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XVII. On the Enumeration of yi-edra having Triedral Summits, and an (x — \)-gonal 
Base. By the Rev, Thomas P. Kirkman, M.A. Communicated by A. Cayley, 
F.R.S. 
Received June 13, — Read June 21, 1855. 
It is easily proved that no x-edron has a face of more than x— I angles, and that, 
if it has an (x— l)-gonal face, it has at least two triangular faces. The object of this 
paper is to determine the number of .r-edra which have an {x — l)-gonal face, and all 
their summits triedral, or, which is the same thing, the number of .r-acra which have 
an (^— l)-edral summit, and all their faces triangular. 
We may call the {x — l)-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 .r-edron A, we have as the result 
an (j:— A:)- 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 (,i — A:)-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 tri- 
angles in B, B, having two contiguous triangles, and all its summits triedral, would 
be a tetraedron. 
If now we suppose the triangular faces of B to vanish, of which there must at least 
be two, there will arise a polyedron C, having only triedral summits, and fewer faces 
than B. In like manner C is reduced by the vanishing of its triangles to a solid of 
still fewer faces, and, by this continual evanescence of triangular faces, we shall 
finally arrive, either at a tetraedron, or a pentaedron with two triangular faces. 
Hence it appears, that every :r-edron having its base (a?— l)-gonal, and all its sum- 
mits triedral, can be cut from one of these two simple solids, by reversing the above 
process, i. e. by cutting away k summits of the base of a polyedron B, having k—h 
triangular faces, so as to leave none of those k—h triangles untouched. And by this 
process no polyedron A can be twice generated. It is to be remembered that we are 
all through handling no polyedra but those whose summits are all triedral. 
The pentaedron on a 4-lateral base, has around that base the faces 3434. This I 
call a doubly reversible polyedron, as it exhibits in the faces about the base the series 
34 repeated, and reads backwards and forwards the same. 
If we cut every base summit of the tetraedron, we obtain a heptaedron having 
around its hexagonal base the faces 353535. This I call a trebly reversible hepta- 
edron. 
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