18 Mr. A. Schuster's Electrical Notes. 



Equation (17) is one of the required boundary conditions. 

 The second is obtained by symmetry. 



Hi dz 2 fi 2 dz 2 ~~ V*i H / d f 



The two last equations complete the solution of the pro- 

 blem, as all other differential coefficients are continuous. 



It is easy to show that the conditions we have found lead 

 to the well-known continuity of the normal components of 

 the induction and of the tangential components of the mag- 

 netic forces. 



We have arrived at the conclusion that 



^tfjr***^*' 



(19) 



with the corresponding equations for G and H will always 

 give us correct expressions for the Vector Potential, u, v, id 

 being the components of the given currents, and u-, v', io' 

 currents distributed over the boundaries between bodies of 

 different magnetic permeabilities. F, G, H are not, however, 

 the only solutions, but others may be obtained by addition of 



j,i j j"> respectively, % heing arbitrary. It appears, how- 

 OjX oiy * CIZ. 



ever, that if it', v', w' in (19) are determined so as to satisfy 

 the necessary boundary conditions, both ^ and its first 

 differential coefficients must be continuous throughout space. 

 The function ^ may therefore be expressed as a potential 

 function of attracting matter distributed throughout space 

 with an arbitrary volume-density, there being no surface 

 distribution anywhere. If % does not vanish it follows from 

 (5) that 



dF dG x d& „ a 



d^ + di + ^ = ~ Vx; 



for it has been shown that whenever currents flow in closed 

 circuits only, 



dx dy dz 



If, then, the additional condition is imposed on the Vector 

 Potential that 



d JL+ <k* + ^1 



dx ' dy dz 



