A Treatise on the Theory of Screws

248 THE THEORY OF SCREWS. [230 relation must subsist between the six screws. This relation may be ex- pressed by equating the determinant of § 39 to zero. The determinant (which may perhaps be called the sexiant) may be otherwise expressed as follows:— The equations of the screw Ak are x~x^.y-yk z-zk We shall presently show that we are justified in assuming for T the equations x V z , . . ä=^ = -(Pltch = p)- The condition that Ak and T be reciprocal is (p + Pfc) (aa* + ßßlc + 77*) + Xk (yßk — ßyk) + yk (ayk _ ydk) + (/3a* - aßk) = o. Writing the six equations of this type, found by giving k the values 1 to 6, and eliminating the six quantities pa, pß, py, a, ß, y, we obtain the result:— aiPi +712/1 ÄPi + Mi - Vi«i, 7iPi + -ajyi, «J, ßlt y1 a2p2 + y2y2- ß2z3, ß2p2 + a3z3- y3x2, y2p2 + ß^-ccpj?, a2, ß2, y2 ^p3 + y3y3-ß3z3, ß3p3 + a3z3-y3x3, y3p3 +ß3x3 - a3y3, a3, ß3, y3 ^Pi + yiyt- ßiZi, ßiPi + a^- yiXi, y4p4 + ß^-a^i, ait ßt, y4 asps + y3y3 - ßfa, ß5p5 + a4Z3 - y3x3, y5ps + ßsxs - a3y3, as, ß5, yä aep3 + y3y3- ß6z3, ßsps + deZf - y3x6, y6p6 + ß6xs - a6y3, a6, ße, ye = 0. By transformation to any parallel axes the value of this determinant is unaltered. The evanescence of the determinant is therefore a necessary condition whenever the six screws are reciprocal to a single screw. Hence we sacrificed no generality in the assumption that T passed through the origin. Since the sexiant is linear in xlt ylt zlt it appears that all parallel screws of given pitch reciprocal to one screw lie in a plane. Since the sexiant is linear in a1; ßlt ylt we have another proof of Möbius’ theorem (§ 110). The property possessed by six screws when their sexiant vanishes may be enunciated in different ways, which are precisely equivalent. (a) The six screws are all reciprocal to one screw.