A Treatise on the Theory of Screws

98 THE THEORY OF SCREWS. [105, To integrate the equations we assume • •• Of =fnil', where...fn are certain constants, which will be determined, and where 11 is an unknown function of the time: introducing also the value of V, given in § 100, we find for the equations of motion : Jan - Mulfl + (-411/1 + -412,/a + • ■ ■ + Aynfn) 12 = 0, czc &c. Jan - Mun*fn + (A./ + Amfy + ... + Annfn) 12 = 0. If the quantity s, and the ratios of the n quantities f, ...fn, be deter- mined by the n equations:— fi (-411 + Mu(-s‘) +fiAVi + ... = 0, &c., &c. fAm +fAnz + ... +fn (-4 mi + Mufs2) = 0; then the n equations of motion will reduce to the single equation: By eliminating f,... fn from the n equations, we obtain precisely the same equation for s2 as that which arose (§ 104) in the determination of the n harmonic screws. The values of f,... fn, which correspond to any value of s2, are therefore proportional to the co-ordinates of a harmonic screw. The equation for Q gives : !2 = H sin (st + c). Let Hi, ... Hn, Cj, ... cn be 2n arbitrary constants. Let fpq denote the value of fq, when the root sp2 has been substituted in the linear equations. Then by the known theory of linear differential equations*, 0/ =fnHi sin (s,f + cj + ... +fmHn sin (snt + c„), 0/ -finHi sin (Syt + Cj) + ... +f,mHn sin (snt + c„). In proof of this solution it sufficient to observe, that the values of 0y,... Of satisfy the given differential equations of motion, while they also contain the requisite number of arbitrary constants. Lagrange’s Method, Routh, Rigid Dynamics, Vol. I., p. 369.