Table of Singularities, and Explanations in regard thereto. 
1. We have on the before-mentioned surfaces respectively certain simple or multiple 
points, right lines, and curves, as shown in the following Table : 
abcdef, 4 
abcdea, 8 
abcda/3, 16 
abca/3y, 24 
obaf3yS, 24 
aa(3y8e, 14 
a(3y8e£, 8 
Points a 
6 x (2) 
5 x (4) 
4 x (8) 
3 x (8) 
2 x (4) 
lx (2) 
0 
Lines ab 
15 x (1), C 
10 x (2), C 
6 x (4), C 
3 x (4), C 
1 x (2), C 
0 
0 
a 
0 
1 x (2), C 
2 x (4), C 
3 x (8), C 
4 x (8), C 
5 x (4), C 
6 X (2), C 
[ab, a, ¡3, y] 
0 
0 
0 
6 x (4), P 
( 3 )8 x (2 + 2), L 
0 
0 
[ a , P, y, 8] 
0 
0 
0 
0 
2 x (8), P 
10 x (2), L 
30x(l), L 
[ab, cd, a, /3] 
0 
0 
6 x (2), P 
0 
0 
0 
0 
abc, def 
10 X (1), P 
0 
0 
0 
0 
0 
0 
abc, de, a 
0 
1—1 
o 
X 
*0 
0 
0 
0 
0 
0 
abc, a, /3 
0 
0 
( 2 )4 x (2 + 2), P 
as 
X 
CO 
0 
0 
0 
Cubic abcdef 
1 x (1), C 
0 
0 
0 
0 
0 
0 
Quadriquadric a[3y, Set, 
0 
0 
0 
0 
0 
0 
(10) x (1), L 
Excuboquartic a/3y, Se, a 
0 
0 
0 
0 
0 
0)10 x (1), L 
0 
J Tangent plane along line is plane abc. 
2 Tacnodal line, each sheet touched along line by plane abc. 
3 Tacnodal line, each sheet of surface touched along line by hyperboloid afiy. 
4 Surface touched along line by hyperboloid a/3y.