Two invariants, two quartics and a **sextic**. They are connected by the relation 212 = 2 i f?0 - D3 -3 jf 3.

The .**sextic** covariant t is seen to be factorizable into three quadratic factors 4 = x 1 x 2, =x 2 1 - 1 - 2 2, 4) - x, which are such that the three mutual second transvectants vanish identically; they are for this reason termed conjugate quadratic factors.

Three quintic forms f; (f, i) 1; (i 2, T)4 two **sextic** forms H; (H, 1)1 one septic form (i, T)2 one nonic form T.

For a further discussion of the binary **sextic** see Gordan, loc. cit., Clebsch, loc. cit.

The complete systems of the quintic and **sextic** were first obtained by Gordan in 1868 (Journ.

The Binary **Sextic**.-The complete system consists of 26 forms, of which the simplest are x2y2z2 + (1 +8 m3) 2 (y3z3 +z3x3 +x3y3).

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.

And for the **sextic** 1 -a3° 1-a 2.1-a 4.1-a'.1 -a io.

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.

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.