﻿Problems 
  connected 
  with 
  the 
  Circular 
  Arc. 
  397 
  

  

  Hence 
  the 
  transformation 
  

  

  sin*(l 
  + 
  ^) 
  + 
  \/sin 
  2 
  ^(l 
  + 
  t~) 
  2 
  -cos 
  2 
  ^(~-f 
  t)" 
  

  

  ?=— 
  ^~ir 
  = 
  (*> 
  

  

  COS 
  sr 
  (^ 
  + 
  

  

  transforms 
  the 
  two 
  sides 
  of 
  the 
  z 
  arc 
  into 
  the 
  real 
  axis 
  of 
  

   the 
  f-plane, 
  the 
  extremities 
  of 
  the 
  arc 
  into 
  the 
  points 
  f= 
  +1. 
  

   This 
  relation 
  may 
  also 
  be 
  written 
  

  

  f 
  2 
  cos^ 
  + 
  2*f 
  sin-^ 
  + 
  cos^ 
  

   ~ 
  £ 
  - 
  ,-«>\ 
  

  

  a 
  a 
  a' 
  

  

  £- 
  cos 
  - 
  — 
  2tf 
  sin 
  2 
  + 
  cos 
  ~ 
  

   It 
  is 
  easily 
  found 
  from 
  (3) 
  that 
  z=ao 
  corresponds 
  to 
  

   f=^sin^±lVcos|, 
  

  

  only 
  one 
  of 
  which 
  is 
  in 
  the 
  upper 
  half 
  of 
  the 
  f-plane. 
  

  

  §3. 
  Suppose 
  that 
  in 
  the 
  f-plane 
  there 
  is 
  a 
  charge 
  at 
  the 
  

   point 
  

  

  g=i\l+ 
  sin^jcos^, 
  

  

  and 
  that 
  the 
  real 
  axis 
  is 
  a 
  conductor. 
  This 
  will 
  correspond 
  

   to 
  a 
  charge 
  at 
  infinity 
  in 
  the 
  r-plaue, 
  with 
  the 
  arc 
  a 
  conductor, 
  

   or, 
  what 
  is 
  practically 
  the 
  same, 
  to 
  the 
  charged 
  conducting 
  

   arc 
  in 
  space. 
  Let 
  w, 
  as 
  usual, 
  denote 
  the 
  potential 
  <£ 
  together 
  

   with 
  1 
  times 
  the 
  conjugate 
  function 
  yfr. 
  Then 
  in 
  the 
  f-plane, 
  

   the 
  method 
  of 
  images 
  gives 
  

  

  or, 
  what 
  is 
  the 
  same 
  thing, 
  

  

  1+ 
  sin 
  

  

  cos 
  ^ 
  

  

  

  (5) 
  

  

  The 
  elimination 
  of 
  f 
  between 
  (3) 
  and 
  (5) 
  then 
  gives, 
  after 
  

   some 
  reduction, 
  

  

  1 
  -f* 
  ""sin 
  I 
  

   l 
  + 
  e 
  w 
  sin^ 
  

  

  