﻿36 
  Dr. 
  L. 
  Silberstein 
  on 
  Light 
  Distribution 
  

  

  the 
  coordinates 
  of 
  P 
  (with 
  \j2tt 
  as 
  unit 
  length). 
  The 
  upper 
  

   limit 
  of 
  the 
  integral 
  stands 
  for 
  # 
  a 
  2 
  , 
  the 
  suffix 
  having 
  been 
  

   dropped. 
  

  

  For 
  certain 
  forms 
  of 
  n 
  the 
  integral 
  (7) 
  can 
  be 
  developed 
  

   into 
  more 
  or 
  less 
  complicated 
  series, 
  as 
  has 
  been 
  done 
  by 
  

   various 
  authors, 
  especially 
  for 
  the 
  focal 
  plane 
  (/3 
  = 
  0). 
  In 
  

   general, 
  however, 
  whether 
  the 
  phase 
  distribution 
  7? 
  be 
  given 
  

   graphically 
  or 
  analytically, 
  the 
  gaol 
  will 
  be 
  reached 
  much 
  

   more 
  easily 
  and 
  quickly 
  by 
  a 
  number 
  of 
  comparatively 
  small 
  

   steps 
  Au 
  or 
  " 
  du" 
  starting 
  from 
  and 
  leading 
  to 
  the 
  

   required 
  u 
  = 
  6i 
  2 
  , 
  either 
  by 
  mechanical 
  quadratures 
  or 
  by 
  

   a 
  graphical 
  construction 
  analogous 
  to 
  that 
  of 
  the 
  famous 
  

   Cornu 
  spiral. 
  The 
  latter 
  method 
  can 
  now 
  and 
  then 
  be 
  

   checked 
  by 
  the 
  former, 
  which 
  is 
  particularly 
  advisable 
  

   for 
  the 
  first 
  stages 
  of 
  the 
  procedure. 
  A 
  curve 
  drawn 
  in 
  

   this 
  manner 
  (for 
  any 
  fixed 
  a, 
  j3) 
  has 
  also 
  the 
  advantage 
  

   of 
  exhibiting 
  the 
  local 
  intensity 
  as 
  a 
  function 
  of 
  the 
  

   aperture 
  ; 
  the 
  process 
  corresponds, 
  in 
  fact, 
  to 
  a 
  gradual 
  

   opening 
  of 
  (the 
  pupil 
  of) 
  a 
  lens, 
  from 
  no 
  to 
  the 
  full 
  

   required 
  aperture. 
  

  

  Consider 
  the 
  plane 
  of 
  the 
  complex 
  variable 
  

  

  2iu 
  = 
  x-\-iy 
  = 
  z, 
  

  

  so 
  that, 
  L 
  being 
  the 
  distance 
  of 
  the 
  point 
  z 
  from 
  the 
  origin, 
  

   the 
  corresponding 
  intensity 
  will 
  be 
  

  

  I 
  = 
  (ttKL/X) 
  2 
  . 
  

  

  To 
  every 
  fixed 
  point 
  P(«, 
  /3) 
  of 
  the 
  luminous 
  region 
  belongs, 
  

   in 
  the 
  ,z-plane, 
  a 
  curve 
  whose 
  element 
  is 
  fully 
  given 
  by 
  

  

  dz 
  = 
  e^-^Jofa^/u) 
  .du. 
  

  

  Thus 
  the 
  sloping 
  angle 
  e 
  at 
  any 
  point 
  of 
  the 
  P-curve 
  will 
  be 
  

  

  6 
  = 
  v-/3u, 
  (8) 
  

  

  the 
  length 
  of 
  an 
  arc 
  element 
  

  

  dl 
  = 
  I 
  J 
  (#\At) 
  I 
  . 
  du, 
  (9) 
  

  

  and, 
  therefore, 
  the 
  curvature 
  

  

  de 
  1 
  /drj 
  

  

  dl 
  

  

  m&->) 
  <10 
  » 
  

  

  By 
  means 
  of 
  these 
  formulae 
  any 
  P- 
  curve 
  can 
  easily 
  be 
  

   drawn 
  step 
  by 
  step, 
  much 
  in 
  the 
  same 
  way 
  as 
  the 
  Cornu 
  

   spiral. 
  For 
  any 
  point 
  P 
  of 
  the 
  focal 
  plane 
  (/3=0) 
  the 
  

   angle 
  e 
  is 
  simply 
  equal 
  to 
  the 
  phase 
  excess 
  97, 
  and 
  for 
  

   points 
  outside 
  the 
  focal 
  plane 
  it 
  is 
  smaller 
  by 
  f$v. 
  The 
  

  

  