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III. — On the triedral partitions of the X-ace, and the tri- 
angular partitions of the X-gon. By the Rnv. Tuo. 
P. Kirxman, A.M., F.R.S., Rector of Croft with 
Southworth, and Honorary Member of the Literary 
and Philosophical Society of Manchester. 
Read November 17th, 1857. 
1. Perhaps the simplest proof that can be given of the 
relation, 
F+S=E+2, 
among the F faces, S summits, and E edges of a polye- 
dron, is the following. Let e be any edge joining the 
summits a 6 and the faces A B, and let e vanish by the 
approach of 6 toa. If A and B are neither of them trian- 
gles, they both remain, though reduced in rank and no 
longer collateral, and the figure has lost one edge e and 
one summit 6. If B is a triangle and A no triangle, B 
vanishes with e into an edge through a, but A remains. 
The figure has lost two edges of B, one face B, and one 
summit 6. If B and A are both triangles, B and A both 
vanish with e, five edges forming those triangles are re- 
duced to two through a, and the figure has lost three 
edges, two faces, and the summit J. In any of these 
cases, whether one edge and one summit vanish, or two 
edges disappear with a face and a summit, or three edges 
with a summit and two faces, the truth or falsehood of 
the equation 
F+S=E+2 
