﻿Wien's 
  Distribution 
  Law. 
  199 
  

  

  this 
  and 
  the 
  Corollary 
  to 
  Theorem 
  I. 
  we 
  have 
  

  

  Theorem 
  II. 
  The 
  molecules 
  having 
  velocities 
  in 
  a 
  given 
  

   interval 
  for 
  = 
  l 
  will 
  have 
  their 
  new 
  velocities 
  in 
  the 
  corre- 
  

   sponding 
  velocity 
  interval 
  for 
  = 
  2 
  . 
  

  

  It 
  the 
  given 
  velocity 
  interval 
  be 
  taken 
  sufficiently 
  small, 
  

   we 
  have 
  

  

  Theorem 
  III. 
  The 
  velocities 
  Vi 
  and 
  v 
  2 
  of 
  a 
  molecule 
  of 
  a 
  

   gas 
  at 
  temperatures 
  1 
  and 
  2 
  respectively, 
  satisfy 
  the 
  relation 
  

  

  Vl 
  ^% 
  

  

  *2 
  V#2 
  

  

  Theorem 
  III. 
  may 
  be 
  restated 
  in 
  this 
  way 
  : 
  — 
  If 
  with 
  

   certain 
  units 
  v 
  1 
  2 
  = 
  K0 
  ii 
  where 
  v 
  l 
  is 
  the 
  velocity 
  of 
  a 
  molecule 
  

   of 
  a 
  gas 
  at 
  temperature 
  1 
  and 
  k 
  is 
  independent 
  of 
  1 
  and 
  v 
  l} 
  

   then 
  the 
  velocity 
  of 
  this 
  molecule 
  when 
  the 
  temperature 
  of 
  

   the 
  gas 
  is 
  2 
  will 
  be 
  v 
  2 
  where 
  v 
  2 
  2 
  = 
  k0 
  2 
  , 
  and 
  this 
  relation 
  

   holds 
  for 
  any 
  velocity 
  v 
  x 
  and 
  any 
  two 
  temperatures 
  1 
  and 
  2 
  . 
  

   Consider 
  two 
  molecules 
  m 
  1 
  and 
  m 
  2 
  of 
  a 
  gas 
  at 
  temperature 
  0i 
  

   having 
  velocities 
  i\ 
  and 
  v 
  2 
  respectively, 
  and 
  producing 
  waves 
  

   of 
  length 
  X 
  2 
  and 
  X 
  2 
  respectively 
  ; 
  if 
  the 
  temperature 
  be 
  

   increased 
  so 
  that 
  the 
  velocity 
  of 
  m 
  ± 
  is 
  v 
  2 
  , 
  the 
  temperature 
  

   now 
  being 
  2i 
  then 
  by 
  Wien's 
  first 
  hypothesis 
  the 
  length, 
  

   X 
  2 
  ', 
  of 
  the 
  waves 
  produced 
  by 
  m 
  x 
  will 
  be 
  the 
  same 
  as 
  the 
  

   length, 
  X 
  2 
  , 
  of 
  the 
  waves 
  produced 
  by 
  m 
  2 
  when 
  the 
  tem- 
  

   perature 
  was 
  lm 
  Now, 
  from 
  (27) 
  and 
  the 
  law 
  X0 
  = 
  a 
  const., 
  

   and 
  since 
  X^X^, 
  

  

  This 
  proves 
  

  

  Theorem 
  IV. 
  The 
  length 
  of 
  the 
  waves 
  produced 
  by 
  a 
  

   molecule 
  in 
  a 
  gas 
  at 
  temperature 
  varies 
  inversely 
  as 
  the 
  

   square 
  of 
  the 
  velocity 
  ; 
  l. 
  e. 
  

  

  v 
  2 
  = 
  — 
  , 
  where 
  b 
  is 
  a 
  constant. 
  . 
  . 
  . 
  (28) 
  

   X 
  

  

  5. 
  A 
  derivation 
  of 
  the 
  Displacement 
  Law 
  from 
  Wiens 
  

   hypotheses. 
  

   Let 
  us 
  consider 
  the 
  distribution 
  of 
  <f>, 
  the 
  intensity 
  of 
  

   radiation 
  from 
  a 
  gas 
  at 
  temperature 
  X 
  with 
  regard 
  to 
  X, 
  the 
  

   wave-length. 
  Denote 
  by 
  K 
  x 
  the 
  curve 
  obtained 
  by 
  taking 
  

   X 
  as 
  the 
  abscissa 
  and 
  as 
  the 
  ordinate. 
  We 
  will 
  fix 
  our 
  

   attention 
  upon 
  the 
  radiation 
  associated 
  with 
  values 
  of 
  X 
  in 
  

   the 
  interval, 
  X^X^Xj 
  -\-B\ 
  u 
  where 
  X 
  l 
  and 
  8\ 
  x 
  are 
  arbitrarily 
  

   chosen 
  (except 
  that 
  8k 
  x 
  is 
  very 
  small). 
  Suppose 
  that 
  the 
  

  

  