﻿ike 
  Echelon 
  Spectroscope. 
  375 
  

  

  lines 
  are 
  those 
  regularly 
  refracted, 
  the 
  dotted 
  wave-fronts 
  are 
  

   drawn 
  perpendicular 
  to 
  the 
  directions 
  in 
  which 
  a 
  maximum 
  

   of 
  order 
  n 
  is 
  formed. 
  The 
  angle 
  between 
  this 
  direction 
  and 
  

   the 
  normally 
  refracted 
  rays 
  is 
  called 
  n/r 
  in 
  the 
  direct 
  case 
  

   and 
  ^/r' 
  in 
  the 
  reversed 
  case. 
  The 
  distance 
  apart 
  in 
  the 
  air 
  of 
  

   regularly 
  refracted 
  rays 
  passing 
  through 
  corresponding 
  points 
  

   E 
  and 
  F 
  of 
  neighbouring 
  step-faces 
  is 
  called 
  / 
  at 
  the 
  end 
  of 
  

   the 
  echelon 
  and 
  e 
  on 
  the 
  step 
  side. 
  The 
  condition 
  for 
  a 
  

   principal 
  maximum 
  is 
  that 
  the 
  sum 
  of 
  the 
  distances 
  of 
  E 
  from 
  

   the 
  incident 
  wave-front 
  on 
  one 
  side 
  and 
  the 
  dotted 
  wave-front 
  

   on 
  the 
  other, 
  shall 
  be 
  n 
  wave-lengths 
  greater 
  than 
  the 
  sum 
  of 
  

   the 
  distances 
  of 
  the 
  corresponding 
  point 
  F 
  from 
  the 
  same 
  

   wave-fronts. 
  

  

  When 
  the 
  angles 
  -^ 
  and 
  y^r' 
  are 
  sufficiently 
  small, 
  we 
  may 
  

   employ 
  Fermat's 
  principle 
  and 
  measure 
  the 
  optical 
  paths 
  

   along 
  the 
  regularly 
  refracted 
  rays. 
  This 
  gives 
  the 
  general 
  

   equations 
  in 
  the 
  form 
  

  

  R-ei/r=?zX, 
  (1) 
  

  

  ^-f^' 
  = 
  n\ 
  (lA) 
  

  

  where 
  R 
  is 
  the 
  retardation 
  produced 
  in 
  a 
  regularly 
  refracted 
  

   ray 
  by 
  its 
  passage 
  through 
  a 
  single 
  plate. 
  Here 
  the 
  equations 
  

   referring 
  to 
  the 
  reversed 
  case 
  are 
  distinguished 
  by 
  a 
  letter 
  A. 
  

   The 
  values 
  of 
  R, 
  e, 
  and 
  / 
  are 
  given 
  below, 
  and 
  are 
  plotted 
  

   in 
  fig. 
  4 
  (p. 
  38U> 
  

  

  R 
  = 
  f(/.A/l-^-^^cos^), 
  .... 
  (2) 
  

  

  e 
  =5 
  cos 
  ^ 
  + 
  ^ 
  sin 
  ^, 
  (3) 
  

  

  J, 
  /) 
  , 
  ^ 
  • 
  /I 
  cos 
  6 
  , 
  -. 
  

  

  / 
  = 
  5 
  cos 
  ^ 
  H 
  — 
  sm 
  ^ 
  . 
  , 
  . 
  . 
  . 
  ^4) 
  

  

  Here 
  5 
  is 
  the 
  width 
  of 
  the 
  step-faces, 
  t 
  the 
  thickness 
  of 
  the 
  

   plates, 
  and 
  6 
  the 
  angle 
  of 
  incidence 
  of 
  the 
  light 
  on 
  the 
  plates. 
  

   This 
  angle 
  is 
  in 
  practice 
  generally 
  kept 
  within 
  the 
  narrow 
  

   limits 
  of 
  + 
  2°, 
  and 
  it 
  is 
  sufficient 
  to 
  retain 
  only 
  the 
  lowest 
  

   powers 
  of 
  6. 
  These 
  expressions 
  then 
  become 
  

  

  ^=^»(l+ 
  £) 
  (5) 
  

  

  e 
  =s 
  + 
  te, 
  (6) 
  

  

  f='+i 
  (7) 
  

  

  Ro 
  (the 
  value 
  of 
  R 
  when 
  ^ 
  = 
  0) 
  is 
  (ft-l)^. 
  The 
  parabolic 
  

   expression 
  for 
  R 
  — 
  Rq 
  in 
  equation 
  (5) 
  gives 
  a 
  very 
  close 
  

  

  2 
  C2 
  

  

  