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

MULTIPLE TANGENTS.

37

~ 7, whence the necessary equations for determining the multiple
tangents to these curves are obtained by substituting in these
equations the values of the coefficients of powers of x and y in (4).
For curves of any given degree, the necessary equations can be
obtained from the equalities which must exist between the roots
of the corresponding equations in one variable. Thus we may find
the conditions that a sextic curve may have (i) a triple tangent,
(ii) a double tangent touching the curve at two points of infiexion,
(iii) a double tangent touching the curve at a point of undulation
and having a contact of the first order at the other point.
63. We shall illustrate this method by finding the double
tangents to the symmetrical quartic curve
Ax4 + 2B.x-y2 f + Cy X + a + by2 = 0............(14).
This curve has a node at the origin, and if we transform to
polar coordinates, it will be found that for every assigned value of
0 there are two equal values of r, one of which is positive and the
other negative. Hence the quartic is uninodal, and it will be
shown in Chapter VIII. that its class is ten and the number of
double tangents is sixteen.
If x= e is a double tangent, it follows that if e be substituted
for x in (14) the two values of y2 must be equal. This gives the
equation
(b + 2Be2)2 = 4Ce2 (Ae2 + a)...............(15),
which shows that there are four double tangents parallel to y. In
the same way it can be shown that there are four double tangents
parallel to x. We have thus accounted for eight double tangents.
To find the remainder, we write down the equation for m which is
im4 (C 4- bre) + 2vm3b:q + m2 (2B + b:2 + arq2) + 2rma: + A + a,2 = 0............(16),
whence, by (16) of ~ 7, the equations of condition are
a2',q2 (C + bq2) = b2e2'2 (A + a:2).17
4b33 ~ + a (C + bq2)2 v = 3btv (C + br2) (2B + b:2 + ac 2))
Dividing out by the extraneous factor ~V, the first equation is
the tangential equation of a central conic, whilst the second
represents a curve of the fourth class. These two curves have
eight common tangents, which are the remaining double tangents
to the quartic.