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+ i[«"(0 + 3gi(f)»(r)]«»(»)t^W. £(#, ÎJ) = 0 is therefore the necessary and sufficient condition that v(z) be factorable as the product of a solution of (5) by a solution of (6). Therefore, if S(v, v) = 0 the number of zeros of v(z) in a given closed interval (a, b) will be limited by Sturm's theorems concerning the zeros of solutions of (5) and (6). If, for example, the equations (5) and (6) have fa and k 2 for their respective indices of oscillation in (a, b) then every solution of (5) [(6)]doi:10.1090/s0002-9904-1920-03304-5 fatcat:g7jttqngpbd4hblsanaituyfke