Non-oscillatoey Lineae Differential Equations of the Second Oedee

and for the sake of simplicity we will assume that the coefficients p and q are, throughout the finite interval a = x = b, continuous real functions of the real variable x. We shall find it convenient to lay down the following definition : The equation (1) is said to be oscillatory or non-oscillatory in the interval a~x~b according as it does or does not have at least one solution (not identically zero) which vanishes more than once in this interval. I t is my object in the present paper to deduce certain conditions (chiefly sufficient conditions) that the equation (1) should be non-oscillatory. Such conditions have been obtained by Picard (Traité d'analyse, volume I I I , pp. 101104); but the method which I use is not only entirely different and, as it seems to me, less artilicial than that of Picard, but yields, besides all of Picard's results, others which Picard's method does not give. My starting point is the special case p = 0 :