512 COLOURS DIFFUSELY REFLECTED FROM SOME COLLODION FILMS [424
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 suppose that % = 0 in the direction backwards along the primary ray, and that % = TT along the primary ray continued. Then with introduction of the value of h0 from (1),
we find
p AJT. 4>7rR3 . ei(nt~kr] ^sin m cosw\
.- K lltf W~)' ............ (5)
where m = 2&./t!cos^ ............................... (6)
The secondary disturbance vanishes with P, viz. when tan m = m, and on these lines the formation of colour may be understood. Some further particulars are given in the paper just referred to.
The solution here expressed may be applied to illustrate the scattering of light by a series of equal spheres distributed at random over a plane perpendicular to the direction of primary propagation. The effect of a reflector will be represented by taking, instead of (1),
/io = eint (eite + g-fc!*+»«>), ........................ (7)*
#0 expressing the distance between the plane of the reflector and that containing the centres of the spheres. The only difference is that
m~s sin m — m~2 cos m is now replaced by
im/ cosm'
m
where m is as before, and m' = 2kR sin |p^. In the special case where, while the incidence is perpendicular, the scattered light is nearly grazing, ^ = |TT, sm i% = cos %X — 1/V2, and m = m' = \/2 .kR; so that (8) becomes
This vanishes if cos 2&»0* = — 1 ; otherwise the reflector merely introduces a constant factor, not affecting the character of the scattering. At other, angles the reflector causes more complication on account of the different values of m and m'.
[* The results given in the original text have been corrected by the substitution of - 2xo for a'o . It is assumed, as apparently in the original, that no change of phase occurs at the reflector. W. F. S.]mainly upon the variation of P, strongly marked effects require an approximate uniformity. If the distribution be at random, the colours due to a large number may then be inferred from, the calculation relating to a single obstacle ; but if the distribution were in regular patterns, complications would ensue from the necessity for taking phases into account, as in the theory of gratings. For the present purpose it suffices to consider a random •distribution, although we may suppose that the centres, or more generally •corresponding points, of the obstacles lie in a plane perpendicular to the •direction of the primary light.