Vortex Theory of Electromagnetic Action. 401 



J = a constant = c say. But the equation of continuity 

 gives us 



2pdt' 



_2ct 



p — p e p- . 



And this is impossible unless c—0, andp = constant ; in this 

 case J = 0. Thus we must suppose our fluid to be incompres- 

 sible. 



In forming equation (7), we have put 



/*X_ __dV 

 2 ~ dx> 

 thus assuming that our forces at all points of the medium had 

 a potential. If this be not so, let 



fiX • . p dY 

 2 . l dx' 



Then the electromotive force P x acting at each point produces 

 a current f 1} given by the equation 



p __4ttt^ • 

 ■fi— — -/* 



if / is the current arising from electromagnetic action, 



and 



P = P X + P', 



where P' is the electromotive force arising from electromag- 

 netic action. Thus 



v ,___dl_cl± 

 dt dx > 

 &c. 



Again, since J = 0, 



t= v +? . 



^r is the potential of the free electricity in the medium ; if the 

 fluid be in equilibrium, the equations give us 



dx ' dy } dz 

 Also 



F = G=H=0. 



Thus, within a conductor in which the fluid is at rest there 

 is no electromotive force at any point. Thus the electrical 

 phenomena that occur in a conducting medium subject to elec- 



