540 
[548 
548. 
ON LISTING’S THEOREM. 
[From the Messenger of Mathematics, vol. u. (1873), pp. 81—89.] 
Listing’s theorem, (established in his Memoir*, Die Census räumlicher Gestalten), 
is a generalisation of Euler’s theorem 8 + F = E + 2, which connects the number of 
summits, faces, and edges in a polyhedron; viz. in Listing’s theorem we have for a 
figure of any sort whatever 
A-B+C-D-(p- 1) = 0, 
or, what is the same thing, 
A + C=B + D + (p- 1), 
where 
A = a, 
B — h — k , 
C — C — K," -f- 7T, 
D=d — k", 
in which theorem a relates to the points; h, tc relate to the lines; c, k", 7r to the 
surfaces; d, k" to the spaces; and p relates to the detached parts of the figure, as 
will be explained. 
a is the number of points; there is no question of multiplicity, but a point is 
always a single point. A point is either detached or situate on a line or surface. 
h is the number of lines (straight or curved). A line is always finite, and if 
not reentrant there must be at each extremity a point: no attention is paid to cusps, 
inflexions, &c., and if the line cut itself there must be at each intersection a point; 
Gott. Abh. t. x. (1862).