﻿Focal 
  Plane 
  of 
  a 
  Telescope 
  with 
  Circular 
  Aperture. 
  11 
  

   Integrating 
  by 
  parts, 
  

  

  _i 
  ifJo'W+Ji'W^ 
  

  

  V4<r 
  — 
  

  

  = 
  i 
  f 
  ?i-Jo'O0-Ji 
  8 
  W)^ 
  a 
  

  

  7T 
  Jo 
  V4a 
  2 
  -r 
  2 
  

  

  The 
  above 
  integral 
  shows 
  that 
  the 
  intensity 
  at 
  the 
  rim 
  

   approaches 
  the 
  value 
  J, 
  as 
  the 
  radius 
  of 
  the 
  disk 
  is 
  indefinitely 
  

   increased. 
  

  

  We 
  have 
  already 
  seen 
  how 
  J 
  2 
  (V) 
  + 
  Ji 
  2 
  (V) 
  can 
  be 
  expanded 
  

   in 
  ascending 
  as 
  well 
  as 
  descending 
  powers 
  of 
  r. 
  For 
  values 
  

   of 
  r 
  smaller 
  than 
  ^', 
  the 
  semiconvergent 
  series 
  cannot 
  be 
  

   used, 
  while 
  for 
  values 
  of 
  r 
  somewhat 
  greater 
  than 
  x 
  x 
  it 
  is 
  

   disadvantageous 
  to 
  use 
  the 
  the 
  ascending 
  series. 
  We 
  shall, 
  

   therefore, 
  have 
  to 
  divide 
  the 
  integral 
  into 
  two 
  parts 
  : 
  namely, 
  

   one 
  extending 
  from 
  to 
  x 
  ly 
  for 
  which 
  the 
  expansion 
  in 
  

   ascending 
  powers 
  of 
  r 
  should 
  be 
  used; 
  and 
  from 
  x 
  x 
  to 
  2a, 
  for 
  

   which 
  the 
  semiconvergent 
  series 
  should 
  be 
  employed. 
  

  

  It 
  is 
  to 
  be 
  noticed 
  that 
  the 
  variation 
  of 
  Jo 
  2 
  (^)+Ji 
  2 
  W 
  is 
  

   very 
  small 
  in 
  the 
  neighbourhood 
  of 
  r 
  = 
  x\, 
  so 
  that 
  it 
  would 
  be 
  

   advantageous 
  to 
  fix 
  the 
  limit 
  of 
  integration 
  at 
  r 
  = 
  Xi( 
  = 
  3*8317), 
  

   in 
  order 
  to 
  diminish 
  the 
  error 
  in 
  integration. 
  

  

  Thus, 
  

  

  j 
  1 
  (* 
  x 
  \.< 
  T 
  o/ 
  v 
  -r 
  o, 
  x 
  v 
  dr 
  

  

  =^r 
  , 
  ( 
  i 
  - 
  j 
  o 
  2 
  w--vw) 
  

  

  Jo 
  

  

  *J 
  .r. 
  

  

  dr 
  

  

  s/Aa'—r*' 
  

  

  For 
  practical 
  application, 
  a 
  is 
  generally 
  large 
  compared 
  to 
  

   x 
  x 
  ; 
  we 
  shall 
  therefore 
  assume 
  a 
  > 
  x 
  x 
  ; 
  then 
  

  

  1 
  

  

  (l-VM-VW) 
  

  

  vv— 
  ^ 
  

  

  J 
  

  

  0-9615 
  1-579 
  1-543 
  

  

  = 
  + 
  — 
  r" 
  + 
  — 
  — 
  + 
  1 
  

  

  a 
  a 
  a 
  

  

  which 
  converges 
  very 
  rapidly 
  when 
  a 
  is 
  large. 
  

  

  