124 The Rev. Thos. P. Kirkman on 



Each of them could be crowned upon its 23 marginal 

 triangles by a different asymmetric propyramidal 23-ace,. 

 and the similar 216TS 23-aces could be by one entry 

 registered as a small fraction of all the asymmetrical pro- 

 pyramidal 23-aces, built by other belts on other irreducibles, 

 that are to be found among the asymmetric summits of the 

 72-acral 63-edra. 



But since, in the solution of the problem of the Polyedra, 

 no asymmetric summits are obtained by coronation, none 

 of these 40-partitioned 71-gons are required in that solution. 



All asymmetric summits, the #-aces, £-aces, &c, are 

 given in that theory by their reciprocal #-gons, b-gons, &c, 

 in the 63-acral 72-edra, which faces are obtained by their 

 edges, constructed in vast numbers by crowning (or imagin- 

 ing so crowned) penesolids with those edges. The only 

 asymmetric reticulations of use in the study of Polyedra 

 are quite elementary ones, by the zoned and zoneless 

 repetition of which round a circle are formed the sym- 

 metricals that alone are crowned, and give the /-zoned 

 and /-pie summits. Those small ones are readily obtained 

 by inspection of previous tables, in which they have been 

 again and again used. 



This general problem of the k-partitions of the R-gon 

 is outside the theory of Polyedra. It may yet find its use 

 in analysis. 



15. We have not above solved this general problem in 

 terms of k and R ; that, I fear, is impossible. But it will 

 be seen that when with k and R are given the list of faces 

 in the irreducible H, with the number of its non-marginal 

 diagonals of which each is in a face that has more than one 

 other non-marginal diagonal among its edges, and when 

 the faces in R that have each two, and only two, non-mar- 

 ginal diagonals for edges are exactly given, whether they 

 carry or do not carry one or more marginal triangles, that 



