104] HARMONIC SCREWS. 95 



must be fulfilled ; but this is precisely the number of arbitrary elements 

 available in the selection of 6. We can thus conceive that for one or more 

 particular screws 6, the two corresponding screws \ and 17 are identical ; and 

 we shall now prove the following important theorem : 



If a rigid body be displaced from a position of equilibrium by a twist 

 about a screw 6, and if the evoked wrench tend to make the body commence to 

 twist about the same screw 6, then if we call 9 an harmonic screw ( 106), we 

 assert that the number of harmonic screws is generally the same as the order 

 of the screw system which defines the freedom of the rigid body. 



We shall adopt as the screws of reference the n principal screws of inertia. 

 The impulsive screw, which corresponds to 6 as an instantaneous screw, will 

 have for co-ordinates 



, 



Pi Pn 



where A is a certain constant which is determined by making the co-ordinates 

 satisfy the condition ( 35). If 6 be an harmonic screw, then, remembering 

 that the screws of reference are co-reciprocal ( 87), we must have n equations, 

 of which the first is ( 102) : 



1 L U *0&amp;gt;- J ^1 



%/ *2 Pl de&amp;gt;- 



j nm 



Assuming -p = Ms&quot;, where M is the mass of the body, and .9 an unknown 

 quantity, and developing F, we deduce the n equations : 

 6, (A n + Ms*u*} + z A K +...+ n A ln = 0, 



t A m + 0*A m +... + n (A nn + Ms 2 u n 2 ) = 0. 



Eliminating 0^, ... n , we have an equation of the nth degree for s&quot;. The 

 n roots of this equation are all real ( 85), 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 H l , . . . H n _ t , be n 1 of the harmonic screws, to which 

 correspond the impulsive screws S lt ... S n ^. Also suppose T to be that one 

 screw of the given screw system which is reciprocal to S lt ...$_! ( 95), 

 then T must form with each one of the screws H lt ... 7/ n _! a pair of con 

 jugate screws of inertia (81). But, since S lt ... /S M _, are the screws on 

 which wrenches are evoked by twists about //,,... #_! respectively, it is 



