﻿2 
  Prof. 
  H. 
  Nagaoka 
  on 
  the 
  Diffraction 
  Phenomena 
  in 
  the 
  

  

  general 
  discussion 
  of 
  Fraunhofer's 
  diffraction-phenomena 
  of 
  a 
  

   circular 
  aperture 
  for 
  a 
  finite 
  source 
  of 
  light, 
  and 
  show 
  how 
  

   the 
  intensity 
  of 
  illumination 
  in 
  the 
  focal 
  plane 
  can 
  be 
  mecha- 
  

   nically 
  evaluated 
  for 
  a 
  luminous 
  source 
  having 
  any 
  given 
  

   shape. 
  I 
  then 
  pass 
  to 
  the 
  consideration 
  of 
  the 
  intensity 
  both 
  

   inside 
  and 
  outside 
  of 
  a 
  circular 
  image 
  ; 
  further, 
  it 
  will 
  be 
  shown 
  

   how, 
  by 
  the 
  superposition 
  of 
  two 
  systems 
  of 
  lines 
  of 
  equal 
  

   intensity 
  we 
  can 
  explain 
  the 
  formation 
  of 
  a 
  ligament 
  during 
  

   the 
  ingress 
  or 
  egress 
  of 
  a 
  dark 
  disk 
  from 
  a 
  luminous 
  source, 
  

   as 
  verified 
  by 
  the 
  experiments 
  of 
  Andre 
  and 
  Angot*. 
  

  

  1. 
  General 
  Expression 
  for 
  the 
  Intensity. 
  

  

  It 
  is 
  well 
  known 
  that 
  a 
  circular 
  aperture 
  gives 
  rise 
  to 
  a 
  

   diffraction 
  pattern 
  which, 
  for 
  a 
  point 
  source 
  of 
  light, 
  consists 
  

   of 
  concentric 
  rings 
  surrounding 
  the 
  image 
  of 
  the 
  luminous 
  

   point. 
  If, 
  instead 
  of 
  a 
  luminous 
  point, 
  we 
  have 
  a 
  finite 
  

   source 
  of 
  light, 
  each 
  element 
  of 
  the 
  source 
  will 
  produce 
  

   similar 
  phenomena 
  ; 
  the 
  illumination 
  in 
  the 
  focal 
  plane 
  of 
  the 
  

   telescope 
  is 
  thus 
  the 
  integral 
  effect 
  of 
  the 
  source 
  of 
  light 
  

   extending 
  over 
  a 
  given 
  geometrical 
  area. 
  The 
  intensity 
  of 
  

   the 
  image 
  of 
  a 
  uniform 
  source 
  is 
  not 
  of 
  a 
  simple 
  character, 
  

   but, 
  as 
  observed 
  in 
  the 
  focal 
  plane 
  of 
  the 
  telescope, 
  will 
  be 
  

   distributed 
  according 
  to 
  a 
  certain 
  law 
  depending 
  on 
  the 
  

   shape 
  of 
  the 
  source 
  and 
  the 
  size 
  of 
  the 
  aperture. 
  

  

  Let 
  the 
  circular 
  opening 
  of 
  the 
  telescope 
  be 
  taken 
  for 
  the 
  

   plane 
  of 
  xy, 
  and 
  denote 
  the 
  cosines 
  of 
  the 
  angles 
  which 
  the 
  

   incident 
  ray 
  makes 
  with 
  xy-axes 
  by 
  a, 
  /3, 
  and 
  those 
  for 
  the 
  

   diffracted 
  ray 
  by 
  «', 
  /3' 
  ; 
  then, 
  putting 
  R 
  = 
  radius 
  of 
  the 
  

   telescope-aperture, 
  \ 
  = 
  wave-length 
  of 
  light, 
  

  

  _ 
  2tt 
  */(*-*')* 
  + 
  (ff-ff)^ 
  

  

  we 
  know 
  that 
  the 
  intensity 
  of 
  the 
  diffracted 
  light 
  in 
  the 
  focal 
  

   plane 
  of 
  the 
  telescope 
  is 
  proportional 
  to 
  

  

  Ji 
  2 
  (r) 
  

  

  where 
  J\(r) 
  is 
  a 
  Bessel 
  function 
  of 
  the 
  first 
  kind 
  and 
  of 
  order 
  1. 
  

   If 
  the 
  source 
  of 
  light 
  be 
  not 
  a 
  geometrical 
  point, 
  we 
  must 
  

   consider 
  a, 
  ft 
  as 
  variable 
  in 
  finding 
  the 
  illumination 
  at 
  points 
  

   corresponding 
  to 
  a', 
  /3', 
  and 
  sum 
  the 
  effects 
  due 
  to 
  all 
  the 
  

  

  * 
  Andr6 
  et 
  Angot, 
  Annates 
  de 
  VEcole 
  normale, 
  [2] 
  torn. 
  x. 
  p. 
  323 
  

   (1881). 
  

  

  