 
  468). 
  

  

  He 
  does 
  not, 
  however, 
  explain 
  the 
  grounds 
  for 
  this 
  

   assertion. 
  The 
  principal 
  directions 
  before 
  incidence 
  are 
  

   independent 
  of 
  the 
  direction 
  Oz 
  ( 
  see 
  figure) 
  of 
  the 
  resonator 
  

   axis. 
  They 
  cannot 
  therefore 
  generally 
  be 
  OP, 
  OH, 
  which 
  are 
  

   determined 
  by 
  the 
  direction 
  0^. 
  Therefore 
  generally 
  K 
  2 
  in 
  

   OP 
  is 
  neither 
  the 
  greatest 
  nor 
  the 
  least 
  intensity. 
  By 
  the 
  

   incidence 
  on 
  the 
  resonator 
  the 
  intensity 
  K 
  x 
  in 
  OR 
  is 
  altered 
  by 
  

  

  a 
  very 
  small 
  quantity 
  proportional 
  to 
  - 
  — 
  . 
  This 
  cannot 
  

  

  generally 
  have 
  the 
  effect 
  of 
  making 
  K 
  2 
  either 
  the 
  greatest 
  or 
  

   the 
  least 
  intensity. 
  

  

  18. 
  The 
  proof, 
  can, 
  however, 
  as 
  seems 
  to 
  me, 
  be 
  completed 
  

   as 
  follows 
  : 
  — 
  For 
  any 
  given 
  wave-front, 
  all 
  directions 
  of 
  the 
  

   axis 
  O2 
  of 
  the 
  resonator 
  are 
  equally 
  probable. 
  Therefore 
  on 
  

   average 
  of 
  all 
  waves 
  and 
  resonators 
  with 
  the 
  same 
  v, 
  

  

  (*!-*)= 
  (V-*')- 
  

  

  And 
  now 
  

  

  = 
  s^ 
  + 
  S' 
  — 
  (Vl 
  + 
  8), 
  on 
  average. 
  

   And 
  this 
  we 
  have 
  proved 
  to 
  be 
  positive. 
  

  

  