304 Mil. S. H. BURBURY ON THE INDUCTION OF ELECTRIC 



where dv \s an element of the normal to S, measured outwards. Then <f> is possible 

 and determinate by known theorems ; and <f> is the strength of the required magnetic 

 shell. 



For let r be the distance of any point from an internal point O. Then at O the 

 potential of the system of magnetic shells whose strength is <f> is 



But, by GREEN'S theorem, applied to the functions <j> and l/r and the infinite space 

 outside of S, 



But V 2 l/r = 0, because is within S, and V 2 < = by definition at all external 

 points. 

 Therefore, 



. : ,, . , = P, , .I''.. 



Corollary. There exists one determinate system of closed electric currents over 



. any closed surface, S, whose magnetic potential, together with that of any arbitrarily 



given external magnetic system, is constant at all points on or within S ; namely, the 



system of currents whose current function is <f>, where <j> is determined as in the 



principal proposition. 



9. The magnetic induction due to the combined systems is, therefore, zero at every 

 point within S. We will define the system of currents on S which has this property 

 to be the magnetic screen on S to the external system. 



Evidently the proposition and its corollary will apply equally to a system of 

 magnetic shells or electric currents on S having at all points in external space the 

 same magnetic effect as that of a magnetic system wholly within S. 



