﻿round 
  the 
  Focus 
  of 
  a 
  Lens, 
  at 
  Various 
  Apertures. 
  41 
  

  

  corresponding 
  coefficient 
  in 
  the 
  second 
  term 
  o£ 
  (15)] 
  . 
  If, 
  for 
  

   instance, 
  ti 
  = 
  1'5, 
  then 
  

  

  2irr 
  9 
  //A 
  4 
  /. 
  1 
  A 
  2 
  \ 
  

  

  and 
  . 
  h 
  / 
  3 
  k 
  2 
  39 
  h*\ 
  

  

  2> 
  \ 
  8 
  r 
  2 
  128 
  r 
  4 
  / 
  

   Passing 
  to 
  the 
  application 
  of 
  formula 
  (18) 
  to 
  our 
  diffrac- 
  

   tion 
  al 
  problem, 
  we 
  can 
  readily 
  limit 
  ourselves 
  to 
  values 
  of 
  - 
  

  

  less 
  than 
  or 
  at 
  any 
  rate 
  not 
  considerably 
  exceeding 
  -j^-. 
  

   Under 
  such 
  circumstances, 
  (17) 
  can 
  be 
  written, 
  with 
  sufficient 
  

   accuracy, 
  

  

  • 
  a 
  • 
  n 
  / 
  i 
  h 
  h 
  

   sin 
  6 
  == 
  6 
  = 
  (n 
  — 
  • 
  1; 
  - 
  = 
  ^ 
  , 
  

  

  and, 
  therefore, 
  

  

  The 
  negative 
  sign 
  can 
  be 
  dropped, 
  which 
  makes 
  no 
  difference 
  

   for 
  points 
  of 
  the 
  focal 
  plane, 
  and, 
  for 
  all 
  other 
  points, 
  requires 
  

   only 
  a 
  sign 
  reversal 
  of 
  their 
  abscissse 
  /3. 
  Thus, 
  in 
  our 
  previous 
  

   notation, 
  

  

  '-«■• 
  6 
  = 
  4^rp 
  < 
  19 
  > 
  

  

  It 
  will 
  be 
  convenient 
  to 
  introduce 
  as 
  the 
  integration 
  

   variable, 
  instead 
  of 
  u, 
  

  

  v 
  = 
  M 
  \/? 
  = 
  u 
  \/w^Yf 
  ' 
  • 
  • 
  • 
  (20) 
  

  

  so 
  that 
  

  

  V 
  = 
  -o"9 
  j 
  = 
  v^27r5, 
  and 
  a\/w 
  = 
  a'\/v, 
  

   z 
  aw 
  

  

  where 
  /.f 
  2^p 
  (2L\ 
  Vi 
  ,„. 
  

  

  Thus 
  the 
  sloping 
  angle, 
  the 
  arc 
  element, 
  and 
  the 
  curvature 
  

   of 
  any 
  P-curve 
  will 
  be, 
  by 
  (8), 
  (9), 
  (10), 
  remembering 
  the 
  

   sign 
  reversal 
  of 
  yS, 
  

  

  7TV 
  2 
  / 
  7T 
  \ 
  1/2 
  / 
  7T 
  \ 
  1/2 
  

  

  1 
  _ 
  

  

  