256 
[451 
451. 
A SECOND MEMOIR ON QUAETIC SURFACES. 
[From the Proceedings of the London Mathematical Society, vol. III. (1869—1871), 
pp. 198—202. Read December 8, 1870.] 
In my Memoir on Quartic Surfaces, ante pp. 19—69, [445], although remarking (see 
No. 79) that the identification was not completely made out, I tacitly assumed that the 
symmetroid and the decadianome (each of them a quartic surface with 10 nodes) were 
in fact identical. There is yet a good deal which I cannot completely explain ; but the 
truth appears to be, that the decadianome includes two cases of coordinate generality, 
say the sextic decadianome, and the bicubic decadianome = symmetroid: viz., in the first 
of these the circumscribed cone, having for vertex any one of the 10 nodes, is a proper 
sextic cone with 9 double lines; in the second it is a system of two cubic cones, 
intersecting, of course, in 9 lines, which are double lines of the aggregate sextic cone: 
or, in the notation of the Table No. 11, in the case of the sextic decadianome, the cir 
cumscribed cones are each of them 6 9 ; in that of the bicubic decadianome = symmetroid, 
they are each of them (3, 3). We thus arrive at a very remarkable sj^stem of 10 points 
in space, viz., giving the name “ ennead ” to any 9 points in piano, which are the 
intersections of two cubic curves, or to any 9 lines through a point which are the 
intersections of two cubic cones; the 10 points in space are such that, taking any one 
whatever of them as vertex, and joining it with the remaining points, the 9 lines form 
an ennead. I purpose in the present short Memoir to consider the theories in question; 
the paragraphs are numbered consecutively with those of the Memoir on Quartic 
Surfaces. 
Plane Sextic Curve with 9 Nodes. 
110. A sextic curve contains 27 constants; and the number of conditions to be 
satisfied in order that a given point may be a node is = 3. Hence it would at first 
sight appear that the curve could be found so as to have 9 given nodes; this would