280 
A THIRD MEMOIR ON QUARTIC SURFACES. 
[454 
symmetroid. That is, a surface having a single node (3, 3) is a symmetroid ; but I 
have shown (Second Memoir, No. 116) that a symmetroid has each of its ten nodes 
(3, 3); wherefore the surface having a single node (3, 3) is the 10(a)-nodal surface, 
nodes 10 (3, 3). 
154. Start from two cubic cones U=0, V = 0, having each the same vertex 
(pc = 0, y = 0, z = 0) ; we may in a variety of ways determine the two cones a U + /3 V = 0, 
yU + 8V = 0, having a common inscribed quadric cone A = 0 (viz., a : /3 being assumed 
at pleasure, then y : 8 will be determined ; not, I believe, uniquely, but I do not 
know what the multiplicity is). This being so, the quadric cone .4=0 is uniquely 
determined ; and then, assuming at pleasure the plane w = 0, the 10 (a)-nodal surface 
Aw 2 + 2i?w + T = 0 is uniquely determined: consequently the remaining nine nodes are 
determinate points on the nine lines U = 0, V = 0 respectively. And we have thus 
a system of ten points in space such that, joining any one of them with the remaining 
nine, the nine lines so obtained are the intersections of two cubic cones, or say that 
they are an ennead of lines. 
Notation for the Cases afterwards considered. 
155. I proceed to further develope the theory of some of the different surfaces. 
The same node-form of equation will, of course, assume different shapes according to 
the actual expressions in terms of the coordinates (pc, y, z) of the several functions 
A, &c., which enter into it. I have found it convenient to attribute to A and B 
certain specific values which are not in every case those of the coefficients of w 2 , w in 
the equation of the surface : this means that we must, in the equation of the surface, 
substitute new symbols for these coefficients, and write the equation say in the form 
A'w 2 + 2B'w + T = 0 ; the change of notation, when it occurs, will be duly explained. 
156. It is in general (but not always) convenient to take the equation of the 
tangent cone to be x 2 -\- y 2 + z 2 — 2yz — 2zx — 2xy = 0 ; for then any plane - + \ + - = 0, 
a P 7 
where a + /3 + 7 = 0, will be a tangent plane; so that six tangent planes may be 
represented by x = 0, y — 0, z = 0, and by three equations of the form just referred to. 
And in reference to this assumed form of the equation of the tangent cone, and to 
what follows, I write 
a + /3 +7=0, 
a.' + /3' + y' = 0, 
a" + ¡3" + y" = 0, 
t, x y z 
P =~ + “I + “ 
« ¡3 y 
P f iAj U ió 
= _ 4- -I 
/ ~ O’ ‘ / > 
a /3 y 
a" + /3" y"