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542                              ON THE SCATTERING  OF  LIGHT BY A                             [430
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
\A+(0-A} cos2 0}2 sin Bdd + |   sin 6dO
3(72 + 4^(7).......(2)
This represents the intensity of that polarized component of the scattered light along OF whose vibrations are parallel to OZ.
For the vibrations parallel to OX the second set of resolving factors is cos UX, cos VX, cos WX.   Now from the spherical triangle UZX,
cos UX = sin (90 + 0) cos < = cos 6 cos <. Also from the triangles VZX, WZX,
cos VX = cos VZX = cos (90 + <) = - sin </>, cos TOT == sin 6 cos 0.
The first set of factors remains as before. Taking both sets into account, we get for the vibration parallel t6 X
 A sin 6 cos 6 cos <f> + C cos 6 sin 6 cos <,
the square of which is
The mean value of cos2 < is . That of cos2 0 is  and that of cos4 6 is |, as above, so that corresponding to (2) we have for the mean intensity of the vibrations parallel to X
The ratio of intensities of the two components is thus
Two particular cases are worthy of notice. If A can be neglected in comparison with C, (5) becomes simply one-third. On the other hand, if A is predominant, (5) reduces to one-eighth.
The above expressions apply when the primary light, propagated parallel to X, is completely polarized with vibrations parallel to Z, the direction of the secondary ray being along 0 F. If the primary light be unpolarized, we have further to include the effect of the primary vibrations parallel to F. The two polarized components scattered along OF, resulting therefrom, both vibrate in directions perpendicular to OF, and accordingly are both represented by (4). In the case of unpolarized primary light we have therefore to.(1)