AMPERE'S THEOREM. 429 



For a magnetic shell of power <, which would be bounded by 

 the same outline (329), we should have 



The two potentials will then be equal, within a constant, 

 provided that 



(4) 7=*- 1. 



the symbol I being a new expression for the strength of the current 

 defined by this condition itself, and which we call the electromagnetic 

 strength. 



The potentials of the current and of the shell for which I = & 

 are not absolutely identical, but they only differ by a constant, and 

 their differential coefficients are the same. Hence the actions 

 exerted by the current and by the shell are the same for each point 

 of the field. We are thus led to Ampere's celebrated theorem : 



The magnetic action of a closed current is equal to that of a 

 magnetic shell of the same contour. 



The positive forces of the current and of the shell correspond, 

 and are on the left of the observer placed in the current, and who is 

 looking towards the interior of the circuit. 



We have deduced this important theorem from Biot and Savart's 

 experiment, but we might consider it as an experimental fact verified 

 in all its consequences, and accept it as a starting point to deduce 

 from it all the magnetic properties of currents. 



451. REMARKS ON THE EQUIVALENCE OF A CLOSED CURRENT 

 AND A MAGNETIC SHELL. It is important to insist on the con- 

 ditions of the equivalence of the current and of the shell. We 

 have seen that with a shell the force is not a continuous function 

 of the co-ordinates ; it is constant in the interior of the shell, and 

 changes its sign when one of the surfaces is passed through ; the 

 lines of force start on each side of the positive face, and are 

 absorbed by the negative. These sudden changes do not take place 

 in the case of a closed current ; the force is a continuous function of 

 co-ordinates, and the lines of force are closed curves which do not 

 touch the circuit, and do not encounter any acting mass. It will be 

 seen that this may be the case, without any contradiction ; for the 

 shell equivalent to the current is only under the condition of being 

 bounded by the same contour, and we may suppose that when a 



