A generalization of the Sturm ~) Separation and Comparison Theorems in n-space

Abstract

The author considers a system of n ordinary, second-order, linear, homogeneous differential equations. The system is supposed self=adjoint and regular. Starting with the theorem tha t any such system may be considered as the Euler equations of an integral of a quadratic form, he shows that the Sturm Separation and Comparison Theorems naturally generalize into theorems concerning the interrelation of conjugate and focal points. Care is taken at every stage to state the final results in a purely analytical form belonging to the theory of differential equations proper. The results obtained were made possible by certain other discoveries already made in connection with the author's theory o/ the calculus o/ variations in the large).