Theory of Electromagnetic Action. 451 



+ 00 



~N k y k = — l 1 1 BHi cos k dx dy dz, . . (16) 



— 00 



where x, y, z are the coordinates of the point at which the 

 induction is B, and dx dy dz is an element of volume. 



Finally, take any point on one of these surfaces, and let B 

 be the induction there. For every such point a surface can 

 be drawn having any one of the circuits as its boundary, and 

 hence by (16) and (3) we have finally for T the equation : 



+« 



T=~m JB(H 1 cos^ l + H 2 cosA+&c.)^^^. . (17) 



— oo 



But if H be the total magnetic force at the point and 6 the 

 angle which it makes with the normal to the surface, we have 



H cos 6 = Hj cos 6i -r H 2 cos 6 2 + &c. 

 Hence (17) becomes 



T=^\\\BK cos 6 dxdydz. . . . (18) 



If we suppose, what is always the case in an isotropic 

 medium, that B and H have the same direction, we have 



+ 00 +00 



T=^(\\BKdxdydz=^((\B 2 dxdydz, . (19) 



— 00 —00 



since B = ftH, if fx be the magnetic inductive capacity or per- 

 meability of the medium. 



•Now let the total induction through the circuit in which 

 the magnetizing current is flowing be increased by an amount 

 d~N produced by increasing the current in that circuit. The 

 energy drawn from the battery is y dN ; and clearly by the 

 investigation given above, we have, if dB be the increase of 

 induction at (x,y,z) y 



+ 00 



yd^=^((RdBdxdydz. . . . (20) 



— 00 



The change of electrokinetic energy dT is given by 



+ 00 



dT = : i- (Tf (EdB + BdR)dx dy dz. . . (21) 



