MK. G. UDNY YULE ON THE THEORY OF CONSISTENCE 
9(; 
§ 7. To obtain the superior conditions, the fourth-order groups must be exj)anded 
{(■/. § 5), and the expansions again put < U, thus giving the systems of limits for 
(ABCD) as below. 
Or the frequency 
given below 
be negative. 
(ABCD) <0. 
< (ACD) + (BCD) - (CD). 
< (ABD) + (BCD) - (BD). 
< (ABC) + (BCD) - (BC). 
< (ABD) + (ACD) - (AD). 
< (ABC) -f (ACD) - (AC). 
< (ABC) + (ABD) - (AB). 
< (ABC) + (ABD) -h (ACD) + (BCD) - (AB) 
- (AC) - (AD) - (BC) - (BD) - (CD) + (A) 
+ (B) + (C) + (D) - (U). 
> (ABC). 
> (ABD). 
> Ugd) . . 
> (BCD). 
> (ABD) + (ACD) + (BCD) - (AD) - (BD) 
- (CD) + (D). 
> (ABC) + (ACD) + (BCD) - (AC) - (BC) 
- (CD) + (C). 
> (ABC) -h (ABD) + (BCD) - (AB) - (BC) 
- (BD) + (B). 
> (ABC) -f (ABD) + (ACD) - (AB) - (AC) 
- (AD) + (A). 
Any fourth-order aggregate will be impossible if any one of the minor limits 
to (ABCD) be greater than any one of the major limits. There are eight of each, or 
sixty-four comparisons to be made ; but thirty-two of these lead only to the already 
known inferior conditions of consistence. The remainino- tliirty-two, involviiur com- 
parisons of }>airs of groups that are contrary as regards three attributes (as in § 5), 
give new conditions all obtainable by cyclic substitution from the eight given in Y. 
below^ 
There are four third-order positive grou})S to be formed from four attributes, and 
therefore four possilde sets of three, giving four sets of the eight ine([ualities Y. 
The remaining three sets may be at once written down by s\ibstituting A, B, or C 
for D (and conversely) in tlie set given below. 
(ABCD) 
[1] 
(a^CD) 
[^] 
(aBvD) 
[3] 
(aBCd) 
[ 4 ] 
(A/3yD) 
[5] 
(A;8CS) 
[6] 
(AByS) 
[7] 
1 
[8] 
(ABCS) 
[9] 
(AByD) 
[10] 
(A^CD) 
[11] 
(aBCD) 
[12] 
}(a/3yD) 
[13] 
1 (ci^CS) 
[ 14 ] 
1 (aByS) 
[15] 
j(A^yS) 
[16] 
