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



as a transformer of aU possible combinations of the elements in threes, and so assembles these 

 combinations into self -evolvent aggregates or trains, analogous to the "path-curves" of a 

 point transformation.* The entire set of such trains, abstractly considered, belonging to any 

 triad system on 15 elements is characteristic for that system and for aU systems equivalent to 

 it. A second method, that of sequences and indices, is described and exemplified by Miss Cum- 

 mings in her dissertation.^ These two methods have given, so far, accordant results, and both 

 lead to the easy discovery of the substitutions which transform two equivalent systems into 

 each other. 



With these two direct and simple methods available for testing triad systems it is reason- 

 able to attempt the complete enumeration of those on 15 elements. In this part, however, 

 I shall undertake an easier task, that of constructing all triad systems on 15 elements whose 

 groups are of order higher than unity ; aU which have gi-oups containing at least one operation 

 different from the identity. In this research the group, and indeed the particular substitution, 

 will furnish the starting point. 



§2. SUBSTITUTIONS ADMISSIBLE IN THE GROUP OF A A15. 



Not every type of substitution can occur in the group of a triad system. Consider in par- 

 ticular a system A15 whose elements are denoted by letters, and S a substitution of its group. 

 Represent S, as is uniquely possible, by a product of mutually exclusive cycles: 



8^{A, A, .... Aa) {B,B, .... Bf,) {C,C, .... C,) 



It is required by the definition of a triad system that every pair of elements shaU occiu' in some 

 triad, and that no pair occur twice. The pair AJi^ must occur in some triad, as A^BiC^. As a 

 special case, the third element 0^ may be in one of the cycles {A) or (J5). We note that in S the 

 order c of cycle {C) must be a factor of the L. C. M. of a and 6; otherwise the pair AJi^, would 

 occm* in two or more triads. Let a = mP, b = ma, where a and /3 are relatively prime; then 



must 



map = yc (7 an integer) . 

 For similar reasons 



a is a factor of the L. C. M. {h, c), ml3 divides mac; 



his & factor of the L. C. M. (a, c), ma divides mfic. 

 Hence, as a is prime to /3, c is a multiple of the product a/3, c = na^, or 



7 c = 7/ia^ = 7na/3, 



therefore m = yiJ.. Now we find, more exactly, that 



ffl( = 7^1/3) divides the L. C. M. {yixa, tia^) 



which is na . L. C. M. (7, P). 



Therefore 



7 /3 divides a. L. C. M. (7, /3) 

 and 



7 o divides /3. L. C. M. (7, a) 



Hence 7 is prime to both a and |3. We have accordingly for the three orders of cycles {A), 



iB), and (C), 



a = M/37, o = nya, c = naff; 



and we have proved this theorem: 



If in the group of a A15 there occurs a substitution containing two mutually exclusive 

 cycles of orders a = pi37, 6 = M7a respectively, then that substitution contains also a 

 cycle (possibly coincident with one of the first two) of order c = txaP; a being prime 

 to /3, and 7 some common factor of a and i but prime to a and /3. 



• H. S. White: Triple systems as transformations, and their paths among triads. Transactions of the American Mathematical Society, vol. 

 14 (1913), pp. 6-13. 



> L. D. Cimunings: On a method of comparison for triple systems. Transactions of the American Mathematical Society, vol. 15 (1914), pp. 

 311-327. 



