If n elements 1, 2, 3, ... n can be distributed in triads in such a way that every pair of elements appears in one and only one triad, the totality of triads forms a triple-system of n elements. Reiss has shown that it is possible to form a triple-system of n elements provided n is of the form 6m + 1 or 6m + 3.
Different methods for constructing triple-systems have been given by Reiss, Netto, Heffter, and E. H. Moore; but methods for testing the non-congruency of these systems when formed are lacking. The group of substitutions that transform a triple-system into itself has hitherto been adopted as the abstract mark of the system, and Zulauf has shown, by a consideration of the groups, that the four systems on 13 elements of Kirkman, Reiss, De Vries, and Netto are reducible to two incongruent systems whose groups are different. In the present paper it is shown however that for n = 15 two incongruent triple-systems, Δ15, may have the same group.
By a simple process non-congruent triple-systems are constructed. These are used to illustrate a new method of comparison, by means of a new sort of abstract marks. This method requires no knowledge of the group but incidentally facilitates its determination. No exhaustive determination of every Δ15 which may be obtained by this process is here undertaken, but the 24 systems discussed include 12 apparently not hitherto constructed.