﻿1054 
  Mr. 
  J. 
  11. 
  Wilton 
  on 
  the 
  

  

  where 
  <j> 
  and 
  -v/r 
  are 
  the 
  velocity 
  potential 
  and 
  the 
  stream 
  

   function 
  , 
  respectively. 
  

  

  Then 
  the 
  motion 
  for 
  which 
  the 
  free 
  surface 
  is 
  given 
  by 
  

   eliminating 
  6 
  between 
  # 
  = 
  X(0), 
  y 
  = 
  Y{6) 
  is 
  represented 
  by 
  

   the 
  equations, 
  

  

  ;=X(fl 
  )+«Y(fl), 
  1 
  

  

  where 
  6 
  is 
  a 
  complex 
  variable, 
  X 
  and 
  Y 
  are 
  functions 
  of 
  6 
  

  

  which 
  are 
  real 
  when 
  6 
  is 
  real, 
  and 
  X' 
  and 
  Y' 
  denote 
  rospec- 
  

  

  ;. 
  . 
  dX 
  . 
  dY 
  

   hvely—and-^. 
  

  

  For 
  the 
  condition 
  to 
  be 
  satisfied 
  on 
  the 
  free 
  surface.,, 
  

   y(r 
  = 
  0, 
  is 
  

  

  q 
  a 
  =G-2gy, 
  

  

  where 
  q 
  is 
  the 
  resultant 
  velocity, 
  i. 
  e. 
  

  

  f 
  = 
  

  

  = 
  (C-2,Y)(X- 
  + 
  Y-)(i,, 
  

  

  ?'. 
  e. 
  q 
  2 
  

  

  But 
  i/r 
  = 
  if 
  is 
  real 
  (provided 
  C 
  — 
  2^Y 
  is 
  positive), 
  

   (C-2,Y)g=ii; 
  r 
  C-2,Y 
  

  

  = 
  C-2#, 
  

  

  for 
  on 
  the 
  surface, 
  since 
  6 
  is 
  real, 
  

  

  ^ 
  = 
  X(<9), 
  # 
  = 
  Y(0)'. 
  

  

  The 
  equation 
  of 
  the 
  free 
  surface 
  may 
  thus 
  be 
  chosen 
  arbi- 
  

   trarily. 
  But 
  the 
  converse 
  problem 
  of 
  determining 
  the 
  form 
  

   of 
  the 
  free 
  surface 
  when 
  some 
  other 
  condition 
  is 
  given 
  can 
  

   only 
  be 
  attacked 
  by 
  tentative 
  methods. 
  

  

  It 
  will 
  be 
  convenient 
  for 
  our 
  purpose 
  to 
  put 
  = 
  ¥(w)? 
  

   and 
  Y(0') 
  = 
  $. 
  This 
  does 
  not 
  in 
  any 
  way 
  limit 
  the 
  generality 
  

   of 
  the 
  result. 
  We 
  then 
  have 
  

  

  and 
  the 
  equation 
  of 
  the 
  free 
  surface 
  is 
  

  

  *=JVod; 
  

  

  gHt) 
  

  

  [F'W]* 
  # 
  

  

  = 
  Jv 
  "Sa/V"" 
  1 
  '^- 
  

  

  