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§ 1. XX.] EEV. T. P. KIEKMAN ON THE THEOET OP THE POLTEDEA. 
Let the r-ple heteroid monaxine P be placed by its polar mr-gonal face A in any way 
on A' the same face of P' identical with P, so that the summits of A and A' may coincide. 
As the two opposite poles aod of the solid PP' are identical, there is an axis |3 of double 
repetition, zoned or zoneless, at right angles to the axis aa', and, as this axis aa' is r-ple, 
there must be r axes /3, having r identical poles jp and r identical poles one of each 
ki nd in each repeated sequence about ««'. 
Let r>2, and let aa! be the only principal zoneless axis of PP'. Then (3 is not 
zoned ; for if it were, it would have at least two zones, neither of which would contain 
the zoneless poles aa'. There would then be at least four poles a, one between each 
pair of hemizones about f3, and aa' would not be the only principal axis. Wherefore 
j3 is zoneless, and there are r zoneless axes j3. 
Let be two like poles of axes j3 most nearly contiguous. They form with the prin- 
cipal poles aa' a repeated sequence; wherefore there is a zoneless pole of double 
repetition between y) and j^', of a different configuration from and there must be r 
zoneless poles j) alternate with r zoneless poles around the axis aa'. 
The solid PP' (r>2) is an r-p/e monarchaxine poly edr on. It has r secondary axes, of 
double repetition, icMch are, according as r is even or odd, all janal and alternately 
different, or all heteroid and alike; and, in either case, the 2r poles present alternate 
configurations. 
In our construction there is a sequence of zonoid edges (in the faces A) in general 
efiaceable ; but we shall find that there are r-ple monarchaxines which have no such 
edges effaced, or effaceable, by effacements which shall preserve all summits. 
Zonoid signature . — Every r-ple monarchaxine has a zonoid signature, which gives an 
exact enumeration of the secondary poles, whether they be faces, poles, or edges, but no 
account of the number of edges in polar faces or summits. This signature has the form 
where 
(°'+/ +«)= 2 , 
and where ever)- solution of the latter equation gives a different system of secondary 
poles. The distinction between the symbols 0 b (II.) in the account of zoneless polar 
edges vanishes. The poles in ^ may he of two names, or of one name only. 
When r=2, there is no principal axis, the demonstration that (3 is zoneless fails, and 
(3 may he a zoned axis. In such case there are as many axes aod as there are zones about 
|3 perpendicular to aci, and there cannot be more, because ad being an axis of repeti- 
tion, must be central in the interzonal space in which it appears ; and the limiting zones 
of that space must evidently be both of the same configuration, for otherwise aod would 
be no axis of repetition. In this case the symmetry is mixed, and will presently he 
discussed. 
When r=2 and (3 is a zoneless axis, it follows that /3 is janal, otherwise aa' would be 
no axis of repetition ; and as the poles of aod and (3 form a repetition, there must he a 
thii’d janal axis y of even repetition at right angles to aa' and to |3- 
