482 



XIV. "On the Enumeration of avedra having an (x l)-gonal 

 Face, and all their Summits Triedral." By the Rev. THO- 

 MAS P. KIRKMAN, A.M., Rector of Croft-with-Southworth. 

 Communicated by A. CAYLEY, Esq., F.R.S. Received 

 June 13, 1855. 



The object of the paper is to enumerate the j--edra which have an 

 (x l)-gonal face, and all their summits triedral ; or, what is the 

 same thing, to find the number of the .r-acra which have an (a: 1)- 

 edral summit, and all their faces triangular. 



Every .r-edron having an (x l)-gonal face has at least two trian- 

 gular faces. Let A be an .r-edron having all its summits triedral, 

 and having about its (x l)-gonal face k triangular faces. Suppose 

 all these triangles to become infinitely small ; there arises an (x k)- 

 edron B, having an (xk l)-gonal face, and all its summits tri- 

 edral. B will have k' triangular faces, k' being not less than two, 

 nor greater than k. And there is no other (x A)-edron but B, 

 which can arise from the vanishing of all the k triangles of A ; i. e. 

 there is no (x A)-edron but B, from which A can be cut by re- 

 moving k of the summits of B in such a way as to leave none of its 

 k' triangles untouched. 



If we next suppose the k' triangles of B to vanish, there will arise 

 an (x k ')-edron C, having an (xkk 1 l)-gonal face, all its 

 summits triedral, and k" triangular faces, k" not <2, nor >A'. And 

 thus we shall at last reduce our .r-edron, either to a tetraedron, or 

 to a pentaedron having triedral summits. 



All .r-edra here considered fall into six varieties, differing in the 

 sequence of the x 1 faces that are collateral with the (x l)-gonal 

 base. They are either irreversible, as the octaedron 6435443, the 

 seven faces about the base reading differently both backwards and 

 forwards from every face ; or doubly irreversible, as the heptaedron 

 543543, whose six faces about the base are a repetition of an irre- 

 versible period of three ; or triply irreversible, as the decaedron 

 643643643, whose faces exhibit a thrice-repeated irreversible period; 

 or they are reversible, doubly reversible, or triply reversible, as the 

 hexaedron 53443, the enneaedron 63536353, or the heptaedron 

 535353, exhibiting a single, double, or triple period, all reading 



