152 
EEV. T. P. KIEKMAN ON THE THEOET OE THE POLTEDEA. [§ 1. XXXIII. 
degrade the symmetry about any higher axis f3, zoned or zoneless, to which a may he 
secondary, and the janal constructions will be all zoneless triaxines. For the entire 
symmetry about a 2r-ple axis j3 cannot be preserved, if r >1, by janal constructions on 
the poles of one only of the r axes a perpendicular to j3 ; that is, (3, by constructions 
on a only, is degraded to a 2-ple repetition. 
When /3 is a 2r-zoned homozone axis, <y and a are identical in configuration ; and any 
janal construction on the poles of a destroys the zones, and (3 becomes a zoneless 2-ple 
axis. 
When the 2-ple axis is contrajanal, and perpendicular to a zone, it is entered thus, 
=m, 
with the zone to which it is perpendicular. The solid on which A is found may be 
either a (2r-l-3)-ple homozone (XXII.), or a 2-ple monaxine monozone. In either case 
the A-gon is the reciprocal of a 2-ple A-ace constructed by our processes as a monaxine 
monozone, having a 2-ple contrajanal axis perpendicular to a given zone, and the entry 
is made as above. The only possible polar janal constructions on the two contrajanal 
polar faces are 2-ple monaxine monozones. 
Janal anypliigrammic 'poles. 
The register of janal poles is completed by the janal polar edges of amphigrammic 
axes, zoned and zoneless. 
The polar edge is registered as the intersection of two A-gons ; and we write 
s'=Q-2A-l-2 
for the number of summits of the P-edron Q-acron not in the A-gons, and 
F'=P-2 
for that of the faces distinct from them. 
(AA)|:^‘^'VF|ZZ'Z"(=?i, 
(AA)jfW| Z^ \^p, 
(AA)j„ s'F =^. 
We read that there are n difierent polar edges of janal A-gons epizonal in Z and zonal 
in Z', Z" being perpendicular to the amphigrammic axis. 
'SA"’e read that there are p polar edges of A-gons, both zonal and epizonal in Z, and 
perpendicular to zoneless axes of given zonoid signature. 
And there are amphigrammic zoneless axes, whose polar edges are intersections of 
A-gons. We have no occasion here to register any zonoid signature of other axes. 
The n edges may be tertiary poles of zoned triarchaxines or hexarchaxines, or secon- 
daries in zoned monarchaxines, or in zoned triaxines. In the first of these four cases 
we shall read Z = Z', in the second, Z=Z'=:Z". 
Thej? edges are either zoned poles of homozone triaxines, or possibly secondaries of 
zoned tetrarchaxines (XL.), if Z is a tetrarchaxine zonal signature. 
