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

FLECNODES AND BIFLECNODES.

I111

The equation of a quartic having a fleenode at A is
a2U1V1 + aCZ V2 + 2t4 = 0.....................(2),
whilst if A is a biflecnode, the equation is
a-2vO + v + U4 = 0.................... (3).
169. A quartic cannot have more than two flecnodes.
The equation of a trinodal quartic whose nodes are A, B and
C cannot contain any powers of a, 3, y higher than the second,
and must therefore be a ternary quadric in I/a, 1//3, 1/y. Hence
the required equation is
X/y2 + ^y2a2 + v232 7 + y + (Ix + a/r3 + ny) = 0...... (4).
If B and C are flecnodes, the coefficients of f and 32 must
have a common linear factor, and similarly for the coefficients of
7 and 72; whence the equation of a trinodal quartic having flecnodes at B and C may be written in the form
n2q/272y + p p22/(p + q)2 + 12pa232
+ ac83y la + In (p + q)/3 + n7} = 0......(5).
The condition that A should be a flecnode is that the coefficient of a should be a factor of that of a2. This requires that
p = q, in which case the quartic becomes a perfect square.
The condition that A should be a cusp is that p + q = ~ 2p.
The upper sign must be rejected for the reason stated above;
taking the lower sign and changing the constants, (5) may be
written
+= ( + ).................. (6),
\ /3 a \a 7
which is the equation of a quartic having a cusp at A and a pair
of flecnodes at B and C.
170. If a trinodal quartic has two biflecnodes, the third node
must also be a bifiecnode. Also two of the bifiecnodes must be real
and the third one complex; or two must be imaginary and the third
real.
It follows from (3) and (4) that if B and C are biflecnodes
the equation of the quartic must be
X/a2 + v/2 + ^/= 0.....................(7),