1. In the linear differential equationx" + f{t)x = 0,let f(t) be a real-valued, continuous function for large positive t, say for l" ^ t < oo. Consider only those solutions x{t) of (1) which are real-valued and distinct from the trivial solution (identically zero). Then, if N(t) denotes the number of zeros of x($) on the interval t0 S s tk t, it is clear from Sturm's separation theorem thatcannot hold for a particular solution x{t) unless it holds for every solution x(t). In this case, (1) is called oscillatory. Thus (1) is called non-oscillatory if one (hence every) non-trivial solution fails to acquire an infinity of zeros, as t -> oo.In the applications, the importance of the classification of the differential equations (1) into the oscillatory and non-oscillatory categories is due to the following well-known fact: A non-trivial solution of (1) must change its sign whenever it vanishes, since x(t) and x'(t) = dx (t)/dt cannot vanish simultaneously.A general criterion has been developed f for the type of stability defined by (2). In what follows, a sufficient criterion that x(t) be oscillatory will be given which is not contained in known explicit tests.2. It is easy to see that, if(1) must be oscillatory whenever the indefinite integral m = f m ds (4) satisfies the condition Fit) -> as t->oo.In fact, (3) and (1) imply that the graph of every solution x = x(t) must always turn its concavities toward the 2-axis of the (t, x)-plane. Hence, if (1) is non-oscillatory, and if the solution is so chosen that x{t) > 0 as t ->oo} it is clear that x(t) ^ c holds for every sufficiently large t and for a positive constant c. Consequently, from (1) and (3) y x'(t) -const. --J x(s)f(s) ds S -c J f(s) ds.Since -c < 0, it now follows from (4) and (5) that, if t is large enough, x'(t) is negative, hence xit) is decreasing. But this contradicts the assumption that x = x(t) is ultimately positive, and concave toward the £-axis, as t -»oo.By a substantial refinement of this argument, it will be shown below that, if (4) satisfies (5), then (1) must be oscillatory, whether (3) is assumed or not.