1918] ON THE SCATTERING OF LIGHT BY SPHERICAL SHELLS 519
In these equations r denotes the distance between the point (a, ft, 7), where the disturbance is to be estimated, and the element of volume (dx dy dz) of the obstacle. The centre of the sphere R will be taken as the origin of coordinates. It is evident that, so far as the secondary ray is concerned, P depends only on the angle (^) which this ray makes with the primary ray. We will suppose that % = 0 in the direction backwards along the primary ray, and that % = TT along the primary ray continued. The integral in (3) may then be found in the form
p denoting the distance of the point of observation from the centre of the sphere. In the paper of 1914 I showed that the integral in (4) can be simply expressed by circular functions in virtue of a theorem given by Hobson, so that
in in
where m
In (5) the optical quality of the sphere, expressed by (K— 1), is supposed to be uniform throughout. In view of an application presently to be considered, it was desired to obtain the expression for a spherical shell of infinitesimal thickness dR, from which could be derived the value of P for a complete symmetrical sphere whose optical quality varies along the radius. The required result is obtained at once from (5) and (6) by differentiation. We find
dP = - (K - 1) . 4,7rR*dR . ei (nt~k^ . sin m/m, ............ (7)
expressing the value of P for a spherical shell of volume 4nrR*dR. The simplicity of (7) suggested that the reasoning by which it had been arrived at is needlessly indirect, and that a better procedure would be an inverse one, in which (7) was established first, and the result for the complete sphere derived from it by integration. And this anticipation was easily confirmed.
Commencing then with a spherical shell of centre 0 and radius OA equal to R, let xO be the direction of the primary and Op that of the secondary ray (Fig. 1). Draw 0% in the plane of Ox, Op, and bisecting the angle between these lines, and let £ be a coordinate measured from 0 in the direction 0%, so that the plane AOA, perpendicular to 0%, is represented by f= 0. The angle xO% is ^-%, as in our former notation. We have now to consider the phases represented by the factor eik (x~^'] in P. For the point 0, x = 0, r = p, and the exponential factor is e~ikt>. As in the ordinary theory of specular reflection, the same is true for every point in the plane AOA and therefore for the element of surface at A A whose volume is 27rRdRd£. For points in a plane
* Given in the 1881 paper.imary light.