CHAP. XIX.] OF STATISTICAL CONDITIONS. 303 



xyz + x (1 - y) z + (1 - x) yz + (1 - x) (1 - y) z 



we should obtain by addition of the first four terms the single 

 term z 9 by addition of the first and fifth term the single term #y, 

 and by addition of the fourth and sixth terms the single term 

 (1 - x) (1 - y) ; and there is no other way in which constituents 

 can be collected into single terms, nor are there are any consti- 

 tuents left which have not been thus taken account of. The 

 three resulting terms give, as the minor limits of the given ex- 

 pression, the values 



n(z\ n(x) + n(y) -n(l), 

 and n (1 - x) + n (1 - y) - n (1), or n (1) - n (x) - n (y). 



8. The proof of the above rule consists in the proper appli- 

 cation of the following principles : 1st. The minor limit of any 

 collection of constituents which admit of being added into a sin- 

 gle term, will obviously be' the minor limit of that single term. 

 This explains the first part of the rule. 2nd. The minor limit 

 of the sum of any two terms which either are distinct constituents, 

 or consist of distinct constituents, but do not admit of being 

 added together, will be the sum of their respective minor limits, 

 if those minor limits are both positive; but if one be positive, and 

 the other negative, it will be equal to the positive minor limit 

 alone. For if the negative one were added, the value of the limit 

 would be diminished, i. e. it would be less for the sum of two 

 terms than for a single term. Now whenever two constituents 

 differ in more than one factor, so as not to admit of being added 

 together, the minor limits of the two cannot be both positive. 

 Thus let the terms be xyz and ( 1 - x) ( 1 - y) z, which differ in 

 two factors, the minor limit of the first is n (x + y + z - 2), that 

 of the second n (1 - x + 1 - y + z - 2), or, 



1st. n{x + y- I -(1 -z)}. 2nd. n (I - x - y - (1-*)}. 



If n (x + y - 1) is positive, n (1 - x - y} is negative, and the se- 

 cond must be negative. If n (x + y - 1 ) is negative, the first is 

 negative; and similarly for cases in which a larger number of 

 factors are involved. It may in this manner be shown that, ac- 

 cording to the mode in which the aggregate terms are formed in 



