No. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 7 



Particular cases under this theorem are: 



a= 1, giving cycles of orders /iff, ixy, nffy; 

 a = /3 = l, giving cycles of orders m7, mt, m; 

 o = ;8 = Y = l, giving cycles of orders fi, ix, n; 

 H = l, giving cycles of orders ^y, ya, a/3. 

 Wlien two of the numbers are equal, they may refer to the same cycle, under conditions which it 

 is not necessary to examine here. 



For our present purpose, the important deduction from this theorem is the simplest case, 

 namely, that if a=l and &=1, then must c = l. In other words, if there are two letters (or 

 elements) A and B invariant under the substitution S of the group, the triad containing these 

 two must contain a third, C, also unaltered by the substitution S. An immediate extension 

 of this corollary gives the following rule: 



AU the elements not altered by a substitution in the group of a triad system (cycles 



of period unity) must constitute a complete triad system contained in the principal 



system, a subordinate system. Hence, for a A15, the number of cycles of period 



imity in any operation in its group can only be 0, 1, 3, 7, or 15. 



The numbers 9, 13 are excluded, since a subordinate system can not contain half as many 



elements as the principal systems. 



If the substitution S has cycles of different periods both higher than unity, a<h, then a 

 power S°- of that operator wiU have invariant a additional elements. If S^ still contains cycles 

 of different periods, another power can be foiuid to dimmish the number of different periods; 

 and ultimately some power of S wiU be fomid with 0, 1, 3, or 7 cycles of period 1, and having 

 the rest of its cycles of equal prime period. 



Every triad s}'stcm on 15 elements, whose group is not the identity, is invariant imder 

 at least one substitution of one (at least) of the following seven types : 



1. (5) (5) (5) 



2. (3) (3) (3) (3) (3) 



3. (1) (7) (7) 



4. (1) (2) (2) (2) (2) (2) (2) (2) 



5. (1) (1) (1) (3) (3) (3) (3) 



6. (1) (1) (1) (2) (2) (2) (2) (2) (2) 



7. (1) (1) (1) (1) (1) (1) (1) (2) (2) (2) (2), 



where the digit in any parenthesis indicates the period of a cycle. 



These seven types of substitution offer a natural means of classifying triad systems on 

 15 elements, provided they admit groups of substitutions. Under each type I shall construct 

 aU possible invariant triad systems, omitting obviously equivalent repetitions. In one case, 

 tj'pe 4, no system can exist; aU the others have actual systems. It wiU still happen that certain 

 systems occur in two or more classes, their groups containing substitutions of two or more of 

 these seven types. Further reduction is uJidertaken by Miss Cummings (Part 2), who furnishes 

 the proof of nonequivalence of the net residue, 44 systems. Of these 44, 24 were known 

 previously, more than haK of them discovered by Miss Cummings. It will be noted that the 

 20 new Ais's contain no A?; they are not of the " odd-and-even" structure; they may be called 

 "headless, " while any A? contained in an earher known A15 is termed its head. One headless 

 system onlj'-, discovered by Heffter, has been known heretofore. The discussion of groupless 

 systems, and their enumeration, is deferred to Parts 3 and 4. 



§3. CLASS I, INCLUDING THE KIRKMAN AND HEFFTER SYSTEMS. 



For each substitution the work of constructing systems of triads must be special; no gen- 

 eral conclusions are to be developed. Three requirements guide us: (1) Every pair of elements 

 shall occur; (2) no pair shall occur twice; and (3) the triads shall be grouped in sets conjugate 

 under the particular substitution (operator). 



Denote by (S, the operator of type I, and take for its three cycles of five, respectively, 

 English and Greek letters and Arabic numerals. 



S,={a bcde) (a /3 7 5 «) (12 3 4 5). 



