As a large part of the difficulty of the theory of curves depends upon the possible intricacy of their multiple points, it seems desirable to shew at once, with the most elementary considerations, that any plane curve is in (1, 1) birational correspondence with a curve, in space of three or more dimensions, which is without multiple points. Some of the ideas involved have already been met with in Chap, 1; but the present chapter is designed to be complete in itself.

Consider a given plane curve f(x, y, z) = 0, of order µ, with any multiple points; the curve is supposed to be irreducible, that is, the polynomial f(x, y, z) cannot be written as the product of other polynomials also rational in x, y, z. A curve whose equation is of the form λ0φ0 + λ1φ1 + … + λr φr = 0, wherein φ0 = 0, …, φr = 0 are definite curves of the same order, which may have common points, upon f = 0, or elsewhere in the plane, determines, by its intersections with f = 0, a set of points thereon. Of these points, some may be the same for all values of the parametric coefficients λ0, …, λr , being common intersections with f = 0 of all the curves φ0 = 0, …, φr = 0; the others will, in general, vary when these coefficients vary.