﻿Elliptic 
  Polarization. 
  199 
  

  

  ^vhich 
  cos 
  2a=± 
  tan 
  2/S, 
  we 
  see 
  that 
  as 
  R 
  increases 
  from 
  

   to 
  TT, 
  corresponding 
  to 
  a 
  change 
  o£ 
  Sfrom 
  2;i7r 
  to 
  2(n±l)7r 
  

   according 
  as 
  y5 
  is 
  positive 
  or 
  negative, 
  the 
  light 
  is 
  initially 
  

   plane 
  polarized 
  in 
  the 
  primitive 
  plane 
  o£ 
  polarization; 
  it 
  

   then 
  becomes 
  elliptically 
  polarized 
  oE 
  a 
  sign, 
  the 
  same 
  as 
  or 
  

   opposite 
  to 
  that 
  o£ 
  the 
  plate 
  according 
  as 
  sin 
  2a 
  is 
  positive 
  

   or 
  negative: 
  the 
  ratio 
  of 
  the 
  axes 
  of 
  the 
  elliptic 
  vibration 
  

   attains 
  its 
  maximum 
  value, 
  the 
  plane 
  of 
  maximum 
  polariza- 
  

   tion 
  being 
  then 
  either 
  parallel 
  or 
  perpendicular 
  to 
  the 
  prin- 
  

   cipal 
  section 
  of 
  the 
  quartz, 
  when 
  tan 
  R 
  = 
  sin 
  2/9 
  cot 
  \/r/2 
  or 
  

   = 
  — 
  sin 
  2y8 
  tan 
  -^12 
  according 
  as 
  yS 
  is 
  positive 
  or 
  negative 
  ; 
  

   -when 
  tan 
  R=— 
  tan2a 
  the 
  light 
  is 
  again 
  plane 
  polarized 
  in 
  

   an 
  azimuth 
  symmetrical 
  to 
  the 
  primitive 
  plane 
  of 
  polarization 
  

   with 
  respect 
  to 
  the 
  principal 
  section 
  of 
  the 
  plate 
  ; 
  the 
  sense 
  

   of 
  rotation 
  then 
  changes, 
  and 
  the 
  ratio 
  of 
  the 
  axes 
  is 
  again 
  

   a 
  maximum 
  when 
  tan 
  R= 
  — 
  sin 
  2/3 
  tan 
  >|r/2 
  or 
  = 
  sin 
  2fi 
  cot'»/r/2, 
  

   the 
  plane 
  of 
  maximum 
  polarization 
  being 
  again 
  either 
  parallel 
  

   •or 
  perpendicular 
  to 
  the 
  principal 
  section 
  ; 
  and 
  finally, 
  when 
  

   R 
  = 
  7r. 
  the 
  light 
  is 
  plane 
  polarized 
  in 
  the 
  original 
  plane*. 
  

  

  4. 
  Turning 
  now 
  to 
  tan 
  2(^ 
  — 
  a), 
  we 
  see 
  that 
  this 
  attains 
  

   its 
  maximum 
  and 
  minimum 
  values 
  tan 
  R;„, 
  when 
  

  

  tanR=tanR„i 
  

  

  _ 
  -sin 
  2« 
  cos 
  2« 
  + 
  >v/(cos^ 
  2ol- 
  tan^ 
  2/8) 
  (sin^ 
  2ci+ 
  tan^ 
  2/3) 
  

  

  cos^2a-sin2 
  2a- 
  tan^ 
  2y8 
  

   This, 
  however, 
  only 
  gives 
  a 
  real 
  value 
  for 
  tan 
  R, 
  if 
  

   cos^ 
  2a 
  > 
  tan^ 
  2y8, 
  and 
  when 
  this 
  is 
  the 
  case, 
  writing 
  

  

  cos^ 
  2cc— 
  tan- 
  2/3= 
  cos^ 
  %, 
  

   where 
  % 
  is 
  a 
  positive 
  angle 
  less 
  than 
  7r/2, 
  we 
  have 
  

   tan 
  R,„= 
  tan 
  {x—2a) 
  or 
  — 
  tan 
  (;^ 
  + 
  2a), 
  

  

  the 
  value 
  of 
  sin 
  2y 
  in 
  these 
  two 
  cases 
  being 
  tan 
  2/3/ 
  cos 
  2a. 
  

  

  Thus 
  if 
  cos^ 
  2a 
  < 
  tan^ 
  2^, 
  the 
  plane 
  of 
  maximum 
  pola- 
  

   rization 
  rotates 
  continuously 
  as 
  R 
  increases 
  ; 
  while 
  if 
  

   cos^ 
  2a 
  > 
  tan^ 
  2(3, 
  it 
  oscillates 
  between 
  two 
  extreme 
  positions, 
  

   making 
  an 
  angle 
  ;)^/2 
  on 
  either 
  side 
  of 
  the 
  principal 
  section 
  

   of 
  the 
  plate 
  or 
  the 
  perpendicular 
  plane 
  according 
  as 
  cos 
  2a 
  

   is 
  positive 
  or 
  negative 
  f- 
  

  

  * 
  The 
  changes 
  in 
  the 
  value 
  of 
  sin 
  2y 
  may 
  also 
  be 
  conveniently 
  traced 
  

   from 
  the 
  formula 
  

  

  sin 
  2y 
  = 
  2 
  sin 
  2a 
  cos 
  2,3 
  sin 
  ^/2 
  sin 
  (\//'+c/2)/sin 
  ^. 
  

  

  t 
  This 
  oscillation 
  of 
  the 
  plane 
  of 
  maximum 
  polarization 
  was 
  deduced 
  

   by 
  Croullebois 
  {Ann. 
  de 
  Ch. 
  et 
  de 
  Phys. 
  [4] 
  xxviii. 
  p. 
  382 
  (1873)) 
  for 
  

   the 
  special 
  case 
  of 
  light 
  initially 
  polarized 
  in 
  the 
  principal 
  section 
  of 
  the 
  

   quartz. 
  lie 
  appears 
  to 
  hav«i 
  overlooked 
  the 
  fact 
  that 
  it 
  requires 
  in 
  this 
  

   case 
  that 
  tan 
  22 
  should 
  be 
  less 
  than 
  unity 
  or 
  /3<:7r/8, 
  a 
  condition 
  found 
  

   by 
  Monnory 
  {J. 
  de 
  PJnjs. 
  [2] 
  ix. 
  p. 
  277 
  (1890)). 
  The 
  condition 
  for 
  the 
  

   general 
  case 
  does 
  not 
  appear 
  to 
  have 
  been 
  previously 
  given. 
  

  

  