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

DISCRIMINANTS.

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The result of eliminating x and y between two binary quantics
is called their eliminant. Eliminants are sometimes called resultants; but the former term is the better one, since it is more
expressive of the precise nature of the process employed.
4. We shall now write down the discriminants A of the
binary quadric, cubic and quartic.
(i) The quadric (a, b, cix, y)2
A = ac - b...........................(5).
(ii) The cubic (a, b, c, dix, y)3
A = (ad - bc)2 - 4 (ac - b') (bd - C2)............(6),
or A = a2d2 - 6abcd + 4ac3 + 4bd -3b2c2............(7).
(iii) The quartic (a, b, c, d, eax, y)4
a = _ -27J2........................(8),
where I = ae-4bd + 3c2
J= ace + 2bcd - ad2 - b2e- c3...............
5. We shall next establish certain propositions concerning the
roots of an equation. These theorems are contained in most
treatises on the Theory of Equations, but it will be convenient
to collect them for future reference.
The condition that the equation F(z)=0 should have r equal
roots is obtained by eliminating z between the r equations.
F(z)=0, F'(z)=0,... Fr-l(z)=0.
Let a be one of the roots of F(z)= 0; and let z-a = h; then
by Taylor's theorem
F() = F(a + h)= F(a)+ hF' (a) + Ih2F" (a) +... = 0.
Since a is a root of the equation, F(a) = 0; whence dividing
out by h, it follows that if a second root is equal to a, F' (a) = 0.
Continuing this process it follows that if r roots are equal to a, all
the differential coefficients of F(a) up to the (r-l)th must
vanish.
6. The condition that the equation (ao, a,... anz, 1)n=0
should have two equal roots is that the discriminant of the binary
quantic (ao, a,... anx, y)n should vanish.
Let
F(z) =(ao, a,... an z,n 1) *.............. (10),
F (x, y) = (a, a,,... a, y).............. (11),
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