The Smallest Characteristic Numbers in a Certain Exceptional Case. by Professor Maxime Bôcher

We have proved this theorem, it is true, only when k > 0. If h = 0 it is, however, merely an obvious consequence of Theorem I. We come now at last to our most important result, though one which is, at bottom, less far reaching than Theorem II, namely THEOREM III. If e is an arbitrarily given positive constant, a continuous, real function g(x) exists such that 0 < g(x) < e and such that the system (5) is incompatible. The proof consists simply in noticing that if we add to the function g(x) determined in Theorem II a sufficiently small function everywhere positive (not zero), the system (5) will, by Theorem I, remain incompatible.* This theorem is useful in making connection, by the method originally given in special cases by Hubert, between the system (1) and an integral equation of the second kind. HARVARD UNIVERSITY, CAMBRIDGE, MASS.