417] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 457 



For these equations must be linear and if X z , X 3 , X^ are all zero then 

 #i&amp;gt; 2, 3, #4 become a;/, # 2 , x 3 , x as they ought to do, and similarly for the 

 others, whence we get 



X,= 



vCo OC*&amp;gt; 



OCn Otif? vu- 



(KA oc* 



We may write this result thus 



Let us now suppose that the vertices of this new tetrahedron are the 

 double points of a homography defined by the equations 



2h = (11) x, + (12) x, + (13) x, + (14) x 4t 

 y, = (21) x, + (22) x 2 + (23) 8 + (24) a? 4 , 

 y 3 = (31) ^ + (32) ^ 2 + (33) x t + (34) # 4 , 

 y 4 = (41) ^ + (42) a; 2 + (43) x a + (44) ar 4 . 



We have to solve the biquadratic 



(11) -p (12) (13) (14) =0. 



(21) (22) -p (23) (24) 



(31) (32) (33) -p (34) 



(41) (42) (43) (44) -p 



Let the roots be p l} p^, p 3 , p t . Then we have 



Pl x{ = (11) / + (12) / + (13) a?, + (14) ar 4 , 

 p a a7 2 = (21) a?/ + (22) # 2 + (23) x, + (24) a?/, 

 /9^ 3 = (31 ) a?/ + (32) a? s + (33) ar, + (34) ar/, 

 P!^ = (41) a;/ + (42) a?/ + (43) &amp;lt;c, + (44) a-/, 



with similar equations for a;/ , a;/&quot;, a^&quot;&quot;, a;/ , &c. 



