374: Prof. Sylvester on the Historical Origin of the 
each satisfying this last condition, which shall together include 
between them all the triads that can be made out of the fifteen 
elements. 
The reader would have reason to be dissatisfied with the 
author’s reticence, were he to leave altogether unmentioned the 
synthematic aggregation of the dinomial triads appertaining to 
the same three triliteral sets or nomes ; but space forbids my doing 
more at present than giving one of these aggregates, and indica- 
ting the number and mode of generation of all from this one. 
It will readily be seen that any such aggregate will be made up 
of two sub-aggregates, which I shall call A and B respectively, 
of which one bears the same relation to the disposition of the 
nomes in the order 123 456 789, as the other to their disposi- 
tion in the order 123 789 456. Thus we may take for our A 
and B the following, which will each contain 9 synthemes, the 
total number of synthemes in the two together being 18* :— 
(A.) (B.) 
124 567 893 127 894. 768 
125 468 739 128 795 436 
126 459 783 129 786 453 
1384 568 279 137 895 246 
185 469 278 188 796 245 
136 457 289 139 784 256 
934 569 187 937 896 154 
935 467 189 238 794 156 
236 458 179 2939 785 146 
The system of triads contained in A may be arranged in 
twelve different aggregates similar to the one given, and the 
same will be true for the triads in the B; ‘so that the total num- 
ber of the combined systems will be 144. All the permutations 
which leave A or B (separately considered) unaltered will form 
a natural group,—the theory of groups in this, as in every other 
case, standing in the closest relation to the doctrine of combina- 
torial aggregation, or what for shortness may be termed syntax. 
I have elsewhere given the general name of Tactic to the third 
pure mathematical science, of which order is the proper sphere, 
* Thus, since there is evidently one mononomial syntheme, the total 
8x 
number of synthemes of all three kinds will be 1+18+9=28= 9 > as 
, ists 
it should be, the total number of triads being 8.f and 3 of them going 
to a syntheme. 
