225, 226] ELECTROMAGNETISM. 437 



= [ I {(ME - NG) cos (not) + (NF - LH) cos (ny) 



+ (LG-MF)cos(nz)}d8 



This important theorem in integration may be abbreviated as 

 (4) jjj(H curl Q-Q curl # } dr =JJV.HQ cos (w, V. 



The integral representing the energy is extended over infinite 

 space, and the surface integral vanishes at infinity. Inserting 

 the value of curl H in terms of the current density, 222 (2), we 

 obtain 



(5) W m = 



and since no portion of space contributes to the integral unless 

 it is traversed by currents, we may take the integral simply 

 through conductors carrying currents. The components of the 

 vector potential are however themselves triple integrals over the 

 same portions of space, so that if we distinguish a second point 

 of integration by an accent, we have the double volume integral 



r* = (x - xj + (y- yj + (z- /) 2 , 



where each point of integration traverses the whole volume 

 occupied by currents. 



This form of the energy corresponds to the form in terms of 

 density given in 117 (5), the integrals being there taken through 

 all distributions of matter. 



If we perform the volume integration by dividing the space 

 up into current tubes, of infinitesimal cross-section S, ds being 

 the length of the generating curve, and I=qS the total current in 

 the tube, we have for the element of volume dr = Sds, so that the 

 integral becomes 



1 ffffrfU'coB (dads') 

 (7) 2JJJJJJ- -IT J . 



