284] DEVELOPMENTS OF THE DYNAMICAL THEORY. 303 



We thus obtain 



1 / / 



and similarly 



whence 7 cos()_ tsr,, 



wiieuut; - ^ -- - -- = - 



COS (0 OT^ y 



or remembering that p a is infinite, 



j-/// 



2L CTg * _ OT i0 



fa &quot; a V,y 



But we had already from (iii), 



whence we deduce that in this case also the formula remains true. We thus 

 obtain the following general theorem. 



If 1), , be three impulsive screws and a, /3, 7 the three corresponding 

 instantaneous screws, then in all cases, no matter how the movements of the body 

 may be limited by constraints, the following formula holds good*: 



It is easily shown that this relation subsists when the correspondence 

 between ?/ and a is of the more general type implied by the equations 



dU 



where U is any homogeneous function of the second degree in the co 

 ordinates. 



284. Case of a Constrained Rigid Body. 



Let 77 and be, as before, a pair of impulsive screws, and let a and ft 

 be the corresponding pair of instantaneous screws. Let p be the screw on 

 which a reaction is contributed by the constraints at the moment when the 

 impulsive wrench is applied on 77. 



The movement of the body twisting about a is therefore the same as if 

 it had been free, and one impulsive wrench had been imparted about 77 

 and another simultaneously about p, so that the following conditions are 

 satisfied : 



* Trans. Eoy. Irish Acad., Vol. xxx. p. 575 (1894). 



