﻿378 
  Lord 
  Rayleigh 
  on 
  the 
  Scattering 
  of 
  Light 
  by 
  a 
  

   A, 
  B, 
  C 
  as 
  before, 
  and 
  the 
  vibration 
  is 
  expressed 
  by 
  

  

  A 
  sin 
  2 
  6 
  cos 
  2 
  f 
  + 
  B 
  sin 
  2 
  6 
  sin 
  2 
  f 
  + 
  C 
  cos 
  2 
  ; 
  . 
  (11) 
  

   while 
  for 
  the 
  intensity 
  

  

  I 
  = 
  A 
  2 
  sin 
  4 
  6 
  cos 
  4 
  ftB 
  2 
  sin 
  4 
  sin 
  4 
  ^ 
  + 
  2 
  cos 
  4 
  (9 
  

  

  + 
  2 
  AB 
  sin 
  4 
  6 
  cos 
  2 
  f 
  sin 
  2 
  ^ 
  + 
  2BC 
  sin 
  2 
  d 
  cos 
  2 
  sin 
  2 
  ^ 
  

  

  + 
  2CAsin 
  2 
  0cos 
  2 
  0cos 
  2 
  ^ 
  (12) 
  

  

  This 
  is 
  for 
  a 
  single 
  particle, 
  and 
  we 
  have 
  now 
  to 
  take 
  the 
  

   mean 
  1'or 
  all 
  orientations. 
  The 
  mean 
  value 
  of 
  sin 
  4 
  i/r, 
  or 
  

   cos 
  4 
  i/r, 
  is 
  | 
  ; 
  that 
  of 
  sin 
  2 
  yjr 
  cos 
  2 
  yjr 
  is 
  J 
  ; 
  and 
  that 
  of 
  sin 
  2 
  i/r 
  

   is 
  \. 
  The 
  averaging 
  with 
  respect 
  to 
  yjr 
  thus 
  yields 
  

  

  I 
  = 
  3. 
  (A 
  2 
  + 
  B 
  2 
  ) 
  sin 
  4 
  + 
  C 
  2 
  cos 
  4 
  6 
  + 
  i 
  AB 
  sin 
  4 
  6 
  

  

  + 
  (A 
  + 
  B)Csin 
  2 
  0cos 
  2 
  6' 
  (13) 
  

  

  Again, 
  the 
  mean 
  value 
  of 
  sin 
  4 
  6 
  is 
  ^-, 
  that 
  of 
  cos 
  4 
  6 
  is 
  \, 
  and 
  

   that 
  of 
  sin 
  2 
  6 
  cos 
  2 
  6 
  is 
  ^. 
  Thus, 
  finally, 
  the 
  mean 
  value 
  

   of 
  I 
  over 
  the 
  sphere 
  is 
  given 
  by 
  

  

  meanI 
  = 
  T 
  « 
  5 
  {3(A 
  2 
  +B 
  2 
  4-C 
  2 
  ) 
  + 
  2(AB 
  + 
  B0 
  + 
  CA)}. 
  . 
  (14) 
  

  

  This 
  refers 
  to 
  the 
  vibrations 
  parallel 
  to 
  Z 
  which 
  are 
  propa- 
  

   gated 
  along 
  OY. 
  

  

  For 
  the 
  vibrations 
  parallel 
  to 
  X, 
  the 
  second 
  set 
  of 
  factors 
  

   is 
  cos 
  XU, 
  cos 
  XV, 
  cos 
  XW, 
  as 
  given 
  above, 
  and 
  the 
  vibra- 
  

   tion 
  is 
  expressed 
  by 
  

  

  — 
  A 
  sin 
  cos 
  yfr{ 
  — 
  sin 
  <f> 
  sin 
  \fr 
  + 
  cos 
  <£ 
  cos 
  -\jr 
  cos 
  6) 
  

  

  + 
  B 
  sin 
  sin 
  i|r 
  ( 
  — 
  sin 
  </> 
  cos 
  o/r 
  — 
  cos 
  <£ 
  sin 
  yfr 
  cos 
  6) 
  

  

  -f 
  C 
  cos 
  6 
  sh\6 
  cos 
  $ 
  (15) 
  

  

  Accordingly 
  for 
  the 
  intensity 
  

  

  I 
  = 
  A 
  2 
  sin 
  2 
  6 
  cos 
  2 
  i/r 
  (sin 
  2 
  </> 
  sin 
  2 
  \jr 
  4- 
  cos 
  2 
  (/> 
  cos 
  2 
  ^ 
  cos 
  2 
  6 
  

  

  — 
  2 
  sin 
  <£ 
  cos 
  # 
  sin 
  yjr 
  cos 
  o/r 
  cos 
  6) 
  

  

  4- 
  B 
  2 
  sin 
  2 
  sin 
  2 
  >/r(sin 
  2 
  <£ 
  cos 
  2 
  ^ 
  + 
  cos 
  2 
  <£ 
  sin 
  2 
  a/t 
  cos 
  2 
  6 
  

  

  4- 
  2 
  sin 
  (f> 
  cos 
  </> 
  sin 
  i/r 
  cos 
  \fr 
  cos 
  0) 
  

   + 
  C 
  2 
  sin 
  2 
  6 
  cos 
  2 
  cos 
  2 
  <f> 
  

  

  — 
  2AB 
  sin 
  2 
  6 
  sin 
  yjr 
  cos 
  yjr 
  (sin 
  2 
  </> 
  sin 
  yjr 
  cos 
  i/r 
  

  

  — 
  cos 
  2 
  $ 
  sin 
  i/r 
  cos 
  -x/r 
  cos 
  2 
  6 
  4- 
  sin 
  </> 
  cos 
  </> 
  sin 
  2 
  -^ 
  cos 
  

  

  — 
  sin 
  (£ 
  cos 
  <f) 
  cos 
  2 
  i|r 
  cos 
  0) 
  

   4- 
  2BC 
  sin 
  2 
  6 
  cos 
  6 
  sin 
  -^ 
  cos 
  ( 
  — 
  sin 
  cos 
  i/r 
  

  

  — 
  cos 
  (£ 
  sin 
  t/t 
  cos 
  0) 
  

  

  — 
  20 
  A 
  sin 
  2 
  6 
  cos# 
  cos-v/r 
  cos 
  (/>( 
  — 
  sin 
  <£ 
  sinifr 
  

  

  + 
  COS 
  <£ 
  COS 
  ^r 
  COS 
  #) 
  . 
  . 
  (lb') 
  

  

  