229] ELECTROMAGNETISM. 447 



the vector potential belonging to the magnetic force is its curl. 



as_8Q 



~ ' 





SQ SP 

 " == ^ -- 



^; ^r~ 

 ex oy 



This leads us to a second manner of obtaining the mutual energy, 

 due to Helmholtz. The expression for the energy of a permanent 

 magnet in a magnetic field L, M, N is, 126 (2), 



(9) W = - ^l + BM + CN) dr, 



where the volume of integration is that occupied by magnets, or 

 it may be extended to infinity, since elsewhere 



4 = = (7=0. 



We may transform the integral into one taken throughout the 

 space occupied by currents. If we introduce the vector potential 

 of intensity of magnetization, we have from (7), if the magnetiza- 

 tion is everywhere finite, 



A = - - AP, 



4-7T 



(10) 



-- AE. 



4>7T 



Introducing these values of A, B, C into the integral (9), 

 (ii) w = (L^P + M AQ + NbR) dr, 



and transforming each term by Green's theorem in its second form, 

 the surface integrals vanishing at infinity, 



(12) W = <f^ + Q&M + R&N) dr. 



00 



Let us now substitute for L, M, N their values in terms of the 

 vector potential belonging to them, noticing that, since the differ- 



