On the Eliminant of a Set of Quadrics, Ternary or Quaternary

Extract

1. There are two methods for expressing in determinant form the eliminant of three ternary quadrics. If u, v, w are the quadrics and J their Jacobian we may eliminate dialytically the ten quantities x³, y³, z³, y²zyz², z²x, zx², x²y, xy², xyz from the ten cubics xu, yu, zu, xv, yv, zv, xw, yw, zw, J. This is in effect the process given by Sylvester in 1841. The other method is to eliminate dialytically the six quantities x², y², z², yz, zx, xy from u, v, w and the three differentials of the Jacobian.

2. The eliminant of four quaternary quadrics may be found in determinant form by a process having points in common with each of these two methods. Multiplying each of the quadrics by the variables x, y, z, w in turn we get sixteen quaternary cubics, and we have also the four differentials of the Jacobian. Thus we have in all twenty quaternary cubics, or just sufficient to eliminate dialytically the twenty expressions x³, …, x²y, … xyz, … The result is a determinant of order 20, and of the correct degree, viz., 8, in the coefficient of each quadric.

It does not appear to be possible to extend this process so as to obtain, in determinant form, free from extraneous factors, the eliminant of a set of quadrics in more than four variables.