54 THE PRINCIPAL SCREWS OF INERTIA. 



a and /3 must be conjugate screws of inertia ( 54), and 

 hence we infer the following theorem : 



If the kinetic energy of a body twisting about a screw 

 a with a certain twist velocity be E ay and if the kinetic 

 energy of the same body twisting about a screw j3 with a 

 certain twist velocity be E ft , then when the body has a 

 motion compounded of the two twisting movements, its 

 kinetic energy will amount to E a + E$ provided that a and 

 j3 are conjugate screws of inertia. 



Since this result may be extended to any number of 

 conjugate screws of inertia, and since the terms E^ &c., 

 are essentially positive, the required theorem has been 

 proved. 



59. Expression for Kinetic Energy. If a rigid body 

 have a twisting motion about a screw a, with a twist 

 velocity d', what is the expression of its kinetic energy 

 in terms of the co-ordinates of a r 



We adopt as the unit of force that force which acting 

 upon the unit of mass for the unit of time will give the 

 body a velocity which would carry it over the unit of 

 distance in the unit of time. The unit of energy is the 

 work done by the unit force in moving over the unit dis- 

 tance. If, therefore, a body of mass w have a movement 

 of translation with a velocity v its kinetic energy ex- 

 pressed in these units is ^wv z . 



The movement is to be decomposed into twisting 

 motions about the screws of reference wi, &c. we, the 

 twist velocity of the component on w m being d'a m . 

 One constituent of the twisting motion about w m con- 

 sists of a velocity of translation equal to ap m a m , and on 

 this account the body has a kinetic energy equal to 

 ^Ma*p m z a m ~. On account of the rotation around the 

 axis with an angular velocity aa m the body has a kinetic 

 energy equal to 



