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

124

QUARTIC CURVES.

Singularities at Infinity.
188. When a quartic has a singularity at infinity, the equation of the curve may be found in the manner explained in ~ 47,
and we shall proceed to find the Cartesian equation when the
singularity lies on the axis of x. To do this we take a triangle of
reference whose angle B is a right angle, and suppose the singularity at A. We then take BA and BC as the axes of x and y
and transform the trilinear equations given in ~ 165 and 168 by
putting
a=x, /=1, y=y,
for since f3 becomes the line at infinity, we may without loss of
generality suppose it equal to unity. The equations are then as
follows, where Un, Vn denote polynomials in y of degree n.
Tacnode.
m2 + 2my (Xx + ) + y2(Vo+ xV + V ) = 0.
Rharphoid-cusp.
(m + XXy)2 + 2my + y2 (,x V + V2) = O.
Oscnode.
(m + Xxy + pfy)2 + y2 (qxy + V) = 0.
Tacnode-cusp.
(m + Xxy + py + cy2)2 + Ey3 +Fy4= 0.
Flecnode.
x2Ulv,+X u,1V2+ U4 = 0.
Biflecnode.
2x2U2X + xU2 V1 U= 0.
Triple point.
xUs+ U4=0.
Point of undulation.
y3+ (xy + Ay + 2By+ C)2= 0,
where S = 0 is the equation of any cubic curve, and the axis of x
is the tangent at the point of undulation.
With the exception of the flecnode, biflecnode and point of
undulation, a quartic curve cannot have more than one singularity
of the preceding character. Hence the discussion of curves having
a pair of imaginary singularities of the latter kind at the circular