﻿546 
  Mr. 
  J. 
  Walker 
  on 
  MacCullagh 
  and 
  

  

  greater 
  or 
  less 
  than 
  7r/2, 
  and 
  the 
  angle 
  6 
  determines 
  the 
  

   plane 
  of 
  maximum 
  or 
  of' 
  minimum 
  polarization 
  according 
  as 
  

   sin 
  aj 
  is 
  greater 
  or 
  less 
  than 
  sin 
  cr-/. 
  

  

  5. 
  As 
  a 
  second 
  illustration 
  of 
  the 
  method, 
  we 
  may 
  take 
  

   Gouy's 
  explanation 
  * 
  of 
  streams 
  of 
  permanent 
  type 
  in 
  active 
  

   crystals, 
  according 
  to 
  which 
  they 
  are 
  to 
  be 
  ascribed 
  to 
  the 
  

   superposition 
  of 
  the 
  effects 
  of 
  ordinary 
  double 
  refraction 
  and 
  

   of 
  an 
  independent 
  rotary 
  power 
  of 
  the 
  crystal. 
  

  

  If 
  this 
  be 
  so, 
  the 
  result 
  of 
  normal 
  passage 
  of 
  the 
  stream 
  

   through 
  unit 
  thickness 
  of 
  a 
  plate 
  of 
  an 
  active 
  crystal 
  is 
  given 
  

   by 
  a 
  rotation 
  through 
  an 
  angle 
  «(/*i 
  — 
  /jl 
  2 
  ) 
  round 
  the 
  axis 
  CA 
  

   corresponding 
  to 
  the 
  principal 
  section 
  of 
  the 
  plate, 
  C 
  being 
  

   the 
  centre 
  of 
  sphere, 
  together 
  with 
  a 
  rotation 
  2p 
  round 
  the 
  

   polar 
  axis 
  CP, 
  where 
  K 
  = 
  2 
  t 
  77/X, 
  ^ 
  and 
  /^ 
  2 
  are 
  the 
  refractive 
  

   indices 
  of 
  streams 
  polarized 
  in 
  and 
  perpendicularly 
  to 
  the- 
  

   principal 
  section 
  of 
  the 
  plate, 
  supposed 
  devoid 
  of 
  rotary 
  

   power. 
  

  

  Regarding 
  these 
  rotations 
  as 
  small 
  and 
  neglecting 
  small 
  

   quantities 
  of 
  the 
  second 
  order, 
  the 
  resultant 
  rotation 
  is 
  

   represented 
  by 
  the 
  sum 
  of 
  the 
  vectors 
  obtained 
  by 
  taking 
  

   along 
  CA, 
  CP 
  lengths 
  proportional 
  to 
  the 
  rotations 
  about 
  

   these* 
  axes, 
  and 
  is 
  therefore 
  a 
  rotation 
  Vtc' 
  2 
  ^— 
  /tt 
  2 
  ) 
  2 
  + 
  4p 
  2 
  

   round 
  the 
  axis 
  CM, 
  where 
  tan 
  AM 
  = 
  2p/{«(/i 
  1 
  - 
  A^)}- 
  

  

  The 
  point 
  M 
  and 
  the 
  diametrically 
  opposite 
  point 
  M'" 
  

   represent 
  the 
  polarizations 
  of 
  streams 
  of 
  permanent 
  type, 
  and 
  

   we 
  see 
  that 
  these 
  streams 
  are 
  oppositely 
  polarized 
  with 
  their 
  

   planes 
  of 
  maximum 
  polarization 
  respectively 
  in 
  and 
  perpen- 
  

   dicular 
  to 
  the 
  principal 
  section, 
  and 
  that 
  the 
  ratio 
  tan 
  /3 
  of 
  

   the 
  axes 
  of 
  the 
  elliptic 
  vibrations 
  of 
  the 
  ends 
  of 
  their 
  polari- 
  

  

  J. 
  de 
  PJnjs. 
  (2) 
  iv. 
  p. 
  149 
  (1885). 
  

  

  