302 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



n (x) + n (1 - y\ and n (x) + n (1 - z), 

 or n (#) + n (1) - TZ (?/), and n(x) + n (1) - H (z). 



If we began with n (y), selected from the first term, and took 

 n (x) from the second, we should have to take n (1 - y) from the 

 third term, and this would give 



n (y) + n (x) + n (1 - y), or w (1) + n (as). 



But as this result exceeds n (1), which is an obvious major limit 

 to every class, it need not be taken into account. 



PROPOSITION III. 



7. To find the minor numerical limit of any class expressed by 

 constituents of the symbols #, y, z 9 having given n(x), n(y), n(z) .. 



-CO- 



This object may be effected by the application of the pre- 

 ceding Proposition, combined with Principle n., but it is better 

 effected by the following method : 



Let any two constituents, which differ from one another only 

 by a single factor, be added, so as to form a single class term 

 as x ( 1 - y) + xy form #, and this species of aggregation having 

 been carried on as far as possible, i. e., there having been selected 

 out of the given series of constituents as many sums of this kind 

 as can be formed, each such sum comprising as many constituents 

 as can be collected into a single term, without regarding whether 

 any of the said constituents enter into the composition of other 

 terms, let these ultimate aggregates, together with those con- 

 stituents which do not admit of being thus added together, be 

 written down as distinct terms. Then the several minor limits 

 of those terms, deduced by Prop. I., will be the minor limits of 

 the expression given, and one only of those minor limits will at 

 the same time be positive. 



Thus from the expression xy + (1 x)y + (1 - x) (1 - y) we 

 can form the aggregates y and 1 - a?, by respectively adding the 

 first and second terms together, and the second and third. 

 Hence n (y) and n(l - x) will be the minor limits of the expres- 

 sion given. Again, if the expression given were 



