HARMONIC SCREWS. 79 



If the quantity s, and the ratios of the n quantities 

 /i, . . . . / n , be determined by the n equations : 



fi(A u - Afufs*) +/+AU + . . . +SnA nl = o, 

 &c., &c. 



fiAm t/ 2 ^ 2 n + . . . + f n (A an ~ Mu n *S*) = O; 



then the n equations of motion will reduce to the sin- 

 gle equation : 



By eliminating / ..... ,/ n from the n equations, 

 we obtain precisely the same equation for s* as that 

 which arose ( 74) in the determination of the n harmonic 

 screws. The values of/,, ....,/, which correspond to 

 any value of s z , are therefore proportional to the co-ordi- 

 nates of a harmonic screw. 



The equation for Q gives : 



O = ffsm (st + c}. 



Let Hi, . . . H ny d, . . . c n be 2n arbitrary constants. 

 ~Let/ pq denote the value of/ ? , when the root s p * has been 

 substituted in the linear equations. Then by the known 

 theory of linear differential equations,* 



0/ =/ n /7i sin fat + d) - . . . . +/mH n sin (s n t + c n ], 



&c., &c. 

 On =finHi sin (sit + d) + . . . . +/ nn H n sin (s*t + c n ). 



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

 the values of Of, . . . satisfy the given differential 

 equations of motion, while they also contain the requi- 

 site number of arbitrary constants. 



* Lagrange's Mecanique Analytique, vol. i., p. 353- 



