§ 2. XLVL-XLYIIl] EEV. T. P. EIEEMAJJf ON THE IHEOEY OF THE POLYEDEA. 163 
0. Asymmetric ^olyedra (18, XXXI.). 
XLVI. Let J be tbe entire number of asymmetric edges on the P-edra Q-acra, which 
are found in Table D (XXXV.). We have shown that the number of repetitions of 
every zoneless non-polar feature is given for all the symmetrical polyedra ; and as the 
signatures give the number of polar and zoned edges on them all, we know how many 
different zoneless and non-polar edges are on them all. 
Let this number be J' ; then 
PTk2(J-J')=0 
is the number of the asymmetric P-edra Q-acra. 
XL VII. e', f'. Monozone faces of Y-edra Qracra (XXXIV.). 
The number d' (XXXIV.) is supposed ahready known (f) (XXXVI.). 
Let be the number of non-polar edges epizonal in the zone Z (XXXV., d) 
(g) (XXXIV.), which are the intersection of an A-gon and a B-gon. 
Let 
be the entire number of different edges of A-gons, epizonal in Z, where we write 27^^^, 
because these edges are each in two different A-gons. 
The Table of polar faces (XXXII.) gives us the entire number of different edges of 
A-gons that are epizonal non-polar in Z in polar faces. A polar face may have either 
one trace or two different traces, whose signature is Z ; for the two zones, though of dif- 
ferent configuration, may have the same signature. In all cases, the number of different 
epizonals in Z of that polar face is given by inspection. 
Let be the whole number of non-polar epizonals in Z of polar A-gons. There 
remain 
^^A(Z) — Pa(Z) 
edges epizonal in Z of A-gons, which edges are in non-polar A-gons. When A is even, 
each A-gon has two of them; when A is odd, each A-gon has only one of them. 
VTierefore for A even, 
i(AA(z)-i?A(Z))=A^^sF{Z} =e' (XXXIV.), 
and for A odd, 
/Wz)-i>Acz) = A-sF{Z} = f (XXXIV.). 
We can therefore enumerate all the monozone faces of the P-edra Q-acra, and of the 
Q-edra P-acra, and register each with its trace and zonal signature. 
XLVIII. Ohjanal monozone faces. — When Z of the preceding article is the signature 
of a repeating zone, certain of the A-gons just enumerated will be the objanal mono- 
zone faces of (e), XXXVI. It is important that the objanal monozone faces should be 
separately registered, and it will be necessary to subtract these A-gons when they exist 
from the number of A-gons just found, in order that the monozone K-gons^ which have 
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