438 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL 



both ds and ds traversing all current tubes. The sextuple in- 

 tegral is here interpreted as a line integral around every current 

 tube and then an integration for the double infinity of tubes for 

 each variable s and s. If the currents consist of two linear 

 circuits, or closed tubes of infinitesimal cross-section and strengths 

 /! and 7 2 , the sextuple integral reduces to a double-line-integral, 

 and since both variables s and s' are to traverse both circuits, we 

 may divide the integral up into four parts according as s or s 

 coincide with s l or s 2 , 



(8) 



The second and third integrals are equal, for it is evidently 

 a matter of indifference which point of integration is associated 

 with either circuit, so that we may write for the sum of these two 

 terms 



TT f [ cos(ds l ds. 2 ) , , 



JlJ 2 f 



where each point of integration goes once around one of the 

 circuits. 



This term is equal to the negative of the mutual potential 

 energy of the electromagnetic forces acting between the two 

 currents, as found in 220 (i i). 



In like manner the first and last terms, where each point of 

 integration goes once around the same circuit, are the negatives 

 respectively of the potential energy of either current in its own 

 field, from which the electromagnetic forces acting between its 

 different parts may be calculated. If we call the integrals 



T _ [ f cos(cfoefe') , , , r _ [ f cos(dsds') 

 l ~iiJi r S S> 2 ~J 2 J 2 r 



we may say then that the magnetic energy of the field due to 



both currents 



(9) W m = \LJf 



