A Treatise on the Theory of Screws

230] FREEDOM OF THE FIFTH ORDER. 249 (b) The six screws are members of a screw-system of the fifth order and first degree. (c) Wrenches of appropriate intensities on the six screws equilibrate, when applied to a free rigid body. (cZ) Properly selected twist velocities about the six screws neutralize, when applied to a rigid body. (e) A body might receive six small twists about the six screws, so that after the last twist the body would occupy the same position which it had before the first. If seven wrenches equilibrate (or twists neutralize), then the intensity of each wrench (or the amplitude of each twist) is proportional to the sexiant of the six non-corresponding screws. For a rigid body which has freedom of the fifth order to be in equilibrium, the necessary and sufficient condition is that the forces which act upon the body constitute a wrench upon that one screw to which the freedom is reciprocal. We thus see that it is not possible for a body which has freedom of the fifth order to be in equilibrium under the action of gravity unless the screw reciprocal to the freedom have zero pitch, and coincide in position with the vertical through the centre of inertia. Sylvester has shown* that when six lines, P, Q, Ji, S, T, U, are so situated that forces acting along them equilibrate when applied to a free rigid body, a certain determinant vanishes, and he speaks of the six lines so related as being in involution^. Using the ideas and language of the Theory of Screws, this determinant is the sexiant of the six screws, the pitches of course being zero. If xm, ym, zm, be a point on one of the lines, the direction cosines of the same line being am, ßm, ym, the condition is «i, Ä, Vi> a2, ß-i, «3, ßä, ?3> at, ßi, 7«, a5, ßs, ys, «6, A> 2/2V2 — Z‘iß‘2> y-sYs ~ z3ß3> y/Yt-ztßi. y^S, — Zäß:i, ys/'i z6ßc, Z-fil3-X3ys, Ztfils - x3ys, zfa-aWi, xißi - y^i x2ß2 - y^ xäß3 - ysas Xißi - y^i Xbßs yyy, xr,ßr, — yy^6 = 0. * Comptes Rendus, tome 52, p. 816. See also p. 741. t In our language a system of lines thus related consists of the screws of equal pitch belonging to a five-system. In the language of Plücker (Neue Geometrie des Raumes) a system of lines in involution forms a linear complex. It may save the reader some trouble to observe here that the word involution has been employed in a more generalised sense by Battaglini, and in quite a different sense by Klein.