﻿On 
  Wood's 
  Criticism 
  of 
  Mien's 
  Distribution 
  Law. 
  411 
  

  

  of 
  energy 
  by 
  a 
  molecule 
  of 
  velocity 
  r, 
  that 
  emitted 
  by 
  all 
  

   the 
  molecules 
  vv-hose 
  velocities 
  lie 
  between 
  v 
  and 
  v 
  + 
  dv 
  is 
  

  

  dU 
  = 
  <j>(\ 
  e)d\ 
  = 
  ^7r- 
  h 
  a- 
  3 
  v 
  2 
  ee- 
  v2 
  ' 
  a2 
  dv, 
  . 
  . 
  (2; 
  

  

  where 
  e 
  is 
  a 
  function 
  of 
  v 
  only. 
  

  

  Now 
  the 
  total 
  radiation 
  for 
  all 
  wave-lengths 
  together 
  is 
  

   obtained 
  bv 
  integrating 
  this 
  from 
  r 
  = 
  to 
  v 
  = 
  co 
  , 
  and 
  must 
  

   by 
  Stefan's 
  law 
  be 
  proportional 
  to 
  6 
  i 
  , 
  and 
  therefore 
  to 
  a 
  8 
  . 
  

   We 
  have 
  thus 
  an 
  integral 
  equation 
  for 
  e. 
  Assume 
  that 
  e(r) 
  

   can 
  be 
  expanded 
  in 
  powers 
  of 
  v 
  for 
  all 
  values 
  of 
  w, 
  so 
  that 
  

  

  e(u) 
  = 
  W«, 
  ...... 
  (3) 
  

  

  o 
  

  

  oo 
  

  

  and 
  that 
  the 
  series 
  Sa 
  n 
  v 
  n+2 
  ^~ 
  r/a2 
  can 
  be 
  integrated 
  term 
  by 
  

   term. 
  ° 
  

  

  Then 
  B, 
  can 
  be 
  expressed 
  as 
  a 
  power 
  series 
  in 
  a, 
  thus 
  : 
  

  

  By 
  hypothesis 
  R 
  is 
  equal 
  to 
  Nc 
  2 
  « 
  8 
  , 
  where 
  c 
  is 
  an 
  absolute 
  

   constant, 
  and 
  therefore 
  all 
  the 
  a's 
  are 
  zero 
  except 
  a 
  8 
  . 
  Hence 
  

  

  A 
  ^ 
  9 
  1.3.5.7.9 
  XT 
  

   Nc 
  2 
  = 
  ^ 
  Na 
  8 
  , 
  

  

  and 
  therefore 
  

  

  rfR=^^-Nc 
  2 
  a-VV 
  eS/a, 
  rft\ 
  ... 
  (4) 
  

   94o 
  

  

  Thus 
  the 
  distribution 
  of 
  energy 
  with 
  regard 
  to 
  the 
  

   molecular 
  velocities 
  is 
  completely 
  found 
  from 
  the 
  first 
  

   assumption 
  alone, 
  and 
  all 
  that 
  is 
  now 
  required 
  is 
  to 
  determine 
  

   the 
  relation 
  of 
  the 
  velocity 
  to 
  the 
  wave-length. 
  

  

  At 
  this 
  stage 
  a 
  further 
  assumption 
  is 
  needed 
  ; 
  and 
  this 
  

   will 
  be 
  that 
  the 
  graph 
  of 
  (f> 
  against 
  X 
  for 
  any 
  temperature 
  

   can 
  be 
  derived 
  from 
  that 
  for 
  any 
  other 
  temperature 
  by 
  

   homogeneous 
  strain 
  in 
  two 
  dimensions. 
  In 
  other 
  words, 
  if 
  

   the 
  temperature 
  be 
  changed 
  from 
  6 
  to 
  k 
  2 
  0, 
  or, 
  what 
  is 
  the 
  

   same 
  thing, 
  if 
  a 
  be 
  replaced 
  by 
  £a, 
  two 
  numbers 
  a 
  and 
  b 
  will 
  

   exist, 
  so 
  that 
  

  

  <£(<i\,P0) 
  = 
  ty(\,0) 
  (5) 
  

  

  for 
  all 
  values 
  of 
  X, 
  where 
  a 
  and 
  b 
  are 
  functions 
  of 
  k 
  alone. 
  

  

  Now 
  it 
  is 
  evident 
  from 
  (4), 
  by 
  putting 
  r 
  2 
  =/(\), 
  that 
  

   </>(\, 
  6) 
  is 
  of 
  the 
  form 
  

  

  *-*F(\)e- 
  AkVa 
  \ 
  (6) 
  

  

  where 
  F 
  and 
  /are 
  at 
  present 
  unknown, 
  but 
  are 
  connected 
  

   by 
  a 
  differential 
  relation. 
  

  

  