464 Prof. Donkiu on certain Questions relating to 



The integral 



(in which Q"~"''j not containing 7;^ may be put outside the sign 



of integration) is a fvmction of /u, and given quantities, suppose 

 <}){fj'), and OT is the sum of the values of (fi{ix) from yLt=0 to fi = m; 

 so that the above expression may be written in the form 



If we were able to calculate the value of this for any proposed 

 ^alue of fx,, we could solve any of the obvious problems connected 

 with the subject ; such as to find the probability that not less 

 than /x out of the m pairs of stars are binary ; or that a proposed 

 paii' is binary, itc. 



13. It remains to indicate the processes which would have to 

 be performed in order to obtain numerical results from the pre- 

 ceding expressions. And first let us consider P" (see art. 13). 



If we knew that k systems were about to be produced, the pro- 

 bability beforehand that /x of them would be binaiy and the rest 

 single, would be the term containing tj/* </*"'' in the development 

 of [p + qY, where q=.\—p. 



If, then, we know that n stars have been produced, without 

 knowing the way in which they are divided into systems, the 

 required probability P" will be proportional, for all values of (i, 



to the term containing jo^g'""^'* in {p + qY~f^ ; and its actual 

 value for an assigned value of fi will be found by dividing this 

 term by the sum of all such tei'ms corresponding to all the posr 

 sible values of fi ; that is, it will be a fraction whose numerator is 



1.2.3... (n-yti) 



,/tiXl.2...(n— 2/i) 



pnqti-in^ 



and denominator the sum of the values of this expression from 

 yLi = to /u.= — or , according as n is even or odd. 



14. The meaning of the expression P", just found, may pcr- 



haps be made clearer by the following problem, of which it would 

 be the solution. A bag contains a nu.mber of balls, some of 

 which are single, and some joined together in couples ; the pro- 

 portion of couples to single balls being such that^; is the pro- 

 bability of drawing a couple in one trial. An unknown number 

 of trials has been made (each drawing being replaced), but it is 

 known that the whole number of balls drawn is n. What is the 



