§ 1. XIL, xiil] rev. t. p. kirkman on the theory of the POLTEDRA. 129 
The secondary zone has eight non-polar zonal edges and no epizonals, and it has eight 
non-polar summits. 
Hence we can subtract, on inspection of the signatures, all the zoned faces, summits, 
and edges from the entme number in the solid, and thus determine the number of 
different zoneless features in each of the eight interzonal spaces. 
A 3-zoned monarchaxine 17-edron 27-acron is 
of which the signatures are 
Z ={(1,+2.1), (2.1, +1,), 0"^ 0-}, 
Z"=3{(1,+2.1), (U 0"'}. 
We read in these signatures that the principal axis is amphiedral, 
because Z has four polar features of which two faces must be principal poles, because 
the principal axis is janal. Either zone shows that the secondary axis is gonoedral. 
A zoned triaxine 30-edi’on 26-acron is 
of which the signatures are 
Z ={(2.1,+4.1) (2.1,) 0"*}, 
Z' ={(..) (2.1,+4.1) 0- 0-}, 
Z''={(2.1,+4.3} ( . . ) or 0"^}. 
Z and Z' have each two polar faces, wherefore their common axis is amphiedral. ZZ" 
have each two polar summits, and have an amphigonal axis. Z'Z" have an amphi- 
grammic axis. 
The non-polar zoned features are four summits, all alike in Z, and four zonal edges 
all alike ; four epizonal edges alike, and four faces alike in Z' ; twelve summits, as also 
twelve zonal edges, of three configurations in Z". 
There is nothing to prevent the three axes of a zoned triaxine having all one signa- 
ture, and axes of any janal character. In every case the zonal signatures record accu- 
rately the configurations. 
XIII. System of ‘princi'pal poles. Zoned polyarcJiaxines. — Let A 1 A 2 A 3 ... be a 
system of poles, zoned or zoneless, of one configuration, not all in one plane, in a poly- 
edron P. If each pole be joined to those nearest it bylines drawn, if necessary, beneath 
the surface of P, it is evident that these lines will form a polyedron, Q, having edges 
all of one configuration, ^. e. each being the intersection of an F-gon and an F-gon. If 
F=F, the polyedron Q is regular. If F<F', let lines be drawn from the centres of 
all the F-gons to all their angles. These lines produced will all pass through centres 
f of other faces F. This is all inevitable, by reason of our hypothesis that the poles 
AjAjAj . .. have the same configuration. Wherefore the system of lines will form 
a regular polyedron, having as many edges as there are poles AjAaAg .... 
Hence we have the 
Theorem. The number of like poles, zoned or zoneless, of any polyedron, which are 
not all in one plane, is equal to that either of the summits or of the edges of a regular 
polyedron. 
MDCCCLXir. s 
