417] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 459 



In general there are four, but only four double points, i.e. points which 

 remain unaltered by the transformation. If however two of the roots of 

 the biquadratic equation be equal, then every point on the ray connecting 

 the two corresponding double points possesses the property of a double 

 point. 



For if p 1 = p 2 , then 



YI : Y 2 :: X 1 : X 2 , 



and hence the point whose co-ordinates are 



X,, Z a , 0, 0, 



being transformed into 



F 1; F 2 , 0, 

 remains unchanged. 



Let us now suppose that a certain quadric surface is to remain unaltered 

 by the homographic transformation. 



At this point it seems necessary to choose the particular character of the 

 quadric surface in the further developments to which we now proceed. The 

 theory of any non-Euclidian geometry will of course depend on whether the 

 surface adopted as the infinite be an ellipsoid or a double sheeted hyperboloid 

 with no real generators or a single sheeted hyperboloid with real generators. 

 We shall suppose the infinite, in the present theory, to be a single sheeted 

 hyperboloid. 



The homographic transformation which we shall consider will transform 

 any generator of the surface into another generator of the same system, for 

 if it transformed the generator into one of the other system, then the 

 two rays would intersect, which is a special case that shall not be here 

 further considered. 



Let three rays R 1} R 2 , R s be generators of the first system on the 

 hyperboloid. After the transformation these rays will be transferred to 

 three other positions jR/, R 2 , R 3 belonging to the same system. 



Let S 1} 83 be two rays of the second system. Then the intersection of 

 R lt RI, R 2 , R 2 &c., with Si give two systems of homographic points. The 

 two double points of these systems on Si give two points through which two 

 rays of the first system must pass both before and after the transformation. 

 Two similar points can also be found on S 2 . These two pairs of Double 

 points on Si and $ 2 will fix a pair of generators of the first system which are 

 unaltered by the transformation. 



