﻿180 
  Prof. 
  J. 
  J. 
  Thomson 
  on 
  a 
  Theory 
  of 
  the 
  Connexion 
  

   when 
  the 
  sphere 
  cuts 
  z 
  = 
  d 
  and 
  not 
  z 
  = 
  — 
  d, 
  we 
  have 
  

  

  [fdS=vj(^) 
  C0S 
  *-^ 
  

  

  cos 
  6 
  

  

  9 
  V 
  2 
  a 
  

  

  =2e 
  Td- 
  

  

  When 
  the 
  sphere 
  cuts 
  both 
  z 
  — 
  d 
  and 
  z=—d, 
  then 
  

  

  1 
  

  

  dt 
  

  

  Thus 
  G> 
  2 
  > 
  the 
  mean 
  value 
  of 
  the 
  initial 
  value 
  of 
  d/3/dt 
  over 
  

   the 
  surface 
  of 
  the 
  sphere, 
  is 
  given 
  by 
  the 
  equation 
  

  

  two= 
  « 
  -r-y, 
  when 
  the 
  sphere 
  cuts 
  z 
  — 
  d 
  and 
  not 
  z= 
  —d, 
  

  

  I 
  bd 
  

  

  = 
  — 
  x 
  7-r 
  when 
  the 
  sphere 
  cuts 
  z= 
  — 
  d 
  and 
  not 
  2 
  = 
  d, 
  

   2 
  6a 
  l 
  

  

  = 
  when 
  the 
  sphere 
  cuts 
  both. 
  

   Hence 
  by 
  Poisson's 
  formula 
  

  

  eY 
  

   6= 
  rr 
  when 
  the 
  sphere 
  cuts 
  z 
  = 
  d 
  and 
  not 
  z=—d, 
  

   bd 
  

  

  = 
  when 
  the 
  sphere 
  cuts 
  z= 
  —d 
  and 
  not 
  z 
  — 
  d, 
  

  

  = 
  when 
  the 
  sphere 
  cuts 
  z 
  = 
  d 
  and 
  also 
  z= 
  —d. 
  

  

  Thus 
  the 
  distribution 
  of 
  magnetic 
  force 
  between 
  the 
  planes 
  

   z= 
  + 
  d 
  is 
  propagated 
  forwards 
  unchanged 
  with 
  the 
  velocity 
  V, 
  

   there 
  is 
  no 
  corresponding 
  pulse 
  propagated 
  in 
  the 
  negative 
  

   direction. 
  

  

  In 
  addition 
  to 
  the 
  plane 
  pulse 
  there 
  will 
  also, 
  as 
  in 
  the 
  

   previous 
  case, 
  be 
  a 
  spherical 
  one, 
  whose 
  thickness 
  is 
  2d 
  ; 
  we 
  

   can 
  calculate 
  the 
  magnetic 
  force 
  at 
  any 
  point 
  in 
  this 
  pulse 
  as 
  

   follows 
  : 
  — 
  Let 
  H 
  be 
  the 
  magnetic 
  force 
  at 
  a 
  point 
  in 
  this 
  

   pulse 
  at 
  a 
  distance 
  b 
  from 
  the 
  axis 
  of 
  z 
  } 
  then 
  the 
  line 
  integral 
  

   of 
  this 
  magnetic 
  force 
  round 
  the 
  circle 
  whose 
  radius 
  is 
  b 
  and 
  

   whose 
  axis 
  is 
  the 
  axis 
  of 
  z 
  is 
  27r5H 
  ; 
  the 
  magnetic 
  force 
  lasts 
  

   for 
  a 
  time 
  2d/V, 
  so 
  that 
  the 
  time 
  integral 
  of 
  the 
  line 
  integral 
  

   is 
  MdR/V. 
  

  

  At 
  any 
  point 
  in 
  front 
  of 
  the 
  particle 
  the 
  time 
  integral 
  

   of 
  the 
  magnetic 
  force 
  due 
  to 
  the 
  plane 
  pulse 
  round 
  the 
  same 
  

   circuit 
  is 
  

  

  o 
  ? 
  eY 
  2d 
  A 
  

   lirb 
  -j—jX^r 
  =47r6. 
  

   bd 
  V 
  

  

  Hence 
  the 
  time 
  integral 
  of 
  the 
  whole 
  magnetic 
  force 
  round 
  

  

  