CHAPTER XXL 



DEVELOPMENTS OF THE DYNAMICAL THEORY. 



279. Expression for the Kinetic Energy. 



Let us suppose that a body of mass M is twisting around a screw a with 

 the twist velocity a. It is obvious that the kinetic energy of the body must 

 be the product of Md 2 and some expression which has the dimensions of the 

 square of a linear magnitude. This expression has a particular geometrical 

 significance in the Theory of Screws, and the symbols of the theory afford a 

 representation of the expression in an extremely concise manner. 



Let 77 be the impulsive screw which corresponds to a as an instantaneous 

 screw, the body being supposed to be perfectly unconstrained. 



As usual p a is the pitch of a and (a?/) is the angle between a and 77. 

 From the formulae of 80 we have, where H is a common factor, 



H^ = + aa^; Hv]2 = a 2 ; 



#773 = + ba 3 ; Hr) 4 = - 6a 4 ; 



Hr) 5 = + ca & ; #77,3 = -ca 6 ; 

 whence 



H [(771 + 7? 2 ) (! + 2 ) + (r; 3 + 774) (a 3 + a 4 ) + (775 + 77 6 ) (a fl + a e )] 



= a (! 2 - a/) + b (a 3 2 - a/) + c (a 5 2 - a 6 2 ) =p a 

 and we obtain 



cos 

 The kinetic energy is 



in 



&quot; 



cos (an) 



cos 

 * Trans. Roy. Irish Acad., Vol. xxxi. p. 99 (1896). 



