﻿318 
  PROFESSOR 
  STOKES 
  ON 
  THE 
  

  

  In 
  the 
  present 
  shape 
  of 
  the 
  integral, 
  we 
  must 
  reserve 
  the 
  integration 
  with 
  

   respect 
  to 
  p 
  and 
  q 
  till 
  the 
  end 
  ; 
  but 
  if 
  we 
  introduce 
  the 
  factor 
  e^«r 
  =f£<i, 
  where 
  

   the 
  sign 
  - 
  or 
  + 
  is 
  supposed 
  to 
  be 
  taken 
  according 
  as 
  p 
  or 
  q 
  is 
  positive 
  or 
  nega- 
  

   tive, 
  we 
  shall 
  evidently 
  arrive 
  at 
  the 
  same 
  result 
  as 
  before, 
  provided 
  we 
  suppose 
  

   in 
  the 
  end 
  a 
  and 
  /? 
  to 
  vanish. 
  When 
  this 
  factor 
  is 
  introduced, 
  we 
  may, 
  if 
  we 
  

   please, 
  integrate 
  with 
  respect 
  to 
  p 
  and 
  q 
  first. 
  We 
  thus 
  get 
  

  

  D* 
  I 
  = 
  limit 
  rtjyjJXft^" 
  1 
  ^ 
  q 
  cos 
  -jx 
  (p 
  * 
  7 
  ~~* 
  + 
  ? 
  ^~y) 
  dxdydtf 
  dy' 
  dp 
  dq. 
  

  

  e^* 
  p 
  cos 
  (kp-q)dp= 
  cosQ 
  / 
  f 
  fa? 
  cos 
  kp 
  dp 
  

  

  -00 
  U 
  — 
  00 
  

  

  /10O 
  

  

  ■f 
  sin 
  Q 
  / 
  € 
  t*p 
  s 
  j 
  n 
  fcp 
  tfp 
  

  

  U 
  — 
  00 
  

  

  n 
  ~ 
  C 
  _«» 
  , 
  , 
  2 
  a 
  cos 
  Q 
  

   = 
  2cosQ 
  / 
  € 
  l 
  cos 
  k 
  p 
  dp= 
  — 
  5 
  — 
  -^ 
  

   ^Jo 
  a 
  2 
  + 
  k 
  2 
  

  

  A 
  similar 
  formula 
  holds 
  good 
  for 
  q, 
  whence 
  

  

  D 
  2 
  I 
  = 
  limit 
  Of 
  1 
  1 
  1 
  l-f— 
  — 
  /0 
  „ 
  f 
  . 
  n 
  v 
  oJ 
  1 
  ^ 
  -to 
  — 
  t~j 
  ntt^- 
  dx 
  dy 
  dx' 
  dy'. 
  

  

  JJJJ 
  ( 
  a 
  * 
  + 
  (?^>) 
  ■} 
  {^ 
  + 
  Qzfcty 
  } 
  

  

  Let 
  now 
  

  

  2 
  tt 
  (x' 
  — 
  x) 
  , 
  » 
  , 
  bXa 
  . 
  

   K 
  — 
  - 
  =« 
  «. 
  whence 
  dx 
  '= 
  —. 
  — 
  du, 
  

  

  6 
  A 
  27T 
  

  

  and 
  the 
  limits 
  of 
  u 
  are 
  ultimately 
  — 
  oo 
  and 
  +oo, 
  since 
  a 
  ultimately 
  vanishes. 
  

   Hence 
  

  

  .. 
  ., 
  „ 
  p 
  2adx' 
  b\ 
  r 
  x 
  du 
  

  

  a 
  + 
  ( 
  6A 
  ) 
  

   A 
  similar 
  formula 
  holds 
  good 
  for 
  y', 
  and 
  we 
  have, 
  therefore, 
  

  

  D 
  2 
  1 
  = 
  b 
  2 
  \ 
  2 
  f/dx 
  dy=b 
  2 
  \ 
  2 
  A, 
  

  

  if 
  A 
  be 
  the 
  whole 
  area 
  of 
  the 
  aperture 
  or 
  apertures. 
  

   Now 
  I 
  ought 
  to 
  be 
  equal 
  to 
  A, 
  and, 
  therefore, 
  

  

  ~D=b\. 
  

  

  Case 
  II. 
  Aperture 
  in 
  front 
  of 
  a 
  screen. 
  

  

  The 
  formula 
  for 
  the 
  illumination 
  is 
  given 
  in 
  Airy's 
  Tract, 
  Art. 
  73. 
  We 
  have 
  

   as 
  before, 
  

  

  ™ 
  - 
  "-* 
  otffMf— 
  « 
  =&? 
  { 
  {<- 
  m 
  " 
  

  

  - 
  (- 
  Hii 
  ' 
  + 
  ('-=5) 
  2 
  - 
  H&) 
  " 
  } 
  ?• 
  " 
  " 
  « 
  d 
  " 
  ** 
  

  

  =1 
  imit 
  tfffffff 
  € 
  ^ 
  p 
  ^ 
  q 
  cos 
  { 
  ^Xa 
  b^ 
  [*' 
  2 
  -^ 
  2 
  +y 
  2 
  -y 
  2 
  ] 
  

  

  