I 



+1 



cos (ej — € 2 + kD/ju)djjL, 



that is 



2 sin&D cos (€ X — e 2 ) 



(18) 



Random Distribution of Luminous Sources. 435 



/, m, ft being the direction cosines of OQ. On the whole 



— ±7rR Q <f> = % cos {pt + €-kR Q + k(Lv + my + nz)}, . (15) 



in which R is a constant for all the sources, but e, x, y, z 

 vary from one source to another. The intensity in the 

 direction (/, m, n) is thus represented by 



l2cos{e + k(lx + my + nz)}] 2 +[2,$m\e + k(lx + my + nz)\] 2 , 



or by 



w + 2Scos[e 1 — € 2 + k{l(x l -x 2 ) + m(y 1 —y 2 ) +n(z 1 — z 2 )}], (16) 



the second summation being for all the \n(n — 1) pairs of 

 sources. In order to find the work done we have now to 

 integrate (16) over angular space. 



It will suffice if we effect the integration for the specimen 

 term ; and we shall do this most easily if we take the line 

 through the points (a? 1? y u z x ) , (# 2 , y 2 , z 2 ) as axis of reference, 

 the distance between them being denoted by D. If (/, m, n) 

 make an angle with D whose cosine is fi, 



~Dfi = l(x 1 — x 2 ) + m{y 1 -y 2 ) + n(z l -z 2 ), . (17) * 

 and the value of the specimen term is 



kJ) .... 



The mean value of (16) over angular space is thus 



oV sin £D cos (e] — € 2 ) , 



n+ 2 kv ■ ■'■••• ( 19 ) 



where e x , e 2 refer to any pair of sources and D denotes the 

 distance between them. If all the sources are in the same 

 initial phase, cos (e! — e 2 ) = l. If the distance between every 

 pair of sources is a multiple of ^X, sinZ:D = 0, and (19) 

 reduces to its first term. 



We fall back upon a former particular case if we suppose 

 that there are only two sources and that they are in the same 

 phase. 



* In the paper referred to, equation (19), (a was inadvertently used in 

 two senses. 



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