﻿Stokes's 
  Elliptic 
  Analyse] 
  

  

  543 
  

  

  Now 
  if 
  6 
  be 
  the 
  angle 
  that 
  the 
  planes 
  of 
  maximum 
  and 
  

   minimum 
  polarization 
  make 
  with 
  the 
  coordinate 
  axes, 
  tan 
  /3 
  

   be 
  the 
  ratio 
  of 
  the 
  axes 
  of 
  the 
  elliptic 
  vibrations 
  and 
  tan 
  <r=b/a, 
  

   we 
  have 
  

  

  tan 
  20 
  = 
  cos 
  A' 
  . 
  tan 
  2<r, 
  sin 
  2/3 
  = 
  sin 
  A 
  f 
  . 
  sin 
  2<r 
  , 
  

  

  which 
  give 
  

  

  u 
  2 
  + 
  v 
  2 
  + 
  2 
  cot 
  20^-1 
  = 
  0, 
  u 
  2 
  + 
  v 
  2 
  — 
  2cosec2£.i? 
  + 
  l 
  = 
  0. 
  

  

  Thus 
  if 
  be 
  constant, 
  the 
  points 
  representing 
  the 
  different 
  

   states 
  of 
  polarization 
  lie 
  on 
  a 
  circle 
  through 
  p 
  and 
  p', 
  and 
  if 
  

   the 
  ratio 
  of 
  the 
  axes 
  of 
  the 
  elliptic 
  vibrations 
  be 
  constant, 
  the 
  

   points 
  corresponding 
  to 
  different 
  orientations 
  of 
  the 
  axes 
  are 
  

   on 
  a 
  circle 
  cutting 
  the 
  first 
  circle 
  orthogonally. 
  

  

  Any 
  point 
  is 
  the 
  intersection 
  of 
  a 
  circle 
  of 
  the 
  one 
  system 
  

   with 
  a 
  circle 
  of 
  the 
  second 
  system 
  : 
  the 
  distance 
  from 
  the 
  

   origin 
  of 
  the 
  point, 
  in 
  which 
  the 
  circle 
  of 
  the 
  first 
  (0) 
  system 
  

   cuts 
  the 
  axis 
  of 
  u, 
  is 
  the 
  tangent 
  of 
  the 
  angle 
  that 
  the 
  plane 
  

   of 
  maximum 
  or 
  of 
  minimum 
  polarization 
  makes 
  with 
  the 
  

   axis 
  of 
  x, 
  according 
  as 
  the 
  representative 
  point 
  is 
  within 
  or 
  

   without 
  the 
  circle 
  of 
  radius 
  equal 
  to 
  unity 
  with 
  its 
  centre 
  at 
  

   the 
  origin. 
  

  

  3. 
  Let 
  us 
  now 
  transform 
  this 
  representation 
  by 
  a 
  stereo- 
  

   graphic 
  projection. 
  

  

  Describe 
  a 
  sphere 
  of 
  unit 
  diameter 
  touching 
  the 
  plane 
  of 
  

   uv 
  at 
  the 
  origin 
  of 
  the 
  coordinates, 
  and 
  let 
  the 
  points 
  of 
  the 
  

   plane 
  be 
  projected 
  on 
  the 
  surface 
  of 
  this 
  sphere 
  by 
  joining 
  

   them 
  to 
  0', 
  the 
  extremity 
  of 
  the 
  sphere 
  through 
  0. 
  

  

  Fio-. 
  2. 
  

  

  Then 
  the 
  axes 
  of 
  u 
  and 
  v 
  project 
  into 
  great 
  circles 
  at 
  right 
  

   angles 
  to 
  one 
  another, 
  the 
  former 
  of 
  which 
  may 
  be 
  called 
  the 
  

   equator; 
  the 
  points 
  p, 
  p' 
  become 
  the 
  poles 
  ; 
  the 
  circles 
  {&) 
  

  

  2 
  02 
  

  

  