﻿Focal 
  Plane 
  of 
  a 
  Telescope 
  with 
  Circular 
  Aperture. 
  15 
  

   whence 
  it 
  follows 
  that 
  

  

  2 
  fV 
  k/ 
  \ 
  d<t> 
  

  

  7T 
  

  

  2 
  

  

  7T 
  2 
  (2< 
  

  

  2 
  /' 
  . 
  E 
  

  

  Similarly 
  

  

  Putting 
  

  

  D 
  

  

  I 
  — 
  =i 
  h 
  = 
  ^ 
  -j 
  '* 
  = 
  e 
  K 
  > 
  

  

  we 
  have 
  

  

  + 
  . 
  . 
  . 
  . 
  

  

  7,1 
  = 
  ¥ 
  +2 
  (l) 
  4+15 
  (l) 
  9+ 
  • 
  • 
  • 
  •' 
  lo 
  S 
  A 
  - 
  lo 
  «*i=^ 
  

  

  and 
  

  

  K 
  = 
  | 
  (1 
  + 
  2A 
  + 
  2A 
  4 
  + 
  2/* 
  9 
  + 
  ....); 
  

  

  further 
  

  

  E=*' 
  2 
  K 
  + 
  ^. 
  

  

  K 
  *1 
  + 
  2h 
  + 
  2A 
  4 
  + 
  2/* 
  9 
  + 
  

   or 
  

  

  ■ 
  b 
  ""4K' 
  l 
  + 
  A 
  1 
  - 
  2 
  + 
  /i 
  2 
  * 
  3 
  + 
  /i 
  3 
  - 
  4 
  + 
  . 
  . 
  . 
  '• 
  

  

  whence 
  the 
  intensity 
  of 
  illumination 
  for 
  both 
  cases 
  can 
  be 
  

   easily 
  calculated. 
  

  

  The 
  following 
  table 
  gives 
  the 
  values 
  of 
  K 
  + 
  -, 
  and 
  

  

  K 
  

  

  V 
  

  

  * 
  In 
  the 
  expansion 
  of 
  J 
  2 
  (p)+Ji 
  (p) 
  in 
  semiconvergent 
  series, 
  only 
  the 
  

   first 
  term 
  was 
  retained 
  ; 
  the 
  effect 
  of 
  the 
  succeeding 
  terms 
  after 
  the 
  

   integration 
  is 
  almost 
  insignificant. 
  

  

  t 
  See 
  Weierstrass-Schwarz, 
  Formeln 
  und 
  Lehrsatze 
  sum 
  Gebrauch 
  der 
  

   elliptischen 
  Functionen, 
  2te 
  Auflage, 
  Berlin, 
  1893, 
  pp. 
  44, 
  61. 
  

  

  