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  122 
  Dr. 
  J. 
  G. 
  Leathern 
  on 
  the 
  

  

  where 
  and 
  D 
  are 
  complex 
  constants, 
  it 
  is 
  known 
  * 
  that 
  

  

  X 
  + 
  iY=-27T 
  / 
  3D, 
  

  

  where 
  p 
  is 
  the 
  density 
  of 
  the 
  liquid, 
  and 
  it 
  is 
  supposed 
  that 
  

   the 
  mean 
  density 
  of 
  the 
  solid 
  is 
  also 
  p. 
  

  

  If 
  formula 
  (6 
  a) 
  be 
  expressed 
  in 
  terms 
  of 
  f 
  by 
  means 
  of 
  

   formula 
  (4), 
  it 
  yields 
  the 
  approximation 
  

  

  = 
  - 
  V/c 
  Q^Jexp 
  i 
  ( 
  70-/4— 
  -y^) 
  + 
  C 
  

  

  + 
  {v<£)exp^-,) 
  + 
  D(£)ex 
  P 
  (- 
  i7o) 
  } 
  eX 
  p(^ 
  

  

  where 
  C 
  is 
  a 
  constant. 
  On 
  comparison 
  of 
  this 
  with 
  

   formula 
  (8), 
  it 
  appears 
  that 
  the 
  coefficient 
  of 
  exp 
  (27nf/\) 
  

   must 
  equal 
  Vfc 
  (i\/27r) 
  exp 
  i(p, 
  — 
  y 
  ). 
  Hence 
  

  

  D 
  = 
  — 
  ( 
  VA 
  2 
  /4tt 
  2 
  ) 
  { 
  *r 
  2 
  exp 
  (ifi) 
  —k 
  k 
  2 
  exp 
  i(y 
  + 
  7 
  2 
  - 
  p,) 
  }, 
  

  

  and 
  therefore 
  

  

  X 
  + 
  iY 
  = 
  (pV\ 
  s 
  /27r){ 
  | 
  c 
  | 
  2 
  exp 
  (») 
  -c 
  c 
  2 
  exp 
  {-ip) 
  }. 
  . 
  (9) 
  

  

  5. 
  Field 
  of 
  flow 
  due 
  to 
  rotation 
  of 
  the 
  boundary. 
  — 
  When 
  

   the 
  boundary 
  has 
  a 
  motion 
  of 
  rotation, 
  the 
  specification 
  of 
  

   the 
  liquid 
  motion 
  presents 
  greater 
  difficulty 
  ; 
  the 
  outline 
  

   of 
  the 
  procedure 
  is 
  as 
  follows. 
  

  

  One 
  motion 
  is 
  known 
  which 
  satisfies 
  the 
  proper 
  condition 
  

   at 
  the 
  moving 
  boundary 
  — 
  namely, 
  a 
  rotation 
  of 
  the 
  whole 
  

   liquid, 
  as 
  if 
  rigid, 
  with 
  the 
  same 
  angular 
  velocity 
  as 
  the 
  

   boundary. 
  This 
  may 
  be 
  called 
  the 
  first 
  motion. 
  It 
  is 
  not 
  

   the 
  required 
  motion 
  because 
  it 
  is 
  rotational, 
  and 
  because 
  it 
  

   has 
  infinite 
  velocity 
  at 
  infinity. 
  

  

  Another 
  motion, 
  which 
  will 
  be 
  called 
  the 
  second 
  motion, 
  

   can 
  be 
  specified. 
  This 
  also 
  is 
  rotational, 
  having 
  the 
  same 
  

   vorticity 
  at 
  every 
  point 
  as 
  the 
  first 
  motion 
  ; 
  but 
  at 
  the 
  

   boundary 
  its 
  normal 
  velocity 
  is 
  zero. 
  It 
  has 
  infinite 
  velocity 
  

   at 
  infinity. 
  

  

  If 
  the 
  second 
  motion 
  be 
  subtracted 
  from 
  the 
  first 
  motion, 
  

   the 
  result 
  is 
  an 
  irrotational 
  motion 
  whose 
  normal 
  velocity 
  

   at 
  the 
  boundary 
  is 
  the 
  same 
  as 
  that 
  of 
  the 
  boundary 
  itself. 
  

   This 
  may 
  be 
  called 
  the 
  difference 
  motion. 
  If 
  the 
  velocities 
  

   of 
  the 
  first 
  and 
  second 
  motions 
  tend 
  to 
  equality 
  at 
  infinity 
  in 
  

   such 
  manner 
  that 
  the 
  difference 
  motion 
  tends 
  to 
  zero 
  at 
  

   infinity 
  and 
  has 
  no 
  circulation 
  round 
  the 
  solid, 
  the 
  difference 
  

  

  * 
  J. 
  G. 
  Leathern, 
  " 
  Some 
  Applications 
  of 
  Conformal 
  TraDsformation 
  

   to 
  Problems 
  in 
  Hydrodynamics," 
  Phil. 
  Trans. 
  Roy. 
  Soc, 
  A, 
  vol. 
  ccxv. 
  

   1915, 
  § 
  17. 
  

  

  