THE PRINCIPAL SCREWS OF INERTIA. 57 



whence the total kinetic energy is found by adding these 

 six terms. 



The difference between the kinetic energy acquired 

 when the body is perfectly free, and when the body is 

 constrained to twist about a, is equal to 



The quantity inside the bracket reduces to the sum of 1 5 

 square terms, of which (p\airii - /2z?i) 2 is a specimen. 

 The entire expression being therefore essentially posi- 

 tive shows that a given impulse imparts greater energy 

 to a quiescent body when free than to the same quiescent 

 body when constrained to twist about a certain screw. 



63. Lemma. If a group of instantaneous screws be- 

 long to a complex of the n th order, then the correspond- 

 ing group of impulsive screws also belong to a complex 

 of the n ih order; for, suppose that n + i twisting motions 

 about n + i screws neutralise, then the corresponding 

 n + i impulsive wrenches must equilibrate, but this would 

 not be possible unless all the impulsive screws belonged 

 to a screw complex of the n th order. 



64. Euler's Theorem. If a free or constrained rigid 

 body receives an impulsive wrench, the body will com- 

 mence to move with a larger kinetic energy when it is 

 permitted to select its own instantaneous screw from the 

 screw complex P defining the freedom, than it would 

 have acquired, had it been arbitrarily restricted to any 

 other screw of the complex. 



Let Q be the reciprocal complex of the (6 - n th ) order, 

 and let P' be the screw complex of the n th order, con- 

 sisting of those impulsive screws which, if the body 

 were free, would correspond to the screws of P as instan- 

 taneous screws. 



