Random Distribution of Luminous Sources. 441 



which finally the imaginary part is to be rejected. The 

 integral over the sphere is easily evaluated, either as it 

 stands, or with introduction of polar coordinates (r, #, &)) 

 which will afterwards be required. Thus with fi written 

 for cos 6, 



ft jV A Ur dy dz = 2tt f* C' e ikr »r 2 dr dfi 



4«7T r E 4?r 



= -y-l sin kr .rdr = -^-(sin&R— &Rcos ffi). . (31) 



Accordingly 



-47rRo^=^ 3 (sinm-fficosffi), . . (32) 



reducing to n simply when JcR is very small. The intensity 

 due to the uniform distribution is thus 



k 6 . 



^(smm-mcos my, .... (33) 



exnctly the n 2 term of (28). The distinction between (28) 

 and (32), at least when &R is very great, has its origin in 

 the circumstance that in the first case the n separate centres, 

 however numerous, are discrete and scattered at random, 

 while in the second case the distribution of the same total is 

 uniform and continuous. 



When we examine more attentively the composition of 

 the velocity-potential (j> in (30), we recognize that it may be 

 regarded as originating at the surface of the sphere R. 

 Along any line parallel to z, the phase varies uniformly, so 

 that every complete cycle occupying a length X contributes 

 nothing. Any contribution which the entire chord may 

 make depends upon the immediate neighbourhood of the 

 ends, where incomplete cycles may stand over. And, since 

 this is true of every chord parallel to z, we may infer that 

 the total depends upon the manner in which the volume 

 terminates, viz. upon the surface. At this rate the n 2 term 

 in (28) must be regarded as due to the surface of the sphere, 

 and if we limit attention to what originates in the interior 

 this term disappears, and (ffi being sufficiently large) (19) 

 reduces to n. 



When we speak of an effect being due to the surface, we 

 can only mean the discontinuity of distribution which occurs 

 there, and the best test is the consideration of what happens 

 when the discontinuity is eased off. Let us then in the 

 integration with respect to r in (31) extend the range beyond 





I 



f 



