A Treatise on the Theory of Screws

396 THE THEORY OF SCREWS. [357- Let Øi,... 0n represent the co-ordinates of the impulsive screw-chain, and let «i,... an be the co-ordinates of the corresponding instantaneous screw- chain, reference being made to the screw-chains of reference just found. Lagrange’s equations have the form d (dT\ dT p dt \ddj da^~ 11 where T is the kinetic energy, and where -PjSaj denotes the work done against the forces by a twist of amplitude 8^. If 0"' denote the intensity of the impulsive wrench, then its component on the first screw of reference is 0"’0lt and the work done is 2p10'" while, since the chains are co-reciprocal, the work done by öal against the components of 0"' on the other chains of reference is zero, we therefore have A = 2p10"'01. We have also L = M (?Zj2åi2 + ... + Itn2««3), when ult ...un are certain constants. We have, therefore, from Lagrange’s equation, whence, integrating during the small time t, during which the impulsive force acts, Mu^aax = - 0xpj0"'dt, in which å is the actual twist velocity about the screw-chain, so that d1 = da1, each being merely the expression for the component of that twist velocity about the screw-chain. We hence obtain 01(... 0n, proportional respectively to Wi2«i Un«n Pi Pn U-l If we make ^- = (11), &c., we have the previous result, 6>1 = (ii) a15 0n = (raw) an. 358. Conjugate screw-chains of Inertia. From the results just obtained, which relate of course only to the chains of reference, we can deduce a very remarkable property connecting instantaneous chains, and impulsive chains in general. Let a and ß be two instantaneous chains, and let 0 and <p be the two corresponding impulsive chains, then when a