﻿Cloud 
  oj 
  similar 
  small 
  Particles 
  of 
  any 
  Shape. 
  375 
  

  

  The 
  coefficients, 
  dependent 
  upon 
  the 
  character 
  of 
  the 
  par- 
  

   ticle, 
  corresponding 
  to 
  U, 
  V, 
  W 
  may 
  be 
  denoted 
  by 
  A, 
  A, 
  

   C 
  ; 
  and 
  we 
  seek 
  the 
  effect 
  along 
  the 
  scattered 
  ray 
  OY, 
  

  

  Fig. 
  1. 
  

   Z 
  

  

  perpendicular 
  to 
  both 
  primary 
  vibrations 
  and 
  primary 
  pro- 
  

   pagation. 
  The 
  ray 
  scattered 
  in 
  this 
  direction 
  will 
  not 
  be 
  

   completely 
  polarized, 
  and 
  we 
  consider 
  separately 
  vibrations 
  

   parallel 
  to 
  Z 
  and 
  to 
  X. 
  As 
  regards 
  the 
  former, 
  we 
  have 
  

   the 
  same 
  set 
  of 
  factors 
  over 
  again, 
  as 
  in 
  (1), 
  so 
  that 
  the 
  

   vibration 
  is 
  A 
  sin 
  2 
  6+C 
  cos 
  2 
  0, 
  reducing 
  to 
  simply, 
  if 
  

   A 
  — 
  0. 
  This 
  is 
  the 
  result 
  for 
  a 
  single 
  particle 
  whose 
  axis 
  is 
  

   at 
  W. 
  What 
  we 
  are 
  aiming 
  at 
  is 
  the 
  aggregate 
  intensity 
  

   due 
  to 
  a 
  large 
  number 
  of 
  particles 
  with 
  their 
  positions 
  and 
  

   their 
  axes 
  distributed 
  at 
  random. 
  The 
  mean 
  intensity 
  is 
  

  

  ("•It 
  fin 
  

  

  \ 
  A 
  + 
  (C- 
  A) 
  cos 
  2 
  6 
  p 
  sin 
  6dd+ 
  \ 
  sin 
  OdO 
  

   %, 
  o 
  Jo 
  

  

  = 
  A 
  2 
  + 
  2(C 
  ~ 
  A)A 
  + 
  Vt^l 
  = 
  _i 
  5 
  . 
  (8A 
  2 
  + 
  3C 
  2 
  + 
  4AC). 
  (2) 
  

  

  This 
  represents 
  the 
  intensity 
  of 
  that 
  polarized 
  component 
  of 
  

   the 
  scattered 
  light 
  along 
  OY 
  whose 
  vibrations 
  are 
  parallel 
  

   to 
  OZ. 
  

  

  For 
  the 
  vibrations 
  parallel 
  to 
  OX 
  the 
  second 
  set 
  of 
  re- 
  

   solving 
  factors 
  is 
  cos 
  UX, 
  cos 
  VX, 
  cos 
  WX. 
  Now 
  from 
  

   the 
  spherical 
  triangle 
  UZX, 
  

  

  cos 
  UX 
  = 
  sin 
  (90° 
  -f 
  6) 
  cos 
  cf> 
  = 
  cos 
  cos 
  <j>. 
  

  

  Also 
  from 
  the 
  triangles 
  VZX, 
  WZX, 
  

  

  cos 
  VX 
  = 
  cosVZW=cos 
  (90° 
  + 
  </>)= 
  -sin 
  </>, 
  

  

  cos 
  WX 
  = 
  sin 
  6 
  cos 
  <f>. 
  

  

  2 
  D2 
  

  

  