

341 



January 8, 1863. 



Major-General SABINE, President, in the Chair. 

 The following communications were read : 



I. " Applications of the Theory of the Polyedra to the Enume- 

 ration and Registration of Results." By the Rev. THOMAS 

 P. KIRKMAN, M.A., F.R.S., and Honorary Member of the 

 Literary and Philosophical Societies of Manchester and 

 Liverpool. Received November 29, 1862. 



The following are a portion of my Tables of polyeclra. They com- 

 prise all the 6-edra, 7-edra, and 8-eclra, with their reciprocals, and 

 all 9-edra of less than 1 7 edges. It is desirable that examples of 

 results should be before the reader of my work on this theory, if it 

 is so fortunate as to be read at all. More results can easily be added, 

 if it is thought necessary, when the entire treatise is before the world. 



The method of computation of these Tables turns to advantage a 

 division of summits and reticulations not mentioned in the theory, as 

 it would have abbreviated nothing, and would have added one more 

 to complications already too numerous, and all inevitable. 



Perfect summits and reticulations, that is, such as have no effaced 

 effaceables, are pyramidal, propyramidal, or metapyramidal. 



A pyramidal perfect summit or reticulation is one of which every 

 effaceable, primary or secondary, is a base edge of a pyramid, by 

 which the A-gonal base of a pyramid can be cut away in the process 

 of reduction of the reticulation. Such a construction is made either 

 by glueing together by their edges A-gonal, B-gonal, C-gonal, &c. 

 pyramidal bases, the vertices being supposed to hang downwards, or 

 by loading marginal triangles of plane reticulations with such bases 

 so posited. 



If a reticulation has no effaceables, it is merely a plane partitioned 

 polygon, and the summit or edge which crowns it completes a poly- 

 edron without the employment of any solid charges. I call such a 

 summit or edge propyramidal. 



If, for one or more of the A-gonal, B-gonal, &c. bases of pyramids 



VOL. XII. 2 C 



