﻿216 
  Prof. 
  J. 
  H. 
  Jeans 
  on 
  the 
  

  

  compared 
  with 
  the 
  time 
  of 
  a 
  complete 
  rearrangement 
  of 
  

   the 
  velocities 
  of 
  the 
  electrons. 
  But 
  when 
  the 
  aether 
  is 
  filled 
  

   with 
  pulses 
  and 
  short 
  waves 
  of 
  sufficient 
  intensity, 
  this 
  latter 
  

   interval 
  may 
  be 
  treated 
  as 
  infinitesimal, 
  and 
  equations 
  (62) 
  

   and 
  (63) 
  may 
  be 
  supposed 
  true 
  for 
  all 
  values 
  of 
  p 
  and 
  

  

  33. 
  Proceeding 
  as 
  before 
  (§ 
  19), 
  we 
  integrate 
  expression 
  

   (59) 
  with 
  the 
  help 
  of 
  relation 
  (63) 
  and 
  obtain 
  for 
  the 
  emission 
  

   per 
  unit 
  volume 
  per 
  unit 
  time, 
  

  

  ^^^^T«;. 
  ^^ 
  ^g^^ 
  

  

  J 
  

  

  For 
  simplicity 
  suppose 
  the 
  enclosure 
  (of 
  volume 
  A) 
  to 
  be 
  

   completely 
  filled 
  with 
  homogeneous 
  matter 
  (K, 
  fi). 
  Let 
  the 
  

   radiant 
  energy 
  per 
  unit 
  volume, 
  analysed 
  into 
  waves 
  of 
  

   different 
  frequencies, 
  be 
  ^Fjpdp. 
  

  

  The 
  total 
  radiant 
  energy 
  of 
  frequency 
  between 
  p 
  and 
  

   p-\-dp 
  in 
  the 
  whole 
  enclosure 
  is 
  AEpdp. 
  The 
  rate 
  at 
  which 
  

   this 
  disappears 
  owing 
  to 
  absorption 
  by 
  matter 
  is 
  as 
  in 
  

   Part 
  I. 
  (§ 
  18), 
  

  

  -^4.7rY'fiAEpdp, 
  .... 
  (65) 
  

  

  _ 
  ..2 
  

   1 
  + 
  

  

  K'^p^ni' 
  

  

  in 
  which 
  YV 
  replaces 
  C7K. 
  The 
  rate 
  at 
  which 
  the 
  energy 
  

   in 
  question 
  is 
  added 
  to 
  by 
  new 
  emissions 
  from 
  the 
  matter 
  is 
  

   at 
  once 
  obtainable 
  from 
  formula 
  (64). 
  Combining 
  loss 
  and 
  

   gain, 
  we 
  find 
  for 
  the 
  complete 
  rate 
  of 
  change 
  in 
  E^, 
  

  

  d^p 
  _ 
  fCfjL 
  f 
  V 
  RT 
  

  

  dt 
  K^p^ 
  

  

  m* 
  

  

  {1^-..Y^E,}. 
  . 
  (66) 
  

  

  34. 
  The 
  steady 
  state 
  is 
  given 
  by 
  dEp/dt 
  = 
  Oj 
  and 
  therefore 
  

   by 
  

  

  E,dp 
  = 
  ^^^,nTp'dp, 
  

  

  = 
  S7rUT\-^dX, 
  (67) 
  

  

  which 
  is 
  exactly 
  the 
  value 
  required 
  for 
  equipartition 
  of 
  

   energy. 
  Moreover, 
  it 
  will 
  readily 
  be 
  found 
  that 
  formula 
  

   (67) 
  exactly 
  expresses 
  the 
  steady 
  state 
  even 
  after 
  allowing 
  

   for 
  the 
  corrections 
  required 
  by 
  the 
  Doppler-effect, 
  which 
  in 
  

   our 
  present 
  analysis 
  have 
  been 
  neglected. 
  Thus 
  the 
  steady 
  

   »state 
  is 
  one 
  of 
  absolute 
  equipartition 
  of 
  energy. 
  

  

  