30 ON POTENTIAL. 



is clearly identical with that arising from a direct consideration of 

 the action of external masses, supposed to be attached to ponderable 

 matter, on the various electrical masses of the conductor. 



42. CONSEQUENCES OF THE DISTRIBUTION OF ELECTRICITY ON 

 THE SURFACE OF CONDUCTORS. The preceding theorem is based on 

 Coulomb's law considered as an experimental fact. We might follow 

 a different course in taking as starting point this other experimental 

 fact, which is more easily verified with strictness, that electricity exists 

 solely on the surface of conductors, and that in the interior there is 

 neither electricity nor electrical force, even when the conductors contain 

 closed cavities. 



Let us consider an electrified surface S (Fig. 10). Through a 

 point in the interior P draw a cone du of infinitely small aperture, 



Fig. 10. 



which intercepts, on the surface, at the distances r and r' two elements, 

 dS and */S', the densities on which are respectively o- and a-'. 



If the electrical forces are proportional to a function of the 

 distance / (r), the action of the element </S on unit mass at the 

 point P may be written, 



Calling / the angle of the radius vector r with the normal to the 

 element dS, we have 



r 2 do) = d$ cos /, 



which gives 



The action of the element d$> is, in like manner, 



and these two actions are directly opposed. 



