﻿Plane, 
  Cylindrical, 
  and 
  Spherical 
  Current- 
  Sheets. 
  395 
  

  

  hydrodynamic 
  problem 
  the 
  portions 
  of 
  the 
  normal 
  velocity 
  

   components 
  d<j>/dr 
  due 
  to 
  the 
  source 
  and 
  its 
  image 
  have 
  to 
  

   cancel 
  one 
  another. 
  

  

  Fig. 
  5. 
  

  

  A 
  _ 
  P 
  

  

  +m 
  -me 
  tR 
  l 
  Za 
  

  

  Tmages 
  due 
  to 
  Generation 
  of 
  Magnetic 
  Poles 
  inside 
  a 
  

   Spherical 
  Sheet. 
  

  

  12. 
  If 
  a 
  pole 
  m 
  be 
  generated 
  inside 
  a 
  sphere 
  at 
  the 
  point 
  

   r=b 
  where 
  b<a, 
  together 
  with 
  an 
  equal 
  and 
  opposite 
  pole 
  

   — 
  m 
  at 
  the 
  centre, 
  the 
  inducing 
  potential 
  takes 
  the 
  form 
  

  

  ft 
  » 
  =m2 
  i:;,-£r 
  p 
  » 
  a*) 
  

  

  (i.) 
  The 
  induced 
  potential 
  outside 
  the 
  sphere 
  takes 
  the 
  form 
  

   il"=-me-'^% 
  K 
  ' 
  e 
  ' 
  P. 
  (19) 
  

  

  This 
  is 
  the 
  potential 
  of 
  a 
  pole 
  of 
  intensity 
  —me~ 
  tli,2a 
  at 
  a 
  

   distance 
  from 
  the 
  centre 
  of 
  be~ 
  mia 
  , 
  together 
  with 
  an 
  equal 
  

   and 
  opposite 
  pole 
  at 
  the 
  centre. 
  

  

  (ii.) 
  The 
  induced 
  potential 
  inside 
  is 
  

  

  XI' 
  =mt 
  7 
  ^- 
  «r<2»+i 
  w»« 
  **L 
  p 
  

   n 
  a 
  2n+l 
  n 
  

  

  If 
  p 
  = 
  e 
  m/a 
  a 
  2 
  /b, 
  this 
  becomes 
  

  

  =ml^ 
  [t^P^^tf^dp'Y 
  . 
  (20) 
  

  

  The 
  images 
  thus 
  consist 
  of 
  a 
  positive 
  pole 
  of 
  strength 
  

   me 
  m 
  l 
  2a 
  aib 
  at 
  a 
  distance 
  from 
  the 
  centre 
  p 
  equal 
  to 
  e 
  iB, 
  ' 
  a 
  a 
  2 
  /b 
  

  

  