﻿certain 
  Problems 
  relating 
  to 
  the 
  Potential. 
  197 
  

  

  Thus 
  

  

  P 
  = 
  r 
  ^~ 
  2a 
  (/? 
  ''" 
  " 
  ^ 
  J 
  + 
  6a 
  2 
  ~ 
  ^ 
  " 
  ?7? 
  ^' 
  

   and 
  by 
  successive 
  approximation 
  with 
  use 
  of 
  (9) 
  

  

  ^^V 
  1.2 
  + 
  a 
  2 
  " 
  1.2.3 
  

  

  a 
  2 
  6 
  rf^W 
  ( 
  ' 
  

  

  The 
  significance 
  of 
  the 
  first 
  three 
  terms 
  is 
  brought 
  out 
  if 
  

   we 
  suppose 
  that 
  r 
  is 
  constant 
  (a), 
  so 
  that 
  the 
  last 
  term 
  

   vanishes. 
  In 
  this 
  case 
  the 
  exact 
  solution 
  is 
  

  

  log— 
  — 
  £=^rlog-^-, 
  . 
  . 
  . 
  (11) 
  

  

  whence 
  

  

  £ 
  /a 
  + 
  *\* 
  _ 
  _ 
  a 
  ^r-1) 
  «» 
  

  

  a 
  V 
  a 
  / 
  Y 
  a 
  + 
  L 
  . 
  2 
  a 
  2 
  

  

  A 
  *tyr 
  -!)(>- 
  2) 
  

  

  1.2.3 
  a 
  : 
  

  

  + 
  (12) 
  

  

  in 
  agreement 
  with 
  (10). 
  

  

  In 
  the 
  above 
  investigation 
  ijr 
  is 
  supposed 
  to 
  be 
  zero 
  

   exactly 
  upon 
  the 
  circle 
  of 
  radius 
  a. 
  If 
  the 
  circle 
  whose 
  

   centre 
  is 
  taken 
  as 
  origin 
  of 
  coordinates 
  be 
  merely 
  the 
  circle 
  

   of 
  curvature 
  of 
  the 
  curve 
  yjr 
  = 
  at 
  the 
  point 
  (6 
  = 
  0) 
  under 
  

   consideration, 
  yfr 
  will 
  not 
  vanish 
  exactly 
  upon 
  it, 
  but 
  only 
  

   when 
  r 
  has 
  the 
  approximate 
  value 
  c6 
  3 
  , 
  c 
  being 
  a 
  constant. 
  

   In 
  (6) 
  an 
  initial 
  term 
  R 
  must 
  be 
  introduced, 
  whose 
  approxi- 
  

   mate 
  value 
  is 
  — 
  c^Rj. 
  But 
  since 
  R 
  " 
  vanishes 
  with 
  0, 
  

   equation 
  (7) 
  and 
  its 
  consequences 
  remain 
  undisturbed 
  and 
  

   (10) 
  is 
  still 
  available 
  as 
  a 
  formula 
  of 
  interpolation. 
  In 
  all 
  

   these 
  cases, 
  the 
  success 
  of 
  the 
  approximation 
  depends 
  of 
  

   course 
  upon 
  the 
  degree 
  of 
  slowness 
  with 
  which 
  y, 
  or 
  *• 
  

   varies. 
  

  

  Another 
  form 
  of 
  the 
  problem 
  arises 
  when 
  what 
  is 
  given 
  

   is 
  not 
  a 
  pair 
  of 
  neighbouring 
  curves 
  along 
  each 
  of 
  which 
  

   (e. 
  g.) 
  the 
  stream- 
  function 
  is 
  constant, 
  but 
  one 
  such 
  curve 
  

   together 
  with 
  the 
  variation 
  of 
  potential 
  along 
  it. 
  It 
  is 
  then 
  

   required 
  to 
  construct 
  a 
  neighbouring 
  stream-line 
  and 
  to 
  

  

  