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4 1. XXXI.] EEY. T. P. KXRKMAJv^ ON THE THEOET OE THE POLTEHEA. 
6. Zoned tetrarcliaxines (XVI.), 
XPQ)yl;y;^{Z}=h. 
The number of edges in the two heteroid principal poles, and in the janal secondary- 
poles, may not always be kno-wn in this Table. 
7. Zoned hexarchaxines (XVII.), 
^(PQ)x^^y„^,y;.uz}=i. 
2. Zoneless Symmetry. 
8. Zoneless x-^le monaxine lieteroids (XVIII.), 
(PQ)YL.=J (r>l), 
where the absence of zonal and zonoid signature, and the heteroid axis, are character- 
istic of the class. 
9. Zoneless r-ple monaxine contrajanals (XIX.), 
(PQ)x;;,,=:K (i->1), 
where the polar feature is exactly registered. 
10. Zoneless triaxines (XX.), 
(PQ)y|«y)«yya=L. 
Here are symbols y of double janal zoneless axes, of which not even the characters (V.) 
are registered. We shall see that it suthces for our problem to Imow the number L of 
all zoneless triaxine P-edra Q-acra. All the poles of them will be found in the follow- 
ing Table, which enumerates -with exact description all janal poles. All that we care to 
know more of these L solids is how many amphigrammic axes they contain, and this we 
shall readily determine in the proper place (XLI.). 
11. Zoneless 2ic-^le, &c. monarcliaxines (XX.), 
(PQ)x;: Y%Y% {0=M,(r>i), 
(PQ)XjrWL {D=N, 
(PQ)X^« yl {D=n', 
(PQ)YJ. y;l,;..o{0=n". 
In all these ^ denotes the zonoid signature common, as well as the other characters 
specified, to all the solids registered in the number M, N, &c. 
The only zoneless monarchaxines, in which there can be any doubt about the number 
of edges in their principal poles, are those which have triple janal axes perpendicular to 
amphiedral, amphigonal, or amphigrammic secondary axes (XXXVIII.). 
12. Zoneless triarcJiaxines (XXI.), 
^(PQ)x^„y;.y;^{0=p, 
in which the principal poles are exactly specified, and where the zonoid signature gives 
the characters of the secondary and tertiary poles. 
u 2 
