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

TRINODAL QUARTICS.

127

Let the nodes be situated at the angular points of the triangle
of reference; then the equation of the quartic is
X227y2 + c27y2a2 + p2a2/3 + 2a/3y (la + m/3 + n) = 0...... (1),
which may be written in the form
(X/3y + loya + va/)2 + 2a37y {(I- av) a + (m - vX) / + (n - X-) y} = 0.........(2),
which shows that the line
(7 - Hv) Ca + (m11 - v) /3 - +( - X\h) y = 0.........(3)
is a double tangent.
Since (1) remains unaltered when the sign of any one of the
quantities X,,u, v is changed, we obtain the equations of the three
other double tangents by writing - X, - /, - v respectively for
X, /t, v in (3).
The equation of the conic passing through the eight points of
contact of the double tangents can be shown to be
(Ila + m + )2 - 2a2 -2X2/32 - X X2U2/y2 = 0......... (4),
for if we multiply the equations of the four double tangents
together and subtract the square of (4), it will be found that the
resulting equation reduces to (1).
When the quartic has three biflecnodes, I = n = n =0, and the
conic (4) is self-conjugate to the triangle formed by joining the
three nodes; and when the quartic is tricuspidal, the coefficients
of a2, /32, 72 vanish, and (4) becomes a conic circumscribing the
triangle in question.
192. We shall add a few miscellaneous propositions concerning trinodal quartics.
The six nodal tangents to a trinodal quartic touch a conic.
From (1) it appears that the equation of the nodal tangents
at A is
2/32 + p y2 + 21l/y = O........)............ (5),
which may be written in the form
(3i/3 + I7)) (92, + ~7) = 0,
1t2 -/2 1 r2 21
where t ~2 ~ ~ _ 21. (6).
where =; + 2.................. (6).
91172 V2 l 91 2 V2