﻿Stokes's 
  Elliptic 
  Analyser. 
  545 
  

  

  stream 
  to 
  be 
  investigated, 
  be 
  determined 
  by 
  its 
  longitude 
  

   ON 
  = 
  20 
  and 
  its 
  latitude 
  NM 
  = 
  2£, 
  then 
  if 
  the 
  arc 
  NA 
  = 
  2(/>, 
  

   the 
  spherical 
  triangle 
  ANM, 
  in 
  which 
  AM 
  = 
  AM 
  / 
  = 
  2<r, 
  gives 
  

  

  sin2<£ 
  = 
  tan2/3cotA, 
  (1) 
  

  

  cos2<j=cos2£cos2(/>, 
  .... 
  (2) 
  

  

  cos 
  A 
  = 
  tan 
  2$ 
  cot 
  2<r 
  (3) 
  

  

  Whence 
  it 
  follows 
  that 
  there 
  are 
  two 
  possible 
  positions 
  of 
  the 
  

   axis, 
  A' 
  A 
  and 
  B'B, 
  such 
  that 
  NA 
  + 
  NB 
  = 
  7r, 
  and 
  that 
  the 
  

   values 
  of 
  a 
  corresponding 
  to 
  these 
  positions 
  are 
  complementary 
  

   to 
  one 
  another. 
  If 
  the 
  polarization 
  be 
  right-handed, 
  A 
  will 
  

   lie 
  within 
  or 
  without 
  the 
  arc 
  ON, 
  according 
  as 
  A 
  is 
  greater 
  

   or 
  less 
  than 
  7r/2; 
  the 
  reverse 
  being 
  the 
  case 
  if 
  the 
  polarization 
  

   be 
  left-banded. 
  

  

  If 
  then 
  xi 
  and 
  90-f^ 
  2 
  be 
  the 
  azimuths, 
  measured 
  from 
  a 
  

   fixed 
  plane 
  of 
  reference 
  in 
  a 
  direction 
  from 
  right 
  to 
  left, 
  of 
  

   the 
  plane 
  of 
  polarization 
  of 
  the 
  most 
  retarded 
  stream 
  in 
  the 
  

   plate 
  of 
  the 
  analyser, 
  when 
  the 
  emergent 
  light 
  is 
  plane- 
  

   polarized, 
  and 
  6 
  be 
  the 
  azimuth 
  of 
  the 
  plane 
  of 
  maximum 
  or 
  

   of 
  minimum 
  polarization 
  of 
  the 
  primitive 
  stream, 
  we 
  have 
  

  

  %i 
  = 
  0±</>, 
  %2=0+<Mnd0=(xi+x 
  2 
  )/2, 
  *=(%r^fc)/ 
  2 
  - 
  W 
  

  

  Again, 
  if 
  o- 
  l5 
  cr 
  2 
  be 
  the 
  azimuths 
  of 
  the 
  plane 
  of 
  polarization 
  

   of 
  the 
  stream 
  emerging 
  from 
  the 
  plate 
  in 
  its 
  first 
  and 
  second 
  

   positions, 
  measured 
  from 
  right 
  to 
  left 
  from 
  a 
  plane 
  of 
  refer- 
  

   ence 
  fixed 
  in 
  the 
  plate, 
  (o^-f 
  o- 
  2 
  )/2 
  gives 
  a 
  direction 
  inclined 
  

   at 
  an 
  angle 
  +7r/4 
  to 
  the 
  plane 
  of 
  polarization 
  of 
  the 
  most 
  

   retarded 
  stream 
  in 
  the 
  plate 
  and 
  o- 
  2 
  -^o-i=7r/2 
  + 
  2<t, 
  whence 
  

  

  cos2/3=sin(cr 
  2 
  ~o- 
  1 
  )sec(x2— 
  ^i). 
  ... 
  (5) 
  

  

  Further, 
  it 
  is 
  easy 
  to 
  see 
  that 
  tan 
  (o" 
  2 
  — 
  o^) 
  aU( 
  i 
  tan 
  (% 
  2 
  ~~ 
  %i) 
  

   have 
  the 
  same 
  or 
  opposite 
  signs 
  according 
  as 
  A 
  is 
  less 
  or 
  

   greater 
  than 
  7r/2, 
  whence 
  

  

  cos 
  2 
  A 
  = 
  tan 
  (c7 
  2 
  — 
  o-j) 
  tan 
  (% 
  2 
  — 
  %i). 
  ... 
  (6) 
  

  

  To 
  complete 
  the 
  specification 
  of 
  the 
  state 
  of 
  polarization 
  of 
  

   the 
  primitive 
  stream, 
  we 
  require 
  to 
  know 
  the 
  azimuths 
  of 
  the 
  

   resulting 
  plane 
  of 
  polarization 
  measured 
  from 
  the 
  plane 
  of 
  

   polarization 
  of 
  the 
  most 
  retarded 
  stream 
  in 
  the 
  plate 
  of 
  the 
  

   analyser, 
  which 
  in 
  the 
  case 
  of 
  a 
  mica 
  plate 
  is 
  the 
  plane 
  

   perpendicular 
  to 
  that 
  of 
  the 
  optic 
  axes, 
  and 
  in 
  selenite 
  plates 
  

   is 
  the 
  plane 
  perpendicular 
  to 
  the 
  first 
  mean 
  line. 
  If 
  cr/ 
  and 
  

   <r 
  2 
  ' 
  be 
  those 
  azimuths 
  measured 
  in 
  a 
  left-handed 
  direction, 
  

   the 
  stream 
  is 
  right- 
  or 
  left-handed, 
  according 
  as 
  <j{, 
  <r 
  2 
  ' 
  are 
  

  

  