Mr. A. Cayley on the Partitions of a Close.- — 425 
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+1+B, 
where B is the number of breaks of contour, as will be explained 
in the sequel. 
The edges of a polygon are right lines: it might at first sight 
appear that the theory would not be materially altered by re- 
moving 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 [ 
term a mere contour; and we have a new theory, which I call 
that of the Partitions of a Close. 
Several definitions and explanations are required. The words 
line and curve are used indifferently to denote any path which 
ean be described currente calamo without lfting the pen from 
the paper. A closed curve, not cutting or meeting itself *, is 
called a contour. An enclosed space, such that no part of it 
is shut out from any other part of it, or, what is the same 
thing, such that any part can be joined with any other part bya 
line not cutting the boundary, is termed a close. The boundary 
of a close may be considered as the limit of a single contour, or 
of two or more contours lying wholly within the close. The 
reason for speaking of a limit will appear by an example. Con- 
sider a circle, and within it, but wholly detached from it, a figure 
of eight; the space imterior to the circle but exterior to the 
figure of eight is a close: its boundary may be considered as the 
limit of two contours,—the first of them interior to the close, 
and indefinitely near the circle (in this case we might say the 
circle itself) ; the second of them an hour-glass-shaped curve, 
interior to the close (that is, exterior to the figure of eight) and 
indefinitely near to the figure of eight. The figure of eight, as 
being a curve which cuts itself, is not a contour; and in the 
case in question we could not have said that the boundary of the 
close consisted of two contours. A similar instance 1s afforded by 
a circle having within it two circles exterior to each other, 
but connected by a line not cutting or meeting itself; or even 
two points, or, as they may be called, summits, connected by a 
lime not cutting or meeting itself; or, again, a smgle summit: 
im each of these cases the boundary of the close may be con- 
sidered as the limit of two contours. But this explanation once 
given, we may for shortness speak of the close as bounded by a 
* Tt is hardly necessary to add, except in so far as any point whatever 
of the curve may be considered as a poimt where the curve meets itself. 
