134 
EEV. T. P. KIEKMAJf ON THE THEOET OE THE POLTEDEA. [§ 1. Xli., XX. 
(r>l), by a 2rm-gonal face upon a mirror, and then turn P through an angle of m sum- 
mits, while the image of P remains unmoved, that image, together with P so turned, 
forms an x-ple monaxine contrajanal jpolyedron. If, for example, 4r, there will be 
in the mirror a repeated sequence of four edges ABCD, and the configurations read by 
opposite eyes in the axis supposed parallel to the page and perpendicular to the lines 
will be (??^=2), 
ABCDABCDAB... 
CDABCDABCD... 
One eye sees A beyond C, and to the right of that B beyond D ; the opposite eye sees 
A beyond C, and to the left of that B beyond D. The configurations are contrajanal (IV.). 
We may take the last-drawn solid for the polyedron P, and conceive it laid by its 
square polar face in a mirror, and then turned through m=-\ summit, while the image 
remains unmoved. The solid with the image will form a 2-ple monaxine contrajanal. 
But it will be found impossible to produce this contrajanal configuration unless by 
employing a 2n?i-gonal polar face, and by turning the solid through m summits, r>I 
being the index of repetition. 
The monaxine heteroids. 
which have each a repeated sequence ABC, will give by this process attempted, no zone- 
less figures but monaxine heteroids. This could be easily demonstrated, but such demon- 
stration would be of no future use to us in our problem. And our object here is simply 
to prove the existence of this class of solids. 
Further, the configurations so obtained are not only contrajanal but monaxine. For 
there is no zoneless axis in the plane of the mhror, because P and its image in our con- 
struction do not form a repeated sequence in revolution about such an axis; and no 
point of P out of the mirror except the given pole of P can be a pole of repetition. 
And as there is evidently still about either of the poles (of P and its image) an ?’-ple 
repetition, the indicated construction is an r-ple monaxine contrajanal polyedron. 
But it is not to be supposed that all these solids have, like the one constructed, a 
closed circle of zonoid edges., viz. the edges in the mirror. But it is evident that such a 
circle is either drawn or drawable, in any janal polyedron, which shall present in the 
faces above and below it, the same repeated sequence of faces, either janal or contra- 
janal, to the two poles. Our object here is not to discuss the form, but to establish the 
existence, of these polyedra. 
XX. One principal zoneless axis. Zoneless x-ple monarcJiaxines. — ^A principal axis of 
repetition, zoned or zoneless, has a higher repetition than another. 
