﻿542 
  Mr. 
  J. 
  Walker 
  on 
  MacCuilagli 
  and 
  

  

  wherein 
  a 
  and 
  b 
  are 
  in 
  general 
  complex, 
  and 
  their 
  ratio 
  

  

  bja 
  = 
  (b/a)e 
  tA 
  ' 
  = 
  (b/a) 
  cos 
  A' 
  + 
  i(b/a) 
  sin 
  A' 
  = 
  u 
  -f 
  vi, 
  say, 
  

  

  a 
  and 
  b 
  being 
  the 
  amplitudes 
  of 
  the 
  components, 
  and 
  A' 
  the 
  

   acceleration 
  of 
  phase 
  of 
  the 
  second 
  relatively 
  to 
  that 
  of 
  

   the 
  first. 
  

  

  This 
  ratio 
  defines 
  the 
  form 
  and 
  orientation 
  of 
  the 
  elliptic 
  

   vibration 
  of 
  the 
  extremity 
  of 
  the 
  polarization-vector 
  of 
  the 
  

   stream 
  ; 
  and 
  we 
  may 
  therefore 
  represent 
  the 
  state 
  of 
  polar- 
  

   ization 
  by 
  a 
  point 
  on 
  a 
  plane, 
  for 
  which 
  the 
  abscissa 
  is 
  u 
  and 
  

   the 
  ordinate 
  is 
  v, 
  the 
  length 
  of 
  the 
  radius-vector 
  to 
  the 
  

   representative 
  point 
  giving 
  the 
  ratio 
  of 
  the 
  amplitudes 
  and 
  

   the 
  angle 
  that 
  it 
  makes 
  with 
  the 
  axis 
  of 
  abscissae 
  being 
  the 
  

   difference 
  of 
  phase. 
  Since 
  the 
  polarization 
  is 
  right- 
  or 
  left- 
  

   handed 
  according 
  as 
  A' 
  lies 
  between 
  and 
  it 
  or 
  between 
  ir 
  and 
  

   27T, 
  the 
  vibrations 
  in 
  the 
  stream 
  will 
  be 
  right- 
  or 
  left-handed, 
  

   according 
  as 
  the 
  representative 
  point 
  is 
  above 
  or 
  below 
  the 
  

   axis 
  of 
  u. 
  

  

  When 
  the 
  point 
  is 
  on 
  the 
  axis 
  of 
  u, 
  the 
  stream 
  is 
  plane- 
  

   polarized 
  in 
  an 
  azimuth 
  tan 
  -1 
  u 
  with 
  respect 
  to 
  the 
  plane 
  of 
  

   xz 
  ; 
  if 
  the 
  point 
  be 
  on 
  the 
  axis 
  of 
  v, 
  the 
  difference 
  of 
  phase 
  

   is 
  7r/2, 
  and 
  the 
  planes 
  of 
  maximum 
  and 
  minimum 
  polarization 
  

  

  

  Fig. 
  1. 
  

  

  

  

  S^X 
  1 
  

  

  **i 
  

  

  o 
  

  

  

  

  ~ 
  

  

  P' 
  

  

  

  are 
  parallel 
  to 
  the 
  axes 
  of 
  x 
  and 
  y. 
  Points 
  p, 
  p' 
  on 
  the 
  axis 
  

   of 
  ordinates 
  at 
  unit 
  distance 
  from 
  the 
  origin 
  represent 
  circular 
  

   polarization. 
  

  

  