C. VIII. 
69 
548] 
ON LISTINGS THEOREM. 
545 
thus also have k diminished, and therefore B increased by 1; and the balance still 
holds. 
Placing on the same bounding curve a second point, a, and therefore A, is 
increased by 1; but the bounding curve is converted into two distinct curves; that 
is, b, and therefore B, is increased by 1; and the balance still holds. And the like 
for each new point on the same bounding curve. 
10. Spherical surface with n points connected in any manner by lines. 
Reverting to the cases 4 and 5, by joining any two points by a line, we increase b t 
and therefore B, by 1 ; but as regards tc" the two united points take effect as a 
single point; that is, k" is diminished, and therefore C increased, by 1 ; the balance 
is therefore undisturbed. 
The case is the same for each new line, if only we do not thereby produce on 
the surface a closed polygon, or partition an existing closed polygon; in each of these 
cases we still increase b, and therefore B, by 1; and instead of diminishing k", we 
increase c, by 1, and therefore still increase C by 1; and the balance continues to 
subsist. 
By continuing to join the several points we at last arrive at a spherical surface 
partitioned into polygons in any manner whatever; or, what is the same thing, 
we have: 
11. Closed polyhedral surface. Here, if S is the number of summits, F the 
number of faces, E the number of edges; then 
a = S, 
II 
Qq 
b =E, 
k =0, 
to 
II 
c =F, 
k" =0, 7T = 0, 
Q 
II 
d = 2, 
k'" = 0, 
D = 
2 
p = 1, 
P- 1 = 
0 
S+F = E + 2, 
so that we have Euler’s theorem. Observe that this theorem (Euler’s) does not apply 
to annular polyhedral surfaces, or to polyhedral shells. For instance, consider a shell, 
the exterior and interior surfaces of which are each of them a closed polyhedral 
surface; S=S'+S", F = F' + F", E = E' + E", where S' + F' = E' + 2, S" + F" = E" + 2, 
and therefore S + F = E + 4. Listing’s theorem, of course, applies, viz. we have 
12. 
a = 8' + S", 
b=E' + E", 
c =F' +F", 
§ d = 3, 
V = % 
A = S' + S", 
B = 
C =F' + F", 
D = 
p-l = 
E' + E" 
1 
S+F 
= E' + E" + 4.