222 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



y + 8y, z + $z. The amount of matter in the fixed infinitesimal 

 parallelepiped dxdydz, dm = pdxdydz is thereby changed, and 

 the work necessary is the same as that required to bring the mass 

 Sdm from infinity to the point x, y, z y where the potential is V, 

 namely, SW = V8dm. We have found in 38, 



Consequently the whole increase of energy is 



(it) 



JJJ ( dx 



Integrating by parts 



- 



dxdydz = -VpSxdydz + p&v dxdydz, 







the integral being over all space, and the surface integrals vanishing 

 at infinity. 



But since p -r- A V, 



47T 



this becomes 



. 



a* "9* ty ty *dz dz a 



so that 



For a third deduction, since in moving a mass dm a distance 

 whose components are Sx, By, $z the energy lost is equal to the 

 work done by the system 



- S JF= dm {XSx + YBy + ZSz] 



, ttV K 9F, 3V 

 (14) = - dm ] ^ 8x + d - oy + r 



