﻿Focal 
  Plane 
  of 
  a 
  Telescope 
  with 
  Circular 
  Aperture. 
  21 
  

  

  the 
  rim 
  towards 
  the 
  centre 
  ; 
  and 
  from 
  that 
  for 
  I 
  c 
  , 
  the 
  decrease 
  

   takes 
  place 
  very 
  rapidly 
  as 
  we 
  pass 
  from 
  the 
  rim 
  outwards. 
  

   Thus 
  the 
  variation 
  of 
  intensity 
  is 
  greatest 
  near 
  the 
  rim, 
  but 
  

   the 
  change 
  does 
  not 
  take 
  place 
  abruptly 
  ; 
  the 
  intensity 
  fades 
  

   away 
  gradually 
  in 
  the 
  neighbourhood 
  of 
  the 
  rim, 
  as 
  illustrated 
  

   in 
  the 
  accompanying 
  diagram* 
  (fig. 
  9). 
  The 
  full 
  line 
  is 
  for 
  

   a 
  = 
  co 
  , 
  and 
  the 
  dotted 
  line 
  for 
  a 
  = 
  40. 
  

  

  The 
  image 
  of 
  a 
  luminous 
  disk, 
  as 
  seen 
  through 
  a 
  telescope, 
  

   is 
  thus 
  not 
  sharply 
  defined 
  at 
  the 
  geometrical 
  rim, 
  where 
  the 
  

   change 
  of 
  intensity 
  takes 
  place 
  continuously. 
  If 
  the 
  intensity 
  

   for 
  the 
  limit 
  of 
  visibility 
  be 
  less 
  than 
  Ir, 
  the 
  image 
  of 
  the 
  disk 
  

   will 
  appear 
  to 
  a 
  slight 
  extent 
  broadened. 
  

  

  7. 
  Lines 
  of 
  Equal 
  Intensity. 
  

  

  For 
  practical 
  purposes 
  it 
  is 
  sometimes 
  convenient 
  to 
  draw 
  

   the 
  lines 
  of 
  equal 
  intensity. 
  For 
  a 
  circular 
  source 
  of 
  light 
  

   they 
  consist 
  of 
  a 
  series 
  of 
  concentric 
  circles, 
  which, 
  if 
  drawn 
  

   for 
  equal 
  difference 
  of 
  intensity, 
  crowd 
  together 
  near 
  the 
  

   geometrical 
  edge. 
  

  

  When 
  there 
  are 
  different 
  sources 
  of 
  light 
  we 
  can 
  superpose 
  

   the 
  separate 
  effects 
  and 
  graphically 
  represent 
  the 
  distribution 
  

   of 
  illumination 
  in 
  the 
  following 
  manner 
  : 
  — 
  

  

  Draw 
  the 
  lines 
  of 
  equal 
  intensity 
  for 
  the 
  image 
  of 
  each 
  

   source 
  ; 
  at 
  the 
  point 
  of 
  intersection 
  of 
  any 
  two 
  lines 
  the 
  

   intensity 
  will 
  be 
  the 
  sum 
  of 
  the 
  two. 
  We 
  thus 
  obtain 
  a 
  

   system 
  of 
  points 
  of 
  equal 
  intensity. 
  By 
  drawing 
  the 
  lines 
  at 
  

   small 
  intervals 
  we 
  can 
  obtain 
  a 
  sufficient 
  number 
  of 
  points 
  to 
  

   draw 
  curves 
  of 
  equal 
  intensity, 
  which 
  will 
  represent 
  the 
  

   distribution 
  of 
  illumination 
  due 
  to 
  different 
  sources. 
  In 
  

   fact, 
  the 
  process 
  of 
  drawing 
  the 
  lines 
  of 
  equal 
  intensity 
  is 
  

   analogous 
  to 
  that 
  of 
  drawing 
  equipotential 
  lines. 
  

  

  Suppose 
  that 
  the 
  luminous 
  source 
  is 
  a 
  circular 
  disk, 
  on 
  

   which 
  there 
  is 
  a 
  small 
  dark 
  circular 
  space 
  touching 
  the 
  rim 
  of 
  

   the 
  disk. 
  The 
  image, 
  as 
  seen 
  through 
  a 
  telescope, 
  will 
  form 
  

   a 
  drop, 
  as 
  the 
  following 
  consideration 
  of 
  the 
  lines 
  of 
  equal 
  

   intensity 
  will 
  show. 
  

  

  We 
  can 
  imagine 
  such 
  a 
  source 
  to 
  be 
  produced 
  by 
  the 
  super- 
  

   position 
  of 
  two 
  different 
  sources, 
  one 
  of 
  which 
  consists 
  of 
  a 
  

   circular 
  disk 
  of 
  uniform 
  intensity 
  without 
  any 
  dark 
  space, 
  

   while 
  the 
  other 
  consists 
  of 
  a 
  circular 
  disk 
  occupying 
  the 
  place 
  

   of 
  the 
  dark 
  space, 
  and 
  of 
  such 
  an 
  intensity 
  that 
  it 
  is 
  equal 
  but 
  

   of 
  opposite 
  sign. 
  This 
  consideration 
  immediately 
  gives 
  the 
  

  

  * 
  The 
  curve 
  will 
  in 
  reality 
  show 
  minute 
  fluctuations, 
  due 
  to 
  the 
  terms 
  

   of 
  the 
  series 
  for 
  J 
  2 
  (/>)+Ji 
  2 
  (p) 
  which 
  we 
  have 
  neglected 
  in 
  finding- 
  (V.) 
  

   and 
  (V.«). 
  

  

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