300 MH. S. II. BUUHURY ON THE INDUCTION OF ELECTRIC 



Then u, v, and w, satisfying these conditions, may be the components of a system of 

 finite currents per unit of area within the space. For they satisfy the condition 



du dv . din _ 

 &c + dy~*~lL = 



They also satisfy the conditions 



rfS dS dS 



u r + v ~j~ + w j 



dx dy dz 



d<h d<f> dd> 



u j + v j + w = ; 

 dx dy dz 



and, therefore, in such a system any surface, S = constant, or (j) = constant, is a current 

 sheet. 



If the surface S = constant be a closed surface within the space the currents upon 

 it are closed currents. 



If we form a shell between two neighbouring surfaces, S = c and S = c + dc, the 

 superficial currents in that shell on S = c are determined by the current function <j> dc, 



so that 



TT , / d<j> <fcfr\ 



U = etc ( w - m -- , &c. 

 \ dy dzj' 



The currents per unit of area being assumed finite, the superficial currents are of 

 course infinitely small in an infinitely thin shell. If h be the thickness of the shell, 



dS _ ndc dS __ mdc rfS _ Idc 



d-~ ~A~ ' dy~ ~h ' ' dx~~dk' 

 And. therefore, 



U = hu, V = hv, W = hw. 



Of the Vector Potential of a System of Superficial Currents. 



5. If S be any current sheet, <f> the current function, we have for the components 

 vector potential 



7C3 



- cZS 

 r 



W 



, , 

 = \\(n ~ m 



d<)>\ 



~ m J ) 

 dy dz / 



