No. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 17 



System V, 4/3 (1 and 2): ABO, d/lj,; 

 Aafii, Acidi, (1) a^a^d^, hfi^c^, CiC^a.^; afi^d^, b^Cid^. 

 BaJ)2, Bcid.^, or 



Ca^c-i, Cb^di, (2) aiO^di, hfi^c.., CiC^a^; afi^d^, b^Cidj. 



System V, Ay (1 and 2): ABC, d,d,d„- 

 Aafii, Ac^di, (1) a^ad3, ^Vu c^c^a^; aj)^^, h^Cid^. 

 Bafi^i Bcid^, or 



CoyCi, Chidi, (2) a^a^du bfixj, CiC^a^; aj)d^_, hiC^d^. 



System V, 43 (1 and 2") : ABC, d^d.^! 

 Aafii, Ac^d^, (1) a^a^^, hfi^Ci, CiC^a^; aiMi, b^Cid^. 

 Bafi^, Bcidg, or 



CttiCj, Ch^di, (2) aiO^dj, hfi^c,, c^c^a^; afi^d^, h^c^d^. 



Equivalent systems, by the substitution {a^a.fl^) (CiC^c^) (2;?) {AB). 



Systems V, 5a (1 and 2) : ABC, dyd^d^; 

 AaJ)^, Acidi, (1) ttjajdi, h^h^d^, c^cjbj^; afi^c^, a^c^d^, 

 BaJ)^, Bc^d^, or 



(7aiC„ CT,di, (2) aia^d^, ifi^d,, c^c.J>:,; aj).,c.^, a^^Cid,. 



These two are equivalent in the same way as V, 4a, 1 and 2. 

 Systems V, 5/3 (1 and 2) : ABC, d,dj,; 

 Aafii, Ac^d^, (1) UjU^di, 6463^3, Cicjb^; afi^c^, a^Cid,, 

 Bafi^, Bcjd^, or 



Ca^c^, Cb^di, (2) a^a^l^, bfi^d^, c^cfi^; afi^c,, a^Cid^. 



Systems V, 57 (1 and 2) : ABC, did2d3; 

 AaJ>i, Ac^d^, (1) a^a^d^, bj)^.^, c^c^bi; afi^c^, a^Cid^, 

 Bafi^, Bcids, or 



CttiC,, Cbidi, (2) a^a^d^, bfi^d^, Cicj)^; afi^c^, a^c^d^. 



Systems V, 55 (1 and 2) : ABC, d^d^d^; 

 Aafii, Acid^, (1) afi^d^, bfi^d^, c^c^b.^; afi^c^, a^Cid^, 

 Bafi^, Bc^d^, or 



Ca^c^, Cb^di, (2) aitt^di, bfi.^^, c^c^b^; afi^Cz, a^Cjd^. 



Two equivalent systems, as under V, 45. 

 Beside the eqxiivalences already noted, one less obvious is that of systems V, 3a and V, lo. 

 which Miss Cummings wiU establish in Part 2. That done, we shall have invariant under this 

 tyi^e of substitutions (1)' (3)*, 21 distinct systems. 



§8. TRIAD SYSTEMS WHOSE GROUP CONTAINS A SUBSTITUTION OF THE TYPE (1)' (2)». 



Operations containing longer single cycles belong to fewer distinct types of triad systems, 

 While the fifth kind of substitution, (I)^ (3)*, gives rise to 21, the sixth, now to be examined, 

 will yield apparently more than 30. Actually the reduced number is the same, 21, for some 

 systems admit two or more substitutions of the same type. Tliis large number of systems might 

 weary the attention, were it not that novel points of difference are developed, in themselves 

 interesting. 



Denote the 15 elements and the operation thus: 



S={A) (5) (CO (a. &.) (a, \) {a, b,) {a, b,) (a, b,) {a, 6.) 



Two triads conjugate imder S we shall call dual to each other; there wiU be six triads self-dual, 

 those containing pairs aj>i, aj)^, etc. According to the principles in section 2, ABC must be 

 one triad and the six self-dual pairs a J}, must be united with A, B, or C to form triads. Denoting 

 the three capitals generically by K, we could specify eight possible types of triads, but for 

 present purposes conjugates combine and form four types. With (or without) the aid of dio- 

 phantine equations, we find three sets of numbers for these four classes, as follows: 

 540G1°— 19 2 



