308] 
ON THE A FACED POLYACRONS &C. 
39 
ap to the octacrons: thus, as regards the examples, stopping at the same point as 
Mr Kirkman, for although perfectly practicable it would be very tedious to carry them 
further, and there would be no commensurate advantage in doing so. The epithet A 
faced will be omitted in the sequel, but it is to be understood throughout that I am 
speaking of such polyacrons only; and I shall for convenience use the epithets tripleural, 
tetrapleural, &c. to denote summits with three, four, &c. edges through them. The 
number of edges at a summit is of course equal to the number of faces, but it is the 
edges rather than the faces which have to be considered. 
An w-acron has 
n summits, 3n — 6 edges, 2n — 4 faces, 
and it is easy to see that there are the following three cases only, viz.: 
1. The polyacron has at least one tripleural summit. 
2. The polyacron, having no tripleural summit, has at least one tetrapleural summit. 
3. The polyacron, having no tripleural or tetrapleural summit, has at least twelve 
pentipleural summits. 
In fact, if the polyacron has c tripleural summits, cl tetrapleural summits, e penti 
pleural summits, and so on, then we have 
11 — c + cZ 6 p -t - h -t- Ac., 
6n — 12 = 3c + 4cZ + 5 e + 6/+ 7 <7 + 8 h + &c., 
and therefore 
12 = 3c+ 2 d + e+Qf— g — 2h — &c., 
or 
3c 2d c = 12 -f- g -f- 2h 4- &c.; 
whence if c = 0 and d = 0, then e = 12 at least. It appears, moreover (since n cannot 
be less than e), that any polyacron with less than 12 summits cannot belong to the 
third class, and must therefore belong to the first or the second class. 
An (n + l)-acron, by a process which I call the subtraction of a summit, may be 
reduced to an ?i-acron; viz., the faces about any summit of the (n + l)-acron stand 
upon a polygon (not in general a plane figure) which may be called the basic polygon, 
and when the summit with the faces and edges belonging to it is removed, the basic 
polygon, if a triangle, will be a face of the w-acron; if not a triangle, it can be 
partitioned into triangles which will be faces of the w-acron. The annexed figures 
exhibit the process for the cases of a tripleural, tetrapleural and pentipleural summit 
respectively, which are the only cases which need be considered ; these may be called 
the first, second and third process respectively. It is proper to remark that for the 
same removed summit the first process can be performed in one way only, the second 
process in two ways, the third in five ways; these being in fact the numbers of ways 
of partitioning the basic polygon. 
We may in like manner, by the converse process of the addition of a summit, 
convert an ?i-acron into an (n 4- l)-acron; viz., it is only necessary to take on the