38 
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308. 
ON THE A FACED POLYACRONS, IN REFERENCE TO THE 
PROBLEM OF THE ENUMERATION OF POLYHEDRA. 
[From the Memoirs of the Literary and Philosophical Society of Manchester, vol. i. (1862), 
pp. 248—256.] 
The problem of the enumeration of polyhedra Q is one of extreme difficulty, and 
I am not aware that it has been discussed elsewhere than in Mr Kirkman’s valuable 
series of papers on this subject in the Memoirs of the Society and in the Philosophical 
Transactions. A case of the general problem is that of the enumeration of the 
polyhedra with trihedral summits; and Mr Kirkman in the earliest of his papers, 
viz. that “ On the representation and enumeration of polyhedra ” (Memoirs, vol. xii. 
pp. 47—70, 1854), has in fact, by an examination of the particular case, accomplished 
the enumeration of the octahedra with trihedral summits. A subsequent paper “ On 
the enumeration of #-edra having trihedral summits and an (%— l)gonal base,” Phil. 
Trans, vol. xlvi. pp. 399—411, 1856), relates, as the title shows, only to a special 
case of the problem of the polyhedra with trihedral summits, and in this particular 
case the number of polyhedra is more completely determined; but the later memoirs 
relate to the problem in all its generality, and the above-mentioned particular problem 
of the enumeration of the polyhedra with trihedral summits is not, I think, any 
where resumed. Instead of the polyhedra with trihedral summits, it is really the 
same thing, but it is rather more convenient to consider the polyacrons with triangular 
faces, or as these may for shortness be called, the A faced polyacrons; and it is 
intended in the present paper to give a method for the derivation of the A faced 
polyacrons of a given number of summits from those of the next inferior number of 
summits, and to exemplify it by finding, in an orderly manner, the A faced polyacrons 
1 I use with Mr Kirkman the expression “ enumeration of polyhedra ” to designate the general problem, 
but I consider that the problem is to find the different polyhedra rather than to count them, and I con 
sequently take the word enumeration in the popular rather than the mathematical sense.