﻿198 
  Lord 
  Rayleigh 
  on 
  the 
  Approximate 
  Solution 
  of 
  

  

  determine 
  the 
  distribution 
  of 
  potential 
  upon 
  it, 
  from 
  which 
  

   again 
  a 
  fresh 
  departure 
  may 
  be 
  made 
  if 
  desired. 
  For 
  this 
  

   purpose 
  we 
  regard 
  the 
  rectangular 
  coordinates 
  x, 
  y 
  as 
  

   functions 
  of 
  f 
  (potential) 
  and 
  77 
  (stream-function), 
  so 
  that 
  

  

  * 
  + 
  *y 
  =/(£ 
  + 
  *?), 
  .... 
  (13) 
  

  

  iii 
  which 
  we 
  are 
  supposed 
  to 
  know 
  /(f) 
  corresponding 
  to 
  

   7] 
  = 
  0, 
  i. 
  e. 
  } 
  x 
  and 
  y 
  are 
  there 
  known 
  functions 
  of 
  f 
  . 
  Take 
  

   a 
  point 
  on 
  77 
  = 
  0, 
  at 
  which 
  without 
  loss 
  of 
  generality 
  f 
  may 
  

   be 
  supposed 
  also 
  to 
  vanish, 
  and 
  form 
  the 
  expressions 
  for 
  

   x 
  and 
  y 
  in 
  the 
  neighbourhood. 
  From 
  

  

  x 
  + 
  iy 
  = 
  A 
  + 
  «B 
  + 
  (A 
  x 
  + 
  tBjXf 
  + 
  h) 
  

  

  + 
  (A 
  2 
  4- 
  iB 
  2 
  ) 
  (f 
  + 
  i 
  v 
  ) 
  2 
  + 
  , 
  

  

  we 
  derive 
  

  

  x 
  = 
  A 
  + 
  A,f 
  - 
  Bn, 
  + 
  A 
  2 
  (f 
  - 
  rj>) 
  - 
  2B 
  2 
  £ 
  V 
  

   + 
  A 
  8 
  (f 
  - 
  ZW) 
  - 
  B 
  3 
  (3f^ 
  - 
  v 
  s 
  ) 
  

   + 
  A 
  4 
  (f 
  - 
  6fV 
  + 
  V*) 
  -4B 
  4 
  (f 
  3 
  , 
  - 
  ^) 
  + 
  . 
  . 
  . 
  ., 
  

  

  y 
  = 
  B„ 
  + 
  Bif 
  + 
  A,, 
  + 
  2A 
  2 
  & 
  + 
  B,(P 
  - 
  *? 
  2 
  ) 
  

  

  + 
  A 
  3 
  (3f, 
  - 
  t, 
  3 
  ) 
  +B 
  3 
  (P-3^ 
  2 
  ) 
  

  

  + 
  4A 
  4 
  (f» 
  1? 
  - 
  £? 
  3 
  ) 
  + 
  B 
  4 
  ({* 
  - 
  6fV 
  +V*) 
  + 
  

  

  When 
  17 
  = 
  0, 
  

  

  .r 
  = 
  A 
  + 
  Af 
  + 
  A# 
  + 
  A 
  3 
  r 
  + 
  A# 
  + 
  . 
  • 
  • 
  

  

  y 
  = 
  B 
  + 
  B,f 
  + 
  B,f» 
  + 
  B 
  3 
  f 
  3 
  + 
  B# 
  + 
  . 
  . 
  . 
  

  

  Since 
  # 
  and 
  2/ 
  are 
  known 
  as 
  functions 
  of 
  £ 
  when 
  rj 
  = 
  0, 
  these 
  

   equations 
  determine 
  the 
  A's 
  and 
  the 
  B's, 
  and 
  the 
  general 
  

   values 
  of 
  x 
  and 
  y 
  follow. 
  When 
  f 
  = 
  0, 
  but 
  77 
  undergoes 
  an 
  

   increment, 
  

  

  x 
  = 
  A 
  - 
  B 
  lV 
  - 
  A 
  2 
  v 
  2 
  + 
  B377 
  3 
  + 
  A 
  4 
  t; 
  4 
  -. 
  . 
  ., 
  (14) 
  

  

  y 
  = 
  B 
  .+ 
  A 
  lV 
  - 
  B^ 
  - 
  A 
  3 
  77 
  3 
  + 
  B 
  4 
  V 
  + 
  . 
  . 
  ., 
  (15) 
  

  

  in 
  which 
  we 
  may 
  suppose 
  97 
  = 
  1. 
  

  

  The 
  A's 
  and 
  B's 
  are 
  readily 
  determined 
  if 
  we 
  know 
  the 
  

   values 
  of 
  x 
  and 
  y 
  for 
  77 
  = 
  and 
  for 
  equidistant 
  values 
  of 
  f 
  , 
  

   say 
  f 
  = 
  0, 
  £ 
  = 
  ±1, 
  f 
  = 
  ±2. 
  Thus, 
  if 
  the 
  values 
  of 
  # 
  be 
  

  

  