﻿( 
  317 
  ) 
  

  

  XIX. 
  — 
  On 
  the 
  Total 
  Intensity 
  of 
  Interfering 
  Light. 
  By 
  Professor 
  Stokes. 
  

  

  [Extracted 
  from 
  a 
  Letter 
  addressed 
  to 
  Professor 
  Kelland.^ 
  

  

  Pembroke 
  College, 
  Cambridge. 
  

  

  My 
  dear 
  Sir, 
  

  

  # 
  # 
  # 
  # 
  

  

  In 
  reading 
  your 
  paper 
  in 
  the 
  Transactions 
  of 
  the 
  Royal 
  Society 
  of 
  Edinburgh, 
  

   vol. 
  xv., 
  p. 
  315, 
  some 
  years 
  ago, 
  it 
  occurred 
  to 
  me 
  to 
  try 
  whether 
  it 
  would 
  not 
  be 
  

   possible 
  to 
  give 
  a 
  general 
  demonstration 
  of 
  the 
  theorem, 
  applying 
  to 
  apertures 
  of 
  

   all 
  forms. 
  I 
  arrived 
  at 
  a 
  proof, 
  which 
  I 
  wrote 
  out, 
  but 
  have 
  never 
  published. 
  As 
  

   I 
  think 
  it 
  will 
  interest 
  you 
  I 
  will 
  communicate 
  it. 
  You 
  may 
  make 
  any 
  use 
  you 
  

   please 
  of 
  it. 
  

  

  Case 
  I. 
  Aperture 
  in 
  front 
  of 
  a 
  lens 
  ; 
  light 
  thrown 
  on 
  a 
  screen 
  at 
  the 
  focus, 
  

   or 
  received 
  through 
  an 
  eye-piece, 
  through 
  which 
  the 
  luminous 
  point 
  is 
  seen 
  in 
  

   focus. 
  

  

  The 
  expression 
  for 
  the 
  intensity 
  is 
  given 
  in 
  Airy's 
  Tract, 
  Prop. 
  20. 
  If 
  the 
  in- 
  

   tensity 
  of 
  the 
  incident 
  light 
  at 
  the 
  distance 
  of 
  the 
  aperture 
  be 
  taken 
  for 
  unity, 
  

   and 
  D 
  be 
  the 
  quantity 
  by 
  which 
  any 
  element 
  of 
  the 
  area 
  of 
  the 
  aperture 
  must 
  be 
  

   divided 
  in 
  forming 
  the 
  expression 
  for 
  the 
  vibration, 
  that 
  expression 
  becomes 
  

  

  \ffa 
  IT 
  ( 
  W 
  " 
  B+ 
  ^— 
  ) 
  dx 
  d 
  * 
  

  

  the 
  integration 
  being 
  extended 
  over 
  the 
  whole 
  aperture. 
  If 
  it 
  should 
  be 
  neces- 
  

   sary 
  to 
  suppose 
  a 
  change 
  of 
  phase 
  to 
  take 
  place 
  in 
  the 
  act 
  of 
  diifraction, 
  such 
  

   change 
  may 
  be 
  included 
  in 
  the 
  constant 
  B. 
  If, 
  then, 
  I 
  be 
  the 
  intensity, 
  

  

  1,1= 
  ijfij-ti^i 
  M 
  ^\ 
  <jf„i«i!+u 
  M 
  ,y 
  i 
  

  

  and 
  if 
  I 
  be 
  the 
  total 
  illumination, 
  

  

  /OO 
  .^00 
  

  

  / 
  Idpdg. 
  

   - 
  oo 
  «/ 
  — 
  oo 
  

  

  NOW, 
  {JJ}(x,y)dxdy} 
  =fffff(x,y)f(x',y')dxdydx 
  , 
  dy', 
  

  

  the 
  limits 
  of 
  x\ 
  y 
  being 
  the 
  same 
  as 
  those 
  of 
  cc, 
  y. 
  Hence, 
  

  

  D 
  2 
  1= 
  / 
  / 
  / 
  /cos 
  -j-j- 
  lpx' 
  — 
  x 
  + 
  qy' 
  — 
  y\ 
  dx 
  dy 
  dx' 
  dy'. 
  

   VOL. 
  XX. 
  PART 
  III. 
  4 
  R 
  

  

  