42 
ON THE V FACED POLYACRONS IN 
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For merely finding the number of the (n + l)-acrons, a more simple process might 
be adopted: say that an тг-acron is p-wise generating when it gives rise to a number 
p of (n + l)-acrons, and that it is q-wise generable when it can be derived from a 
number q of (n + l)-acrons ; and assume that a given w-acron is (y 1 + + Уз + &c.)- 
wise generating, viz. that it gives rise to a number y x of (n + l)-acrons which are 
1-wise generable, a number y 2 of (n + l)-acrons which are 2-wise generable, and so on ; 
these forming the sum 
2(yi + |у а + £у«+—) 
where 2 refers to the entire series of the n-acrons, it is clear that every m-wise 
generable (n + l)-acron will in respect of each of the ?i-acrons from which it is derivable 
be reckoned as —, that is, it will be in the entire sum reckoned as 1, and the sum 
m 
in question will consequently be the number of the (n + l)-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 tripleural 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 inspection 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 octacrons from the heptacrons will 
further illustrate the method: 
1. The heptacron 3335556 has three kinds of faces, viz. 3550, 356, 555, 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 therefore 3 octacrons. The heptacron has only two tripleural 
summits, and they are disposed in the proper manner; the second process gives there 
fore 1 octacron. 
3. The heptacron 3344556 has five kinds of faces, viz. 345, 346, 356, 456 and 
455, and the first process consequently gives 5 octacrons. The heptacron has two 
tripleural summits, but they are not disposed in such manner as to render the second 
process applicable. 
1 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.