Theory of Electromagnetic Action. 447 



Here part of the impressed electromotive forces E b E 2 in 

 each case is employed in working against the inductive elec- 

 tromotive force, and thus in increasing the electrokinetic 

 energy. By the electromagnetic forces the electrokinetic 

 energy is increased by the amount y x y 2 dM., which also is the 

 amount of work spent in moving the circuits. 



If the conductors are displaced from rest to rest again, so 

 that y v y 2 , have resumed their steady values dy 1 =Q, dy 2 = 0, 

 the energy furnished by the batteries is 2i/iy 2 <^M, of which 

 one half is accounted for in dT, the other by the work 

 done in moving the circuits, which has its equivalent in work 

 done against the external resistance by which the conductors 

 were brought to rest. This result was obtained by Sir William 

 Thomson so long ago as 1851. 



(2) Assuming, according to Ampere, that a magnetic shell 

 is equivalent to a current round its edge, and putting y 2 for 

 this current, we have, since E 2 =0, K 2 = 0, instead of (7), 



E 1 -|(L,2/ 1 + My 2 )=R^ 1 



d 



|(L^ 3 + M^)=0. 



As before we get 



dT = Ly^ + M {y 2 dy x + y x dy 2 ) + L 2 y 2 dy 2 + y^dM, "j 

 dW = y 1 y 2 dM. J ' {li) 



The energy furnished by the battery is, as before, 



L^dyt + My^y 2 + y^dM. 



By the other circuit no energy is given. But multiplied bv 2/ 3 

 the second of (10) gives 



Ls2/2^ 2 + M?/^ + y^dM = 0. 

 Thus 



dT^y^ + My^ly,, 



dT + dW = Uyidy! + Myidfo + ymM, 



(12) 



which again, of course, is the energy furnished by the battery. 

 Here, if the changes are estimated for the system when 

 brought to rest, dy x = 0. The other current y 2 does not, 

 however, remain constant, and we have to inquire what is its 

 effect upon the magnet. We know that the moment of a hard 

 magnet is not seriously altered by a displacement produced by 

 the mutual forces if y l is moderate ; and the small alteration 



