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

130

QUARTIC CURVES.

which shows that the nodal tangents at A, whose equation is
2 + r + I = o..................... (1o),
intersect the quartic at the two points D, D' where it is cut by
the conic
y + a (nm/ + ny) =..................(11).
This conic circumscribes the triangle of reference, and therefore passes through the three nodes which make up the remaining
six points of intersection of the conic and the quartic.
From (10) and (11) it can be shown that the equation of the
line DD' is
kla + 3/mn + y/n = 0..................... (12),
where k1 = m/n + n/m - I........................(13).
By cyclical interchanges of the letters (a,,/, y) and (1, m, n)
the corresponding results for the nodal tangents at B and C and
the corresponding points of intersection E, E' and F, F' can be
obtained.
The equation of the quartic may also be written in the form
(/32 + 72 + 1t7)(72 + a2 + mya) - y72 {72 + /3 y + mya + (Inm - n)a3} = 0......(14),
the first term of which is the product of the equations of the
nodal tangents at A and B. The form of (14) shows that the
conic
y2 + l/3, + mnya + (lm - n) a/3 = 0............(15)
passes through the points of intersection D, D' and E, E' of the
nodal tangents at A and B respectively; accordingly we obtain
the following theorem:A conic can be described through any two nodes of a trinodal
quartic and the four points at which the tangents at these nodes
intersect the quartic.
Let S = (P, Q, R, P', Q', ',/, y)2= 0......... (16)
be the equation of the proposed conic which is assumed to pass
through the six points D, D'; E, E'; F, F' in which the nodal
tangents at A, B and C intersect the quartic. Then the equation
S + (ca + //m n ry/n) (a/I + k2,3 + Y7/n) = 0.........(17)