139-141] General Theory 141 



so that the equations of equilibrium have the common integral 



</>(= + C ......................... .....(396), 



where C is a constant. At any point inside the mass, 



V 2 F T =0, 

 so that, from equation (390), 



V 2 H = 2ft) 3 - 4777) ........................... (397). 



Thus on operating on equation (396) with V 2 , we obtain 



= 2ft> 2 ........................ (398), 



the differential equation which must be satisfied by p for equilibrium to be 

 possible. 



From equation (396) we can obtain p in the form 



P 



and equation (397) now becomes 



= 2ft) 2 ........................ (399), 



the differential equation which must be satisfied by O for equilibrium to be 

 possible. 



141. Let P be any point inside the mass, and let R denote the distance 

 from P to a variable point x, y, z inside the mass ; let dS' be an element of 

 surface of a small sphere surrounding P, and let dS be an element of surface 

 of the boundary. Then for the value of V M at the point P we have 



so that 



The integral on the right is of course the potential of a Green's equivalent 

 stratum; it is a known theorem that the potential of this together with 

 that of external masses has a constant value inside the surface. 



