﻿210 
  Prof. 
  J. 
  H. 
  Jeans 
  on 
  the 
  

  

  Mx, 
  My, 
  M^) 
  be 
  the 
  moment 
  of 
  the 
  doublets 
  in 
  dv 
  at 
  any 
  

   instant, 
  then 
  in 
  the 
  notation 
  already 
  used, 
  

  

  —7—^ 
  = 
  Xeu 
  = 
  ix 
  dv 
  (46) 
  

  

  Here 
  X, 
  which 
  in 
  the 
  previous 
  paper 
  denoted 
  summation 
  

   over 
  a 
  great 
  number 
  of 
  electrons, 
  will 
  now, 
  on 
  account 
  of 
  

   the 
  smallness 
  of 
  dv, 
  represent 
  (except 
  in 
  special 
  cases) 
  a 
  sum 
  

   over 
  zero 
  or 
  one 
  electron 
  only. 
  

  

  Taking 
  the 
  element 
  dv 
  as 
  origin, 
  the 
  components 
  of 
  mag- 
  

   netic 
  force 
  at 
  cv\ 
  y\ 
  z' 
  arising 
  from 
  the 
  motion 
  parallel 
  to 
  

   O.t' 
  of 
  any 
  electron 
  or 
  electrons 
  which 
  may 
  happen 
  to 
  be 
  

   in 
  dv^ 
  will 
  be 
  * 
  

  

  

  {., 
  

  

  z' 
  y\ 
  d 
  ndW\ 
  

   r' 
  r 
  / 
  dr\r 
  dt 
  ) 
  

  

  ', 
  by 
  equation 
  

  

  (46), 
  

  

  

  

  («. 
  

  

  ->mr-") 
  

  

  ■ 
  ■ 
  - 
  (47) 
  

  

  where, 
  in 
  calculating 
  the 
  field 
  at 
  time 
  t, 
  M^ 
  and 
  ix 
  are 
  to 
  be 
  

   evaluated 
  at 
  time 
  t—r/V. 
  

  

  26. 
  Consider 
  now 
  a 
  small 
  prism 
  of 
  infinitesimal 
  cross 
  

   section 
  di/ 
  dz 
  extending 
  from 
  .«=— 
  Z 
  to 
  a; 
  = 
  l. 
  The 
  small 
  

   piece 
  da; 
  dy 
  dz 
  of 
  this 
  prism 
  is 
  to 
  replace 
  the 
  former 
  dv. 
  

  

  The 
  total 
  current 
  parallel 
  to 
  Ok 
  which 
  flows 
  through 
  this 
  

   prism 
  at 
  any 
  point 
  will 
  be 
  i^ 
  dx 
  dy. 
  Let 
  us 
  call 
  this^',. 
  

  

  The 
  value 
  0I 
  j^. 
  at 
  any 
  point 
  of 
  the 
  prism 
  can 
  be 
  expanded 
  

   in 
  the 
  form 
  

  

  . 
  . 
  . 
  (48) 
  

  

  here 
  

  

  j^ 
  = 
  I 
  (A 
  cos 
  go; 
  + 
  B^ 
  sin 
  qx) 
  dq, 
  . 
  . 
  . 
  

  

  TTj 
  <? 
  

  

  q=0 
  

  

  Ag 
  = 
  \ 
  j^ 
  cos 
  qx 
  dx, 
  B? 
  = 
  1 
  jx 
  sin 
  qx 
  dx. 
  

  

  Ultimately, 
  when 
  the 
  elements 
  of 
  current 
  are 
  identified 
  

   with 
  individual 
  electrons, 
  these 
  equations 
  become 
  

  

  Kq 
  = 
  %eu 
  cos 
  qx, 
  B^ 
  = 
  2^i^ 
  sin 
  qx, 
  . 
  . 
  (49) 
  

  

  the 
  summation 
  being 
  through 
  the 
  prism 
  of 
  length 
  2Z. 
  

  

  The 
  values 
  of 
  A^, 
  B^ 
  of 
  course 
  vary 
  with 
  the 
  time. 
  Again 
  

   using 
  Fourier's 
  theorem, 
  we 
  put 
  

  

  A 
  = 
  - 
  \ 
  {«-qp 
  COS 
  pt 
  + 
  a'gp 
  sin 
  pt) 
  dp, 
  

  

  2 
  TTJ 
  

  

  p=0 
  

  

  * 
  Larmor, 
  ' 
  ^ther 
  and 
  Matter,' 
  p. 
  223. 
  

  

  (50) 
  

  

  