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which is satisfied by the potential V of an electric field in an isotropic medium, in the so-called " static case " of hysteresis.t By solving this equation, regarded as an integral equation, for V2V, we are led to the differential equation of Poisson to determine V. We may however have the case that an integro-differential equation whose solution is subject to certain boundary conditions is reducible to one or more problems in equations of more elementary character only when account is taken of these conditions. This is the point of view to be developed in the present paper. It is a point of view which can be extended to more general types of functional equations. We shall treat, in the following pages, the reduction of boundary value problems in the case of the integro-differential equation of parabolic type, and quite briedy that of hyperbolic type, to problems in linear diBerential and integral equations separately. It is worth while perhaps to specially mention § 2, in which the generalization of the integro-differential equation of the second order to an equation of the first order involving integration

(1957). Calculation of Moments for a Cantor-Vitali Function. The American Mathematical Monthly: Vol. 64, No. 8P2, pp. 22-27.

A recent contribution to potential theory is characterized by the names of Lebesgue, Wiener, Kellogg, Vasilesco and Bouligand. Central features of this contribution are the notions of capacity and of regular boundary point, which are related to each other by Kellogg's hypothetical lemma.t A recent memoir by de la Vallee Poussin reinterprets these theories in the light of potentials of positive mass and the Poincare sweeping out process.: In the present memoir, the author's aim is to push on the development of these central problems of mass distribution, regularity, capacity, and approximation, and to answer definitely some of the questions which have become important. Fortunately there is already at hand, in the analysis of the general integral or linear functional of Radon, Daniell, and F. Riesz, the precise mathematical tool which is necessary.? 1. Integrals and potentials. Let F be a closed bounded point set, T the infinite domain lying in the complement of F, whose boundary t consists entirely of points of F. We consider an arbitrary distribution of positive mass f(e) on F, finite in total amount.II The potential of this mass, at a point M, is given by the generalized Stieltjes integral

There are many things that make the present position of the high school teacher unsatisfactory and of doubtful portent for the future. A particular misfortune dogs the teacher of mathematics. Perhaps only a little can be done about it, but we may at least look the situation in the face. Moreover, even if merely small things may be done at present, an awakened public conscience and love of parents for their children may eventually furnish the lever which will change a small movement into a considerable reform. Is there any doubt

In this equation the function K(w, t ) is called the kerllel; the desired function is u(z) . VVhen the functions K(w, {) and +(.z) are continuous there is no diSculty in finding a continuous 6(z) that shall satisfy equation (1). This general case of the equation (1), under the conditions solelythat +(z) be continuouswhen a-z-b and that X(x, t:) be continuous in the triangular region a-t:-z _ b, was first investigated by VOLTERRA, t who showed that there is one and only one continuous solution in the interval a _ z c b . His method applies without essential change to equations whose kernels are finite in the region a c t _ z _ b and have discontinuities, provided the discontinuities are regularly t distributed. Let us consider, however, a certain equation to which we are led by a hydrostatical problem. Suppose we are given a tube lying in a vertical plane along a curve of arbitrary shape, s t6(.z), where s is the distance along the curve and x the altitude. Let us fill this tube with a liquid of variable linear density v, and then regulate its height .z in the tube by allowing various amounts to flow through the bottom. Let us then legard zo as an analytic function of the depth in the liquid, i. e. zo zo ( Z t: ) . The average linear density is given by the formula

(1924). The Dynamics of Monopoly. The American Mathematical Monthly: Vol. 31, No. 2, pp. 77-83.

(1947). A Necessary and Sufficient Condition of Wiener. The American Mathematical Monthly: Vol. 54, No. 3, pp. 151-155.

Functionals and their applications. Selected topics, including integral equations by G. C. Evans Analysis situs by O. Veblen.