HARMONIC SCREWS. 75 



Art" 



Assuming = - Ms 1 , where M is the mass of the body, 



and s an unknown quantity, and substituting for Fits 

 value, we deduce the n equations : 



+ Mu + . . . + O n A ln = o, 

 &c., &c. 



. . . + 6 n (A nn ~ Ms*U n *) = O. 



Eliminating Q ly . . . . , O n , we have an equation of the 

 n th degree for s 2 . The n roots of this equation are all 

 real, and each one substituted in the set of n equations 

 will determine, by a system of n linear equations, the 

 ratios of the n co-ordinates of one of the harmonic screws. 



It is a remarkable property of the n harmonic screws 

 that each pair of them are conjugate screws of inertia, 

 and also conjugate screws of the potential. Let HI, ..., 

 H n -\, be n- i of the harmonic screws, to which corres- 

 pond the impulsive screws ,5*1, . . . . , S n . i. Also sup- 

 pose Tto be that one screw of the given screw complex 

 which is reciprocal to Si, . . . . , S n - 1 ( 65), then T must 

 form with each one of the screws HI, . . . . , H n . \ a pair of 

 conjugate screws of inertia ( 54). Also, since Si, .... 9 

 S n ^ are the screws on which wrenches are evoked by 

 twists about H l9 . . . . , H n . \ respectively, it is evident 

 that Tmust form with each one of these screws//!, . . . ., 

 H n . i a pair of conjugate screws of the potential ( 70). 

 It follows that the impulsive screw, corresponding to T 

 as the instantaneous screw, must be reciprocal to Hi, .... 

 H n .i', and also that a twist about Jmust evoke a wrench 

 on a screw reciprocal to H ly .... H n . i. But as we can 

 only have one screw of the screw complex reciprocal to 

 HI, . . . H n . it follows that the impulsive screw, which 

 corresponds to T as an instantaneous screw, must also be 

 the screw on which a wrench is evoked by a twist about 



