﻿402 
  Mr. 
  W. 
  G. 
  Bickley 
  on 
  Two-Dimensional 
  Potential 
  

  

  a 
  known 
  result. 
  Also 
  by 
  making 
  a 
  = 
  7r, 
  we 
  have 
  the 
  case 
  of 
  

   a 
  complete 
  cylinder, 
  supposed 
  to 
  be 
  filled 
  with 
  liquid 
  of 
  the 
  

   same 
  density 
  as 
  surrounds 
  it 
  : 
  — 
  

  

  X 
  = 
  2<7rpa 
  2 
  Ucos/3, 
  Y 
  = 
  27r 
  / 
  oa 
  2 
  Usin#, 
  . 
  . 
  (17) 
  

  

  another 
  known 
  result. 
  For 
  the 
  kinetic 
  energy, 
  we 
  easily 
  

   derive 
  from 
  (15') 
  

  

  T 
  = 
  7rpa 
  2 
  U 
  2 
  |cos 
  2 
  /3sm 
  4 
  |+ 
  si.i 
  2 
  /5siii 
  2 
  ^l+ 
  cos 
  2 
  ^)|. 
  (18) 
  

  

  Equations 
  (15') 
  and 
  (18) 
  show 
  that 
  the 
  lamina 
  behaves 
  as 
  

   if 
  it 
  had 
  a 
  mass 
  depending 
  on 
  the 
  direction 
  of 
  motion, 
  

   compounded 
  of 
  masses 
  m 
  x 
  , 
  m 
  y 
  , 
  the 
  coefficients 
  of 
  Ucos/3, 
  

   Usin/3, 
  respectively 
  in 
  (If/). 
  

  

  § 
  7. 
  Next 
  to 
  determine 
  r,he 
  resultant 
  of 
  the 
  pressures 
  on 
  

   the 
  lamina. 
  Substituting 
  z— 
  —ie 
  lQ 
  in 
  (14), 
  w 
  is 
  found 
  to 
  be 
  

   real 
  (as 
  it 
  should 
  be) 
  when 
  \6\ 
  < 
  a, 
  and 
  so 
  we 
  have 
  

  

  <£=-2 
  -j 
  sin-cos[^-/3j 
  ± 
  sin(- 
  -/3J^sin 
  2 
  | 
  - 
  sin 
  2 
  ^ 
  |, 
  (19) 
  

  

  the 
  ambiguous 
  sign 
  referring 
  to 
  opposite 
  faces 
  of 
  the 
  lamina, 
  

   -f 
  to 
  the 
  concave 
  side. 
  The 
  velocity 
  q 
  at 
  any 
  point 
  is 
  given 
  

  

  bv 
  — 
  ^ 
  , 
  so 
  that 
  

  

  q=] 
  cos(6>-/3)±cos^-^Y/sin' 
  2 
  |-sin 
  2 
  | 
  

  

  . 
  e 
  e 
  . 
  /$ 
  

  

  sin 
  - 
  cos 
  ^ 
  sin 
  

  

  + 
  V— 
  ::— 
  =jt 
  1- 
  • 
  • 
  • 
  (20; 
  

  

  v 
  sin2 
  l 
  

  

  2 
  # 
  

   sm 
  2 
  J 
  

  

  But 
  p 
  = 
  const— 
  ^pq 
  2 
  , 
  so 
  denoting 
  the 
  excess 
  of 
  pressure 
  on 
  

   the 
  concave 
  face 
  at 
  any 
  point 
  Ap, 
  we 
  have 
  

  

  Ap 
  __ 
  

  

  P 
  

  

  (0-/3) 
  4 
  cos(--/3Ysin 
  2 
  |- 
  sin 
  2 
  9 
  J- 
  sin 
  -cos 
  ^ 
  sin 
  (^ 
  -Z 
  3 
  ) 
  J 
  

  

  1 
  cos 
  

  

  V 
  

  

  sin^- 
  — 
  snr 
  ^ 
  

  

  (21) 
  

  

  