﻿196 
  Lord 
  Rayleigh 
  on 
  the 
  Approximate 
  Solution 
  of 
  

   or 
  by 
  use 
  of 
  (3) 
  

  

  *?-W 
  + 
  6 
  dxAy)^ 
  36 
  l^ty/J 
  

  

  ^"" 
  36 
  ^ 
  2 
  \ 
  y 
  dxAy)) 
  120 
  ^ 
  4 
  \?/| 
  w 
  

  

  The 
  evanescence 
  of 
  i/r 
  when 
  y 
  = 
  may 
  arise 
  from 
  this 
  axis 
  

   being 
  itself 
  a 
  boundary, 
  or 
  from 
  the 
  second 
  boundary 
  being 
  

   a 
  symmetrical 
  curve 
  situated 
  upon 
  the 
  other 
  side 
  of 
  the 
  axis. 
  

   In 
  the 
  former 
  paper 
  expressions 
  for 
  the 
  " 
  resistance 
  " 
  and 
  

   u 
  conductivity 
  " 
  were 
  developed. 
  

  

  We 
  will 
  now 
  suppose 
  that 
  ^ 
  = 
  along 
  a 
  circle 
  of 
  radius 
  a, 
  

   in 
  substitution 
  for 
  the 
  axis 
  of 
  x. 
  Taking 
  polar 
  coordinates 
  

   ■a 
  + 
  r 
  and 
  0, 
  we 
  have 
  as 
  the 
  general 
  equation 
  

  

  («+^+^)? 
  + 
  sr=o.. 
  . 
  ( 
  5) 
  

  

  Assuming 
  

  

  ^ 
  = 
  R 
  1 
  r 
  + 
  R 
  2 
  r- 
  2 
  + 
  R 
  3 
  ^M-..., 
  .... 
  (6) 
  

  

  where 
  R 
  l5 
  R 
  2 
  , 
  &c, 
  are 
  functions 
  of 
  0, 
  we 
  find 
  on 
  substitution 
  

   in 
  (5) 
  

  

  2a 
  2 
  R 
  2 
  + 
  oRi 
  = 
  0, 
  "1 
  

  

  6a 
  2 
  R 
  3 
  + 
  6aR 
  2 
  + 
  R 
  2 
  + 
  R/' 
  = 
  0;/ 
  

   so 
  that 
  

  

  t=ftr 
  _ 
  ¥+ 
  «^v. 
  .. 
  (8) 
  

  

  is 
  the 
  form 
  corresponding 
  to 
  (2) 
  above. 
  

   If 
  yfr 
  = 
  1, 
  (8) 
  yields 
  

  

  ^r 
  2rc 
  12a 
  s 
  + 
  6a 
  a 
  d0 
  2 
  W 
  ' 
  " 
  [) 
  

  

  expressing 
  R 
  x 
  as 
  a 
  function 
  of 
  0, 
  when 
  r 
  is 
  known 
  as 
  such. 
  

   To 
  interpolate 
  a 
  curve 
  for 
  which 
  p 
  takes 
  the 
  place 
  of 
  r, 
  we 
  

   Jaave 
  to 
  eliminate 
  R 
  T 
  between 
  

  

  (7) 
  

  

  Btt* 
  (2R 
  t 
  - 
  B 
  /'fr 
  

   2a 
  6a 
  s 
  

  

  l 
  = 
  B„-^ 
  + 
  

  

  + 
  = 
  K 
  ,_M 
  + 
  <»^>' 
  

  

  