^{page 27 note *} The discriminant may always be written in either of the forms:—

where χ, χ′ are determinants whose first columns contain respectively only the coefficients A, B, D, and N, P, Q; and Ψ, Ψ′ are determinants whose first columns contain respectively only A, B and P, Q.

Differentiating (α) we obtain:—

It is evident from (α) that Δ = 0 always passes through the points common to A = 0, B = 0, and that if these co-efficients have a common factor Δ contains this factor. In this case (c) shows that Δ_B, Δ_D, etc., vanish.

If A, B, and D have a common factor, χ also contains this factor, and the same factor is by (a) repeated in Δ; (c) shows that in this case all the derived functions vanish. If one of the functions B, D vanish identically, it may be considered as divisible by the factors contained in A.

Similar propositions hold with respect to N, P, Q, and in fact all the results obtained above may also be obtained in this way.

^{page 31 note *} Note also here that the vanishing of the first three or the last three coefficients, in virtue of their containing a common factor, leads to a repeated factor in the discriminant. This case has already been disposed of under (1).