52 REY. J. P. KIRKMAN ON THE 
a portion of the whole number I(4v +m-+1,2h), of (4a+ 
m-+1)-edra singly irreversible having 2 triangles. 
11. Prosiem d. To determine the number of triply 
irreversible (67-+-m-+1)-edra having 3 triangles, which 
are reducible (Art. 3) to triply irreversible (67+ 1)-edra 
having 3w triangles. 
We must operate on the third part only of the 62-3 
edges k (Art. 6), and draw 4m lines j. This gives us 
2v—2 | 1 
ae 
122 {1 
results. If we repeat these operations in direct order on 
the other two thirds of the subject, we shall, as is evident 
from the reasoning of Art. 9, have all possible configura- 
tions and none twice repeated among them. The number 
of triangles must of course be 32. 
If then [°I°(6z+m1,3z) be the triply irreversibles, thus 
constructible from the whole number [°(62+1,3z) of tri- 
ply irreversibles with 3 triangles, then is 
m 22—2|1 
a ) m 
PR (62 +m+1,37)=1(6r 4+ 1,32) - r= rer 2°. 
a portion of I°(6v+m-+1,3x), the triply irreversible 
(62-+m-+1)-edra having 3z triangles. Others will here- 
after be found. 
No doubly irreversible can be constructed on a triply 
irreversible subject, for a plane bisecting the base does not 
here show the configuration on one side of it repeated 
exactly on the other. We proceed then to 
12, Proptem e. To determine the number of singly 
irreversible (62+ m+ 1,)-edra having 3 triangles reducible 
to) 6v+1)-edra triply irreversible having the same num. 
ber. 
If we plant m points 7 on our 67-3 edges k in every 
possible way, and draw the lines 7, we obtain 
