342 [Jan. 8, 



about a pyramidal summit or edge, we substitute an A-gonal, B-gonal, 

 &c. face of any polyedron which is not a pyramidal base, we have a 

 perfect summit or edge of a solid of a greater number of edges, which 

 may or may not have another symmetry. Such a summit or edge is 

 metapyramidal. 



The most expeditious method of computing the polyedra of N or 

 fewer edges, is first to form and to crown all possible propyramidal 

 and pyramidal perfect reticulations which can be reduced by efface- 

 ment of effaceables to N or to fewer edges : these are to be registered 

 in tables of perfect edges and summits, which show at a glance what 

 pyramidal bases enter into the constructions registered. Having 

 determined, by inspection of these tables and by effacements, the 

 lower polyedra, we form tables of metapyramidal edges and sum- 

 mits, by merely conceiving the substitution of other A-gons, B-gons, 

 &c. of solids thus far determined, for the A-gonal, B-gonal, &c. py- 

 ramidal bases. The edge (MN), or the^-ace, considered, is at once 

 entered as an edge (MN), or as a jp-ace, of a polyedron of more 

 edges, iii the metapyramidal tables. 



Rules are easily laid down for the result of this conceived substi- 

 tution, as to symmetry, signatures, and the tabular value (of enu- 

 meration) . We thus escape the enormous toil of separately construct- 

 ing and crowning the metapyramidal reticulations. These rules 

 will be given in the supplement of applications, of which this abstract 

 exhibits a few results. 



I observe that a case is unprovided for in art. XLVII. of my 

 second section, namely, the case of a zone which exhibits in some of 

 its forms the symbol 0^ of an epizonal polar edge. Such a zone will of 

 course occur about an amphigrammic, about an edrogrammic, or gono- 

 grammic zoned axis. This polar epizonal is to be included, as part 

 of the number AAA> in the sinister of the equation first read, since 

 this edge Q p is generally the epizonal edge of a monozone A-gon, 

 never of two different A-gons. 



This XLVIIth article is sufficiently corrected by the effacement of 

 the word non-polar in the 2nd, 9th, and 13th lines, and by writing 

 when for because in the 7th line. So read, all cases are covered. 



It may appear to the reader at first sight that the Table A, or at 

 most the Tables A, B, and C, would comprise a sufficient solution of 

 the problem of the polyedra. The truth is, that it is impossible to 



