The tetrahedral complex of lines in space of three dimensions may be defined as the aggregate of lines which meet four given planes in a range of points which is related to some fixed range. In the present paper we consider the more general system of lines, in space of r dimensions ([r]), which meet a certain number k of given primes (or [r−1]) in a range of points which is related to some given range. The paper is divided into three parts. In the first part, after considering successively the cases r = 2, 3, 4, we proceed by an induction argument to the general case. The section following treats the matter from an algebraical point of view, and in it we obtain in a concise form expressions for the locus (when such exists) generated by the lines. In the final section the systems of lines are shown to arise naturally as projections of known configurations in higher space.