﻿Plane, 
  Cylindrical, 
  and 
  Spherical 
  Current- 
  Sheets. 
  389 
  

  

  cylindrical 
  two-dimensional 
  Arago's 
  disk, 
  and 
  it 
  is 
  to 
  be 
  

   noticed 
  that 
  the 
  trails 
  of 
  images 
  lie 
  along 
  equiangular 
  spirals 
  

  

  Fiz. 
  3. 
  

  

  (as 
  may 
  readily 
  be 
  verified), 
  the 
  intensity 
  of 
  magnetization 
  

   of 
  the 
  trail 
  decreasing 
  toward 
  the 
  centre 
  and 
  increasing 
  

   outward 
  and 
  being, 
  in 
  fact, 
  proportional 
  to 
  the 
  distance 
  from 
  

   the 
  centre. 
  

  

  Part 
  III. 
  Spherical 
  Sheets. 
  

  

  Image 
  due 
  to 
  Generation 
  of 
  a 
  Small 
  Magnet 
  whose 
  Axis 
  is 
  

   Radial 
  outside 
  a 
  Spherical 
  Sheet. 
  

  

  8. 
  We 
  now 
  consider 
  the 
  problem 
  presented 
  by 
  the 
  currents 
  

   induced 
  in 
  a 
  spherical 
  sheet 
  of 
  radius 
  a 
  when 
  a 
  magnet 
  is 
  

   generated 
  at 
  a 
  distance 
  b 
  from 
  the 
  centre, 
  the 
  axis 
  of 
  the 
  

   magnet 
  passing 
  through 
  the 
  centre 
  of 
  the 
  sphere. 
  Taking 
  

   this 
  axis 
  as 
  the 
  axis 
  of 
  zonal 
  spherical 
  harmonics, 
  and 
  starting 
  

   in 
  the 
  first 
  instance 
  with 
  a 
  magnet 
  of 
  variable 
  moment 
  f(t), 
  

   (the 
  positive 
  pole 
  being 
  furthest 
  from 
  the 
  centre), 
  the 
  magnetic 
  

   potential 
  O 
  at 
  a 
  point 
  (r, 
  6) 
  is 
  given 
  by 
  

  

  n 
  ° 
  = 
  f 
  {£) 
  db 
  sj 
  (r 
  2 
  + 
  b 
  2 
  - 
  2rb 
  cos 
  6) 
  ' 
  

   When 
  r 
  < 
  b 
  this 
  becomes 
  

  

  «»= 
  /wi€I^ 
  p 
  »( 
  cos 
  *) 
  

  

  = 
  -At) 
  X 
  

  

  «=o 
  

  

  (n 
  + 
  l}r> 
  1 
  

   b 
  n+2 
  

  

  P 
  • 
  

  

  (9) 
  

  

  