PARTITIONS OF THE X-ACE AND THE X-GON. 45 
3. Let P be an (#+1)-edron in an #-gonal base, having 
only triedral summits. P has f triangular faces, f<¢2. If 
f=, no edge meets the base which is not the side of a 
triangle. If f<4v, there are x—2f edges (j) meeting 
the base which are not sides of triangles. And f cannot 
be > 3a, for if it were, there would be two triangles 
about the base having a common vertex out of it, which 
would be no triace, unless the figure were a tetraedron. 
Each of these v— 2f edges (j) connects a summit of the 
base with one (2) out of it. If we suppose that edge 7 to 
evanesce by the coincidence of the two faces about it, a 
summit disappears of the base, and a summit (2) disappears 
among those out of the base. Let all the x-—2f edges 
then disappear ; the base is now 2/-gonal, and has the same 
f triangles collateral with it, the greatest number possible, 
as has just been shown. And all the 2 -2/f summits 2 out 
of the base that have vanished have disappeared from 
edges & that do not meet the base. 
4. It follows that all (#+1)-edra having f (< 42) tri- 
angles, can be constructed from the (2/+1)-edra having 
f triangles without touching those triangles, by joining 
points () in edges not meeting the base and not in tri- 
angles, to point in base edges, the face in which any such 
joining line (7) is drawn being supposed fractured into two 
faces thereby, and every such operation adding a face, two 
summits, and three edges, but altering and adding no 
triangles. Every (2f+1)-edron having / triangles is evi- 
dently wade by cutting away every base summit of a 
certain (f+1)-edron on an f-gonal base, which may have 
any number of triangles from 2 to $f, or 3(f-1), as f is 
even or odd. 
If the number of these (f+1)-edra is known, the num- 
ber of (2f+1)-edra having f triangles is also known; and 
if the whole number of ways be known in which m join- 
ing lines (7) above described can be drawn upon these 
