198 MATHEMATICS: COLE, CUM MINGS AND WHITE 
known and twenty new ones. The orders of the groups are 2, 3, 4, 
5, 6, 8, 12, 21, 24, 32, 36, 60, 168, 192, 288, and 20160. 
Groupless systems on 31 elements were next found (see these Pro- 
ceedings, 1, 1915, 4), and the question was raised whether possibly 
groupless systems could exist in 15 elements. This question is answered 
by the actual construction of several such systems by Mr. White and 
Miss Cummings. But while their empirical method is productive, it 
is not deductive and so could not be shown to be exhaustive. A new 
starting-point and a new method are requisite, to insure a complete 
survey of groupless systems as well as the better known kinds. This 
method was furnished, and its tedious and difficult execution under- 
taken, by Mr. Cole. 
Starting out with the four possible openings : 
11011 11213 11415 
21112 21314 2515 
11011 11213 11415 
21012 21114 21315 
11011 11213 11415 
2912 21114 21315 
11011 11213 11415 
2911 21214 21315 
which, from the mode of interlacing of the triads containing 1 with 
those containing 2, may be called dodekad, hexad, single tetrad, and 
triple tetrad types, respectively, he began by proving that no system 
could be built up with interlacings of type I alone. Then it was found 
that with types I and II alone only one system was possible: that al- 
ready found by Heffter. With this one exception every triad system 
in 15 letters has an interlacing of type III or IV. 
The census was next continued by working out all the systems con- 
taining type IV. These included most of the systems with a 7-head 
(triad system in 7 of the 15 letters); a brief excursus covered the re- 
mainder of the 7-head systems. There were in all 23 systems with 
7-head and 38 without 7-head. 
There remained the systems with ty^^ III but not type IV. These 
were divided into two classes: (1) those with a 'semi-head:' 123 145 
167; 246 257; 347; and (2) those without semi-head. Of the former 
there were 3, of the latter 15. 
The total number of types of triad systems in 15 letters therefore 
proves to be 80. 
Proof that the 44 systems with groups are different is based on the 
set of trains belonging invariantively to each system. There are over 
1 123 
II 123 
III 123 
IV 123 
145 
167 
189 
246 
278 
2910 
^145 
167 
189 
246 
258 
279 
^145 
167 
189 
246 
257 
2810 
145 
167 
189 
246 
257 
2810 
