A Treatise on the Theory of Screws

428] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 475 We thus learn the remarkable fact, that if a right (left) vector be followed by a left (right) vector, the effect produced is the same as if the order of the two vectors had been interchanged. This is not true for two right vectors or two left vectors. The theorems at which we have arrived may be thus generally enunciated:— In the composition of vectors the order of two heteronymous vectors does not affect the result, but that of two homonymous vectors does affect the result. In the composition of two homonymous vectors the result is also an homony- mous vector. In the composition of two heteronymous vectors the result is not a vector at all. The theorems just established constitute the first of the fundamental principles relating to the Theory of Screws in non-Kuclidian Space referred to in § 396. Their importance is such that it may be desirable to give a geometrical investigation. 428. Geometrical proof that two Homonymous Vectors com- pound into one Homonymous Vector. Left vectors cannot disturb any right generators of the infinite quad- ric. Take two such generators, AB and A'B' (Fig. 48). Let AA', BB’, GC be three left generators which the first vector conveys to A-lAi, Bfff, GfJf and the second vector further conveys to A.,A.f B2B.f C2Cf Let X and Y be the double points of the two homo- graphic systems defined by A, B, G and A*, B.2, G2. Then we have (ABGX) = (A2BAX), and (ABGY) = (AMY). As anharmonic ratios cannot be altered by any rigid displacement, it follows that X and Y must each occupy the same position after the second vector which they had before the first, similarly, X' and Y' will remain unchanged, and as the two rays, AB and A'B‘ are divided homo- Fig. 48.