﻿1 
  wo- 
  Dimensional 
  Motion 
  of 
  Infinite 
  Liquid. 
  129 
  

  

  a 
  higher 
  order 
  of! 
  greatness 
  at 
  one 
  of 
  the 
  limits 
  of 
  inte- 
  

   gration, 
  while 
  v' 
  = 
  n 
  at 
  another 
  of 
  the 
  limits. 
  These 
  con- 
  

   siderations 
  reduce 
  the 
  important 
  terms 
  in 
  the 
  equivalent 
  

   of 
  the 
  square 
  bracket 
  in 
  formula 
  (26) 
  to 
  

  

  -"" 
  -'» 
  Jo 
  >1 
  

  

  + 
  2«r 
  w/x 
  « 
  * 
  2 
  cosh 
  ^ 
  {i?-t(f- 
  f)} 
  cos 
  (^r 
  + 
  7i~7o)] 
  <*W 
  

  

  —ii\ 
  \ 
  2K 
  K 
  2 
  sinh(-~-^j 
  cos 
  (-^f 
  + 
  72— 
  7o) 
  

  

  xex 
  P 
  ^{i(f-r)-^irfr^', 
  

  

  which 
  reduces, 
  after 
  omission 
  of 
  some 
  negligible 
  elements, 
  

   to 
  a 
  form 
  whose 
  limit, 
  for 
  tr*>x>, 
  on 
  substitution 
  in 
  (26) 
  

   yields 
  the 
  formula 
  

  

  v 
  2 
  + 
  iu 
  2 
  00^ 
  ~^o 
  2 
  exp(^J-|- 
  2^o« 
  2 
  sin 
  (-^ 
  + 
  72— 
  7oJ 
  |- 
  

  

  (27) 
  

  

  11. 
  The 
  difference 
  motion. 
  — 
  If 
  formula 
  (27) 
  be 
  compared 
  

   with 
  formulae 
  (13) 
  and 
  (14) 
  it 
  is 
  seen 
  that 
  

  

  v 
  2 
  + 
  iu 
  2 
  — 
  (vi 
  + 
  iiii) 
  >0 
  (28) 
  

  

  Hence 
  if 
  (u, 
  v) 
  specify 
  the 
  difference 
  motion, 
  so 
  that 
  

  

  u 
  = 
  Ui 
  — 
  u 
  2 
  > 
  v 
  = 
  Vi 
  — 
  v 
  2 
  , 
  

  

  u 
  and 
  v 
  tend 
  to 
  zero 
  at 
  infinity. 
  

  

  Thus 
  the 
  difference 
  motion 
  is 
  an 
  irrotational 
  motion, 
  

   vanishing 
  at 
  infinity 
  and 
  having 
  at 
  the 
  boundary 
  a 
  normal 
  

   velocity 
  corresponding 
  to 
  rotation 
  about 
  the 
  point 
  /cexp 
  (17). 
  

   It 
  is 
  free 
  from 
  circulation, 
  as 
  a 
  circulation 
  would 
  involve, 
  

   for 
  7] 
  infinite, 
  a 
  definite 
  limit 
  value 
  of 
  u 
  different 
  from 
  zero. 
  

   It 
  therefore 
  constitutes 
  the 
  solution 
  of 
  the 
  problem 
  of 
  motion 
  

   due 
  to 
  the 
  rotation 
  of 
  the 
  boundary. 
  

  

  12. 
  Forms 
  of 
  boundary 
  to 
  ichich 
  the 
  method 
  applies. 
  — 
  The 
  

   applicability 
  of 
  this 
  method 
  to 
  solving 
  the 
  problems 
  of 
  

   motion 
  due 
  to 
  translation 
  and 
  rotation 
  depends 
  upon 
  the 
  

   knowledge 
  of 
  a 
  periodic 
  conformal 
  transformation 
  which 
  

   will 
  make 
  any 
  particular 
  form 
  of 
  boundary 
  correspond 
  to 
  

   the 
  real 
  axis 
  in 
  the 
  f 
  plane. 
  That 
  a 
  considerable 
  variety 
  of 
  

   such 
  transformations 
  and 
  their 
  corresponding 
  boundaries 
  is 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  35. 
  No. 
  205. 
  Jan. 
  1918. 
  K 
  

  

  