§ 1. XilX., XXX.] EEV. T. P. EIEKMAN ON THE THEOEY OP THE POLYEDEA. 145 
The edge ah makes with the zoneless pole a of P a configuration which is the reflected 
image of that made by the opposite edge ap^ with the opposite pole a of a janal axis. 
The only zoneless polyedron whose opposite janal poles can have such a configuration 
is the r-ple monaxine contrajanal polyedron (XIX.). 
And we see, by considering the repeated sequence of faces above and below any closed 
line drawn or drawable on the polyedron equidistant from both poles, as, for example, 
(XIX.), 
ABCD ABCD 
C D A B C D A B, for f even ( =2), 
and 
ABCDABCDABCD 
CDABCDABCDAB, for r odd (=3), 
A A . C 
that the edge q diametrically opposite to q in the former does not, but that the edge ^ 
diametrically opposite to q in the latter, does present, to two eyes either in the plane of 
A 
the sequence or at the poles, a contrajanal configuration as compared with the first edge q- 
The same thing is easily proved by taking any sequence of 2m faces, and comparing 
the cases of r odd and r even. 
The same thing is also proved thus. If ap^ are a janal anaxine pair about a zone- 
less axis of r-ple repetition, the line bisecting the two edges is a diameter (XXV.), and 
the plane containing it and the axis is a diametral plane. There are about the axis r—1 
other similar janal anaxine diameters symmetrically disposed; and such symmetry is 
evidently impossible unless r — 1 be even, that is, unless r be odd. 
The x-ple mmaxine contrajanal polyedron has janal anaxine pairs ifx he odd, hut not if r 
he even. 
In all other zoneless polyedra, the opposite poles of a janal axis have configurations 
which are exact repetitions of each other to opposite eyes, wherefore the configuration 
read by one eye along an edge ah, which configuration includes the pole, cannot be the 
reflected image of that read along ap^ by an opposite eye. 
No polyedron of zoneless symmetry, except the monaxine contrajanal, has janal anaxine 
edges. 
XXX. We have proved that janal anaxine edges are found in the polyedra following : — 
1. The zoned triaxines (XI.) ; 
2. The 2??i-zoned monarchaxines (XII.) ; 
3. The (27?i-f 3)-zoned homozones (XXII.); 
4. The zoned triarchaxines (XV.); 
5. The zoned hexarchaxines (XVII.) ; 
6. The 2m-ple monaxine monozones (XXIII.) ; 
7. The 2m+l-ple monaxine contrajanals (XIX.). 
Every zoneless and non-polar edge on these solids is a janal anaxine edge. 
MDCCCLXII. u 
