A Treatise on the Theory of Screws

372] THE THEORY OF PERMANENT SCBEWS. 409 T is For, if #/,... 0f be the co-ordinates of the displacement, the change in 0i ,dT -d-^+ - + d0->, which may be written z,, 1 dT , /> / 1 df Pi 01 — + • • • + pn dn ~ ^7) > ‘ p1 d0i r pn d0n but this will be zero if, and only if, the screw-chain 0/, ... 0n' bo reciprocal to the screw-chain 1 dT 1 dT Pl 00, Pn d0n 371. The Accelerating Screw-chain. When the mass-chain has forsaken the instantaneous screw-chain 0, and is twisting about another instantaneous screw-chain </>, there must be a twist velocity about some screw-chain p, which, when compounded with the twist velocity about 0, gives the twist velocity about f When </> and 0 are indefinitely close, then p is the accelerating screw-chain. Taking the n principal screw-chains of inertia as the screws of reference and assuming that external forces are absent, we have ,, .n dT MU1 01 “ døl' ’ 20 __ 1V1 Un Un — ! It is plain that the co-ordinates of the accelerating screw-chain are 0lt... 0H whence we have the following theorem:— If a mass-chain be twisting around a screw-chain 0, and if external forces are absent, the co-ordinates of the corresponding accelerating screw-chain are proportional to 1 dT 1 dT Ui døl ’ ” ’ Un d0f ’ 372. Another Proof. It is known from the theory of screw-chains (§ 357) that if a quiescent mass-chain receive an impulsive wrench-chain with co-ordinates Wi2 uf — pl, ••• -Pn, Pl Pn r the mass-chain commences to twist about the screw-chain, of which the co-ordinates are Pi, ••• Pn-