CHAP. XIX.] OF STATISTICAL CONDITIONS. 317 



/ 



The minor limit of Prob. x l will be identical with the minor 

 limit of the expression 



81 - 8 l S 2 . . S n + (1 - S^ (1 - S 2 ) (1 - S n ). 



A little attention will show that the different aggregates, 

 terms which can be formed out of the above, each including the 

 greatest possible number of constituents, will be the following, 



*! (1 - S 2 ), 5i (1 - S 3 ), . . Si (1 - ), (1 - * 2 ) (1 - * 3 ) . . (1 - *). 



From these we deduce the following expressions for the minor 

 limit, viz. : 



Pi-p*, Pi -Pz PI -Pn, l-p z ~p z . . -p n . 



The value of Prob. x l will, therefore, not fall short of any of 

 these values, nor exceed the value of pi . 



Instead, however, of employing these conditions, we may 

 directly avail ourselves of the" principle stated in the demon- 

 stration of the general method in probabilities. The condition 

 that Si, 2 , . .s n must each be less than unity, requires that X 



should be less than each of the quantities , , . . . And 



Pi P* Pn 



the condition that s l9 s 2 $n, must each be greater than 0, re- 

 quires that X should also be greater than 0. Now pi p z . . p n 

 -being proper fractions satisfying the condition 



Pi + j2 + Pn > 1, 



it may be shown that but one positive value of X can be deduced 

 from the central equation (10) which shall be less than each of 



the quantities , , . . . That value of X is, therefore, the 



Pi P* Pn 

 one required. 



To prove this, let us consider the equation 



(1 - Pl \) (1 -/> 2 X) (1 - jp.X) - 1 + X = 0. 



When X = the first member vanishes, and the equation is 

 satisfied. Let us examine the variations of the first member 



between the limits X = and X = , supposing//! the greatest of 



PI 

 the values pi p 2 . . p n . 



