! 



m 

 so that (5) becomes 



ej m fg-W^-DK 



2^)1—^— f (6) 



I 



Random Distribution of Luminous Sources. 431 



phase are combined, what precisely is the significance to be 

 attached to this result ? As has already been suggested, we 

 must look to what is likely to happen when we have to do 

 with a large number m of independent trials, in each of which 

 the n phases are redistributed at random. By (1) the chance 

 of the separate intensities I 1? I 3 , . . . I m > lying between Ij + ^Iir 

 I 2 + dI 2 , &c is ■ 



w -^-di+i 2 +...)/^T 1 ^i 2 .,.^i wl . 



and we may inquire what is altogether the chance of the 

 sum of intensities, represented by J, lying between J and 

 J + e?J. Over the range concerned the factor e~ J/n may be 

 treated as constant, and so the question is reduced to finding 

 the value of 



f T . . . . dli dl 2 .... dl m 



under the condition that I x + I 2 + . . . . lies between J and 

 J + dJ. This is* 



TWl-1 



( TTT^ 5 



(m — 1) I 



so that the chance of l! + I 2 -h. ... lying between J and 

 J -\-dJ is 



g -J/nJm-l^J 



n m .(m-l)! ; [) 



or, if we employ the mean value of the I's instead of the 

 sum, the chance of the mean, viz. (I x + 1 2 + . . . .)/w, lying 

 between K and K-f t?K is 



n" l .m ! ^ ' 



We may compare this with the corresponding expression 

 when m=l, where we have to do with a single I, to which K 

 then reduces. The ratio 



R= W : w= — ^r^i — . • • • (5) 



When we treat m as very large, we may take 

 \=m m </(2irm).e- m , 



See for example Todhunter's Int. Calc. § 272. 

 2G 2 



