228] ELECTROMAGNETISM. 443 



so that the vector potential is not related simply to the current, 

 but the magnetic forces still occur in the differential equation. 

 We have the same difficulty as occurs when we undertake to find 

 the potential of the field when the inductivity varies. As we 

 there made use of an apparent density, so here we may define an 

 apparent current as 



dy dz 

 so that the vector potentials are 



'-//* -//* *- 



If each magnetizable body is homogeneous, 



fix dy dz 



and the apparent currents are the true currents multiplied by the 

 inductivity, except at the surface separating two media, where the 

 derivatives of fi, and consequently the values of &F, A6r, A77, are 

 infinite. We have the above form for F, G, H only when AF, etc. 

 are finite, and when they are infinite at a surface we must proceed 

 as in the case of a surface distribution of matter, that is we must 

 consider an apparent current-sheet between the two media. Con- 

 sidering two surfaces infinitely near each other and situated on 

 opposite sides of a surface of discontinuity of ^ at a distance dn 

 from each other, and integrating the equation (3) over the volume 

 of the thin sheet between them we obtain* 



-^ cos (nz) I dndS 

 dn 'J 



- 91) cos (ny) - (3R' - 3) cos (n*)} d 



* The second integrand in ( 4 ) is equal to the first since n is the direction of 

 most rapid (infinitely rapid) change in the functions 9t, 2ft, in the infinitely thin 

 sheet. 



