A Treatise on the Theory of Screws

430] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 477 the result obtained is a displacement of the most general type called a motor. We now prove Clifford’s great theorem that a right vector and a left vector can be determined so as to form any motor, i.e. to accomplish any required homographic transformation that conserves intervene. For, if we identify the several coefficients at the foot of p. 474 with those of § 420, we obtain equations of the type (11) = aa' — ßß' — 77' — S8', (21) = — aß' — ßa + y8' — 3y', (31) = — ay — ß8' — ya + 8/3', (41) = — a8' + ßy' — yß' — 8a'. These can be simply reduced to a linear form ; for multiply the first by a', and the second, third, and fourth by - ß’, - y, - S', respectively, and add, we obtain (11) a' - (21) ß' - (31) 7' - (41) 8' = a; for a'2 +/3'2 + ?'2 + 8'2= 1. In a similar manner we obtain a number of analogous equations, which are here all brought together for convenience— (11) a’ - (21) ß' - (31) y - (41) 8' = a, - (21) a' - (11) ß' + (41) 7' - (31) S' = ß, - (31) a' - (41) ß' - (11) 7' + (21) 8' = y, - (41) a! + (31) ß’ - (21) y' - (11) 8' = 8. + (22) a' + (12) ß' - (42) y' + (32) 8' = a, + (12) a' - (22) ß' - (.32) y' - (42) 8' = ß, + (42) a — (32) ß' + (22) y' + (12) 8’ — y, - (.32) a’ - (42) ß’ - (12) y' + (22) 8' = 8. + (33) a' + (43) ß' + (13) y’ - (23) 8' = a, + (43) a + (33) ß’ - (23) y' - (13) 8' = ß, - (13) a’ - (23) ß‘ - (33) y - (43) 3' = y, + (23) a' + (13) ß’ — (43) y + (33) 8' - 8. + (44) a - (34) ß' + (24) y' + (14) 8' = a, + (34) a! + (44) ß' + (14) 7' - (24) 8' = ß, — (24) a — (14) ß' + (44) y — (34) 8' = y, + (14) a' — (24) ß' — (34) y - (44) 8' = 8. These will enable a, ß, y, 8 and a, ß', y', 8' to be uniquely determined.