PARTITIONS OF THE X-ACE AND THE X-GON. 49 
in the edge k containing the point 7, the whole number of 
arrangements of the m new lines 7 is 
(m a5 1D ie : 9m 
jer [1 
If then I1(2x+1,x7) be the number of singly irrever- 
sible (27+ 1)-edra having 2 triangles, and 
Il(2e2+m-+1,2) 
denote the number thus constructible of (27+m-+1)- 
edra singly irreversible that have x triangles, we have 
proved that 
1 
Il(2¢-+m+1,2)=1(22+1,2)-2" —— 
For any one of the 1(2z%+1,z) being taken for the 
subject of operation, will give the same number of results 
as our subject P. 
The number II(2z+m-+1,z) is only part of I(24+m 
+1,#), the whole of the singly irreversible (27+m-+1)- 
edra that have 2 triangles, namely that part which reduces, 
by the process of Art. 3, to the irreversibles I(2¢+ 1,2). 
The remainder of the number I(22+m-+1,z) we shall 
find in the proper place. 
9. Prosiem b. To determine the number of doubly ir- 
reversible (4¢-+m-+1)-edra having 2z triangles reducible 
(Art. 3) to doubly irreversible (4z+1)-edra having 2z tri- 
angles, 
It is to be understood here and everywhere in this 
memoir, that we are considering only (#+1)-edra which 
. have an x-gonal base, and all their summits triedral. 
A doubly irreversible must have an even number 2w of 
triangles, because any sequence of faces about the base is 
twice read in the circuit; that is, the triangles and the 
configurations about them are corresponding pairs. 
Let P be the doubly irreversible (4a +1)-gon, the sub- 
ject of our operations. That our results be also doubly 
VOL, XV. H 
