THE THEORY OF SCREWS TN NON-EUCLIDIAN SPACE. 435 



The fifth property of common space which we desire to generalize is 

 one which is especially obscured by the conventional coincidence of the two 

 points at infinity on every straight line. We prefer, therefore, to adduce 

 the analogous, but more perfect, theorem relative to two plane pencils of 

 homographic rays in ordinary space, which is thus stated. If the two rays 

 to the circular points at infinity in one pencil have as their correspondents 

 the two rays to the circular points in the other pencil, then it is easily 

 shown that the angle between any two rays equals that between their two 

 correspondents. 



We now write the five correlative properties which the intervene is to 

 possess. They may be regarded as the axioms in the Theory of the Content. 

 Other axioms will be added subsequently. 



399. First Group of Axioms of the Content. 



(I) If three objects, P, Q, R on a range be ordered in ascending para 

 meter ( 400), then the intervenes PQ, QR, PR are to be so determined that 



(II) The intervene between two objects cannot be zero unless the objects 

 are coincident, or unless the intervene between every pair of objects on the 

 same range is also zero. 



(III) Of the objects on a range, two either distinct or coincident are at 

 infinity, i.e. have each an infinite intervene with all the remainder. 



(IV) An infinite object on any range has an infinite intervene from every 

 object of the content. 



(V) If the several objects on one range correspond one-to-one with the 

 several objects on another, and if the two objects at infinity on one range 

 have as their correspondents the two objects at infinity on the other, then 

 the intervene between any two objects on the one range is equal to that 

 between their correspondents on the other. 



400. Determination of the Function expressing the Intervene 

 between Two Objects on a Given Range. 



Let x l , x 2 , x s , x 4 , and y 1} y z , y s , y 4 be the co-ordinates of the objects by 

 which the range is determined. Then each remaining object is constituted 

 by giving an appropriate value to p in the system, 



Let A. and /* be the two values of p which produce the pair of objects of 

 which the intervene is required. It is plain that the intervene, whatever it 

 be, must be a function of x lt x 2 , # 3 , # 4 and y j} y 2 , y s&amp;gt; y t , and also of X and /z. 

 So far as objects on the same range are concerned, we may treat the co- 



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