400] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 437 



upon the specific value of 8, but must be such that, when one of the 

 quantities is given, the other shall be determined by a linear equation. It 

 is therefore assumed that X and //, must be related by the equation 



where the ratios of the coefficients A, B, C, D shall depend, to some extent, 

 upon 8. If X and // be a given pair of parameters belonging to objects 

 at the required intervene, then 



by which the disposable coefficients in the homographic equation are reduced 

 to two. 



The converse of Axiom II., though generally true, is not universally so. 

 It will, of course, generally happen that when two objects coincide their 

 intervene is zero. But on every range two objects can be found, each 

 of which is truly to be regarded as two coincident objects of which the 

 intervene is not zero. 



Let us, for instance, make X = yu, in the above equation ; then we have 



This equation has, of course, two roots, each of which points out an object 

 of critical significance on the range. We shall denote these objects by 

 and . Each of them consists of a pair of objects which, though actually 

 coincident, have the intervene 8. The fundamental property of and is 

 thus demonstrated. 



Let X be any object on the range ; then (Axiom I.) 



XO + 8 = XO; 



and as 8 is not zero, we have 



XO = infinity. 



Therefore every object on the range is at an infinite intervene from 0. A 

 similar remark may be made with respect to ; and hence we learn that 

 the two objects, and , are at infinity. 



We assume, in Axiom in., that there are not to be more than two objects 

 on the range at infinity : these are, of course, and . We must, therefore, 

 be conducted to the same two objects at infinity, whatever be the value of 

 the intervene 8, from which we started.* We thus see that while the 

 original coefficients A , B, C, D do undoubtedly contain 8, yet that 8 does not 

 affect the equation 



A\* + (B + C) X + D = 0. 



* My attention was kindly directed to this point in a letter from Mr F. J. M Aulay. 



