﻿130 
  Two-. 
  Dimensional 
  Motion 
  of 
  Infinite 
  Liquid. 
  

  

  available 
  is 
  demonstrated 
  in 
  the 
  writer's 
  paper 
  on 
  the 
  

   subject 
  referred 
  to 
  above. 
  In 
  particular, 
  mention 
  may 
  be 
  

   made 
  of 
  polygonal 
  boundaries 
  (I. 
  c. 
  § 
  8) 
  ; 
  and 
  it 
  may 
  be 
  

   noticed 
  that 
  for 
  a 
  regular 
  polygon 
  of 
  n 
  sides 
  the 
  trans- 
  

   formation 
  is 
  

  

  i- 
  i 
  r£-x(f-*)f-"{-- 
  ! 
  ?} 
  5 
  - 
  (29 
  » 
  

  

  ""■* 
  ,. 
  = 
  i-[ 
  2 
  {«„^-co,^}]", 
  . 
  . 
  ,30, 
  

  

  K 
  being 
  a 
  constant 
  ; 
  the 
  latter 
  expression, 
  with 
  accented 
  

   letters, 
  would 
  be 
  the 
  first 
  factor 
  under 
  the 
  sign 
  of 
  inte- 
  

   gration 
  in 
  formula 
  (25). 
  

  

  In 
  all 
  cases 
  where 
  the 
  periodic 
  transformation 
  is 
  known 
  

   the 
  solution 
  of 
  the 
  hydrody 
  mimical 
  problems 
  is 
  reduced 
  to 
  

   quadratures. 
  

  

  In 
  certain 
  cases 
  the 
  integrations 
  can 
  be 
  completed 
  ; 
  this 
  

   is 
  noticeably 
  the 
  case 
  when 
  /(£"), 
  and 
  therefore 
  also 
  A 
  2 
  , 
  is 
  the 
  

   sum 
  of 
  a 
  finite 
  number 
  of 
  terms 
  harmonic 
  in 
  £. 
  The 
  inte- 
  

   gration 
  may 
  be 
  accurately 
  effected 
  by 
  the 
  method 
  used 
  for 
  

   approximation 
  in 
  article 
  10 
  above. 
  Of 
  the 
  terms 
  arising 
  

   from 
  the 
  multiplication 
  of 
  li 
  2 
  into 
  the 
  series 
  [22] 
  and 
  [23] 
  

   there 
  are 
  only 
  a 
  finite 
  number 
  which 
  do 
  not 
  yield 
  zero 
  result 
  

   when 
  integrated 
  with 
  respect 
  to 
  (■' 
  through 
  a 
  range 
  X, 
  and 
  

   each 
  of 
  these 
  can 
  be 
  integrated 
  separately 
  with 
  respect 
  

  

  to 
  y. 
  

  

  The 
  simplest 
  example 
  is 
  the 
  ellipse, 
  for 
  which 
  the 
  trans- 
  

   formation 
  is 
  

  

  2= 
  ccosh{a 
  — 
  (2iri/\)^}, 
  .... 
  (31) 
  

   so 
  that 
  

  

  » 
  = 
  ^{oo.b2(. 
  + 
  £,)-o« 
  (£f)}.. 
  (32) 
  

  

  The 
  working 
  out 
  of 
  this 
  case 
  may 
  be 
  used 
  to 
  test 
  the 
  

   method, 
  as 
  the 
  results 
  are 
  otherwise 
  known. 
  

  

  Another 
  simple 
  integrable 
  case 
  corresponds 
  to 
  a 
  boundary 
  

   whose 
  polar 
  equation 
  is 
  

  

  r 
  = 
  a 
  + 
  2b 
  cos 
  26, 
  (a>2b). 
  . 
  . 
  . 
  (33) 
  

  

  The 
  transformation 
  is 
  

  

  z 
  = 
  6 
  exp 
  ( 
  - 
  27nf/X) 
  + 
  a 
  exp 
  {jtiri^/X) 
  + 
  b 
  exp 
  (6?n 
  f/\). 
  (34) 
  

  

  12th 
  November, 
  1917. 
  

  

  