A Treatise on the Theory of Screws

413] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 453 413. Representation of Objects by Points in Space. The several objects in a content are each completely specified when the four numbers, xlt x2, xit xit corresponding to each are known. It is only the ratios of these numbers that are significant. We may hence take them to be the four quadriplanar co-ordinates of a point in space. We are thus led to the construction of a system of one-to-one correspondence between the several points of an Euclidian space, and the several objects of a content. The following propositions are evident: One object in a content has for its correspondent one point in space, and one point in space corresponds to one object in the content. The several objects on a range correspond one to one with the several points on a straight line. The several objects in an extent correspond one to one with the several points in a plane. Since the objects at infinity are obtained by taking values of xlt x2, xs, x4, which satisfy a quadric equation, we find that— The several objects at infinity in the content correspond with the several points of a quadric surface. This surface we shall call the infinite quadric. The following theorem in quadriplanar co-ordinates is the foundation of the metrics of the objects in the content by the points in space. If «i, x2, xs, x4 and ylt y2, y3, be the quadriplanar co-ordinates of two points P and Q respectively, and if 0lt 02, 03, be any other four points on the ray PQ whose co-ordinates are respectively x2 + X^y2, x3lXiy3, x3 + ^ä + ^2?/2> «b + ^s3/3> Xi-lX^yi, ®i + X32/i, x2JrX3y2, x3-\rX3y3, Xi + X-jT/j, æi + \3/i> Æ2 + Å.4y2, ®3 + K2/s> Xi + XtPi, then we have the following identity Xi ■— Xg x2 — x3 Off.020i X*, — A/4 X2 X, • 0203 Remembering the definition of the anharmonic ratio of four objects on a range (§ 404), we obtain the following theorem :— 771 e anharmonic ratio of four objects on a range equals the anharmonic ratio of their four corresponding points on a straight line.