170 



STATICS. 



[278. 



Fig. 85. 



278. The attraction at the point P exerted by a single particle 

 Q of mass m (Fig. 85) is =m/r z if the units be so selected as 



to make the constant K= i (Art. 

 262 and Art. 263, Ex. 2). (This 

 assumption is generally made 

 in theoretical investigations, as 

 there is nothing to be gained by 

 carrying the constant factor K 

 through all the formulae. The 

 factor K can always be re-intro- 

 duced when numerical results 

 are required.) 



Let the particle P be displaced through an infinitesimal dis- 

 tance PP' = ds in any direction, and let <f> be the angle between 

 QP = r and PP'. The element of work done by the force m/fl 

 in this displacement is 



jji/ m ij m dr , d(ni 



aW=- cos$ds= ds = ( 



1T r*> ds ds\r 



The quantity m/r occurring in the last expression is called the 

 potential of the mass m at the point P] it is usually denoted 

 by V. 



279. If the particle continue to move along some curve from 

 its initial position P to some final position P v the total work 

 done by the attraction of Q is evidently 



TTT J T 7" TT 



where F=7/2/ris the potential at P, and V^ = m/r^ is the poten- 

 tial at P v Hence, the difference of the potentials at any two 

 points is equal to the work done by the attraction, whatever may 

 have been the path along which the displacement has taken 

 place. 



As the potential V=m/r becomes zero when r=oo, it appears N 

 that the potential V 1 at any point P x is the work that would be 



