The considerations set forth in the present paper are valid for the most part in parallel form for integration in space of any number of dimensions, and in the widest measure for any modification of the process that may be devised naturally: integration of functions not necessarily one-valued, with respect to functions of sets, or functions of ∑, not necessarily unchanged by subdivision, in fields as wide or as restricted as may be convenient, as long as we maintain the possibility of forming subdivisions of arbitrarily small “norm,” and of splitting up such a subdivision arbitrarily into two parts, each of which may be subdivided independently in any manner, to give a subdivision of the whole. These conditions are essential for the proper working of the theory, and in particular for the validity in more dimensions of Hyslop's method referred to in 5.