OF LOGICAL CLASS-FKEQCENCIPIR, ETC. 
‘)5 
(ABC) <0. 
< (AB) + (AC) - (A). 
< (AB) + (BC) -(B). 
< (AC) + (BC) -(C) . 
> (AB). 
> (AC). 
> (BC). 
> (AB) + (AC) + (BC) - (A) - (B) - (C) + (U) . 
Or the fi'equftiicy 
given l)elo\v 
wiil 1)6 negative. 
. (ABC) 
[1] 
• (A^y) 
[2] 
■ (X) 
[3] 
. (ay8C) 
[4] 
. (ABy) 
[5] 
. (A^C) 
[6] 
. (aBC) 
[7] 
• (a/3y) 
[8] 
IT. 
Now evidently, any third-order aggregate whatever is impossible if amj one of the 
minor limits [l]-[4] greater than any one. of the major limits [5]-[8]. If the 
second-order freqnencies he such as to create this condition they mnst be impossible 
within one and tlie same universe, re., they are inconsistent or incongrnent. There 
are sixteen comparisons to be made, taking each of the major limits in turn with each 
of the minor limits, but the majority of these comparisons, viz., 12, only lead back 
to the inferior conditions. Tlie four comparisons of expansions due to contrary 
frequencies alone lead to new cf)nditions— 
II. [IJ [8] (AB) 4- (AC) A (BC) < (A) -f (B) + (C) - (U) . . [1]] 
II. [2] [7] - (AB) -f (AC) + (BC) >(C).[2] ^ 
II. [3] [6] (AB) - (AC) + (BC) >(B) .[3] 
II. [4] [5] (AB) -h (AC) - (BC) XA).[4] ^ 
III. 
These are the four superior conditions ot consistence for the congruence of the 
third degree. In order that the second-order frequencies may be consistent with 
each other and with the given frequencies of the first order, they must fulfil all the 
conditions of type I. and the conditions of type III. These inequalities are highly 
interesting ; but discussion is best deferred till after we have obtained the similar 
conditions for the congruences of higher degree. 
Congruence of the Fourth. Degree. 
§ 6. Two congruences of this degree are possible, viz., those of the second and 
third orders. The third-order congruence will be taken first, as the conditions of 
consistence for the second-order congruence may be obtained directly from the 
third-order conditions. 
The inferior conditions need not be written down as they have been given already 
(II. of § 5); similar C)3nditions hold of course for all four groups (ABC), (ABD), 
(ACD),’(BCD). 
