§ 1. The object of this note is to establish the above inequality in as general a form as possible, and to prove by means of it two of the principal propositions in the theory of inequalities, one of which is usually proved by means of infinite series. The logical advantage in making the theory of inequalities independent of that of infinite series is obvious, when it is remarked that the discussion of the convergency of infinite series is strictly speaking a part of the theory of inequalities.