165 
§ 2. XLIX., L.] EEV. T. P. KIEKMAN ON THE THEOET OE THE POLYEDEA. 
is the number of the janal anaxine A-gons on all the P-edra Q-acra, or of janal anaxine 
A-aces in the Q-edra P-acra. 
i. Asymmetric A-gons of V-edra Q-acra (XXXIV.). 
L. We know by (g), XXXVI., the number of asymmetric edges of A-gons of P-edra 
Q-acra, which are all entered under the number y' (XXXV.). We have 
2(AA^)+(AB^)-1-(AC^)-1- . . =:A^ 
for this entire number. By inspection of our Table C (XXXIV.), we see that there 
are in aU the polar and zoned A-gons entered under a, b, c, d, e, f, d', e', f (XXXIV.), 
which comprise all the symmetric A-gons of Table C, the number A^^ of different 
zoneless non-polar edges in these A-gons. The remaming A^— Al^ must be all edges in 
asymmetric A-gons ; wherefore 
(XXXIV.) 
is the number of asymmetric A-gons of the P-edra Q-acra. If we would know the 
number i" of these which are not janal anaxine A-gons, we have 
4// A(js Aa^ Aj'.ad' Aj_a 
1 _ - .. 
As the Tables A, B, C, D are supposed to contain the same account of the features both 
of P-edra Q-acra and of Q-edra P-acra, we can obtain all the results of XXXVII L. 
ahke for both P-edra Q-acra and for Q-edra P-acra ; and with every face enumerated, 
we have also its reciprocal summit. 
Thus we have demonstrated, in this second section, that the data of art. XXXVI. are 
sufficient for the entire completion of the Tables A, B, C, D (XXXI XXXV.), for 
faces and for summits. 
All that remains for the complete solution of our problem of classification and 
enumeration of the P-edra Q-acra and Q-edra P-acra, is that we show how these data 
(XXXVI.) can be obtained and registered without ambiguity or repetition. We shall 
consider first the reciprocals of the faces (d) (f) (XXXVI.), and the edges (g) (XXXVI.). 
Note ox Aeticles XLIX. axd L. — The number Kas in art. L. is intended to comprise all edges of 
zoneless non-polar A-gons, and should have been defined as including the numbers V and # of XXXV. ; 
for these, although symmetric edges, are often edges of asymmetric A-gons. The reader will therefore con- 
ceive that the numbers {AJC)az.po.=V and (AA)^o= J', for every zonal signature, are included in the third 
line of L. And by the first-named edges in the seventh line of L. are meant edges not epizonal in those 
A-gons. In like manner the number (A^-.a.) ha XLIX. is intended to comprise all zoneless polar edges, and 
aU zonal edges in aU zones, of A-gons on the solids of XXX.; for these are often edges of janal anaxine 
A-gons. The polars are in our signatures ; the zonals ought to have been registered in a separate table in 
XXXV. as objanal zonal edges, i. e. edges (AAjoj.^o. (of A-gons) in a repeating zone (XXVII.). These are 
either diagonal traces drawn in objanal monozone (2A— 2)-gons, or crowning edges of (2A— 2)-gonalo^ffl«a/ 
monozone penesolids, of which we shall obtain in the sequel an accurate account. This will be all made clear 
in our applications. 
