CURRENTS IN CONDUCTING SIIKI.I.S u| SMALL THICKNESS. 301 



where r is the distance from a jx>int in the shell to the point at which F, G, H are 

 required. If the current sheet be a closed surface, or if it be a bounded surface, 

 and f/> be zero at the boundary, these expressions can be put in another form, as 

 follows : 



Applying STOKES'S theorem to any bounded surface S, and the function <j>/r, we 

 have 



in which the integral on the left hand side is round the bounding curve. Therefore 

 for any closed surface, or auy unclosed surface, provided that <f> is continuous and 

 vanishes at the boundary, 



and, therefore, 



F is therefore a linear function of all the <'s with coefficients functions of the 

 coordinates ; G and H have corresponding values. Given S and <j>, F, G, and H are 

 completely determinant, and are independent of h. 



Corollary. The vector potential due to any spherical current sheet is tangential 

 to every spherical surface concentric with the sheet, as shown by MAXWELL, 671. 



Of the Energy of a System of Current Sheets. 



6. The electrokinetic energy of a system of currents over the surface or system of 

 surfaces S is 



T = JJ (Fhu + Ghv + Uhw) dS 



We can transform this by STOKES'S theorem in the same manner as we transformed the 

 integral 



For, if the surfaces be closed, or if (ft be continuous and vanish at the boundary, 



