﻿94 
  Mr. 
  Nalinimohan 
  Basu 
  on 
  the 
  Diffraction 
  of 
  

  

  fringes 
  seen 
  in 
  the 
  field 
  are 
  due 
  only 
  to 
  the 
  interference 
  

   of 
  the 
  direct 
  and 
  reflected 
  wave-trains. 
  The 
  phenomena 
  

   noticed 
  as 
  the 
  focal 
  plane 
  is 
  advanced 
  towards 
  the 
  source 
  of 
  

   light, 
  represent 
  a 
  gradual 
  transition 
  from 
  this 
  stage 
  to 
  one 
  

   in 
  which 
  Airy's 
  theory 
  becomes 
  fully 
  applicable. 
  In 
  the 
  

   transition-stages 
  the 
  field 
  of 
  illumination 
  is 
  a 
  continuous 
  

   whole, 
  of 
  which, 
  however, 
  the 
  different 
  parts 
  present 
  distinct 
  

   characteristics. 
  First, 
  within 
  the 
  geometrical 
  edge 
  of 
  the 
  

   shadow, 
  we 
  have 
  a 
  finite 
  number 
  of 
  fringes 
  (one, 
  two, 
  or 
  

   more 
  according 
  to 
  the 
  position 
  of 
  the 
  plane 
  of 
  observation, 
  

   but 
  not 
  an 
  indefinitely 
  large 
  number 
  as 
  contemplated 
  by 
  

   Airy's 
  theory) 
  ; 
  these 
  may 
  be 
  regarded 
  as 
  the 
  interference- 
  

   fringes 
  in 
  the 
  neighbourhood 
  of 
  the 
  caustic 
  doe 
  to 
  the 
  

   reflected 
  light 
  alone. 
  Following 
  these 
  we 
  have 
  a 
  long 
  train 
  

   of 
  fringes 
  due 
  to 
  the 
  interference 
  of 
  the 
  direct 
  and 
  the 
  

   reflected 
  pencils. 
  The 
  first 
  few 
  of 
  these 
  should 
  evidently 
  be 
  

   modified 
  by 
  the 
  diffraction 
  which 
  the 
  direct 
  rays 
  suffer 
  at 
  the 
  

   edge 
  C 
  before 
  they 
  reach 
  the 
  observing 
  microscope. 
  Finally, 
  

   we 
  may 
  also 
  have 
  a 
  part 
  of 
  the 
  field 
  in 
  which 
  the 
  illumination 
  

   is 
  due 
  only 
  to 
  the 
  direct 
  pencil, 
  the 
  reflected 
  rays 
  not 
  entering 
  

   the 
  objective 
  of 
  the 
  microscope 
  owing 
  to 
  their 
  obliquity. 
  

   This 
  part 
  of 
  the 
  field 
  should 
  appear 
  less 
  brightly 
  illuminated 
  

   than 
  the 
  rest. 
  

  

  19. 
  A 
  complete 
  theoretical 
  treatment 
  of 
  the 
  transition- 
  

   stages 
  described 
  in 
  the 
  preceding 
  paragraph 
  is 
  somewhat 
  

   difficult, 
  and 
  has 
  to 
  be 
  deferred 
  to 
  some 
  future 
  occasion. 
  

   There 
  is 
  no 
  difficulty, 
  however, 
  in 
  calculating 
  the 
  positions 
  

   of 
  the 
  fringes 
  due 
  to 
  the 
  interference 
  of 
  the 
  direct 
  and 
  the 
  

   reflected 
  pencils 
  when 
  the 
  focal 
  plane 
  is 
  in 
  advance 
  of 
  the 
  

   edge, 
  provided 
  the 
  diffraction-effect 
  due 
  to 
  the 
  edge 
  is 
  

   neglected. 
  It 
  is 
  easily 
  shown 
  that 
  the 
  path-difference 
  

   between 
  the 
  direct 
  and 
  reflected 
  rays 
  at 
  a 
  point 
  x' 
  is 
  given 
  by 
  

  

  S'=2a0 
  3 
  -2^ 
  2 
  , 
  1 
  

  

  ,x' 
  = 
  3a6 
  2 
  /2-2d6j 
  ' 
  * 
  * 
  ' 
  * 
  ^ 
  ; 
  

  

  where 
  x' 
  is 
  measured 
  from 
  C 
  f 
  and 
  d 
  = 
  CC. 
  By 
  putting 
  

   $' 
  =rik 
  and 
  eliminating 
  0, 
  the 
  positions 
  of 
  the 
  minima 
  of 
  

   illumination 
  may 
  be 
  calculated. 
  A 
  complete 
  agreement 
  

   of 
  the 
  results 
  thus 
  obtained 
  with 
  those 
  found 
  in 
  experiment 
  

   cannot, 
  however, 
  be 
  expected, 
  as 
  the 
  fringes 
  are 
  narrow 
  and 
  

   the 
  modifications 
  due 
  to 
  diffraction 
  are 
  not 
  negligible. 
  As 
  

   regards 
  the 
  fringes 
  alongside 
  the 
  caustic 
  due 
  to 
  the 
  reflected 
  

   rays, 
  we 
  cannot 
  expect 
  to 
  find 
  a 
  complete 
  agreement 
  between 
  

   their 
  widths 
  and 
  those 
  found 
  from 
  Airy's 
  theory, 
  so 
  long 
  as 
  

   the 
  latter 
  is 
  not 
  fully 
  applicable. 
  The 
  divergence, 
  if 
  any, 
  

  

  