﻿Theory of Photographic Exposure. 201 



following one. Of the n quanta the total number falling 

 each upon some of the N grains will be * 



m = Na n, 



JS r a being the total area of silver halide, with S as unit area. 

 Thus the problem is reduced to finding the distribution of m 

 quanta among JS 7 grains. Now, let p m (i) be the probability 

 of affecting, in a single trial, a number i of the JV grains, by 

 the m quanta. By a combinatorial discussion which may 

 be omitted here, I find 



P»W = jy»(]y-i)V ( } 



where a mi may be most shortly described as identical with 

 the number of ways in which a product of m different 

 primes can be decomposed into i factors. These numbers, 

 which will be known to many readers from combinatorial 

 algebra, have the obvious properties 



a m i=a mm —l, for any m, 



a mi = for i > m, 

 and satisfy the general recurrency formula 



which enables us to write down successively without trouble 

 any number of them. 



Thus, up to wi = 10, we have the following- table which the reader 

 may continue to extend at his leisure. Columns correspond to con- 

 stant m, and rows to constant i. 



1111111 1 1 1 



1 3 7 15 31 63 127 255 5 LI 



1 6 25 90 301 966 3025 9330 



1 10 65 350 1701 7770 34105 



1 15 140 2401 13706 76300 



1 21 266 3997 37688 



1 28 462 7231 



1 36 750 



1 45 



1 



We may mention in passing that any a m i can be represented by f 



«»••= n D'"'- (i) {i ~ 1)m+ (») ('- 2 >*"- ± (!)]■ 



But for any numerical applications the table will be found more 

 convenient. 



* In a large number of trials of the same experiment. 

 t Cf. e. g. Lehrbuch d. Kombinatorik by E. Netto, Leipsig, 1901, 

 p. 170. 



