THE PRINCIPAL SCREWS OF INERTIA. 55 



where r denotes the perpendicular from the element dM 

 on w m . Remembering that p m is the radius of gyration 

 this expression also reduces to J Md^p^m a m 2 , and hence 

 the total kinetic energy of the twisting motion about w m 

 is M&pJaJ. 



We see, therefore (58), that the kinetic energy due 

 to the twisting motion about a is 



Ma z (/xW + &c. + /.W). 



The quantity inside the bracket is the square of a 

 certain linear magnitude which is determined by the dis- 

 tribution of the material of the body with respect to the 

 screw a. It will facilitate the kinetic applications of the 

 present theory to employ the symbol u a to denote this quan- 

 tity. It is then to be understood that the kinetic energy 

 of a body of mass M, animated by a twisting motion 

 about the screw a with a twist velocity a is represented by 



60. Twist Velocity acquired by an Impulse. A body of 

 mass My which is only free to twist about a screw o, is 

 acted upon for a short time e by a wrench of intensity tj" 

 on a screw r\. It is required to find the twist velocity a 

 which is acquired. 



Let the initial reaction of the constraints consist of a 

 wrench of intensity X" on a screw X. Then the body 

 moves as if it were free, but had been acted upon by a 

 wrench of which the component on w m had the intensity 

 "n"i\m + X^Xm. This component would generate a velocity 



of translation parallel to w m and equal to -^>(j/ / 7 + X // A m ). 



The twist velocity about w m produced by this component 

 is found by dividing the velocity of translation by p m . 

 On the other hand, since the co-ordinates of the screw 



