422] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 467 



seen the necessary characteristic of the homographic transformation which 

 preserves intervene. The infinite quadric which the transformation fails 

 to derange can be written at once, for we have 



x* + x? + # 3 2 + # 4 2 = yi 2 + / 2 2 + y? + y? = 0. 



It is also easily seen that the expression 



is unchanged by the orthogonal transformation. We thus have the following 

 quadric, which remains unaltered : 



+ (24K] 

 ^ 

 or, writing it otherwise, 



If this be denoted by U, and x? + x + x + # 4 2 by H, then, more generally, 

 U AH is unaltered by the transformation. 



We now investigate the intervene 6, through which every object on 



V-MI = Q 



is conveyed by the transformation. 



If we substitute x 1 + \y 1 &c. for x l &c. in the infinite quadric we have 



and, accordingly, the intervene 6, through which an object is conveyed by 

 the orthogonal transformation is defined by the equation 



cos ^ = n ; 



hence the locus of objects moved through the intervene is simply 



u - n cos e = o. 



422. Proof that U and H have Four Common Generators. 



The equation in p has four roots of the type 



These correspond to the vertices of the tetrahedron (fig. 47). Symmetry 

 shows that the conjugate polars as distinguished from the generators will be 

 the ray joining the vertices corresponding to 



and p , * 



302 



