CHAPTER XII. 



PLANE REPRESENTATION OF DYNAMICAL PROBLEMS CONCERNING A BODY 

 WITH TWO DEGREES OF FREEDOM*. 



133. The Kinetic Energy. 



If a rigid body of mass M twist about a screw 0, with the twist velocity 

 0, then the kinetic energy of the body may be written in the form 



Mufffi, 

 where u e is a linear magnitude appropriate to the screw 6 ( 89). 



The function uf is the arithmetic mean between the square of the radius 

 of gyration and the square of the pitch, for the kinetic energy of the body 

 when twisting about 9 is the sum of two parts : one, the kinetic energy of 

 the rotation ; the other, of the translation. The energy of the rotation 

 is simply 



this being in accordance with the definition of the radius of gyration p e . 

 The kinetic energy due to the translation is, of course, 



whence the total kinetic energy is 

 and therefore 



134. Body with two Degrees of Freedom. 



The movements are under these circumstances restricted to twists about 

 the screws of a cylindroid, and we shall now examine the law of distribution 



* Royal Irish Academy, Cunningham Memoirs, No. 4, p. 19 (1886) ; see also Proceedings of the 

 Royal Irish Academy, 2nd Series, Vol. iv. p. 29 (1883). 



