Suppose in general that ϕ = 0, ψ = 0, &c. denote a system of (n − 1) equations between the n variables (x, y, z, …), where the functions ϕ, ψ, &c. are quantics (i.e. rational and integral homogeneous functions) of the variables. Any values (x1y1z1 …) satisfying the equations, are said to constitute a set of roots of the system; the roots of the same set are, it is clear, only determinate to a common factor près, i.e. only the ratios inter se and not the absolute magnitudes of the roots of a set are determinate. The number of sets, or the degree of the system, is equal to the product of the degrees of the component equations. Imagine a function of the roots which remains unaltered when any two sets (x1, y1, z1, …) and (x2, y2, z2, …) are interchanged (that is, when x1 and x2, y1 and y2, &c. are simultaneously interchanged), and which is besides homogeneous of the same degree as regards each entire set of roots, although not of necessity homogeneous as regards the different roots of the same set; thus, for example, if the sets are (x1, y1), (x2, y2), then the functions x1x2, x1y2 + x2y1, y1y2 are each of them of the form in question; but the first and third of these functions, although homogeneous of the first degree in regard to each entire set, are not homogeneous as regards the two variables of each set.