544 on listing’s theorem. [5 48 
Imagine that besides the n points there is an aperture (bounded by a closed curve); 
the case is: 
6. Spherical surface with n points (n > 2) and aperture. 
a — n, A — n, 
b = 1, K =1 B = 0 
C = 1, k" = 11 , 7T — 0, G = 1 — n, 
d = 1, k" = 0 , D = 1 
p = 1, p — 1=0 
1 =1: 
viz. 6 and « are each increased by 1, and therefore B is unaltered; k" is increased, 
and therefore G is diminished, by 1; but d is diminished, and therefore D also 
diminished, by 1. 
7. Spherical surface with n points (n > 2) and two apertures. 
a — n, 
A = 
n, 
6=2, 
k =2 , 
5=0 
c =1, 
K =71+1, 
4 
II 
© 
II 
- n, 
d — 1, 
/// -1 
1C = 1 
b 
II 
© 
P = 1» 
1 
i—* 
II 
© 
0 
= 0 
viz. b and k are each increased by 1, and thus B is still unaltered; k" is increased, 
and therefore G diminished, by 1; k" is increased, and therefore D diminished, by 1, 
and each new aperture produces the like effect. Thus we have: 
8. Spherical surface with m apertures (m > 2). 
a — 0, 
6 = in, 
/ 
K 
= VI 
> 
© 
II 
© 
II 
cq 
c = 1, 
// 
K 
= m — 
1, 
7r = 0, G = 2 — m, 
d = 1 , 
/// 
K 
= in — 
1, 
b 
ii 
to 
i 
p = l, 
2 — in 
p- 1=0 
= 2—771 
where, comparing with case 5, we see the different effects of a point and an aperture. 
9. Spherical surface with m apertures (m > 2) and a point or points on each or 
any of the bounding curves of the aperture. 
If on the bounding curve of any aperture we place a point, this increases a, and 
therefore A, by 1; the bounding curve is no longer a simple closed curve, and we