﻿Focal 
  Plane 
  of 
  a 
  Telescope 
  with 
  Circular 
  Aperture. 
  19 
  

  

  I 
  have 
  calculated 
  these 
  two 
  integrals 
  1/ 
  and 
  LJ 
  1 
  by 
  

   mechanical 
  quadrature, 
  for 
  which 
  Gauss's 
  method 
  seems 
  

   specially 
  suited. 
  The 
  course 
  of 
  1J 
  and 
  1/ 
  is 
  shown 
  in 
  

   fig. 
  7. 
  

  

  The 
  effect 
  due 
  to 
  the 
  remaining 
  part 
  of 
  the 
  circular 
  source 
  

   can 
  be 
  easily 
  calculated 
  by 
  assuming 
  the 
  semiconvergent 
  

   series 
  for 
  J 
  2 
  (p) 
  + 
  J 
  x 
  2 
  (p) 
  . 
  Retaining 
  the 
  first 
  term 
  of 
  the 
  

   series 
  only, 
  we 
  get 
  for 
  the 
  illumination 
  at 
  an 
  internal 
  

   point 
  due 
  to 
  the 
  part 
  external 
  to 
  the 
  sector 
  A 
  OB, 
  

  

  it 
  7T 
  2 
  (2a-S)J 
  I 
  1-lfsin^/^l^ 
  

  

  k 
  2 
  sin 
  2 
  (f> 
  

  

  7T 
  7r 
  2 
  (2a-6)V 
  ri9lj+ 
  U 
  2/c/v/l-^sin 
  2 
  ^/ 
  

  

  where 
  F 
  and 
  E 
  denote 
  elliptic 
  integrals 
  of 
  the 
  1st 
  and 
  

   2nd 
  kind 
  respectively. 
  

  

  For 
  an 
  external 
  point 
  we 
  shall 
  have 
  

  

  a 
  2 
  f 
  E(&) 
  hi 
  sin 
  20 
  t 
  \ 
  

  

  le 
  _ 
  7T 
  + 
  7T\2a-h)V 
  {9lJ 
  kj 
  + 
  *£.' 
  ^1-^sin^r'* 
  

  

  Thus 
  we 
  obtain, 
  for 
  the 
  total 
  intensity 
  of 
  illumination 
  at 
  an 
  

   internal 
  point 
  in 
  the 
  very 
  neighbourhood 
  of 
  the 
  rim, 
  

  

  T 
  1 
  2 
  fYfrh 
  \ 
  . 
  E 
  (»■> 
  tf 
  SiD 
  ^ 
  ^ 
  

  

  1? 
  '~ 
  i 
  ^(2a-8)l 
  ri9lj+ 
  fc' 
  2* 
  4 
  Vl-^sin 
  2 
  ^/ 
  

  

  I.". 
  . 
  . 
  (V.) 
  

  

  2>/«(a— 
  8) 
  

  

  At 
  an 
  external 
  point, 
  

  

  T- 
  2 
  / 
  Ff 
  , 
  , 
  E(fr) 
  *, 
  2 
  sin 
  20t 
  \ 
  

  

  ie 
  -7r 
  2 
  (2«-S)\ 
  ri9lj 
  */ 
  " 
  i 
  "2/:/ 
  v 
  /l-^ 
  2 
  sin 
  2 
  (/, 
  1 
  / 
  

  

  Z 
  wa{a 
  — 
  o) 
  

  

  The 
  expression 
  within 
  the 
  bracket 
  can 
  be 
  calculated 
  by 
  

   means 
  of 
  Legend 
  re's 
  table 
  ; 
  the 
  course 
  of 
  the 
  function 
  for 
  

   different 
  values 
  of 
  <f) 
  and 
  8 
  is 
  shown 
  in 
  fig. 
  8. 
  The 
  curves 
  on 
  

   the 
  right-hand 
  side 
  apply 
  to 
  It, 
  and 
  those 
  on 
  the 
  left 
  to 
  I 
  e 
  . 
  

  

  Having 
  found 
  the 
  values 
  of 
  these 
  different 
  integrals, 
  we 
  can 
  

   now 
  discuss 
  the 
  illumination 
  near 
  the 
  rim 
  of 
  the 
  circular 
  

   image. 
  

  

  We 
  have 
  already 
  found 
  that 
  the 
  intensity 
  of 
  light 
  at 
  the 
  

  

  C2 
  

  

  