quartic's Usage Examples:

The simplest form to which the quartic is in general reducible is +6mxix2+x2, involving one parameter m; then Ox = 2m (xi +x2) +2 (1-3m2) x2 ix2; i = 2 (t +3m2); j= '6m (1 - m) 2; t= (1 - 9m 2) (xi - x2) (x21 + x2) x i x 2.

For it is easy to establish] the formula (yx) 2 0 4 = 2f.4-2(f y 1) 2 connecting the Hessian with the quartic and its first and second polars; now a, a root of f, is also a root of Ox, and con se uentl the first polar 1 of of q y p f?

The quartic will have two pairs of equal roots, that is, will be a perfect square, if it and its Hessian merely differ by a numerical factor.

On the one hand, assuming the quartic to have the form 4xix 2, we find i=j=o, and on the other hand, assuming i=j=o, we find that the quartic must have the form a o xi+4a 1 xix 2 which proves the proposition.

The vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three equal roots.

The quartic has four equal roots, that is to say, is a perfect fourth power, when the Hessian vanishes identically; and conversely.

For the quartic (ab) 4 = (aib2-a2b,) alb2 -4a7a2blb2+64a2 bib2 - 4a 1 a 2 b 7 b 2 + a a b i = a,a 4 - 4ca,a 3 +6a2 - 4a3a3+ aoa4 = 2(a 0 a 4 - 4a1a3 +e3a2), one of the well-known invariants of the quartic.

The expression (ab) 4 properly appertains to a quartic; for a quadratic it may also be written (ab) 2 (cd) 2, and would denote the square of the discriminant to a factor pres.

Xnyviaicos, ribbon), a quartic curve invented by Jacques Bernoulli (Acta Eruditorum, 1694) and afterwards investigated by Giulio Carlo Fagnano, who gave its principal properties and applied it to effect the division of a quadrant into 2 2 m, 3.2 m and 5.2 m equal parts.

The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.

Synonyms:

biquadrate, number, biquadratic, fourth power,

Antonyms:

Arabic numeral, Roman numeral, minority, majority,