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

THE QUARTIC.

5

Combining this with (12) we obtain
a/b = b/c = c/d.....................(13),
which are the required conditions that three roots should be
equal.
The Quartic.
The conditions that three of the roots of the quartic
(a, b, c, d, e7z, 1)4 = 0
should be equal is that the discriminants (6) and (8) should
vanish. From (6) and (9) we obtain
I= ae - C2 - 4 (bd - c2)
(ad - bc)2
= ae - c2 -
ac - b2
aJ
ac - b2
which by (8) requires that
1 = 0, J= 0........................(14).
The condition that the quartic should have four equal roots
involves the additional equation ac - b2 = 0. In combination with
(14), this leads to the three equations
a/b = b/c = c/d = d/e..................(15).
The conditions that a quartic should have two pairs of equal
roots, which are the conditions that the quartic should be the
square of a quadratic factor, can be readily obtained by means of
the relations which exist between the coefficients and the two
pairs of equal roots a and /3. These four relations, after a and /
have been eliminated, lead to the results
ad2= b~e 2
2b3 + a2d = 3abc..................(16).
8. The condition that the product of two of the roots of the
equation F (z) = 0 should be equal to - 1 is the condition that the
elirninant of the equations F (z) = 0 and F (- z-1) = 0 should
vanish. This eliminant is required in finding the orthoptic locus
of a curve, and we shall show how it can be obtained in the case