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A MEMOIR ON QUARTIC SURFACES. 
[445 
there are not any 8 nodes in regard to which the surface is octadic), the 10th node 
is then any one of a system of 22 [should be 13] points determined by means of the 
9 nodes, and called the dianodal system of these 9 points. But the quartic surface is 
now completely determined; viz., starting with any 7 given points as nodes, we have a 
dianome with 8 nodes, 9 nodes, or 10 nodes, say, an octodianome, enneadianome, or 
decadianome, but not with any greater number of nodes; these can only present them- 
selves when particular conditions are satisfied in regard to the 7 given nodes, and to 
the 8th and ,9th node; and the consideration of the quartic surfaces with more than 
10 nodes would thus form a separate branch of the subject. 
The case of the decadianome (or quartic surface with 10 nodes formed as above 
with 7 given points as nodes) is peculiarly interesting. I identify this with the surface 
which I call a symmetroid; viz., the surface represented by an equation A = 0, where 
A is a symmetrical determinant of the 4th order the several terms whereof are linear 
functions of the coordinates (x, y, z, w); this surface is related to the Jacobian surface 
of 4 quadric surfaces (itself a very remarkable surface), and this theory of the symmetroid 
and the Jacobian, and of questions connected therewith, forms a large portion of the 
present Memoir. 
The theory of the Jacobian is connected also with the researches in regard to 
nodal quartic surfaces in general; and, for greater clearness, it has seemed to me 
proper to commence the Memoir with certain definitions, &c., in regard to this theory. 
It will be seen in what manner I extend the notion of the Jacobian. 
I remark that the present researches on Quartic Surfaces were suggested to me 
by Professor Kummer’s most interesting Memoir “Ueber die algebraischen Strahlen- 
systeme u.s.w.,” Berl. Abh. 1866, in which, without entering upon the general theory, he 
is led to consider the quartic surfaces, or certain quartic surfaces, with 16, 15, 14, 13, 12, 
or 11 nodes; the last of these, or surface with 11 nodes, being in fact a particular 
case of the symmetroid. 
Considerations in regard to the Jacobian of four, or more or less than four, Surfaces. 
1. In the case of any four surfaces, P = 0, Q = 0, R = 0, S = 0, the differential 
coefficients of P, Q, R, S in regard to the coordinates (x, y, z, w) may be arranged 
as a square matrix in either of the ways 
P 
Q 
R 
S 
P, Q, R, S ; B x , By, S z , B w