﻿Plane, 
  Cylindrical, 
  and 
  Spherical 
  Current- 
  Sheets. 
  3$ 
  7 
  

  

  together 
  with 
  a 
  return 
  current 
  — 
  \m 
  down 
  the 
  axis 
  expands 
  

   in 
  the 
  series 
  

  

  1 
  b 
  n 
  

  

  m2, 
  sin 
  nQ. 
  

  

  n 
  r 
  n 
  

  

  These 
  expressions 
  differ 
  from 
  those 
  used 
  in 
  the 
  previous 
  

   articles 
  by 
  the 
  substitution 
  of 
  sines 
  for 
  cosines 
  only, 
  and 
  it 
  

   follows 
  that 
  when 
  we 
  make 
  similar 
  changes 
  in 
  the 
  assumed 
  

   expressions 
  for 
  the 
  potentials 
  of 
  the 
  induced 
  currents, 
  the 
  

   work 
  of 
  substituting 
  in 
  the 
  boundary 
  conditions 
  will 
  be 
  

   unaltered. 
  For 
  convenience, 
  however, 
  it 
  will 
  be 
  well 
  to 
  again 
  

   state 
  the 
  final 
  conclusions, 
  which 
  are 
  as 
  follows 
  : 
  — 
  

  

  (i.) 
  If 
  a 
  current 
  of 
  strength 
  I 
  be 
  suddenly 
  generated 
  

   parallel 
  to 
  the 
  axis 
  of 
  the 
  cylinder 
  at 
  A 
  outside 
  the 
  cylinder, 
  

   the 
  image 
  representing 
  the 
  effect 
  of 
  the 
  induced 
  currents 
  for 
  

   inside 
  points 
  at 
  time 
  t 
  will 
  be 
  a 
  current 
  of 
  strength 
  — 
  I 
  at 
  P, 
  

   and 
  the 
  images 
  representing 
  their 
  effect 
  for 
  outside 
  points 
  

   will 
  be 
  a 
  current 
  +1 
  at 
  Q 
  and 
  a 
  return 
  current 
  —I 
  in 
  the 
  

   axis, 
  where 
  OP 
  =.t 
  /<z 
  OA, 
  and 
  Q 
  is 
  the 
  inverse 
  point 
  of 
  P, 
  so 
  

   thatOQ 
  = 
  ^ 
  R 
  'V/OA. 
  

  

  (ii.) 
  If 
  a 
  current 
  of 
  strength 
  I 
  be 
  suddenly 
  generated 
  at 
  

   B 
  inside 
  the 
  cylinder, 
  accompanied 
  by 
  a 
  return 
  current 
  —I 
  

   in 
  the 
  axis, 
  the 
  image 
  representing 
  the 
  effect 
  of 
  the 
  induced 
  

   currents 
  for 
  outside 
  points 
  will 
  be 
  a 
  current 
  I 
  at 
  P, 
  and 
  the 
  

   images 
  representing 
  their 
  effect 
  for 
  inside 
  points 
  will 
  be 
  currents 
  

   — 
  I 
  at 
  Q, 
  and 
  I 
  in 
  the 
  axis, 
  where 
  OQ 
  = 
  e~ 
  m/a 
  OB, 
  and 
  P 
  is 
  

   the 
  inverse 
  point 
  of 
  Q, 
  so 
  that 
  OP=£ 
  /a 
  a 
  2 
  /QB. 
  

  

  The 
  effect 
  of 
  breaking 
  a 
  current 
  of 
  intensity 
  I 
  will 
  of 
  course 
  

   be 
  the 
  same 
  as 
  that 
  of 
  superposing 
  a 
  current 
  —I 
  and 
  the 
  

   images 
  will 
  be 
  equal 
  and 
  opposite 
  to 
  those 
  above 
  obtained. 
  

  

  Images 
  due 
  to 
  sudden 
  Generation 
  of 
  Lines 
  of 
  Magnets, 
  and 
  

  

  Moving 
  Poles. 
  

  

  6. 
  It 
  is 
  now 
  easy 
  to 
  buildup 
  the 
  images 
  due 
  to 
  the 
  sudden 
  

   generation 
  or 
  destruction 
  of 
  lines 
  of 
  magnets 
  parallel 
  to 
  the 
  

   axis 
  of 
  the 
  cylinder, 
  the 
  motion 
  of 
  lines 
  of 
  poles 
  and 
  so 
  forth. 
  

   The 
  methods 
  are 
  almost 
  identical 
  with 
  those 
  used 
  for 
  plane 
  

   sheets, 
  and 
  it 
  will 
  be 
  sufficient 
  here 
  to 
  call 
  attention 
  to 
  one 
  

   or 
  two 
  salient 
  features 
  of 
  the 
  images 
  in 
  cylindrical 
  sheets. 
  

   As 
  in 
  the 
  previous 
  articles 
  we 
  shall 
  take 
  A, 
  B 
  to 
  be 
  any 
  two 
  

   inverse 
  points 
  inside 
  and 
  outside 
  the 
  cylinder, 
  P 
  and 
  Q 
  two 
  

   other 
  inverse 
  points 
  such 
  that 
  OP 
  = 
  e 
  a 
  OA, 
  OQ 
  = 
  £~ 
  a 
  OB, 
  

   and 
  similar 
  relations 
  will 
  be 
  supposed 
  to 
  hold 
  when 
  the 
  letters 
  

   are 
  accented. 
  

  

  (i.) 
  If 
  a 
  line 
  of 
  magnets 
  parallel 
  to 
  the 
  axis 
  at 
  AA' 
  be 
  gene- 
  

   rated 
  suddenly, 
  having 
  their 
  positive 
  poles 
  at 
  A 
  and 
  their 
  

  

  