- And so if D =2, then the transformed curve is a nodal quartic; 4 can be expressed as the square root of a
**sextic**function of 0 and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of 0, and of the square root of a**sextic**function of 0. - Observe that the radical, square root of a quartic function, is connected with the theory of elliptic functions, and the radical, square root of a
**sextic**function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic function. - Descartes used the curve to solve
**sextic**equations by determining its intersections with a circle; mechanical constructions were given by Descartes (Geometry, lib. - John Wallis utilized the intersections of this curve with a right line to solve cubic equations, and Edmund Halley solved
**sextic**equations with the aid of a circle. - From the invariant a2 -2a 1 a 3 -2aoa4 of the quartic the diminishing process yields ai-2a 0 a 21 the leading coefficient of the Hessian of the cubic, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the
**sextic**. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2.

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