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  LXIII. 
  On 
  MacCullagh 
  and 
  Stokes's 
  Elliptic 
  Analyser, 
  and 
  

   other 
  Applications 
  of 
  a 
  Geometrical 
  Representation 
  of 
  the 
  

   State 
  of 
  Polarization 
  of 
  a 
  Stream 
  of 
  Light. 
  By 
  James 
  

   Walker, 
  M.A.* 
  

  

  1. 
  TN 
  1843 
  MacCullagh 
  f 
  showed 
  that 
  an 
  imperfectly 
  

   X 
  adjusted 
  Fresnel's 
  rhomb 
  can 
  be 
  employed 
  within 
  

   certain 
  limits 
  for 
  the 
  investigation 
  o£ 
  a 
  stream 
  of 
  elliptically- 
  

   polarized 
  light, 
  and 
  in 
  the 
  same 
  paper 
  says 
  : 
  — 
  " 
  In 
  making 
  

   experiments 
  on 
  elliptically-polarized 
  light, 
  a 
  plate 
  of 
  mica, 
  

   or 
  any 
  other 
  doubly-refracting 
  crystal, 
  placed 
  perpendicular 
  

   to 
  the 
  ray, 
  may 
  be 
  used 
  instead 
  of 
  Fresnel's 
  rhomb 
  . 
  . 
  . 
  The 
  

   two 
  cases 
  are 
  precisely 
  similar 
  ; 
  and 
  if 
  it 
  is 
  necessary 
  not 
  to 
  

   neglect 
  the 
  errors 
  of 
  the 
  rhomb, 
  it 
  is 
  certainly 
  not 
  less 
  

   necessary 
  to 
  take 
  into 
  account 
  those 
  which 
  may 
  arise 
  from 
  a 
  

   want 
  of 
  accuracy 
  in 
  the 
  thickness 
  of 
  the 
  plate, 
  considering 
  

   how 
  difficult 
  it 
  is 
  to 
  make 
  the 
  thickness 
  correspond 
  exactly 
  

   to 
  the 
  particular 
  ray 
  which 
  w 
  T 
  e 
  wish 
  to 
  observe." 
  

  

  These 
  remarks 
  seem 
  to 
  have 
  escaped 
  attention; 
  and 
  Sir 
  G. 
  G. 
  

   Stokes 
  J, 
  in 
  1851 
  , 
  suggested 
  the 
  use 
  of 
  an 
  imperfect 
  quarter- 
  

   wave 
  plate 
  combined 
  with 
  a 
  Nicolas 
  prism 
  as 
  a 
  New 
  Elliptic 
  

   Analyser, 
  giving 
  formulae 
  for 
  its 
  use 
  that 
  are 
  substantially 
  

   the 
  same 
  as 
  those 
  published 
  by 
  MacCullagh. 
  

  

  In 
  both 
  cases 
  the 
  formulae 
  are 
  stated 
  without 
  proof, 
  but 
  

   this 
  is 
  of 
  course 
  easily 
  supplied. 
  The 
  reason 
  for 
  referring 
  to 
  

   the 
  subject 
  in 
  the 
  present 
  paper 
  is 
  that 
  the 
  method 
  of 
  proof 
  

   herein 
  employed 
  affords 
  a 
  good 
  illustration 
  of 
  the 
  advantages 
  

   of 
  a 
  geometrical 
  representation 
  of 
  the 
  state 
  of 
  polarization 
  of 
  

   a 
  stream 
  of 
  light 
  that 
  Poincare 
  § 
  has 
  employed 
  for 
  explaining- 
  

   Mallard's 
  theory 
  of 
  Rotary 
  Polarization, 
  without 
  using 
  it, 
  as 
  

   far 
  as 
  I 
  am 
  aware, 
  for 
  the 
  purpose 
  of 
  obtaining 
  numerical 
  

   results. 
  

  

  2. 
  It 
  may 
  perhaps 
  be 
  as 
  well 
  to 
  recall 
  the 
  main 
  features 
  of 
  

   this 
  geometrical 
  representation. 
  

  

  Taking 
  the 
  axis 
  of 
  z 
  in 
  the 
  direction 
  of 
  propagation, 
  a 
  stream 
  

   of 
  polarized 
  light 
  may 
  be 
  represented 
  by 
  its 
  components 
  

   polarized 
  in 
  planes 
  parallel 
  respectively 
  to 
  the 
  axes 
  of 
  x 
  and 
  y 
  

   with 
  the 
  polarization-vectors 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  t 
  Proc. 
  R. 
  I. 
  Acad. 
  ii. 
  p. 
  384 
  (1843) 
  ; 
  Collected 
  Works, 
  pp. 
  238-242. 
  

   X 
  B. 
  A. 
  Report 
  for 
  1851, 
  Part 
  ii. 
  p. 
  14 
  ; 
  Collected 
  Works, 
  iii. 
  p. 
  197. 
  

   § 
  TMorie 
  Math, 
  de 
  la 
  Lumihre, 
  ii. 
  ch. 
  xii. 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  3. 
  No. 
  17. 
  May 
  1902. 
  20 
  

  

  