﻿212 
  Prof. 
  J. 
  H. 
  Jeans 
  on 
  the 
  

  

  in 
  which 
  q 
  may 
  now 
  be 
  regarded 
  as 
  simply 
  an 
  abbreviation^ 
  

   iovVpjY, 
  

  

  We 
  have 
  in 
  effect 
  imagined 
  the 
  motion 
  of 
  the 
  electrons 
  

   parallel 
  to 
  Ox 
  to 
  be 
  resolved 
  into 
  a 
  doubly 
  infinite 
  series 
  

   o£ 
  regular 
  waves 
  corresponding 
  to 
  all 
  values 
  o£ 
  j? 
  and 
  q. 
  

   Analysis 
  has 
  now 
  shown 
  that 
  the 
  only 
  waves 
  which 
  con- 
  

   tribute 
  at 
  all 
  to 
  the 
  radiation 
  are 
  those 
  for 
  which 
  p 
  and 
  q 
  are 
  

   connected 
  by 
  the 
  relation 
  q 
  = 
  rpjY. 
  Thus 
  the 
  only 
  waves 
  in 
  

   the 
  prism 
  parallel 
  to 
  Ox 
  from 
  which 
  the 
  radiation 
  is 
  appre-- 
  

   ciable 
  are 
  those 
  with 
  an 
  exponential 
  factor 
  g^>^-i^Wv. 
  It 
  

   follows 
  that 
  when 
  the 
  whole 
  three-dimensional 
  motion 
  of 
  

   electrons 
  is 
  resolved 
  into 
  waves 
  of 
  electrons 
  travelling 
  in 
  all 
  

   directions 
  with 
  all 
  possible 
  frequencies 
  and 
  wave-lengths, 
  

   the 
  only 
  waves 
  which 
  contribute 
  to 
  the 
  radiation 
  in 
  the 
  

   direction 
  l\ 
  ??/, 
  n^ 
  are 
  those 
  having 
  the 
  exponential 
  factor 
  

  

  ^ipU- 
  {l'x-\-m'y+n'z)[Y 
  \ 
  , 
  

  

  These 
  are 
  waves 
  of 
  electrons 
  travelling 
  in 
  the 
  direction 
  

   /', 
  w', 
  n' 
  with 
  the 
  velocity 
  of 
  light. 
  In 
  fact 
  it 
  can 
  easily 
  be 
  

   shown, 
  by 
  direct 
  physical 
  reasoning, 
  that 
  the 
  disturbances, 
  

   sent 
  out 
  in 
  the 
  direction 
  l\ 
  ni\ 
  n' 
  by 
  the 
  different 
  elements 
  

   of 
  a 
  wave 
  of 
  electrons 
  which 
  is 
  not 
  of 
  this 
  kind, 
  must 
  

   annihilate 
  one 
  another 
  by 
  interference. 
  

   2^. 
  Putting 
  

  

  we 
  see 
  that 
  equa4;ion 
  (54) 
  can 
  be 
  expressed 
  in 
  the 
  form 
  

  

  X=l 
  00 
  

  

  J 
  Jx^"^' 
  =- 
  J 
  ypGOspit-e) 
  dp. 
  

  

  x=-l 
  

  

  On 
  replacing 
  q 
  hy 
  pl'/Y, 
  we 
  have 
  {cf. 
  equations 
  ^51). 
  

   ^nd 
  (49)) 
  

  

  dqp 
  = 
  \ 
  2, 
  eu 
  cos 
  pt 
  cos 
  -y- 
  "S 
  

  

  so 
  that, 
  from 
  this 
  and 
  similar 
  equations, 
  

  

  u^p 
  + 
  ^\p 
  = 
  r 
  S 
  eu 
  cos 
  p(^t-~^ 
  dt 
  

  

  Jt 
  / 
  y^\ 
  

  

  l^eusinplt^^jdt. 
  

  

  Squaring 
  and 
  adding, 
  we 
  obtain 
  

  

  7/= 
  r 
  Cxt 
  ehi.ii', 
  cos 
  p(h^t, 
  - 
  IXfiTlfi)^ 
  dt, 
  dt„ 
  . 
  (56) 
  

  

  the 
  summation 
  extending 
  over 
  all 
  possible 
  pairs 
  of 
  electrons,, 
  

   one 
  being 
  taken 
  at 
  instant 
  t^ 
  and 
  the 
  other 
  at 
  instant 
  ^2- 
  

  

  