‘Mr. A. Cayley on the Partitions of a Close. 427 
the number of edges of the contour. If, on the other hand, the 
summit added be an isolated one, then the addition of such 
summit causes a break of contour, or B is increased by unity. 
Hence the addition of a summit increases by unity S; and it 
also increases by unity E or else B, that is, it leaves the equation 
undisturbed. The effect of the addition of a mere contour is to 
increase P by unity, and also to increase B by unity: it is easy 
to see that this is the case, whether the new mere contour 
does or does not contain withim it any contour or contours. 
Hence the addition of a mere contour leaves the equation undis- 
turbed. The effect of drawing an edge is first to increase E by 
unity; if the edge is drawn from a summit to itself, or from a 
summit on a contour to another summit on the same contour, 
then the effect is also to increase P by unity; if, however, the 
edge is drawn from a summit on a contour to a summit ona 
different contour, then P remains unaltered, but B is diminished 
by unity. There are a few special cases, which, although appa- 
rently different, are really included in the two preceding ones: 
thus, if the edge be drawn to connect two isolated summits, 
these are in fact to be considered as summits belonging to two 
distinct contours, and the like when a summit on a contour is 
joined to an isolated summit. And so if there be two or more 
summits connected together in order, and a new edge is drawn 
connecting the first and last of them, this is the same as when 
the edge is drawn through two summits of the same contour. 
The effect of drawing a new edge is thus to increase EK by unity, 
and also to increase P by unity, or else to diminish B by unity ; 
that is, it leaves the equation undisturbed. Hence the equation 
P+8+8=E+1-+B, which subsists for the unpartitioned close, 
continues to subsist in whatever manner the close is partitioned, 
or it is always true. 
In particular, if 8=0, that is, if the original close be bounded 
by a mere contour, P+S=H+1+B; and if, besides, B=O, 
. then P+S=E+1, which is the ordinary equation in the theory 
of the partitions of a polygon. 
If we consider the surface of a plane as bounded by a mere 
contour at infinity, then for the infinite plane, @=0, or we have 
P+S=EH+1+B: in the case where the infinite plane is parti- 
tioned by a mere contour, P=2, S=0, E=0, B=1 (for the 
exterior part is bounded by the contour at infinity, and the par- 
titioning contour, that is, for it, B=1), and the equation is thus 
satisfied. And so for a contour having upon it 2 summits, 
P=2, S=n, E=n, B=1, and the equation is still satisfied: 
this is the case of the plane partitioned into two parts by means 
of a single polygon. 
The case of a spherical surface is very interesting: the entire 
