﻿174 
  Prof. 
  J. 
  J. 
  Thomson 
  on 
  a 
  Theory 
  of 
  the 
  Connexion 
  

  

  of 
  a 
  sphere 
  whose 
  centre 
  is 
  at 
  P, 
  and 
  whose 
  radius 
  is 
  Yt 
  ; 
  

   o) 
  2 
  is 
  the 
  mean 
  value 
  of 
  d(j)/dt 
  when 
  t 
  = 
  over 
  the 
  surface 
  of 
  

   the 
  same 
  sphere. 
  

  

  Let 
  £ 
  = 
  0be 
  the 
  time 
  when 
  the 
  particle 
  is 
  suddenly 
  brought 
  

   to 
  rest. 
  Take 
  the 
  centre 
  of 
  the 
  particle 
  when 
  it 
  is 
  brought 
  to 
  

   rest 
  as 
  the 
  origin 
  of 
  coordinates, 
  and 
  the 
  line 
  of 
  motion 
  of 
  

   the 
  centre 
  of 
  the 
  particle 
  as 
  the 
  axis 
  of 
  z. 
  Then 
  a, 
  /3, 
  7, 
  the 
  

   components 
  of 
  the 
  magnetic 
  force 
  when 
  the 
  particle 
  is 
  stopped, 
  

   are 
  for 
  all 
  points 
  outside 
  the 
  particle 
  given 
  by 
  the 
  equation* 
  

  

  eYw 
  y 
  da 
  _ 
  doe. 
  ~\ 
  

  

  R 
  _ 
  eYw 
  a? 
  dJ3_ 
  d/3^' 
  

  

  P~ 
  (VW)W 
  V* 
  \f. 
  s 
  dt~ 
  W 
  dz 
  

  

  \ 
  x 
  +y 
  + 
  Y*-ic^ 
  ) 
  

   7=0, 
  

  

  At 
  all 
  points 
  inside 
  the 
  particle, 
  which 
  we 
  shall 
  take 
  to 
  be 
  

   a 
  sphere 
  of 
  radius 
  a, 
  

  

  a 
  = 
  /3 
  = 
  7 
  = 
  0. 
  

  

  In 
  these 
  equations 
  V 
  is 
  the 
  velocity 
  of 
  light 
  through 
  the 
  

   dielectric, 
  iv 
  the 
  velocity 
  of 
  the 
  charged 
  sphere 
  before 
  it 
  was 
  

   stopped, 
  e 
  the 
  charge 
  on 
  the 
  sphere. 
  To 
  get 
  the 
  values 
  of 
  

   a, 
  /•?, 
  7 
  at 
  any 
  time 
  after 
  the 
  particle 
  is 
  stopped, 
  we 
  have 
  by 
  

   Poisson's 
  method 
  to 
  integrate 
  the 
  values 
  just 
  given 
  over 
  the 
  

   surfaces 
  of 
  certain 
  spheres 
  ; 
  in 
  the 
  general 
  case 
  this 
  integration 
  

   leads 
  to 
  complicated 
  elliptic 
  integrals. 
  We 
  shall 
  get 
  a 
  clearer 
  

   idea 
  of 
  the 
  physical 
  nature 
  of 
  the 
  disturbance 
  if 
  we 
  consider 
  

   two 
  special 
  cases, 
  (1) 
  when 
  we 
  can 
  neglect 
  the 
  square 
  and 
  

   higher 
  powers 
  of 
  w/Y; 
  (2) 
  when 
  w>/V 
  is 
  very 
  nearly 
  unity. 
  

  

  In 
  the 
  first 
  case, 
  when 
  we 
  neglect 
  w*/Y*, 
  a, 
  J3, 
  dec/dt, 
  d/3/dt 
  

   when 
  £ 
  = 
  all 
  satisfy 
  Laplace's 
  equation, 
  hence 
  the 
  mean 
  

   value 
  of 
  any 
  of 
  these 
  quantities 
  over 
  the 
  surface 
  of 
  a 
  sphere 
  

   which 
  does 
  not 
  enclose 
  the 
  origin, 
  nor 
  cut 
  through 
  any 
  part 
  

   of 
  the 
  electrified 
  sphere, 
  is 
  equal 
  to 
  the 
  value 
  of 
  this 
  quantity 
  

   at 
  the 
  centre 
  of 
  the 
  sphere 
  ; 
  we 
  can 
  easily 
  see, 
  too, 
  that 
  when 
  

   the 
  sphere 
  entirely 
  surrounds 
  the 
  electrified 
  sphere 
  the 
  mean 
  

   value 
  of 
  any 
  of 
  these 
  quantities 
  over 
  its 
  surface 
  is 
  zero. 
  

   Thus 
  we 
  have, 
  by 
  Poisson's 
  solution, 
  the 
  following 
  values 
  for 
  

   the 
  components 
  of 
  the 
  magnetic 
  force 
  after 
  a 
  time 
  t 
  from 
  the 
  

   stoppage 
  of 
  the 
  electrified 
  sphere, 
  

  

  * 
  Heaviside, 
  Phil. 
  Mag. 
  April 
  1889; 
  J. 
  J. 
  Thomson, 
  'Recent 
  Re- 
  

   searches/ 
  p. 
  19. 
  

  

  