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

4

INTRODUCTION.

then we have already shown that the condition that (10) should
have a pair of equal roots is that the eliminant of F(z)=0
and F'(z)= 0 should vanish; and consequently the eliminant of
F' (z)= 0 and nF(z)- zF' (z)= O must vanish. But on writing out
these expressions in full, it will be found that these conditions
are the same as
dF I dF
=0 =0,
dx ' dy
which are the conditions that the discriminant of F(x, y) should
vanish.
If F(x, y) is a binary n-tic given by (11), it follows that
drF/dx" divided by the numerical factor n (n- 1)... (n- r + 1) is a
binary (n - r)-tic; in other words if
F(x, y)=(ao, a,... ancx, y)n,
1 drF
then (= (ao, a,... anrx, y)n-r
whence the theorem of ~ 5 may be otherwise stated:-If
F(z) = (ao, a,... anZ, 1)n,
the condition that the equation F (z) = 0 may have r equal roots is
that the discrimninants of F (z) and all its differential coefficients up
to the (r - 2)th should vanish.
7. We shall proceed to find the conditions for equalities
between the roots of cubic and quartic equations.
The Cubic.
The condition that two of the roots of the cubic
(a, b, c, dz, 1)3 = 0
should be equal is that the discriminant should vanish; whence
by (6) the required condition is
(ad - bc)2 - 4 (ac - b2) (bd - c2) = 0.........(12).
If a third root is equal the discriminant of the quadratic
(a, b, c3z, 1)2= 0 must also vanish; whence by (5) we have the
second condition
ac - b2 = 0.