CHAP. XIX.] OF STATISTICAL CONDITIONS. 305 



by that constituent are found in w. It is evident that n (10) will 

 have its highest numerical value when all the members of the 

 class denoted by (1 - s) (1 -t) are found in w. Moreover, as 

 none of the individuals contained in the classes denoted by 

 s (1 - 1) and (1 - s) t are found in w> the superior numerical limits 

 of 10 will be identical with those of the class st + (1 - s) (1 - t). 

 They are, therefore, 



ns + n (I - t) and nt + n (1 - s). 



In like manner a system of superior numerical limits of the 

 development A + QB + - C + - D, may be found from those of 

 A + Cby Prop. 2. 



Again, any minor numerical limit of w will, by Principle n., 

 be given by the expression 



n (1) - major limit of n (1 - w) 9 



but the development of w being given by (1), that of 1 - w will 

 obviously be 



l-w = OA + B + ^C + i Z>. 



This may be directly proved by the method of Prop. 2, Chap. x. 

 Hence 



Minor limit of n(w) = n(l) - major limit (B + C) 



= minor limit of (J. + JD), 



by Principle n., since the classes A + D and B + C are supple- 

 mentary. Thus the minor limit of the second member of (2) 

 would be n (t) 9 and, generalizing this mode of reasoning, we have 

 the following result : 



A system of minor limits of the development 

 A + QB + .C + ^D 



will be given by the minor limits of A + D. 



This result may also be directly inferred. For of minor nu- 

 merical limits we are bound to seek the greatest. Now we ob- 

 tain inj general a higher minor limit by connecting the class D 



x 



