﻿$86 
  Dr. 
  Gr. 
  H. 
  Bryan 
  on 
  Electromagnetic 
  Induction, 
  in 
  

   distribution 
  expands 
  where 
  r>b 
  in 
  the 
  form 
  

  

  fl 
  o 
  =/(O:£~co 
  S 
  »0 
  (8) 
  

  

  The 
  analysis 
  is 
  almost 
  identical 
  with 
  that 
  of 
  the 
  preceding 
  

   case, 
  and 
  passing 
  to 
  the 
  case 
  when 
  

  

  f(t) 
  = 
  when 
  t<0, 
  

   f(t) 
  = 
  m 
  when 
  t>0, 
  

  

  the 
  following 
  results 
  may 
  be 
  obtained, 
  taking 
  B 
  (fig. 
  1) 
  to 
  

   now 
  represent 
  the 
  point 
  (6, 
  0). 
  

  

  (i.) 
  If 
  lines 
  of 
  poles 
  of 
  strengths 
  m 
  and 
  — 
  m 
  be 
  instantaneously 
  

   generated 
  at 
  B 
  and 
  the 
  axis 
  respectively, 
  the 
  images 
  repre- 
  

   senting 
  the 
  magnetic 
  potential 
  of 
  the 
  induced 
  currents 
  outside 
  

   the 
  cylinder 
  after 
  a 
  time 
  t 
  are 
  lines 
  of 
  poles 
  of 
  strengths 
  — 
  m 
  

   and 
  +m 
  at 
  Q 
  and 
  the 
  axis, 
  where 
  OQ=<?~' 
  '%, 
  Hence 
  the 
  

   total 
  magnetic 
  potential 
  at 
  outside 
  points 
  due 
  to 
  the 
  poles 
  and 
  

   the 
  induced 
  currents 
  at 
  time 
  t 
  is 
  that 
  of 
  lines 
  of 
  poles 
  of 
  

   strengths 
  + 
  ra 
  and 
  — 
  m 
  at 
  B 
  and 
  Q 
  respectively. 
  

  

  (ii.) 
  At 
  points 
  inside 
  the 
  cylinder, 
  the 
  magnetic 
  potential 
  due 
  

   to 
  the 
  induced 
  currents 
  alone 
  is 
  that 
  which 
  would 
  be 
  produced 
  

   by 
  a 
  line 
  of 
  poles 
  of 
  strength 
  -f 
  m 
  at 
  P, 
  where 
  P 
  is 
  the 
  

   inverse 
  point 
  of 
  Q, 
  so 
  that 
  OF 
  = 
  e 
  tn,a 
  a 
  2 
  /b. 
  

  

  Images 
  due 
  to 
  Generation 
  of 
  Straight 
  Currents 
  Parallel 
  

   to 
  the 
  Axis 
  of 
  the 
  Cylindrical 
  Sheet. 
  

  

  5. 
  The 
  cases 
  where 
  the 
  currents 
  are 
  induced 
  by 
  the 
  sudden 
  

   generation 
  of 
  a 
  straight 
  current 
  parallel 
  to 
  the 
  axis 
  of 
  the 
  

   cylinder 
  probably 
  have 
  a 
  more 
  practical 
  physical 
  interest 
  than 
  

   those 
  considered 
  in 
  the 
  previous 
  articles. 
  If 
  we 
  suppose 
  

   all 
  the 
  lines 
  of 
  poles 
  of 
  strengths 
  ±m 
  of 
  the 
  preceding 
  

   sections 
  to 
  be 
  replaced 
  by 
  currents 
  of 
  intensities 
  +1 
  the 
  

   results 
  regarding 
  the 
  images 
  will 
  be 
  identically 
  the 
  same 
  as 
  

   before. 
  To 
  satisfy 
  ourselves 
  that 
  such 
  is 
  the 
  case 
  we 
  have 
  

   only 
  to 
  notice 
  that 
  while 
  the 
  magnetic 
  potential 
  of 
  a 
  line 
  

   of 
  poles 
  of 
  line-density 
  m 
  at 
  (&, 
  0) 
  is 
  the 
  real 
  part 
  of 
  the 
  

   expression 
  

  

  m 
  log 
  (re 
  i() 
  — 
  b), 
  

  

  the 
  magnetic 
  potential 
  of 
  a 
  current 
  perpendicular 
  to 
  the 
  

   plane 
  of 
  reference 
  and 
  of 
  intensity 
  ^m 
  is 
  the 
  imaginary 
  part 
  

   of 
  the 
  same 
  expression. 
  Where 
  r<b, 
  the 
  potential 
  of 
  the 
  

   current 
  expands 
  therefore 
  in 
  the 
  series 
  

  

  1 
  r 
  n 
  

   m%- 
  7- 
  sin 
  n6. 
  

   n 
  b 
  n 
  ■ 
  

  

  and 
  where 
  r>b 
  the 
  potential 
  of 
  a 
  current 
  \m 
  at 
  (6, 
  0) 
  

  

  