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

2

INTRODUCTION.

denotes a p-ary n-tic, in which the different terms are multiplied
by the coefficients of the corresponding term in the expansion of
(x1 + x2 +.. xcp).
2. If F be a quantic, the result of eliminating the variables
xi, x,... x, between the equations
dF dF dF
-= 0 = = O0
dx1 ' dx2 ' dxn
is called the discriminant of the quantic.
The discriminant of every quadric can be at once written
down in the form of a symmetrical determinant; for in this case
dF/dxl &c. are linear functions of the variables. Thus the discriminant of the ternary quadric
ax2 + by2 + cz + 2fyz + 2gzx + 2hxy............(3)
is a, h, g
h, b, f
g, f, c
or abc + 2fgh - af2 bg ch2.................(4),
which expresses the condition that the quadric should be resolvable into two linear factors. When the quantic is not a
quadric, the elimination must be performed by the methods
explained in treatises on Algebra; but for binary cubics and
quartics, the elimination may easily be performed by the following
process.
3. Let
(ao, a,,... anx, y) =0 and (b0, b,,... bn3x, y)n = 0
be two binary n-tics. If both these equations be divided by yn
and z = xy, they become two equations of the nth degree in z.
Multiply the first equation by bn, and the second by an, subtract
and divide out by z, and the resulting equation is one of degree
n-1. Multiply the first equation by b0 and the second by a0,
and subtract, and the resulting equation will also be of degree
n-1. We have therefore replaced the two equations of degree n
by two other equations of degree n-1, and the process may be
continued until we arrive at two simple equations from which z
can be eliminated.