238 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



and the potential V experiences a discontinuity of 4?r (fa fa) 

 in passing through the shell from the positive to the negative 

 side. 



The discontinuity may be also explained 

 by considering the solid angles subtended 

 at points 1 and 2 approaching a point on 

 the surface from opposite sides. If the 

 solid angles have different signs on opposite 

 sides, as the points come together the sum 

 of the absolute values of the two angles 

 approaches 4-Tr, so that at the surface 



If the thickness of the shell is e, the polarization is (fa fa)/6, 

 and the moment of the equal and opposite charges on the element 

 of surface dS on the opposite sides of the shell is, since the volume 

 of the element is edS, equal to 



(fa fa) dS. 



Thus the surface density times the thickness, or the moment 

 of polarization per unit of surface of a simple polarized shell, is 

 constant. The value of the constant <!> = fa fa is called the 

 strength of the shell, and it is this strength that is multiplied by 

 the solid angle in the expression for the potential.* Suppose now 

 that the intensity of polarization increases without limit, so that 

 the strength of the shell fa fa is finite, instead of infinitesimal. 

 Then the difference of potential on the two sides of the shell is 

 finite, or the potential is discontinuous in crossing the shell, by 

 the amount 



The derivative, dV/dn, is however continuous. We may prove 

 the converse of this proposition. If a function satisfies Laplace's 

 equation, vanishes at infinity, and is continuous everywhere ex- 

 cept at a certain surface, its first derivatives being everywhere 

 continuous, the function represents the potential of a double 

 distribution on the surface of discontinuity. If the function were 

 uniform and continuous, it must, by Dirichlet's principle, vanish 

 everywhere. The demonstration will be given in 210. 



* Gauss. " Allgemeine Theorie des Erdmagnetismus," 38, 1839. Werke, Bd. 

 v., p. 119. 



