﻿376 
  Dr. 
  H. 
  8tansfield 
  on 
  

  

  approximation 
  to 
  the 
  value 
  given 
  by 
  (2), 
  the 
  difference, 
  which 
  

   depends 
  on 
  ^*, 
  being 
  only 
  1 
  part 
  in 
  800 
  when 
  6 
  = 
  5°, 
  The 
  

   slight 
  deviations 
  o£ 
  e 
  and 
  / 
  from 
  the 
  linear 
  expressions 
  (6) 
  

   «nd 
  (7) 
  may 
  be 
  detected 
  in 
  fig. 
  4. 
  

  

  Substituting 
  the 
  approximate 
  expressions 
  for 
  R, 
  e, 
  and 
  /in 
  

   equations 
  (1) 
  and 
  (1 
  A), 
  we 
  obtain 
  the 
  general 
  equations 
  in 
  

   the 
  form: 
  

  

  Bo(n-|^)-5t(l+^^) 
  = 
  nX, 
  . 
  . 
  . 
  (8) 
  

  

  R„(l+Q_^-(l+i^')=„X.. 
  . 
  . 
  (8A) 
  

  

  Equation 
  (8) 
  is 
  in 
  agreement 
  with 
  Galitzin^s 
  calculated 
  

   values 
  and 
  has 
  the 
  support 
  o£ 
  his 
  measurements 
  which 
  were 
  

   made 
  on 
  orders 
  close 
  to 
  the 
  position 
  of 
  greatest 
  brightness 
  

   so 
  that 
  the 
  angle 
  yfr 
  was 
  always 
  small. 
  

  

  n 
  the 
  principal 
  maxima 
  several 
  places 
  removed 
  from 
  the 
  

   direction 
  of 
  the 
  regularly 
  refracted 
  rays 
  are 
  considered, 
  it 
  

   becomes 
  necessary 
  to 
  take 
  into 
  account 
  the 
  terms 
  depending 
  

   on 
  -v/r^, 
  which 
  were 
  neglected 
  in 
  applying 
  Fermat's 
  principle. 
  

   "When 
  the 
  path 
  of 
  the 
  diffracted 
  light 
  is 
  measured 
  along 
  the 
  

   diffracted 
  rays, 
  the 
  equation 
  for 
  the 
  direct 
  case 
  may 
  be 
  

   written 
  

  

  ^— 
  -e 
  sin 
  '\lr-\-(t 
  cos 
  6 
  — 
  s 
  sin 
  6) 
  (1 
  — 
  cos 
  ylr) 
  = 
  n\, 
  . 
  (9) 
  

  

  If 
  we 
  now 
  suppose 
  the 
  light 
  to 
  retrace 
  its 
  path, 
  the 
  angle 
  

   of 
  incidence, 
  6', 
  on 
  the 
  step-faces 
  is 
  equal 
  to 
  6— 
  '\jr 
  (see 
  fig. 
  2), 
  

   and 
  the 
  angle 
  6' 
  -f 
  'yfr' 
  at 
  which 
  the 
  diffracted 
  rays 
  leave 
  the 
  

   end 
  plate 
  is 
  equal 
  to 
  0. 
  Hence 
  the 
  equation 
  for 
  the 
  reversed 
  

   case 
  may 
  be 
  found 
  by 
  substituting 
  6' 
  + 
  -yjr' 
  for 
  6 
  and 
  -v/r' 
  for 
  yjr 
  

   in 
  equation 
  (9) 
  after 
  substituting 
  for 
  E- 
  its 
  value 
  from 
  

   equation 
  (2). 
  The 
  equation 
  thus 
  obtained 
  may 
  be 
  written 
  in 
  

   the 
  form: 
  

  

  t(\//M^-sin^ 
  ((9' 
  + 
  ^0-cos 
  (9')-4cos 
  6' 
  sin 
  ^|r' 
  

  

  — 
  sin 
  eXl- 
  COS 
  ^lr')} 
  = 
  nX. 
  . 
  (9A) 
  

  

  If 
  the 
  terms 
  whose 
  order 
  in 
  6 
  and 
  '\jr 
  is 
  higher 
  than 
  the 
  

   second 
  are 
  neglected, 
  equations 
  (9) 
  and 
  (9 
  A) 
  reduce 
  to 
  

  

  