﻿where 
  

  

  i. 
  e. 
  X 
  /Z 
  

  

  V 
  2b 
  

  

  42 
  Dr. 
  L. 
  Silberstein 
  on 
  Light 
  Distribution 
  

  

  It 
  will 
  be 
  enough 
  to 
  consider 
  in 
  detail 
  the 
  intensity 
  * 
  at 
  

   points 
  P 
  of 
  the 
  focal 
  plane 
  only. 
  Then 
  the 
  required 
  data 
  

   are 
  

  

  e 
  = 
  T' 
  dl 
  = 
  (iJ 
  l2 
  WS-v)\dv, 
  *=^, 
  (22a) 
  

  

  where 
  now 
  a! 
  defines 
  the 
  distance 
  of 
  P 
  from 
  the 
  optical 
  axis, 
  

   and 
  v 
  the 
  aperture 
  at 
  any 
  stage. 
  If 
  L 
  be 
  the 
  chord 
  from 
  the 
  

   origin 
  to 
  the 
  point 
  v 
  of 
  the 
  P-curve, 
  then 
  the 
  intensity 
  at 
  P 
  

   is, 
  as 
  in 
  (12), 
  

  

  *-<$)'»*v£i?»- 
  ■ 
  ■ 
  ■ 
  ™ 
  

  

  This 
  is 
  the 
  graphical 
  equivalent 
  of 
  the 
  explicit 
  expression 
  

  

  I 
  = 
  ( 
  ! 
  r) 
  2 
  i( 
  A2+B2 
  )' 
  • 
  • 
  • 
  < 
  23 
  "> 
  

  

  A, 
  B 
  = 
  f 
  cos, 
  sin 
  {^) 
  • 
  Jo(" 
  W) 
  dv, 
  . 
  (24) 
  

  

  times 
  the 
  previous 
  x 
  and 
  y, 
  respectively. 
  

  

  The 
  angle 
  e 
  is 
  independent 
  of 
  a', 
  i. 
  e. 
  of 
  p. 
  Thus 
  the 
  tangents 
  

   to 
  the 
  various 
  curves 
  at 
  corresponding 
  points 
  (same 
  v, 
  i. 
  e. 
  

   same 
  aperture) 
  are 
  parallel 
  to 
  one 
  another, 
  while 
  their 
  

   curvature 
  is 
  inversely 
  proportional 
  to 
  |J 
  |. 
  For 
  v 
  = 
  l, 
  

   v2, 
  \/3, 
  etc., 
  e 
  becomes 
  equal 
  to 
  one, 
  two, 
  three, 
  etc., 
  right 
  

   angles. 
  For 
  the 
  central 
  point 
  or 
  focus, 
  A, 
  B 
  become 
  

   identical 
  with 
  the 
  usual 
  Fresnelian 
  integrals, 
  

  

  A 
  = 
  G(v) 
  = 
  I 
  cos 
  — 
  dv, 
  B 
  = 
  S(v) 
  = 
  I 
  sin 
  —z-dv 
  ; 
  

  

  the 
  corresponding 
  curve 
  is 
  the 
  Cornu 
  spiral 
  f 
  , 
  well-known 
  

   in 
  connexion 
  with 
  the 
  straight-edge 
  problem. 
  With 
  in- 
  

   creasing 
  distance 
  p 
  from 
  the 
  focus, 
  i. 
  e. 
  with 
  increasing 
  a', 
  

   the 
  curves 
  change 
  rapidly, 
  and 
  soon 
  lose 
  their 
  simple 
  spiral 
  

   character, 
  owing 
  to 
  the 
  Bessel 
  factor 
  of 
  the 
  integrand. 
  

   Starting 
  from 
  v 
  = 
  and 
  ascending 
  by 
  steps 
  "dv" 
  = 
  - 
  l, 
  the 
  

   desired 
  P-curves, 
  or 
  .^/-curves, 
  are 
  at 
  first 
  more 
  conveniently 
  

   drawn 
  by 
  quadratures, 
  the 
  rectangular 
  coordinates 
  A, 
  B 
  

   being 
  calculated 
  by 
  (24) 
  as 
  far 
  as 
  i> 
  = 
  T5 
  or 
  2*0 
  ; 
  after 
  that 
  

  

  * 
  In 
  applying 
  the 
  diffraction 
  formula 
  to 
  the 
  present 
  case 
  we 
  disregard, 
  

   of 
  course, 
  the 
  inequalities 
  of 
  amplitude 
  at 
  points 
  of 
  the 
  sphere 
  s 
  due 
  to 
  

   the 
  slightly 
  different 
  incident 
  angles 
  i 
  aud 
  glass 
  thicknesses 
  traversed 
  by 
  

   the 
  rays. 
  

  

  t 
  In 
  the 
  present 
  case, 
  only 
  one 
  of 
  its 
  two 
  branches 
  comes 
  into 
  play. 
  

  

  