﻿260 Dr. L. Silberstein on a Quantum 



following reasoning. The total area of silver halide being 

 a fraction Na of the area S of the plate, each of the nJSa 

 light-quanta will fall upon some grain. Of these nNa 



quanta one, say the first, will hit one grain, the next ^ 



N . 

 quanta will fall upon another grain, the next _ will hit 



yet another, and so on, up to ^ 7 _ , .. quanta for the Irth 



grain *, Thus the required relation between k and n 

 will be 



_1 1 1 l_ 



an ~ JX + N-l "" + N-k + 2 + N-k + r 



Now by a well-known theorem of Analysis 



J + i + l+ •••• +-=0 + logm + e(m), . . (2) 

 JL o nx 



where C is Euler's constant and 0<e(«i) < 1/m. Thus in 

 our case 



lo gyZTk= an + £ 



where f lies between — 1/A 7 and l/(J¥—k) or practically, 

 since iVis at any rate a large number, 



e<?= £ (ir-*)<^. 



Whence, 



fc=JV(l— *-**-£) (3) 



In practicable experimental tests (counts of affected grains 

 of a given, narrow size-class) the role of the correction f 

 whose value f=e(iV r — k) can at any time be found by (2), 

 may become perceptible only when the contemplated grain 

 class is near exhaustion. 



Thus, apart from such extreme cases, we have again, 

 as in (A), the simple formula 



k = A\l-e- na ), (4) 



which, though approximate only, will turn out to be 

 accurate enough even for moderate values of N. 



A thoroughly rigorous treatment of the probability 

 problem valid for any numbers iV, ?i, seems to be the 



* It is scarcely necessary to say that statements such as U N/N— 1 

 quanta hit another grain " are to be taken statistically as relating to 

 averages over a group of many trials. 



