is the t-value satisfying q (t) S. The assumption of the existence of a third derivative, as pointed out by Milne, was unessential to his proof. Titchmarsh [5] gave a simplified proof of (4) retaining all of Milne's conditions, except that concerning the existence of a third derivative. It has been shown by Wintner and the author [3] that (4) holds under the mere assumption that the graph of q = q (t) is convex upwards; for example, if q (t) has a non-negative second derivative. It will be seen below, however, that the assumption of convexity in all of these proof is quite artificial. In fact, convexity implies that the first derivative q'(t) exists, except possibly