448 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. 



ential operator A is commutative with any partial differentiation, 

 we may write 



. fbH dG^ 



But if the vector potentials F, G, H are those of the currents u, v, w, 



so that finally 



which by the theorem of 226 (3), (4) is equal to 



dP d 



(.,) w-Iff{ 



--///[ 



Comparing with 226 (5) we find a difference in sign, W being 

 mutual potential energy, W m electrokinetic energy, while the 

 factor J is omitted in mutual energy. 



The integral may now be restricted to the space occupied by 

 currents. The form involving the curl of P, Q, R is that used by 

 Helmholtz*, who writes L, M, N instead of P, Q, R. Replacing 

 P, Q, R by their values (7) we obtain the double volume-integral 



,/r 



(/~\ TTT" I 1 I I I I I / T\/ XV/\ V * > 



1 6) 



+ (uC f - wA') 



which differs from the result of substituting (5) in 226 (5) in 

 the same way as (15), above. 



We have thus seen how we may replace every magnet by an 

 apparent current 



* Helmholtz, Ges. Abh. Bd. i. p. 619. 



