546 
on listing’s theorem. 
[548 
As another group of examples, consider a plane rectangle, for instance, a sheet of 
paper bounded by its four edges; here 
13. 
a = 4, 
A= 4, 
6=4, 
K 
= 0, 
B = 4 
c = 1, 
k" 
= 0, 
7T = 0, (7=1, 
d = 1, 
/// 
K 
= 0, 
D = 1 
jo = 1, 
© 
II 
rH 
l 
5 
= 5. 
Let the paper be formed into a tube by uniting two opposite sides, the suture not 
being obliterated, but continuing as a line drawn lengthwise from one extremity of 
the tube to the other: here 
14. 
a = 2, 
II 
to 
6=3, 
«' =0, 
B = S 
c = 1, 
k = 0, 
7T = 0, (7=1, 
rH 
II 
k" = 1, 
D = 0 
p = l, 
1 
M1 
II 
O 
3 
= 3. 
Let the suture be obliterated, so that we have simply a tube open at each end; 
here 
a = 0, 
6=2, 
II 
JnS 
© 
II 
to 
II 
o 
rH 
II 
O 
II 
1 
3 
II 
o 
0=0, 
rH 
II 
= 1, 
o 
II 
p — 1, 
0 
1 
Il II 
o o 
Let the tube be formed into an annulus by bending it round and joining the 
two extremities, the suture not being obliterated, but continuing as a closed curve 
round the tube; here 
cT 
II 
e 
II 
p 
6=1, 
II 
y-1 
c = 1, 
tH 
II 
3 
II 
p 
Q 
II 
p 
d = 2, 
= 2, 
II 
i—i 
16.