298 MR. S. H BURBURY ON THE INDUCTION OF ELECTRIC 



at every point. We will suppose that a closed surface can be described completely 

 enclosing all the currents of the system. Then the stream lines must, in order that 

 the above condition may be everywhere fulfilled, be in closed curves, and the currents 

 are then said to be closed currents. 



Of Current Sheets and Sliells and Superficial Currents. 



2. Any surface to which the resultant currents or stream lines are everywhere 

 tangential is defined to be a current sheet. The space between any two current 

 sheets infinitely near each other may be defined to be a current shell. If a line be 

 drawn oh a current sheet perpendicular to the stream line at any point P, and dc be 

 any element of that line at P, and h be the distance at P between the two current 

 sheets forming the shell, then the ratio of the quantity of electricity which flows 

 through the area hdc in unit of time to dc is the superficial current,* or current 

 referred to unit of length, at P. 



If the current sheet be a closed surface, S 0, the stream lines form cloBsd curves 

 upon it, and no two of them intersect each other. Let there be given on S any system 

 of superficial currents. We can suppose the stream lines traced out on the surface. 

 Then the superficial current flowing along the belt between any two of them is 

 inversely proportional at any point in its course to the distance between the two 

 stream lines at the point. 



Of the Current Function. 



3. For any system of superficial currents which can be formed on a closed surface 

 S = there exists a function <, called the current function, such that the component 

 superficial currents at any point are 



Ud& d<i> 



= n - m " " 



- 

 dz ax 



Wrf<l> j !/<)> 



= m- -- I -r- 



dx ay 



where I, m, n afe the direction cosines of the normal to the surface. We will use 



U, V, W for the superficial currents ; u, v, w for the currents referred to unit of area. 



For there exists a function, <f>, of x, y, and z, which has any arbitrarily assigned 



* It is usual to employ the term tuperficial current only in Cases Where the quantity of electricity in 

 question is infinitely great compared with h by analogy to the definition of a superficial distribution 

 of electricity in electrostatics. But the definition above given is unambiguous, and includes as a 

 particular case the case of a finite current in an infinitely thin shell. 



