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 system of the quadratic and cubic, consisting of 15 forms, and that of two cubics, consisting of 26 forms, were obtained by Salmon and Clebsch; that of the cubic and quartic we owe to Sigmund Gundelfinger (Programm Stuttgart, 186 9, 1 -43); that of the quadratic and quintic to Winter (Programm Darmstadt, 1880); that of the quadratic and sextic to von Gall (Programm Lemgo, 3873); that of two quartics to Gordan (Math.
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.
Thus the general curve of three bar-motion (or locus of the vertex of a triangle, the other two vertices whereof move on fixed circles) is a tricircular sextic, having besides three nodes (m = 6, 6 = 3+3+3, = 9), and having the centres of the fixed circles each for a singular focus; there is a third singular focus, and we have thus the remarkable theorem (due to S.
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.