104 
MR. G. UDNY YULE ON THE THEORY OF CONSISTENCE 
-f (ABCD . . . M) — (terms of lower order than m)^ 
— (ABCD . . . M) + (terms of lower order than 7n).i. 
We took this in the previous sections as giving 
(ABCD . . . M) < (terms of lower order than 
(ABCD . . . M) > (terms of lower order than 7 / 1 ) 0 , 
and deduced 
(terms of lower order than m).^ <L (terms of lower order than 7 / 7 )^. 
But precisely the same result is arrived at by adding the two expansions and 
putting the sum not less than zero,* and this is a much more convenient conception 
from which to ol^tain the general conditions. 
Let the two groups contain positive attrilnites K^Ko . . . and negative 
attriljutes a^Xo . . X^ in common; positive attributes M^Mo . . . M,. in the one, with 
their negatives p-jpo . . . p,,. in the other ; and negative attributes in the 
one, witli their positives in the other. Using the symbol [J to denote 
“ the continued operator-|)roduct of all quantities like ...” the expansions of tlie 
two frequencies may be written 
/>=/> 'i = <i 
77[KJ 77[U - LJ 
= 1 5=1 
’ //[M,.] /7[U - NJ 
)• = I 5=1 
'"}7[U - L,] 
31 = I 5 = 1 
’/7[U-M,.] ^~77[NJ. 
!■ = 1 S = 1 
Til these expressions it must lie remembered tliat as the one group is to contain 
an even, the other an odd, numlier of negatives, if // -j- r Tie odd <] + .s must he 
even, and vice vcvsci. Tdence v A s must in any case he odd, i.e., the two groiq^s 
must he contrary as regai’ds an odd number of attributes, foi‘ if 
q A .S' — 2a' 
q A r = 2p + 1 
r A = ^(.c A y ~ q) A 1 , 
wliicli is necessarily odd. The general condition of consistence for a cono-ruence 
of degree 
m = 2J + q + r + s, 
Every frequency must lie greater than zero, and a foiiiori the sum of an}' tivo. Rut it is onlv hy 
taking the sum of two frequencies, the one containing an even, and the other an odd, uumher of negatives 
that an expression is olitained from which the ///th-order term is eliminated. 
