82 THE THEORY OF SCREWS. [90- 



other hand, since the co-ordinates of the screw a are a 1; ... 6 , the twist 

 velocity about &amp;lt;u n may also be represented by ay n ( 34), whence 



Iiii 

 r\ 



If we multiply this equation by p n 2 w , add the six equations found by giving 

 n all values from 1 to 6, and remember that a and X are reciprocal, we 

 have ( 39) 



ftua z = M -n&quot;^a\ 



whence a is determined. 



This expression shows that the twist velocity produced by an impulsive 

 wrench on a given rigid body constrained to twist about a given screw, varies 

 directly as the virtual coefficient and the intensity of the impulsive wrench, 

 and inversely as the square of u a . (See Appendix, Note 3.) 



91. The Kinetic Energy acquired by an Impulsive Wrench can 



be easily found by 89 ; for, from the last equation, 



1///0 

 fj ^ 



Mr/I ll 2 - __ - _ 7TT - 



(Ua ~ M ?C &quot;&quot; 



hence the kinetic energy produced by the action of an impulsive wrench on 

 a body constrained to twist about a given screw varies directly as the product 

 of the square of the virtual coefficient of the two screws and the square of 

 the intensity of the impulsive wrench, and inversely as the square of u a . 



92. Formula for a free body. 



We shall now express the kinetic energy communicated by the impulsive 

 wrench on TJ to the body when perfectly free. The component on eo n of 

 intensity r!&quot;t] n imparts a kinetic energy equal to 



///o 



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 15 square terms, of 

 which (&amp;gt;!! 770 j9 2 02?7i) 2 is a specimen. The entire expression being therefore 

 essentially positive shows that a given impulsive wrench imparts greater 

 energy to a quiescent body when free than to the same quiescent body when 

 constrained to twist about a certain screw. 



