PARTITIONS OF THE X-ACE AND THE X-GON. 73 
a 
oS ay a2 29 
the number of triedral partitions of the (13+ m)-ace 
having six angles of the polyace undisturbed, or that of 
the triangular partitions of the (13+ 2)-gon in which are 
always six marginal triangles. 
Ge.4 P(15,7)=P(8,2) + P(8,3) 
=R(8, 2) Boks 2) +1(8,3) 
en a lage oe Ue by (2567) 
= R(15,7) +1(15,7) 
kax=1. 
= 2 + 2. (Vide p, and Art. 4.) 
dd. RR(15+m,7)=2:- 2, by (13,cc). 
kax=1. 
aL bot 
m+1 10{1 G 
ee, IR(5+m,7)=1- FP —.2m—1 : page yas ea . oF 
by (14,cc). 
101 
if. Ts oy a ee by (8,cc). 
Adding these three quantities, we obtain, 
(5 +1) 
10|1 m 
F. P(154+m,7)=3-2"- —— : 42,72? —_—— 3 
the number of these partitions of the (15+ 7)-ace, or of 
the (15 + m)-gon. 
And thus we can, without difficulty, proceed to find for 
any value of x and m, P(2Q2+m-+1,#), the triedral par- 
titions of the (2a -+m-+1)-ace having # pairs of rays united 
pairs, or the triagular partitions of the (2a -+m-+1)-gon 
having 2 marginal triangles. 
In the Philosophical Transactions for 1856, in a memoir 
“On the enumeration of the #-edra having triedral 
summits and an (#—1)-gonal base,” I have given a 
VOL. XV. L 
