SOLUTION OF THE PROBLEM OF THE POLYEDRA. 103 
acra haying only triangular faces. But I do not see any 
way to the solution of the problem, even in this shape, 
more direct than that given in the pages preceding. In 
fact, the question of the #-edra having only triedral sum- 
mits is identical with this problem in Tactics : 
To form triplets of three different letters with A A’s, 
B B’s, C C’s, &c., so that all these symbols shall be once, 
and once only, employed, and that every possible duad, 
AB, AC, &c., shall be either not or twice employed. 
The analysis above given supplies a complete solution 
hereof; and this is, I believe, the first instance in which 
a problem of this nature has found either an entire solu- 
tion, or even a translation into algebraical language. I 
have my hopes that the method here opened will suggest 
the proper expression and solution of a multitude of 
tactical problems, which have hitherto bidden defiance to 
our analysis. 
