590 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 



where p is the distance of any point from ds. Hence 



dz 



where is the angle between the directions of the two elements ds, (/*', and 

 p is the distance between them, and the integration is performed round both 

 circuits. 



In this method we confine our attention during integration to the two linear 

 circuits alone. 



(110) Second Method. M is the number of lines of magnetic force which 

 pass through the circuit B when A carries a unit current, or 



M = 2 (pal + pfim + pyn) d&, 



where pa, p/3, py are the components of magnetic induction due to unit current 

 in A, S' is a surface bounded by the current B, and I, m, n are the direction- 

 cosines of the normal to the surface, the integration being extended over the 

 surface. 



We may express this in the form 



M= /i2 - t sin e sin ff sin ^dS'ds, 



where dS' is an element of the surface bounded by B, ds is an element 

 of the circuit A, p is the distance between them, 6 and ff are the angles 

 between p and ds and between p and the normal to dS? respectively, and <f> is 

 the angle between the planes in which 6 and ff are measured. The integration 



O 1 O 



is performed round the circuit A and over the surface bounded by B. 



This method is most convenient in the case of circuits lying in one plane, 

 in which case sin 0=1, and sin^=l. 



(111) Third Method. M is that part of the intrinsic magnetic energy of 

 the whole field which depends on the product of the currents in the two 

 circuits, each current being unity. 



