156 EEV. T. P. KIRKMAJf ON THE THEOET OE THE POLTEDEA. [§ 2. XXXY -XXZVli: 
It is necessary to determine what data are required, and how they are to he employed, 
for the completion of these Tables ; and next to find the means of obtaining these data. 
This will all be discussed, and the requisite general formulae will be recorded, in the 
following sections. 
Section 2. — Problem of Classification and Enumeration of the V-edra Qrocra. 
XXXVI. (a) Let us suppose given for all polyedra of fewer than P+Q— 2 edges, 
as far as we require them, the Tables A, B, C, D (XXXI. . . XXXV.). 
(b) Let us suppose given for all the P-edra Q-acra, and for all the Q-edra P-acra, P 
and Q being definite numbers, all the polyarchaxine polyedra, with their poles and 
signatures, as entered in XXXI., XXXII., XXXIV. 
(c) We suppose known also for P-edra Q-acra and for Q-edra P-acra all janal poles^ 
with their signatures, as entered in Tables B (XXXII., XXXIII.). 
(d) Let us conceive that all polar faces and summits of the same solids are known, as 
entered in Tables C (XXXIV.), and all polar edges, as entered in Tables D (XXXV.). 
(e) Likewise all obianal monozone faces of the same solids, as entered in Tables C 
(XXXIV.). 
(f) And also all monozone faces which have a diagonal trace of a single zone, ^. e. the 
number d' of Tables C (XXXIV.), for all signatures A and Z. 
(g) And, finally, let us suppose that all the edges are given of Table D (XXXV.), 
for P-edra Q-acra, and for Q-edra P-acra. 
We suppose given in the data (a), (b), (c), (d), (e), (f), (g), the numbers GHIPQE 
of Tables A (XXXI.), and all the Tables B, C, D, except only the numbers e', f', h, i 
of Tables C. 
All the remaining numbers entered in Table A, and the numbers e', f ', h, i of Table C, 
can be determined by the data (a), (b), (c), (d), (e), (f), (g). 
When this has been proved, and when we have shown that we can obtain the data 
(a) . . . (g), our problem will be solved. 
XXXVII. We have first to show that the data (a), (b) . . . (g) suffice for the determi- 
nation of the sought numbers in Tables A and C. Tnese numbers we shall consider in 
the order following:— EFTT'UKMNN'N"S'SDWLBCJAHe in Table A (XXXI.), and 
e', f ', h, i in Tables C (XXXIV.). 
One difficulty of our problem lies in the danger incurred of enumerating the same 
solid more than once. Thus, all janal and objanal principal secondary or tertiary poles 
of zoned or zoneless polyarchaxines will be constructed by our processes simply as janal 
poles of a single axis : and we cannot be certain, without consideration, when we con- 
struct a 5-ple, 4-ple, 3-ple, or 2-ple axis, zoned or zoneless, that we do not thereby com- 
plete a polyarchaxine. 
For example, if we load all the principal faces but two opposite ones of a regular 
12-edron with 5-gonal pyramids (pentaces, aK'/j), we have in the two void faces an amphi- 
edral axis of a 5-zoned homozone polyedron. If in our processes we were to charge this 
