548] on listing’s theorem. 547 
69—2 
Let the suture be obliterated, so that we have simply a tubular annulus; here 
17. 
a = 0, 
h 
p 
6=0, 
*7 = 0, 
/7 = 0 
c=l, 
II 
to 
4 
II 
i—* 
(7 = 0, 
d = 2, 
k'" = 2, 
/7 = 0 
P = 1> 
p — 1 = 0 
0 
= 0. 
We 
may compare herewith the case of a 
simple annulus or closed curve. 
18. 
a = 0, 
A= 0, 
6=1, 
K =1, 
/7 = 0 
c — 0, 
k" =0, 7T = 0, 
(7=0, 
d = 1, 
k" = 1, 
o 
II 
P= 1. 
p — 1 = 0 
0 
= 0. 
Add 
to such an annulus 
, for instance, three radii 
meeting in the centre ; 
19. 
a = 4, 
A = 4, 
6=6, 
k =0, 
5= 6 
c = 0, 
?5 x 
II 
© 
II 
O 
(7 = 0, 
d = l, 
m o 
= 3, 
Z7 = - 2 
p = 1, 
1 
i—i 
II 
o 
4 
= 4. 
Let the last-mentioned figure become tubular, all sutures being obliterated; then 
20. 
a = 0, 
A = 
0, 
6=0, 
=0, 
5= 0 
c =1, 
= 6, 7T = 1, (7 = 
-4, 
d = 2, 
m n 
fC =0, 
Z) = — 4 
P = 1» 
p — 1 = 0 
-4 
= — 4. 
And so 
if instead of 
the tubular figure, 
annulus with three radii, we had 
tubular figure, 
annulus with 
diameter, then 
21. 
a = 0, 
A = 
0, 
6=0, 
K =0, 
5= 0 
c = 1, 
k = 4, 7T = 1, (7 = 
-2, 
d = 2, 
= 4, 
/> = -2 
P = 1, 
p — 1 = 0 
- 2 
= — 2 ; 
and the like in other cases.