226] ELECTROMAGNETISM. 439 



is the negative of the total potential energy. But the potential 

 energy tends to decrease, and if the current strengths are constant, 

 while the circuits are moved or deformed, their position and form 

 being specified by a certain number of geometrical parameters q s , 

 the forces P s according to these parameters are given by 



(10) 2,P,fy. = - B W = 8 W m = 



(11) ^-W&^ 



oq 8 s s 



The magnetic energy of the field then tends to increase, and 

 we find the system behaving in the same manner as a cyclic system 

 during an isocyclic motion, 70. The energy which must be 

 furnished to the system during a motion caused by the electro- 

 magnetic forces must be double the amount of work done by 

 the electromagnetic forces, which is equal to the loss of potential 

 energy, and must be furnished by the impressed electromotive 

 forces that maintain the currents. We have already seen that 

 in the case of concealed motions we cannot always tell whether 

 energy is potential or kinetic, and that in cyclic systems the 

 kinetic energy has the properties of a force function for either 

 isocyclic or adiabatic motions. We are therefore led naturally 

 to consider a system of currents as a cyclic system, and, instead 

 of considering W as potential energy, to consider W m = W as 

 kinetic energy. We shall henceforth call it the electrokinetic 

 energy, and denote it by T. 



These considerations, assimilating an electrical system to a 

 mechanical system, are due principally to Maxwell, and by means 

 of them we shall in the next chapter be able to deduce the laws of 

 induction of currents. 



If in the integral (5) we integrate over current-tubes in the 

 manner just explained, for udr we must put 



q cos (qx) Sds = Idx etc., 

 so that we obtain for each current 



(12) T= (Fdx + Gdy + Hdz\ 



where the integral is around its own circuit, but F t G, H are 

 the definite integrals over all currents, as previously used. Apply- 

 ing Stokes's theorem to the above line-integral, we obtain 



