﻿round 
  the 
  Focus 
  of 
  a 
  Lens, 
  at 
  Various 
  Apertures. 
  33 
  

  

  General 
  Formula?. 
  

  

  Let 
  the 
  portion 
  s 
  of 
  the 
  surface 
  of 
  a 
  fixed 
  sphere, 
  of 
  

   centre 
  and 
  radius 
  P, 
  which 
  we 
  will 
  take 
  as 
  our 
  reference 
  

   sphere, 
  be 
  the 
  seat 
  of 
  monochromatic 
  luminous 
  oscillations 
  

   of 
  constant 
  amplitude 
  a, 
  but 
  of 
  different 
  phases 
  rj, 
  

  

  >H^r 
  — 
  vy 
  

  

  /2irt 
  

  

  Given 
  the 
  distribution 
  of 
  77 
  over 
  s, 
  find 
  the 
  intensity 
  of 
  light 
  

   at 
  the 
  centre 
  and 
  at 
  points 
  P 
  near 
  0. 
  Let 
  ds 
  be 
  an 
  element 
  

   of 
  the 
  reference 
  surface, 
  r 
  its 
  distance 
  from 
  the 
  point 
  P 
  in 
  

   question, 
  c 
  the 
  light 
  velocity 
  and 
  \ 
  = 
  cl 
  the 
  wave-length, 
  in 
  

   vacuo. 
  Then 
  the 
  usual 
  way 
  of 
  applying 
  Huyghens' 
  principle 
  

   gives, 
  for 
  the 
  luminous 
  vibration 
  at 
  P, 
  

  

  -Hlsm(nt-^r- 
  V 
  )ds, 
  . 
  . 
  . 
  (1) 
  

  

  where 
  n 
  = 
  2ir/T. 
  Let 
  N 
  (fig. 
  1) 
  be 
  the 
  pole 
  of 
  angular 
  

   coordinates, 
  i. 
  e. 
  ON 
  the 
  axis, 
  6 
  and 
  <£ 
  the 
  pole 
  distance 
  

   and 
  the 
  longitude 
  of 
  an 
  element 
  ds. 
  Let 
  <j) 
  P 
  be 
  the 
  longitude 
  

  

  Fig. 
  1. 
  

  

  of 
  the 
  point 
  P, 
  further 
  p 
  its 
  distance 
  from 
  the 
  axis 
  and 
  

   a 
  its 
  axial 
  distance 
  from 
  0, 
  away 
  from 
  N. 
  Then, 
  neglecting 
  

   the 
  squares 
  of 
  p/P, 
  cr/P, 
  

  

  r 
  = 
  P 
  1 
  — 
  ~ 
  sin 
  6 
  . 
  cos 
  (</>-- 
  <£ 
  P 
  ) 
  + 
  ~ 
  cos 
  . 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  35. 
  No. 
  205. 
  Jan. 
  1918. 
  D 
  

  

  