THE PRINCIPAL SCREWS OF INERTIA. 53 



twisting motion on any n screws of the complex defining 

 the freedom. If the screws of reference be a set of con- 

 jugate screws of inertia, the expression for the kinetic 

 energy of the body consists of n square terms. This will 

 now be proved. 



If a free or constrained rigid body be at rest in a po- 

 sition A, and if the 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 ju, and 

 that if the body had been at rest in the position A, it 

 would have commenced to twist about a screw /3, with a 

 kinetic energy E$. 



We are now to consider how the amount of energy 

 acquired by the second impulse is affected by the circum- 

 stance tjiat the body is then not at rest in A, but is 

 moving through A in consequence of the former im- 

 pulse. The amount will in general differ from E ft , for 

 the movement of the body may cause it to do work 

 against the wrench on ^ 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 ju will be dimi- 

 nished. Under other circumstances the motion through 

 A might be of such a character that the impulsive wrench 

 on [i 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 p. When this is the case 



