§ 1. XXXV.] EEV. T. P. KIEOIAi^ ON THE THEOET OF THE POLTEHEA. 
155 
Begistration of edges of Y-edra Q-acra and Q-acra Y-edra. 
Tables D. 
Polar edges. 
xxxy. {AA)fY'{ZZ^ =«', 
(AA)yF:. =h\ 
■where x is am.gr, go.gr, or ed.gr (V.). Here az. means azone or zoneless. 
P=F'+2, 
Q=s'+2.A-2. 
The amphigrammic polar edges may be either janal or heteroid. When they are 
janal, they will of com’se be found both in this Table and in the Table B of janal poles 
(XXXII., XXXIII.). 
Non-gwlar zoned edges. 
(AA) ,,sT'-[Z}=6Z', 
which record d edges, each the intersection of a B-gon and A-gon, and each epizonal in 
the zone Z, and d' edges of A-gons zonal in Z. 
The number d!, and also the number d, when B=:A, will not include polar edges of 
A-gons above entered in the zone Z, a signature which may belong equally to polar and 
to monozone polyedra (IX.). 
Janal anaxine edges. 
(AB) ,,.„„s'F=e' (B^A), 
showing that there are d different janal anaxine edges of intersection of an A-gon and a 
B-gon, on all the P-edra Q-acra. 
Asymmetric edges. 
{KY,)J¥=f (B^A), 
which records the entire number f, including d last registered, of asymmetric intersec- 
tions of an A-gon with a B-gon on all the P-edra Q-acra. 
One at least of the faces about each of they' edges here registered is zoneless and 
non-polar, otherwise the edge would not be zoneless and non-polar. One of the faces 
may be polar or zonal non-polar. Thus we see that asymmetric or janal anaxine edges 
are found in faces which are not asymmetric nor janal anaxine faces (XXVI.). 
Our Tables D of P-acra Q-edra would give us a Table D of the above edges in terms 
of the summits of the edges ; but such a Table is of no use to us. 
When the Tables A, B, C, D (XXXI. to XXXV.) are completed for P-edra Q-acra and 
Q-edra P-acra, so that no two features are recorded of which one is the reflected image 
of another, our problem of enumeration and classiflcation of P-edra Q-acra is perfectly 
solved ; and we shall see that we have the power of continuing such Tables to higher 
values of P and Q. 
X 2 
