68 REV. T. P. KIRKMAN ON THE REPRESENTATION 



The fourth and seventh of these have each two hexagons, 

 two pentagons, two quadrilaterals, and two triangles; but 

 they are different polyedra, their crowns being different 

 when they are made to stand on hexagonal faces. Thus 

 we see that a polyedron is not determined, when the 

 angular rank of its faces and summits is given. The triplets 

 which represent the summits of these octaedra will of course 

 equally denote and describe the faces of those 12-edrons 

 whose faces are all triangular, and which have no summit of 

 more than six edges ; if the letters a, 6, c, &c. be taken for 

 summits, not faces. 



21. The remainder of the octaedra having triedral summits, 

 namely, those on a 7-gonal base, I shall content myself wdth 

 writing out as follows; premising that the symbols stand for 

 edges only, and exhibit, if read vertically, the faces, if read 

 horizontally, the summits, of the solid ; so that each figure 

 may be viewed either as an 8-edron with triedral summits, or 

 as a 12-edron with triangular faces. All polyedra can be 

 presented to the eye in nearly the same manner, and this is 

 perhaps the most elegant way of exhibiting them. 

 abcdefg----- ab-cde-fg--~ 



m - - - ' - a h i j k I k-~-~^--ahij 



^-dr'---~kn- -cnr-------- 



g 1^ o f p I . ^ . . 



-^-ep-'-jr-- --rdq--'-'-jo 



----fo-ip--- ----epm--iq- 



-cn-'-----lq bno--------k 



bq-'------'-m ------Ighm-- 



