﻿Light 
  by 
  Cylinders 
  of 
  Large 
  Radius. 
  93 
  

  

  surface 
  at 
  C. 
  We 
  have 
  

  

  6d 
  v 
  3 
  6d 
  v 
  \d 
  v 
  2 
  J' 
  

  

  where 
  ( 
  -~ 
  2 
  ) 
  is 
  the 
  measure 
  of 
  the 
  convergence 
  or 
  divergence 
  

  

  of 
  the 
  normals 
  to 
  the 
  wave-front 
  in 
  the 
  neighbourhood 
  of 
  the 
  

   point 
  of 
  inflexion. 
  Substituting 
  the 
  values 
  obtained 
  from 
  

   the 
  formulae 
  of 
  geometrical 
  optics, 
  it 
  is 
  found 
  that 
  

  

  . 
  _ 
  1 
  3a 
  la 
  

  

  6'T'f 
  3 
  ~8f*' 
  

  

  The 
  equation 
  of 
  the 
  wave-front 
  is 
  accordingly 
  

  

  The 
  illumination 
  in 
  the 
  fringe-system 
  alongside 
  the 
  caustic 
  is 
  

   then 
  given 
  by 
  Airy's 
  formula 
  

  

  r- 
  /*°° 
  — 
  i 
  

  

  ! 
  = 
  4 
  I 
  cos 
  -x 
  (w 
  3 
  + 
  mw) 
  dw 
  

  

  where 
  ??i 
  = 
  4 
  .2*. 
  a 
  *.\ 
  3 
  .#i, 
  

  

  x 
  x 
  being 
  the 
  distance 
  of 
  any 
  point 
  in 
  the 
  focal 
  plane 
  measured 
  

   from 
  the 
  point 
  of 
  intersection 
  with 
  the 
  caustic. 
  The 
  integral 
  

   gives 
  a 
  series 
  of 
  maxima 
  of 
  which 
  the 
  first 
  is 
  the 
  largest, 
  and 
  

   the 
  rest 
  gradually 
  converge 
  to 
  zero. 
  The 
  minima 
  of 
  illumi- 
  

   nation 
  are 
  zeroes*. 
  As 
  the 
  focal 
  plane 
  is 
  moved 
  further 
  and 
  

   further 
  towards 
  the 
  source 
  of 
  light, 
  the 
  fringe-system 
  moves 
  

   inwards 
  along 
  the 
  caustic, 
  but 
  remains 
  otherwise 
  unaltered. 
  

  

  18. 
  The 
  foregoing 
  treatment 
  of 
  the 
  reflected 
  fringe-system 
  

   in 
  terms 
  of 
  Airy's 
  theory 
  ceases 
  to 
  be 
  valid 
  when 
  the 
  focal 
  

   plane 
  is 
  not 
  sufficiently 
  in 
  advance 
  of 
  the 
  edge, 
  and 
  the 
  arc 
  

   CPi 
  of 
  the 
  caustic 
  is 
  therefore 
  not 
  large 
  enough. 
  For 
  the 
  

   reflected 
  wave-front 
  on 
  one 
  side 
  of 
  the 
  point 
  of 
  inflexion 
  then 
  

   becomes 
  limited 
  in 
  extent, 
  and 
  its 
  equation 
  cannot 
  with 
  

   sufficient 
  accuracy 
  be 
  assumed 
  to 
  be 
  of 
  the 
  simple 
  form 
  

   f 
  =At7 
  3 
  , 
  extending 
  to 
  infinity 
  in 
  either 
  direction. 
  In 
  fact, 
  

   when 
  the 
  focal 
  plane 
  is 
  at 
  the 
  edge 
  of 
  the 
  cylinder 
  and 
  CPi 
  

   is 
  zero, 
  the 
  point 
  of 
  inflexion 
  coincides 
  with 
  the 
  extreme 
  edge 
  

   of 
  the 
  reflected 
  wave-front. 
  At 
  this 
  stage, 
  of 
  course, 
  the 
  

  

  * 
  Graphs 
  of 
  Airy's 
  integral 
  and 
  references 
  to 
  the 
  literature 
  will 
  be 
  

   found 
  in 
  an 
  interesting 
  paper 
  by 
  Aichi 
  and 
  Tanakadate 
  (Journal 
  of 
  the 
  

   College 
  of 
  Science, 
  Tokyo, 
  vol. 
  xxi. 
  Art. 
  3). 
  

  

  