SOLUTION OF THE PROBLEM OF THE POLYEDRA. 93 
ace, a b-ace---a p-ace, e being the number of its edges, 
the equation following will be satisfied. 
(A)* A+B+4C+4+---+Q=a+b+e4-+-4+p 
=2e=2(p'-14+Q’-1); 
and there are in general many p’-acral Q’-edra, having a 
common description in this equation. But to determine 
whether this equation, when values of AB, &c. are chosen, 
really describes polyedra, and how many it includes, all 
differing in the mutual arrangement of the faces and sum- 
mits specified, has hitherto been found a problem of insu- 
perable difficulty. This difficulty arises from the fact that 
the problem is chiefly tactical; and we have had no ¢actic 
calculus, nor any means of expressing tactical conditions 
in algebraic language. It is easy enough to multiply 
theorems in combination, when these are symmetrical and 
exhaustive; but to assign the number of combinations or 
systems of combinations that can be selected, so as to 
fulfil given unsymmetrical conditions, and to be all essen- 
tially distinct, has thus far been a task beyond the powers 
of our algebra. 
The question, How many z-edra are there? is not with- 
out interest for its own sake; for every x-edron is one out 
of a precise number of permanent facts in nature: its 
principal charm, however, has lain in its difficulty, and in 
the consideration that the answer has been part of the 
sublime unknown. It is precisely on problems like this 
that the modern analysis delights to try the edge and 
temper of her tools; or rather to exercise herself in the 
handling of instruments which, though lent to human 
workmen, are not of human workmanship, but whose 
fabric and finish are divine. 
The obstacle to research in this direction has been 
* Vide my paper “On the representation and enumeration of the Poly- 
edra,”’ — Manchester Memoirs, vol. xii. 
