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VII. On the Theory of the Polyedra. By the Bev. Thomas P. Kiekman, M.A,, 
Honorary Member of the Literary and Philosqphical Societies of Manchester and 
Liverpool. 
Eeceived January 3, — Eead January 23, 1862. 
The following memoir contains a complete solution of the problem of the classification 
and enumeration of the P-edra Q*acra. The actual construction of the solids is a task 
impracticable from its magnitude ; but it is here shown, that we can enumerate them 
with an accurate account of their symmetry, to any values of P and Q. 
Section first discusses fully the symmetry of polyedra., and gives a sketch of the Tables 
which it is required to form for the solution of our problem, viz. Tables of the classified 
P-edra Q-acra and Q-edra P-acra, and of their faces, summits, and edges, symmetrical and 
unsymmetrical. 
Section second proves that all these Tables are given, if certain data are first 
obtained. The rest of the memoir is occupied by the investigation and construction of 
these assigned data. 
Section third contains the analysis of a polar or monozone summit of a P-edron Q-acron : 
the deltotomous effacedbles of the summit are defined and restored: the reticulation 
which is laid bare by the removal of the rays of the summit is analysed and reduced. 
Section fourth is devoted to the construction of polar and monozone reticulations^ and 
to their registration with their signatures of form and symmetry, in groups which suffice 
for our problem. 
Section fifth gives formulae for the zoned and zoneless coronation of a polar or mono- 
zone perfect reticulation, whereby it becomes a polyedron ; and Tables of the perfect 
summits thus obtained are sketched out. 
Section sixth enumerates and registers the results of deltotomous effacements about 
perfect summits, that is, summits about which all deltotomous effaceables have been 
restored, or which have no effaced effaceables, whereby the summit analysed in section 
third is obtained and registered. 
Section seventh analyses the polar summits of a janal axis ; the rhombotomous efface- 
ables of the opposite summits are defined, and, as well as their deltotomous ones, 
restored, about either pole; the janal reticulation laid bare by 'the removal of the 
summits is analysed and reduced. 
Section eighth constructs, enumerates, and registers, with their signatures of symmetry, 
fundamental primitive janal reticulations. 
Section ninth contains the construction, enumeration, and registration of janal subnu- 
clextjS reticulations. 
Section tenth gives the like concerning nucleus reticulations. 
MDCCCLXII. E 
