where r E P[a, 03), r > 0, and q E C[a, co), are classified by the behavior of their real solutions, as oscillatory or nonoscillatory. In the first instance, one, and thereby every, solution vanishes at an infinite number of isolated points in [a, co); in the second instance each solution has only a finite number of zeros in [a, co). By solution is always meant a function which is not identically zero. A special instance of nonoscillation is the disconjugate case in which every solution has at most one zero in [a, 00). Although there are many results concerning the classification of equations of the form (1.1) with respect to these properties, no completely satisfactory answer has yet been obtained. The purpose of this paper or survey is to identify the known results, to relate the new results and old results to one another, and to unify some aspects of the known theory. For the sake of completeness, we will mention most of the results included in the excellent survey of Rb 19.591. There is a further justification of this duplication in that we will develop the known theory in a different manner than did Rib. The qualitative study of second order linear equations originated in the classic paper of Sturm [81; 18361. However, the general importance and usefulness of Sturm’s work was not properly recognized until the end of the 19th and the beginning of the 20th centuries. At that time the work of B&her [4-71 had a considerable influence in getting recognition of Sturm’s work. For the problem of classifing the solutions of (1. l), Sturm’s main result is his famous comparison theorem: