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

SINGULARITIES AT INFINITY.

27

has a double point at infinity lying on the axis of x. The equation of the tangents at the double point is
X12 + 2//y + vy2 = 0,
and the latter will be a node, a cusp or a conjugate point according
as 2 > or = or < Xv.
When v = 0, the line at infinity is one of the tangents at the
double point; and when /u = v = 0, the double point is a cusp and
the line at infinity is the cuspidal tangent.
49. To find the equation of a curve having a point of inflexion
at infinity on the axis of x.
The general equation of a curve having a point of inflexion at
B is
M-2 u, (pa + q, + ry) + /-3u3 +... = 0.....(35);
whence if B is at infinity, the trilinear equation is found by writing
I for a; whilst the Cartesian equation is found as in the last
article by substituting the values of a, fi, y from (33). Whence if
ul = Xa + vy, the required equation is
x-2 (XI + vy) (pI + qx + ry) + xn-3 U3 +... Un = 0...(:36).
The equation of the inflexional tangent is
XI + vy = 0........................(37),
and is therefore parallel to the axis of x, excepting in the case
in which v = 0, when it becomes the line at infinity.
50. To find the condition that the line at infinity should touch
the curve.
If the line a = 0 is the tangent at C, the equation of the curve
is
yn- + -lay22 +... n = 0.................(38),
where un is a binary quantic in a and /. Let
a=I, B3=y, y=ax+by,
then (38) becomes
(ax + by)-l + (ax + by)-2 U2 +... U, = 0,