(1) In the Cambridge Math. Journal, ii. p. 233, Sylvester showed how to eliminate x, y, z from the set of equations

His method consisted in deriving other three equations involving the variables x2, y2, z2, yz, zx, xy, and then eliminating these six variables from the six equations. The result obtained was

—a result which, on account of its complexity, it is impossible to rest satisfied with.

The manifest fact that when A = M = R = 0, the eliminant takes the form

and the further fact, that when a = m = r = 0, the equations become linear, with the eliminant

raise the presumption that Sylvester had not hit upon the simplest mode of performing the elimination.