Random Distribution of Luminous Sources. 449 



# l5 2 . . . . n -i already fixed. If 5 = 0, every term such as 



... cos (0 T -0 a )d0 1 .... d0 n -i- ii( ^, .... d0„ 



= ( fcOS^r-^^cr^r-f-rr^^r 



^o-fsin (2?r — o-) + sin o-}-4-4tt 2 = 0, 



|' 2 



.Jo 



nndfthe expectation is n simply, as we have already seen. 

 In the next approximation the correction to n will be of 

 order 6\ and we neglect 8 2 . 



In evaluating (40) there are \n(n— 1) terms under the 

 sign of summation, but these are all equal, since there is 

 really nothing to distinguish one pair from another. If we 

 put cr=l, t = 2, we have to consider 



Jff cos (0 2 -0 1 )d0 1 d0 2 dd n 



+ §§§.... d0 1 d0 a ....d0 u , . (43) 



The integration with respect to 6 n extends over the range 

 from to 2tt with avoidance of the neighbourhood of 

 #!, 2 ,'... 6 n -i> For each of these there is usually a range 

 28 to be omitted, but this does not apply when any of them 

 happen to be too near the ends of the range or too near one 

 another. This complication, however, may be neglected in 

 the present approximation. Then 



jcos (0 2 -0 1 )d0 n =cos (0,- 0i). {27r-2S(n~l)}, 



and in like manner 



§d0 n = 27r-28(n-l) i 



so that this factor disappears. Continuing the process, we 

 get approximately 



jjcos (0 2 -0 1 )d0 l d0 2 +§\d0 x d0 2 , 



as when there were only two phases to be regarded. 



Accordingly, the expectation of intensity for n phases is 



n{l— (n-l)8/w}, (44) 



less than when S = 0, as was to be expected, since the cases 

 excluded are specially favourable. But in order that this 

 formula may be applicable, not merely 8, but also 7iS, must 

 be small relatively to 2tt. 



A similar calculation is admissible when the whole range 

 is}2ra7r, instead of 27r, where m is an integer. 



Terling' Place, Witham. 

 Nov. 1. 



