where pn is a polynomial in n variables and Un(z) is a sum of integrals; the convergence is uniform in any compact set of functions z. This result has been obtained by another method by Frechet: Comptes Rendus, 18th January 1909 and 1st February 1909. Construction of pn. Let zp be a value of z(α) such that p−1 n ≤ α ≤ p n . Let us replace z(α) by the piecewise linear function ζ (α) of which the heights are defined by ζ ( p n ) = zp , the first section being parallel to Oα. We have U(ζ ) = un(z1,. . . zn), a continuous function which we can replace by a polynomial pn(z1,. . .zn) such that |pn − un| < 1 n .