66 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



displacements which take place in the actual motion of the system 



in the time dt, 



dx r dy r dz r 



we obtain 



r \dt* ~dt~ ~*~ ~W ~~dt~ + ~dt r ~di. 



dx r dy r dz\ 



Now since 



mr ~d~dT = ^dt\ mr \dT } I' 

 the sum of the first three terms is the derivative of the sum 



and the equation may be written, omitting the factor dt, 



The expression 



dXr\* ( d y r \* / d! *r\ 2 } 1 



) + ( -W) + U) ) = 



the half -sum of the products of the mass of each particle by the 

 square of its velocity, is called the Kinetic Energy of the system. 

 It is one of the most important dynamical quantities. If we denote 



-j fj-j 



it by T, equation 20) has on the left -^- Since 



X r dx r H- Y r dy r + Z r dz r 



is the work done upon the r th particle, the terms under the summa- 

 tion sign on the right denote the total work done by the impressed 

 forces in unit time, or the Activity ^) of the forces. The equation 20) 

 is called the equation of activity, and states that the rate of increase 

 of kinetic energy of the system is equal to the activity of the im- 

 pressed forces. 



1) The word "adio" is used by Newton, in a scholium on the third law, 



where he says, "If the activity of an agent (force) be measured by its amount 



and its velocity conjointly; . . . activity and counteractivity, in all combinations 



of machines, will be equal and opposite." The activity will sometimes be denoted 



dA 



