﻿r* 
  

  

  124 
  

  

  Dr. 
  J. 
  G. 
  Leathern 
  on 
  the 
  

  

  7. 
  The 
  second 
  motion. 
  — 
  The 
  second 
  motion, 
  a 
  rotational 
  

   motion 
  having 
  zero 
  normal 
  velocity 
  at 
  the 
  boundary, 
  is 
  got 
  

   by 
  an 
  imaging 
  of 
  the 
  vortex 
  distribution 
  ; 
  and 
  the 
  utility 
  of 
  

   the 
  periodic 
  conformal 
  transformation 
  consists 
  in 
  the 
  fact 
  

   that 
  it 
  makes 
  this 
  imaging 
  process 
  possible. 
  

  

  The 
  specification 
  of 
  the 
  motion 
  may 
  be 
  by 
  a 
  stream- 
  

   function 
  -\|r 
  2 
  , 
  or 
  by 
  functions 
  (u 
  2 
  , 
  v 
  2 
  ) 
  equal 
  to 
  ^yfr^drj, 
  

   — 
  C^s/d 
  f> 
  t 
  ne 
  corresponding 
  velocity 
  components 
  in 
  the 
  

   z 
  plane 
  being 
  u 
  2 
  /h, 
  v 
  2 
  /h. 
  

  

  It 
  is 
  convenient, 
  for 
  a 
  moment, 
  to 
  think 
  of 
  yfr 
  2 
  as 
  the 
  

   stream-function 
  of 
  a 
  motion 
  in 
  the 
  % 
  plane, 
  of 
  which 
  (w 
  2 
  , 
  ^2) 
  

   would 
  be 
  the 
  velocity. 
  If 
  such 
  a 
  motion, 
  unhampered 
  by 
  

   any 
  rigid 
  boundary, 
  were 
  due 
  to 
  a 
  line- 
  vortex 
  representing 
  

   a 
  circulation 
  m, 
  situated 
  at 
  ?=£', 
  and 
  periodically 
  repeated 
  

   at 
  £'±r\, 
  (rsl, 
  2, 
  3,...), 
  it 
  is 
  known* 
  that 
  the 
  corre- 
  

   sponding 
  stream-function 
  would 
  be 
  

  

  m 
  , 
  

  

  sin 
  % 
  (?-?') 
  

  

  When 
  the 
  line 
  77 
  = 
  is 
  a 
  rigid 
  boundary 
  an 
  image 
  must 
  be 
  

   introduced 
  at 
  the 
  point 
  £=£", 
  where 
  f" 
  is 
  the 
  complex 
  

   conjugate 
  to 
  f 
  ', 
  and 
  the 
  stream-function 
  is 
  

  

  m 
  . 
  

  

  s 
  log 
  

  

  sin|(r-r)/sin^(t-r) 
  

  

  Instead 
  of 
  single 
  vortices 
  a 
  continuous 
  periodic 
  distri- 
  

   bution 
  of 
  vorticity 
  may 
  be 
  postulated 
  over 
  the 
  whole 
  area 
  

   between 
  r) 
  = 
  and 
  rj 
  = 
  t, 
  the 
  former 
  line 
  being 
  still 
  a 
  rigid 
  

   boundary; 
  and 
  if 
  m 
  be 
  replaced 
  by 
  o-(f')dS', 
  where 
  d$' 
  is 
  

   an 
  element 
  of 
  area 
  and 
  <r 
  a 
  density 
  of 
  distribution, 
  the 
  

   stream-function 
  is 
  

  

  &jW)log 
  

  

  sin 
  I 
  (?-?)/ 
  sin 
  £«-{") 
  

  

  dS', 
  (15) 
  

  

  the 
  area 
  integral 
  being 
  taken 
  over 
  a 
  rectangle 
  of 
  length 
  t 
  

   and 
  breadth 
  X. 
  

  

  If 
  d8 
  in 
  the 
  z 
  plane 
  and 
  dS' 
  in 
  the 
  f 
  plane 
  be 
  corre- 
  

   sponding 
  elements 
  of 
  area, 
  

  

  so, 
  if 
  the 
  circulation 
  round 
  the 
  contour 
  of 
  any 
  area 
  in 
  the 
  

   f 
  plane 
  is 
  to 
  be 
  equal 
  to 
  2o> 
  times 
  the 
  corresponding 
  area 
  in 
  

   the 
  z 
  plane, 
  it 
  is 
  necessary 
  that 
  

  

  <r(0 
  = 
  2a,^(r')p. 
  .... 
  (16) 
  

  

  * 
  Proc. 
  Koyal 
  Irish 
  Academy, 
  /. 
  c. 
  § 
  13. 
  

  

  