﻿Wien's 
  Distribution 
  Law. 
  201 
  

  

  Therefore, 
  the 
  ratio 
  of 
  corresponding 
  radiation 
  elements 
  is 
  a 
  

   function 
  of 
  0i 
  and 
  6 
  2 
  respectively 
  ; 
  in 
  particular 
  this 
  ratio 
  

   does 
  not 
  depend 
  upon 
  the 
  value 
  of 
  X 
  about 
  which 
  one 
  element 
  is 
  

   chosen. 
  

  

  Since 
  the 
  area 
  of 
  each 
  radiation 
  element 
  under 
  K 
  2 
  is 
  

   equal 
  to 
  a 
  constant 
  times 
  the 
  area 
  of 
  the 
  corresponding 
  

   radiation 
  element 
  under 
  K 
  1? 
  then 
  the 
  total 
  area 
  under 
  K 
  2 
  is 
  

   equal 
  to 
  the 
  total 
  area 
  under 
  K^ 
  multiplied 
  by 
  this 
  constant 
  ; 
  

  

  but 
  bv 
  the 
  Stefan-Boltzmann 
  law 
  this 
  constant 
  is 
  ( 
  -£ 
  \ 
  . 
  

   Therefore 
  W 
  

  

  

  (32) 
  

  

  and 
  therefore 
  

  

  This 
  is 
  another 
  form 
  of 
  equation 
  (5), 
  and 
  our 
  derivation 
  of 
  

   the 
  displacement 
  law 
  is 
  complete. 
  

   From 
  (31) 
  and 
  (32) 
  it 
  follows 
  that 
  

  

  

  

  

  g{ 
  K 
  d 
  1 
  ) 
  

  

  g(fcd 
  2 
  ) 
  

  

  2 
  4 
  ' 
  

  

  from 
  which 
  

  

  

  9 
  ("Oil 
  9 
  

   OS 
  ~ 
  

  

  (k6 
  2 
  ) 
  

  

  where 
  

  

  c 
  is 
  a 
  

  

  con 
  

  

  stant. 
  Then 
  

   gUd) 
  

  

  = 
  c0\ 
  

  

  (33) 
  

  

  and 
  from 
  (29) 
  it 
  follows 
  that 
  e 
  = 
  cV 
  ; 
  i. 
  e. 
  the 
  intensity 
  of 
  

   radiation 
  given 
  off 
  by 
  a 
  molecule 
  varies 
  directly 
  as 
  the 
  eighth 
  

   power 
  of 
  its 
  velocity. 
  

  

  * 
  In 
  Wien's 
  first 
  article 
  upon 
  the 
  displacement 
  law 
  he 
  assumed 
  that 
  

   (32) 
  followed 
  immediately 
  from 
  the 
  Stefan-Boltzmann 
  law 
  ; 
  in 
  his 
  

   second 
  article 
  the 
  error 
  is 
  corrected, 
  but 
  the 
  derivation 
  given 
  is 
  difficult, 
  

   even 
  though 
  rigorous, 
  so 
  most 
  authors 
  have 
  followed 
  Wien's 
  first 
  

   method 
  and 
  have 
  not 
  remedied 
  the 
  errors 
  : 
  cf. 
  Wood, 
  Physical 
  Optics, 
  

   p. 
  620 
  ; 
  Sachur, 
  Thermochemie 
  und 
  Thermodynamik, 
  p. 
  302 
  ; 
  Winkel- 
  

   mann, 
  Hanihuch 
  der 
  Physik, 
  iii. 
  pp. 
  379-80. 
  

  

  