A Treatise on the Theory of Screws

260 THE THEORY OF SCREWS. [239- If a, b, c be the radii of gyration, then the instantaneous screw cor- responding to y has for co-ordinates , Vi % , % Vi . Vs Vs a a b b c c The condition that y and its instantaneous screw shall be parallel to a pair of conjugate diameters of the momental ellipsoid is a2 (.Vi + %) - + b2 (y3 + %) — ~ + c3 (% - vs) Vs—-~ = 0; tv 0 c or 2p1i712=p, = O. But if the impulsive wrench on 17 be a force, then the pitch of y is zero, whence the theorem is proved. 240. Theorem. When an impulsive wrench acting on a free rigid body produces an instantaneous rotation, the axis of the rotation must be perpendicular to the impulsive screw. Let i?!, ... ys be the axis of the rotation, then %PiVi = 0, or a (vi - V2) (Vi + V2) + b (i?3 - t?4) (% + 7/4) + c (% - Vs) (vs + Vs) = 0, whence the screw of which the co-ordinates are + (iylt — aVz> + bys, ... is perpendicular to y, and the theorem is proved. From this theorem, and the last, we infer that, when an impulsive force acting on a rigid body produces an instantaneous rotation, the direction of the force, and the axis of the rotation, are parallel to the principal axes of a section of the momental ellipsoid. 241. Principal Axis. If 7) be a principal axis of a rigid body, it is required to prove that ^Pi3Vi = 0, reference being made to the absolute principal screws of inertia. For in this case a force along a line f) intersecting y, compounded with a couple in a plane perpendicular to 77, must constitute an impulsive wrench to which y corresponds as an instantaneous screw, whence we deduce (§ 120), h and k being the same for each coordinate, n h dR . ^dy^^’ h dR Ps dys + kp6y6.