110 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II 



represented by the partial derivatives of a single function of the 

 coordinates 



7 



? '~* 



In this case the expression 



2 r {X r dx r + Y r dy r + Z r dz r ] 



is the exact differential of the function U, and the integral 



[ tl Z(X r dx r + Y r dy r + Z r dz r ) = U tl - U to . 

 J t 



The equation of energy then is 



The function U is called the force-function, and its negative 

 W = U is called the Potential Energy of the system. Inserting 

 W in (25) we have 



(26) T tl +W tl =T,.+ W ta . 



The sum of the kinetic and potential energies of a system 

 possessing a force-function is the same at all instants of time. 

 This' is the principle of Conservation of Energy. 



Systems for which the conditions (24) are satisfied are accord- 

 ingly called conservative systems. 



The potential energy, being defined by its derivatives, contains 

 an arbitrary constant. Conservative systems possess the property, 

 since W depends only on the coordinates, and T + W is constant, 

 that T, the kinetic energy, depends only on the coordinates, or if 

 in the course of the motion all the points of the system pass 

 simultaneously through positions that they have before occupied, 

 the kinetic energy will be the same as at the previous instant, 

 irrespective of the directions in which the points may be moving. 

 For instance, a particle thrown vertically upwards, or a pendulum 

 swinging, have the same velocity when passing a given point 

 whether rising or falling. 



The principle of virtual work, 56, may evidently be expressed 

 by saying that for equilibrium the potential energy of the system 



