﻿> 
  g- 
  ZLO& 
  

  

  Two-Dimensional 
  Motion 
  of 
  Infinite 
  Liquid. 
  

   The 
  stream-function 
  is 
  then 
  

  

  125 
  

  

  tyt 
  

  

  

  { 
  A(?')} 
  2 
  log- 
  

  

  sin 
  £(?-r) 
  

  

  S 
  in|(r-r') 
  

  

  <*?w, 
  (17) 
  

  

  and 
  this 
  specifies 
  a 
  motion 
  in 
  the 
  z 
  plane 
  which 
  has 
  vorticity 
  a> 
  

   at 
  all 
  points 
  for 
  which 
  £>77>-0, 
  and 
  has 
  the 
  curve 
  corre- 
  

   sponding 
  to 
  77 
  = 
  as 
  a 
  fixed 
  boundary. 
  

   The 
  corresponding 
  u, 
  v 
  functions 
  are 
  

  

  u 
  t 
  

  

  

  sinh-^(i;-V) 
  

  

  cosh 
  ^ 
  (17 
  -*/)- 
  cos 
  -^ 
  ((■-£') 
  

  

  sinh— 
  (77 
  + 
  77') 
  

  

  cosh 
  — 
  (77 
  + 
  77') 
  — 
  cos 
  — 
  - 
  (f 
  — 
  f 
  ') 
  

  

  df'<V, 
  (18) 
  

  

  v<= 
  — 
  

  

  

  nWff)Y 
  

   

  

  sinh^(f-r) 
  

  

  cosh-^(ij— 
  i/O-cos-^ 
  (f-f) 
  

  

  sin^(f-r) 
  

  

  iTT 
  

  

  27T 
  

  

  cosh 
  — 
  (rj 
  -f 
  77O 
  — 
  cos 
  — 
  (f 
  - 
  f 
  ') 
  

  

  <*£'<&/. 
  (19) 
  

  

  It 
  is 
  to 
  be 
  noticed 
  that 
  if 
  £is 
  inside 
  the 
  area 
  of 
  integration 
  

   the 
  subjects 
  of 
  integration 
  in 
  (17), 
  (18), 
  and 
  (19) 
  have 
  

   infinities 
  at 
  ?' 
  = 
  £; 
  these 
  infinities, 
  however, 
  are 
  not 
  suffi- 
  

   ciently 
  powerful 
  to 
  make 
  the 
  integrals 
  divergent. 
  

  

  8. 
  These 
  formulae 
  may 
  be 
  checked 
  by 
  noting 
  directly 
  what 
  

   conditions 
  u 
  t 
  , 
  v 
  t 
  must 
  satisfy 
  if 
  they 
  are 
  to 
  represent 
  the 
  kind 
  

   of 
  motion 
  in 
  the 
  z 
  plane 
  which 
  has 
  been 
  described 
  in 
  the 
  

   previous 
  article. 
  

  

  £ 
  and 
  77 
  are 
  curvilinear 
  coordinates 
  in 
  the 
  z 
  plane, 
  and 
  the 
  

   corresponding 
  velocity 
  components 
  are 
  TJ 
  = 
  u 
  t 
  /h 
  and 
  V=v 
  t 
  /h. 
  

   The 
  boundary 
  condition 
  V 
  = 
  or 
  v*=0, 
  when 
  77 
  = 
  0, 
  is 
  clearly 
  

  

  