146 EE7. T. P. IvIEKMAN ON THE THEOEY OE THE POLYEDEA. [§ 1. XXX., XXXI.’ 
The signatures, zonal and zonoid, give the number of polar, zonal, and epizonal edges of 
the polyedron, whereby that of the zoneless and non-polar is accurately known ; and as 
the number of repetitions of every feature is known by what precedes, the exact niimler 
of different janal cmaxine pairs upon all these solids is given hy our signatures. 
Registration of R-edra Q-acra and Q-edra R-acra. 
XXXI. It has been sho^vn that aU possible symmetrical P-edra Q-acra are comprised 
in the preceding classes. They are thus registered. 
Tables A. 
1. Zoned Symmetry. 
1. Monozone P-edra Q-acra (HI.), 
(PQ){Z}=A, 
where A is the number of monozones which have the zonal signature Z. 
2. Zoned monaxine heteroids (VIII., IX.), 
(PQ)YL {ZZ'}=B, 
(PQ)Yr.r{ z}=c, 
where B is the number of these solids having the zonal signatures ZZ' and a heteroid- 
2r-zoned axis, of which Y expresses the character and nothing more (V.). In the same 
way, we record that there are C (2r+3)-zoned heteroids, having a given character of axis, 
and a given zonal signature Z. 
We are content to know about these solids what is here registered, without asking 
what are the exact polar features ; for we have a separate table of polar summits and 
faces, in which every polar A-ace and A-gon is recorded, with its zones, and with the 
character of its axis. If the question should arise, which, however, never does arise in 
our problem, what is the exact feature opposite to a given heteroid pole, it can easily be 
determined by reference to our processes of construction. 
3. Zoned triaxines (XII.), 
(PQ)(y^y):y;^){zz'z"}=d, 
where the zonal signature and characters of the janal axes of the D polyedra are 
recorded. 
4. Zoned monarchaxines (XII.), 
(PQ)xj:y^.y;uzz'z"}=e, 
(PQ)Xfr^YL{ ZZ" }=F, 
w/here X denotes an axis whose polar features are exactly registered, the principal poles 
being always known on these solids, and recorded with their zonal traces. 
5. Zoned triarchaxines (XV.), 
W)x^y„^,y;^{ZZ,}=g. 
The principal janal poles are always registered with their traces. The secondary and 
tertiary poles may not be exactly given. 
