1918] CLOUD OF SIMILAR SMALL PARTICLES OF ANY SHAPE 543 double (4) for the secondary vibrations parallel to Z, and to add together (2) and (4) for the vibrations parallel to Z. The latter becomes and for the ratio of intensities of the two components V When A = Q, this ratio is one-half. For a more general treatment, which shall include all forms of particle, we must introduce another angle -^ to represent the inclination of WU to ZW Fig- 2' produced, Fig. 2. The direction cosines of either set of axes with respect to the other are given by the formulae* cos X U = — sin </> sin ^ + cos <£ cos ^ cos 6" cos YU ~ cos <£ sin ^ + sin ^> cos ty cos 6 cos ZU = — sin 6 cos ty cos XV =•— sin <f> cos ^ — cos $ sin ^ cos 6 cos YV = cos <£ cos ^ —'sin (/> sin ty cos 0 cos #F = siri d sin -^ cos X W = sin d cos </> \ cos lrW = sin. 0 sin 0 I.....................................(9) cos ZW = cos 0 j Supposing, as before, that the primary vibration is parallel to Z, we have us the first set of factors (10) = — sin. 6 cos -v/r = sinflsin^ = C080 For the vibrations propagated along OF which are parallel to Z, we have the same factors over again with coefficients A, B, G as before, and the vibration is expressed by A sin2 0 cos2 ^ + B sin2 Q sin2 ^ + Ccos20; ............ (11) while for the intensity I == J.2 sin* 0 cos4 ^ + JS2 sin4 0 sin4 ^ + tf2 cos4 0 + 2AB sin4 0 cosa ^ sin2 ^ + 2BG sin2 0 cos8 0 sin2 -f + 204 sin3 0 cos8 (9 cos8 ^ ........................................ (12) * See, for example, Routh's Rigid Dynamics, Part I. § 258, 1897. ^ and 0 are interchanged.ry 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)