§ 1. XXII.] EEV. T. P. KIEKMAN ON THE THEOEY OF THE POLTEDEA. 
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and there will be 2r of these zoneless poles ; one central between every two hemizones 
about the r-zoned axis. And as the same sequence is read about them all, the 
2r-zoneless poles have all one configuration, or configurations of which one is the 
reflected image of another. 
Let PP' be the only principal axis of the solid ; then, as the pole P can recur only 
twice in a revolution about the zoneless poles, they are all of double re'petition. Where- 
fore there are r-janal and similar axes of zoneless double repetition in a plane perpendi- 
cular to the r-zoned axis PP'. 
1. Let r be odd; then every zone of the axis PP' is perpendicular to one of the 
r zoneless axes, or symmetry would be impossible ; and it is consequently a repeating zone, 
of which every non-polar feature is an ohjanal monozone feature, i. e. a feature y* diame- 
trically opposite to another which is to an opposite eye the inverted image off (art. IV.). 
Further, when r is odd, the poles aa! of any one of the r zoneless axes are contrajanal 
poles', for, if not, the axis aa' will be strictly j anal, such that two opposite eyes in the 
axis will read exactly the same configurations from left to right : therefore the axis PP' 
perpendicular to aa' will be an axis of even repetition, since the same pole a recurs 
exactly in half a revolution about PP'; which is absurd, because, r being odd, PP' is an 
axis of odd repetition (XVIII.). 
Therefore aa' is, w'hen r is odd, a contrajanal axis. 
2. Let r be even ; then because the axis PP' is of even repetition, the configurations 
read by two opposite eyes in the axis aa' are strictly identical, and aa' is a zoneless 2-ple 
strictly janal axis. 
When r is even, either zone of M conspires to form the zone of the solid (MM'), as is 
evident from the position of the trace t upon the trace t'. For this reason the solid is 
called homozone, the two zones of M being confounded together. And this name is con- 
veniently used to designate the solid (MM'), whether r be odd or even. 
The axis PP' is an x-zoned homozone axis. 
When r>2, the axis PP' is a principal axis, and the solid is a homozone monarchaxine 
folyedron. When r=2, there is no principal axis (XX.), as the zoned axis has, like the 
two zoneless ones, but a 2-ple repetition. The 2-zoned homozone is a triaxine homozone 
pohjedron. 
The zonal and zonoid signatures of the r-zoned homozone polyedron, for r odd or even, 
are 
Z={(2«-,+2yX2/,+2G) O; O; 0“ 0“}, 
where 
ffp-f-/j,-l-a=I, and a=0, if r>2, 
K>—^r{gp-\-(Pp-\-0p}, 
where 
describes the zoneless pole. 
Every nonpolar zoned feature is read on the solid 2r times, namely, once in each hemi- 
zone about the zoned axis. 
T 2 
