The following investigation of Prof. Sylvester's theory of Residuation may be compared with that given in Salmon's Higher Plane Curves, 2nd Edition (1873), pp. 133–137:

If the intersections of a cubic curve Us with any other curve Vn are divided in any manner into two systems of points, then each of these systems is said to be the residue of the other; and, in like manner, if starting with a given system of points on a cubic curve we draw through them a curve of any order Vn, then the remaining intersections of this curve with the cubic constitute a residue of the original system of points.

If the number of points in the original system is = 3p, then the number of points in the residual system is = 3q; and if we again take the residue, and so on indefinitely, the number of points in each residue will be = 0 (Mod. 3); viz. we can never in this way arrive at a single point. But if the number of points in the original system be 3p ±1, then that in the residual system will be 3q ∓1; and we may in an infinity of different ways arrive at a residue consisting of a single point; or say at a “residual point,” viz. after an odd number of steps if the original number of points is = 3p -1, but after an even number of steps if the original number of points is =3p+l.