102 
^lE. G. UDXY YULE OX THE THEOEY OF COXSISTEXCE 
§ 10. The conditions of superior congruence given in YII. only become impossible 
if either of the minor limits for any one set of fourth-order frequencies be gTeater 
than either of the major limits [c.g., [3] or [4] greater than [1] or [2]). But on 
expressing the condition that each minor limit must Ije less than tlie major, we see we 
, are sinqd}^ led back to limits of the form A"., § 7. That is to say—“A congruence 
of the fifth deo'ree and third order is self-consistent, if each of the five conoimences 
of the fourth degree, into which it can 1je resolved, is self-consistent.” But we have 
seen that comparison of the minor and major limits of conditions A", again leads 
back simply to conditions III. I'herefore we must have also—“ A congruence of the 
filth degree and second order is self-consistent, if each of the ten conoTuences of the 
third degree, into which it can be resolved, is self-con.sistent.” The conditions 
ol' consistence for the congruence of an}’' degree are then, so far as we have gone at 
all events, thrown Ijack on the conditions for the simple congruence, the degree 
of which only exceeds its order by unity. 
IiEilARKS ox THE PRECEDING SeCTIOX.S. 
General Solution. 
§ 11. The elementary metliod employed in tiie preceding .sections is the one best 
adapted for exhiliiting clearly the })hysical meaning of the conditions of consistence. 
It is perfectly adapted for finding the conditions for a congruence of any degree, 
though the number of coniparisous of limits to be made appears at first sight to 
make the work extremely lengthy. A few considerations, however, rapidly reduce 
the number of necessary comparisons. Thus all comparisons of expansions due to 
two groups that are contrary to each other in one term only give luferior conditions. 
Again, all comparisons of expansions due to groups that are contiary to each other 
in three terms, give conditions simply derivable from those of the congruence of the 
tliird degree. Take for instance the first condition of § 5, III. 
(AB) -f (AC) + (BC) < (A) + (B) + (C) - (U). 
The universe in which this inequality is to hold good is not specified at all. Let 
it be a unl^else m vliich all things are 13. Then the condition becomes 
(ABD) + (ACT)) -h (BCD) < (AD) + (BD) + (CD) - (D). 
But tills is simply the second condition of superior consistence for a congruence 
of the fourth degree (Ah, § 7). Again let the universe be not D, but S. Then the 
condition becomes 
(ABS) -f (ACd) + (BC8) < (A8) + (BS) + (CS) - (8), 
