142 EEV. T. P. KIEKMAN ON THE THEOEY OE THE POLTEDEA. [§ 1. XXIIL, XXIV. 
The former is a 4-ple monaxine monozone reticulation, which becomes a 4-ple monaxine 
monozone polyedron, if crowned in both the opposite polar faces by octaces upon the 
eight marginal triangles. 
The latter is a 4-zoned monarchaxine reticulation, which becomes a 4-ple monaxine 
monozone polyedron, if crowned both above and below by octaces through the same four 
points 12 3 4 and through the marginal triangles. By such coronation the principal 
zones of the reticulation are destroyed, but the 4-ple repetition is preserved, and the 
secondary zone of the reticulation in the plane of the page is preserved also. 
XXIV. Theorem. There cannot he more than one principal axis in a mixed sym- 
metry. 
For we have proved (XIII., XIV.) that the only systems of principal axes are those 
of the regular polyedra. 
In a zoned polyarchaxine there can be no zoneless poles ; for if there were one, there 
would be one at least in every interzonal region, i. e. there would be 24, 48, or 120; 
and such a number of similar poles has been proved impossible in XIII. 
For a like reason there can be no zoned pole about any of the zoneless poles of a 
polyarchaxine, and consequently no zone ; for if there were one zone there would be 
many, and therefore zoned axes and poles. Hence the theorem is proved. 
From this theorem it follows that there cannot be in the solid PP' of the preceding 
article any other axis than ««'. For if there were another, either (1) aa' would be a prin- 
cipal axis, or (2) there would be one principal axis of more than (3-|-r)-ple repetition, 
or (3) there would be no principal axis. 
1. If aal be a principal axis, every secondary axis /3 will be at right angles to aa', 
otherwise there would be more than one axis [aod) in the sequence repeated about j3, 
which is impossible by the preceding theorem ; and (3 will be a 2m-ple axis, because 
the pole a occurs in half a revolution about (3 ; but there is no axis of even repetition at 
right angles to aa' by the reasoning of XXII. because aa' is contrajanal. Therefore aa' 
is no principal axis. 
2. If there be a principal axis A different from aa', since it is not at right angles to 
the zone FF', there will be, by the definition of a zone (II.)^ more than one such axis A, 
which is impossible by the preceding theorem. 
3. If there be no principal axis, there* will be at least about, the pole of aa', 3+r 
