328 
ON THE THEORY OF GROUPS. 
[690 
To explain the theory, I introduce the notion of a hemipolyhedron, or say a 
hemihedron, viz. this is a figure obtained from a polyhedron by the removal of 
certain faces. In a polyhedron each edge occurs twice (more properly it occurs in 
the two forms ab and ba), as belonging to two faces; but in a hemihedron one of 
these faces must always be removed, so that the edge may occur once only; and 
again (what is apparently, although not really, a different thing), we may remove two 
intersecting faces, leaving their edge of intersection; this edge is, in fact, then considered 
as a bilateral face ab = ab. ba, just as abc is a trilateral face abc = ab .be. ca. Thus, if 
in a prism we remove the lateral faces, leaving the lateral edges, and leaving also the 
terminal faces, we have a hemihedron: thus, the prism being trilateral, say the faces 
of the hemihedron are abc, clef, ad, be, cf where ad, be, cf are the edges regarded as 
bilateral faces. And, for the present purpose, abc denotes the cyclical substitution a 
into b, b into c, c into a; and ad denotes in like manner the cyclical substitution 
(or interchange) a into d, d into a. 
But the hemihedron about to be considered has no bilateral faces; it is, in fact, 
the figure composed of the 8 triangular faces of the octo-hexahedron or figure obtained 
by truncating the summits of a hexahedron (or of an octahedron) so as to obtain a 
polyhedron of 8 triangular faces and 6 square faces, representing the faces of the 
octahedron and the hexahedron respectively. The faces of the octo-hexahedron may 
be taken to be 
abc, def glii, jkl, 
ajg, bif, cek, dhl, 
cbfe, fihd, hgjl, jack, aegib, klde, 
(where I observe in passing that the symbols are written in such manner that each 
edge ab occurs under the two opposite forms ab in abc and ba in agib). And then, 
omitting the square faces, represented by the third line, we have the hemihedron, 
wherein as before abc denotes the cyclical substitution a into b, b into c, c into a; 
and so for the other faces. 
I represent this by a diagram, the lines of which were red and black, and they 
will be thus spoken of, but the black lines are in the woodcut continuous lines, and 
the red lines broken lines: each face indicates a cyclical substitution, as shown by 
the arrows. The figure should be in the first instance drawn with the arrows, but 
without the letters, and these may then be affixed to the several points in a perfectly 
arbitrary manner; but I have in fact affixed them in such wise that the group given