62 
[312 
312. 
ON THE PARTITIONS OF A CLOSE. 
[From the Philosophical Magazine, vol. xxi. (18G1), pp. 424—428.] 
If F, S, E denote the number of faces, summits, and edges of a polyhedron, then, 
by Euler’s well-known theorem, 
F 4- S = E + 2 ; 
and if we imagine the polyhedron projected on the plane of any one face in such 
manner that the projections of all the summits not belonging to the face fall within 
the face, then we have a partitioned polygon, in which (if P denote the number of 
component polygons, or say the number of parts) F = P +1, or we have 
P + S = E+ 1, 
where S is the number of summits and E the number of edges of the plane figure. 
I retain for convenience the word edge, as having a different initial letter from summit. 
The formula, however, excludes cases such as that of a polygon divided into two 
parts by means of an interior polygon wholly detached from it; and in order to 
extend it to such cases, the formula must be written under the form 
P + S= E+l + B, 
where B is the number of breaks of contour, as will be presently explained. 
The edges of a polygon are right lines: it might at first sight appear that the 
theory would not be materially altered by removing this restriction, and allowing the 
edges to be curved lines ; but the fact is that we thus introduce closed figures bounded 
by two edges, or even by a single edge, or by what I term a mere contour; and we 
have a new theory, which I call- that of the Partitions of a Close.