22 ON POTENTIAL. 



33. DISTRIBUTION OF ELECTRICITY ON THE SURFACE OF CON- 

 DUCTORS. In any conductor in equilibrium, electricity is present 

 on the surface only. 



For we have seen that the force is zero in a conductor in 

 equilibrium, and that therefore the potential has a constant value 

 for the whole surface of the conductors. All orders of differentials 

 of the potential are zero at each point, and we have then 



AV = 0, or p = 0. 



Hence, in the interior of a conductor in equilibrium, not only 

 is there no electrical force, but there is no electricity at all. The 

 distribution is exclusively on the surface. 



34. GREEN'S FORMULA. The formula relative to the flow of 

 force gave, for a closed surface, the equation 



- . 



I on 



J 



The mass M, inside the surface, is equal to the sum of the 

 masses pdv comprised in the different elements of volume ; taking 

 equation (9) into account, we have then, 



(n) </S= - \47rpdv= AWz>. 



This equation is a particular case of a more general formula 

 due to Green. 



Let U and V be two finite and continuous functions of x, y, 

 and z. Let us also put 



AV = - + - + -, 

 and consider the integral 



J 



IV 



UAW?;= | | | U( ^ + ^ + T r ) dxdydz, 



ox oy oz I 

 / 



extended to the volume enveloped by a closed surface S. This 

 integral consists of three terms of the form 



U dxdyd*=\ \dydz\V~dx. 



