166 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



distributing 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 

 distribution. Such a layer is called an equipotential layer. 

 (Definition A superficial layer which coincides with one of its 

 own equipotential surfaces.) Reversing the sign of this density 

 will give us a layer which will, outside, neutralize the effect of the 

 bodies within. 



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), 



1 ff I ir d r 1 9M j' 1 fffAP, 

 (12) K = --T V F- --- ^ JdS --:-- dr, 

 4<7rJJ s \ dn e r dn e / 4>7rJJJ r 



(P inside S). 



Note that both formulae (5) and (12) are identical if the 

 normal is drawn into the space in which P lies. 



Hence within a closed surface a holomorphic function is 

 determined 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 func- 

 tion of a surface distribution. 



Now if the surface S is equipotential, the function V cannot 

 be harmonic everywhere within unless it is constant. For since 

 two equipotential surfaces cannot cut each other, the potential 

 being a one-valued function, successive equipotential surfaces 

 must surround each other, and the innermost one, which is reduced 

 to a point, will be a point of maximum or minimum. But we 

 have seen ( 34) that this is impossible for a harmonic function. 

 Accordingly a function constant on a closed surface and harmonic 

 within must be a constant. 



If however there be matter within and without S, the volume 

 integral, as before, denotes the potential due to the matter in the 



