1. Mention what form, of given relation ϕ (a, b, c, …) = 0 between the roots of a given equation will in general serve for the rational determination of the roots; explain the case of failure; and state what information as to the roots is furnished by a given relation not of the form in question.

In the given relation, φ (a, b, c,…) must be a wholly unsymmetrical function of the roots; that is, a function altered by any permutation whatever of the roots; or, what is the same thing, by any interchange whatever of two roots.

For this being so, if α, β, γ, … be the values of the roots, then for some one order, say α, β, γ, …, of these values the given relation φ (a, b, c,…) = 0 will be satisfied by writing therein a = α, b = β, c = γ, &c.; but it will in general be satisfied for this order only, and not for any other order whatever (viz. it will not be satisfied by writing a = β, b = α, c=γ, &c, or by any other such system).