542 
on listing’s theorem. 
[548 
7T. It might be possible to find an analogous definition, but the most simple one 
is that 7r denotes the number of ovoids (unblocked ovoids) or other surfaces not 
bounded by any point or line. 
d is the number of spaces, reckoning as one of them infinite space. 
k" is the sum, for the several spaces, of the number of circuits in each space: 
the word circuit here signifying a path in the space from a point to itself; all 
circuits which can by continuous variation be made to coincide being considered 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 a repetition or combination of these. Thus for infinite space, or 
for the space within an ovoid, there is only the evanescible circuit, or there is no 
circuit to be counted; and the same is the case if within such space we have any 
number of ovoidal blocks (the term will, I think, be understood without explanation); 
but if within the space we have an oval, ring, or other ring-block of any kind 
whatever, then there is (besides the evanescible circuit) a circuit interlacing the ring- 
block, and we count one circuit; and so if there are n ring-blocks, either separate or 
interlacing each other in any manner, then there are, and we accordingly count, 
n circuits. So for the space inside a ring there is (besides the evanescible circuit), 
and we count, one circuit; and the case is the same if we have within the ring any 
number of ovoidal blocks whatever; but if there is within the ring an oval ring or 
other ring-block, then there is one new circuit, and we count in all (for the space 
in question) two circuits. 
p is the number of detached parts of the figure; or, say the number of detached 
aggregations of points, lines, and surfaces. Observe, that rings interlacing each other 
in any manner (but not intersecting) are considered as detached; so also two closed 
surfaces, one within the other, are considered as detached. The figure may be infinite 
space alone; we have then p = 0. 
The examples which follow will further illustrate the meaning of the terms and 
nature of the theorem; and will also indicate in what manner a general demonstration 
of the theorem might be arrived at. 
1. Infinite space. 
a = 
0, 
A 
= 0, 
b = 
o, 
/ 
K 
= o, 
B = 
c — 
0, 
// 
fC 
= 0, 
3 
II 
© 
Q 
= 0, 
d = 
1, 
m 
fC 
= 0, 
D = 
P = 
0, 
P~ 1 = 
0 
0.