128, 129] EQUIPOTENTIAL LAYERS. 373 



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

 123, 68), equal to M, the mass within 8. 



Accordingly we may enunciate the theorem, due to Chasles and 

 Gauss 1 ): 



We may produce outside any equipotential surface of a distribu- 

 tion M the same effect as the distribution itself produces, by dis- 



tributing over that surface a layer of surface -density equal to - 



times the outward force at every point of the surface. The mass of 

 the whole layer will be precisely that of the original internal dis- 

 tribution. Such a layer is called an equipotential layer. (Definition 

 A superficial layer which coincides with one of its own equi- 

 potential surfaces.) Reversing the sign of this density will give us 

 a layer which will, outside, neutralize the effect of the bodies within. 



The above theorem has an important application in determining 

 the attraction of the earth at outside points. Equation 10) shows that 

 the potential and therefore the attraction is determined at all outside 

 points if F, which is connected with g as in 123, is known at all 

 points of an equipotential surface. It will be shown later that the 

 surface of the sea is an equipotential surface. Consequently if the 

 value of g is known from pendulum observations at a sufficient 

 number of stations distributed over the surface of the earth the 

 attraction at all points outside the earth can be calculated. 



Let us now suppose the point P is within S. In this case, we 

 apply Green's theorem to the space within S, and we do not have 

 the integrals over the infinite sphere. The normal in the above 

 formulae is now drawn inward from S, or if we still wish to use the 

 outward normal, we change the sign of the surface integral in 5), 



12) v p = - r _ I T A8 _ _L 



r 



(P inside 8). 



Note that both formulae 5) are 12) are identical if the normal 

 is drawn into the space in which P lies. 



Hence within a closed surface a holomorphic function is deter- 

 mined at every point solely by its values and those of its normally 

 resolved parameter at all points of the surface, and by the values of 

 its second parameter at all points in the space within the surface. 



A harmonic function may be represented by a potential function 

 of a surface distribution. 



1) Chasles, Sur I' attraction d'une couche ellipsoidale infiniment mince, Journ. 

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



