346 On Harmonic Ratios in the Spectra of Gases. [Jan. 27, 



bility that the sum of all the quantities, a, shall lie between st and 

 st + dst, then the whole expectancy is 



23* r 



ptstdst. 



The integral represents the expectancy for the sum of t quantities all 

 equally probable between - and A, and this expectancy we know to 



a, 



be— 



IK) 



Hence the required expression is — 



A« + l 



rtX 



1 



In the actual case neither r nor t are given, we only know their 

 sum n ; hence we must add up a number of expressions of the form 

 we have found, varying r and t, and multiplying each with the proba- 

 bility that the particular distribution actually exists. 



The probability that there should be t out of n values in the first 

 compartment is — 



nl («A-l)*(l-«)' 

 t \ (n-ty. a »(A-l)« ' 



giving t successively all values from 1 to n— 1 we find for the whole 

 expectancy — 



.Aa + l 1 t-n-\ t(n— t)n\ , i -i wi \ v -t 



C — ■ V 1 (oiA — l) t (I — a.) nt , 



(l-oc) ^(A-iyzi ti (n-t) r Jy J 



and adding up under the summation sign, the expression reduces to — 



n.n—1 AV-1 g 

 (A -1)2 ^ 



which is the complete expectancy. 



We have assumed that A is not larger than the reciprocal of the 

 square of «, and we may now extend the formula to larger values 

 of A. 



Imagine a quantity B smaller than and larger than — , and let 



. a 2 at, 



33 



A gradually increase from B to — . Divide the whole range A to 1 into 



a. 



two compartments, one from A to B and the second from B to 1, then 

 if a given number of quantities is in each compartment, I can calcu- 

 late the whole expectancy by knowing : — 



