122 EEV. T. P. KIKKMAN ON THE THEOEY OE THE POLYEDEA. [§ i. 
Section eleventh gives a similar account oi 'perfect janal mixed reticulations. 
Sections tivelfth and thirteenth are devoted to i\iejanal coronation of janal reticulations, 
and to the enumeration and registration oi perfect janal summits. 
Sections fourteenth and fifteenth enumerate and register the results of deltotomous 
and rhombotomous effacements about perfect janal summits. 
Section sixteenth analyses a polyarchipolar summit ; effaceables are restored about all 
like archipoles ; the polyarchaxine reticulation laid bare by the removal of these polar 
summits is reduced, and afterwards constructed with enumeration and registration of 
results : the formulae for polyarchaxine coronation are given, and the results of efface- 
ment about the principal axes are enumerated and registered. 
Section seventeenth gives the analysis, construction, and enumeration of contrajanal 
anaxine pairs of edges., which have neither polar nor zoned symmetry, but of which one 
edge is the reflected image of the other, and diametrically opposed to it. 
The rest of the memoir is devoted to the enumeration of plane reticulations, i. e. par- 
titioned polygons, the knowledge of which is taken for granted in all that precedes. 
Section eighteenth enumerates and registers the symmetrical and asymmetric plane 
penesolids, i. e. plane reticulations laid bare by the removal of an edge of a polyedron. 
Section nineteenth enumerates and registers the primary plane reticulations, symmetric 
and asymmetric, with their signatures of symmetry. 
Section ticentieth constructs, enumerates, and registers with their signatures, the 
zoned plane reticulations. 
Section twenty-first gives the like account of the zoneless plane reticulations. 
It is unfortunate that my previous labours on the partitions of the R-gon, that is, on 
plane reticulations, are of little utility for this problem of the polyedra, by reason of 
their too great generality, and of their not giving the number of marginal triangles in each 
partition. Yet the fundamental theorem on the I:-di visions of the R-gon* has been the 
key to the greatest difficulty in this theory, which is to find the number of the asymmetric 
plane reticulations which have a given marginal s'ignature. 
Of the two sections here presented to the public, the first (arts. I. . . . XXXV.) is in 
itself a complete treatise on the symmetry and classification of Polyedra ; and the second, 
together with the preceding introduction, puts the reader clearly in possession of the 
main outline of the argument. Much remains of the entire work, which is, however, 
completely written, and in the possession of the Royal Society, as well as extensive 
applications of the method to the enumeration of polyedra. These will be intelligible 
only when the general methods and formnlee of this memoir have been investigated and 
exhibited. 
Section 1. — On the Symmetry of Polyedra. 
I. The symmetry of polyedra is — 
1. Zoned symmetry; 
2. Zoneless axial symmetry ; 
* PbilosopHcal Transactions, 1857, p. 225. 
