444 THE ELECTROMAGNETIC FIELD. [FT. III. CH. XI. 



', S 3ft', 91', being the components of the induction on the side 

 toward which n is drawn, S, 93t, $1, on the side from which it is 

 drawn. We must now, as shown in 85 (18), add to the volume 

 integral already found for F the surface integral 



J^f/W dF\dS 

 + ~' 



which is the effect of an apparent current whose ^-component per 

 unit of surface is l/4?r times 



(W - 91) cos (ny) - (W - W) cos (nz) 



= 33' {cos (33V) cos (ny) cos (33'y) cos (nz)} 



33 {cos (33^) cos (wy) cos (33?/) cos (ws)}. 



Now the normal component of the induction is continuous, its 

 tangential component being Discontinuous, while the tangential 

 component of the force is continuous. The normal plane tangent 

 to the line of force is the same in both media, and the amount of 

 the discontinuity in the tangential component of the induction is 



33' sin (S3 n) - 33 sin (33n). 



Referring now to the definition of a vector product, we see that 

 the first parenthesis above is the ^-component of the vector product 

 of the induction and a unit vector in the direction of the normal, 

 which vector product has the magnitude 33' sin (33'ft). The apparent 

 current is accordingly in the surface, perpendicular to the normal 

 plane containing the line of force where it crosses the surface, and 

 its magnitude per unit of surface is l/4?r times the discontinuity 

 in the tangential induction. If the lines of force are normal to 

 the surface, the apparent surface current vanishes*. If, however, 

 there is a surface carrying a true current-sheet, by the same 

 reasoning, applied to equations 222 (2), we find a discontinuity 

 in the component of the force tangent to the surface and perpen- 

 dicular to the current of amount 4?r times the current density. 



229. Mutual Energy of Magnets and Currents. If we 



have permanent magnets and currents situated in a homogeneous 

 medium of unit inductivity, we may represent their mutual energy 

 in two ways. We may in the first place consider the magnets to 



* This apparent current-sheet was overlooked by Maxwell, and it was not until 

 the appearance of the Third Edition of his Treatise that the correction was made 

 by J. J. Thomson. 



