A Treatise on the Theory of Screws

96 THE THEORY OF SCREWS. [104, evident that T must form with each one of these screws H., ... Hn^ a pair of conjugate screws of the potential (§ 100). It follows that the impulsive screw, corresponding to T as the instantaneous screw, must be reciprocal to H.,... and also that a twist about T must evoke a wrench on a screw reciprocal to H., ... Hn_.. As in general only one screw of the system can be reciprocal to it follows that the impulsive screw, which corresponds to Tas an instantaneous screw, must also be the screw on which a wrench is evoked by a twist about 1'. Hcucc, T must bo a harmonic screw, and as there are only n harmonic screws, it is plain that T must coincide with Hn, and that therefore Hn is a conjugate screw of inertia, as well as a conjugate screw of the potential, to each one of the remaining n - 1 harmonic screws. Similar reasoning will, of course, apply to each of the harmonic screws taken in succession. 105. Equations of Motion. We now consider the kinetical problem, which may be thus stated. A free or constrained rigid body, which is acted upon by a system of forces, is displaced by an initial twist of small amplitude, from a position of equi- librium. The body also receives an initial twisting motion, with a small twist velocity, and is then abandoned to the influence of the forces. It is required to ascertain the nature of its subsequent movements. Let T represent the kinetic energy of the body, in the position of which the co-ordinates, referred to the principal screws of inertia, are 0/,... Of. Then we have (§ 97):— _ \ dt J \dt) while the potential energy which, as before, we denote by V, is an homo- geneous function of the second order of the quantities 0., ... 0n. By the use of Lagrange’s method of generalized co-ordinates we are enabled to write down at once the n equations of motion in the form :— Substituting for T we have:— d? d0. ’ with (n —1) similar equations. Finally, introducing the expression for K (§ 100), we obtain n linear differential equations of the second order. The equations which we require can be otherwise demonstrated as follows.