418] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 461 



Reverting, then, to a space of which the several points correspond to the 

 objects of a content, we find, that for every homographic transformation 

 which corresponds to a displacement in ordinary geometry a singly infinite 

 family of quadrics is to remain unchanged, and the infinite quadric itself is 

 to form one member of this family. 



Let us now suppose a range in the content submitted to this description 

 of homographic transformation. Let P, Q be two objects on the range, and 

 let X, Y be the two infinite objects thereon. This range will be transformed 

 to a new position, and the objects will now be P , Q , X , Y . Since infinite 

 objects must remain infinite, it follows that X and Y must be infinite, as 

 well as X and Y. Also, since homographic transformation does not alter 

 anharmonic ratio, we have 



(PQXY) = (P Q X Y }; 



whence, by Axiom v., we see that the intervene from P to Q equals the 

 intervene from P to Q ; in other words, that all intervenes remain unchanged 

 by this homographic transformation. 



Every homographic transformation which possesses these properties must 

 satisfy a special condition in the coefficients. This may be found from 

 the determinantal equation for p (p. 458), for then the following symmetric 

 function of the four roots p lt p. 2 , p :i , p 4 must vanish : 



(Pi Pz ~ Pap*) (pip 3 - p 2 pi) (pip* - p 2 p s ). 



418. The Geometrical Meaning of this Symmetric Function. 



We may write the family of quadrics thus : 



JVXs+ : xr t x-.a, 



All these quadrics have two common generators of each kind : 

 , = 0, ^ 3 = fZ 1 = 0, X, = 



and and 



For the rays ^ = 0, X a = Q, and X 1 = 0, Z 4 = 0, are both contained in the 

 plane X 1; and therefore intersect, and, accordingly, belong to the opposed 

 system of generators. 



The geometrical meaning of the equation 



PiPz Psp* 



can be also shown. 



The tetrahedron formed by the intersection of the two pairs of generators 

 just referred to remains unaltered by the transformation. Any point on the 

 edge, X l = 0, X 3 = 0, of which the co-ordinates are 



0, *, 0, Z 4 , 



