﻿Wien's 
  Distribution 
  Law. 
  193 
  

  

  Now 
  if 
  we 
  make 
  the 
  transformation 
  i/'=.y, 
  \'=\, 
  this 
  

   •curve 
  with 
  equation 
  (6) 
  coincides 
  with 
  Ci, 
  and 
  so 
  

  

  x». 
  J 
  *-<2W£)'-*' 
  w 
  - 
  ■ 
  • 
  <» 
  

  

  This 
  is 
  an 
  identity 
  for 
  all 
  values 
  of 
  1? 
  # 
  2 
  , 
  and 
  X. 
  

   Now 
  (7) 
  can 
  be 
  written 
  in 
  the 
  form 
  

  

  \ej 
  F(x) 
  ~ 
  • 
  ■ 
  ■ 
  • 
  w 
  

  

  If 
  #! 
  and 
  # 
  2 
  be 
  replaced 
  by 
  kO 
  x 
  and 
  &0 
  2 
  , 
  where 
  k 
  is 
  an 
  

   arbitrary 
  constant 
  different 
  from 
  zero, 
  the 
  value 
  of 
  the 
  left- 
  

   hand 
  member 
  of 
  (8) 
  is 
  unchanged, 
  and 
  therefore 
  the 
  value 
  

   of 
  the 
  right 
  member 
  is 
  also 
  unchanged. 
  This 
  gives 
  

  

  which 
  is 
  an 
  identity 
  for 
  all 
  values 
  of 
  ^^0. 
  Therefore 
  

  

  i/(x)">= 
  ' 
  • 
  ' 
  • 
  • 
  (9) 
  

  

  ^r^ 
  1 
  ("» 
  

  

  Substituting 
  (9) 
  in 
  (8) 
  gives 
  

  

  \0 
  2 
  ) 
  \6 
  2 
  J 
  m 
  x 
  

  

  F(X) 
  ~ 
  l 
  ; 
  

  

  In 
  order 
  to 
  obtain 
  the 
  form 
  of 
  F(X) 
  and 
  /(X) 
  we 
  will 
  

   prove 
  the 
  Lemma 
  : 
  The 
  most 
  general 
  solution 
  of 
  the 
  functional 
  

   equation 
  

  

  k^(kx) 
  _ 
  n2 
  . 
  

  

  "*W 
  _1 
  (12) 
  

  

  c 
  

  

  is 
  </>(#) 
  = 
  -^, 
  wfore 
  A:(|A|gfcO), 
  C, 
  and 
  a 
  cm? 
  constants, 
  and 
  <f> 
  

   x 
  

  

  is 
  a 
  continuous 
  function 
  of 
  x. 
  

  

  Let 
  A 
  (*)=*>(*) 
  5 
  (13) 
  

  

  then 
  A 
  (a?) 
  is 
  a 
  continuous 
  function 
  of 
  x. 
  From 
  (12) 
  and 
  

   (13) 
  it 
  follows 
  that 
  

  

  A(kx)=AU) 
  (14) 
  

  

  