§ 1. XXXII., XXXIII.] EEV. T. P. KIEKMAN ON THE THEOET OE THE POLYEDEA. 151 
Eomozone janal 'polav faces (XXII.). 
Kjy sF{ZO=5', 
where y stands for the traces ag, di, mo, agdi, as the case may be. Z is the zonal, and 
^ the zonoid signature. 
When r=l, the g homozones recorded are homozone triaxines (XXII.). 
The number h will contain every secondary polar A-gon of all triarchaxines, and every 
primary and secondary polar A-gon of all hexarchaxines ; for all these faces are in 
repeating zones to which an axis of 2 -ple (zoned) repetition is perpendicular. 
The number ^(r=l) will contain every secondary polar A-gon of a tetrarchaxine, for 
this sohd has homozone secondary axes (XVI.). 
XXXIII. 
Zoneless final polar faces. 
a;+^sF{ ^ }=i, 
A::,isF{Ko.m.}=f 
\’'+?sF 
mo.co 
=Jc. 
The first are monarchaxine janals (XX.), with zonoid signature; the second are mon- 
axine monozones, with the signature of the zone perpendicular to the zoneless axis 
(XXIII.); the thud are monaxine contrajanals of (r4-2)-ple repetition (XIX.). 
Zoneless janal polyarchipolar faces. 
Badical zoneless janal polyarchipoles (§ 16.). 
The register of janal polar faces is completed by the 2-ple janal and contrajanal polar 
faces, the contrajanal axis being perpendicular to a zone. The janal are entered thus, 
A?„sF=^, 
without zonoid signature ; for since we can construct no janal symmetry on such a pair 
of opposite A-gons (as we shall see), but that of a zoneless triaxine, we require no 
account of the axes perpendicular to a that terminated by the A-gon. 
This axis a being janal, must be perpendicular to two other 2m-ple janal axes, and 
7 (XX.); and the polyedron may be either a 2r-ple monarchaxine, of which a is a 
secondary axis, or it may be a zoneless triaxine, of which a, (3, y (XX.) are three axes; 
or it may be, when /3 or 7 is zoned, a 2 r-zoned homozone (XXII.) (r>l). 
Whatever the polyedron may be, any constructions on the opposite 2-ple A-gons will 
