208 
EEV. T. P. KIEKMAI^ OX AITTOPOLAE POLTEDEA. 
axis of symmetry, we had drawn any pair of diagonals so as to preserve symmetry, 
such as two (e, r—e) and r+ej, or the pair (r, e) (r, r — e\ or the pair (e, c+/), 
{r—e, v—e—f) ; of which the first pair are parallels, the second pair meet in the pentace 
7\ and the third pair are equidistant from the centre of the axis of symmetry on opposite 
sides of it ; or suppose that we had drawn any number 2^, of such lines, making X’ pairs, 
each preserving the symmetry about the axis through r. The gamic operations being 
performed in the faces opposite the afiFected summits, the sjunmetry would remain, and 
the result would be autopolar. If, next, the summits e and (r—e), {e-\-f) and (r — e—f), 
&c., be exchanged in the parallels to (k, ^+1), the arrangement will become nodal and 
remain autopolar. The pair (e, r — e) and (e^ r — do not change their names; the 
pair (r, e) and (r, r—e) make a simple exchange of names with each other, as do also 
the pan- (e, e-\-f) and (r—e, r—e—f). The line (r, e) of the enodal arrangement, by 
changing its name to (r, r — e) in the nodal arrangement, is no longer the gamic to (E, E) ; 
but to (E, E — E) an edge symmetrically placed with (E, E) about the axis of symmetry 
through r. From these considerations, which need not be further dwelt upon, it is plaia 
that all the enodally autopolar (r+3)'6<lra, generable by symmetrically drawn pairs of 
generators in the enodal Q, are also nodal, and are consequently among the class (Q) 
which have been already enumerated. For every one of them, in its nodal shape, can 
be reduced to the nodally-signed pyramid from which the (Q) were generated. 
As any bisector of the enodal Q from any summit is an axis of symmetr)', it is dear 
that all the parallel or symmetric pairs of generators that can be drawn in Q, are among 
those just considered about the axis through r. And, for the same reason, of any pair 
whatever that can be drawn in the enodal Q, one may be always assumed to be drawn 
from r ; or, if it be more convenient, perpendicular to the axis through r ; and if two 
are drawn from one point, or to meet in a point, this point may be assumed to be r. 
It may be hereafter convenient to define autopolarity as of three kinds, nodal, enodal, 
and utral ; understanding by the first, purely nodal, or incapable of signature without 
two nodal summits, and by the second, purely enodal, or incapable of signature with a 
nodal summit or summits. The autopolars just reviewed, which are capable of both 
nodal and enodal signature, are then utrally autopolar. 
XXXIII. Now let two non-parallel and non-symmetrical generators in the enodal Q 
be drawn. We may assume one (r, e) to be drawn fr-om r; the other is not parallel to 
(r, e), nor of the same length with it, whether it meets it at r or not. If this other be 
{r, f), whether y" be on the same side or not of the axis of symmetry "uith (r, e), it is 
impossible that by exchanging the signatures of the diagonals perpendicular to that axis, 
the result should be autopolar. For e and y being 4-aces not in the same perpendicular, 
will after the exchanges be triaces, while E and F remain 4-laterals. And if (r, e) (k, 1) 
be the two generators, whether (k, 1) is or is not perpendicular to the axis of symmetry, 
e, k and I cannot all, after the exchanges, be 4-aces ; but one of them (e or k) a^EI appear 
as a triace, while E or K remains a 4-lateral. The enodal Q made by this (r, e) and 
(k, f ), or by this (r, e) and (k, 1), is then not capable of being made nodal, and thus 
