§ 2. XXXYIII.-XL.] EEV. T. P. KIEKMAN OX THE THEOET OE THE POLTEDEA. 159 
N. For r=5 (^, XXXIII.), every janal pole not archipolar on a zoneless hexarchaxine 
(XXXI.), or not having poles in ^ of one name only, gives a 5-ple zoneless monarchaxine 
of the number N (XXXI.). When 5-ple zoneless A-gons are constructed which have 
poles of one name only in and which are found also in hexarchaxines having tertiary 
poles of that name, we obtain N in the same way as above by subtraction for 2r-\-5 = 5. 
W. For r=3, the Table of triple zoneless poles (XXXIII.) which have not amphiedral, 
amphigonal, or amphigrammic secondary poles described in and which axes therefore 
cannot be tertiary axes in any triarchaxine or hexarchaxine (for these have all janal ter- 
tiary axes), gives exactly the number of triple zoneless monarchaxines of the signature. 
N". Let_p be the number of triple janal zoneless poles, of all terminating features, of 
an axis of given character, perpendicular to amphiedral secondary axes, ^. e. showing in 
^ only polar faces; and let^' be the number of zoneless hexarchaxines, and^" that of 
zoneless triarchaxines whose secondary and tertiary axes have the same characters (given 
by the signatures under the numbers P and E- (XXXI.)) with those ^ axes and these 
secondaries: we have 
^-y-/'=(PQ)YjXU{0=N:, 
wFere Xj^ can be written for Y|„, if j)'=p"=0. 
In the same manner w*e can obtain Nj for amphigonal and Ng for amphigrammic 
secondary axes. Hence 
n"=n;'-i-n;+N 3' (ii, xxxi.) 
is given, and all zoneless monarcliaxines can be enumerated both for P-edra Q-acra, and 
for Q-edra and P-acra.^ 
SS', monaxine monozones. 
XXXIX. S. Every (3+r)-ple pole {j, XXXIII.) of a monaxine monozone gives a 
polyedron of the number S (XXXI.), with the proper signature Z. 
S' (XXXI.). Let s be the number of 2-ple contrajanal poles (a) of given character 
(V.), under the number m (XXXIII.), and having the zonal signature Z"; and let s' be 
the entire number of the (2r-f-l)-zoned homozones (XXXII. A) which have the zone Z", 
and in their zonoid signature ^ a pole of the name a. Then 
s-s'=(PQ).„..X«>{Z"} =S' (XXXI.); 
which is thus known for every zone Z" and for every character of axis. 
D. Zoned triaxines (3, XXXI.). 
XL. The Table B gives {a, h, c, XXXII.; w, XXXIII.) I for the number of 2-zoned 
heterozone poles of all names which are in zonal signatures TIZITI'^ TJTf'Z or TJ'T/L', which 
are aU the same, the last-written zone in Table B being perpendicular to the axis carry- 
ing the janal pole recorded. If this be identical with one of the first two ^vritten zones, 
the pole may he tertiary on a zoned triarchaxine (XV.); if all the three signatures are 
alike, the pole may he tertiary on a hexarchaxine (XVII.); or the pole may he secondary 
on a 2r-zoned monarchaxine (XII.), whatever be the signatures. 
