412] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 451 



hitherto assumed will entitle us to draw any inference as to the connexion, 

 much less as to the actual identity, between the critical systems related to 

 intervene, and those related to departure. We have already assumed five 

 properties for the intervene, and five like properties for the departure. These 

 are, in fact, the axioms by which alone the functions of intervene and of 

 departure could be constructed. But another axiom of quite a distinct type 

 has now to be introduced. 



There are objects of infinite intervene, and objects of zero departure. 

 There are ranges of infinite departure, and ranges of zero intervene. A 

 range generally contains two objects of infinite intervene, and two of zero 

 departure. A star generally contains two ranges of infinite departure, and 

 two ranges of zero intervene. On a range of zero intervene the two objects 

 of infinite intervene coalesce, and their intervene from other objects on the 

 range becomes indeterminate. In a star of zero departure the two ranges 

 of infinite departure coalesce, and their departure from other ranges in 

 the same star becomes indeterminate. We have thus the following state 

 ment : 



On a range of zero intervene, the intervene between every pair of objects 

 is zero, except where one particular object is involved, in which case the 

 intervene is indeterminate. 



In a star of zero departure, the departure between every pair of ranges 

 is zero, except where one particular range is involved, in which case the 

 departure is indeterminate. 



The new axiom to be now introduced will be formed as the others have 

 been by generalization from the conceptions of ordinary geometry. In that 

 geometry we have two different aspects in which the phenomenon of paral 

 lelism may be presented. Two non-coincident lines are parallel when the 

 ansle between them is zero, or when their intersection is at an infinite 



o 



distance. Without entering into any statement about parallel lines, we may 

 simply say, that when two different straight lines are inclined at the angle 

 zero, their point of intersection is at infinity. Generalizing this proposition, 

 we assume the following axiom or property, which we desire that our systems 

 of measurement shall possess. 



412. The Eleventh Axiom of the Content. 



This axiom, which is the first to bring together the notions of intervene 

 and of departure, is thus stated : 



(xi) If two ranges in the same extent have zero departure, their common 

 object will be at infinity, and conversely. 



The vertex of every star of zero departure will thus be at infinity, and 

 hence we deduce the important result that all the objects of infinite inter- 



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