§ 1. XXIV., XXV.] EEY. T. P. KIEKMAN ON THE THEOEY OF THE POLTEDEA. 143 
identical poles of (34-^)-ple repetition, and these will form by the reasoning of XIII. 
a polyarchipolar system, which is absurd, if there be no principal axis. 
It is therefore impossible that there can be any axis in PP' except aa', unless 
r^=2 = 3+r in the preceding article. 
If we had taken a a double axis, we might or might not have completed in that 
article a monaxine monozone. Nothing prevents the existence of other double axes a, 
each perpendicular to a different zone (F). 
The solid (PP') constructed might thus have been a(2m-|-l)-zoned homozone, having 
2w+l contrajanal double axes (XXII.). 
4. Neuter Symmetry. 
XXV. The polyedra which have a neuter symmetry are of one species only, viz., 
Contrajanal anaxine polyedra. — Let F be any 2m-gonal zoneless non-polar face of a 
polyedron P, and let P be placed on a mirror by the face F, and turned through two 
right angles, while the image P' remains unmoved. The solid PP' constructed by P so 
turned, and by the image P', may be a contrajanal anaxine polyedron. It will be such, 
if it has neither pole nor zone. 
It is evident that to any edge ah, on one side of the mirror, there corresponds another 
diametrically opposite edge ap^, on the other side, in the image, such that the configura- 
tion seen along ah to the right is that read along ap^ to the left. The edges of the solid 
(PP') in all cases form janal anaxine pairs [ah, apj of edges diametrically opposite, 
unless P be such that by our construction we have completed a zoned or polar symmetry, 
in which case certain of the pairs will be zoned or polar. 
Every feature f is twice read on a contrajanal anaxine polyedron, namely, f and its 
reflected image. 
Here is a contrajanal anaxine lO-edron lO-acron, where the 
dotted lines are supposed below the page. The configuration 
read along ah to the left is that read by an opposite eye along 
to the right, and the same is true of any other pair he, h'd. 
This solid is formed by crowning with edges ah, a!V the contra- 
ja,nal penesolid below on the left. 
The same solid is constructed by drawing the janal anaxine pair. 
A 
)CC 
AB', A'B, in the quadrilaterals A'c'Ba, Aa'B'c, of the 2-ple monaxine monozone poly- 
edron on the right, whereby both the zone and the axis are destroyed. 
