In this note we consider a relatively ancient stability problem: the behaviour of solutions of the second order differential equation ẍ + f(s) x = 0, where f(s) tends to plus infinity as s tends to plus infinity. An extensive survey of the literature concerning this problem and a resume of results may be found in [ l ]. More recently McShane et a l. [2] have shown that the additional assumption f(s) ≥ 0 is not sufficient to guarantee that all solutions tend to zero as s tends to infinity. Our aim is to demonstrate a new criterion for which all solutions do have the above property. This criterion overlaps many of the cases heretofore considered.