148 
EEV. T. P. KIEKMAN ON THE THEOET OE THE POLTEDEA. [§ 1. Yxyt. 
13. Zoneless tetrarchaxines (XXI.), 
^(PQ)yly;uO=Q- 
We may not know always the edges in the pole opposed to a given tetrarchipole. 
14. Zoneless Jiexarchaccines (XXI.), 
There are but few values of P and Q for which polyarchaxine polyedra are possible ; and 
the ambiguities above denoted with respect to their principal poles can only exist for 
high values of P and Q, such as will not be calculated for a millennium or two. 
3. Mixed Symmetry. 
15. {x-\-2>)-]ple monaxine monozones (XXIII.), 
(PQ) 
(PQ) 
mo.mo 
mo.mo 
x:+^{Z}=s, 
YL;4Z}=S', 
where the exact polar features are always known, except for 2-ple axes. This is, how- 
ever, sufficient for our purpose, as we shall see (XXXIX.). 
16. Eomozones 2Y-zoned and {2Y-{-l)-zoned (XXII.). 
Y% { zo=T,(r^i), 
ZO=T' (r>l), 
(PQ).o.Y^.y.Y;^{^>«zo=u, 
(PQ)nomY% Yl { ^ZO=W. 
In all these entries the first axis is zoned, the second zoneless. In the third, U, ®’®Z 
denotes a zonal signature of the triarchaxine form, or hexarchaxine, or possibly of both 
forms. When such a zone occurs in homozones along with a zonoid signatm’e show- 
ing poles of a name not excluded from Z, we may not always know the exact edges of 
the zoned pole (vide XXXVII., XXXIX.). 
When this ambiguity does not exist, the numbers U and W (where ^Z denotes a 
signature of the tetrarchaxine form) will not be found in our Table ; and they can only 
present themselves for high values of P and Q, which will not be calculated for the next 
thousand years. But it is enough for us that the numbers U W are exactly known, as 
this suffices for our problem. 
4. Neuter Symmetry. 
17. Janal anaxine jpolyedra (XXV.), 
(PQ)ya.a»— 
in v. hich there is neither zone nor pole, E being the entire number of the solids. 
