A Treatise on the Theory of Screws

227] FREEDOM OF THE FOURTH ORDER. 241 Hence we deduce that the screw with co-ordinates otl, > • • • #6 > and the screw with co-ordinates proportional to i dU IdU IdU p1 da* ’ p2 daf pe da6 ’ in which U is the expression /»i«!2 + P^2 ... + peaf + (®i2 + a2a • •• + 2«!^ cos (12)...) must be collinear, and this is true for all values of X. We hence see that the co-ordinates of a screw collinear with a must be proportional to X dR X dR 0(1 + 2/>! do^ ’ a'2 + 2p2 da± " where R = ax2 + a22 + ... + 2^^ cos (12) + ... Thus we obtain the results of § 47 in a different manner. 227. Dynamical application of Polar Screws. We have seen (§ 97) that the kinetic energy of a body twisting about a dd' velocity and belonging to a n-system screw 0 with a twist is /JØ'V >( ,, ) («i2^2+...+Wn2M> \ Uju J the screws of reference being the principal screws of inertia. If we make u1201‘2+...+izn2ön2 = 0, then 0 must belong to a quadratic «-system. This system is, of course, imaginary, for the kinetic energy of the body when twisting about any screw which belongs to it is zero*. The polar y of the screw 0, with respect to this quadratic ^-system, has co-ordinates proportional to ^-0lt ... — 0„. Pl Pn Comparing this with § 97, we deduce the following important theorem : A quiescent rigid body is free to twist about all the screws of an enclosing (n + V)-system A. If the body receive an impulsive wrench on a screw y • In a letter to the writer, Professor Klein pointed out many years ago the importance of the above screw system. He was led to it by expressing the condition that the impulsive screw should be reciprocal to the corresponding instantaneous screw. B. 16