286] DEVELOPMENTS OF THE DYNAMICAL THEORY. 305 



Multiplying similarly by p^P^, ...p n /3 n , and adding, 



Eliminating H, we find 



UaaVTpr, - Uafi Srar, = 0. 



We may also prove this formula by physical consideration. Let a, /3 be 

 the two screws which correspond, as instantaneous screws, to 77 and , as 

 impulsive screws. 



Let us take on the cylindroid a, @, a screw 9, which is conjugate to a 

 with respect to inertia ( 81). Then, by known principles, the screw 6 so 

 defined must be reciprocal to 77. 



Hence Prf& + ... +p n r) n n = Q. 



As, however, a and 6 are conjugate, we have 



also, since 6 is co-cylindroidal with a and , there must be relations of the 

 kind 



Substituting these in the two previous equations, we get 



= ; 



= ; 

 whence, as before, 



UaatVpT, U a ptff ayl = 0. 







286. Twist Velocity acquired by an Impulse. 



From the fact that the twist velocity a. acquired by a free body in 

 consequence of an impulsive wrench of unit intensity on a screw 77 is 

 expressed ( 280) by the equation 



cos (otr?) 

 a = - ^- L 



Pa 



we see that the second of the two formulae of 281 may be expressed thus : 



The proof thus given of this expression has assumed that the body is quite 

 free. 



It is however a remarkable fact that this formula holds good whatever 

 be the constraints to which the body is submitted. If the body receive the 

 unit impulsive wrench on a screw 77, the body will commence to twist about 



. 20 



