﻿384 
  Dr. 
  G. 
  H. 
  Bryan 
  on 
  Electromagnetic 
  Induction 
  in 
  

  

  account 
  of 
  changes 
  that 
  have 
  been 
  taking 
  place 
  from 
  an 
  

   infinite 
  time 
  back. 
  

  

  To 
  analyse 
  the 
  images, 
  consider 
  now 
  the 
  case 
  in 
  which 
  a 
  

   line 
  of 
  poles 
  of 
  strength 
  m 
  is 
  suddenly 
  generated 
  at 
  A 
  at 
  the 
  

   time 
  0, 
  so 
  that 
  

  

  /(0=0 
  whenf<0, 
  

   f(t) 
  =m 
  when 
  t 
  > 
  0, 
  

  

  f'(t) 
  vanishes 
  except 
  when 
  £ 
  = 
  0, 
  and 
  the 
  time-integral 
  of 
  

   f'(t) 
  over 
  a 
  very 
  small 
  interval 
  containing 
  the 
  time 
  is 
  equal 
  

   to 
  m. 
  We 
  thus 
  obtain 
  

  

  A 
  n 
  = 
  me-"™?"f 
  n 
  . 
  

  

  Substituting 
  in 
  the 
  expressions 
  for 
  XI' 
  and 
  Q" 
  we 
  find 
  

   1 
  v 
  n 
  

  

  == 
  -m[\ogbe 
  m/a 
  -\og 
  {r 
  2 
  + 
  (be 
  m 
  /") 
  2 
  -2rbe 
  m/a 
  cos 
  6}*] 
  . 
  (6) 
  

  

  1 
  a 
  2n 
  

   XI" 
  = 
  m 
  2 
  - 
  tt^trt-cos 
  nO 
  

  

  =m[log 
  r-log{r« 
  + 
  (j^j 
  -%■ 
  (jj^) 
  cos 
  o] 
  *]. 
  (7) 
  

  

  Hence 
  we 
  have 
  the 
  following 
  results 
  : 
  — 
  

  

  (i.) 
  At 
  points 
  inside 
  the 
  conductor, 
  the 
  magnetic 
  potential 
  due 
  

   to 
  the 
  induced 
  currents 
  initially 
  neutralizes 
  that 
  produced 
  by 
  

   the 
  sudden 
  generation 
  of 
  the 
  line 
  of 
  poles 
  at 
  A, 
  and 
  is 
  equiva- 
  

   lent 
  to 
  that 
  of 
  a 
  line 
  of 
  poles 
  (P, 
  fig. 
  1) 
  of 
  constant 
  strength 
  

   — 
  m, 
  starting 
  at 
  the 
  point 
  A 
  and 
  moving 
  away 
  from 
  the 
  

   centre 
  so 
  that 
  its 
  distance 
  from 
  the 
  centre 
  at 
  time 
  t 
  is 
  e 
  m 
  ! 
  u 
  b. 
  

  

  (ii.) 
  At 
  points 
  outside 
  the 
  conductor 
  the 
  magnetic 
  potential 
  is 
  

   equivalent 
  to 
  that 
  of 
  two 
  lines 
  of 
  poles, 
  one 
  of 
  constant 
  

   strength 
  —m 
  at 
  the 
  centre, 
  and 
  the 
  other 
  (Q) 
  of 
  strength 
  

   + 
  m, 
  starting 
  at 
  B 
  the 
  inverse 
  point 
  of 
  A 
  {a 
  2 
  /b, 
  0), 
  and 
  moving 
  

   towards 
  the 
  centre 
  so 
  that 
  at 
  time 
  t 
  its 
  distance 
  from 
  the 
  

   centre 
  is 
  e" 
  tR 
  l 
  a 
  a 
  2 
  jb. 
  We 
  observe 
  that 
  P 
  and 
  Q, 
  the 
  positions 
  

   of 
  the 
  inside 
  and 
  outside 
  images, 
  move 
  so 
  as 
  always 
  to 
  remain 
  

   inverse 
  points. 
  

  

  The 
  velocities 
  with 
  which 
  the 
  images 
  P, 
  Q 
  move 
  away 
  

   from 
  and 
  towards 
  respectively 
  are 
  R/a 
  times 
  their 
  distances 
  

   from 
  0, 
  and 
  their 
  accelerations 
  are 
  W/a 
  2 
  times 
  these 
  distances 
  

   respectively. 
  

  

  3. 
  The 
  current 
  function 
  at 
  any 
  point 
  M 
  of 
  the 
  cylinder 
  is 
  

   given 
  by 
  

  

  d>= 
  — 
  (Xl 
  2 
  — 
  -Xli) 
  +a 
  constant 
  ; 
  

  

  47T 
  

  

  