164 £EV. T. P. KIEKMAJ^ ON THE THEOET OE THE POLYEDEA. [§ 2. XLVIII., xt.tx . 
a trace of Z, may be understood as not being ol^anal monozone. We shall find that 
objanal monozone summits are obtained by construction, and by these we know the 
reciprocal faces. 
h, XXXIV. Janal anaxine A-gons of the Y-edra Q-acra. 
XLIX. By (g) XXXVI. and e\ XXXV., we know the number 
2(AAj3.„„)+(ABy<,,^„)+(ACj«.a„)+ . . =(Ay,o) 
of janal anaxine edges of A-gons, where we write 2(AAja,j„), because these edges are 
each in two difierent A-gons. 
Some of these A-gons are polar and some zoned non-polar, the rest being janal 
anaxine A-gons. 
The polar zoned A-gons are exactly those found on the first five of the seven classes 
of polyedra named in art. XXX. ; and they are all either heterozone 2r-zoned A-gons 
(r>l), or homozone (2r+3)-zoned A-gons (r>0), or hexarchaxine 2-zoned A-gons, which 
can only improperly be called heterozone, since their (tertiary) axis is perpendicular to 
a zone identical with its own two zones (XXXII). Now these are precisely the polar 
A-gons enumerated under the numbers «, 5, c, XXXII. (r>0), and under the number 
h (XXXII.) for r^O. 
Wherefore we know by inspection of signatures the number of different zoneless edges 
in these zoned polar A-gons. 
The polar zoneless A-gons of the third and last but one of the seven classes of art. 
XXX. are all either principal poles of (2r-|-4)-ple monaxine monozones, which are 
found under the number / (XXXIII.), for all odd values of r; or they are 2-ple poles 
of monaxine monozones, or 2-ple poles of (2r+3)-zoned homozones, which 2-ple A-gons 
are all found entered under the number m (XXXIII.). 
The polar zoneless A-gons of the last of the seven classes of XXX. are all entered 
under the number ^ (XXXIII.), for r=2^^+l• 
Inspection of the signatures of repetition in all these polar A-gons, zoned or zoneless, 
gives us the number of different zoneless non-polar edges, that is of janal anaxine edges, 
in them all. Let this number be (Aj^^). 
The zoned non-polar A-gons are all given by (e) XXXVI., under the numbers (j, k, 1) 
XXXIV. ; whence that of their different zoneless edges is given also. Let this be (A)^). 
Then all the remaining 
(a,j-(a;j-(a;j 
janal anaxine edges of A-gons must be found in janal anaxine A-gons (XXVI.); and if 
we consider, as we may without confusion, the 2(AA)j^ „„ above written as double edges, 
we can say that all these remaining edges are found in no A-gons which are not janal 
anaxine. Each A-gon has A of them ; wherefore 
(XXXIV.) 
