76 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



34) U.= - 



1* '2s 



and the symmetrical function U will be 



In 



m 9 m, m.m, m 9 m 



+ -r + -r + -- + -J;- 



35) Wo m, m, m 9 m,, m 



+ -^ + -^ + + -4-^ 



'3n 



| .4 ( .i | | ..-i ^ 



^7Z 1 * 7Z 2 *'w, W 1 



or more briefly, 



understanding that terms in which r = s are to be omitted. 



The factor is introduced because in the above summation 



every term appears twice. But in U each pair of particles is to 

 appear only once. 



If no constant be added to U as defined above, both it and the 

 potential energy 



36) W= 



will vanish when every r rt is infinite , that is when no two particles 

 are within a finite distance of each other. This furnishes a con- 

 venient zero configuration for the potential energy, and is the one 

 generally adopted. We may accordingly define the potential energy 

 of the system in any given configuration as the work that must be 

 done against the mutual repulsions or attractions of the particles in 

 order to bring them from a state of infinite dispersion to the given 

 configuration. In the case of attracting forces like those of gravitation, 

 we shall, with the notation of this section, put y negative 1 ), so that 

 the potential energy of finite systems is negative, or in the terminology 

 of Thomson and Tait, the exhaustion of potential energy, - - W, is 

 positive. 



1) For the reason for the adoption of this convention see 119. 



