Introduction. In a recent unified theory originated by Einstein and Straus [l], the gravitational and electromagnetic fields are represented by a single nonsymmetric tensor gy which is a function of four coordinates xr(r = 1, 2, 3, 4). In addition a non-symmetric linear connection Γjki is assumed for the space and a Hamiltonian function is defined in terms of gij and Γjki. By means of a variational principle in which the gij and Γjki are allowed to vary independently the field equations are obtained and can be written

(0.1)

(0.2)

(0.3)

(0.4)

1. Introduction. Pour la détermination des orbites infiniment voisines de l'équateur, dans le probléme de Störmer, une équation de Hill est à résoudre. Les méthodes sont expliquées d'abord sur l'équation générale, puis appliquées au probléme de Störmer. Signalons les résultats suivants: les orbites équatoriales, considérées du point de vue de leur perturbation dans le plan méridien sont successivement stables, instables impaires, stables, instables paires et cela indéfiniment quand γ1 se rapproche de un; quelques orbites limites entre les zones de stabilité et d'instabilité sont obtenues avec une méthode qui permet n'importe quelle précision désirée.

1. Introduction. A curve on an ordinary surface is a union curve if its osculating plane at each point contains the line of a specified rectilinear congruence through the point. The author has obtained the differential equations of union curves on a metric surface in ordinary space and has exhibited certain generalizations for union curves of known results concerning geodesic curves on a surface. It is the purpose of the present paper to develop the differential equations of the union curves of a hypersurface Vn immersed in a Riemannian manifold Vn+1 of n + 1 dimensions. The osculating plane to a curve on a surface is generalized to a totally geodesic surface the straight lines of which are geodesies in the space Vn+1. A formula is given for the union curvature vector of a curve in Vn.

Introduction. One standard method of studying the incidences of a system of sets A 1, A2, .… , An is to consider the nerve of the system. However, this gives no direct information as to the numbers of components of the various intersections of the sets—information which would be desirable in several geometrical problems. The object of the present paper is to modify the definition of the nerve so that these numbers of components can be taken into account, and to study this modified nerve for systems of sets in a connected, locally connected, normal T1 space S of a given degree of multicoherence r(S). The principal result (Theorem 6, 6.4) is a refinement of a theorem of Eilenberg [4, p. 107], and asserts that, if then under suitable hypotheses we have

(1)

The notion of convexity in (m-dimensional Euclidean spaces) can be generalized to apply to non-connected sets as follows.

A set is said to be C-convex if each of its components is convex. If the number of components of such a set is n, it is called a Cn-convex set.

Introduction. Let I be the closed real number interval: Any subset Δ of I containing at least one number interior to I, will be called a quasiconvexity generating set. To each quasiconvexity generating set Δ we associate as follows a generalized notion of convexity, here called quasiconvexity or Δ convexity. Two numbers α and β, one of which belongs to Δ, the other being determined by the relation a α + β = 1, are called complementary ratios of Δ. A set Q in a real vector space is said to be A convex if for every pair of complementary ratios α and β in Δ and every pair of points a and b lying in Q the point αa +β b also lies in Q.

In this note we investigate certain properties of a set formed as t he vector sum of continua. Our interest in this subject arose in connection with the preceding paper Quasiconvex sets where we use, but do not prove, item 3 below.