1. In the complete algebraic system of concomitants of a pair of quaternary quadrics given by Turnbull† there appear sixteen covariant line-complexes, eight of the second order and eight of the third. In a later paper‡, the same author gave some geometrical interpretations of these and other concomitants. None the less, no systematic geometrical account of the relations of these complexes seems to have been made. The purpose of the present paper is to discuss the geometrical relations systematically, and make their origin clearer. Incidentally, the syzygies which exist between the complexes are obtained explicitly.