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451] A SECOND MEMOIR ON QUARTIC SURFACES, 
so that the equation is 
(S,V-^V)*+(g ?l V )* = (); 
a sextic cone breaking up into the two cubic cones 
S C V -8 ( *V ±»S n V= 0, 
so that the cone is (3, 3). And since clearly the point (0, 0, 0) may be regarded as 
representing any one whatever of the 10 nodes, it follows that for any node whatever 
of the symmetroid, the circumscribed cone is (3, 3), so that, as stated above, bicubic 
decadianome = symmetroid. 
Deductions from the foregoing theory. 
119. Referring to No. 85 of the original memoir, it appears that, with 6 given 
points as nodes, we can actually find for the symmetroid an equation containing 6 con 
stants. I cannot discover any ground for doubting that 3 of these may be determined 
so as to give to the symmetroid a seventh given node; and I therefore assume that 
with 7 given points as nodes, an equation can be found with 3 constants. The 
symmetroid is certainly not octadic, hence the eighth node must lie on the dianodal 
surface of the 7 given points. I can discover no ground for doubting but that two 
of the constants may be determined so that the eighth node shall be any given point 
whatever on the dianodal surface of the 7 points; and (this being so) that further 
the remaining constant may be determined so that the ninth node shall be any given 
point whatever on the dianodal curve of the 8 points. But if all this be so, the 
consequence is very remarkable; the tenth node is not any one whatever of the 22 
dianodal centres of the 9 points, but it is a uniquely determinate “ enneadic centre,” 
viz., we must have the following theorem : 
120. “Take any 7 points; an eighth point at pleasure on the dianodal surface 
of the 7 points; a ninth point at pleasure on the dianodal curve of the 8 points. 
In the system of 9 points so determined, take any one as vertex, and joining it with 
the remaining 8, construct the ninth line of the ennead. Performing this construction 
with each of the 9 points successively as vertex, we obtain 9 lines passing through 
the 9 points respectively. These 9 lines meet in a point which is the ‘ enneadic 
centre ’ of the 9 points: and further, the 10 points form a completely symmetrical 
system, so that each one of them is the enneadic centre of the remaining 9.” 
121. Assuming that the 9 lines do intersect so as to give rise to an enneadic 
centre, there is no difficulty in conceiving that the loci, which by their intersection 
determine the dianodal centres, do each of them pass through the enneadic centre; 
so that this enneadic centre counts once or more among the dianodal centres, and the 
number of proper dianodal centres, instead of being = 22, will be suppose = 22 — &>, 
and if, further, the 9 points, together with the enneadic centre, are the nodes of a 
symmetroid, but the 9 points together with any one of the 22 — &> dianodal centres 
are the nodes of a sextic decadianome, then we must also have as follows: 
33—2