308] 
REFERENCE TO THE ENUMERATION OF POLYHEDRA. 
41 
[308 
308] REFERENCE TO THE ENUMERATION OF POLYHEDRA. 41 
polygon of the 
tees within the 
i the sides of 
•it the process 
ectively, which 
adjacent faces. In the last-mentioned two cases respectively it is only necessary to 
consider the basic quadrangles which pass through the single tripleural summit and' the 
basic quadrangle which passes through the two tripleural summits; for with any other 
basic quadrangle the derived (n + l)-acron would retain a tripleural summit, and would 
consequently be of the first class. The condition is more simply expressed as follows, 
viz.: The second process need only be employed when there is on the w-acron a basic 
quadrangle the summits of which are at least of the number of edges shown in the 
3 
4 
3 
annexed figure, and all the other summits are at least 4-pleural. Again, by the third 
process (as already mentioned) we seek only to obtain the (n + l)-acrons of the third 
class ; the process need only be applied to the ?i-acrons for which there exists a basic 
pentagon the summits of which are at least of the number of edges shown in the 
6 
d. It may be 
a unique one; 
g as the new 
4 ^ * 
5 5 
to three classes, 
ion be reduced 
n derived from 
al summit, may 
nversely it can 
n like manner, 
nay be by the 
may be by the 
annexed figure, all the other summits being at least 5-pleural; for it. is only in this 
case that the derived (n + l)-acron will be of the third class. The condition just 
referred to obviously implies that the w-acron is of the second or third class. It is 
to be noticed that in applying the foregoing principles to the formation of the 
polyacrons as far as the 11-acrons we are only concerned with the first and second 
processes. 
Consider the entire series of w-acrons, say A, B, C, &c., and suppose that the 
?i-acron A gives rise to a certain number, say P, Q, B, S of (n + l)-acrons, the (n + 1)- 
acron P is of course derivable from the w-acron A, but it may be derivable from 
other «-acrons, suppose from the «-acrons B and G. Then in considering the (n + 1)- 
rd processes of 
d that all the 
second process 
nd these being 
'or finding the 
j be made use 
one tripleural 
summits of two 
acrons derived from B, one of these will of course be found to be the (« + l)-acron P, 
and it is only the remaining (« +1)-acrons derived from B which are or may be 
(n + l)-acrons not already previously obtained as (« + l)-acrons derived from A. And 
if in this manner, as soon as each (« + l)-acron is obtained, we apply to it the 
process of subtraction so as to ascertain the entire series of «-acrons from which it is 
derivable, and, in forming the (n + l)-acrons derived from these, take account of the 
(n + l)-acrons already previously obtained and found to be derivable from these, we 
should obtain without any repetitions the entire series of the (n + l)-acrons. 
C. V. 6 
6