﻿34 
  Dr. 
  L. 
  Silberstein 
  on 
  Light 
  Distribution 
  

  

  Introduce 
  this 
  into 
  (1), 
  writing 
  in 
  the 
  denominator, 
  r==M, 
  

  

  t 
  ) 
  is 
  constant 
  all 
  

  

  over 
  s. 
  Then 
  the 
  integral 
  will 
  split 
  into 
  two 
  others 
  

  

  multiplied 
  by 
  sin, 
  and 
  by 
  cosnl* 
  ). 
  Thus, 
  taking 
  the 
  

  

  intensity 
  at 
  the 
  sphere 
  s 
  as 
  our 
  unit, 
  i. 
  e. 
  putting 
  ^a 
  2 
  =l, 
  the 
  

   intensity 
  of 
  light 
  at 
  P 
  will 
  be 
  

  

  I 
  = 
  _^(C>+ 
  n 
  ..... 
  (2) 
  

   where 
  

  

  C, 
  S 
  = 
  1 
  cos, 
  sin 
  7j 
  -^ 
  sin 
  . 
  cos 
  (cj>— 
  <£ 
  P 
  ) 
  

  

  + 
  ^ 
  cos 
  0~].ds. 
  (3) 
  

  

  These 
  formulae 
  are 
  valid 
  for 
  any 
  distribution 
  of 
  phase, 
  

   rj=r)(0, 
  <£), 
  and 
  for 
  any 
  form 
  of 
  the 
  edge 
  of 
  s 
  (diaphragm). 
  

   If, 
  as 
  corresponds 
  to 
  the 
  subject 
  of 
  the 
  present 
  paper, 
  r) 
  is 
  

   a 
  function 
  of 
  alone 
  *, 
  and 
  if 
  the 
  edge 
  is 
  itself 
  a 
  circle 
  of 
  

   latitude, 
  — 
  x 
  =. 
  const., 
  then 
  we 
  have 
  axial 
  symmetry 
  round 
  

   ON, 
  so 
  that 
  C, 
  S 
  become 
  independent 
  of 
  the 
  longitude 
  of 
  

   the 
  point 
  P, 
  and 
  we 
  can 
  put 
  <£ 
  P 
  = 
  0. 
  The 
  most 
  convenient 
  

   integration 
  variables 
  being 
  now 
  6 
  and 
  <f> 
  themselves, 
  take 
  

  

  ds 
  = 
  R 
  2 
  sin 
  6 
  .d0.d<j> 
  

  

  and 
  integrate 
  over 
  <£ 
  = 
  to 
  27rand 
  over 
  = 
  to 
  X 
  . 
  Develop 
  

   (3) 
  and 
  introduce 
  the 
  abbreviations 
  

  

  2irp 
  Q 
  2ira 
  

  

  a 
  =x' 
  P^-x 
  (4) 
  

  

  Then, 
  after 
  some 
  easy 
  transformations, 
  the 
  expression 
  for 
  the 
  

   light 
  intensity 
  at 
  the 
  point 
  T(p, 
  a) 
  will 
  be 
  

  

  I^(I^)Vp, 
  (5) 
  

  

  where 
  [ 
  w 
  | 
  is 
  the 
  absolute 
  value 
  of 
  the 
  complex 
  integral 
  

  

  = 
  j 
  V'c 
  + 
  /3 
  cos 
  *>J 
  (a 
  sin 
  0). 
  sin 
  0d0. 
  . 
  . 
  (6) 
  

  

  For 
  the 
  focal 
  plane, 
  as 
  we 
  shall 
  henceforth 
  call 
  the 
  plane 
  

  

  * 
  That 
  is, 
  if 
  the 
  " 
  contour 
  lines 
  " 
  exhibited 
  by 
  the 
  Lens 
  Interfero- 
  

   meter 
  are 
  circles 
  of 
  latitude. 
  

  

  w 
  

  

  