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be 9 x 3, = 27 conditions, or the curve would be completely determinate. But observe 
that through the 9 given points we have a determinate cubic curve U = 0 ; we have 
therefore U 2 = 0 a sextic curve, and the only sextic curve with the 9 given nodes ; 
that is, there is not in a proper sense any sextic curve with the 9 given nodes. The 
number of given nodes is thus = 8 at most. 
111. The sextic curve with 8 given nodes should contain 27 — 3.8 = 3 constants. 
We may through the 8 given points draw the two cubics P = 0, Q = 0 ; and we have 
then (a, b, c\P, Q) 2 = 0, a bicubic, or improper sextic curve having the 8 nodes, and 
also a ninth node, viz., the remaining point of intersection of the two cubic curves, 
or say the remaining point of the ennead. Hence if V = 0 be any particular sextic 
curve having the 8 given nodes, we have 
(a, b, c$P, Q) 2 + dV =0 
a proper sextic curve having the 8 given nodes ; and this, as containing the right 
number (=3) of constants, will be the general sextic curve having the 8 given nodes. 
112. There will be a ninth node if 0 = 0; viz., the curve is then (a, b, c]£P, Q) 2 — 0, 
a bicubic, or improper sextic curve, having for nodes the 9 points of the ennead. 
Observe that the ninth node is here a point completely and uniquely determined by 
means of the given 8 nodes. Moreover the number of constants is = 2, so that we 
have here a general (improper) solution of the question of finding a sextic curve with 
9 nodes, 8 of them given. 
113. But if 6 is not = 0, then the ninth node must be a point on the curve 
J(P, Q, V) = 0; viz., this is a curve of the order 9, determined by means of the 
given 8 points ; say it is the “ dianodal curve ” of these 8 points, and, as is easy to 
see, it has each of these 8 points for a node. The ninth node of the sextic may be 
any point whatever on the dianodal curve ; and regarding it as a given point, the 
sextic will still contain 1 constant ; that is, we have the general solution of the 
problem of finding a sextic curve with 9 nodes, 8 of them given, and the 9th a given 
point on the dianodal curve. 
114. So long as the 8 points are arbitrary, the dianodal curve does not pass 
through the 9th point of the ennead, and the two cases above considered are mutually 
exclusive. It will be noticed how closely analogous this theory of the plane sextic 
with 9 nodes, is to that of the quartic surface with 8 nodes. 
115. Of course, instead of the plane sextic curve, we may have a sextic cone; 
such a cone has at most 8 given double lines ; and if there be a 9th double line, 
then there are the two cases of coordinate generality ; viz., (1), the new double line 
is the ninth line of the ennead, the cone being in this case not a proper sextic cone, but 
a bicubic cone ; (2), the new double line may be any line whatever on the dianodal 
cone, (cone of the order 9 determined by the 8 given lines, and having each of these 
for a double line,) and regarding it as a given line on the dianodal cone, the sextic 
cone contains 1 constant. 
C. VII. 
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