150 EEV. T. P. KIEKMAN ON THE THEOEY OE THE POLYEDEA, [§ 1. XXXII. 
where Z" always denotes the zone perpendicular to the axis of the registered janal 
pole. 
The number s is that of the summits of the solid not in the A-gon, e. inferior to the 
A-gon ; and F is the number of faces different from it, that is, inferior to it. 
We have here 
P=F+1, Q=A+s. 
We read that there are a A-gonal janal poles ha\dng r agonal traces of the zone Z, and 
r agonal traces of Z'; and a zone Z" perpendicular to the axis of the A-gons. We read 
also c A-gonal janal poles having r agonal traces of Z and r diagonal traces of Z', vrith 
the secondary zone Z". 
When 2r=4, the numbers a, i, c, when ZZ'Z are the zoned signatures, will comprise 
the archipolar A-gons of triarchaxines having the zones ZZ'. 
When 2r=2, the numbers a, b, c enumerate the polar A-gons secondary in zoned 
monarchaxines, those of zoned triaxines, the tertiary polar A-gons of zoned triarchaxines 
and hexarchaxines, for the signatures may be ZZ'Z, or ZZZ, the case of the hexarch- 
axines. In this latter case only the term heterozone is improperly applied to any polar 
face above registered. 
All (2r+3)-zoned heterozone A-gons are registered thus: 
Ajr+^)“^sF{ZZ"} = 6Z, 
Ajf+'^‘"sF{ZZ"}=6, 
Ajr+3>«05P|2:Z"}=:/. 
Princi])al janal ^olar faces of zoned Polyarchaccines. 
^a;„^5F{ZZ"}=^„ 
^A%sF{ Z } = 4 
where 4_y means 4*2^ or Mi, and 5y means bag, bdi or bmo, as the case may be. 
Padical janal 'polar faces of zoned Polyarchaxines. 
'Af,,,.,sF|ZZ"}=4i, 
It is necessary, as we shall hereafter learn (§ 16.), to have a separate Table of all the 
radical archipoles of the polyarchaxines, which are, however, supposed to be included 
in the numbers 6^ and ^2 above written. 
A radical polyarchipole janal or heteroid, whether face or summit, is a principal pole 
of a polyarchaxine collateral with other principal poles of the solid. 
The triarchipoles will be found also under the numbers a, h, c (r=2); for nothing 
prevents them being constructed and handled as monarchaxine poles. 
The hexarchipoles may be constructed and handled as homozone poles, and they will 
consequently be found also in the Table following. This will, however, create no con- 
fusion in our results. There are no janal tretrarchipoles ^A. 
