An introduction to the modern theory of equations, by Florian Cajori.

ELIMIN ATION

97

The discriminants of the general quadratic, cubic, and
quartic are, respectively, as follows:
Quadratic disc. (bl2 - bob,);
bo
Cubic disc., ~ 35 - 27 (G + 4 H);
b06
Quartic disc., ~ 51, = 6(13 - 27 J2).
77. Discriminant expressed as a Symmetric Function of the
Roots. Since the discriminant of the equation f(x) == 0 vanishes
always when at least two roots are equal, but under no other
conditions, it follows that a, - c2 must be a factor of the discriminant. For if a, and ac are the equal roots, a, - (a2 is the
only simple factor which will vanish because of this equality.
But an interchange of any two roots, say a, and tl must not
alter the numerical value or the sign of the discriminant, since
the discriminant is a constant when the coefficients of the
equation are constants. Hence the lowest positive power to
which the factor aC - ac can occur in the discriminant is the
second power. In other words, (aI - a,)2 is a factor of the
discriminant.
Since this reasoning applies to any two roots whatever,
(a1 - aQ)2 is a factor; also (a1 - C4)2; and so on.
Hence the product
D n (, - a)2 ( (a1 - a )... (a- )2
is a factor of the discriminant. If the multiplications indicated
in this product were carried out, each term would be of the
n(n - 1)th degree in the roots.
The resultant of f(x) = 0 and f'(x) = 0 may be expressed by
~ 73 as f'(cl). f'(a) **f'(9f )
where (,, a2,., an are the roots of.f(x) =. One term of this
product is (na0o2)m(ja]2 * 9,)"-l1; the degree of tis term in the roots
H