RELATIVE ENERGY OF A MAGNETIC SYSTEM. 431 



be continuous, for an extraneous energy necessarily comes into play 

 in the phenomena, such as that of the chemical actions which take 

 place in batteries. 



If Q be the flow of force of the magnetic system, which traverses 

 the surface of the current, entering by the negative face, the relative 

 energy of the two systems is expressed, to within a constant, by 



(5) W=-IQ. 



When the magnetic system is left to itself, the work ^T of the 

 magnetic forces, for any infinitely small displacement, is equal and of 

 opposite sign to the change in the energy, and we have 



or 



The motion of the system takes place in such a way that the 

 value of Q tends towards a maximum. 



For two successive positions characterized by the indices i and 2, 

 the work, to within a constant, will be 



It is important to remark that in a general way the difference 

 Q 2 Qj depends not only on the final and initial positions of the 

 magnetic system, but also on the path pursued by each mass; for 

 the product ;#47rl should be added to the work of all those which 

 should have surrounded one of the branches of the current. If a 

 long and flexible uniform magnet, for instance, were placed near a 

 current, the positive pole would turn in one direction indefinitely 

 about the current, and the negative pole in the contrary direction. 



Nevertheless, if the contour of the current is rigid, as well as the 

 magnetic system, all the masses which constitute the magnet will 

 necessarily make the same number of turns, and in the same time, 

 and the corresponding work is equal to n^irl^m. As the total mass 

 of a magnet is always zero, the work of any given displacement only 

 depends on the initial and final position of the system, and not of 

 the path traversed ; in this case the work is null when the magnet 

 reverts to its original position. 



It is therefore impossible to obtain the continuous motion of a 

 magnet by a current which traverses a rigid system ; the reciprocal 



