248 Mil. A. CAYLEY ON THE A FACED POLYACRONS IN 



XIX. — On the A faced Polyacrons, in reference to the 



Problem of the Enumeration of Polyhedra. 



By Mr. A, Cayley. 



Read October 2nd, i860. 



The problem of the enumeration of polyhedra* 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 poly- 

 hedra 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 par- 

 ticular case, accomplished the enumeration of the octa- 

 hedra with trihedral summits. A subsequent paper "On 

 the enumeration of ^-edra having triedral summits and 

 an (<«'- i)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 re- 

 late to the problem in all its generality, and the above- 



* 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 consequently take 

 the word enumeration in the popular I'ather than the mathematical sense. 



