﻿5U 
  

  

  Mr. 
  J. 
  Walker 
  on 
  MacCullagh 
  and 
  

  

  will 
  be 
  represented 
  by 
  the 
  meridians 
  26, 
  and 
  the 
  circles 
  (j3) 
  

   by 
  the 
  parallels 
  of 
  latitude 
  2/3 
  ; 
  and 
  if 
  tan 
  a 
  be 
  the 
  length 
  of 
  

   the 
  radius-vector 
  Om 
  and 
  the 
  angle 
  mOu 
  = 
  A' 
  } 
  the 
  point 
  m 
  

   will 
  be 
  projected 
  into 
  M 
  where 
  the 
  arc 
  OM 
  — 
  2cr 
  and 
  the 
  

   angle 
  MOO' 
  = 
  A'. 
  

  

  Thus 
  any 
  point 
  on 
  the 
  sphere 
  will 
  represent 
  the 
  state 
  

   of 
  polarization 
  of 
  a 
  stream 
  of 
  light, 
  the 
  azimuth 
  of 
  its 
  plane 
  

   of 
  maximum 
  polarization 
  being 
  half 
  the 
  longitude, 
  and 
  the 
  

   ratio 
  of 
  the 
  axes 
  of 
  the 
  elliptic 
  vibrations 
  being 
  the 
  tangent 
  

   of 
  half 
  the 
  latitude 
  of 
  the 
  point 
  ; 
  and 
  the 
  polarization 
  is 
  

   right- 
  or 
  left-handed 
  according 
  as 
  the 
  point 
  is 
  in 
  the 
  Northern 
  

   or 
  Southern 
  Hemisphere. 
  

  

  It 
  is 
  now 
  easy 
  to 
  represent 
  the 
  effect 
  on 
  the 
  polarization 
  

   produced 
  by 
  the 
  passage 
  of 
  the 
  stream 
  through 
  a 
  crystalline 
  

   plate. 
  

  

  Let 
  a 
  be 
  the 
  angle 
  between 
  the 
  axis 
  of 
  x 
  and 
  the 
  plane 
  of 
  

   polarization 
  of 
  the 
  most 
  retarded 
  stream 
  in 
  the 
  plate, 
  A 
  the 
  

   relative 
  retardation 
  of 
  phase 
  introduced 
  by 
  the 
  plate 
  ; 
  then 
  

   the 
  effect 
  of 
  the 
  transmission 
  is 
  represented 
  by 
  three 
  suc- 
  

   cessive 
  rotations 
  : 
  — 
  (1) 
  round 
  the 
  polar 
  axis 
  PP' 
  through 
  an 
  

   angle 
  —2a, 
  (2) 
  round 
  the 
  diameter 
  00' 
  in 
  a 
  left-handed 
  

   direction 
  through 
  the 
  angle 
  A, 
  (3) 
  round 
  the 
  polar 
  axis 
  

   through 
  an 
  angle 
  2a 
  ; 
  and 
  these 
  three 
  operations 
  are 
  clearly 
  

   equivalent 
  to 
  a 
  single 
  rotation 
  in 
  a 
  left-handed 
  direction 
  

   through 
  an 
  angle 
  A 
  round 
  an 
  equatorial 
  diameter 
  A.' 
  A, 
  where 
  

   the 
  arc 
  0A 
  = 
  2a. 
  

  

  Fig. 
  3. 
  

  

  If 
  the 
  resulting 
  polarization 
  be 
  plane, 
  this 
  rotation 
  must 
  

   bring 
  the 
  representative 
  point 
  into 
  the 
  plane 
  of 
  the 
  equator 
  

   to 
  the 
  point 
  M', 
  say, 
  and 
  the 
  arc 
  AM' 
  is 
  twice 
  the 
  angle 
  a 
  

   that 
  the 
  resulting 
  plane 
  of 
  polarization 
  makes 
  with 
  that 
  of 
  

   the 
  most 
  retarded 
  stream 
  in 
  the 
  plate. 
  

  

  4. 
  The 
  application 
  to 
  the 
  elliptic 
  analyser 
  follows 
  at 
  once. 
  

  

  Let 
  the 
  point 
  M, 
  representative 
  of 
  the 
  polarization 
  of 
  the 
  

  

  