448 THE THEORY OF SCREWS. [406- 



Two linear equations in # 1} a? 2 , # 3 , x 4 determine a range, and the simul 

 taneous solution of these equations with 



gives the infinite objects on that range but there can only be two and 

 hence we have the following important theorem : 



The co-ordinates a; 1} # 2 , x 3 , x 4 of the infinite objects in a content satisfy an 

 homogeneous equation of the second degree. 



We denote this equation by 



407. The Departure. 



Let the ranges x^ x 2 , 0, be formed when the parameter x 1 -r x 2 has every 

 possible value, then the entire group of ranges produced in this way is called 

 a star. In ordinary geometry the most important function of a pair of rays 

 in a pencil is that which expresses their inclination. We have now to create, 

 for our generalized conceptions, a function of two ranges in a star which 

 shall be homologous with the notion of ordinary angular magnitude. 



We shall call this function the Departure. Its form is to be determined 

 by the properties that we wish it to possess. In the investigation of the 

 departure between two ranges, we shall follow steps parallel to those which 

 determined the intervene between two objects. 



If OP, OQ, OR be three rays in an ordinary plane diverging from 0, then 



In general the angle between two rays is not zero unless the rays are coin 

 cident ; but this statement ceases to be true when the vertex of the pencil is 

 at infinity. In this case, however, the angle between every pair of rays in 

 the pencil is zero. 



Every plane pencil has two rays (i.e. those to the circular points at 

 infinity), which make an infinite angle with every other ray. 



408. Second Group of Axioms of the Content. 



We desire to construct a departure function which shall possess the 

 following properties : 



(VI) If three ranges, P, Q, R, in a star, be ordered in ascending parameter, 

 and if the departure between two ranges, for example, P and Q, be expressed 

 by PQ, then 



PQ + QR = PR. 



