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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.1. Introduction. In a recent paper 1 we initiated a theory of symbolic dynamics. In this theory we consider unending sequences of symbols or symbolic trajectories and devote attention to those properties of symbolic trajectories which are suggested by dynamical considerations. A symbolic trajectory is formed from symbols taken from a finite set of generating symbols subject to certain rules of admissibility. In SD admissibility conditions were formulated of such generality that the resulting symbolic trajectories include in particular those which arise in the geodesic problem on surfaces which satisfy the condition of uniform geodesic instability.However, no surface of the topological type of a torus satisfies the conclition of uniform geodesic instability and the admissibility conditions of SD do not include those which arise in the case of the torus.In the present paper we consider a class of symbolic trajectories formed from two generating symbols subject to admissibility conditions defined by a simple comparison property. These are the symbolic trajectories which characterize the geodesics on a flat torus. They may be used to characterize the distribution of the zeros of the solutions of a differential equation of the form y" + f(x)y = 0, where f(x) is a periodic function of x. We term the trajectories of this class Sturmian. A first fundamental result is as follows:Sturmian trajectories possess certain numerical characteristics, namely, a frequency, a pole, and a type index, and admit mechanical constructions uniquely determined by these characteristics.There are three types of Sturmian trajectories,-irrational, skew and periodic. The trajectories of irrational type are recurrent but not periodic; those of skew type are not recurrent. The recurrency functioll of a recurrent Sturmian trajectory is completely determined by the frequency a of the trajectory and may be denoted by R(n, a). We introduce the variable y = 2 (1 + a)-'. Let CG/Dv be the convergents in a continued fraction representation of y. We have the following fundamental theorem:

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