OF LOGICxiL CLASS-FREQUENCIES, ETC. 
90 
Or the fi’equency 
given Ijclow is 
negative. 
(aBySE) 
(ABCDE) > (ABCE) -f (ACDE) + (BGDE) ~ (ACE) G 
- (BCE) - (CDE) + (CE) . . . 
> (ABCD) A (ACDE) + (BCDE) - (ACD) i 
- (BCD) - (CDE) + (CD).r ^ ^ 
> (ABCE) + (ABDE) + (BCDE) - (ABE) 
- (BCE) - (BDE) + (BE) .... 
> (ABCD) + (ABDE) + (BCDE) ~ (ABD) 1 , p .. . 
- (BCD) - (BDE) + (BD) .... ^ 
> (ABCD) + (ABCE) + (BCDE) - (ABC!) 1 
- (BCD) ~ (BCE) + (BC) . . . . ■ j 
> (ABCE) + (ABDE) -f (ACDE) - (ABE) 9 
~ (ACE) - (ADE) + (AE) . . . . j 
> (ABCD) + (ABDE) + (ACDE) - (ABD) i 
- (ACD) - (ADE) + (AD) .... 
> (ABCD) 4- (ABCE) + (ACDE) - (ABC) 
~ (ACD) - (ACE) + (AC) .... 
> (ABCD) + (ABCE) (ABDE) — (ABC) 
[23] 
[24] 
[25] 
[ 20 ] 
[27] 
[28] 
[29] 
[30] 
(ABU) - (ABE) + (.\B) . . . 
> (ABCD) + (ABCE) + (ABDE) + (ACDE) | 
+ (BCDE) - (ABC) - (ABD) - (ABE) | 
- (ACD) - (ACE) - (ADE) - (BCD) i 
- (BCE) - (BDE) - (CDE) + (AB) (aJBySe) [32] 
(A,5yDe) 
J8e) 
+ (AC) + (AD) + (AE) + (BC) + (BD) 
+ (BE) + (CD) + (CE) + (DE) - (A) 
— (B) — (C) — (D) — 'E) + (U) . . 
) VI. 
Any fifth-order aggregate will be impossible if any one of the minor limits to 
(ABCDE) be greater than any one of the major limits. Since there are sixteen 
of each, there are 250 comparisons to be made. Eighty of these lead only to the 
already known inferior conditions. Of the remaining 17(), 100 may be obtained l)y 
cyclic substitution from the sixteen conditions iiiYolving oidy tliree fourth-order 
groups each, as given in VII. below, the remaining sixteen, each involving all 
five fourth-order positive groups, being given in VIII. 
