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 



Fig. 50. 



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 Y, 



H log (XX 00 ) = # log ( YY PP). 



