4.4 REV. J. P. KIRKMAN ON THE 
remains unaltered. By causing all the edges which do not 
meet any face to vanish, we reduce the figure to a pyramid 
upon that face. Now the relation is true of the pyramid ; 
therefore it is true of the undiminished polyedron. 
TuroreM A. In every (#@+1)-edron P having an x-gona, 
face X, there are at least two triangular faces. 
For there is no face of the (w+1)-edron not collateral 
with X: wherefore the edges of P which do not meet X 
can enclose no space, but must form a broken line whose 
terminations are p, P,..., two or more. From each 
point p, p,... two or more edges must pass to meet the 
base, since all the summits are at least 3-edral. That is 
Py; Pz... ave vertices each of one or more triangles collate- 
ral with X. Q. H. D. 
2. The partitions of an w-ace, (w rays diverging from a 
point,) correspond to those of the #-gon. The diagonal 
joining the m™ and 2 summits of the latter answers to 
the intersection of the m‘ and n™ faces about the former. 
The number of triedral partitions of the #-ace is that of 
the triangular partitions of the z-gon, in which of course 
no two diagonals cross each other. A triangular partition 
of a non-planar or skew 2-gon, crowned by a simple 2-ace 
whose rays pass through its # summits, is an (v+1)-acron 
with only triangular faces. A triedral partition of an 
g-ace, standing on an unpartitioned x-gon, is an (#+1)- 
edron whose summits are all triaces. 
If we can enumerate the (v+1)-edra on an #-gonal base 
which have all their summits triaces, we shall have the 
solution of the problem of the triangular partitions of the 
x-gon, or of the triedral partitions of the x-ace. The ob- 
ject of this memoir is to show that this problem is very 
conveniently attacked by the consideration of these (v¥+1)- 
edra, and that this mode of solution is, where all modes 
seem difficult, as easy as any. And the view here taken 
of the question is entirely new. 
