4.26 Mr. A. Cayley on the Partitions of a Close. 
single contour, or by two or more contours ; and I shall throngh- 
out do so, instead of using the more precise expression of the 
boundary being the limit of a contour, or of two or more contours. 
The excess above unity of the number of the contours which 
form the boundary of a close is the break of contour for such 
close ; in the case of a close bounded by a single contour, the 
break of contour is zero. 
Any point whatever on a curve may be considered as the point 
of meeting of two curves, or, in the case of a closed curve, as the 
point where the curve meets itself, but it is not of necessity so 
considered. A point where a curve cuts or meets itself or any 
other curve, is a summit; each point of termination of an un- 
closed curve is also a summit; any isolated point may be taken 
to be a summit. It follows that, in the case of a closed curve 
not cutting or meeting itself (that is, a contour), any point 
or points on the curve may be taken to be summits; but 
the contour need not have upon it any summit: it is in this case 
termed a mere contour. The curve which is the path from a 
suminit to itself, or to any other summit, is an edge: the former 
case is that of a contour having upon it a single summit, the 
latter that of an edge having, that is, terminated by, two sum- 
mits, and no more. It is hardly necessary to remark that a 
contour having upon it two or more summits consists of the 
same number of edges, and, by what precedes, a contour having 
upon it a single summit is an edge; but it is to be noted that a 
contour without any summit upon it, or mere contour, is noft an 
edge, It may be added that an edge does not cut or meet itself 
or any other edge except at the summit or summits of the edge 
itself. : 
Consider now a close bounded by 8+1 mere contours: if for 
any partitioned close we have P the number of parts, S the 
number of summits, EK the number of edges, B the number 
of breaks of contour; then, for the unpartitioned close, we have 
P=1, S=0, E=0, B= 8, and therefore 
P4+S+@=E+14B; 
and it is to be shown that this equation holds good in whatever 
manner the close is partitioned. The partitionment is effected 
by the addition, in any manner, of summits and mere contours, 
and by drawing edges, any edge from a summit to itself or to 
another summit. The effect of adding a summit is first to in- 
crease S by unity: if the summit added be on a contour, E will 
be thereby increased by unity ; for if the contour is a mere con- 
tour, it is not an edge, but becomes so by the addition of the 
summit ; if it 1s not a mere contour, but has upon it a summit 
or summits, the addition of the summit will increase by unity 
