§ 1. XVI., XVII.] EEV. T. P. KIEKMAJ!^ ON THE THEOET OE THE POLYEDEA. 131 
The secondary poles are 
Each of the six zones has ^+G+a-j-§>0 non-polar features all of different configura- 
tion. 
Every zoned non-folar feature occurs 12 times on the solid. Every zoneless feature is 
found 24 times, in as many interzonal regions. 
XVII. A zoned hexarchaccine jpolyedron has six objanal (IV.) principal 5-zoned axes, 
ten objanal secondary 3-zoned axes, and fifteen janal tertiary 2-zoned axes, which may 
have any terminations. 
There are fifteen identical zones, whose signature is 
z={2(«,+s;+s;+2y), 2(/,+/;+/;+2G), oror'0“o“}. 
The principal poles of the solid are 
6(2s,+2/,)=6.2, (s,+/;=l); 
the secondary poles are 
io(24+2/;)=io.2. (y+/'=i); 
the tertiary poles are 
15(2<+2;^'+2«-)=16.2, (s"+/"+<»''=l). 
Every zoned non-polar feature is read four times in each zone, i. e. sixty times on the 
solid. Every zoneless feature is read 120 times, in as many interzonal spaces. 
Observe that two zoned signatures are numerically identical, if they differ only in p 
subscribed, and in the mode of exhibiting the factors of a number. There is nothing to 
prevent any two of the signatures having the same number of summits, face, and edges, 
of art. (II. . . . XVII.) from being spoken of as the same signature Z ; but the polarity 
and repetition of the features differ with the symmetry. 
For example, the two solids 
have, the former the zones 
Z = {(..)(4.U 0-}, 
Z'={(4.1,)(2.1,), Of}, 
and the latter the zones 
Z,= {(..)(2,+2.1)0“} 
Z:={(2.2)(2,)0"'}. 
We see that, neglecting jp subscribed, Z and Z^ are the same signature, as are Z' and Z'. 
The second solid is a 4-zoned monaxine heteroid, having only two polar zoned features. 
In the former every feature is polar. 
The following will suffice for examples of zoned polyarchaxines. Here are five zoned 
s2 
