﻿Two- 
  Dimensional 
  Motion 
  of 
  Infinite 
  Liquid. 
  12 
  S 
  

  

  motion 
  satisfies 
  all 
  the 
  requirements 
  of 
  the 
  problem, 
  and 
  is 
  

   the 
  motion 
  due 
  to 
  the 
  rotation 
  of 
  the 
  solid. 
  

  

  6. 
  The 
  first 
  motion. 
  — 
  The 
  first 
  motion 
  will 
  be 
  taken 
  to 
  be 
  

   a 
  rotation, 
  as 
  if 
  rigid, 
  with 
  angular 
  velocity 
  g>, 
  about 
  the 
  

   point 
  z 
  = 
  k 
  exp 
  (iy) 
  . 
  There 
  is 
  no 
  loss 
  of 
  generality 
  in 
  this 
  

   choice, 
  as 
  the 
  substitution 
  of 
  another 
  centre 
  of 
  rotation 
  can 
  

   always 
  be 
  effected 
  by 
  superposing 
  a 
  motion 
  due 
  to 
  trans- 
  

   lation 
  of 
  the 
  boundary 
  as 
  explained 
  in 
  article 
  3. 
  

  

  The 
  first 
  motion 
  may 
  be 
  specified 
  by 
  its 
  stream 
  function 
  yjr 
  u 
  

   namely 
  

  

  +i 
  = 
  — 
  i© 
  |*-*exp(ty) 
  | 
  2 
  , 
  . 
  . 
  . 
  (10) 
  

   which 
  is 
  also 
  expressible 
  as 
  a 
  function 
  of 
  f 
  and 
  n. 
  Another 
  

   specification 
  is 
  by 
  w 
  1? 
  Vi, 
  where 
  

  

  ui='dir 
  1 
  fdv, 
  vi 
  =— 
  3^/3?, 
  . 
  . 
  (11) 
  

   these 
  also 
  being 
  regarded 
  as 
  functions 
  of 
  f 
  and 
  n. 
  It 
  is 
  to 
  

   be 
  noted 
  that 
  u 
  x 
  and 
  vi 
  are 
  not 
  velocity 
  components, 
  but 
  that 
  

   the 
  velocity 
  components 
  in 
  the 
  directions 
  corresponding 
  

   respectively 
  to 
  f 
  and 
  v 
  increasing 
  are 
  ujh 
  and 
  vi/h. 
  

   From 
  formula 
  (4) 
  it 
  follows 
  that 
  

  

  x 
  — 
  /CC0S7+X--J 
  /c 
  e 
  2nr 
  '^ 
  sin 
  (-r~? 
  — 
  To) 
  

  

  + 
  £ 
  JL.-M.-,* 
  sin 
  ^jKirDf 
  +7s 
  ) 
  j, 
  

  

  y 
  = 
  /esiny-f- 
  — 
  -J 
  ^^cos^^f 
  — 
  y 
  ) 
  

  

  so 
  that, 
  when 
  rj 
  is 
  great, 
  

  

  ^i=-g^2- 
  J 
  * 
  2 
  ^''*- 
  2*0*2 
  cos 
  (-yf-f- 
  72-70) 
  

  

  -/W^cos 
  (^f 
  + 
  78-7o)+ 
  • 
  • 
  . 
  } 
  , 
  (12) 
  

   and 
  

  

  .... 
  (13) 
  

   », 
  = 
  ^ 
  J 
  2* 
  KjSin 
  I 
  -- 
  f 
  +72-70 
  J 
  

  

  + 
  | 
  W 
  -2»»A 
  sin 
  (£?f 
  +78-7o) 
  +•••}, 
  (14) 
  

   the 
  terms 
  being 
  arranged 
  in 
  descending 
  order 
  of 
  importance. 
  

  

  