548] 
on listing’s theorem. 
543 
2. Spherical surface. 
a — 0, A. = 0, 
6 = 0, k! = 0, B = 0 
C = 1, K." = 0, 7T = 1, G =2, 
d = 2, = 0, D = 2 
p — 1, p — 1 = 0 
2 =2: 
viz. the effect is to increase G by 2 and D and p — 1 each by 1. 
3. Spherical surface, with point upon it. 
a = 1, 
6 = 0, 
II 
o 
pH 
II 
to 
II 
o 
c = 1, 
?5^ 
II 
O 
3 
II 
o 
Q 
II 
j—* 
d = 2, 
/// f\ 
K, — (J, 
D = 2 
II 
i—i 
2 
p — 1 = 0 
= 2 
viz. the effect is to increase a and diminish ir each by 1; that is, A is increased 
and G diminished each by 1. 
4. Spherical surface with two points. 
a — 2, 
6=0, 
k =0, 
II 
B = 0 
c=l, 
k" = 1, 
3 
II 
O 
Q 
II 
p 
d= 2, 
k"'= 0, 
<N 
II 
II 
i—i 
2 
p — 1 = 0 
= 2 
viz. the second point increases a and tc" each by 1, that is, it increases A and 
diminishes G each by 1. 
And for each new point on the spherical surface there is this same effect; so 
that we have, for the next case: 
5. Spherical surface with n points {n > 2). 
a = 
n, 
A = 
= w. 
6 = 
o, 
K 
= 0, 
B 
= 0 
c = 
1, 
// 
fC 
= n — 1, 
7T = 0, G = 
= 2 — 71, 
d = 
: % 
m 
K, 
= 0, 
D 
= 2 
p = 
1, 
P- 1 
= 0 
2 
= 2.