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It is easily proved that nox -edron has a face of more thanx -1 angles, and that, if it has an (x —1)-gonal face, it has at least two triangular faces. The object of this paper is to determine the number ofx -edra which have an (x -1)-gonal face, and all their summits triedral, or, which is the same thing, the number ofx -acra which have an (x — 1)-edral summit, and all their faces triangular. We may call the (x —1)-gonal face the base of thex -edron. All the faces will be collateral with that base, andk of them will be triangular faces. If we suppose thosek triangles to become infinitely small, in any.x -edron A, we have as the result an (x —k )-edron B, having only triedral summits, none of whose triangular faces was a triangle of A. And it is evident that there is only one (x —k )-edron B from which A can be cut by sections that shall remove no edge entirely, and shall leave untouched no triangle of B. It is plain also that B cannot have more, but may have fewer, triangles than A; for if the vanishing of a triangle of A gave rise to two tri­angles in B, B, having two contiguous triangles, and all its summits triedral, would be a tetraedron.

XVII. On the enumeration of x-edra having triedral summits, and an (x—1)-gonal base