It is easily proved that no x -edron has a face of more than x -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 of x -edra which have an ( x -1)-gonal face, and all their summits triedral, or, which is the same thing, the number of x -acra which have an ( x — 1)-edral summit, and all their faces triangular. We may call the ( x —1)-gonal face the base of the x -edron. All the faces will be collateral with that base, and k of them will be triangular faces. If we suppose those k 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.