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NOTES ON POLYHEDRA. 
[375 
3. For any one of the regular polyhedra, E being the number of edges, then 
the number of axes or Si is = i?+l, and their weight or 1 + 2 (q — 1) is = 2E. In 
fact, if as usual S denote the number of summits, F the number of faces, and if 
there be to edges to a face, and n edges to a summit, then S + F=E + 2, mF=nS = 2E. 
Now in all the polyhedra except the tetrahedron, we have a number \F of TO-axes 
passing through the centres of opposite faces (amphihedral axes as Mr Kirkman has termed 
them) and a number of n-axes passing through opposite summits (amphigonal axes); 
and we have besides a number \E of 2-axes passing through the mid-points of opposite edges 
(amphigrammic axes): the entire number of axes is thus ^ (S + F+ E), which is = E-f 1: 
and the weight is 1 + \F(m — 1) + (n — 1) -f- \E, which is = 1 + ^mF+^nS —-| (F + S — E), 
= 1+E + E— 1, =2E. In the case of the tetrahedron $=F=4, m = n = 3, and the 
only difference is that instead of the ^F amphihedral TO-axes and the amphigonal 
■w-axes, we have a number (F = S =) ^ (F + S) of (n =) TO-gonal axes each through a 
summit and the centre of an opposite face (gonohedral axes). 
4. The theorem that the weight 1 + 2 (q — 1) = 2E, or say 1 + 2 (q — 1) = niF, may 
be extended so as to apply to any polyhedron whatever. In fact considering any face 
A of the polyhedron, let F be the number of faces homologous to (and inclusive of) 
A; and, taking a any edge of the face A, let to be the number of edges of A 
homologous to (and inclusive of) a: then we have 1 + X(q— 1) =mF. This is almost 
a truism when the signification of the term “ homologous ” is explained. Imagine the 
polyhedron placed on a plane, say the table, and draw on the table a polygon equal 
to the polygonal face A, and in this polygon select some one edge corresponding to 
the edge a. The polyhedron may be placed on the table with the face A coinciding 
with the polygon, or say the face A may be superimposed on the polygon, and that 
in to different ways, viz. any one of the edges homologous to a may be made to 
coincide with the assumed edge: and in like manner there are F different faces (viz. 
the faces homologous to A) which may be superimposed on the polygon, each of them 
in w different -ways; that is there are in all mF different positions of the polyhedron 
for each of which it occupies the same portion of space. And we have thus the 
required theorem 1 +2 (q— 1) = mF. 
5. As an example, take the regular pyramid on a square base; there is here a 
single axis, viz. a 4-axis, and we have 1 + 2 (q — 1) = 1 + 3 = 4. If for the face A we 
take the square base, then there is no other face homologous thereto and therefore 
F = 1; but the four sides are homologous to each other or to = 4, and we have 
mF = 4. Similarly taking for A one of the triangular faces, since these are homologous 
to each other, then F= 4; and if we take for the side a the base of the triangle, 
then there is no other side homologous to this, or m — 1 ; and therefore mF= 4. It 
might at first sight appear that the two equal sides of the triangle were homologous 
to each other, and therefore that taking for the edge a one of these sides we should 
have to = 2 ; but in fact although the two sides in question are homologously related 
to the pyramid, yet according to the definition they are not homologous sides of the 
triangular face, and we still have to = 1, and therefore mF = 4.