Random Distribution of Luminous Sources. 



437 



remains constant in the various trials for each pair, and we 

 have to deal with the mean value of sinAD-f-AJ) when the 

 spatial distribution is at random. We may begin by sup- 

 posing two sources constrained to lie upon a straight line of 

 limited length Z, where, however, I includes a very large 

 number of wave-lengths (X). 



If the first source occupies a position sufficiently remote 

 from the ends of the line, so that the two parts on either 

 side (/x and l 2 ) are large multiples of X, the mean required, 

 represented by 



h ( '■ sin IcD dT) U T^ sin kV dD 

 /J kD U J J kD l 2 ' ' 



- (21) 



may be identified with ir/kl, since both upper limits may be 

 treuted as infinite. Moreover, 7rjkl may be regarded as 

 evanescent, kl being by supposition a large quantity. 



So far positions of the first source near the ends of the 

 line have been excluded. If the neglect of these positions 

 can be justified, (20) reduces to 2 simply. 



It is not difficult to see that the suggested simplification 

 is admissible under the conditions contemplated. If a?, x' be 

 the distances of the two sources from one end of the line, 

 the question is as to the value of 



dx{ l dx' sin k(x' — x) 

 T k(x'-x) 



C l dxL l aV 



Jo I Jo ' 



(22) 



where the integration with respect to ce' may be taken first. 

 Let X denote a length large in comparison with X, but at 

 the same time small in comparison with I. If x lie between 

 X and I — X, the integral with respect to x' may be identified 

 with irjkl, and neglected, as we have seen. We have still 

 to include the ranges from # = to # = X, and from x — l — X 

 to x = l, of which it suffices to consider the former. The 

 range for x' may be divided into two parts, from to x, and 

 from x to /. For the latter we may take 



C 



l dx' sin k(x f — x) 

 ~T k(x'-x) 



IT 



2kV 



so that this part yields finally after integration with respect 

 to x, 



X 



7T 



/ ' 2kl ' 



(23) 





