CONDUCTING SHELLS. 557 



Let us consider an infinitely thin homogeneous conductor, which 

 may be supposed reduced to a surface, and in which exist, for any 

 reason, electric currents which are not brought from without by elec- 

 trodes these currents are necessarily closed, and the stream-lines 

 cannot intersect each other. The annular space comprised between 

 two infinitely near currents, may be regarded as a linear circuit 

 traversed by a current of strength d&. This current may be replaced 

 by a magnetic shell of the same strength and the same contour. 



If the surface of the conductor is thus cut into infinitely thin 

 bands by the stream-lines, it is seen that for an external point, the 

 sum of the currents will be equivalent to a complex shell (333), the 

 magnetic strength < of which is equal at each point to the sum of 

 that of the superposed shells. On the positive face of the shell 

 the currents are in the opposite direction to the hands of a watch, 

 about the spaces where the value of 4> is a maximum. This value 

 is null at the edge if the plate is bounded. 



Along a current line the value of < is constant ; the current lines 

 are equipotential lines of the function <. An element dn of a 

 line , at right angles to the current line, is cut perpendicularly by 



rffc 



a current of strength dn, in the direction of the right of an 

 dn 



observer, who, along this line, would move towards points where 

 the function < increases. Finally an element ds of any given curve 



t3> 

 is cut by a current of strength ds' t but which is no longer per. 



pendicular. 



The expression for the magnetic potential on the outside is 



= Utfco 



This function is discontinuous on traversing the surface ; the 

 two values V 2 and V 1 which it assumes on either side of the shell, 

 on the positive and on the negative face (330) are connected by 

 the equation, 



The component of the magnetic force perpendicular to the 

 surface is continuous, for it represents, on either side, the flow of 

 induction for unit surface. This is the case also with the tangential 



d<l> 



component along a stream-line, for then we have - = . 



Of 



