REFERENCE TO THE ENUMERATION Of POLYHEDRA. 249 



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 poly- 

 acrons 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 up 

 to the octacrons : thus, as regards the examples, stopping 

 at the same point as Mr. Kirkman, for although perfectly 

 practicable it would be very tedious to carry them further, 

 and there would be no commensurate advantage in doing so. 

 The epithet A faced will be omitted in the sequel, but it 

 is to be understood throughout that I am speaking of such 

 polyacrons only ; and I shall for convenience use the epi- 

 thets tripleural, tetrapleural, &c. to denote summits with 

 three, four, &c. edges through them. The number of edges 

 at a summit is of course equal to the number of faces, but 

 it is the edges rather than the faces which have to be con- 

 sidered. 



An w-acron has 



n summits, 3^-6 edges, 2w - 4 faces, 

 and it is easy to see that there are the following three cases 

 only, viz. : 



1. The polyacron has at least one tripleural summit. 



2. The polyacron, having no tripleural summit, has at 



least one tetrapleural summit. 



3. The*polyacron, having no tripleural or tetrapleural 



summit, has at least twelve pentipleural summits. 

 In fact, if the polyacron has c tripleural summits, d 

 tetrapleural summits, e pentipleural summits, and so on, 

 then we have 



SER. III. VOL. I. K K 



