548] 
on listing’s theorem. 
541 
and in general a point placed on a line constitutes a termination or boundary of the 
line. Thus a line is either an oval (that is, a non-intersecting closed curve of any 
form whatever), a punctate oval (oval with a single point upon it), or a biterminal 
(line terminated by two distinct points). For instance, a figure of eight is taken to 
be two punctate ovals; an oval, placing upon it two points, is thereby changed into 
two biterminals. 
k. The definition, analogous to the subsequent definitions of k" and k", would be 
that k is the sum, for all the lines, of the number of circuits for each line; but 
inasmuch as for an oval the number of circuits is = 1, and for any other line (punctate 
oval, or biterminal) it is = 0, k' is in fact the number of ovals. 
c is the number of surfaces. A surface is always finite, and if not reentrant 
there must be at every termination thereof a line: no attention is paid to cuspidal 
lines, &c., and if the surface cut itself there must be at each intersection a point or 
a line; and in general a point or a line placed on a surface constitutes a termination 
or boundary thereof. It may be added that if a line intersects a surface there must 
be at the intersection a point, constituting a termination or boundary as well of the 
line as of the surface. Thus a surface is either an ovoid (simple closed surface, such 
as the sphere or the ellipsoid), a ring (surface such as the torus or anchor-ring), or 
other more complicated form of reentrant surface; or else it is a surface in part 
bounded by a point or points, line or lines. We may in particular consider a 
blocked surface having upon it one or more blocks: where by a block is meant a 
point, line, or connected superficial figure composed of points and lines in any manner 
whatever, the superficial area (if any) included within the block being disregarded as 
not belonging to the surface, or being, if we please, cut out from the surface. Thus 
an ovoid having upon it a point, and a segment or incomplete ovoid bounded by an 
oval, are each of them to be regarded as a one-blocked ovoid; the boundary being 
in the first case the point, and in the second case the oval; and so in general the 
blocked surface is bounded by the boundary or boundaries of the block or blocks. It 
will be understood from what precedes, and it is almost needless to mention, that for 
any surface we can pass along the surface from each point to each point thereof; 
any line which would prevent this would divide the surface into two or more distinct 
surfaces. 
k" is the sum, for the several surfaces, of the number of circuits on each surface. 
The word circuit here signifies a path on the surface from any point to itself: all 
circuits which can by continuous variation be made to coincide being regarded as 
identical; and the evanescible circuit reducible to the point itself being throughout 
disregarded. Moreover, we count only the simple circuits, disregarding circuits which 
can be obtained by any repetition or combination of these. Thus for an ovoid, or 
for a one-blocked ovoid, there is only the evanescible circuit, that is, no circuit to 
be counted; but for a two-blocked ovoid there is besides one circuit, or we count 
this as one; and so for a ?i-blocked ovoid we count n — 1 circuits. For a ring it is 
easy to see that (besides the evanescible circuit) there are, and we accordingly count, 
two circuits; and so in other cases.