80 THE THEORY OF SCREWS. [88- 



body receive an impulsive wrench, the body will commence to twist about a 

 screw a with a kinetic energy E a . Let us now suppose that a second 

 impulsive wrench acts upon the body on a screw /*, and that if the body had 

 been at rest in the position L, it would have commenced to twist about a 

 screw /?, with a kinetic energy E$. 



We are to consider how the amount of energy acquired by the second 

 impulse is affected by the circumstance that the body is then not at rest in 

 L, but is moving through L in consequence of the former impulse. The 

 amount will in general differ from Ep, for the movement of the body may 

 cause it to do work against the wrench on JJL during the short time that it 

 acts, so that not only will the body thus expend some of the kinetic energy 

 which it previously possessed, but the efficiency of the impulsive wrench on 

 /A will be diminished. Under other circumstances the motion through A 

 might be of such a character that the impulsive wrench on p acting for a 

 given time would impart to the body a larger amount of kinetic energy than 

 if the body were at rest. Between these two cases must lie the intermediate 

 one in which the kinetic energy imparted is precisely the same as if the body 

 had been at rest. It is obvious that this will happen if each point of the 

 body at which the forces of the impulsive wrench are applied be moving in a 

 direction perpendicular to the corresponding force, or more generally if the 

 screw a. about which the body is twisting be reciprocal to /A. When this is 

 the case a and ft must be conjugate screws of inertia ( 81), 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 a , and if the kinetic energy of the same body twisting 

 about a screw /3 with a certain twist velocity be Ep, then when the body has 

 a motion compounded of the two twisting movements, its kinetic energy will 

 amount to E a + Ep provided that a and /3 are conjugate screws of inertia. 



Since this result may be extended to any number of conjugate screws of 

 inertia, and since the terms E a , &c., are essentially positive, the required 

 theorem has been proved. 



89. Expression for Kinetic Energy. 



If a rigid body have a twisting motion about a screw a, with a twist 

 velocity a, what is the expression of its kinetic energy in terms of the 

 co-ordinates of a ? 



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 distance. If, therefore, 



