Random Distribution of Luminous Sources. 445 



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 on 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 eutirely on how it terminates*. In forming an 

 integral analogous to that of (30), in addition to the factor 

 e ikz 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, /3, 7, the factor in ques- 

 tion is e ik ^ ax+ ^ + y z \ 7 being —1 when the directions of the 

 primary and secondary rays are the same. The complete 

 exponent in the phase-factor is thus 



= iV(2+27) "H-Av + 0*+l> 



v/{* 2 +/3 2 + (Y+in' 



The fraction on the right represents merely a new co- 

 ordinate (£*), measured in a direction bisecting the angle 

 between the primary and secondary rays, so that the phase- 

 factor may be written g l V(2+2y).-^ y being the cosine of the 

 angle (^) between the rays. In integrating for the sphere 

 the only change required in the integrand is the substitution 

 of 2k cos ^x ? or ^- With this alteration equations (31), 

 (32), (33) are still applicable. When the secondary ray is 

 perpendicular to the primary, 



2k cos ±x = \/2-k. 



Phil Mag. S. 6. Vol. 36. No. 216. Dec. 1918. 2 H 



|P 



In order to find the mean intensity in all directions we 

 have to integrate (33) over angular space and divide the 

 result by 47T. It may be remarked that although cos 6 -^ 

 appears in the denominator of (33), it is compensated when 



* 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 

 from the interior of such a body. For X rays this condition is not 

 satisfied, and regular reflexions from the interior are possible. Com- 

 parison may be made with the behaviour of a grating referred to below. 



\ 



