﻿Cloud 
  of 
  similar 
  small 
  Particles 
  oj 
  any 
  Shape. 
  379 
  

  

  In 
  taking 
  the 
  mean 
  with 
  respect 
  to 
  $, 
  the 
  terms 
  which 
  are 
  

   odd 
  in 
  sin 
  (f>, 
  or 
  cos 
  <£>, 
  disappear, 
  while 
  the 
  mean 
  value 
  of 
  

   sin 
  2 
  <£, 
  or 
  cos 
  2 
  <£, 
  is 
  J. 
  We 
  get 
  for 
  the 
  mean 
  

  

  1 
  = 
  ?>A 
  2 
  sin 
  2 
  6 
  cos 
  2 
  f 
  (sin 
  2 
  ^ 
  + 
  cos 
  2 
  ^ 
  cos 
  2 
  6) 
  

   + 
  JB 
  2 
  sin 
  2 
  d 
  sin 
  2 
  f 
  (cos 
  2 
  f 
  + 
  sin 
  2 
  ^ 
  cos 
  2 
  6) 
  

   + 
  iC 
  2 
  sin 
  2 
  0cos 
  2 
  

  

  — 
  AB 
  sin 
  2 
  6 
  sin 
  yfr 
  cos 
  i/r 
  . 
  sin 
  ijr 
  cos 
  t/t 
  sin 
  2 
  6 
  

  

  — 
  BC 
  sin 
  2 
  # 
  cos 
  6 
  sin 
  i|r 
  . 
  sin 
  yjr 
  cos 
  # 
  

  

  — 
  CA 
  sin 
  2 
  6 
  cos 
  # 
  cos 
  yjr 
  . 
  cos 
  t/t 
  cos 
  6. 
  . 
  . 
  . 
  (17) 
  

  

  The 
  averaging 
  with 
  respect 
  to 
  yjr 
  now 
  goes 
  as 
  before, 
  and 
  

   we 
  obtain 
  

  

  I 
  r(A 
  2 
  + 
  B 
  2 
  ) 
  sin 
  2 
  (I 
  + 
  f 
  cos 
  2 
  6) 
  + 
  JC 
  2 
  sin 
  2 
  6 
  cos 
  2 
  (9 
  

  

  - 
  |rAB 
  sin 
  4 
  (9 
  - 
  J 
  (A 
  + 
  B)C 
  sin 
  2 
  (9 
  cos 
  2 
  0; 
  .... 
  (18 
  j 
  

  

  and, 
  finally, 
  the 
  averaging 
  with 
  respect 
  to 
  6 
  gives 
  

  

  T 
  A 
  2 
  + 
  B 
  2 
  C 
  2 
  AB 
  (A 
  + 
  B)C 
  

  

  mean 
  1= 
  — 
  X— 
  (1— 
  l 
  + 
  A) 
  + 
  _ 
  _v 
  — 
  _^_ 
  

  

  16 
  ° 
  ' 
  ' 
  lo 
  Id 
  Id 
  

  

  = 
  r 
  V{A 
  2 
  + 
  B 
  2 
  + 
  C 
  2 
  -AB-BC-CA}. 
  . 
  . 
  . 
  (19) 
  

  

  This 
  represents 
  the 
  intensity 
  of 
  the 
  vibrations 
  parallel 
  to 
  X 
  

   dispersed 
  along 
  OY, 
  due 
  to 
  primary 
  vibrations 
  parallel 
  to 
  Z. 
  

   It 
  vanishes, 
  of 
  course, 
  if 
  A 
  = 
  B 
  = 
  0; 
  while, 
  if 
  A 
  = 
  B 
  merely, 
  

   it 
  reduces 
  to 
  (4) 
  . 
  

  

  The 
  ratio 
  of 
  the 
  two 
  polarized 
  components 
  is 
  

  

  A 
  2 
  + 
  B 
  2 
  + 
  C 
  2 
  -AB-BC-CA 
  ( 
  

  

  3(A 
  2 
  + 
  B 
  2 
  + 
  C 
  2 
  ) 
  + 
  2(AB 
  + 
  BC+OA) 
  5 
  * 
  

  

  reducing 
  to 
  (5) 
  when 
  B=A. 
  

  

  If 
  the 
  primary 
  light 
  travelling 
  in 
  direction 
  OX 
  is 
  un- 
  

   polarized, 
  we 
  have 
  also 
  to 
  include 
  primary 
  vibrations 
  

   parallel 
  to 
  Y. 
  The 
  secondary 
  vibrations 
  scattered 
  along 
  

   Y 
  are 
  of 
  the 
  same 
  intensity 
  whether 
  they 
  are 
  parallel 
  to 
  Z 
  

   or 
  to 
  X. 
  They 
  are 
  given 
  by 
  (19), 
  where 
  all 
  that 
  is 
  essential 
  

   is 
  the 
  perpendicularity 
  of 
  the 
  primary 
  and 
  secondary 
  vibra- 
  

   tions. 
  Thus, 
  in 
  order 
  to 
  obtain 
  the 
  effect 
  along 
  OY 
  of 
  

   unpolarized 
  primary 
  light 
  travelling 
  along 
  OX, 
  we 
  have 
  

   merely 
  to 
  add 
  (19) 
  to 
  both 
  components. 
  The 
  intensity 
  of 
  

  

  