﻿38 
  Dr. 
  L. 
  Silberstein 
  on 
  Light 
  Distribution 
  

  

  at 
  once 
  be 
  derived 
  from 
  (9). 
  Up 
  to 
  the 
  first 
  cusp, 
  i. 
  e., 
  as 
  

   long 
  as 
  the 
  first 
  root 
  of 
  J 
  is 
  not 
  exceeded, 
  we 
  have, 
  in 
  virtue 
  

   of 
  that 
  formula, 
  

  

  I 
  _ 
  hLlLj^ay/u), 
  a*/u 
  < 
  2*4048. 
  

  

  (10') 
  

  

  If 
  a\/u 
  is 
  contained 
  between 
  the 
  first 
  two 
  roots 
  x 
  l<t 
  x 
  2 
  of 
  

   J 
  (V) 
  = 
  0, 
  then 
  

  

  I 
  = 
  ~2 
  \ 
  Jo(n)tida: 
  - 
  2 
  I 
  J 
  (x)xdx 
  = 
  -yJiC^i) 
  * 
  Ji(a 
  ? 
  )> 
  

  

  a 
  " 
  a 
  J 
  Xl 
  a 
  a 
  

  

  7 
  2-4969 
  2 
  x 
  /u 
  J/ 
  /-. 
  _ 
  /- 
  nA 
  . 
  

  

  i.e. 
  t= 
  g 
  Ji(«vy), 
  x 
  1 
  "^avu^.x 
  2i 
  (10 
  a) 
  

  

  and 
  so 
  on. 
  

  

  The 
  arc 
  length 
  (10') 
  has, 
  as 
  well 
  as 
  the 
  chord 
  L, 
  a 
  note- 
  

   worthy 
  physical 
  meaning. 
  In 
  fact, 
  it 
  represents 
  the 
  value 
  

   of 
  2m*, 
  for 
  any 
  point 
  of 
  the 
  focal 
  plane, 
  for 
  a 
  perfect 
  wave 
  

   (?7 
  = 
  0), 
  so 
  that 
  the 
  wormaHntensity, 
  due 
  to 
  such 
  a 
  wave, 
  can 
  

   be 
  written, 
  by 
  (5), 
  

  

  N=(^YV, 
  ax/^2-4048. 
  . 
  . 
  . 
  (11) 
  

  

  On 
  the 
  other 
  hand, 
  the 
  intensity 
  due 
  to 
  the 
  defective 
  wave 
  

   has 
  been 
  

  

  f)*L> 
  (12) 
  

  

  More 
  generally, 
  for 
  a 
  ring-shaped 
  aperture 
  contained 
  

   between 
  6 
  a 
  and 
  #&, 
  the 
  chord 
  L 
  in 
  (12) 
  is 
  to 
  be 
  replaced 
  by 
  

   the 
  chord 
  ab 
  joining 
  the 
  corresponding 
  pair 
  of 
  points 
  u 
  a 
  , 
  u 
  b 
  of 
  

   the 
  curve. 
  Thus, 
  up 
  to 
  the 
  first 
  cusp, 
  the 
  ratio 
  of 
  the 
  two 
  

   intensities, 
  at 
  the 
  same 
  point 
  of 
  the 
  focal 
  plane, 
  is 
  

  

  I:N 
  = 
  Z 
  2 
  :Z 
  2 
  (13) 
  

  

  In 
  words, 
  the 
  defective 
  intensity 
  is 
  to 
  the 
  normal 
  intensity 
  

   as 
  the 
  squared 
  chord 
  to 
  the 
  squared 
  arc 
  of 
  the 
  P-curve, 
  from 
  

   the 
  origin 
  to 
  the 
  point 
  u 
  in 
  question. 
  The 
  latter 
  will 
  obviously 
  

   be 
  the 
  longer 
  of 
  the 
  two, 
  the 
  more 
  so 
  the 
  greater 
  the 
  value 
  

   attained 
  by 
  n 
  at 
  the 
  aperture 
  u. 
  More 
  generally, 
  for 
  a 
  ring- 
  

   shaped 
  aperture 
  a 
  , 
  b 
  , 
  the 
  ratio 
  of 
  the 
  two 
  intensities 
  will 
  be 
  

   equal 
  to 
  the 
  square 
  of 
  the 
  ratio 
  of 
  the 
  chord 
  L 
  a 
  b 
  to 
  the 
  arc 
  4 
  6 
  

   of 
  the 
  curve. 
  The 
  simple 
  relation 
  (13) 
  will 
  enable 
  us 
  to 
  see 
  

   at 
  a 
  glance 
  on 
  the 
  P-curves 
  the 
  relative 
  value 
  of 
  the 
  intensity 
  

   due 
  to 
  the 
  defective 
  wave, 
  for 
  various 
  apertures. 
  The 
  

   " 
  definition 
  " 
  of 
  the 
  image 
  will 
  be 
  exhibited 
  by 
  the 
  mutual 
  

   position 
  of 
  curves 
  corresponding 
  to 
  various 
  points 
  of 
  the 
  

   field. 
  

  

  -( 
  

  

  