128 EEV. T. P. KIEKMAN ON THE THEOEY OF THE POLTEDEA. [§ 1. XII. 
ip denoting a polar summit, and ^p a polar face, show the secondary poles, viz. r of either 
configuration (XI.). The numbers gGah, g'G'aJh' record all the different zoned non-polar 
features of Z and Z', as g"G''cd record those of Z". The secondary polar edges are zonal 
in one zone and epizonal in another. 
Every non-polar zoned feature of the r zones Z, or of the r zones Z', occurs four times in 
each zone Z or 71. Every non-polar zoned feature of Z” occurs in it 4r times. Thus all 
mch features are read 4r times. 
Every zoneless feature of the solid is read in it 8r times, in as many interzonal regions. 
The principal axis is either amphiedral or amphigonal. The characters of the secon- 
dary axes, all j anal, vary with the solutions of the equations enumerating their poles. 
When r=l, the solid is a zoned triaxine, whose three signatures have one form, and 
may or may not be identical signatures. But the three configurations of the zones are 
always different. 
The zonal signatures of a (2r-f-l)-zoned monarchaxine are 
Z —{{2(rp-\-gp-\-2g), {2fp-{-<pp-{-2G), 0^ 0^ 0^*}, 
Z"=(2r+l){?,+2/), (^,+2G"), 0^ 0“ 0^^, 
where 
describes the principal pole, and 
s>+?’p+Q5+/3=2 
describes the secondary poles, of different configurations. Thus ?p=2 gives an amphi- 
gonal, Cp—a—\ gives a gonogrammic axis, &c. 
Every non-polar feature in the principal zone Z is read twice in the solid in each of the 
2r-l-l zones. Every non-polar feature in the secondary 7J' is read 4r+2 times in Z". 
Every zoneless feature of the solid is read 8r-j-4 times, once in each of 8r-f-4 interzonal 
regions. 
A (2?'=)4-zoned monarchaxine janal] 22-edron 36-acron is here figured, of which 
the zonal signatures are 
Z={(..), 2(1,+1,), 0-}, 
ap=gp—g=^, b=:ffz=z(pp=\, G=a=(5=a=0, 
Z'={(2.1,+4.1), (2.1J, 0"'}, 
<^p = fp—Q, 4=5'—/?=!=®'’ G'=a'=ft'=b'=0, 
Z"=4{(1,+2.1), (Ip), 0^-'}, 
?p=l=g"=<Pp=c. 
yVe see, on inspection of these signatures, that the principal axis is amphiedral, 
because Z has four polar faces, of which only two can be secondary poles. We see that 
Z" has four polar summits in the two zones Z', and four polar faces in the two zones Z. 
Hence the secondary axes are alternately amphigonal and amphiedral. 
The two zones Z have each four non-polar epizonal, and no zonal edges. The two 
zones Z' have each four non-polar zonal edges, and no epizonals. 
