﻿Plane, 
  Cylindrical, 
  and 
  Spherical 
  Cur 
  rent- 
  Sheets. 
  385 
  

   therefore 
  <f> 
  differs 
  by 
  a 
  constant 
  from 
  

  

  -^(logPM 
  + 
  logQM). 
  

  

  The 
  arbitrary 
  constants 
  in 
  </> 
  may 
  be 
  functions 
  of 
  the 
  time, 
  

   but 
  they 
  do 
  not 
  affect 
  the 
  distribution 
  of 
  currents 
  in 
  the 
  

   cylinder, 
  and 
  therefore 
  need 
  not 
  be 
  considered. 
  Since 
  P 
  and 
  

   Q 
  are 
  inverse 
  points 
  

  

  PM 
  : 
  QM 
  = 
  OP 
  : 
  a 
  = 
  a 
  : 
  OQ 
  = 
  be 
  n 
  « 
  a 
  :a; 
  

  

  log 
  PM 
  and 
  log 
  QM 
  therefore 
  differ 
  by 
  a 
  term 
  which 
  may 
  be 
  

   included 
  in 
  the 
  arbitrary 
  constant 
  of 
  <£, 
  so 
  that 
  for 
  determining 
  

   the 
  currents 
  we 
  may 
  take 
  

  

  <t> 
  £ 
  ] 
  °g 
  PM 
  > 
  or^.= 
  -£logQM, 
  

  

  whichever 
  is 
  most 
  convenient. 
  

  

  The 
  current 
  at 
  M 
  is 
  evidently 
  parallel 
  to 
  the 
  axis 
  (since 
  ^> 
  

   is 
  independent 
  of 
  the 
  third 
  cylindrical 
  coordinate 
  z), 
  and 
  the 
  

   intensity 
  of 
  the 
  current 
  is 
  equal 
  to 
  

  

  d<j> 
  m 
  OP 
  sin 
  6 
  _ 
  m 
  OQ 
  sin 
  6 
  

  

  a~dd^~'¥ir 
  PM 
  2 
  ~~~2^ 
  QM 
  2 
  * 
  

  

  Images 
  due 
  to 
  Generation 
  of 
  Lines 
  of 
  Poles 
  inside 
  the 
  

   Cylindrical 
  Sheet. 
  

  

  4. 
  If 
  we 
  were 
  to 
  suppose 
  the 
  currents 
  to 
  be 
  induced 
  by 
  a 
  

   line 
  of 
  poles 
  of 
  strength 
  /(£) 
  at 
  the 
  point 
  (b, 
  0) 
  where 
  b<a,we 
  

   should 
  have 
  to 
  expand 
  the 
  potential 
  in 
  descending 
  powers 
  of 
  

   v 
  in 
  order 
  to 
  obtain 
  a 
  convergent 
  series 
  at 
  the 
  surface 
  of 
  the 
  

   sheet. 
  We 
  should 
  thus 
  obtain 
  for 
  the 
  inducing 
  potential 
  

  

  f 
  1 
  b 
  n 
  "i 
  

  

  &<>=/(*) 
  |const. 
  - 
  logr 
  + 
  S 
  - 
  - 
  cos 
  n0 
  j 
  . 
  

  

  Now 
  the 
  portion 
  

  

  /(0{consfc.-logr} 
  

  

  represents 
  the 
  potential 
  of 
  a 
  line 
  of 
  poles 
  of 
  strength 
  f(t) 
  

   along 
  the 
  axis, 
  for 
  investigating 
  the 
  effects 
  of 
  which 
  the 
  

   image 
  method 
  is 
  inconvenient. 
  To 
  remove 
  these 
  terms 
  we 
  

   shall 
  suppose 
  the 
  inducing 
  system 
  to 
  consist 
  of 
  a 
  line 
  of 
  poles 
  

   through 
  the 
  point 
  (b, 
  0) 
  together 
  with 
  an 
  equal 
  and 
  opposite 
  

   line 
  of 
  poles 
  along 
  the 
  axis 
  : 
  in 
  other 
  words 
  of 
  a 
  line 
  of 
  

   magnets 
  having 
  their 
  positive 
  poles 
  f(t) 
  at 
  (b, 
  0) 
  and 
  their 
  

   negative 
  poles 
  —f(f) 
  at 
  the 
  axis, 
  The 
  potential 
  of 
  such 
  a 
  

  

  