V. On the locus of singular points and lines which occur in connection the theory of the locus of ultimate intersections of a system of surfaces

In a paper “On the c - and p -Discriminants of Ordinary Integrable Differential Equations of the First Order,” published in vol. 19 of the ‘Proceedings of the London Mathematical Society,' the factors which occur in the c -discriminant of an equation of the form f ( x, y, c ) = 0, where f ( x, y, c ) is a rational integral function of x, y, c , are determined analytically. It is shown that if E = 0 be the equation of the envelope locus of the curves f ( x, y, c ) = 0; if N = 0 be the equation of their node-locus; if C = 0 be the equation of their cusp-locus, then the factors of the discriminant are E, N 2 , C 3 .