MATHEMATICS: H. S. WHITE 
5 
Although Reiss in 1859 had answered this question in the affirmative, 
constructing one system for every value of t, his work was overlooked 
until the same question was settled independently in 1893 by E. H. 
Moore. Moore's methods, based on more penetrating analysis than 
Reiss's, established at least two distinct sorts of systems for every t 
above 13, and led him to forecast definitely that the number of such 
systems would be found a rapidly increasing function of the number of 
elements, /. As to the sole doubtful number, / = 13, Zulauf, a pupil 
of Netto, found that there are two different systems, and others soon 
proved that there are no more than two. 
After the lapse of ten years or more, Miss L. D. Cummings has now 
shown conclusively (1914) the existence of at least 24 distinct triad 
systems in 15 elements. These 24 include all that had been found before 
and as many more new systems, with their differences now for the first 
time rigorously demonstrated. All but one (viz., Heffter's) of these 24 
exhibit what I call odd-and-even structure. The odd part are the ele- 
ments appearing in seven triads that constitute an included triad system 
At, which may be termed the head in its A15. Heffter's A15 is at present 
the only headless system in 15 elements whose description has been 
published. 
Since the appearance of Miss Cummings' dissertation, I have applied 
a new method for constructing all possible Aib's which can be trans- 
formed into themselves by any substitution among their elements, — 
all whose group is above the identity. By this means I find a con- 
siderable additional number of systems, all headless. These new Ai5 s 
I now employ in attacking the question, how many distinct systems A31 
are there in 31 elements? If there were but few, then it would be de- 
sirable to compile a complete census of them as further substratum for a 
general theory. But what I find is that even the restricted class selected 
for this study are far too numerous for detailed exhibition, their number 
being greater than 10^^. This result is attained through a new theorem, 
whose generality is significant of further possibilities. 
The theorem, specialized for application, is this. If among the triads 
of a system A31 there occur two complete systems A15 and A^5, then there is a 
A 7 whose seven triads, and no other triads or elements, are common to A15 and 
A\5. Conversely, if a A31 contains a headless A15, it can contain no other 
triad system A^s; nor indeed any other larger than a A 7, and even such a 
A 7 will have one triad from the A15. 
Odd-and-even structure in a A31 consists in this: its 155 triads include 
35 that form a A15 in 15 elements, and of the remaining 16 elements two 
are found in each of the other 120 triads, along with one element from 
