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  ] 
  

  

  XII. 
  On 
  the 
  Approximate 
  Solution 
  of 
  certain 
  Problems 
  

   relating 
  to 
  the 
  Potential. 
  — 
  II. 
  By 
  Lord 
  Rayleigh, 
  O.M., 
  

   F.R.S* 
  

  

  THE 
  present 
  paper 
  may 
  be 
  regarded 
  as 
  supplementary 
  to 
  

   one 
  with 
  the 
  same 
  title 
  published 
  a 
  long 
  while 
  ago 
  f. 
  

   In 
  two 
  dimensions, 
  if 
  </>, 
  yfr 
  be 
  potential 
  and 
  stream-functions, 
  

   and 
  if 
  (e. 
  g.) 
  yjr 
  be 
  zero 
  along 
  the 
  line 
  y 
  = 
  0, 
  we 
  may 
  take 
  

  

  (i) 
  

  

  ^ 
  = 
  ^-I^3/" 
  + 
  OT3.475^-' 
  • 
  & 
  

  

  / 
  being 
  a 
  function 
  of 
  x 
  so 
  far 
  arbitrary. 
  These 
  values 
  satisfy 
  

   the 
  general 
  conditions 
  for 
  the 
  potential 
  and 
  stream-functions, 
  

   and 
  when 
  y 
  = 
  make 
  

  

  d<j>/dx=f, 
  ^ 
  = 
  0. 
  

  

  Equation 
  (2) 
  may 
  be 
  regarded 
  as 
  determining 
  the 
  lines 
  of 
  

   flow 
  (any 
  one 
  of 
  which 
  may 
  be 
  supposed 
  to 
  be 
  the 
  boundary) 
  

   in 
  terms 
  of 
  /. 
  Conversely, 
  if 
  y 
  be 
  supposed 
  known 
  as 
  a 
  

   function 
  of 
  x 
  and 
  ty 
  be 
  constant 
  (say 
  unity), 
  we 
  may 
  find 
  ; 
  

   by 
  successive 
  approximation. 
  Thus 
  

  

  y 
  * 
  6 
  alxAy) 
  36 
  dx 
  2 
  \ 
  y 
  dx 
  2 
  \y) 
  J 
  120dx*\yJ 
  K 
  } 
  

  

  We 
  may 
  use 
  these 
  equations 
  to 
  investigate 
  the 
  stream- 
  

   lines 
  for 
  which 
  -v|r 
  has 
  a 
  value 
  intermediate 
  between 
  and 
  1. 
  

   If 
  Tj 
  denote 
  the 
  corresponding 
  value 
  of 
  y, 
  we 
  have 
  to 
  

   eliminate 
  / 
  between 
  

  

  i 
  =*/-£/" 
  + 
  ,4/*-..., 
  

  

  120 
  

  

  ♦ 
  -*-*/" 
  + 
  &* 
  1 
  ' 
  — 
  .! 
  

   whence 
  

  

  f" 
  /' 
  iv 
  

  

  v 
  = 
  t/y 
  + 
  y 
  W 
  - 
  vf) 
  - 
  f2o 
  Cyr-VX 
  

  

  * 
  Commuuicated 
  by 
  the 
  Author. 
  

  

  t 
  l'roc. 
  Lond. 
  Math. 
  Soc. 
  vii. 
  p. 
  75 
  (1870) 
  ; 
  Scientific 
  Papers, 
  vol. 
  i. 
  

   p, 
  272. 
  

  

  02 
  

  

  