﻿Light 
  by 
  Cylinders 
  oj 
  Large 
  Radius. 
  89 
  

  

  due 
  to 
  divergence 
  in 
  the 
  direction 
  of 
  the 
  incident 
  rays, 
  and 
  

   less 
  rapid 
  attenuation 
  in 
  other 
  directions. 
  In 
  any 
  plane 
  

   C'P', 
  therefore, 
  the 
  effect 
  of 
  the 
  reflected 
  light 
  is 
  negligible 
  

   in 
  the 
  immediate 
  neighbourhood 
  of 
  the 
  point 
  C, 
  and 
  would 
  

   be 
  most 
  perceptible 
  at 
  points 
  farthest 
  removed 
  from 
  C'*, 
  

   On 
  the 
  other 
  hand, 
  the 
  fluctuations 
  of 
  intensity 
  due 
  to 
  the 
  

   diffraction 
  of 
  the 
  direct 
  rays 
  are 
  most 
  marked 
  in 
  the 
  neigh- 
  

   bourhood 
  of 
  C, 
  that 
  is, 
  for 
  the 
  smallest 
  values 
  of 
  6. 
  We 
  

   should 
  accordingly 
  expect 
  to 
  find 
  that 
  when 
  d 
  is 
  not 
  zero, 
  

   the 
  first 
  few 
  bands 
  are 
  practically 
  identical 
  in 
  position 
  with 
  

   those 
  due 
  to 
  simple 
  diffraction, 
  and 
  those 
  following 
  are 
  due 
  

   to 
  simple 
  interference 
  between 
  the 
  direct 
  and 
  the 
  reflected 
  

   rays. 
  The 
  formulae 
  given 
  above 
  satisfy 
  both 
  of 
  these 
  re- 
  

   quirements. 
  For 
  it 
  is 
  obvious 
  from 
  the 
  manner 
  in 
  which 
  

   they 
  have 
  been 
  deduced 
  that 
  they 
  satisfy 
  the 
  second 
  

   requirement. 
  The 
  first 
  requirement 
  is 
  also 
  satisfied, 
  as, 
  by 
  

   putting 
  6 
  small, 
  the 
  formulae 
  reduce 
  to 
  n\ 
  = 
  2d0 
  2 
  and 
  

   x' 
  = 
  2dd 
  ; 
  or, 
  in 
  other 
  words, 
  x' 
  = 
  \/2nd\ 
  for 
  the 
  minima 
  of 
  

   illumination, 
  which 
  is 
  also 
  the 
  usual 
  approximate 
  diffraction 
  

   formula. 
  Accordingly, 
  the 
  complete 
  formulas 
  n\=2d0 
  2 
  -f 
  2a0 
  s 
  

   and 
  x' 
  = 
  2dd-\-3a0 
  2 
  /2 
  would 
  (on 
  eliminating 
  0) 
  give 
  the 
  posi- 
  

   tions 
  of 
  the 
  minima 
  over 
  the 
  entire 
  field 
  with 
  considerable 
  

   accuracy. 
  

  

  14. 
  The 
  statements 
  made 
  in 
  the 
  preceding 
  paragraph 
  are, 
  

   however, 
  subject 
  to 
  an 
  important 
  qualification. 
  The 
  validity 
  

   of 
  the 
  formula 
  obtained 
  rests 
  on 
  the 
  basis 
  that, 
  for 
  large 
  

   values 
  of 
  d, 
  the 
  positions 
  of 
  the 
  minima 
  of 
  illumination 
  are 
  

   given 
  by 
  the 
  simple 
  relation 
  x 
  / 
  = 
  \/2nd\. 
  This, 
  however, 
  is 
  

   only 
  an 
  approximation, 
  as 
  the 
  accurate 
  values 
  are 
  to 
  be 
  

   found 
  from 
  Schuster's 
  formula 
  (see 
  Table 
  II., 
  above), 
  when 
  

   the 
  effect 
  of 
  the 
  reflected 
  light 
  is 
  negligible. 
  When 
  d 
  is 
  so 
  

   large 
  that 
  the 
  formulas 
  n\ 
  = 
  2dd 
  2 
  + 
  2a6 
  s 
  and 
  x' 
  = 
  2dd 
  + 
  3ad 
  2 
  /2 
  

   give 
  nearly 
  the 
  same 
  positions 
  for 
  the 
  minima 
  as 
  the 
  simple 
  

   relation 
  x'= 
  V2nd\, 
  they 
  should 
  therefore 
  cease 
  to 
  be 
  strictly 
  

   valid. 
  The 
  actual 
  positions 
  of 
  the 
  minima 
  for 
  such 
  values 
  

   of 
  d 
  should 
  agree 
  more 
  closely 
  with 
  those 
  given 
  by 
  Schuster's 
  

   formula, 
  and 
  should, 
  when 
  d 
  is 
  very 
  large, 
  agree 
  absolutely 
  

   with 
  the 
  same. 
  This 
  qualification 
  is, 
  however, 
  of 
  importance 
  

   only 
  with 
  reference 
  to 
  the 
  first 
  two 
  or 
  three 
  bands 
  obtained 
  

   for 
  fairly 
  large 
  values 
  of 
  d. 
  The 
  differences 
  in 
  respect 
  of 
  the 
  

   other 
  bands 
  would 
  be 
  negligibly 
  small. 
  

  

  15. 
  To 
  test 
  the 
  foregoing 
  results, 
  measurements 
  were 
  made 
  

   of 
  the 
  widths 
  of 
  the 
  bright 
  bands 
  for 
  a 
  series 
  of 
  values 
  of 
  d 
  

  

  * 
  Debye's 
  formula 
  {loc. 
  cit.) 
  shows 
  that 
  the 
  intensity 
  of 
  the 
  reflected 
  

   light 
  becomes 
  very 
  small 
  as 
  $ 
  approaches 
  tr. 
  

  

  