A Treatise on the Theory of Screws

432] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 479 that when P is situated on either of two rays then the directions of dis- placement are identical. To determine these two rays we draw the two pairs of generators corresponding to the two vectors. As these generators belong to opposite systems, they will form four edges of a tetrahedron. The two remaining edges are a pair of conjugate polars, and they form the two rays of which we are in search. The proof is obvious: a point P on one of these rays must be displaced along the same ray by either of the vectors, for this ray intersects both of the generators which define that vector. Let a right vector consist of rotations + a, + a about two conjugate polars, and let a left vector consist of rotations + a, —a, also about two conjugate polars. Without loss of generality we may take the two conjugate polars in both cases to be the pair just determined. Let 00' and PP' be two conjugate polars (fig. 50). The right vector is appropriate to the generators OP and O'P'. The left vector to the generators OP' and O'P. If we take the intersections with the quadric in the order 00' for A, then we must take them on B in the order PP’ if we are considering a right vector, and in the order P'P if we are considering a left vector. This is obvious, for in the first case we take the intersections of the conjugate polar with the generators OP and O'P'. In the second case we take the intersection of the conjugate polar with OP' and O’P. If, therefore, the vector be right, we have for the displacements of X and F, II log (XX’OO') = H log (YY'PP').