A Treatise on the Theory of Screws

410 THE THEORY OF SCREWS. [372- If, by imposition of a restraining wrench-chain, the mass-chain continues to twist about the same screw-chain 6, the restraining wrench-chain has neutralized the acceleration. It follows that the restraining wrench-chain, regarded as impulsive, must have generated an instantaneous twist velocity on the accelerating screw-chain, equal and opposite to the acceleration that would otherwise have taken place. The co-ordinates of this impulsive wrench-chain are proportional to fdT 1 dT Pi d0f" pn d0n" The corresponding instantaneous screw-chain is these expressions severally by £1 PJL Un ’ and thus we find, as before, for the co-ordinates of the accelerating screw- chain Id?' 1 dT uj d0j ’ Un2 ddn 373. Accelerating Screw-chain and instantaneous Screw-chain. We have, from the expressions already given, M(uj0Ä + ... + uMn) = 0, + ... 0n . (lui (Lu lti But the right-hand side is the emanant which we know to be zero, whence u20f1 + ... + un20n0n == 0. This shows that 0lt... 0n, and ... 0n are on conjugate screw-chains of inertia, and hence we deduce the following theorem:— Whenever a mass-chain is moving without the action of external forces, other than from constraints restricting the freedom, the instantaneous screw- chain and the accelerating screw-chain are conjugate screw-chains of inertia. 374. Permanent Screw-chains. Reverting to the general system of equations (§ 366) we shall now in- vestigate the condition that 0 may be a permanent screw-chain. It is obvious that if Øj, ... 0n are all zero, then dT dT døj’ døn must each be zero. If, conversely, the differential coefficients just written are all zero, then the quantities 0lt ... 0n must each also vanish.