﻿40 
  Dr. 
  L. 
  Silberstein 
  on 
  Light 
  Distribution 
  

  

  and 
  will 
  continue 
  its 
  path 
  in 
  air 
  along 
  BO 
  f 
  , 
  piercing 
  the 
  

   sphere 
  s 
  at 
  B. 
  Write 
  AB 
  = 
  ?n. 
  Then, 
  taking 
  77 
  = 
  at 
  N, 
  

   the 
  phase 
  at 
  B, 
  corresponding 
  to 
  the 
  angle 
  NOB 
  = 
  #, 
  

  

  will 
  be 
  

  

  ^ 
  = 
  ¥{ 
  m 
  ~ 
  nr 
  ( 
  1 
  "V 
  1 
  "■?)}• 
  • 
  • 
  {i6) 
  

  

  The 
  required 
  length 
  m 
  will 
  be 
  determined, 
  for 
  every 
  h 
  *, 
  by 
  

   the 
  following 
  trigonometric 
  set 
  : 
  

  

  . 
  . 
  h 
  . 
  ., 
  nh 
  ., 
  

  

  sin 
  1 
  = 
  -, 
  sin 
  1 
  = 
  — 
  : 
  ay 
  = 
  1 
  

   r 
  r 
  

  

  / 
  = 
  O'N 
  = 
  r 
  (^i'-lV 
  A 
  S 
  ' 
  = 
  ,'-*„' 
  = 
  ,'_R, 
  

  

  \ 
  sin 
  ft) 
  /' 
  ° 
  J 
  

  

  and, 
  finally, 
  

  

  6 
  = 
  &)— 
  i 
  2 
  , 
  where 
  sini 
  2 
  = 
  ^ 
  | 
  As' 
  I 
  sin&>, 
  

  

  r 
  sini 
  — 
  It 
  sin 
  

  

  m 
  = 
  : 
  , 
  

  

  sin 
  &) 
  

  

  By 
  successive 
  series 
  developments 
  I 
  find, 
  up 
  to 
  (-) 
  

   inclusively, 
  the 
  spherical 
  aberration 
  f 
  

  

  r 
  n* 
  MV/i 
  , 
  n 
  2 
  -n 
  + 
  l/7A 
  2 
  

  

  + 
  n(n» 
  + 
  l)(n-l) 
  + 
  lgy| 
  t 
  (lfi) 
  

   and 
  E 
  = 
  s 
  '= 
  ~—j, 
  of 
  course 
  ; 
  next, 
  for 
  the 
  angle 
  #, 
  

  

  n(n-l)q 
  + 
  3n-n 
  2 
  ) 
  /fc\* 
  j 
  (1?) 
  

  

  and 
  ultimately, 
  for 
  the 
  phase, 
  as 
  defined 
  by 
  (15), 
  

  

  2<7rr 
  n 
  2 
  (n-l) 
  /hV 
  { 
  - 
  n 
  2 
  -n-l 
  /A\ 
  2 
  \ 
  , 
  1Q 
  . 
  

   ^ 
  = 
  "X- 
  8— 
  W 
  1 
  %~\r) 
  )' 
  (18) 
  

  

  a 
  series 
  starting 
  with 
  the 
  fow 
  th 
  power 
  of 
  the 
  relative 
  aperture, 
  

   as 
  has 
  been 
  expected 
  [the 
  coefficient 
  of 
  the 
  second 
  power 
  in 
  

   the 
  series 
  of 
  m 
  is 
  \nr, 
  and 
  this 
  is 
  exactly 
  cancelled 
  by 
  the 
  

  

  * 
  Subject 
  to 
  the 
  obvious 
  condition 
  h 
  < 
  -, 
  beyond 
  which 
  there 
  is 
  total 
  

   reflexion. 
  n 
  

  

  t 
  Which, 
  although 
  not 
  needed 
  ultimately, 
  may 
  be 
  of 
  some 
  interest. 
  

  

  