OF LOGICAL CLASS-FREQUENCIES, ETC. 
105 
and order [m — I) may then l)e written 
77[K,] 77[U - Ljji7[MJ 77[U - NJ -}- 77[U - M,] i7[NJ j < 0 . (I). 
11 1 1 1 J 
All the conditions of consistence, whether inferior or superior, given in the pre¬ 
ceding sections may he readily verified from this general expression.'^ 
§ 13. If the two groups compared be contrary in c terms (c = r s), the expan¬ 
sions will give rise to c terms of the {ni — l)th order, viz., s from the first term, and 
r from the second term within the curly bracket. Thus the conditions VII., § 9, with 
three fourth-order frecpiencies on the left, were all obtained liy comjjarison of 
expansions of fifth-order frequencies contrary in three attributes; the conditions VIII. 
on the other hand by comparison of expansions due to fifth-order frequencies 
contrary in all five attributes. 
The term outside the bracket in (1) may be regarded as a mere specification of 
the universe within which the simple condition 
/7[MJ i7[U - NJ + /7[U - M,] i7[NJ <0.(2), 
11 1 1 
is to hold good. For consider the conditions of the general form (1) in whicli the 
contrary terms (those within the bracket) are the same, but in which the universe- 
terms outside the bracket are contrary as regards one attribute, say K^,. Then tlie 
universes are specified by 
k /'. 77[Iv] . i7[U- L,,]. 
1 1 
[U - KJ ' 77[KJ . /7[U - LJ, 
1 1 
and if the corresponding conditions (1) be added, the term in goes out, leaving a 
condition of one degree lower. By addition of successive pairs of conditions in this 
way it is evident that the universe-terms may be entirely eliminated, and only 
condition (2) left. By the converse process of specification of the universe tlie 
conditions involving c terms of tlie {in — l)th order may always be obtained from 
the conditions for the congruence of the cth degree and (c — l)th order, a projierty 
on wliich we have remarked while considering the congruences of low orders inves¬ 
tigated in § 4—§ 10 ; the conditions (2) merely require to be specified for all possible 
universes. 
If instead of proceeding to the entire elimination of the uni verse-terms we stop 
short at conditions of degree n {n < c), the whole series of conditions so olitained 
may be grouped into sets arranged according to the attriliutes that have not been 
* They were actually fir.st ohtained by the method there described. 
VOL. CXCVII.—A. 
P 
