A Treatise on the Theory of Screws

368] Neglecting the small quantities 0/ ... &c. we have JuT dPP ~=2Mu1i, ... ^4 = 2Mun\ døl døn 407 THE THEORY OF PERMANENT SCREWS. ---;---- = 0, &C. dØydØ-i Introducing these values we obtain Mui &i — Pi \Vi" + 1 dT\ pi d0i) ’ ,, x / „ 1 dT\ MUnØn — PniVn + in / J • \ pn ^un / These may be regarded as the generalization for any material arrangement whatever of th© well-known Eulerian equations for the rotation of a rigid body around a fixed point. If there are no externa] forces then are all zero, and the equations of movement assume the simple form „X dT MU1-0i - ; ng _dT 1U lln Un — ^0 , . 368. The Restraining Wrench-chain. If a mass-chain be twisting about an instantaneous screw-chain 0, the mass-chain will, in general, presently forsake 0 and gradually adopt one instantaneous screw-chain after another. It is however possible, by the application of a suitable wrench-chain, to compel the mass-chain to continue twisting about the same screw 0 with unchanged twist velocity. We now proceed to the discovery of this restraining wrench-chain when no other external forces act on the mass-chain. As all the accelerations of 0 must vanish, the co-ordinates of the wrench- chain required are obtained by imposing the conditions 01 = O; 02 = O,... 0n = O. We therefore infer from the general equations of §366, that if ?/",... t?,," are the co-ordinates of the restraining wrench-chain we must have „ 1 dT Vi 4----j/i / = Pi d0! "4-1 dT =0 Vn +pnd0n'