560 PHENOMENA OF INDUCTION IN NON-LINEAR CONDUCTORS. 



These equations are satisfied by putting 



F JQ r JQ 



~ G= ' 



577. If, at the same time, there is an external system of magnets 

 or of currents, we know (498) that its action on a surface 

 surrounding it is equivalent to that of a suitable system of surface 

 currents. .The magnetic potential of this system, in the positive 

 surface of the shell under consideration, might then be expressed 

 by a function Q', analogous to the function Q ; we shall have then 



which gives 



Putting = R, equations (4) become 



27* 





Integrating the first with respect to y, or the second with respect 

 to #, we get 



For a complete integration we should add an arbitrary function of 

 /, but it must be observed that this function will disappear whenever 

 we take a partial differential in respect of x or y to calculate the 

 components of the current ; it need not then be taken into account. 

 578. Let us suppose, at first, that there is no external magnetic 

 body that is to say that Q' is null. This would be the case of a 

 system of currents set up in the shell and left to themselves ; these 

 currents would act upon each other by their mutual induction, and 



