418] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 463 



The theorem can otherwise be shown by drawing Figure 46. 



Fig. 46. 



1 4 and 3 2 are to be generators of the infinite quadric. This will show 

 that 4 (and not 2) is the correspondent to 1, and that 2 (and not 4) is the 

 correspondent to 3, and thus the statement of anharmonic equality, 



(1 QQ 3) = (4 PP 2), 

 becomes perfectly definite. 



1 2 and 3 4 are, of course, not generators; they are two conjugate polars 

 of the infinite quadric. 



We can now see the reason of the anharmonic equality. Let PQ be a 

 generator of the infinite quadric, as is clearly possible, for 1, 3 and 2, 4 are 

 both generators of the opposite system. Then, since a generator of the 

 infinite quadric must remain thereon after the displacement, it will follow 

 that P Q , to which PQ is displaced, must also be a generator ; and thus 

 we have four generators, 4 1, PQ, P Q , 2 3, on a hyperboloid of one system 

 intersecting the two generators of another, and by the well-known property 

 of the surface, 



We also see why the infinite quadric is only one of a family which remains 

 unaltered. For, if PQ be a generator of any quadric through the tetrahedron, 

 1, 2, 3, 4; then, since P and Q are conveyed to P and Q , and since the 

 anharmonic equality holds, it follows that P Q will also be a generator of 

 the quadric, i.e. a generator of the quadric will remain thereon after the 

 displacement. 



It is a remarkable fact that, when the linear transformation is given, the 

 infinite quadric is not definitely settled. We have seen how, in the first 

 place, the linear transformation must fulfil a fundamental condition; but 

 when that condition is obeyed, then a whole family of quadrics present 

 themselves, any one of which is equally eligible for the infinite. 



