228, 229] ELECTROMAGNETISM. 445 



be traversed by apparent currents and current-sheets, as in the 

 preceding section in the case of temporary magnets. We there 

 introduced the discontinuity in the induction, but we might have 

 introduced the intensity of magnetization. In the case of the 

 permanent magnet this will be more convenient in either case 

 the form of the vector potential will be the same. 



We have for the potential due to a magnet in a homogeneous 

 medium of unit inductivity, 122 (3), 



< o 



where A', B', C f are the values at a, 6, c, 

 dr' = dadbdc, 



r* = (x - of + (y- by + (z- c) 2 . 

 The field at any point x, y, z has the component 



< L . 



d 



Now the derivatives of 1/r with respect to a, b, c are the negatives 

 of its derivatives with respect to x, y, z, so that we may write 



L 



^ - ^ o 



Bo 2 dxdy 



But since 1/r is harmonic we may put 



I 



= _rw 



I 9y 2 

 and thus the integral becomes 



'-) *(- 



r) ,, \r 



v r 



da) dy / dz\ dz dx 



(4) 





' 9 ^!1, Lf/rfr.' ,- a L 



-sr- * r T -^JJJ\ c is ^ irr T - 



