OF LOGICAL CLASS-FREQUENCIES, ETC. 
103 
getting rifl of the negative terms by expansion, that is 
(ABD) + (ACD) (BCD) > (AB) + (AC) + AD) + (BC) + (BD) + (CD) 
- (A) ~ (B) - (C; ~ (D) + (U). 
But this is again simply the first condition of V., § 7. The whole of the conditions V. 
for the congruence of fourth degree may in fact he derived by writing down the 
conditions III. for each aggregate ABC, ABD, ACD, BCD, inserting the universes 
D or S in the first case, C or y in the second, and so on. Evidently this will give the 
right number of conditions, there being four aggregates and four conditions for each, 
while each condition must hold good in two universes, total 4X4X2=32. 
A precisely similar theorem holds good for the 160 conditions of § 9, VII., for the 
congruence of the fifth degree. The congruence of second order and fifth degree 
may be resolved into ten congruences of tlie third degree. The four conditions 
of III. must hold good for each of these congruences in four universes, c.y., for the 
aggregate ABC, the universes DE, De, oE, 8e. The whole numl^er of conditions so 
derived is 10 X 4 X 4, or 160. The sixteen conditions of YIIL, § 9, cannot ])e so 
derived; they involve five, not three, fifth-order frequencies each, and are quite new 
conditions derived from the comparison of expansions of groups contrary to each 
other in five attributes. 
The results suggest, howevei', that to ol)taiii the new conditions for a congruence 
of degree 2ni + 1, order '2m, we have only to consider the 2'-'" possible comparisons 
of contraries. A congruence of even degree, say 2/a, is subject to no conditions 
beyond those immediately derivaljle from the congruence of degree 2/a — 1. 
But this result is only suggested, not })roved, by the few cases taken ; nor are the 
general theorems corresponding to those given at the end of § 8 and § 10 demonstrable 
from the mere jjarticular cases. 
§ 12. By slightly clianging the point of view, and remembering thaf any frequency 
maybe exp.inded l)y considering A, B, 0, U, &c., as “elective operators,” sidject to 
the ordinary laws of multiplication, and to the special laws 
U.A == A 
A" = A, 
the conditions of congruence may Ire olrtained in a simple general form. 
Pteferring hack to the earlier sections (§ 5, § 7, or § 9), it will be seen that in 
considering the congruence of degree m order m — 1, all the groups of the mth 
order, containing an even number of negative terms (or attributes), were first taken 
and expanded, and the expansion put not less than zero, thus giving a system 
of minor limits for (ABCD . . M). The remaining groups, containing an odd numlrer 
of negatives, were similarly expanded, giving a system of major limits. The 
expansions in the two cases were of the forms 
