74 HARMONIC SCREWS. 



the two systems of correspondence, it will of course usually 

 happen that the two screws X and TJ are not identical, 

 But a little reflection will enable us to foresee what we 

 shall afterwards prove, viz., that when has been appro- 

 priately chosen, then A and 17 may coincide. For since 

 n - i arbitrary quantities are disposable in the selection 

 of a screw from a screw complex of the n th order ( 43], it 

 follows that for any two screws (for example X and j) to 

 coincide, n - i conditions 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 0, the two corresponding screws 

 X and jj are identical ; and we shall now prove the fol- 

 lowing important theorem : 



If a rigid body be displaced from a position of eqtiili- 

 brium by a twist about a screw 0, and if the evoked wrench 

 tend to make the body commence to twist about the same 

 screw 9, then may be called an harmonic screw, and the 

 number of harmonic screws is the same as the order of the 

 screw complex which defines the freedom of the rigid body. 



We shall adopt as the screws of reference the n prin- 

 cipal screws of inertia. The impulsive screw, which 

 corresponds to as an instantaneous screw, will have 

 for co-ordinates 



where h is a certain constant which is determined by 

 making the co-ordinates satisfy the condition ( 37). If 

 6 be a harmonic screw, then, remembering that the 

 screws of reference are co-reciprocal ( 57), we must have 

 n equations, of which the first is ( 72) : 



