The substance of this paper is contained in Chrystal, chap, xxv., §§13, and 15 to 20, with some applications thereof occurring in chap. xxvi. But it is treated here in a fresh manner which would seem simpler on several points. This mode of presentation was, in the start, suggested by Peano's method given by Prof. Gibson in his “Note on the Fundamental Inequality Theorems Connected with ex and xm,” in Vol. XVIII. of the Proceedings.