﻿262 Dr. L. Silberstein on a Quantum 



By formula (5) we have for instance the probability of 

 hitting but one grain 



Pm\ Ij = jy^-l -> 



which is an obvious result. The probability of hitting two 

 grains will be 



i?m (2) = (2— i-l)(j\r-l) . jy*- 1 , 



which is (approximately) 2 m N times as large as the preceding 

 probability, and so on. If m be kept constant, p m (i) will 

 increase with growing i up to its largest value for i = m, 

 if ra<iV, or for i — N, if m > N. But it would be futile to 

 expect, on an average, that distribution to which corresponds 

 the greatest probability *. For all other distributions have 

 some generally non-negligible probabilities and these are by 

 no means symmetrically spaced with respect to the largest 

 one. The only reasonable way of determining the number k 

 is to define it as the average of i taken over a large number 

 M of trials. Out of these M trials a number M p m (i) t of 

 trials will give each i grains affected, and the total number 

 of grains affected in all M trials will be XM^p m (^), to be 

 summed from i = l to i = m, if m-<N and to i — N\i m >iV r ; 

 but since a m i = for i > m, we can as well extend the sum 

 in each case from i=l to i = N. 



Dividing this sum by M we shall have the average number 

 of grains hit in one trial, i. e., by (5) 



7Vf N in - 



This, with m — Nan, is the required rigorous formula for 

 the number of grains affected, i. e., hit once or more. In 

 order to see how this complicated formula degenerates into 

 (4), which, of course, will be our working formula, develop 

 the sum in (6). Collecting the terms in i^ -1 , N~ 2 etc., and 

 taking- account of the values of a m ,\ it will be found that 



l mij 



, 1 (m\ 1 fm\ ( 1 V" -1 



* If, say, ?n<:N, the most probable distribution is the equipartition 



(a quantum per grain), corresponding- to p m (m)= T^ m /j^^_ \ t ■ This 



would give as the number of grains affected k = m = Nna, or just the 

 first term in the series development of (4), which would be hopelessly 

 wrong- unless mN were very small. 



t "With a deviation ruled by Bernoulli's law. 



