A Treatise on the Theory of Screws

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 angle between them is zero, or when their intersection is at an infinite distance. Without entering into any statement about parallel lines, we may simply sav, 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- 29—2