434 The Magnetic Field produced by Electric Currents [OH. xm 



On reducing the thickness of the shell indefinitely, S becomes infinite, for 

 at any point of the shell, 



(Sdn = - (difference of potential between the two forces of shell) 



so that S becomes infinite when the thickness vanishes. 

 Thus on passing to the limit, the first integral 



becomes infinite. This quantity is, however, a constant, for it represents the 

 energy required to separate the shell into infinitesimal poles scattered at 

 infinity. 



The second integral vanishes on passing to the limit, and so need not be 

 further considered. 



The third integral can be simplified. We have 



i //(/ 



** 



)dS. 



Now I S dn = 4>7T(f), while II adS is the integral of normal induction 



over the shell, and may therefore be replaced by N, the number of unit tubes 

 of induction from the external field, which pass through the shell. Thus the 

 third integral is seen to be equal to 



In calculating expression (424) when the energy is that of a system of 

 currents, the contribution from the space occupied by the equivalent mag- 

 netic shells is infinitesimal. Thus all the terms which we have discussed 

 represent differences between the energies of shells and of circuits. 



Terms such as the first integrals of scheme (427) represent merely that the 

 energies are measured from different standard positions. In the case of the 

 shells, we suppose the shells to have a permanent existence, and merely to 

 be brought into position. The currents, on the other hand, have to be 

 created, as well as placed in position. Beyond this difference, there is an 

 outstanding difference of amount $N for each circuit, and this exactly 

 accounts for the difference between expressions (425) and (426). 



503. Let us suppose that we have a system of circuits, which we shall 

 denote by the numbers 1, 2, .... Let us suppose that when a unit current 

 flows through 1, all the other circuits being devoid of currents, a magnetic 



