1914] . DIFFRACTION OF LIGHT BY SPHERES OF SMALL RELATIVE INDEX 221
to throw some light upon the general course of the phenomenon. It has already been treated up to a certain point, both in the paper cited and the earlier one* in which experiments upon precipitated sulphur were first described. It is now proposed to develop the matter further.
The specific inductive capacity of the general medium being unity, that of the sphere of radius R is supposed to be K, where K — 1 is very small. Denoting electric displacements by /, g, h, the primary wave is taken to be
so that the direction of propagation is along co (negatively), and that of vibration parallel to z. The electric displacements (/1; yl} 7^) in the scattered wave, so far as they depend upon the first power of (K — 1), have at a great distance the values
in which P = - (K - 1) . Qint \\\eik (x~r] dx dy dz ................... (3)
In these equations r denotes the distance between the point (a, /9, 7) where the disturbance is required to be estimated, and the element of volume (dxdydz) 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 upon 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) rnay then be found in the form.
cos
f^71"
</! (ZIcR cos AY . cos 6) cos2 d> d6, ............ (4)
Jo v ^ r/ r r v '
r now denoting the distance of the point of observation from the centre of the sphere. Expanding the Bessel's function, we get
__ (K - 1) g*(»«-
JL — ~~™ f\ ' "\ X ~~ rt — *T~
2.5^2.4.5.7 2.4.6.5.7.9
m8
2 : 4^678757779711
in which m is written for 27dS cos fa. It is to be observed that in this solution there is no limitation upon the value of R if (K — I)2 is neglected absolutely. In practice it will suffice that (K— 1) R/\ be small, X (equal to 2?r/&) being the wave-length.
* Phil. Mag. Vol. xir. p. 81 (1881) ; Scientific Papers, Vol. I. p. 518.
kDn' (ka) _ hJn' (ha)