CONSERVATION OF ENERGY 171 



so that the total work performed by the particle is P ds. Thus 

 the theorem can be stated in the following alternative form : 



During the motion of a particle under any system of forces, the 

 decrease in kinetic energy is equal to the total work done by the 

 particle against external agencies. 



142. If the system of forces acting on the particle is a conserv- 



C Q 

 ative system, the value of / Pds, the total work performed by 



Jp 

 the particle on external agencies, is equal, by 132, to W Q W P . 



Thus equation (37) becomes 



or again W Q + \mv%=W P + ^mv p> (38) 



so that the sum of the potential and kinetic energies is the same 

 at Q as at P, proving the theorem. 



The sum of the potential and kinetic energies is called the total 

 energy of the particle. 



CONSERVATION OF ENERGY 



143. The kinetic energy of a system of bodies is obviously equal 

 to the sum of the kinetic energies of the separate particles. The 

 potential energy of the system, as has been seen, is the sum of 

 the potential energies of its particles. 



Thus the total energy of a system is equal to the sum of the 

 total energies of the separate particles. Since the total energy of 

 each particle remains constant, it follows that the total energy of 

 the system remains constant. 



The fact that the total energy remains constant is spoken of as 

 the Conservation of Energy. An equation expressing that the total 

 energy at one instant is equal to that at any other instant is 

 spoken of as an equation of energy. 



144. As an illustration, let us consider the firing of a stone from a 

 catapult. 



Work is performed in the first place in stretching the elastic of the 

 catapult, and the work is stored as potential energy of the stretched elastic. 

 As soon as the catapult is released, the stone is acted on by the tension of 



