160 
EEV. T. P. KIRKIVM ON THE THEOET OF THE POLTEDEA. [§ 2. XL., XLI. 
The number I comprises d tertiary poles of hexarchaxines, tertiary poles of triarch- 
axines, and 2d^^ secondary poles (XI.) of monarchaxines, which all have these zonal sig- 
natures. All other poles must be in zoned triaxines having these signatures ; whence, 
as each (XII.) has three poles, we obtain, since d^ d^, and d,i are given, 
T>=^l-d-d,-2d,) (XXXI.). 
W. Homozone triaxines (16, XXXI.). 
There is never any ambiguity about the zoned poles of homozone triaxines, except 
when the zone, save inp subscribed, is of the form of a tetrarchaxine signature (^Z) 
(XVI.), and when at the same time the pole named in ^ has the name of the secondary 
pole of the tetrarchaxine, named in (^Z). 
There are under the numbers g, p (XXXII., XXXIII.), Jc^ 2-zoned homozone poles 
of all terminating features of a given name, which have the zone (^Z) containing, 
though of course without^ subscribed (XVII.), the feature named as secondary pole in 
and there are (6, XXXI.) tetrarchaxines hawg this zonal signature, and the 
secondary pole named in 
Wherefore 
is the number of homozone triaxines which have these signatures 
It is worth while to show how the ambiguity spoken of can arise. 
If we charge all the faces of a tetraedron P with tetraedra, and then efface two oppo- 
site edges of P, we obtain the solid here figured, where 12 3 5 are 
the summits of P, and 52 and 13 are the effaced edges. This is a 
homozone triaxine, in which the edges 32, 21, 15, 53 of P have 
become zoneless polar edges. 
This homozone will be registered in our Table ; and if, as will 
inevitably happen in our processes, we draw the polar edges 52, 
13, we construct a homozone amphigrammic axis to which two 
2-ple axes are perpendicular. But it would be an error to register the construction as a 
homozone triaxine ; for it has a tetrarchaxine signature, and the same zonoid poles 
which the figured homozone has, and which can be made to appear \>y p subscribed in 
the zonal signature. We have simply completed a zoned tetrarchaxine. And this 
ambiguity may occur in far more complex constructions, in which the secondary poles 
may be faces or summits. 
If, however, when we have completed a 2-zoned homozone by coronation either until 
lines as above, or with a pair of summits, we find that we have completed a zonal signa- 
ture of tetrarchaxine form, which has no zonal faces, while our zonoid signature ^ shows 
a polar face, we have completed no tetrarchaxine. 
L. Zoneless triaxines (10, XXXI.). 
XLI. The Table B (XXXIII.) of janal poles [Imq) gives the entffe number of 2-ple 
