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

OSCNODES.

107

must therefore be satisfied, (i) the two tangents at the origin
must coincide, (ii) the coincident tangent must have a contact of
the third order with the curve.
Equation (2) shows that the first condition requires that
B2b2 = Att, whence (1) may be written
{Ax (x - a) - Bby}2 + 2ABx2y (x - a) + Ay2 (21 + i,) =...(6),
also by (2) the tangent at the origin is
Aax +Bby = 0........................(7).
Let u2 = ax2 + 213xy + 7y2
tux= 2ex + 2fy } ** * *..................
To find where (7) intersects (6), substitute the value of y from
(7), and it will be found that the resulting equation will reduce to
= 0, provided
B2b + Ae - A2af/Bb = 0..................(9).
Putting a = 0 in (6) and (9) and substituting the value of e from
(9), (6) becomes
(Ax2 - Bby + Bxy)2 + y2 (2Afy + Au - B2x2) = 0......(10),
which is the general equation of a quartic having an oscnode at
the origin and the axis of x as the oscnodal tangent.
Tacnode Cusps.
164. A tacnode cusp is formed by the union of a tacnode and
a cusp.
The figures show the forms of the curve just before and after
coincidence. The curve belongs to species VIII., which has two
double tangents; and since the tangent at a tacnode counts twice,
A
0 O