:8o.] PRINCIPLE OF KINETIC ENERGY. 43 



78. The dynamical meaning of the existence of a force-function 

 U lies mainly in the fact that, if a force-function exists, the work 

 done by the forces as the particle passes from its initial to its 

 final position depends only on these positions, and not on the 

 intervening path. This is at once apparent from equation (5), 

 in which U (7 represents this work. 



It follows that the work of conservative forces is zero if the 

 particle returns finally to its original position, that is, if it 

 describes a closed path, provided that the force-function U is 

 .single-valued, an assumption which will here always be made. 



79. In the case of central forces inversely proportional to 

 the square of the distance, for which a force-function can always 

 be shown to exist (see Part II., Arts. 278-281), the force- 

 function is usually called the potential. The negative of the 

 force-function, say 



v=-u, 



is called the potential energy. If this quantity be introduced, 

 and the kinetic energy be denoted by T (as in Art. 57), the 

 equation (5) assumes the form 



T+ v= r + r a> (8) 



which expresses the principle of the conservation of energy for a 



particle : the total energy, i.e. the sum of the kinetic and potential 

 energies, remains constant throughout the motion whenever there 

 exists a force-function. In other words, whatever is gained in 

 kinetic is lost in potential energy, and vice versa. 



80. The name force-function is due to Sir William Rowan Hamilton. 

 Some authors use it for V U, and not for U. With regard to the 

 term potential, the usage is still less settled. Some writers use it for U, 

 others for U, nor is its use always restricted to Newtonian forces. 

 Green was the first to give the name potential function to the function 

 U\ Gauss brought the expression potential into common use. Clausius 

 uses " potential function " for what is called above " potential," reserv- 

 ing the latter name for the potential of a system on another system, or 

 on itself. He also uses the term ergal for what is called above " poten- 

 tial energy." Several writers have followed him in this terminology. 



