A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 557 



Equations of Currents. 



(62) It is known from experiment that the motion of a magnetic pole 

 in the electromagnetic field in a closed circuit cannot generate work unless the 

 circuit which the pole describes passes round an electric current. Hence, except 

 in the space occupied by the electric currents, 



adx + fidy + ydz = d<f> (31) 



a complete differential of <, the magnetic potential. 



The quantity <f> may be susceptible of an indefinite number of distinct values, 

 according to the number of times that the exploring poini passes round electric 

 currents in its course, the difference between successive values of < corre- 

 sponding to a passage completely round a current of strength c being 4-n-c. 



Hence if there is no electric current, 



dy dz 

 but if there is a current p', 



dy dfi _ , 

 dy dz * 



dp _ da 

 dx dy 



We may call these the Equations of Currents. 



Electromotive Force in a Circuit. 



(63) Let be the electromotive force acting round the circuit A, then 



,. 

 s ........................... (32) ' 



where ds is the element of length, and the integration is performed round the 

 circuit. 



N 



