ion 
MK. G. UDNY YULE ON THE THEORY OF CONSISTENCE 
eliniiiiatecl, one such set correspoiicUng to each possiljle congruence of the nth degree 
and {n — 1 )th order. Hence the general theorem—“A congruence of the {n — l)th 
order and ndh degree is self-consistent if all the possible congruences of the nth 
degree, into udiich it may Ije resolved, are self-consistent.” This is the generalisation 
of the theorems given in § 8 and § 10. 
§ 14. Tlie iiumljer of conditions to wliich any congruence of the 7??th degree is 
subject is readily o1)tained. Consider first the congruence of order {rii — 1). The 
numijer of combinations of c (positive) attributes that can he selected from rn is 
rn{m. — 1) . . . (no — r -f 1) 
(C! ■ 
From each of these comljinations can Ije generated 2'"“^ contrary pairs, by negativing 
one or more of the attributes: e.g., from (ABC) can be formed the four contrary 
})airs 
ABCl aBCl ABCl ABy] 
a/3y I A^yJ aBy | a/3C | 
From the remaining (p-f y) or {)n — c) attrilmtes can Ije formed 2'"~^ different 
universes. Hence tlie numijer of conditions involving c terms of the {ui — 1 )th 
order is 
ri(m - 1) . . . (m - r + 1) 
(r) ! 
Tlie whole number of such conditions (including those of inferior congruence) will 
lie given by inserting all possible values of c* in the above expression (viz., all odd 
numbers not greater tlian m), and summing. That is 
, ')»(»*- l)(wi - 2) , w(m - l)(m - 2)(jn - 2 )(jh - 4)(j» - 6) , \ 
^ lT3 + L2HI^ +•••)> 
or. 
stituting 1 + ju — 1 for m and rearranging 
O m ■ 
1(1 + (,„ + 1 . '"- -1><” -gw - o j = 
1.2 
i.2.;3 
The number of conditions is thus (piadrupled fiir every unit rise in the degree of 
the congruence, and grows with extreme rajiidity. The actual numhei's, up to the 
congruence of nintli degree and eighth order are given in Table I. Ijelow. For moiu 
than five or six attributes the actual arithmetic discussion of any particular case 
seems to pass the bounds of practical possiliilities. 
