§ 1. XVIII., XIX.] EEV. T. P. KIEKMAN ON THE THEOEY OF THE POLYEDEA. 133 
2. Zoneless Symmetry. 
XVIII. The polyedra which have a zoneless symmetry are — 
a. r-ple monaxine heteroid polyedra ; 
h. r-ple monaxine contrajanal polyedra; 
c. r-ple zoneless monarchaxine polyedra ; 
d. zoneless triaxine polyedra ; 
e. zoneless polyarchaxine polyedra, which have the axial 
system of the regular polyedra. 
An axis zoned or zoneless is said to be of m-fle repetition, if the same feature presents 
itself m times in the same posture in a revolution of the solid about that axis. 
Zvery r-zoned axis is an axis of r-ple repetition ; for though it has 2r like interzonal 
spaces about it, reflecting each other, no zoneless feature presents itself more than r times 
in the same posture in a revolution about the zoned axis. 
Every amphigrammic, edrogrammic, or gonogrammic axis, zoned or zoneless, is of 
necessity an axis of 2-ple repetition. 
By an x-ple axis, when zoned is not added, we understand a zoneless axis of r-ple 
repetition. 
Heteroid x-ple monaxine polyedron . — A polyedron which has no zone, and but one 
zoneless r-ple heteroid axis, is an x-ple monaxine heteroid polyedron. The axis may be 
of any character. In this solid every non-polar feature is read x times, and no more, 
namely, once in each repeated sequence which presents itself in revolution about the 
axis. 
Such solids are the following : — 
G 
A has a 2-ple gonoedral axis ; B has a 4-ple amphiedral axis ; C has a triple gonoedral, 
D has a 2-ple amphigrammic, and E a 2-ple edrogrammic axis. 
XIX. Monaxine contrajanal polyedron . — If we place any r-ple heteroid monaxine P, 
