83, 84] NEWTONIAN POTENTIAL FUNCTION. 165 



represents, since everywhere 



J_ 

 ~4^ 



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

 outside S. 



84. 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= V s = const., and the constant may be taken out from the first 

 integral, 



r 



47rJJ r 



Now by Gauss's theorem 1 1 - -^ d$ = ; accordingly, 



r n r 



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

 of surface-density 



4?r dn 4?r 

 The whole mass of the equivalent surface distribution is 



*** * 8 



a 



which, being the flux of force outward from S, is by Gauss's 

 theorem, 77 (i), equal to M, the mass within 



Accordingly we may enunciate the theorem, due to Chasles and 

 Gauss* : 



We may produce outside any equipotential surface of a dis- 

 tribution M the same effect as the distribution itself produces, by 



* Chasles, " Sur 1' attraction d'une couche ellipsoidale infiniment mince." Journ. 

 EC. Polytec., Cahier 25, p. 266, 1837 ; Gauss, Allgemeine Lehrsatze, 36. 



