254 MR. A. CAYLEY ON THE A FACED POLYACRONS IN 



reckoned as i, and the sum in question will consequently 

 be the number of the {n+ i)-acrons. 



The figures of the polyacrons comprised in the annexed 

 Tables show the application of the method to the genesis 

 of the polyacrons as far as the octacrons, in which the 

 numbers indicate the nature of the different summits^ 

 according to the number of edges through each summit, 

 viz., 3 a triplcural summit, 4 a tetrapleural summit, and 

 so on. It will be noticed that there is only a single case 

 in which this notation is insufficient to distinguish the 

 polyacron, viz. among the octacrons there are two forms 

 each of them with the same symbol 33445566; the inspec- 

 tion of the figures shows at once that these are wholly 

 distinct forms, for in the first of them, viz. that derived 

 from 3344555, each of the tripleural summits stands upon 

 a basic triangle 456, while in the other of them, that from 

 3444555; each of the tripleural summits stands upon a 

 basic triangle 566. But the symbol is merely generic, 

 and of course in the polyacrons of a greater number of 

 summits it may very well happen that a considerable 

 number of polyacrons are comprised in the same genus. 



The following remarks on the derivation of the octa- 

 crons from the heptacrons will further illustrate the 

 method : 



1. The heptacron 3335556 has three kinds of faces, viz. 



355j* 356^ 555j the first process consequently gives 

 rise to 3 octacrons. As the heptacron has more 

 than two tripleural summits the second process is 

 not applicable. 



2. The heptacron 3344466 has three kinds of faces, 



viz. : 366, 346 and 446, and the first process gives 



* It is hardly necessary to remark that it must not be imagined that in 

 general all the faces denoted by a symbol such as 355 (which determines 

 only the nature of the summits on the face) are faces of the same kind, but 

 this is so in the cases referred to in the text. 



