279, 280] DEVELOPMENTS OF THE DYNAMICAL THEORY. 297 



which is the required expression for the kinetic energy. It is remarkable 

 that the co-ordinates of the rigid body are introduced by the medium of the 

 impulsive screw alone. 



280. Expression for the Twist Velocity. 



If an impulsive wrench of unit intensity on a screw 77 be applied to a 

 quiescent rigid body of unit mass which in consequence commences to twist 

 about an instantaneous screw a, it is required to find the initial twist 

 velocity a. 



The impulsive wrench may be replaced by component impulsive forces 

 T]I, ,.. rj 6 on the six principal screws of inertia and component impulsive 

 couples with moments 0%, atj^, br} 3 , 6774, crj 5 , cij s about those screws 

 of inertia. 



The force ^ is expressed by the velocity it produces in the unit mass 

 parallel to the direction of 77. The component twist velocity of a is di 

 about the first principal screw, and accordingly the velocity of translation 

 parallel to that screw is adaj. Hence we have 



but ! = ~-. T?I, 



cos (a?;) 



whence we obtain* 



cos (OCT?) 



f* . . _ v / 



P 



It will be noted that in this expression the co-ordinates of the rigid body 

 are introduced through the medium of the impulsive screw alone. 



A special case arises when the impulsive wrench is a couple, in which 

 case of course p^= oo . As the effect of an impulsive couple is to produce 

 a pure rotation only, we must under these circumstances have p a = 0. 

 Poinsot s well-known construction exhibits the axis of the initial rotation 

 as the diameter of the mornental ellipsoid conjugate to the plane of the 

 impulsive couple. As three conjugate diameters of an ellipsoid could not 

 lie in the same plane, it follows that in the case of p^ = oo we can never 

 have a and 77 at right angles. As p a is zero while cos (777) is not zero, we 

 must have d infinite. 



This might have been inferred from the fact that as the intensity of the 

 impulsive wrench was not zero while the pitch of the screw on which it lay 

 was infinite, the moment of the impulsive couple was infinite and conse 

 quently the initial twist velocity must be infinite. 



* Trans. Roy. Irish Acad., Vol. xxxi. p. 100 (1896). 



