OF LOGICAL CLASS-FREQUEJICIPIS, ETC. 
07 
(ABD) -L (ACD) A (BCD) > (AB) + (AC) + (AD) + (BC) + (BD) 
A (CD) - (A) - (B) - (C) - (D) 
+ (U). 
< (AD) A (BD) A (CD) - (D) . . . 
(ABD) A (ACD) - (BCD) > (AD). 
< (AB) A (AC) A (AD) - (BC) - (A) [4] [ 
(ABD) - (ACD) A (BCD) > (BD).[5] 
< (AB) - (AC) A (BC) A (BD) - (B) [O] 
-(ABD) A (ACD) A (BCD) > (CD).[7] 
< - (AB) A (AC) A (BC) A (CD) - (C) [8] ^ 
§ 8. The superior conditions of consistence for the secourZ-order congruence of the 
fourth degree may be at once obtained from V. For in order that the congrueirce 
of the third order may lie self-consistent, it is clearly necessary that tlie congruence 
of the second order should l)e so ; if this condition do not hold the conditions of 
consistence V. will prove themselves impossible. But the limits V. only become 
impossible if, in either of the four pairs of limits ( [1] [2], [3] [4] &c.), the minor 
limit be greater than the major. If we ex|)ress tlie condition tliat each minor limit 
must be the less we have, 
from V [1] [2] (AB) A (AC) A (BC) > (A) A (B) A (C) - (U). 
[3] [4] (AB) A (AC) - (BC) < (A). 
[5] [6] (AB) - (AC) A (BC) A (B). 
[7] [8] - (AB) A (AC) A (BC) < (C). 
But these are simply tlie conditions III. of § 5, the superior conditions of consistence 
for the second-order congruence of the third degree. Similar conditions for the 
aggregates ABD, ACT), BCT), will of course be derived trom the unwritten sets 
of inequalities correspon(.ling to (V.). The theorem therefoi'e holds—“ A congruence 
of the fourth degree and second order is self-consistent, if each of the three con¬ 
gruences of the third degree, into which it can Ije resolved, is self-consistent.” 
Congruence of the Fifth Degree. 
§ 9. Congruences of the fifth degree may Ije of either the second, third, or fourtli 
order. The congruence of the fourth order may Ije taken first and tlie others 
derived from it. 
The inferior conditions of consistence for fourth-order groups have been already 
given in § 7, IV. The similar conditions for fifth-order groups, obtained of course in 
precisely the same way, are as follows :— 
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