﻿Plane, 
  Cylindrical, 
  and 
  Spherical 
  Current- 
  Sheets. 
  393 
  

  

  are 
  in 
  the 
  same 
  ratio. 
  Thus 
  finally, 
  if 
  V 
  and 
  I" 
  be 
  the 
  inten- 
  

   sities 
  of 
  the 
  image-currents 
  at 
  time 
  t 
  in 
  PP'and 
  QQ', 
  we 
  have 
  

  

  i 
  ' 
  = 
  -(sr 
  i 
  =-^ 
  K2 
  " 
  i 
  > 
  

  

  * 
  w 
  — 
  (ST 
  1 
  —*; 
  1 
  - 
  

  

  It 
  will 
  be 
  noticed 
  that 
  V 
  increases 
  as 
  its 
  circuit 
  QQ' 
  

   approaches 
  the 
  centre, 
  while 
  I' 
  decreases 
  as 
  its 
  circuit 
  PP 
  7 
  

   recedes 
  to 
  infinity. 
  

  

  (ii.) 
  Next 
  taking 
  the 
  inducing 
  current 
  I 
  to 
  be 
  suddenly 
  

   generated 
  in 
  BB' 
  inside 
  the 
  sphere, 
  and 
  taking 
  0B 
  = 
  £. 
  

   OQ=e- 
  tR 
  ! 
  a 
  b, 
  and 
  OT 
  = 
  e 
  m 
  f 
  a 
  a 
  2 
  /b, 
  we 
  find 
  that 
  at 
  time 
  t, 
  the 
  

   image 
  representing 
  the 
  potential 
  of 
  the 
  induced 
  currents 
  at 
  

   points 
  outside 
  the 
  sphere 
  is 
  a 
  current 
  in 
  QQ' 
  of 
  intensity 
  

  

  e 
  tR 
  2a 
  J 
  

  

  and 
  at 
  points 
  inside 
  the 
  sphere 
  a 
  current 
  in 
  PP 
  r 
  of 
  intensity 
  

  

  a 
  

  

  Images 
  due 
  to 
  Generation 
  of 
  Magnetic 
  Pole 
  outside 
  a 
  

   Spherical 
  Sheet. 
  

  

  11. 
  A 
  magnetic 
  pole 
  m 
  generated 
  at 
  time 
  outside 
  a 
  sphere 
  

   at 
  the 
  point 
  r 
  = 
  b 
  where 
  h 
  >a 
  might 
  be 
  regarded 
  as 
  equivalent 
  

   to 
  a 
  line 
  of 
  elements 
  uniformlv 
  magnetized 
  in 
  the 
  direction 
  of 
  

   the 
  radius-vector 
  extending 
  along 
  that 
  radius 
  from 
  r 
  = 
  b 
  to 
  

   infmitv 
  ; 
  and 
  similarly 
  a 
  positive 
  pole 
  m 
  inside 
  the 
  sphere 
  at 
  

   r 
  = 
  b 
  when 
  b<a 
  together 
  with 
  a 
  negative 
  pole 
  — 
  m 
  at 
  the 
  

   centre 
  might 
  be 
  regarded 
  as 
  equivalent 
  to 
  a 
  line 
  of 
  elements 
  

   uniformly 
  magnetized 
  in 
  the 
  line 
  joining 
  these 
  points. 
  In 
  

   this 
  way 
  the 
  images 
  might 
  be 
  obtained 
  by 
  integration. 
  

   Practically 
  as 
  short 
  a 
  way 
  of 
  obtaining 
  the 
  result 
  as 
  any 
  is 
  by 
  

   utilizing 
  the 
  work 
  of 
  §§ 
  8, 
  9, 
  taking 
  into 
  account 
  the 
  differ- 
  

   ences 
  introduced 
  into 
  the 
  coefficients 
  of 
  the 
  various 
  harmonics 
  

   wmen 
  for 
  the 
  inducing 
  potential 
  of 
  a 
  magnet 
  there 
  is 
  sub- 
  

   stituted 
  that 
  of 
  a 
  pole. 
  

  

  When 
  the 
  pole 
  is 
  outside 
  the 
  sphere 
  the 
  inducing 
  potential 
  

   after 
  its 
  generation 
  becomes 
  

  

  O 
  =mS^iP„ 
  (r<b) 
  (15) 
  

  

  The 
  coefficient 
  of 
  P„ 
  is 
  — 
  b/(n 
  + 
  l) 
  of 
  the 
  corresponding 
  co-, 
  

   efficient 
  for 
  a 
  magnet 
  of 
  moment 
  m. 
  

  

  