139 
445] 
A MEMOIR ON QUARTIC SURFACES. 
Circumscribed Sextic Cone. 
Nodes of 
Surface. 
1 
i 6 
2 
6, 
3 
6 2 
4 
6 3 
5 
6 4 
6 
6 5 ; 
5 , 1 
7 
6 6 ; 
5,, 1 
8 
6 7 ; 
5s, 1 
9 
b 8 ; 
5a. 1; 
4 , 2 
10 
6 9 ; 
5 4 , 1; 
4, 2; 
4, 1, 1; 
3, 3 
11 
610; 
5„ 1; 
4 2 , 2; 
4i, l, l; 
3:, 3 
12 
5«, 1; 
4a, 2; 
4 2 , 1, 1; 
8,, 3, 
13 
4a, 1, 1; 
... 
14 
15 
16 
3 , 2, 1 
3r, 2, 1; 3, 1, 1, 1; 2 , 2, 2 
3 1? 1, 1, 1; 2, 2, 1, 1 
2i, 1, 1, 1, 1 
1,1, 1,1, 1,1; 
and moreover, in the cases where there are two or more forms of the sextic cone, 
then the ¿ sextic cones may be of the different forms in various combinations. The 
total number of cases primd facie possible is thus very great; but only a comparatively 
small number of them actually exist. 
12. In the case where there is a plane 1, the sextic cone breaks u}3 into this 
plane, and into a (proper or improper) quintic cone intersecting the plane in 5 lines; 
that is, there will be in the plane 6 nodes; the plane is, in fact, a singular tangent 
plane meeting the surface in a conic twice repeated; and the 6 nodes lie on this 
conic. Taking any one of these nodes as vertex, the corresponding sextic cone breaks 
up into the plane, and into a (proper or improper) quintic cone. 
13. In the cases ¿ = 1, 2, 3, 4, 5, and ¿ = 15, 16, there is only one form of sextic 
cone; so that each node (at least so far as appears) stands in the same relation to 
the surface. Considering the last mentioned two cases; ¿=16, each of the 16 nodes 
gives 6 singular tangent planes, but each of these passes through 6 nodes, theiefore 
the number of planes is =16: similarly, ¿=15, the number of singular tangent planes 
is 15 x 4 4- 6, = 10.