DETERMINATION OF THE FICTIVE LAYER. 653 



The terms of the second member relative to the surface S are 



null, for we have U = and ^r = ; this differential being counted 



on 



towards the interior or towards the exterior of the surface, according 

 as the point P is itself on the outside or inside. 



If the point P is outside, AV = within the limits of the integral, 

 except perhaps on the surface S, but in that case U = ; we have 

 then 



I 



On the other hand, AU = in the same conditions, but we must 

 allow for the layer on the surface S, when V is not null. The former 

 member of the equation then becomes 



V<fS. 



J J 



It follows that 



v*-- 



The external potential of the magnet is then expressed solely as 

 a function of the potential on the surface. 



If the point P is within the surface S, we shall replace the 

 magnet by the fictive surface layer. The same reasoning will then 

 apply without any modification, and the expression for the potential 

 is the same. 



In both cases the density u will be defined by the aid of potential 

 U by the condition 



If a be the perpendicular distance of the point P to the surface, 

 the density u on the element ^S is a function of 0, and we have 



The values of this expression, inside and outside, for a = 0, will 

 give the intensity o- by equation (8). 



