126 EEV. T. P. KIBKMAN ON THE THEOEY OF THE POLYEDEA. [§ 1. IX. 
When m is odd, the zonal signature is 
0 “ 0 *}, 
where 
shows the number of polar summits, andj^ of polar faces, beside g different non-polar 
zonal summits, G different non-polar zonal faces, a different non-polar zonal edges, and 
h different non-polar epizonal edges. 
The axis may be amphiedral, amphigonal, or gonoedral. 
When m is even, the two zonal signatures are 
Z={(<r,+2^), (/,+2G), 
Z,= {(<r,+2^J, (/,+2GJ, 
where 
(a=0=j3 if w>2 (art. V.)) 
describes the poles, which may give six different characters to the axis (V.), one for every 
solution of the equation where a=2 differs not from j3=2 in form. 
We consider two zoned features, of which one is the reflected image of the other, to 
be the same in configuration, and enumerate the two as one in our Tables of zoned 
features. 
The above signatures show g, G, a, b, and g^, G^, a^, h, for the number of their different 
zoned non-polar features. For each zone has two identical hemizones, when m is even, 
but not when m is odd (VII. ). 
The polar edge which (c£+(3>0) is zonal or epizonal in Z, is epizonal or zonal in Z^. 
There is nothing to prevent the two zones Z Z^ from having the same signature ; but 
they cannot have the same configuration. And we always consider two zonal signatures 
which differ only in p subscribed, and in the ways of writing the factors of the same 
number of features, as numerically the same zonal signature. This is important to be 
remembered, when we inspect our Tables in considering the zone Z. 
Whether m he odd or even, every dff event non-polar feature is read 2m times on the solid, 
namely, once in each of 2m interzonal regions. 
A 2'Zoned monaxine heteroid 6-edron 8-acron is . . 
The zonal signatures are, 
Z={(2.1) (2.1) 0;0;}; 
^=1, G=l, a=l=/3, «=0=5; 
Z'={(2.1) (2.2), 0;0;}; 
9,='^^ G^=2, (3=l=ci, a,—0=h,. 
The axis is amphigrammic. 
A 3-zoned monaxine heteroid 7-edron 9-acron is . . 
whose signature is 
z={(l), (2,+2), 03}; 
<^p=0, ^=1, /^=2, G=2, a=0, h=S. 
