﻿Motion 
  of 
  an 
  Infinite 
  Liquid. 
  501 
  

  

  complete 
  and 
  general. 
  When 
  the 
  paper 
  appeared, 
  the 
  author 
  

   of 
  the 
  present 
  paper 
  was 
  engaged 
  in 
  an 
  attempt, 
  since 
  com- 
  

   pleted, 
  to 
  solve 
  a 
  particular 
  case 
  of 
  the 
  above 
  problem, 
  and 
  

   was 
  led 
  to 
  a 
  general 
  method 
  of 
  attack 
  which 
  seems 
  to 
  give 
  

   results 
  more 
  immediately, 
  and 
  with 
  a 
  less 
  complicated 
  pro- 
  

   cedure, 
  and 
  it 
  is 
  thought 
  that 
  an 
  outline 
  of 
  the 
  method 
  may 
  

   be 
  of 
  interest. 
  

  

  § 
  2. 
  Needless 
  to 
  mention, 
  conformal 
  representation 
  plays 
  a 
  

   large 
  part. 
  Instead 
  of 
  the 
  periodic 
  transformation 
  advocated 
  

   by 
  Dr. 
  Leathern, 
  it 
  was 
  found 
  more 
  convenient 
  to 
  use 
  one 
  

   whereby 
  the 
  doubly 
  connected 
  region 
  of 
  the 
  2-plane 
  becomes 
  

   the 
  upper 
  half 
  of 
  the 
  f-plane, 
  the 
  boundary 
  in 
  the 
  ^-plane 
  

   becoming 
  the 
  real 
  axis 
  in 
  the 
  f-plane. 
  If 
  the 
  periodic 
  

   transformation 
  is 
  known, 
  this 
  may 
  be 
  effected 
  by 
  taking 
  as 
  

  

  the 
  auxiliary 
  variable 
  tan— 
  in 
  the 
  notation 
  of 
  Dr. 
  Leathern. 
  

  

  A, 
  

  

  This 
  will 
  for 
  the 
  present 
  be 
  denoted 
  by 
  f 
  (=f-H?7). 
  The 
  

  

  transformation 
  may 
  be 
  written 
  

  

  --=/(?) 
  (i) 
  

  

  In 
  particular 
  for 
  the 
  ellipse, 
  of 
  semiaxes 
  c 
  cosh 
  a, 
  csinh 
  a, 
  

   ^ 
  = 
  c-{2fcosh« 
  + 
  t(l-? 
  2 
  )sinh 
  a 
  }/(l+^). 
  

  

  § 
  3. 
  The 
  method 
  now 
  depends 
  on 
  the 
  fact 
  that, 
  except 
  as 
  

   to 
  a 
  constant, 
  the 
  value 
  of 
  the 
  stream 
  function 
  is 
  known 
  on 
  

   the 
  boundary, 
  and 
  therefore 
  on 
  the 
  axis 
  of 
  £. 
  In 
  particular, 
  

   if 
  the 
  motion 
  is 
  one 
  of 
  uniform 
  translation 
  with 
  velocity 
  U 
  

   in 
  the 
  direction 
  inclined 
  at 
  an 
  angle 
  /3 
  to 
  the 
  #-axis, 
  we 
  

   have 
  on 
  the 
  boundary 
  

  

  f=iUxI{^} 
  + 
  const 
  (2) 
  

  

  For 
  the 
  case 
  of 
  rotation 
  about 
  the 
  point 
  z 
  with 
  angular 
  

   velocity 
  a>, 
  on 
  the 
  boundary 
  

  

  yjr 
  a 
  =z\a) 
  U— 
  r 
  j 
  2 
  + 
  const 
  (3) 
  

  

  On 
  the 
  boundary 
  z 
  = 
  f(jj) 
  since 
  77 
  = 
  0, 
  therefore 
  we 
  have 
  

   as 
  the 
  values 
  of 
  yjr 
  on 
  the 
  real 
  axis 
  in 
  the 
  f-plane, 
  

  

  ih=«UxI{/(f).<T*} 
  + 
  const. 
  . 
  . 
  . 
  (2') 
  

  

  *2 
  =i« 
  |/(5) 
  -*o| 
  2 
  + 
  const 
  (30 
  

  

  Hence, 
  as 
  the 
  corresponding 
  values 
  of 
  to 
  ( 
  = 
  <f>-\- 
  n/r), 
  

  

  ■>-?£«*> 
  -^» 
  & 
  

  

  "'■ 
  = 
  &„L 
  l/<e 
  ~* 
  , 
  "F?r 
  

  

  provided 
  these 
  Integrals 
  converge. 
  This 
  may 
  always 
  be 
  

   secured, 
  since 
  the 
  boundary 
  in 
  the 
  --plane 
  is 
  by 
  hypothesis 
  

  

  