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EEV. T. P. KIRKMAJf ON THE THEORY OE THE POLTEDRA, [§ 1. XIV.-XVI. 
XIV. If there be a system of collateral polar summits AiAgAg ... of the same con- 
figuration in any polyedron P, it can be reduced by sections passing only through 
edges AiAg, A2A3, A3A4, ... to a regular polyedron having only the edges AjAa, A2A3, 
-^ 3 -^ 4 ? • • ' 
If a system of principal polar summits be not collateral, sections of the solid can be 
made, removing all the edges of those summits, and laying bare a system of as many 
principal faces of a solid having fewer edges. The reciprocal of this has a system of 
principal polar summits, which can be treated like the preceding one ; and thus we shall 
inevitably arrive finally at a solid having collateral principal summits, which reduces by 
a set of simple sections, as above shown, to a regular polyedron. 
It is thus proved that the only systems of principal summits, zoned or zoneless, are 
those of the regular polyedra. 
The following description of zoned polyarchaxines is easily verified by considering the 
regular solids. 
XV. A zoned triarchaxine polyedron has three 4-zoned janal principal axes, four 
3-zoned objanal secondary axes, and six janal 2-zoned tertiary axes. The axes may have 
various characters (V.). 
The zonal signatures of the zoned triarchaxine are 
z={(4s,+44+8yX¥,+v;'+8G)ororo“0"‘}, 
Z,={(2s,+44+2s;+4y,)(2/,+4/;+2/;+4GJ0ror0“'O‘*'}. 
There are six zones and three zones Z. 
The principal poles are 6(s -]-/’)= 6.1. 
The secondary poles are 8(5'+/’') = 8.1. 
The tertiary poles are 12(s"-j-y^'+“^^+/3^0=12.1. 
The secondary poles, as well as the tertiary, are of one name only. 
The numbers G, «, h, b^, See., enumerate the non-polar zoned features 
which have all different configurations. The sum of these numbers >0. 
Every non-polar zoned feature is read 24 times on the solid. Every zoneless feature is 
read 48 times on the solid, in as many interzonal regions. 
XVI. Def. A janal zoned axis is homozonal, when, the axis being horizontal, two 
opposite eyes can see in the poles at the same time, one the trace t vertical between t'H, 
and the other t' vertical between tt. When there are but two traces in the pole, one 
eye will see the trace t vertical and ^ horizontal, while the opposite eye sees t' vertical 
and t horizontal ; and this is the configuration seen by opposite eyes in the secondary 
axis of a zoned tetrarchaxine. 
A zoned tetrarchaxine polyedron has four heteroid principal 3-zoned axes, and three 
secondary homozonal 2-zoned axes. It has six identical zones, whose signature is 
z={ 2 (s,+s,+^), 2(/,+/;+g) 0;' O;' 0“ 0“}- 
The principal poles are 
