Ths envelope of a family of plane curves f (x, y, c) = 0 is defined in many text-books as the locus of the ultimate intersections of f(x, y, c) = 0 and f(x, y, c + δc) = 0. This definition has one serious disadvantage; for, it is easily prove J that if the family possess any double points, these will appear in the c-discriminant.