§ 2. xxxyil] eev. t. p. kiekman on the theoet of the POLTEDEA. 157 
on its two pentagon poles Avith pentaces, we should certainly thereby construct an amphi- 
gonal 5-zoned homozone axis. But it would be an error to enumerate the construction 
among the homozones, for we have evidently here only completed a hexarchaxine. 
In like manner, in constructing a 2-zoned janal axis, we cannot be sure that we have 
not been completing, by secondary or tertiary poles, a monarchaxine or a polyarchaxine. 
E. F. 
E. 2r-zoned monarchaxines (XXXI.). — When r>2, every 2r-zoned heterozone polar 
face of Table B (XXXII.) gives a distinct 2r-zoned monarchaxine of the number E ; for 
no polyarchaxine has a (6+2??^)-ple pole. Wherefore E is given for r>2, by (a, 5, c) 
Table B (XXXII.), and the exact poles of the 2r-zoned axis are known and can be 
registered for either polar faces or for their reciprocal summits. 
Let r=2, 2r=4; and let be the entme number of 4-zoned A-gonal poles having 
given traces (XXXII.) and the zonal signatures {ZZ'Z"}. Some of these poles may 
be principal poles of triarchaxines. When Z" is not identical with one of the zones 
ZZ', this cannot happen, because no triarchaxine has three zonal signatm*es. 
In the case then of ZZ'Z" all different, every one of the poles gives a different 
4-zoned monarchaxine, and they can be registered thus in Table A, 
(PQ)a;.w|„y;uzz'Z"} = j,=E, 
an entry which gives all that is indicated under the number E ; for the principal pole is 
seen, and the signatures ZZ'Z" give exactly the characters of the other poles and axes. 
Here y denotes the traces of the A-gonal pole. 
Let next be the number of 4-zoned A-gonal polar faces in Table B which have 
Z"=Z' and definite traces and signatures; and let^' be the number of triarchaxines in 
5, Table A, Avhich have a principal A-gonal face with the same traces and signatures 
{ZZ'} (a) XXXYI. We have the number E in this case thus: — 
J; -i=(PQ)iV"YJ„Y;^{ZZ'Z'} =E. 
F. {2r-\-\)-zoned monarchaxines. — No zoned polyarchaxine (XV., XVI., XVII.) has 
(2/’-l-I)-gonal poles properly janal. Therefore every janal (2r+I)-zoned pole of Table 
B gh’es one of the F polyedra required, with its signatures. 
TT'. Ix-zoned and {2r-\-\)-zoned Homozones (XXXI.). — No polyarchaxine has a 
2r-zoned homozone axis, if r>I. Therefore all the poles registered under the number 
Table B, give polyedra here required with all their signatures, of the number T, for 
r>I. 
No polyarchaxine has a (2r-{-I)-zoned homozone axis, if r>2 ; therefore all the poles 
entered under the number h in Table B, give polyedra to be entered under T' in Table A, 
with their signatures, for 2r+I >5. And the numbers TT' are thus completed for the 
above values of r. 
Let A%sY{^Zr,)=c - . • 
be the number of 5-zoned janal poles entered under the number h in Table B, having 
