140-144] INTERNAL FORCES. 129 



The work done in any displacement of the system is therefore 



where m, m' are the masses of two particles, r Q their initial distance, and r l 

 their final distance, and the summation extends to all the pairs of particles. 



For calculating the work function of such a system the most convenient 

 position to choose as the standard position is one in which all the distances, 

 such as r , are indefinitely great ; then the work function is -yS (mm'/r) where 

 r is the distance between the particles m and m'. 



143. Potential Function. The potential function of a gravitating 

 system at a point is the work function for the system and a unit mass at the 

 point, when the standard positions of all the particles of the system are the 

 positions they occupy, and that of the unit mass is at an infinite distance. 



If m is the mass of any particle of the system, and r the distance of this 

 particle from the point, the potential function of the system at the point is 

 y2 (m/r). 



If there is a particle of the system at the point that particle is to be 

 omitted from the sum. 



Let V be the potential at P, and V the potential at P' \ then V Fis 

 the work done by the forces in a displacement in which the particle of unit 

 mass is displaced from P to P', and the particles of the system are not 

 displaced. 



Let F be the resultant force on the particle of unit mass at P, i.e. the 

 intensity of the field at P, and let Jf, T, Z be its resolved parts parallel to the 

 axes of reference ; then, if PP' is infinitesimal, and &p, dy, dz are its projections 

 on the axes of reference, and if V'= V+8V, we must have 



If PP' is parallel to the axis of #, 8 V is the change made in V by changing 

 x into x+toc without altering y or z t and thus we have 



3F 



-A ~ . 



OX 



In the same way F= ~ , and Z=-~-. 



9y ' oz 



This shows that the resolved part in any direction of the force on the 

 particle is the rate of variation of the potential per unit of length in that 

 direction. 



144. Potential energy of gravitating system. Let m^ m 2 ,... be the 

 masses of the particles, r 12 the distance between the positions of m l and m 2 , 

 and similarly for the others, and let the standard position be that in which all 

 the distances, such as r 12 , are indefinitely great. 



L. 9 



