94 
MR. G. UDNY YULE OX THE THEORY OF COXSISTEXCE 
of attributes specified, m, will Ije termed the degree of the congruence, its order 
being the order of the component aggregates.^ 
A congruence of degree m and order n may be regarded as built up of a series 
of congruences of degree lower than m. Thus the congruence of the fourth degree 
and second order containing the aggregates AB, AC, AD, BC, BD, CD, may be split 
up into four congruences of the third degree oiily, viz., those containing the 
aggregates (AB, AC, BC^) (AB, AD, BD) (AC, AD, CD) and (BC, BD, CD) 
respectively. In general, any congruence of the ??ith degree may be divided into 
m (ni - 1) . . . (ill — r + 1) 
r I 
congruences of the rth degree, one to each positive group of the rth order that can be 
formed from the m attributes. These component congruences evidently, as in the 
above example, overlap : any one aggregate occurs in two or more congruences. 
In the following sections (§ 4-§ 10) 1 proceed to the discussion of congruences 
of the third, fourth, and fifth degrees ; §§ 11-13 deal with the general case, and the 
remainder of the jraper consists of a discussion of the geometrical representation, by 
means of polyhedra, of the conditions of consistence, together with a few numerical 
illustrations. 
Congruence of the Third Degree. 
§ 4. In terms of the definition § 2, p. 93, the inferior conditions of consistence are 
given at once by expanding all second-oi'der frequencies in terms of the positive 
groups only, and putting the resulting expression < 0. Thus, retaining for con¬ 
venience the second-order terms only on the left of the inequality, we must have 
(AB) < 0 
or 
(AB) will he negative 
< (A) -h (B) - (U) 
or 
(a^) 
> (A) 
or 
(A/3) „ 
> (B) 
or 
(aB) „ 
Similar conditions hold of course for (AC) and (BC). 
§ 5. To find the superior conditions of consistence {cf. again the definition), write 
down the inferior conditions for the (ABC) aggregate. These are of course given 
similarly by expanding the third-order frequencies in terms of the positive groups, 
and putting the resulting expansion < 0. Thus— 
* In any S3^stematic tabulation of frequencies I think the grouping sliould be made Iw aggregates. 
Such an arrangement would be distinctfy better than that adopted by me in the sample table on p. 318 
of my paper on “ Association.’’ he. cit. 
