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§ 1. VI.-IX.] EEV. T. P. KIEKMAN ON THE THEOET OE THE POLTEDEA. 
For, 1°, there cannot be fewer than two; because if two contiguous hemizones had 
exactly the same configuration, they would be each the reflected image of the other, and 
therefore not contiguous, but on opposite sides of a different zone. 
And 2°, there cannot be more than two ; for every hemizone A has on either hand, 
by definition of a zone, the same hemizone B, and B has on either hand the same hemi- 
zone A. 
The propositions of the following article are easy deductions from the above theorem. 
AVe use the term perpendicular in a wide sense, in consideration of art. II. ; that is, 
under the assumption of the greatest possible symmetry. 
VII. AVhen there is an even number of zones about a zoned axis, each is perpen 
dicular to another, and has two identical hemizones. 
If a 2m-zoned axis is the only one of the solid, there are two diflferent entire zonal 
cmjiejurations about the axis, viz. m zones Z alternating with the m zones Z'. The 
signatures of Z and TI may or may not be different ; the configurations cannot be the 
same. 
When there are 4m zones, each is perpendicular to one of the same configuration. 
AV^hen the number of zones is 4m-l-2, each is perpendicular to one of a different con- 
figuration. 
AVhen the number of zones is odd, none is perpendicular to another. 
AATien the axis is 2m-zoned, there are, in each polar feature, m traces t alternating 
with m traces t'. 
AYhen the axis is (2m-l-l)-zoned, the polar face or summit has identical traces. 
About a (2m+l)-zoned axis there is hut one entire zonal configuration, and conse- 
quently but one signature for all the zones. These zones have not each two identical 
hemizones. 
AVhether the r zones be odd or even, the 2r semi-traces present alternate configura- 
tions in the circuit of the pole. 
Each trace has two like terminations, if r be even, and two different terminations, if 
r be odd. 
A"III. Theorem. If a zoned axis is the only axis of the jiolyedron, that axis is 
heteroid. 
For if not, the opposite poles will be either identical, or one the reflected image of 
the other. 
If they are identical, there is an axis of even rejpetition perpendicular to the zoned 
axis, i. e. an axis about which in revolution of the solid, the same configuration is 2m 
times repeated to the eye ; which is contrary to hypothesis. 
If they are one the reflected image of the other, there is, at right angles to the zoned 
axis, a zone, whose intersections with those of the zoned axis are other axes of the solid ; 
which is contrary to hypothesis. AVherefore the theorem is proved. 
IX. b. Zoned monaxine heteroids. — A polyedron whose only axis is an m-zoned axis, 
is an ra-zoned monaxine heteroid jpolyedron. 
