A Treatise on the Theory of Screws

400] THE THEORY OF SCREWS TN NON-ETTCLIDIAN 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 PQ + QR = PR. (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 «1; «a, xs, x4, and ylt y2, y3, y4 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, a^ + pyi, x2 + py2, x3 + pys, x4 +py4. Let Å. and y 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 a\, x2, x3, x4 and y2, y3, y3, y4, and also of X and y. So far as objects on the same range are concerned, we may treat the co- 28—2