j(.58 EEV. T. P. TCTR K MAN ON THE THEOET OE THE POLTEDEA. [§ 2. XXX^'II., XXXVIII. 
the zone ®Z of hexarchaxine signature (XVII.), y denoting the traces ; and let ^ found 
in the c poles with ®Z have a zonoid pole of name not excluded from ®Z, Then every 
principal polar A-gon of an hexarchaxine having those traces and the zone ®Z is included 
in the number c. The Tables of XXXI. give us the number d of the hexarchaxines 
which have this pole and zone by our hypothesis (a) XXXVI. Wherefore for this value 
>•=2, <;-<;'=T'=(PQ)A3,.Yj.{«Z5) 
is the number T' required for these signatures. 
U. (16. XXXI.). Let be the entire number of 3-zoned homozone polar faces 
AjAjAs... under the number h (XXXII.), which have the zonal signature Z^^ of the tri- 
archaxine form (XV.) or of the hexarchaxine form (XVII.), or of both, a thing quite 
possible (XVII.); and let ^ in these homozone poles have a pole of name not excluded 
from Z^j. 
Let me be the entire number of hexarchaxines having any secondary polar faces and 
this signature and let m.^ be the entire number of triarchaxines having any secondary 
polar faces and this signature Z^^ ; then 
Mil— ?7^3— m6=U ; 
for all of the 3-zoned homozone poles which are not secondary polar faces of hexarch- 
axines or triarchaxines, are to be enumerated as poles of 3-zoned homozone polyedra. 
K. Y-jyle monaxine contrajanals (9, XXXI.). 
XXXVIII. There can be no ambiguity in the enumeration of the r-ple monaxine con- 
trajanals ; for by its name and definition the solid can have no pole out of its single axis 
(XIX.). Wherefore every pole registered under the number Jc, Table B (XXIII.), gives 
a distinct polyedron of the number K under consideration. 
MNN'N" (11, XXXI.). 
Zoneless monarcliaxines. 
No polyarchaxine has more than a 5-ple repetition. Hence for (r+3)>5, every pole 
registered under the number ^ in Table B (XXXIII.) gives one of the M solids with its 
signature when r+3 (Table B) is even, and one of the N solids if r+3 is odd. 
M. For r=4, every janal pole registered in Table B under ^, which is not archipolar 
on a zoneless triarchaxine, gives a 4-ple zoneless monarchaxine of the number M. 
The only principal triarchaxine poles in the number i (XXXIII.) are among those 
4-ple poles in which ^ has poles of one name only (XXI.). Let n, be the entire number 
of 4-ple janal A-gons (?, XXXIII.) having poles of one given name only; and let be 
the number of zoneless A-gonal triarchipoles in Table A (5, XXXI.) which have tertiary 
poles of that name. Then n^—n^^ is the number of 4-ple zoneless monarchaxine A-gons 
ha\ing this zonoid signature and this gives the entry 
(XXXL). 
