﻿396 
  Mr. 
  W. 
  Gr. 
  Bicklej 
  on 
  Two- 
  Dimensional 
  Potential 
  

   A., 
  B, 
  I 
  l5 
  J 
  1? 
  I 
  2 
  , 
  J 
  2 
  , 
  I 
  3 
  , 
  J 
  3 
  , 
  I 
  4 
  , 
  J 
  4 
  , 
  &c, 
  equal 
  respectively 
  to 
  

  

  Vab 
  Yah 
  Xla 
  2 
  b 
  Yah 
  2 
  \Ja 
  2 
  b 
  2 
  

  

  \Ja, 
  Yb, 
  

  

  c 
  ' 
  cAI 
  x 
  ' 
  cBJ 
  x 
  ' 
  6-.BI2.AIx' 
  

   Ya 
  2 
  b 
  2 
  TJa*b 
  2 
  Ya 
  2 
  b* 
  

  

  c.AJs.BJi' 
  cAI3.BI2.Al!' 
  cBJ3.AJ2.BJi 
  

  

  which 
  are 
  the 
  image 
  charges 
  in 
  the 
  usual 
  way 
  of 
  treating 
  

   the 
  subject. 
  

  

  Note. 
  — 
  The 
  above 
  paper 
  was 
  written 
  before 
  the 
  one 
  that 
  

   appeared 
  in 
  the 
  March 
  number 
  of 
  the 
  Philosophical 
  Magazine 
  

   on 
  the 
  same 
  subject. 
  It 
  is, 
  perhaps, 
  unfortunate 
  that 
  the 
  

   word 
  " 
  image 
  " 
  lias 
  been 
  used 
  for 
  " 
  inverse 
  point/' 
  The 
  

   method 
  has, 
  of 
  course, 
  nothing 
  to 
  do 
  with 
  electrical 
  images. 
  

  

  XLV. 
  Some 
  Two-Dimensional 
  Potential 
  Problems 
  connected 
  

   with 
  the 
  Circular 
  Arc. 
  By 
  W. 
  Gr. 
  Bickley, 
  B.Sc* 
  

  

  § 
  1. 
  TN 
  this 
  paper 
  a 
  method 
  of 
  dealing 
  with 
  potential 
  

   A 
  problems 
  in 
  two 
  dimensions, 
  depending 
  on 
  the 
  

   use 
  of 
  functions 
  of 
  a 
  complex 
  variable 
  and 
  of 
  the 
  method 
  of 
  

   images, 
  is 
  applied 
  to 
  the 
  solution 
  of 
  problems 
  connected 
  with 
  

   an 
  infinitely 
  long 
  lamina, 
  the 
  section 
  of 
  which 
  is 
  a 
  circular 
  

   arc. 
  The 
  results 
  obtained 
  are 
  interpreted 
  in 
  terms 
  of 
  

   electricity 
  and 
  hydrodynamics. 
  

  

  § 
  2. 
  The 
  first 
  step 
  in 
  the 
  investigation 
  is 
  the 
  determination 
  

   of 
  the 
  Transformation 
  by 
  which 
  the 
  two 
  sides 
  of 
  the 
  arc 
  in 
  

   the 
  z-plane 
  become 
  the 
  real 
  axis 
  in 
  the 
  plane 
  of 
  an 
  auxiliary 
  

   variable 
  f 
  ( 
  = 
  f 
  +417). 
  The 
  arc 
  is 
  taken 
  as 
  that 
  part 
  of 
  the 
  

   circle 
  z= 
  — 
  ie 
  l 
  ° 
  for 
  which 
  — 
  a<0<a, 
  so 
  that 
  the 
  angle 
  sub- 
  

   tended 
  at 
  the 
  centre 
  is 
  2a. 
  For 
  any 
  point 
  on 
  this 
  circle 
  the 
  

  

  a 
  

   ratio 
  (1 
  + 
  iz) 
  / 
  (z 
  + 
  c) 
  is 
  purely 
  real, 
  its 
  value 
  being 
  cot-, 
  so 
  

  

  that 
  when 
  6 
  lies 
  within 
  the 
  above 
  limits, 
  the 
  values 
  of 
  

  

  , 
  al 
  + 
  iz 
  , 
  /, 
  9 
  a/l 
  + 
  iz\ 
  2 
  n 
  ,- 
  N 
  

  

  ^STtT+v^iti+r)- 
  1 
  • 
  • 
  (1) 
  

  

  are 
  purely 
  real, 
  but 
  become 
  complex 
  when 
  6 
  lies 
  outside 
  

   them. 
  Also, 
  when 
  z^ 
  — 
  c, 
  the 
  expression 
  (1) 
  tends 
  to 
  or 
  qo 
  

   according 
  as 
  the 
  root 
  is 
  taken 
  negatively 
  or 
  positively 
  ; 
  and 
  

   when 
  0= 
  +a, 
  the 
  expression 
  has 
  the 
  values 
  4;1. 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  