AND ENUMEBATION OF POLYBDKA. G 1 



We have to complete the system 



Aa6, Khc, Acd, Ade, Aef, A/y, Agh Ahi 

 abc cde efg ghi 



Aij Ajk Akl Aim Awa, gik, akg, aeg^ ace, 

 ijk klm ma, mk, ak; 



which can be effected but in one way, by the addition of akm. 

 When a (2w4-l)-edron on a 2w-gonal base has n trian- 

 gles, the system consists of no triplets besides those denoting 

 the summits of the base and of the triangles, and the ridge- 

 summits. 



13. It would perhaps not be a very difficult matter to 

 discover an algebraic expression for the number of /z-edrons 

 which have a (n — l)-gonal face, with all their summits 

 triedral. 



All that is necessary, is,, first, to find the number N of 

 solutions of (vid. 9) 



c-hc,+e,+ .... 4-^A-2Zw— A— 1, 

 6, e,, &c., being positive, and k not less than 2, nor greater 

 than i(n — 1), and every different order of the numbers cc,c,, 

 &c., counting as a solution ; for every such permutation gives 

 a different polyedron. This number N is known and readily 

 found. Then it would be required to find the number R 

 of ways in which k— 2 ridge-summits could be selected, under 

 the restrictions of article (10). The sum SNR, from k=2 

 to fc=the greatest integer in ^{n — I), is the number of specie*. 

 Each of these is then to be multiplied by the number of its 

 varieties V, which arise in the manner of those above dis- 

 cussed in (7) and (12). The full number required is SNR V 

 taken within the proper limits. 



14. Leaving the pleasure of this discovery to the learned 

 reader, I shall proceed to consider the construction of n-edra 

 which have all their summits triedral and no face of more 

 than n — e — 1 angles. 



