fuAW* - f VAU<& = f (u - V \ JS, 



J J J \ ^n TM) 



650 CONSTANTS OF MAGNETISATION. 



If, in Green's formula (34), the letters U and V are interchanged, 

 and if the two equations are subtracted, we get 



(5) 



the second expression being extended to the entire surface S, and 

 the first to the volume which it contains. 



We shall apply this equation to the volume of the magnet, 

 assuming first that V represents its potential, and p the magnetic 

 density at a point M. Consider an external point P, at the distance 

 r from the point M : we shall represent by U the potential - at M 

 of a mass equal to unity at the point P ; we shall have then, within 

 the limits of the integral, 



AU = 0, AV= - 



The former member of the equation (5) reduces to 

 j UAWz; = - 477 f = - 4 7rV p , 



Vp denoting the potential at the point P; it follows that 



(6) 



a J r a* 



The value of V p only depends then on the potential V, and on its 

 differential perpendicular to the exterior of the surface S. 



It is sufficient if we know, to within a constant, the potential on 

 the surface, for, by calculating the external potential, this constant 

 would be determined by the condition that the potential is null at a 

 great distance from the magnet. 



There is, moreover, a relation between the potential V on the 

 surface and its perpendicular differential. Let us, in fact, consider 

 a point in the interior, and, the function U being defined in the same 

 way, apply equation (5) to the volume bounded by the surface S and 



