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578                       •   • -   -   ON THE LIQHT EMITTED  PROM A        - •                       [436
radial velocity, we find that when the radii are small enough, the obstacle acts as a source whose strength is independent of neighbours.
The operation of a cloud of similar particles may now be deduced without 'much difficulty from what has already been prpved. We suppose that the individual particles are so small that 'the cloud has no sensible effect upon the progress of the primary waves. Each particle then acts as a source of given strength. But the initial phase for the various particles is not constant, being dependent upon the situation along the primary rays. This is, in fact, the only new feature of which we have to take account.
Perhaps the most important difference thence arising is that there is no longer equality of radiation in various directions, even from a spherical cloud, and that, whatever may be the shape of the cloud, the radiation in the direction of the primary rays produced is specially favoured. In this direction any retardation along the primary ray is exactly compensated by a corresponding acceleration along the secondary ray, so that oil arrival at a distant point the phases due to all parts are the same. But, except in this direction and in others approximating to it, the argument that the effect may be attributed to the surface still applies. If in a continuous uniform distribution we take chords in the direction, for example, of either the incident or the scattered rays, we see as before that the effect of any chord depends entirely on how it terminates*. In. forming an integral analogous to that of (30), in addition to the factor eikg expressive of retardation along the secondary ray, we must include another in respect of the primary ray. If the direction cosines of the latter be a, IB, 7, the factor in question is eik ("x+Pv+yz^ j being — 1 when the directions of the primary and secondary rays are the same. The complete exponent in the phase-factor is thus
The fraction on the right represents merely a new coordinate (£), measured in a direction bisecting the angle between the primary and secondary rays, so that the phase-factor may be written el v {2+2^ • *£, 7 being the cosine, of the angl.e (%) between the rays. In integrating for the sphere the only change required in the integrand is the substitution of %k cos |% for lc. With this alteration equations (31), (32), (33) are still applicable. When the secondary ray is perpendicular to the primary,
•      .....             . 2/ccos-|-% = v/2.&.
In order to find the mean intensity in all directions we have to integrate (33) over angular space and divide the result by 4?r.    It may be remarked
* It may be remarked that the same argument applies to the particles of a crystal forming a regular space lattice. If the wave-length be large in comparison with the molecular distance, no light can be scattered fr«m the .interior of such a body. For X rays this condition is not satisfied, and regular reflexions from the interior are'possible. Comparison may be made with the behaviour of a grating referred to below. ......                              ........           • - 'ature, Vol. L. p. 223 (1894).918).stadt's.