﻿374 
  Lord 
  Rayleigh 
  on 
  the 
  Scattering 
  of 
  Light 
  hy 
  a 
  

  

  perpendicularly, 
  and 
  seem 
  to 
  call 
  for 
  a 
  calculation 
  of 
  what 
  is 
  

   to 
  be 
  expected 
  from 
  particles 
  of 
  arbitrary 
  shape. 
  

  

  As 
  a 
  preliminary 
  to 
  a 
  more 
  complete 
  treatment, 
  it 
  may 
  be 
  

   well 
  to 
  take 
  first 
  the 
  case 
  of 
  particles 
  symmetrical 
  about 
  an 
  

   axis, 
  or 
  at 
  any 
  rate 
  behaving 
  as 
  if 
  they 
  were 
  such, 
  for 
  the 
  

   calculation 
  is 
  then 
  a 
  good 
  deal 
  simpler. 
  We 
  may 
  also 
  limit 
  

   ourselves 
  to 
  finding 
  the 
  ratio 
  of 
  intensities 
  of 
  the 
  two 
  polarized 
  

   components 
  in 
  the 
  light 
  scattered 
  at 
  right 
  angles, 
  the 
  prin- 
  

   cipal 
  component 
  being 
  that 
  which 
  vibrates 
  parallel 
  to 
  the 
  

   primary 
  vibrations, 
  and 
  the 
  subordinate 
  component 
  

   (vanishing 
  for 
  spherical 
  particles) 
  being 
  that 
  in 
  which 
  the 
  

   vibrations 
  are 
  perpendicular 
  to 
  the 
  primary 
  vibrations. 
  

   All 
  that 
  we 
  are 
  then 
  concerned 
  with 
  are 
  certain 
  resolving 
  

   factors, 
  and 
  the 
  integration 
  over 
  angular 
  space 
  required 
  to 
  

   take 
  account 
  of 
  the 
  random 
  orientations. 
  In 
  virtue 
  of 
  the 
  

   postulated 
  symmetry, 
  a 
  revolution 
  of 
  a 
  particle 
  about 
  its 
  own 
  

   axis 
  has 
  no 
  effect, 
  so 
  that 
  in 
  the 
  integration 
  we 
  have 
  to 
  deal 
  

   only 
  with 
  the 
  direction 
  of 
  this 
  axis. 
  It 
  is 
  to 
  be 
  observed 
  

   that 
  the 
  system 
  of 
  vibrations 
  scattered 
  by 
  a 
  particle 
  depends 
  

   upon 
  the 
  direction 
  of 
  primary 
  vibration 
  without 
  regard 
  to 
  

   that 
  of 
  primary 
  propagation. 
  In 
  the 
  case 
  of 
  a 
  spherical 
  

   particle 
  the 
  system 
  of 
  scattered 
  vibrations 
  is 
  symmetrical 
  

   with 
  respect 
  to 
  this 
  direction 
  and 
  the 
  amplitude 
  of 
  the 
  

   scattered 
  vibration 
  is 
  proportional 
  to 
  the 
  cosine 
  of 
  the 
  angle 
  

   between 
  the 
  primary 
  and 
  secondary 
  vibrations. 
  When 
  we 
  

   pass 
  to 
  unsymmetrical 
  particles, 
  we 
  have 
  first 
  to 
  resolve 
  the 
  

   primary 
  vibrations 
  in 
  directions 
  corresponding 
  to 
  certain 
  

   principal 
  axes 
  of 
  the 
  disturbing 
  particle 
  and 
  to 
  introduce 
  

   separate 
  coefficients 
  of 
  radiation 
  for 
  the 
  different 
  axes. 
  

   Each 
  of 
  (he 
  three 
  component 
  radiations 
  is 
  symmetrical 
  with 
  

   respect 
  to 
  its 
  own 
  axis, 
  and 
  follows 
  the 
  same 
  law 
  as 
  obtains 
  

   for 
  the 
  sphere 
  *. 
  

  

  In 
  fig. 
  1 
  the 
  various 
  directions 
  are 
  represented 
  by 
  points 
  

   on 
  a 
  spherical 
  surface 
  with 
  centre 
  0. 
  Thus 
  in 
  the 
  rect- 
  

   angular 
  system 
  XYZ, 
  OZ 
  is 
  the 
  direction 
  of 
  primary 
  

   vibration, 
  corresponding 
  (we 
  may 
  suppose) 
  to 
  primary 
  pro- 
  

   pagation 
  parallel 
  to 
  OX. 
  The 
  rectangular 
  system 
  UVW 
  

   represents 
  in 
  like 
  manner 
  the 
  principal 
  axes 
  of 
  a 
  particle, 
  

   so 
  that 
  UV, 
  VW, 
  WQ 
  are 
  quadrants. 
  Since 
  symmetry 
  of 
  

   the 
  particle 
  round 
  W 
  has 
  been 
  postulated, 
  there 
  is 
  no 
  loss 
  

   of 
  generality 
  in 
  taking 
  U 
  upon 
  the 
  prolongation 
  of 
  ZW. 
  

   As 
  usual, 
  we 
  denote 
  ZW 
  by 
  0, 
  and 
  XZW 
  by 
  <£. 
  

  

  The 
  first 
  step 
  is 
  the 
  resolving 
  of 
  the 
  primary 
  vibration 
  Z 
  

   in 
  the 
  directions 
  U, 
  Y, 
  W. 
  We 
  have 
  

  

  cosZU=-sin0, 
  cosZV 
  = 
  0, 
  cosZW 
  = 
  cos#. 
  . 
  (1) 
  

  

  * 
  Phil. 
  Mag. 
  vol. 
  xliv. 
  p. 
  28 
  (1897) 
  ; 
  Sci. 
  Papers, 
  vol. 
  iv. 
  p. 
  305. 
  

  

  