﻿398 
  Mr. 
  W. 
  G. 
  Bickley 
  on 
  Two-Dimensional 
  Potential 
  

  

  w= 
  — 
  log 
  -tcosec^ 
  J 
  (z 
  + 
  i) 
  +\/z 
  2 
  -\-2iz 
  cos 
  a— 
  1 
  M. 
  (7) 
  

  

  With 
  regard 
  to 
  the 
  branch 
  of 
  the 
  two-valued 
  function 
  to 
  

   be 
  taken, 
  the 
  arc 
  may 
  be 
  considered 
  as 
  the 
  branch 
  line 
  

   joining 
  the 
  two 
  branch 
  points 
  (its 
  extremities) 
  and 
  that 
  

   branch 
  is 
  to 
  be 
  taken 
  which, 
  inter 
  alia, 
  is 
  equal 
  to 
  + 
  * 
  when 
  

   z 
  = 
  0, 
  and 
  tends 
  to 
  +z 
  as 
  £->co 
  . 
  

  

  Equations 
  (6) 
  and 
  (7) 
  then 
  give 
  the 
  potential 
  due 
  to 
  the 
  

   charged 
  arc. 
  Separating 
  the 
  real 
  and 
  imaginary 
  parts 
  of 
  

   (6), 
  after 
  writing 
  — 
  (<£+ 
  *^r) 
  for 
  id 
  (since 
  the 
  real 
  part 
  is 
  to 
  

   be 
  negative) 
  we 
  obtain 
  

  

  2 
  sin 
  - 
  cosh 
  <f> 
  sin 
  yjr 
  + 
  sin 
  2 
  = 
  sin 
  2yjr 
  

   1 
  + 
  2<? 
  _£P 
  sin 
  | 
  cos 
  yfr 
  + 
  *- 
  2< 
  P 
  sin 
  2 
  - 
  

   1 
  + 
  2 
  sin 
  =■ 
  cosh 
  <j> 
  cos 
  i/r 
  + 
  sin 
  2 
  9 
  cos 
  2i/r 
  

   1 
  + 
  2e 
  -Q 
  sin- 
  cos 
  ifr 
  + 
  <? 
  _2cp 
  sin 
  2 
  - 
  

  

  Kg. 
  1. 
  

  

  1 
  

  

  j>. 
  . 
  (8) 
  

  

  Using 
  (8), 
  the 
  equipotentials 
  and 
  lines 
  of 
  force 
  are 
  easily 
  

   plotted. 
  In 
  fig. 
  1 
  they 
  are 
  drawn 
  for 
  the 
  case 
  of 
  a 
  semicircle, 
  

  

  