138 REV. T. P. KIRKMAN ON THE THEORY OE THE POLTEDRA, [§ 1. XXL, XXII. 
The principal poles may be of one name, or of two names. 
Every non-polar feature is read twelve times on the solid, viz. thrice about the pole^ of 
every principal axis. 
A zoneless hexarchaxine polyedron has six quintuple janal principal axes, ten triple 
janal secondary axes, and fifteen double janal tertiary axes. Its zonoid signature is 
^={20(s+/,), 30(.;+/,)+or'}, 
where 
2 'ives the secondary, and 
'^P+/p+a" = l, 
the tertiary poles. 
Every oion-polar feature is read sixty times on the solid, viz. five times about each 
extremity of every principal axis. 
Zoneless polyarchaxines are easily constructed on zoned ones, by drawing lines in 
every principal, or secondary, or tertiary face, by which the zones are destroyed, and the 
repetition preserved ; or by crowning like polar faces with polyedra which have a zoneless 
repetition of equal rank. 
We shall see, by our processes of construction, that we always know the name of the 
pole opposite to any zoned or zoneless pole that we may be handling ; for we always 
know the character (V.) of the axis ; but the number of edges in that opposite pole is 
not thereby given. 
We know of course the edges of every pole that we construct, but when the axis is 
heteroid, we do not always know the exact edges of the pole opposed. 
Hence it may happen, in high values of P and Q, that we do not always know the 
exact feature opposite to a given principal pole of a tetrarchaxine. But this is not of 
the least consequence in our problem ; and if we should wish to know what is the exact 
feature so opposite, a question that never arises in our argument, we can easily determine 
the point by other considerations. 
3. Mixed symmetry. 
XXII. The polyedra which have a mixed symmetry are — 
a. r-zoned homozone polyedra. 
h. r-ple monozone monaxine polyedra. 
Homozone axes . — Let M M' be two identical r-zoned polyedra having a 2r??^-gonal polar 
face F. There are m edges of F between two contiguous traces (VII.), wherefore the 
traces of F are either all agonal, or all diagonal. Let the identical polar faces FF' of M 
and M' be so united that the trace t of F shall cover the trace t of F' of different con- 
figuration from t (VII.). Letjpy?' be two contiguous terminations of traces in the united 
faces FF' ; and let PP' be the poles of the r-zoned janal axis of the solid MM'. The 
sequence Py>Py is a repetition; there is therefore di, ^ole of even repetition hetvfeeii p and 
p', which is a zoneless pole, because no zone by hypothesis intervenes between andjp'; 
