144 REV. T. P. KIRKMAN ON THE THEORY OE THE POLYEDRA. [§ 1. XXV.-XXIX. 
Observe that the ierrajanal along with anaxine always means contrajanal. 
XXVI. Def. Ajanal anaxine 'pair on any poly edron are any two edges ah, a^b^ diametri- 
cally opposite, non-polar, and zoneless, such that the configuration read along ab to the 
right is exactly that read hy an opposite eye along a^b^ to the left. 
Kjanal anaxine A-gon (or A-ace) is any zoneless non-polar face [or summit) whose A 
edges form with those of an opposite A-gon (or A-ace) Ajanal anaxine pairs. 
For example, A' and A are janal anaxine triaces in the last-drawn polyedron. 
Thus w^e see that there are janal anaxine edges in solids which are not janal anaxine 
polyedra. 
XXVII. It is important that we should here determine what kinds of polyedra have 
janal anaxine pairs. 
1. Let P be a polyedron of zoned or mixed symmetry which has janal anaxine pairs. 
The janal anaxine pair ah, af will either meet on a zone Z, or ah will meet a’h' and ap, 
will meet dh\ on a zone Z (II.). The pair ah, ap^ being diametrically opposite may be 
supposed parallels. Then the zoneless ah meets its reflected image dV , and ap^ meets 
dV^ on Z, and the angle [ah, dV) has exactly the configuration of [dV^, apj, and is dia- 
metrically opposite thereto. 
Hence Z must be a repeating zone, to which an axis of even repetition is perpendicular, 
for the same configuration recurs about that axis in half a revolution. 
It is then requisite and sufficient, in order that a zoned polyedron have janal anaxine 
pairs, that it have a zone perpendicular to an axis, zoned or zoneless, of even repetition. 
XXVIII. The (2m-l-I)-zoned monarchaxine (XII.) has none of its 2m + 1 secondary 
axes perpendicular to a zone; for each of these axes is in one of the 2m-l-I zones. 
The [2m-\-V)-zon€d monarchaxine has no janal anaxine edge. 
The 2?>i-zoned monarchaxine polyedron has every secondary axis in one zone and 
perpendicular to another (XI., VII.). 
The 'Tm.-zoned monarchaxine has janal anaxine pairs. 
The {2m-\-\)-ple monaxine monozone has them not, the axis being of odd repetition. 
The Tm-ple monaxine monozone has such edges. 
The [2m-\-l)-zoned homozone has them (XXII.). 
The 2m-zoned homozone has them not. 
The zoned triarchaxine polyedron has each of its six tertiary axes in two of its nine 
zones, and perpendicular to one of them, otherwise symmetry would be impossible. 
The zoned triarchaxine has janal anaxine edges. 
No secondary axis of a zoned tetrarchaxine is perpendicular to any of its six zones, 
for this axis is in two of them. 
The zoned tetrarchaxine has no janal anaxine edges. 
Each of the fifteen tertiary axes of a zoned hexarchaxine is in two of the fifteen 
zones, wherefore it is perpendicular to one of them, or symmetry would be impossible. 
The zoned hexarchaxine has janal anaxine edges. 
XXIX. 2. Let P be a polyedron of zoneless symmetry which has janal anaxine edges. 
