544 . ON THE SCATTERING OF LIGHT BY A [430
This is for a single particle, and we have now to take the mean for all orientations. The mean value of sin4^ or cos4i/r, is f ; that of sin2-\|r cos-ty is £ ; and that of sin2^ is %. The averaging with respect to -^ thus yields
I = |(^2 + ^)sm4^-h(72cos^ + iJ.5sm^ + (A-fS)C/sin2^cos2^ ...(13)
Again, the mean value of sin4 d is -&%, that of cos4 6 is •£, and that of sin2 6 cos2 0 is T2-. Thus, finally, the mean value of / over the sphere is given by
mean/=TV (3 (A2 + 52-f (72) + 2 (AB + 5(7+ GA)} ....... (14)
This refers to the vibrations parallel to Z which are propagated along OY.
For. the vibrations parallel to X, the second set of factors is cosXU, cos XV, cos X W, as given above, and the vibration is expressed by
— A sin 6 cos ^ (— sin <£ sin ^r + cos $ cos ty cos 0) + 5 sin 0 s,in i/r (— sin $ cos -v/r — cos $ sin i/r cos 6) + (7 cos 0 sin 0 cos <£ ................................. . ............ (15)
Accordingly for the intensity
I — A* sin2 6 .cos2 ^ (sin2 $ sin2 ty + cos2 $ cos- \jr cos2 0
— 2 sin </> cos 0 sin i/r cos -^ cos 0)
4- .fi2 sin2 0 sin2 ^ (sin2 <£ cos2 -v/r + cos2 <£ sin2 ^ cos2 0
+ 2 sin <£ cos <£ sin i/r cos -^ cos 0)
4- (72 sin2 0 cos2 0 cos2 <£
— %AB sin2 0 sin -^ cos t/r (sin2 ^ sin ^ cos i/r — cos2 </> sin i/r cos ^ cos2 0
4- sin <£ cos $ sin2 -v//- cos 0 — sin <£ cos (/> cos2 ^ cos 0) 4- 2BC sin2 0 cos 0 sin \|r cos 0 (— sin <^> cos ^ — cos ^ sin ^ cos 0)
— 2(L4 sin2 0 cos 0 cos -^ cos $ (— sin 0 sin ty + cos ^6 cos -v/r cos 0). . . .(16)
In taking the mean with respect to <J3, the terras which are odd in sin (f), or cos<£, disappear, while the mean value of sin2</>, or cos2<£, is £. We get for the mean
/ = | A'2 sin2 0 cos2 ^ (sin2 -^ + cos2 ^ cos2 0) 4- £jB2 sin2 0 sin2 ty (cos2 i/r + sin2 ^ cos2 0)
— AB sin2 0 sin ^ cos -v^ . sin ^ cos ^ sin2 0
— JSC sin2 0 cos 0 sin ^ . sin \|r cos 0
- CA sin2 0 cos 0 cos ^ . cos -^ cos 0 ................... (17)
The averaging with respect to T/T now goes as before, and we obtain | (A2 + £2) sin2 0 (| 4- f cos2 0) 4- \ C2 sin2 0 cos2 0
sin4 0 - %(A + B) C sin2 0 cos2 0 ; . . .(18)on. 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 sin2 6 + C cos2 6, reducing to G simply, if A = G. This is the result for a single particle whose axis is at W. What we are aiming