A Treatise on the Theory of Screws

474 THE THEORY OF SCREWS. [427, and, from the second vector, = 4- ay. + ß'y2 + y'ys + S'y4, Zz = ~ ß'yi + a'y2 + 8’y3 - y'y„ z3 = - yy.- + ay3 + ß'yt, Zi = - 8'y. + yy3-ß'y3 + a'y,. Substituting for y., y.2, y3, yit we obtain the following values for z., z2, z3, zt. The right vector, a, ß, y, 8, followed by the right vector, a, ß', y, 8'— z1= + aa' - ßß' - yy' - 33' xT + + aß' + ßa' + yd' - Sy' x2 + ay' - ß8' + ya + 8ß' x3 + + ct3z + ßy' - yß' + 8a x4, z2 — — aß' ~ ßa' ~ yd' + dy' x±+ + aa' - ßß' - yy - 88' x2 + aS' + ßy' - yß' + 8a' x3 H— ay' + ß8' - ya' - 8ß' x4, z3= - ay' 4- ß8' - ya' - 8ß'xT H— a8' - ßy' + yß' - 8a'x,2 + aa' — ßß' - yy' - 88' x3 + 4- aß' + ßa' + yd' - 8y' x4, z4— - aS' - ßy' + yß' - 8a'x^ + 4- ay' - ßS' + ya' 4- 8ß'x2 4— aß' - ßa' - y8' + 8y' x3 + + aa' - ßß' - yy' - 88' x4. The right vector, a', ß', y', 8', followed by the right vector, a, ß, y, 8— z4 = 4- aa' - ßß' - yy' - 88' x4 + + aß' + ßa - y8' + 8y' x2 + + ay' + ß8' + ya' - 8ß' #3 + + aS' - ßy' + yß' + da' x4, z2= - aß' - ßa + y8' - 8y' + + aa' - ßß' - yy' - 88'x2 + + a8' - ßy' + yß' + da' x3H— ay' — ß8' - ya' 4- 8ß' x4i z3= - ay' - 8ß' - ya' + 8ß'x1 H— aS' + ßy' - yß' -8a' x2 + + aa' - ßß' - yy' - 88'x3 4- + aß' + ßa' - yS'4- Sy' x4i z4= - a8' 4-ßy' — yß' - 8a' x4 + + ay' + ßS' + 7a - Sß' x2-i— aß' - ßa' + yS' - 8y' x3 + + aa' - ßß' - yy' - 88' x4. We thus learn the important truth, that when two or more homonymous vectors are compounded, the order of their application must be carefully specified. For example, if the object æ be first transposed by the vector a. and then by a!, it attains a position different from that it would have gained if first transposed by a! and then by a. We see, however, that in either case two homonymous vectors compound into a vector homonymous with the two components. We now study two heteronymous vectors, i.e. one right and one left. The right vector, a, ß, y, 8, followed by the left vector, a', ß', y', 8'— z1 = 4- aa' - ßß' - yy' - 88' + + aß' + ßa' 4- y8' - 8y' x2 + + cry' - ß8' + ya' + 8ß' x3 + + aS' + ßy' - yß' 4- 8a x4, z2= — aß' - ßa' + y8' - 8y'x14- + aa' - ßß' 4- yy' + 88' x2 4— aS' — ßy' - yß' + 8a'x3 + + ay' - ß8' - ya' - 8ß' x4i z3— - ay' — ß8' - ya' + 8ß'3^+4- aS' - ß8' - yß' - 8a' x2+ + aa’ 4- ßß' - yy' + 88'x3-\— aß' + ßa' - y8' - 8y' x4i z4= - a8' + ßy' - yß' - 8a'x4 4— ay' - ß8' + ya' - 8ß' x2 + + aß' - ßa' - yb' - 8y' x3 + + aa' + ßß' + yy' - 88' x4. The left vector, a', ß', y', 8', followed by the right vector, a, ß, y, 3— z4 = 4- aa' - ßß' - yy' - 88' x4 + 4 aß' + ßa' + yS' - 8y' x2 + + ay' - ß8' + ya' 4- 8ß' x3 + + aS' + ßy' - yß' + 8a' x4i z2~ - aß' — ßa +y8' - 8y' ^+4- aa' - ßß' + yy' -f 53'— a8' - ßy' - yß' + 8a x3 + +ay' - ß8' - ya' - 8ß' x4i z3= - ay' - ß8' - ya' + 8ß' ^ + + aS' - ßy' - yß' - 8a' x2 + + aa' + ßß? - yy' + 88' x3 H— aß' 4- ßa' - y8’ - 8y' x4, z4— — a8' + ßy' - yß' - 8a' ^4- - ay' - ß8' + ya' - 8ß'x2+ +aß' - ßa' - y8' - Sy1 x3 + +aa' + ßß' + yy' - 88'x4.