A Treatise on the Theory of Screws

404] THE TH KOP,V OF SCREWS IN NON-EUCLIDIAN SPACE. 445 one range equals that between their correspondents in the other. But this homography is only possible when a critical condition is fulfilled. In the first place, an infinite object on one range must have, as its correspondent, an infinite object on. the other. For if X be an infinite object on one range, it has an infinite intervene with every other object on that range (Axiom IV.); therefore X', the correspondent of X, must have an infinite intervene with every other object on the second range. If, then, X and Y are the infinite objects on one range, and X' and Y' the infinite objects on the other, and if A and B be two arbitrary objects on the first range, and A' and B' their correspondents on the other; then, using the accustomed notation for anharmonic ratio, (ABXY) = (A'B'X'Y'). But, if H' be the factor (p. 440) for the second range, which II is for the first, we have, since the intervenes are equal, H log (ABXY) = H' log (A'B'X'Y'); and, since the anharmonic ratios are equal, we obtain H=H'. If, then, it be possible to order two homographic systems of objects, so that the intervene between any two is equal to that between their correspondents, we must have H and H' equal; and conversely, when II and H' are equal, then equi-intervene homography is possible. We have therefore assumed Axiom v. (§ 399) which we have now seen to be equivalent to the assumption that the metric constant H is to be the same for every range of the content. Nor is there anything in Axiom V. which constitutes it a merely gratuitous or fantastic assumption. Its propriety will be admitted when we reduce our generalized conceptions to Euclidian space. It is an obvious notion that any two straight lines in space can have their several points so correlated that the distance between a pair on one line is the same as that between their correspondents on the other. In fact, this merely amounts to the statement that a straight line marked in any way can be conveyed, marks and all, into a different situation, or that a foot-rule will not change the length of its inches because it is carried about in its owner’s pocket. In a similar, but more general manner, we desire to have it possible, on any two ranges, to mark out systems of corresponding objects, such that the intervene between each pair of objects shall be equal to that between their correspondents. We have shown in this Article that such an arrangement is possible, when, and only when, the property V. is postulated. We may speak of such a pair of ranges as equally graduated.