﻿A 
  Criticism 
  of 
  Wien's 
  Distribution 
  Law. 
  191 
  

  

  Wien's 
  distribution 
  law 
  is 
  

  

  C) 
  e 
  c/\e 
  

  

  *(M)=-£-, 
  • 
  (i) 
  

  

  where 
  cf>(X, 
  6)dX 
  represents 
  the 
  intensity 
  of 
  radiation 
  of 
  a 
  

   black 
  body 
  at 
  a 
  temperature 
  6 
  produced 
  by 
  waves 
  whose 
  

   lengths 
  lie 
  between 
  X 
  and 
  X-\-dX, 
  and 
  where 
  C 
  and 
  care 
  

   constants 
  *. 
  M. 
  Planck 
  f 
  obtained 
  this 
  same 
  formula 
  (and 
  

   also 
  another 
  {) 
  but 
  from 
  considerations 
  entirely 
  different 
  

   from 
  those 
  used 
  by 
  Wien. 
  The 
  criticism 
  of 
  this 
  article 
  does 
  

   not 
  apply 
  to 
  Planck's 
  work 
  ; 
  however, 
  it 
  has 
  been 
  suggested 
  

   that 
  a 
  similar 
  criticism 
  might 
  apply 
  to 
  Planck's 
  derivation 
  

   of 
  this 
  same 
  law. 
  Other 
  formulas 
  for 
  c£(\, 
  6) 
  have 
  been 
  

   obtained 
  by 
  Callendar 
  § 
  and 
  Rayleigh 
  || 
  ; 
  although 
  these 
  

   formulas 
  may 
  be 
  in 
  closer 
  accord 
  with 
  experimental 
  results, 
  

   still 
  the 
  Wien 
  formula 
  has 
  considerable 
  importance 
  due 
  to 
  

   its 
  use 
  by 
  Drude 
  and 
  many 
  other 
  investigators. 
  

  

  2. 
  A 
  derivation 
  of 
  the 
  Distribution 
  Law 
  along 
  the 
  lines 
  

   proposed 
  by 
  Wien. 
  

  

  Wien 
  takes 
  a 
  gas 
  as 
  the 
  black 
  body, 
  and 
  uses 
  Maxwell's 
  

   law 
  that 
  the 
  number 
  of 
  molecules 
  whose 
  velocities 
  lie 
  between 
  

   ■v 
  and 
  v 
  + 
  dv 
  is 
  proportional 
  to 
  

  

  v 
  2 
  e-^ 
  2 
  dv, 
  (2) 
  

  

  where 
  a 
  2 
  = 
  §£ 
  2 
  , 
  and 
  v 
  is 
  the 
  root-mean-square 
  velocity 
  ; 
  a 
  2 
  is 
  

   proportional 
  to 
  0, 
  the 
  absolute 
  temperature 
  of 
  the 
  gas. 
  

   Wien 
  makes 
  the 
  hypotheses 
  : 
  

  

  (a) 
  That 
  the 
  length 
  of 
  the 
  wave 
  sent 
  out 
  by 
  a 
  molecule 
  

   depends 
  only 
  upon 
  the 
  velocity 
  of 
  that 
  molecule 
  : 
  — 
  then 
  v 
  is 
  

   a 
  function 
  of 
  X 
  only. 
  

  

  (b) 
  That 
  the 
  intensity 
  of 
  the 
  radiation 
  for 
  wave-lengths 
  

   between 
  X 
  and 
  X-\-dX 
  is 
  proportional 
  to 
  the 
  number 
  of 
  

   molecules, 
  as 
  given 
  by 
  Maxwell's 
  law 
  (2), 
  which 
  send 
  out 
  

   waves 
  with 
  lengths 
  between 
  X 
  and 
  X-f 
  dX. 
  

  

  Wien 
  states 
  that 
  it 
  follows 
  from 
  these 
  two 
  hypotheses 
  that 
  

  

  _fVO 
  

   4>(\,0)=¥(\)e~ 
  ° 
  , 
  (3) 
  

  

  where 
  F(\) 
  and 
  /(A.) 
  are 
  two 
  unknown 
  functions^. 
  

  

  * 
  Wied. 
  Ann. 
  lviii. 
  p. 
  662 
  (1896). 
  

  

  t 
  Wied. 
  Ann. 
  i. 
  pp. 
  69, 
  719 
  (1900). 
  

  

  t 
  Verh. 
  d. 
  Deutsch. 
  Phys. 
  Ges. 
  ii. 
  p. 
  202 
  (1900). 
  

  

  * 
  Phil. 
  Mag. 
  xxvi. 
  pp. 
  787 
  (1913) 
  ; 
  xxvii. 
  p. 
  870 
  (1914). 
  

   || 
  Phil. 
  Mag. 
  xlix. 
  p. 
  539 
  (1900). 
  

  

  4f 
  In 
  our 
  development 
  of 
  Wien's 
  distribution 
  law, 
  it 
  is 
  assumed 
  that 
  

   F(X) 
  and/(X) 
  are 
  continuous 
  functions. 
  Just 
  what 
  physical 
  significance 
  

   these 
  assumptions 
  have 
  can 
  be 
  seen 
  from 
  § 
  6 
  of 
  this 
  article. 
  

  

  P2 
  

  

  