﻿Problems 
  connected 
  with 
  the 
  Circular 
  Arc. 
  399 
  

  

  with 
  intervals 
  '25 
  of 
  #, 
  and 
  — 
  ~ 
  of 
  yfr. 
  These 
  show 
  distinctly 
  

  

  the 
  physical 
  features, 
  which 
  of 
  course 
  were 
  anticipated 
  in 
  

   a 
  general 
  way. 
  The 
  charge 
  resides 
  for 
  the 
  most 
  part 
  on 
  

   the 
  outer 
  face, 
  the 
  surface-density 
  becomes 
  infinite 
  at 
  the 
  

   edges, 
  and 
  at 
  a 
  distance 
  the 
  equi 
  potentials 
  are 
  approximately 
  

   circular. 
  One 
  point 
  of 
  some 
  interest 
  appears 
  in 
  the 
  figure, 
  

   namely, 
  the 
  approximate 
  uniformity 
  of 
  the 
  field 
  on 
  the 
  axis, 
  

   just 
  beyond 
  the 
  centre. 
  

  

  If 
  we 
  expand 
  (6) 
  on 
  the 
  supposition 
  that 
  | 
  w 
  | 
  (and 
  conse- 
  

   quently 
  | 
  z 
  |) 
  is 
  large, 
  we 
  obtain 
  

  

  sin 
  g 
  -r-^'cos 
  2 
  ^ 
  ) 
  3 
  

  

  l 
  ' 
  e 
  " 
  9« 
  ,r 
  • 
  a 
  ,r^^ 
  

  

  c4-icos^ 
  ie~ 
  w 
  sm-z, 
  (9) 
  

  

  showing 
  that 
  the 
  field 
  at 
  a 
  great 
  distance 
  is 
  approximately 
  

  

  that 
  due 
  to 
  an 
  equal 
  charge 
  at 
  the 
  point 
  — 
  ^cos 
  2 
  ^. 
  It 
  is 
  

  

  seen 
  that 
  this 
  point, 
  the 
  "centre 
  of 
  charge/' 
  is 
  the 
  mid 
  

   point 
  of 
  that 
  portion 
  of 
  the 
  central 
  radius 
  cut 
  off 
  by 
  the 
  

   chord 
  of 
  the 
  arc. 
  This 
  result 
  may 
  also 
  be 
  obtained 
  by 
  

   integration 
  from 
  the 
  expressions 
  for 
  the 
  surface-density 
  now 
  

   to 
  be 
  obtained. 
  

  

  §4. 
  On 
  the 
  lamina, 
  iv 
  = 
  iijr, 
  while 
  z=—ie' 
  d 
  , 
  and 
  from 
  (6) 
  

   or 
  (7) 
  we 
  obtain 
  the 
  relation 
  

  

  sin 
  ^-gj 
  sin 
  | 
  = 
  sin^ 
  (10) 
  

  

  Now 
  the 
  surface-density 
  is 
  proportional 
  to 
  --— 
  , 
  and 
  so 
  

  

  cos 
  g 
  \ 
  1 
  

  

  cr 
  x 
  J 
  — 
  r 
  + 
  1 
  1 
  convex 
  face, 
  

  

  sin 
  2 
  *- 
  sin 
  2 
  ^ 
  

  

  

  6 
  

  

  cos- 
  

  

  — 
  - 
  — 
  llconcave 
  face. 
  

  

  11) 
  

  

  The 
  values 
  of 
  cr, 
  multiplied 
  by 
  the 
  factor 
  sin 
  x 
  (which 
  makes 
  

   the 
  mean 
  of 
  the 
  values 
  at 
  the 
  pole 
  equal 
  to 
  1, 
  and 
  so 
  

  

  