152 THEORY OF NEWTONIAN FORCES. [FT. I. CH. IV. 



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

 to 



(3) 



The surface S may be drawn inside the attracting mass, 

 providing that we take for the potential only that due to matter 

 in the space r within S. 



Accordingly for r we may take any part whatever of the 

 attracting mass, and 



(4) JjJ(AF+ 



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

 ever, we must have everywhere (by 23) 



(5) AF+4wy> = 0. 



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

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

 4-7T times the density at that point. Outside the attracting 

 bodies, where p = 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 a?, y, z construct a small rectangular 

 parallelepiped whose faces have the coordinates 



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

 normal to the X-axis whose x coordinate is x let the mean value of 



the force be -x = P x - 



ox 



The area of the face is 97 f, so that this face contributes to the 

 integral 1 1 P cos (Pri) dS the amount ^ 17 f. 



At the opposite face, since is continuous, we have for its 



ox 



value 



dV .9 /9F\ ,_. , , . ,. 



- h cs-'i ^ + terms of higher order in f, 



