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1917]                      REFLECTED   FROM.  SOME  COLLODION  FILMS                         511
susceptibility is supposed to be uniform throughout ; the specific inductive capacity to be K, altered within the obstacle to K + AAr. The suffixes 0 and 1 refer respectively to the primary and scattered waves. The direction of propagation being supposed parallel to x and that of vibration parallel to z, we have /0 = g0 = 0, and
h0 = einteilcsc,  ................................. (1)
eint being the time factor for simple progressive waves. For the scattered vibration at the point (a, ft, 7) distant r from the element of volume (dxdydz) of the obstacle, we have
fi, 9i, AI =         [ay, £7, -(a2 + /32)}, .................. (2)
where                           P = -- ~-       h^e~'ikrdxdydz,    ..................... (3)
and the integration is over the volume of the obstacle. If the obstacle is very small in comparison with the wave-length (X) of the vibrations, h0e~ikr may be removed from under the integral sign and
T A/T It P~ikr
P —         "*•   '           ' ' °6                                                          fA.\
1  - -- jf - > ........................... (4)
T denoting the volume of the obstacle. In the direction of primary vibration a = y3= 0, so that in this direction there is no scattered vibration. It will be understood that our suppositions correspond to primary light already polarized. If, as usually in experiment, the primary light is unpolarized, the light scattered perpendicularly to the incident rays is plane polarized and can be extinguished with a nicol.
The formation of colour depends upon other factors. When the obstacle is very small, P is constant, and the secondary vibration varies as &2, so that the intensity is as the inverse fourth power of the wave-length, as in the theory of the blue of the sky. In this case ifc is immaterial whether the obstacles are of the same size or not, but for larger sizes when the colour depends 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.
When the obstacle is a sphere, the integral in (3) can be evaluated*. The •centre of the sphere, of radius R, is taken as the origin of coordinates. It is
* Proc. Roy. Soc. A, Vol. xo. p. 219 (1914).    [This volume, p. 220.] h are the electric displacements. The magnetic