CUHRENTS IN CONDUCTIM! SIIKLLS O SMALL THICKNESS. '"' 



we see that T can be expressed as a quadratic function of all the <f> with coefficients 

 functions of the coordinates. 



Evidently, if we have two systems of currents on different sheets or on the same sheet, 

 their energy of mutual action is ^ JJ <f> dil <lv c/S = ^ JJ <f> dfl'fdv dS, where <j> and 11 

 relate to one system, and <f> and SI' to the other. 



Comparison with Magnetic Shells. 



7. If a system of magnetic shells be formed over the surface S, and <f> be the strength 

 at any point, regarded as positive when the positive face is outwards, the components 

 of vector potential of magnetic induction due to the system at any point not within 

 the substance of the shells are (MAXWELL, 41G) 



They are, therefore, the same as the components of vector potential of the system 

 of electric currents over S, determined by <f> as current function. 



It follows that the components of magnetic force or magnetic induction, namely, 

 dH/'dy dG/dz, &c>,are at any point not within the substance of the shells the same 

 for the system of shells whose strength is <f> as for the system of currents whose 

 current function is <f> over the same surface ; or, as we may otherwise express it, the 

 magnetic potential due to the system of shells differs from that due to the system of 

 currents by some constant at all points external to the sheet, and by some, but not 

 necessarily the same, constant at all points within the sheet, the particular constants 

 depending on the definition we choose to adopt of the magnetic potential due to a 

 current shell. 



8. Proposition. There exists a determinate system of magnetic shells over any 

 closed surface, S, which has magnetic potential at each point on or within S equal 

 to that of any arbitrarily assigned external magnetic system. 



For let P be the potential of the external system. 



Let q be the density of a distribution of matter over S, the potential of which is 

 equal to P at all points on S, and, therefore, also at all points within 8. 



Then q is possible and determinate by known theorems. 



Let (f> be that function of x, y, and z of negative degree which satisfies the 

 conditions V 2 <= at all points outside of S, and d<f>/dv = q at all points on S, 



