An elementary treatise on cubic and quartic curves, by A. B. Basset.

TRINODAL QUARTICS.

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193. From each node of a trinodal quartic two tangents can be
drawn to the curve, and these six tangents touch a conic.
Let / = kIy be one of the tangents drawn to the quartic from
the node A. Substitute in (1), divide out by y2 and express the
condition that the resulting quadratic in a/y should have equal
roots; this gives a quadratic equation for k, and on substituting
/3/7 for k we obtain
(X2v2 - 2) 2 + 2 (2 (2 - rmn) / + (X2L2 - n2) 72 = 0,
which is the equation of the two tangents drawn from the node A.
Let o1 = /2v2 — 12, 2 = P2X2-m2, -2 3 = X2/ 2- n2;
then proceeding as in ~ 192 we shall find that the tangential
equation of the conic which touches the six tangents is
0-03a2 + o-3-2l + O1C2 2 + 2o1 (mn - lX2) vr
+ 202 (nl - m,2) t + 20-3 (Im - nv2) f = 0,
and the trilinear equation is
o-2X2a2 + o-22A/2/2 7 + 0-32P272 + 2120-3-/y + 2mo-3oyya + 2no-2a/3 = 0.
194. The following additional properties of trinodal quartics
may be mentioned*.
(i) The six points of inflexion lie on a conic.
(ii) The six points of contact of the tangents drawn from the
nodes lie on a second conic.
(iii) The six points in which the nodal tangents intersect the
quartic lie on a third conic.
(iv) The three conics pass through two points P and Q on the
quartic, which lie on the conic
X21/3y + u2mya + z2na3 = 0.
We shall prove the third theorem as an example of the mode
of dealing with such questions.
If in (1) we choose three new coordinates a', 3', y' such that
a/X = a', &c., and then change the constants 1, mn, n; the equation
of a trinodal quartic may be written in the form
a2 (/32 + y2 + 1/73) + /y {/3 + a (ns3 + n)} = 0,
* Brill, Math. Annalen, Vol. xII. p. 90; xIII. p. 175; F. Meyer, Apolaritdt und
Rationale Curven, pp. 283-7.

B. C.

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