﻿Wien's 
  Distribution 
  Law, 
  197 
  

  

  4. 
  A 
  proof 
  of 
  some 
  auxiliary 
  theorems 
  in 
  the 
  Kinetic 
  theory 
  

   of 
  Gases. 
  

  

  Before 
  giving 
  oar 
  derivation 
  of 
  Wien's 
  displacement 
  law, 
  

  

  it 
  will 
  be 
  necessary 
  to 
  obtain 
  some 
  laws 
  in 
  the 
  kinetic 
  theory 
  

  

  of 
  gases. 
  Let 
  us 
  consider 
  the 
  distribution 
  of 
  the 
  molecules 
  

  

  of 
  a 
  given 
  homogeneous 
  gas 
  at 
  a 
  temperature 
  Y 
  with 
  regard 
  

  

  to 
  their 
  velocities, 
  v; 
  if 
  we 
  plot 
  v 
  2 
  as 
  absciss;;, 
  and 
  y, 
  the 
  

  

  corresponding 
  number 
  of 
  molecules, 
  as 
  ordinate*, 
  then 
  the 
  

  

  curve 
  l5 
  which 
  gives 
  the 
  distribution, 
  will 
  by 
  Maxwell's 
  f 
  

  

  kv 
  2 
  - 
  ~ 
  

   law 
  have 
  the 
  equation 
  y— 
  -^e 
  6l 
  , 
  where 
  k 
  and 
  I 
  are 
  inde- 
  

  

  pendent 
  of 
  v 
  and 
  0^ 
  The 
  equation 
  of: 
  the 
  corresponding 
  

   curve 
  Oj 
  for 
  the 
  temperature 
  6 
  2 
  >6 
  1 
  (this 
  restriction 
  is 
  

   convenient, 
  but 
  unnecessary) 
  upon 
  the 
  (V 
  2 
  , 
  Y) 
  plane 
  will 
  be 
  

   v 
  *V* 
  - 
  = 
  

  

  Y 
  = 
  p 
  "a 
  

  

  1 
  — 
  AS/2 
  e 
  ' 
  __ 
  

  

  The 
  transformation 
  V 
  2 
  =-^v 
  2 
  , 
  Y=\/^ 
  : 
  y 
  takes 
  2 
  into 
  

  

  fQ- 
  "i 
  v 
  #2 
  

  

  Of 
  Let 
  Vi 
  and 
  \/ 
  ^Vi 
  be 
  called 
  corresponding 
  velocities 
  for 
  

   the 
  temperatures 
  6 
  1 
  and 
  6 
  2 
  respectively 
  ; 
  and 
  the 
  intervals 
  

   v 
  1 
  <v<v 
  2 
  and 
  \/ 
  7r 
  v 
  i— 
  w 
  — 
  \/ 
  zr 
  r 
  2 
  De 
  called 
  corresponding 
  

  

  velocity 
  intervals 
  for 
  the 
  temperatures 
  # 
  x 
  and 
  d 
  2 
  respectively. 
  

  

  Now 
  

  

  C^kv 
  2 
  _ 
  Z 
  J* 
  2N 
  /OA^kV 
  2 
  -?^™ 
  ,„- 
  

  

  \,w 
  e 
  6ld(v) 
  =vi)e^w 
  e 
  Hd(J) 
  > 
  (5) 
  

  

  and 
  (25) 
  holds 
  for 
  all 
  finite 
  values 
  of 
  v 
  1 
  and 
  v 
  2 
  and 
  also 
  for 
  

   v 
  1 
  = 
  0, 
  3 
  =oo, 
  i. 
  £. 
  

  

  J 
  ^ 
  ^ 
  2 
  )=WJ 
  y 
  ^ 
  72 
  )- 
  < 
  2b 
  > 
  

  

  It 
  follows 
  from 
  (26) 
  that 
  the 
  total 
  area 
  under 
  the 
  curve 
  

   C 
  x 
  does 
  not 
  equal 
  the 
  total 
  area 
  under 
  the 
  curve 
  C 
  2 
  , 
  as 
  one 
  

   might 
  expect 
  from 
  the 
  fact 
  that 
  in 
  each 
  case 
  the 
  area 
  is 
  

   proportional 
  to 
  the 
  total 
  number 
  of 
  molecules 
  in 
  the 
  gas, 
  a 
  

   number 
  which 
  is 
  unchanged 
  by 
  the 
  rise 
  in 
  temperature. 
  

  

  * 
  Strictly 
  speaking, 
  there 
  will 
  be 
  in 
  general 
  no 
  molecule 
  of 
  the 
  gas 
  

   having 
  a 
  yiven 
  velocity 
  v 
  at 
  a 
  given 
  instant 
  ; 
  however, 
  if 
  N 
  molecules 
  

   have 
  their 
  velocities 
  within 
  a 
  very 
  small 
  interval 
  dv 
  about 
  v, 
  where 
  the 
  

   width 
  of 
  dv 
  is 
  preassigned, 
  then 
  we 
  shall 
  say 
  for 
  brevity 
  that 
  N 
  molecules 
  

   have 
  the 
  velocity 
  v. 
  

  

  t 
  Scientific 
  Papers 
  of 
  James 
  Clerk 
  Maxwell, 
  i. 
  p. 
  381. 
  

  

  