﻿Plane, 
  Cylindrical, 
  and 
  Spherical 
  Current 
  -Sheets. 
  391 
  

  

  Substituting 
  in 
  the 
  potentials 
  of 
  the 
  induced 
  currents, 
  

   we 
  have 
  

  

  Of 
  = 
  M2(n 
  + 
  1)«-(2»+ 
  W*> 
  _i_ 
  p 
  n 
  

  

  = 
  M^«S 
  ( 
  ^±^P,. 
  .... 
  (13) 
  

  

  a 
  

  

  2n+l 
  

  

  Q»- 
  -IC&w-^+w* 
  ^+i 
  P.. 
  

  

  = 
  -M^<?>**l 
  ' 
  l{e 
  £["> 
  P.. 
  . 
  (14) 
  

  

  Comparing 
  the 
  first 
  of 
  these 
  results 
  with 
  (9) 
  and 
  the 
  

   second 
  with 
  (10), 
  we 
  have 
  the 
  following 
  conclusions: 
  — 
  

  

  (i.) 
  At 
  time 
  t 
  after 
  the 
  generation 
  of 
  the 
  magnet, 
  the 
  potential 
  

   due 
  to 
  the 
  induced 
  currents 
  is, 
  at 
  points 
  inside 
  the 
  sphere, 
  

   equivalent 
  to 
  that 
  produced 
  by 
  a 
  magnet 
  of 
  moment 
  — 
  M<? 
  3m/2,T 
  

   at 
  a 
  distance 
  from 
  the 
  centre 
  of 
  e 
  tRa 
  b. 
  

  

  (ii.) 
  At 
  points 
  outside 
  the 
  sphere, 
  the 
  potential 
  is 
  equivalent 
  

   to 
  that 
  of 
  a 
  magnet 
  of 
  moment 
  — 
  Me 
  _3m2a 
  a 
  3 
  /^ 
  3 
  a 
  ^ 
  a 
  distance 
  

   from 
  the 
  centre 
  of 
  e~ 
  tRcl 
  a 
  2 
  /b. 
  The 
  positions 
  P, 
  Q 
  occupied 
  by 
  

   these 
  images 
  at 
  time 
  t 
  are 
  always 
  inverse 
  points. 
  

  

  It 
  is 
  easily 
  seen 
  that 
  the 
  intensities 
  of 
  the 
  images 
  at 
  P 
  and 
  

   Q 
  are 
  to 
  each 
  other 
  and 
  to 
  that 
  of 
  the 
  inducing 
  magnet 
  as 
  

   OP2 
  : 
  OQi 
  : 
  OAI, 
  that 
  is 
  in 
  the 
  sesquiplicate 
  ratio 
  of 
  their 
  

   distances 
  from 
  the 
  centre. 
  

  

  Images 
  due 
  to 
  Generation 
  of 
  a 
  Small 
  Magnet 
  inside 
  a 
  

   Spherical 
  Sheet. 
  

  

  9. 
  If 
  b<a 
  or 
  the 
  magnet 
  is 
  generated 
  within 
  the 
  sphere, 
  we 
  

   must 
  expand 
  the 
  potential 
  in 
  the 
  second 
  form 
  (10) 
  in 
  order 
  to 
  

   substitute 
  in 
  the 
  surface-conditions. 
  When 
  this 
  is 
  done, 
  it 
  

   will 
  be 
  found 
  that 
  

  

  A 
  n 
  = 
  Me-( 
  2n+1 
  )' 
  R 
  < 
  2a 
  

  

  a" 
  

  

  and 
  the 
  conclusions 
  finally 
  arrived 
  at 
  are 
  as 
  follows 
  : 
  — 
  

  

  (i.) 
  If 
  a 
  small 
  magnet 
  of 
  moment 
  M 
  be 
  instantaneously 
  gene- 
  

   rated 
  at 
  a 
  point 
  B 
  inside 
  the 
  sphere 
  at 
  a 
  distance 
  b 
  from 
  the 
  

   centre, 
  the 
  axis 
  of 
  the 
  magnet 
  being 
  radial 
  ; 
  then 
  the 
  image 
  

   representing 
  the 
  magnetic 
  potential 
  of 
  the 
  induced 
  currents 
  

   outside 
  the 
  sphere 
  at 
  time 
  t 
  is 
  a 
  magnet 
  of 
  moment 
  — 
  ~M.e~ 
  3tR/2a 
  

   at 
  a 
  point 
  Q 
  such 
  that 
  OQ 
  = 
  be~ 
  tR/a 
  . 
  

  

  (ii.) 
  At 
  points 
  inside 
  the 
  cylinder 
  the 
  potential 
  is 
  that 
  which 
  

  

  