Of the literature of the geometry of hyperspace that has accumulated in recent years the following papers are cited as having points of contact with the ideas here set forth: CLIFFORD: Prelinminary Sketch of Biquaternions in P r o c e e d i n g s o f t h e London Mathematical Society, vol. 4 (1873), pp. 381-395. CLIFFORD'S theory of parallels in elliptic space is identical with the theory of isoclinal systems of planes in four-dimensional space; namely, planes that pass through a fixed point and make equal dihedral angles with any transversal plane through the same point. (See ?? 30-3 2 of this paper.) CHARLES S. PEIRCE: Reprint of the Linear Associative Algebra of BENJAMIN PEIRCE in the American Journal of Mathematics, vol. 4 (1881). In the foot-note of page 132 attention is called to the fact that in four-dimensional space two planes may be so related to one another that every straight line in the one is perpendicular to every straight line in the other. (See ? 28 (3) of this paper.) I. STRINGHAM: (1) On a Geometrical interpretation of the Linear Bilateral Quaternion Equation; (2) On the Rotation of a Rigid Systenm in Space of Four Dimensions; (3) On the Measure of 4Inclination of two Planes in Space qf Four Dimensions. Papers presented to Section A of the American Association for the Advancement of Science, the first two at the Philadelphia neeting of 1884, the third at the Cleveland meeting of 1888. Abstracts printed in Proceedings of the Association, 1884, pp. 54-56, and privately, 1888. These papers form the nucleus of the present investigation. A. BUTCHHEIM: A Mfemoir on Biqataternions, in the A m e r i c a n J o u r n al