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  XXI. 
  A 
  Criticism 
  of 
  Wien's 
  Distribution 
  Law. 
  

   By 
  Fkank 
  Edwin 
  Wood 
  *. 
  

  

  1. 
  Introduction. 
  

  

  THE 
  purpose 
  of 
  this 
  article 
  is 
  to 
  criticise 
  Wien's 
  distri- 
  

   bution 
  law 
  from 
  a 
  mathematical 
  point 
  of 
  view 
  : 
  — 
  To 
  

   show 
  (1) 
  that 
  although 
  the 
  derivation 
  of 
  Wien's 
  distribution 
  

   law 
  is 
  generally 
  made 
  by 
  steps 
  which 
  are 
  not 
  mathematicall}'- 
  

   justified, 
  or 
  for 
  which 
  no 
  rigorous 
  justification 
  is 
  given, 
  still, 
  

   by 
  using 
  Wien's 
  assumptions 
  t, 
  a 
  rigorous 
  derivation 
  is 
  

   possible 
  ; 
  (2) 
  that 
  the 
  law 
  obtained 
  by 
  Wien 
  is 
  inconsistent 
  

   with 
  other 
  results 
  which 
  follow 
  from 
  the 
  same 
  assumptions 
  ; 
  

   and 
  (3) 
  that 
  this 
  inconsistency 
  is 
  eliminated 
  and 
  a 
  new 
  law 
  \ 
  

   obtained 
  if 
  Wien's 
  implicit 
  assumption 
  be 
  replaced 
  by 
  a 
  

   simpler 
  and 
  more 
  probable 
  one. 
  

  

  Incidentally 
  several 
  theorems, 
  new 
  so 
  far 
  as 
  the 
  author 
  

   knows, 
  in 
  the 
  kinetic 
  theory 
  of 
  gases 
  will 
  be 
  obtained 
  from 
  

   the 
  Maxwell 
  law 
  for 
  the 
  distribution, 
  with 
  respect 
  to 
  their 
  

   velocities, 
  of 
  the 
  molecules 
  in 
  a 
  gas 
  ; 
  a 
  simple 
  proof 
  of 
  the 
  

   Wien 
  displacement 
  law 
  will 
  be 
  obtained 
  from 
  these 
  theorems 
  

   and 
  the 
  Wien 
  assumptions, 
  and 
  two 
  interesting 
  relations 
  

   regarding 
  the 
  dependence 
  of 
  the 
  radiation 
  of 
  a 
  molecule 
  upon 
  

   its 
  velocity 
  will 
  be 
  given. 
  Also 
  there 
  will 
  be 
  found 
  some 
  

   criticisms 
  of 
  the 
  treatment 
  of 
  the 
  distribution 
  law 
  and 
  of 
  

   the 
  displacement 
  law 
  as 
  given 
  in 
  the 
  standard 
  treatises. 
  

  

  Mendenhall 
  and 
  Saunders 
  §, 
  Waidner 
  and 
  Burgess 
  ||, 
  

   Rayleigh 
  % 
  and 
  others 
  have 
  criticised 
  from 
  a 
  physical 
  point 
  

   of 
  view 
  the 
  assumptions 
  used 
  by 
  Wien 
  to 
  prove 
  his 
  distri- 
  

   bution 
  law, 
  while 
  Lummer 
  and 
  Pringsheim 
  **, 
  Paschen 
  ft, 
  

   and 
  others 
  have 
  considered 
  the 
  agreement 
  of 
  this 
  law 
  with 
  

   experimental 
  results. 
  So 
  far 
  as 
  the 
  author 
  knows, 
  no 
  

   criticism 
  from 
  a 
  mathematical 
  point 
  of 
  view 
  has 
  been 
  

   published. 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  t 
  It 
  will 
  be 
  necessary 
  to 
  include 
  under 
  Wien's 
  assumptions 
  one 
  which 
  

   he 
  has 
  made 
  implicitly, 
  but 
  not 
  explicitly 
  ; 
  or 
  else 
  to 
  assume 
  that 
  he 
  

   has 
  made 
  a 
  fundamental 
  error. 
  

  

  % 
  I 
  am 
  indebted 
  to 
  Professor 
  Lunn, 
  of 
  the 
  University 
  of 
  Chicago, 
  for 
  

   this 
  new 
  formula, 
  and 
  for 
  the 
  observation 
  that 
  Wien 
  either 
  made 
  a 
  

   mistake 
  or 
  an 
  unstated 
  assumption. 
  

  

  S 
  Astrophysical 
  Journal, 
  xiii. 
  p. 
  25 
  (1901). 
  

   || 
  Bull. 
  Bur. 
  of 
  Standards, 
  i. 
  p. 
  189 
  (1904). 
  

  

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

  

  ** 
  Verh. 
  d. 
  Deutsch. 
  Phys. 
  Ges. 
  i. 
  p. 
  1 
  (1900). 
  

  

  tt 
  Astrophysical 
  Journal, 
  x. 
  p. 
  40 
  (1899) 
  ; 
  xi. 
  p. 
  288 
  (1900). 
  

  

  