18 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv. 



The doubles of these numbers, plus the 1 and 6 above mentioned, give the total of 35 triads for 

 a system. The second kind will be taken as standard ; the other two wiU be found to be reducible 

 to this. 



By definition, each of the elements A, B, Cmust appear with six pairs of small letters a, h. 

 Since those not self-dual, as AaJ)],^ or ^aifflk, must occiu* in pairs, Aaih•^^, Aa\,hi, the self-dual 

 triads containing A (or B, C) must also be an even number G, 4, 2, or 0. We therefore divide 

 systems of this section into thi'ee principal classes. 



In class VI, 1 : Six self-dual triads contain A; 



In class VI, 2: Four self-dual triads contain A, two have B; 



In class VI, 3 : Two self-dual triads contain A, two have B, two have C. 



We can fix, for each class, these six self -dual triads and stiU retain freedom to exchange symbols 

 a, h in each pair; also to exchange certain subscripts. Beside ABC, assign to each class these 

 fundamental triads: 



Class VI, 1 : Aafii, Aajb.^, AaJ}^, Aaja^, Aa^hr,, AaJ)^. 



Class VI, 2 : Aajb^, Aafij Bajb^, Ba^h^. 



Class VI, 3 : Aa^bi, AaJ)^; BaJ)^, Bafit,- CaJ)^, Ca^hg. 



In class VI, 1 we are free to arrange that pairs with the symbol B shall be either aiaj^ or iib^ 

 and that the pairs of subscripts shall be 12, 34, 56. Quite similar is the choice permitted in 



VI, 2. 



Class VI, 1: Ba^a^, Bb^b^,; Ba/i^, BbJ)j Ba^a^, BbJ)^. 

 Class VI, 2: i?ffl,a2, Bb^b^; Ba^a^, BbJ)j Aa^a^, AbJ)^. 



In both classes it is still optional to exchange the Oi, bi of any conjugate pair as Ba^a^, 

 Bb^b^. Hence it residts that triads in Ccan all be put into a standard form Cuib],. Thus in classes 

 VI, 1 and VI, 2 the numbers of triads can be brought to agree with the second column in our 

 tabulation; that is, there will be two triads of type aiajay,, and six of type aiOjb^. 



For both these classes, therefore, we \\Tite down at once the possible sets of triads containing 

 G, leaving for separate discussion the class VI, 3. 



Class VI, 1 — Triads in C. 



No. VI, li: CaJ>2, Oa^b^, Oa^b^, CaJ)^, Ca^b^, Ca^b^. 

 No. VI, la". CaJ)^, Ca^bi, Oa^bi, Cajb^, Cafi^, Ca^b^. 

 No. VI, I3: Oafi^, CaJ)^, Cajb^, Ca^h, CaJ)^, CaJ)^. 



Class VI, 2 — Triads in C 



No. VI, 2,: CaJ)^, Cajb^, CaJ>^, Cajb^, CaJ)„, CaJ)^. 

 No. VI, 22: CaJ)^, CaJ)^, Cajb^, Cap^; Ca^b^, CaJ)^. 

 No. VI, 23: CaJ)^, Cajb^; Oa^b^, Cajy^, Bajj^, OaJ)^. 

 No. VI, 24: CaJ)^, OaJ)^, Cajb^, Cafi^, Cafi^, Cajb^. 



Class VI, 3 has two self-conjugate triads in A, two in B, and two containing C. There are 

 yet to be formed for each, four triads or two pau"s of conjugates. That is, for each of these 

 three letters we must combine four subscripts into two pairs. Notice that the six pairs are to 

 contaui each subscript twice. These may be grouped into one or more c}' cles ; for example, if 12, 

 23, 31 are among them, they constitute a cycle of thi-ee. Possible are apparently 



Three cycles of two pairs; 



One cycle of two, one cycle of four; 



Two cycles of tluee; 



One cycle of six. 



