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

36

TANGENTIAL COORDINATES.

in the tangential equation, the line at infinity is a multiple
tangent of order k, and the curve is the reciprocal polar of another
curve with respect to a multiple point of the same order.
Multiple Tangents.
62. We shall now employ equation (4) of ~ 53 to find the
multiple tangents to a curve. This equation determines the
vectorial angle of the points in which the straight line
x + Y=1 = 1.........................(12)
cuts the curve 2'0 u. = 0, and we shall denote it by
F (n)= 0...........................(13),
where m = tan 0.
(i) If three of the roots of (13) are equal, (12) has a contact
of the second order with the curve. The conditions for this are
that the discriminants A, A' of F(mn) and F'(m) should vanish.
This leads to two equations of the form A (I, V) = 0, A' (~, ) = 0,
which are the tangential equations of the original curve and of a
second one, such that every line which has a contact of the second
order with the original curve is a tangent to the latter curve.
(ii) If two pairs of roots of (13) are equal, (12) has a contact
of the first order with the curve at two distinct points.
(iii) If four roots are equal, (12) has a contact of the third
order with the curve.
The preceding method does something more than determine
the multiple tangents to curves. In the case of a cubic the two
nodal tangents, as well as the stationary tangents, have a contact
of the second order with the cubic. Hence if the origin is not a
node, this method will determine the nodal as well as the stationary
tangents. So also in the case of a quartic, every ordinary tangent
drawn from a double point to the curve, and also every line
joining a pair of double points, has a contact of the first order with
the curve at two distinct points; hence this method will not only
determine the double tangents, but also the tangents drawn from
each double point to the curve, together with the lines joining
each pair of double points.
The conditions for the different equalities which can exist
between the roots of cubic and quartic equations are given in