372 VIII. NEWTONIAN POTENTIAL FUNCTION. 



we may write 6) 



7) Vp=~ A 



Consequently, we may produce at all points outside of a closed 

 surface S the same field of force as is produced by any distribution 

 of masses lying inside of S, whose potential is F, if we distribute 

 over the surface S a surface distribution of surface -density, 



1 (Fcos(wr) d_V\ 



8) 6 = 



In the general expression, 5) the surface integral represents the 

 potential due to the masses within S, while the volume integral 



-mm 



since everywhere 



is equal to 



that is, the potential due to all the masses in the region T, viz., 

 outside S. 



129. Equipotential Layers. As a still more particular case 

 of 7), if the surface S is taken as one of the equipotential surfaces 

 of the internal distribution, we have all over the surface V= Vs const., 

 and the constant may be taken out from the first integral, 



9) Fp= _ d8 _ 



*J J r ^JJ r $ n 



Now by Gauss's theorem / / C ^f^dS = Q, accordingly, 



so that V P is represented as the potential of a surface distribution 

 of surface -density 



l dV IF ,-v N +1F 

 G = - - -5-- = - - cos (Fn) = ~f --- 

 4:7C on 4:7f y kit y 



The whole mass of the equivalent surface distribution is 



