96 THE THEORY OF SCREWS. [104, 



evident that T must form with each one of these screws H lt ... H n ^ a pair 

 of conjugate screws of the potential ( 100). It follows that the impulsive 

 screw, corresponding to T as the instantaneous screw, must be reciprocal to 

 7/i , ... ff n -i , and also that a twist about T must evoke a wrench on a screw 

 reciprocal to H lt ... H n ^. As in general only one screw of the system can 

 be reciprocal to H l ,...H n ^, it follows that the impulsive screw, which 

 corresponds to T as an instantaneous screw, must also be the screw on which a 

 wrench is evoked by a twist about T. Hence, T must be a harmonic screw, 

 and as there are only n harmonic screws, it is plain that T must coincide 

 with H n , and that therefore H n is a conjugate screw of inertia, as well as a 

 conjugate screw of the potential, to each one of the remaining n 1 harmonic 

 screws. Similar reasoning will, of course, apply to each of the harmonic 

 screws taken in succession. 



105. Equations of Motion. 



We now consider the kinetical problem, which may be thus stated. A 

 free or constrained rigid body, which is acted upon by a system of forces, is 

 displaced by an initial twist of small amplitude, from a position of equi 

 librium. The body also receives an initial twisting motion, with a small 

 twist velocity, and is then abandoned to the influence of the forces. It is 

 required to ascertain the nature of its subsequent movements. 



Let T represent the kinetic energy of the body, in the position of which 

 the co-ordinates, referred to the principal screws of inertia, are #/, . . . n . 

 Then we have ( 97) : 



while the potential energy which, as before, we denote by V, is an homo 

 geneous function of the second order of the quantities #/, . . . # . 



By the use of Lagrange s method of generalized co-ordinates we are 

 enabled to write down at once the n equations of motion in the form : 



Substituting for T we have : 



with (n 1) similar equations. Finally, introducing the expression for V 

 ( 100), we obtain n linear differential equations of the second order. 



The equations which we require can be otherwise demonstrated as follows. 



