360 VIII. NEWTONIAN POTENTIAL FUNCTION. 



thin 8 contributes to the potential at 



to the flux through the surface S. Therefore the whole 



situated within 8 contributes to the potential at any point and 



mass 



; when the potential is F>= / / I - > contributes to the flux 



K 



rrr 

 = " **rJJJ ' 



K 



and 



S K 



Now the surface integral is, by the divergence theorem, equal to 



70) 



The surface 8 may be drawn inside the attracting mass, provid- 

 ing that we take for the potential only that due to matter in the 

 space V within S. 



Accordingly for r we may take any part whatever of the attract- 

 ing mass, and 



71) 



As the above theorem applies to any field of integration what- 

 ever, we must have everywhere 



72) z/F+ 4^0 = 0. 



This is Poisson's extension of Laplace's equation, and says that 

 at any point the second differential parameter of F is equal to 

 - 4# times the density at that point. Outside the attracting bodies, 

 where Q = 0, this becomes Laplace's equation. 



In our nomenclature, the concentration of the potential at any 

 point is proportional to the density at that point. 



A more elementary proof of the same theorem may be given 

 as follows. At a point x,y,2 construct a small rectangular parallel- 

 epiped whose faces have the coordinates 



x, x -f g, y, y -f vj, 0, z + 6, 



and find the flux of force through its six faces. At the face normal 

 to the a;- axis whose x coordinate is x let the mean value of the 



force be - d ~ = - P x . 



dx x 



