﻿Plane, 
  Cylindrical, 
  and 
  Spherical 
  Current- 
  Sheets, 
  383 
  

  

  where 
  fi, 
  the 
  magnetic 
  permeability 
  of 
  the 
  dielectric, 
  will 
  be 
  

   taken 
  to 
  be 
  unity, 
  or 
  included 
  under 
  f 
  K 
  t). 
  

  

  Fig. 
  1. 
  

  

  The 
  above 
  potential 
  may, 
  for 
  values 
  of 
  r 
  less 
  than 
  £, 
  be 
  

   expanded 
  in 
  the 
  form 
  

  

  n 
  =f(t) 
  | 
  const. 
  - 
  log 
  b 
  + 
  2 
  - 
  £ 
  cos 
  n6 
  \ 
  . 
  . 
  (3) 
  

  

  Consistently 
  with 
  the 
  surface 
  condition 
  of 
  continuity 
  of 
  

  

  magnetic 
  induction, 
  we 
  can 
  assume 
  for 
  the 
  potentials 
  of 
  the 
  

  

  induced 
  currents 
  inside 
  and 
  outside 
  the 
  cylinder 
  expressions 
  

  

  of 
  the 
  form 
  

  

  A 
  r 
  n 
  

   H'= 
  — 
  2— 
  -r 
  cos 
  nO 
  (inside), 
  . 
  . 
  . 
  (4) 
  

   n 
  a 
  n 
  v 
  

  

  A 
  n 
  n 
  

  

  H"=2— 
  - 
  cos 
  nO 
  (outside), 
  . 
  . 
  . 
  (5) 
  

  

  since 
  these 
  make 
  dCl'/dr 
  = 
  d£l 
  n 
  /dr 
  when 
  r 
  = 
  a 
  ; 
  here 
  the 
  

   coefficients 
  A 
  n 
  are 
  functions 
  of 
  the 
  time 
  t. 
  

  

  Substituting 
  the 
  expressions 
  thus 
  obtained, 
  O 
  -f-O 
  ; 
  for 
  I2j, 
  

   and 
  X2 
  + 
  n" 
  for 
  fl 
  2 
  in 
  the 
  surface- 
  conditions 
  (1), 
  we 
  have, 
  or- 
  

   equating 
  coefficients 
  of 
  cos 
  n0, 
  

  

  — 
  +n 
  -A 
  K= 
  f 
  { 
  t)y, 
  

   .-. 
  |{A„e»«/«| 
  =/ 
  '(O 
  e 
  »'W«|' 
  ; 
  

  

  =j> 
  

  

  tf 
  

  

  ( 
  T 
  ) 
  e 
  -»«-T)B/«rf 
  T 
  " 
  

  

  the 
  limits 
  of 
  integration 
  being 
  £ 
  and 
  -co 
  if 
  we 
  have 
  to 
  take 
  

  

  2 
  D 
  2 
  

  

  