98 THE THEORY OF SCREWS. [105, 



To integrate the equations we assume 



tf^/n, ...#,/=/ n ; 



where f lt ...f n are certain constants, which will be determined, and where H 

 is an unknown function of the time : introducing also the value of V, given 

 in 100, we find for the equations of motion : 



- MI*/I fj + (4,1/1 + 4 M / + + 4i/) n = o, 



&c. 



c? 2 n 



- Mu n % ^ + (A m f, + An.fi + ...+ Annfn) 11 = 0. 



If the quantity s, and the ratios of the n quantities f 1} .../, be deter 

 mined by the n equations : 



(A u + Mu*s-) +f*A lz + ... +f n A m = 0, 

 &c., &c. 



+f,A nn _ +... +f n (A nn + Mu^ff) = J 



then the n equations of motion will reduce to the single equation : 



By eliminating /j, ... f n from the n equations, we obtain precisely the 

 same equation for s 2 as that which arose ( 104) in the determination of the 

 n harmonic screws. The values of fj, ... f n , which correspond to any value 

 of s 2 , are therefore proportional to the co-ordinates of a harmonic screw. 



The equation for Q gives : 



H = H sin (st + c). 



Let H l} ... H n , Cj , ... c n be 2n arbitrary constants. Let f pq denote the 

 value of/ 9 , when the root s p z has been substituted in the linear equations. 

 Then by the known theory of linear differential equations*, 



ff n sin (s n t + c n ), 



0n =fmffi Si&quot; (*i* + C,) + . +fnnH n Sin (s n t + C M ). 



In proof of this solution it is sufficient to observe, that the values of 

 #/,... n 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. 



