in the Theory of Probability. 751 



Hence if X B =1, X„_i<l, we have 



i+ n - 



v 



m 1 n(n — l)^ ()m 



.'. v>n(n — 1). 

 On the other hand, if we assume 



-^n =z 1} X„ + 1 >1, 



the equation 



(x^-±l)(x-3 =1 



gives 



v<?i(n + l). 



This suggests that the value of v for which X n (y) = l lies 

 between n(n — 1) and n(n-\-l). 



By using Lodgers tables * for the functions l r (v) I have 

 been able to make a rough estimate of the position of the 

 roots of X 1 (v) = l and X 2 (v) = l. The values found are 



1-58 for X 1 (v) = l 9 



4-58 for X s (v) = l. 



The ratio of the two numbers is thus about three. 



It was thought that the ratio might turn out to be the 

 ratio of the atomic weights of two elements such as hydrogen 

 and helium, whose atoms sometimes carry charges of one and 

 two units respectively. This idea was based on the supposition 

 that the average number of particles required for the for- 

 mation of a particular atom may be such as to make the chance 

 of getting a given valency charge as large as possible f. The 

 idea need not be altogether abandoned because we have left out 

 of account the occurrence of neutral particles (e. g. doublets) 

 and particles carrying more than one unit charge. This 

 brings us to the consideration of a more general problem. 



* British Association Reports (1880). To solve X r (*0=l we must 

 find a value of v such that Ir=|(Ir-i+tr+l). The tables do not go far 

 enough to enable me to carry the calculations any further. 



f It should be noticed, however, that the probable number of particles 

 v is equal to the probable value of /•-. 



