﻿Focal 
  Plane 
  of 
  a 
  Telescope 
  with 
  Circular 
  Aperture. 
  3 
  

  

  elements 
  of 
  the 
  luminous 
  source. 
  In 
  other 
  words, 
  the 
  illumi- 
  

   nation 
  is 
  proportional 
  to 
  

  

  m^. 
  

  

  ■where 
  da 
  represents 
  an 
  element 
  of 
  the 
  luminous 
  area. 
  

  

  For 
  effecting 
  the 
  integration 
  we 
  can 
  conveniently 
  put 
  for 
  

   nearly 
  normal 
  incidence 
  

  

  2ttRA 
  

  

  where 
  A 
  denotes 
  the 
  angular 
  interval 
  between 
  the 
  incident 
  

   and 
  the 
  diffracted 
  ray. 
  It 
  is 
  to 
  be 
  remarked 
  that 
  for 
  

   \ 
  = 
  0*589^, 
  r=l, 
  and"A 
  = 
  l", 
  R=l*93 
  centims. 
  Thus, 
  ex- 
  

   pressed 
  in 
  polar 
  coordinates, 
  r, 
  6, 
  the 
  present 
  problem 
  reverts 
  

   to 
  the 
  evaluation 
  of 
  

  

  •) 
  

  

  ffSH 
  

  

  drdd. 
  

  

  the 
  integration 
  extending 
  over 
  the 
  whole 
  luminous 
  source. 
  

  

  « 
  • 
  • 
  • 
  

  

  The 
  intensity 
  of 
  illumination 
  at 
  « 
  , 
  /3' 
  in 
  the 
  focal 
  plane 
  is 
  

  

  consequently 
  given 
  by 
  

  

  =K 
  KW 
  d) 
  , 
  cW! 
  

  

  K 
  being 
  a 
  constant 
  to 
  be 
  afterwards 
  determined. 
  

   Since 
  

  

  -^(J.'W+J 
  l 
  -(0)=^*, 
  

  

  we 
  can 
  write 
  

  

  i=-fjy^(J«r)+j 
  1 
  «(r))*«»: 
  

  

  If 
  the 
  luminous 
  source 
  be 
  a 
  plane 
  of 
  infinite 
  extent, 
  

  

  since 
  

  

  J 
  (0) 
  = 
  1, 
  JilO) 
  = 
  0, 
  J 
  (qo)=0, 
  J 
  1 
  (oo)=0. 
  

  

  We 
  shall 
  henceforth 
  assume 
  the 
  intensity 
  for 
  an 
  infinite 
  

   source 
  to 
  be 
  unity 
  ; 
  thus, 
  on 
  this 
  assumption, 
  

  

  1 
  

  

  K= 
  

  

  7T 
  

  

  and 
  

  

  * 
  See 
  Lommel, 
  Mathematische. 
  Annakn, 
  Ed. 
  xiv. 
  p. 
  510 
  (1879) 
  ; 
  Eay- 
  

   leigh, 
  Phil. 
  Mag. 
  [5] 
  vol. 
  xi. 
  (1881). 
  

  

  B 
  2 
  

  

  