EQUATIONS OF THE MAGNETIC FIELD. 549 



product of the electromotive force P, which acts at a given time 

 on unit length parallel to the axis, by the time dt, is equal to the 

 corresponding diminution of the value of F, which gives 



or 



P- ^ 



5? 



In like manner, if Q and R are the analogous components 

 along the other axes, we shall have, for the determination of the 

 electromotive force at each instant, the three equations 



p- 3F 



Tt' 



. 



dt 



The function - P expresses the difference of potential, which is 

 produced at the point in question between the two ends of a 

 length equal to unity, parallel to the x axis, in consequence of 

 the variations which at the same time take place in the value of 

 the flow of magnetic induction. 



If there were also electrical masses producing the potential ^ 

 at the same point, the total difference of potential dx will be equal 



D D\l/ 



to -dx - ^-dx : but this latter part is an exact differential which 

 Dt Dx 



itself disappears when the formula is applied to a closed circuit 

 569. EQUATIONS OF CURRENTS. Let z/, v, w be the com- 



ponents of the current at a point that is to say, the quantities of 



electricity which in unit time traverse unit surface, perpendicularly 



to each of the axes. 



We know that the work of a current I on a unit pole, movable 



on a closed curve which traverses the plane of the current (452), 



is equal to 4?rl for each complete turn of the pole about the 



current. 



If we apply this property to the case of a pole which traverses 



a rectangular circuit dydz, the centre of which is at the origin of 



