432 CURRENTS AND MAGNETIC SHELLS. 



action of the two systems is then identical with that of two magnets. 

 The maximum and minimum values of the flow of force Q correspond 

 to positions of relative equilibrium, stable in the former case and 

 unstable in the latter. The motion may be continuous, on the 

 contrary, if the circuit can be deformed ; if it contains, for instance, 

 liquid portions, sliding contacts, or if it can be broken in certain 

 parts while the magnet is being displaced. ' 



453. RECIPROCAL ACTION OF Two CLOSED CURRENTS. It 

 may be asked whether a closed current and a shell, which are 

 equivalent with respect to any magnetic system, are so towards 

 another current? Thus the current Cj and the shell Sj of the 

 same contour, are equivalent in their action upon the magnetic 

 system M 2 ; suppose that this magnetic system is a shell S 2 ; the 

 reciprocal action which is exerted between S T and S 2 is identical 

 with that which is exerted between S x and the current C 2 , which 

 is equivalent to S 2 ; but is this latter action the same as that which 

 would be exerted between the two currents C} and C 2 ? The 

 affirmative seems probable ; but this is only an induction, and it 

 would be easy to find examples, for which the same mode of 

 reasoning would lead to conclusions which are manifestly erroneous. 

 Thus, under conditions suitably chosen, it might happen that the 

 actions exerted upon a magnet by a magnet and by a piece of soft 

 iron are the same ; we could not conclude from this that the soft 

 iron and the magnet would be equivalent towards another piece of 

 soft iron. 



It is therefore as an experimental result, and not as a necessary 

 deduction from theory, that we shall assume the following theorem of 

 Ampere : 



The reciprocal action of two closed currents is identical with that of 

 two magnetic shells respectively equivalent to each of them. 



454. RELATIVE ENERGY OF Two CURRENTS. The value of 

 the potential energy of two magnetic shells (341) is 



W= -**'M. 



From Ampere's theorem, that of two closed currents will be ex- 

 pressed, to within a constant, by the formula 



(6) W=-II'M, 



in which I and I' are the strength of the two currents, and M the flow 

 of force which, starting from one of the circuits, traverses the other by 

 its negative face, the strength in each of them being equal to unity. 



