§ 1. XX., XXI.] EEV. T. P. KIEKMAJ^ ON THE THEOET OF THE POLTEDEA. 
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polar face with the hexace 123456, and in the lower polar face with the hexace 
1^2^3^4^5^6p the solid completed will be a triple monarchaxine. If we crown the opposite 
polar faces of the latter with tessaraces upon the marginal triangles, we form a zoneless 
triaxine polyedron. 
XXI. System of principal zoneless axes. Zoneless polyarchaxines. — The following 
propositions have been sufficiently established by the proof in arts. XIII., XIV., that 
every system of principal axes is that of a regular polyedron. 
We give an account in the zonoid signature of a polyarchaxine, of its secondary and 
tertiary poles. 
A zoneless triarchaxine has three principal 4-ple janal axes, four secondary triple janal 
axes, and six double janal tertiary axes. Its zonoid signature is 
^=: { 8(o-^+/p)+12((rp4-/^')+0^^“"}, 
where 
describes the secondary, and 
the tertiary poles. 
The poles of the same rank are of one name only. Tkery non-polar feature is read 
twenty four times in the solid, viz. four times about each pole of every principal axis. 
Such a solid is here half-drawn, the portion unseen being identical with that seen, 
and below the page. 
/ 
This triarchaxine 72-edron 62-acron has am phigonal principal and secondary 
axes, and amphigrammic tertiary axes. If we efface the twelve polar edges of the 
tertiary axes, we obtain a 60-edron 62-acron having amphiedral tertiary axes. 
A zoneless tetrarchaxine polyedron has four triple heteroid principal axes, and three 
double janal secondary axes. Its zonoid signature is 
^— { + 6(®'p+/y) 4- 
where 
<^p+/p+«')=l 
describes the secondary poles. 
MDCCCLXII. T 
