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In this paper, the oscillatory properties of the eigenfunctions of an elastically constrained beam are studied. The method is as follows. The eigenfunction and its first three derivatives are considered as a fourdimensional vector, (u, u', pu", (pu")'). This vector is projected onto two independent planes and polar coordinates are introduced in each of these two planes. The resulting transformation is then used to study the oscillatory properties of the eigenfunctions and their derivatives in a manner analogous to the use of the Pruffer transformation in the study of second order Sturm-Liouville systems. This analysis yields, for a given set of boundary conditions, the number of zeros of each of the derivatives, u', pu", (pu")' and the relation of these zeros to the n 1 zeros of the nth eigenfunction. The method also can be used to establish comparison theorems of a given type. 1. The main purpose of this paper is to study the oscillatory properties of the solutions and their derivatives of the eigenvalue problem (1.1) @(x)u")" Ap(x)u = 0, X > 0, x E (0, ), (1 .2a) u'(O) cos a (pu")(O) sin a = 0, (1.2b) u(O) cos , + (pu")'(O) sin 3 = 0, (1.2c) u'(0 cos y + (pu")(l) sin y = 0, (1.2d) u(l) cos 6 (pu")'(l) sin 6 = 0, where p and p are piecewise continuous, positive functions on [0, 1] and 0 < oa, 3, y, 6 < ir/2 unless specified otherwise. We consider throughout the nonzero eigenvalues and their corresponding eigenfunctions. It is know that the Received by the editors April 16, 1973 and, in revised form, December 17, 1973. AMS (MOS) subject classifications (1970). Primary 34B25, 34C10; Secondary 34C20, 70J 10.

A Prüfer transformation for the equation of the vibrating beam