﻿Light 
  by 
  Cylinders 
  of 
  Large 
  Radius. 
  

  

  87 
  

  

  pencil 
  while 
  the 
  incident 
  rays 
  are 
  parallel, 
  the 
  effect 
  of 
  the 
  

   former 
  at 
  any 
  point 
  sufficiently 
  removed 
  from 
  the 
  cylinder 
  

   would 
  be 
  negligible 
  in 
  comparison 
  with 
  the 
  effect 
  of 
  the 
  

   latter. 
  If 
  d, 
  the 
  distance 
  of 
  the 
  plane 
  of 
  observation 
  from 
  

   the 
  edge 
  of 
  the 
  cylinder, 
  be 
  sufficiently 
  large, 
  the 
  problem 
  

   thus 
  practically 
  reduces 
  to 
  one 
  of 
  simple 
  diffraction 
  of 
  the 
  

   incident 
  waves 
  by 
  the 
  straight 
  edge 
  C. 
  The 
  positions 
  of 
  

   the 
  minima 
  of 
  illumination 
  with 
  reference 
  to 
  the 
  geometrical 
  

   edge 
  of 
  the 
  shadow 
  would 
  then 
  be 
  given 
  approximately 
  by 
  

   the 
  simple 
  formula 
  

  

  x' 
  = 
  v 
  / 
  2nd\ 
  = 
  y/±ny/d\/2 
  9 
  

   where 
  a?' 
  = 
  C'P' 
  and 
  d=GC, 
  

  

  or 
  with 
  great 
  accuracy 
  by 
  Schuster's 
  formula. 
  

  

  The 
  two 
  formulse 
  give 
  results 
  which 
  do 
  not 
  differ 
  materially 
  

   except 
  in 
  regard 
  to 
  the 
  first 
  two 
  or 
  three 
  bands, 
  as 
  can 
  be 
  

   seen 
  from 
  Table 
  II. 
  

  

  Table 
  II. 
  

  

  1. 
  

  

  n. 
  

  

  2. 
  

  

  3. 
  

  

  4. 
  

  

  Proportionate 
  

  

  widths 
  of 
  bands 
  

  

  as 
  per 
  column 
  2. 
  

  

  5. 
  

  

  Proportionate 
  

   widths 
  of 
  bands 
  

   as 
  per 
  column 
  3. 
  

  

  V(8»-l)/2. 
  

  

  1. 
  

  

  2000 
  

  

  1-871 
  

  

  2 
  000 
  

  

  1-871 
  

  

  2. 
  

  

  2-828 
  

  

  2-739 
  

  

  0-828 
  

  

  0-868 
  

  

  3. 
  

  

  3464 
  

  

  3-391 
  

  

  0-636 
  

  

  0652 
  

  

  4. 
  

  

  4-000 
  

  

  3-937 
  

  

  0-536 
  

  

  0546 
  

  

  5. 
  

  

  4-472 
  

  

  4416 
  

  

  0-472 
  

  

  0-479 
  

  

  6. 
  

  

  4-899 
  

  

  4-848 
  

  

  0-427 
  

  

  0-432 
  

  

  7. 
  

  

  5-292 
  

  

  5-244 
  

  

  0-393 
  

  

  0-396 
  

  

  12. 
  If 
  d 
  be 
  not 
  large, 
  the 
  intensity 
  of 
  the 
  reflected 
  rays 
  is 
  

   not 
  negligible. 
  The 
  following 
  considerations 
  enable 
  us 
  to 
  

   find 
  a 
  simple 
  formula 
  for 
  the 
  positions 
  of 
  the 
  minima 
  of 
  

   illumination 
  which 
  takes 
  both 
  diffraction 
  and 
  interference 
  

   into 
  account. 
  We 
  may, 
  to 
  begin 
  with, 
  find 
  the 
  positions 
  of 
  

   the 
  minima 
  assuming 
  the 
  case 
  to 
  be 
  one 
  of 
  simple 
  inter- 
  

   ference 
  between 
  the 
  direct 
  and 
  the 
  reflected 
  rays. 
  The 
  

   expression 
  for 
  the 
  path 
  difference, 
  8', 
  of 
  the 
  rays 
  arriving 
  

   at 
  the 
  point 
  P' 
  is 
  readily 
  seen 
  from 
  fig. 
  2 
  to 
  be 
  given 
  by 
  the 
  

   formula 
  

  

  S' 
  = 
  (d 
  + 
  asin0)(sec20-l). 
  

  

  