﻿400 
  Mr. 
  W. 
  G. 
  Bickley 
  onTwo-Dimemional 
  Potential 
  

  

  facilitates 
  comparison), 
  are 
  tabulated 
  for 
  the 
  values 
  10°, 
  30°, 
  

   90°, 
  and 
  170°, 
  of 
  a, 
  for 
  the 
  central 
  point 
  and 
  for 
  four 
  more 
  

   dividing 
  the 
  semi-arc 
  into 
  five 
  equal 
  parts. 
  The 
  fractions 
  of 
  

   charge 
  on 
  the 
  two 
  faces 
  are 
  added, 
  and 
  also 
  the 
  values 
  for 
  

   a 
  plane 
  lamina. 
  The 
  third 
  column 
  in 
  each 
  case 
  gives 
  the 
  

   mean 
  of 
  the 
  first 
  two, 
  and 
  even 
  for 
  « 
  = 
  30° 
  shows 
  no 
  great 
  

   difference 
  from 
  the 
  values 
  for 
  the 
  plane. 
  

  

  Table 
  of 
  Surface 
  Densities. 
  

  

  Plane 
  J 
  

  

  1-000 
  

   I 
  1-020 
  

   1-091 
  

   1-250 
  

   1-667 
  

  

  10°. 
  

  

  •913 
  

  

  •933 
  

  

  1-003 
  

  

  1-162 
  

  

  1-577 
  

  

  •472 
  

  

  1-087 
  

   1-107 
  

   1177 
  

   1-336 
  

   1-753 
  

  

  •528 
  

  

  1-000 
  

   L-020 
  

   1-090 
  

   1-249 
  

  

  *=3G°. 
  

  

  1-664 
  jl'383!l-901 
  1642 
  '749 
  

  

  •74111-259 
  1-000 
  

   •761 
  1279 
  L-020 
  

   •828 
  ! 
  1-346 
  1-087 
  

   •981 
  ! 
  1-500 
  1-240 
  

  

  06 
  = 
  90°. 
  

  

  •293 
  

  

  •305 
  

   •358 
  

   •454 
  

  

  •417 
  

  

  •583 
  

  

  •25 
  

  

  1-707 
  1-000 
  

   1-720 
  1013 
  

   1-772 
  1-065 
  

   1-868 
  1-161 
  

   2-163 
  [1456 
  

  

  •75 
  : 
  

  

  «=170°. 
  

  

  •0038 
  

  

  1-9962 
  

  

  1-0000 
  

  

  •0042 
  

  

  1-9962 
  

  

  100021 
  

  

  •0055 
  

  

  1-9966 
  

  

  1-0010 
  

  

  0097 
  

  

  2-0017 
  

  

  1-0057 
  

  

  •0281 
  

  

  2-0198 
  

  

  1-024 
  

  

  •0278 
  

  

  i 
  

  

  •9732 
  

  

  

  These 
  may 
  be 
  compared 
  with 
  the 
  values 
  obtained 
  by 
  

   Lord 
  Kelvin 
  for 
  the 
  analogous 
  problem 
  of 
  the 
  spherical 
  

   bowl. 
  

  

  § 
  5. 
  Of 
  course, 
  the 
  above 
  results 
  may 
  also 
  be 
  interpreted 
  

   as 
  applying 
  to 
  the 
  motion 
  of 
  an 
  incompressible 
  fluid 
  circu- 
  

   lating 
  round 
  the 
  lamina, 
  but 
  this 
  is 
  passed 
  over 
  as 
  of 
  relatively 
  

   small 
  importance. 
  The 
  more 
  interesting 
  problem 
  of 
  the 
  

   flow 
  past 
  such 
  a 
  lamina 
  when 
  placed 
  in 
  a 
  steady 
  stream, 
  or 
  

   the 
  motion 
  in 
  an 
  incompressible 
  fluid 
  due 
  to 
  a 
  motion 
  of 
  

   translation 
  of 
  the 
  lamina, 
  may 
  be 
  solved 
  in 
  a 
  similar 
  manner, 
  

   by 
  taking 
  a 
  double 
  source 
  in 
  the 
  f-plane. 
  In 
  view 
  of 
  the 
  

   final 
  hydrodynamical 
  interpretation 
  it 
  will 
  be 
  convenient 
  to 
  

   make 
  w 
  purely 
  real 
  along 
  the 
  real 
  axis 
  and 
  to 
  choose 
  the 
  

   strength 
  and 
  inclination 
  of 
  the 
  source, 
  so 
  that 
  w->~ze~*P 
  as 
  

  

  This 
  gives 
  the 
  value 
  of 
  w 
  as 
  a 
  function 
  of 
  f 
  : 
  — 
  

  

  w=. 
  — 
  

  

  2 
  sin 
  ^ 
  I 
  1 
  -4- 
  sin 
  -x 
  \ 
  

  

  cos- 
  

  

  »/3 
  

  

  >*|8 
  

  

  + 
  

  

  .1 
  

  

  1+ 
  ^ 
  I 
  

  

  1+si,1 
  9 
  

  

  1 
  ?-' 
  — 
  - 
  r+«- 
  

  

  a 
  a. 
  

  

  cos 
  2 
  cos- 
  j 
  

  

  + 
  2sin^sin/3, 
  (12) 
  

  

  here 
  the 
  last 
  term 
  has 
  been 
  added 
  to 
  give 
  the 
  final 
  result 
  a 
  

  

  