COULOMB^ THEOREM. 25 



from which it follows that the force at each point of the tube is inversely 

 as the section. 



36. COULOMB'S THEOREM. J^et us consider any elementary 

 surface </S, on a conductor in equilibrium, and let us draw the 

 corresponding tube on the outside until it meets an infinitely 

 near equipotential surface S t . Let us prolong the tube inside the 

 conductor in any manner, terminate it by any given surface S 2 , and let 

 us apply the theorem to the volume bounded by the surfaces Si and 

 S 2 , and the lateral surface of the tube. The force is zero on the 

 whole surface S 2 which forms part of the conductor, and the 

 perpendicular component is zero on the lateral surface of the tube ; 

 there would therefore only be flow of force for the exterior surface 

 </S. If we denote by a-, the surface density of the electricity on the 

 element */S, the total mass comprised in the volume in question is 

 ov/S. We get thus 



The two surfaces being infinitely near, and parallel, we have 

 ^S 1 = ^/S, and 



F = 47TCT. 



Thus, the electrical force at any point infinitely near a conductor 

 in equilibrium, whatever be the masses in action, is equal to the electrical 

 density close to this point multiplied by 471-. 



Since, moreover 



it follows that the density at the surface of a conductor may be 

 expressed as a function of the external force, or of the potential, by 

 the ratio, 



-__ _L^X 



4?r 477 dn 



37. CORRESPONDING ELEMENTS. Let us lastly consider an 

 infinitely narrow tube of force placed between the surfaces of two 

 conductors, at which it terminates perpendicularly. 



The two surface elements ^S and */S', cut by the tube, are called 

 corresponding elements. 



Let us prolong the tube on both sides in the interior of the 

 conductor, where we will suppose it terminated by any two surfaces. 

 The flow of force is zero on the whole surface of the volume thus 

 determined ; for the force is tangential along the sides, and is zero at 



