﻿92 
  Mr. 
  Nalinimohan 
  Basu 
  on 
  the 
  Diffraction 
  of 
  

  

  from 
  the 
  surface 
  when 
  produced 
  backwards 
  would 
  touch 
  the 
  

   enveloping 
  surface 
  which 
  lies 
  within 
  the 
  cylinder. 
  This 
  

   surface, 
  which 
  is 
  virtually 
  the 
  caustic 
  of 
  the 
  reflected 
  rays, 
  

   terminates 
  at 
  the 
  edge 
  C 
  of 
  the 
  cylinder, 
  and 
  when 
  the 
  focal 
  

   plane 
  of 
  the 
  observing 
  microscope 
  is 
  moved 
  forward 
  from 
  

   CP 
  to 
  a 
  position 
  P'C'Px 
  in 
  front 
  of 
  the 
  edge, 
  the 
  boundary 
  

   of 
  the 
  field 
  on 
  the 
  right-hand 
  side 
  would 
  shift 
  into 
  the 
  region 
  

   of 
  the 
  shadow, 
  and 
  would, 
  in 
  fact, 
  lie 
  on 
  the 
  surface 
  of 
  the 
  

   caustic 
  at 
  the 
  point 
  P 
  lm 
  If 
  the 
  focal 
  plane 
  P'C'Pi 
  is 
  consi- 
  

   derably 
  forward 
  of 
  PC, 
  the 
  field 
  is 
  seen 
  divided 
  into 
  two 
  

   parts. 
  The 
  first 
  part 
  P'C 
  consists 
  of 
  the 
  direct 
  rays 
  alone 
  

   (the 
  reflected 
  rays 
  meeting 
  P'C 
  being 
  too 
  oblique 
  to 
  enter 
  into 
  

   the 
  field 
  of 
  the 
  microscope), 
  and 
  should 
  obviously 
  be 
  bounded 
  

   at 
  C 
  by 
  a 
  few 
  diffraction-fringes 
  of 
  the 
  ordinary 
  Fresnel 
  

   type. 
  The 
  second 
  part 
  of 
  the 
  field 
  PxC 
  is 
  due 
  to 
  the 
  reflected 
  

   rays 
  alone, 
  and 
  requires 
  separate 
  consideration. 
  

  

  17. 
  In 
  the 
  case 
  considered 
  above, 
  that 
  is, 
  when 
  the 
  focal 
  

   plane 
  is 
  considerably 
  in 
  advance 
  of 
  the 
  edge, 
  the 
  fringe- 
  

   system 
  within 
  the 
  shadow 
  due 
  to 
  the 
  reflected 
  light 
  is 
  of 
  the 
  

   same 
  type 
  as 
  that 
  found 
  by 
  Airy 
  in 
  his 
  well-known 
  investi- 
  

   gation 
  on 
  the 
  intensity 
  of 
  light 
  in 
  the 
  neighbourhood 
  of 
  a 
  

   caustic. 
  For 
  the 
  elementary 
  pencils 
  into 
  which 
  the 
  reflected 
  

   rays 
  may 
  be 
  divided 
  up 
  diverge 
  from 
  points 
  lying 
  along 
  the 
  

   caustic, 
  and 
  if 
  the 
  point 
  P 
  x 
  at 
  which 
  the 
  focal 
  plane 
  intersects 
  

   the 
  caustic 
  is 
  sufficiently 
  removed 
  from 
  the 
  edge 
  C 
  at 
  which 
  

   the 
  latter 
  terminates, 
  Airy's 
  investigation 
  becomes 
  fully 
  

   applicable, 
  but 
  not 
  otherwise. 
  The 
  rays 
  emerging 
  from 
  the 
  

   point 
  P 
  x 
  after 
  passage 
  through 
  the 
  objective 
  of 
  the 
  micro- 
  

   scope 
  become 
  a 
  parallel 
  pencil, 
  while 
  pencils 
  emerging 
  from 
  

   points 
  on 
  either 
  side 
  of 
  Pi 
  become 
  convergent 
  and 
  divergent 
  

   respectively. 
  The 
  reflected 
  wave-front 
  after 
  passage 
  through 
  

   the 
  objective 
  has 
  thus 
  a 
  point 
  of 
  inflexion 
  on 
  either 
  side 
  of 
  

   which 
  it 
  may 
  be 
  taken 
  to 
  extend 
  indefinitely, 
  provided 
  the 
  

   arc 
  CPx 
  be 
  long 
  enough. 
  Assuming 
  the 
  focal 
  length 
  to 
  be/ 
  

   and 
  the 
  equation 
  of 
  the 
  wave-front 
  to 
  be 
  % 
  — 
  Ar) 
  3 
  , 
  the 
  value 
  of 
  

   A 
  may 
  be 
  readily 
  found. 
  The 
  equation 
  of 
  the 
  caustic 
  is 
  

  

  (±x 
  2 
  + 
  4rf-a 
  2 
  ) 
  s 
  -27a\v 
  2 
  = 
  Q. 
  

  

  Prom 
  this, 
  or 
  directly 
  by 
  an 
  approximate 
  treatment, 
  it 
  may 
  

   readily 
  be 
  shown 
  that 
  the 
  radius 
  of 
  curvature 
  of 
  the 
  caustic 
  

   at 
  the 
  point 
  C 
  is 
  f 
  the 
  radius 
  of 
  the 
  cylinder. 
  For 
  our 
  

   present 
  purpose, 
  it 
  is 
  thus 
  sufficient 
  to 
  treat 
  the 
  caustic 
  as 
  

   equivalent 
  to 
  a 
  cylinder 
  of 
  radius 
  3a/4 
  touching 
  the 
  reflecting 
  

  

  