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§ 1. IX.-XII.] EEY. T. P. KIEKMAN ON THE THEOET OF THE POLYEDEA. 
The polar triangle has three monogonal traces ; the polar hexagon has three agonal 
traces. 
X. Principcd zoned axis. Secondary axes. — A principal axis has more zones or not 
fewer zones than any other. The zones which intersect in it are principal zones. 
Theorem. If there be but one principal zoned axis., A, it is janal ; the solid has one., 
and but one., zone besides those of the axis, and this zone is perpendicular to the axis A. 
For Z", a zone not containing the principal axis, cannot divide the solid into halves 
of which one is the reflected image of the other, unless the axis is janal and at right 
angles to TJ'. As the axis can be perpendicular to but one zone, the truth of the theo- 
rem is evident. 
This Z" is called, when m>2, the secondary zone of the polyedron, and its intersec- 
tions with the zones of the principal axis are the secondary zoned axes of the polyedron. 
XI. Theorem. A 2m-zoned principal axis has 2m 2-zoned secondary axes, all janal; 
the 4m secondary poles of these axes are in the secondary zone, of two alternate configura- 
tions, and the alternate secondary cures are different. 
The 2m axes are 2-zoned, being each the intersection of the secondary with a prin- 
cipal zone. The rest is evident, if we make a section of the solid in its secondary zone ; 
for the 2m traces of this section are the 2m axes (VII.). 
Theorem. A {2m.-{-V)-zoned principal axis has 2m-]- 1 secondary cures. Their 4m -{-2 
secondary poles are in the secondary zone, in the circuit of ichich they present two alter- 
nate configurations. The secondary axes are all 2-zoned, all alike, and all heteroid. 
XII. d. Zoned monarchaxines. — A polyedron having only one principal m-zoned axis 
is an m-zoned monarchaxine janal polyedron, which is also sufflciently described as an 
m.-zoned monarchaxine (X.), wFere the number m does not include the secondary zone. 
When m=2 no axis is principal, and there are three 2-zoned axes, any one of which 
is perpendicular to the other two. These three axes are of three difierent configura- 
tions ; for if two semi-axes at right angles to each other were identical, they would 
reflect each other, and a zone would pass between them ; whence it would follow that 
the third axis were no 2-zoned axis. 
c. The solid in this case is called a zoned triaxine polyedron. It has three zones of 
difierent configurations. 
The zonal signatures of a 2r-zoned monarchaxine (r > 1) are thus written, ZZ' being 
principal zones, and Z" being secondary : 
Z=[2{u^+;^+2g), 2(/,-b^,+2G), or, Of, 0^ 0^^}, 
Z’ = [2{a^+;^+2cJ), Or, Of, 0^ 0^^’}, 
Z"=2r{(^,-h?;+2/), (9 ^-f^;+ 2G"), Or"', 0;"“’, Of 
where 
^pA'fp 1 
is the principal pole, summit or face, in the janal axis ; and 
