350]
ON THE CLASSIFICATION OF CUBIC CURVES.
395
As to the Groups of the Divergent Parabolas. Article Nos. 117 and 118.
117. Taking the equation under the form
act? + by~z + kz 2 (Xx -+- vz) = 0,
we find for a critic centre
3acc 2 + kz . Xz =0,
2 byz = 0,
by* + kz (2Xx + Svz) = 0 ;
hence there is a single critic centre y = 0. 2Xx + 3vz = 0, the critic value of k is
k — —> and with this value of k the equation in fact is
4A, 3 (a# 3 + by 2 z) — 27av 2 (Xx + vz) z 2 = 0,
that is
a ( 4A 3 « 3 — 2lX 2 vx 2 z — 27v 2 z~) + 4X 3 by 2 z = 0,
or, as it may be written,
a (2Xx + 3vz) 2 (Xx — 3vz) + 4X 3 by 2 z = 0,
which puts in evidence the critic centre or node of the curve. But, as there is here
only a single critic centre, there is of course no further theory of the two-fold centre, &c.
118. The groups are as given a priori by the relation of the satellite line \x + vz — 0,
to the semicubical parabola ax 3 + by-z = 0, viz. writing z = 1 and changing the constants,
Divergent Parabolas, the semicubical parabola y 2 = x 3 , which is
LV.
Divergent Parabolas, Asymptotic Semicubical Parabola of seven-pointic contact,
viz. equation of the asymptotic parabola being y 2 — x 3 = 0, and writing for convenience
k (Xx + v) = — Sax + 2b = 0 for the equation of the satellite line, the equation of the curve
is y- = a? — Sax + 2b. And the groups are
LVI. Satellite line cuts asymptotic parabola.
LVII. „ does not cut „ „ .
LVIII. „ passes through vertex of parabola.
And further
Divergent Parabolas. Asymptotic Semicubical Parabola of nine-pointic intersection.
The satellite line is at infinity, and the equation is y 2 = x 3 + 2b. This is group
LIX.
As to the Trident Curve and the Cubical Parabola. Article Nos. 119 and 120.
119. For the Trident Curve, equation is x (x 2 + Xy) + n = 0, or the satellite line is
at infinity, and there is no distinction of groups; we have only group
LX.
120. For the Cubical Parabola this is x 3 + gy = 0, there is no distinction of groups,
and the curve is group
LXI.
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