[312 
312] ON THE PARTITIONS OF A CLOSE. 65 
does not cut or 
edge itself. 
If we consider the surface of a plane as bounded by a mere contour at infinity, 
then for the infinite plane, /3=0, or we have P + S=E + 1+ B: in the case where 
? any partitioned 
S' the number of 
3d close, we have 
the infinite plane is partitioned 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 partitioning contour, that 
is, for it, B — 1), and the equation is thus satisfied. And so for a contour having upon 
it n summits, P = 2, S = n, E = n, B—l, and the equation is still satisfied: this is the 
case of the plane partitioned into two parts by means of a single polygon. 
liner the close is 
inner, of summits 
b to itself or to 
S by unity: if 
unity; for if the 
! addition of the 
or summits, the 
of the contour, 
the addition of 
lity. Hence the 
aity E or else B, 
of a mere con- 
b is easy to see 
t contain within 
,ves the equation 
>y unity; if the 
Ltour to another 
P by unity; if, 
it on a different 
Hi ere are a few 
ied in the two 
mmits, these are 
rs, and the like 
so if there be 
Irawn connecting 
iwn through two 
thus to increase 
by unity; that 
+ /3 = P+1+P, 
ever manner the 
The case of a spherical surface is very interesting: the entire surface of the sphere 
must be considered as a close bounded by 0 contour, or we have ¡3 = — 1, and the 
equation thus becomes P + S = E+ '2+B. Thus, if the sphere be divided into two 
parts by a mere contour, P = 2, $ = 0, E= 0, B = 0, and the equation is satisfied. And 
in general, when B= 0, then P + S — E + 2 ; or writing F for P, then F + S = E + 2, 
which is Euler’s equation for a polyhedron. 
2, Stone Buildings, W.G., March 8, 1861. 
a mere contour, 
is the ordinary 
C. V. 9