162 EEV. T. P. KIEKMAN ON THE THEOET OF THE POLTEDEA. [§ 2. XLin.-XLV, 
Thus all polar polyedra, zoned or zoneless, can be exactly registered, and as the signa- 
tures give the number of 'polar and zoned edges on them all^ we know upon them all the 
number of different zoneless and non-polar edges. 
A. Monozone V-edra Q-acra. 
XLIV. Let Z be any zonal signature having b epizonal (or zonal) edges, without p 
subscribed. Let E be the number of non-polar epizonal (or zonal) edges which are 
found in the zone under the number d (or d') in Table D (KKKY.) ; and let E' be the 
number of different non-polar epizonal (or zonal) edges read in the zone Z, which will of 
course here show^ subscribed (IX.) upon all the zoned polar P-edra Q-acra. This E' is 
given by inspection of signatures, and the instalment of it contributed by different polar 
polyedra will vary according to the manner in which Z is written in their signatures 
(IX.) (XVII.). 
The number of epizonals (or zonals) in Z found upon monozones will be E— E', and 
as each solid has (HI.) b of these, we obtain 
^(E-E)=A(1, XXXI.) 
for the number of monozone P-edra Q-acra having the signature Z. 
But it may be that 5=0, or that Z has neither zonal nor epizonal edges. In that case 
Z must have only diagonally traced summits. Let s be the number of these summits, 
which are by hypothesis non-polar. We have in the datum (f) (XXXVI.) the number 
d! of diagonally traced summits non-polar in the, zone Z, that are found on the P-edra 
Q-acra which have or have not zoned axes. And inspection of the signatures of all zoned 
axes pre\iously enumerated tells us how many different non-polar summits in Z these 
axial solids contain. Let this be d'. Then 
is the number of monozone P-edra Q-acra which have the zone Z. 
Thus aU monozone P-edra Q-acra and Q-edra P-acra of all zonal signatures can be 
registered, and the number of their zoned edges being given by their signatures^ that of their 
different zoneless edges is known. 
E. Janal anaxine V-edra Q-acra (17, XXXI.). 
XLV. Let G be the entire of janal anaxine pairs entered in Table D (XXXV.) of 
the P-edra Q-acra ; and let GT be the number of janal anaxine pairs on all the solids 
enumerated in art. XXX. G' is known by inspection of signatures of polyedra already 
constructed. Then, as every janal anaxine polyedron has -KP-f-Q— 2) of these pairs, 
p^|r2(G-G')=S (17, XXXI.) 
is the number of those polyedra. 
