﻿'220 
  Prof. 
  J. 
  H. 
  Jeans 
  on 
  the 
  

  

  the 
  accelerations 
  of 
  the 
  electrons. 
  Hence, 
  as 
  a 
  condition 
  that 
  

   equation 
  (72) 
  may 
  be 
  true, 
  we 
  must 
  have 
  

  

  39. 
  We 
  wish 
  to 
  evaluate 
  both 
  sides 
  o£ 
  this 
  equation 
  in 
  

   terms 
  of 
  electron-motion. 
  Let 
  u^ 
  r, 
  lo 
  be 
  the 
  components 
  of 
  

   velocity 
  of 
  an 
  electron 
  at 
  any 
  point 
  .i", 
  y, 
  z 
  at 
  which 
  the 
  

   potential 
  is 
  V. 
  The 
  equations 
  of 
  motion 
  of 
  the 
  electron 
  are 
  

  

  du 
  e. 
  BV 
  . 
  .^. 
  

  

  so 
  that 
  

  

  ^=5{(S)"-(S)"*{ST} 
  

  

  Hence, 
  if 
  E^ 
  denote 
  at 
  any 
  instant 
  the 
  mean 
  of 
  the 
  squares 
  

   of 
  the 
  electric 
  intensities 
  at 
  the 
  points 
  occupied 
  by 
  electrons 
  

   we 
  have 
  

  

  /^=-^ 
  (76) 
  

  

  The 
  evaluation 
  of 
  the 
  right-hand 
  member 
  of 
  equation 
  (74) 
  

   is 
  more 
  difficult, 
  as 
  it 
  involves 
  the 
  evaluation 
  of 
  /c. 
  

   If 
  we 
  denote 
  by 
  •& 
  the 
  operator 
  

  

  B 
  

   we 
  obtain 
  from 
  (75) 
  

  

  Ot 
  C^ 
  oy 
  O^ 
  

  

  ^^ 
  = 
  _-^^-i|X, 
  (77) 
  

  

  where, 
  on 
  the 
  right, 
  ^- 
  operates 
  only 
  on 
  w, 
  v, 
  w, 
  and 
  

  

  B 
  B 
  B 
  , 
  r 
  ^T 
  

  

  B^' 
  ^-.^, 
  operate 
  only 
  oirV. 
  

  

  Let 
  the 
  electrons 
  in 
  unit 
  volume 
  move 
  so 
  that 
  the 
  small 
  

   departures 
  from 
  Maxwell's 
  law 
  produce 
  an 
  infinitesimal 
  

   current 
  ix 
  parallel 
  to 
  0^, 
  this 
  being 
  given 
  (c/. 
  equation 
  (2)) 
  

  

  by 
  ■ 
  . 
  ^ 
  

  

  The 
  summation 
  of 
  equation 
  (77) 
  over 
  all 
  electrons 
  in 
  unit 
  

   volume 
  leads 
  to 
  

  

  f^Y^.= 
  -^2s»-|^ 
  (78) 
  

  

  \dtj 
  m 
  ox 
  ^ 
  

  

  If 
  we 
  take 
  the 
  expectation 
  of 
  value 
  of 
  each 
  term 
  on 
  the 
  

  

  