﻿Characteristic Curve of a Photo<jrapliic Emulsion. 357 



When the a grains in the volume V of silver halide are 

 subjected to an intensity I t , every grain which happens to 

 have at least one of these N : nuclei will be made developable. 

 We have, therefore, to find the chance of a grain containing 

 at least one of the N\ nuclei when they are distributed hap- 

 hazard amongst a grains. This can easily be obtained from 

 the theory of probability. 



If p denotes the very small probability that an event will 

 happen on a single trial, the probability P r that it will happen 

 r times in a very great number, say n trials, is (Mellor, 

 1 Higher Mathematics/ p. 502) 



V r =(n P y.e-»P/rl (3) 



Let the volume of a single grain be v, then since the volume 

 of every grain is the same the total volume V is an. Let p 

 be the very small probability that a volume dv will contain a 

 nucleus, then 



p = ^ 1 .dvlav (4) 



To obtain the probability of the volume v containing a nucleus, 

 we may suppose each dv to be a trial, so that the number of 

 trials n is 



n = vjdv (5) 



Therefore the value of np in equation (3) is Nj/a, which is 

 equal to N . If in this number of trials the event (i. e. v 

 containing a nucleus) happens once, a grain will contain one 

 nucleus; if it happens r times it will contain r nuclei, so 

 that from (3), (4), and (5) we see that the probability of a 

 grain containing r nuclei is 



p r =(N y*-^/H, (6) 



which is the same equation as was obtained independently 

 and first published by Svedberg. The probability of a grain 

 containing no nuclei is the value of this expression when 

 r = ; i. e. 



Now, since it is certain that a grain must contain either zero 

 or at least one nucleus, the probability P x that a grain will 

 have at least one is 



P 1 = 1_,-n (7) 



