418 
REV. T. P. KIRKMAN ON THE REPRESENTATION OF POLYEDRA. 
of the k-gons, the base being one of these. The whole number of summits will 
therefore be 
053+ 2054+ 3a5+ — 2)a*+2 = 5 ; 
for we have counted all the summits of one k-go\i, viz. of the base. 
In order, then, that such a 5 ^-gon should be possible, it is necessary that among 
the p faces of our ;?-edral g'-acron, there should be triangles, quadrilaterals, 
pentagons, &c., of which the above equation can be affirmed. Now if q should be 
odd, and all the p faces even-angled, this equation becomes 
2o54-l-4a6H-6a8-}-&c. = 2r-l- 1, 
which is impossible. Hence it appears, that if the number of summits of a ^-acron 
be odd, while the faces are all even-angled, the closed ^'-gon cannot be drawn through 
its summits. I find exceedingly few polyedra on which the closed jo-gon and q-gon 
cannot be drawn. In fact, it is far from being necessary to their existence, that all 
the conditions of the theorems A and B should be fulfilled. 
If we cut in two the cell of the bee by a section of its six parallel edges, we have a 
13-acron, whose faces are one hexagon and nine quadrilaterals. The closed 13-gon 
cannot be drawn. But if a line be drawn from the triedral vertex to the opposite 
angle of one of the quadrilaterals about that vertex, and this quadrilateral supposed 
broken into two triangles having that line for their common edge, we shall then have 
a 13-acron whose faces are one hexagon, eight quadrilaterals, and two triangles; 
and whose summits are nine triaces and two tessaraces. Of this figure the paradigm 
can be constructed. Here I would fain beg the reader’s permission to call a 5-edral 
summit a pentace, a 6 - or 7-edral summit a hexace or a heptace. The words are at 
least convenient in speaking of the summits of polyacra. 
As authorities and analogy are alike divided about the spelling of the word poly- 
edron, I have pleased myself herein. Why polyhedron of necessity, and yet not peri- 
hodic ? 
