MATHEMATICS: COLE, CUM MINGS AND WHITE 197 
constituent or constituents presumably belonging to the group of hitherto 
unrecognized but essential components of an adequate diet. 
In the essential features the pathological manifestations described 
in this investigation closely resemble those which may be observed in 
human pellagra. 
THE COMPLETE ENUMERATION OF TRIAD SYSTEMS IN 15 
ELEMENTS 
By F. N. Cole, Louise D. Cummings, and H. S. White 
Read before the Academy, November 14, 1916 
If any set of 15 points are joined by all the possible 105 connecting 
lines, then these may be combined in sets of three to form 35 triangles. 
The marks or ^elements' designating the three vertices are a sufhcient 
description of any one triangle, and a list of all 35 triangles constitutes 
a Hriad system on 15 elements.' Two such sets of triangles are essen- 
tially alike if a renaming of its points turns the one list into the other; 
if that cannot be done, the two lists or systems are essentially differ- 
ent. How many essentially different triad systems can be formed of 
a given number of elements, is a question of much difficulty, never 
before answered when the number of elements is 15 or more. For 13 
elements there are but 2 different systems, for 9 or 7, only one. The 
present paper shows that for 15 elements, there are exactly 80 different 
systems. This conclusive result is established by Mr. Cole. 
Three years ago the dissertation of Dr. Cummings increased the number 
of known triad systems on 15 elements from 10 to 24, and furnished a defi- 
nite method for comparing systems and verifying their difference or equiv- 
alence. All the new systems found by Miss Cummings contained a ' head' 
— a triad system on 7 of the 15 elements, while one exceptional system 
among the 10 previously known was 'headless,' a system constructed by 
Heffter. But all of them admitted groups of transformation into them- 
selves, and it was suspected that the possession of a group might be a 
necessary property of triad systems. Mr. White takes the group for a 
starting-point; finds seven types of substitutions, one or more of which 
must occur in the group ; and constructs all the district triad systems for 
each of those seven typical substitutions. This gives as a gross result 83 
systems. By two methods, that of Miss Cummings (by sequences) and 
one introduced by Mr. White (by trains) these are tested, duplicates 
are eliminated, and the net result is found to be 44 systems. Of these, 
exactly 23 exhibit heads and are equivalent to those in Miss Cummings' 
list, while 21 are headless, including the one such (Heffter 's) previously 
