318 CONSTITUTION OF MAGNETS. 



In this case, the expression A.dx + >dy + Cdz is no longer an 

 exact differential. We may again eliminate the function 3> between 

 these equations, and we get 



This is the condition which must be satisfied to have a complex 

 lamellar magnetisation. 



Conversely, if equation (14) is satisfied, the magnet is formed 

 of complex magnetic shells, for the lines of magnetisation are 

 orthogonal to a system of surfaces ; unless each of the expressions 

 in the parenthesis is separately zero, in which case the magnetisation 

 would be lamellar, from equations (12). 



334. POTENTIAL OF A SOLENOIDAL MAGNET. The general value 

 of the potential of a magnet is 



' P -dv. 



If the magnet is solenoidal, the density p is everywhere zero, and 

 the potential is reduced to 



The potential of a solenoidal magnet at any internal or external 

 point only depends then on the surface density, or on the per- 

 pendicular component of the strength of magnetisation at every 

 point of the surface. This potential is independent of the manner 

 in which the internal magnetisation varies, or in other words, on the 

 internal form of the solenoidal filaments which terminate at the 

 surface, as well as of the existence of closed filaments. 



We may suppose, for instance, that the magnetism of the earth 

 is produced by solenoidal filaments, maintained in the surface 

 rocks at a low temperature, and terminating on the surface in such 

 a way as to produce a distribution equivalent to that of a uniform 

 magnetisation. 



335. POTENTIAL OF A LAMELLAR MAGNET. If the magnet is 

 lamellar, it consists of closed magnetic shells, and of open shells with 

 their contour on the surface. The force outside only depends then 



