374 VIII. NEWTONIAN POTENTIAL FUNCTION. 



Now if the surface S is equip otential, 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 ( 116) 

 that this is impossible for a harmonic function. Accordingly a func- 

 tion 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 

 space" r (within 8), and the surface integral that due to the matter 

 without. If the surface is equipotential, the surface integral 



&/#& 



**, *J J r d n e 



( *^-dS + ~ 



The first integral is now equal to 4jr, so that 



V s being constant contributes nothing to the derivatives of F, so that 

 the outside bodies may be replaced by a surface layer of density 



14) tf = - = _ .Pcos (Fn e ) = + 



4tt dn g kny - 



The mass of the surface distribution, 

 15) 



n e being the outward normal, is the inward flux of force through S, 

 which is equal to minus the mass of the interim 1 matter, and is not, 

 as in the former case, equal to the mass which it replaces. 



13O. Gauss's Mean Theorem. As an example of equation 6) 

 let us make the surface S a sphere with center at P. Then in the 

 first term of the integral we have 



which is constant and may be taken outside the integral. In the 

 second term J being similarly taken outside the integral, we have 



