POTENTIAL OF A MAGNET. 299 



312. UNIFORM MAGNET. A magnet of finite dimensions, which 

 is formed of identical filaments placed parallel to each other, 

 might be called a uniform magnet; the poles of the elementary 

 filaments being placed at the ends, on the surface of the body, it 

 will be seen that the action of the whole magnet would reduce to 

 that of two magnetic layers, distributed on the surface according to 

 a simple law. 



313. ANY GIVEN MAGNET. At a point P within any given 

 magnet the magnetic axis has a determinate direction, and this 

 direction varies continuously; hence, inside a magnet we may 

 draw lines tangential at every point to the magnetic axis, and 

 we may imagine magnetic filaments directed along these lines of 

 magnetisation. 



The magnet would thus be subdivided either into non-uniform 

 magnetic filaments closed or terminating at the surface, or into 

 uniform filaments, some closed, and others terminating on the 

 surface, and others, lastly, terminating in the interior. So long as 

 no hypothesis is made as to the form of the filaments, this con- 

 ception is a pure and simple translation of facts, and has nothing 

 hypothetical. It leads to considering any given magnet as formed of 

 a magnetic layer distributed on the surface, and of magnetic masses 

 disseminated throughout the interior. We may accordingly consider 

 a surface density of magnetisation and a volume density. 



The density of the free magnetism at a point is the limit of the 

 ratio of the magnetic mass contained in a volume element taken 

 about this point to the volume itself; the surface density is the 

 quotient of the quantity of magnetism which exists on an element 

 of surface about this point, by the area of the element 



314. POTENTIAL OF A MAGNET. This being admitted, it is 

 clear that the value of the potential of the magnet at any external 

 point P will be 



(i) V 



=/>/' 



In the first integral, which will extend to the entire surface of 

 the magnet, o- denotes the surface density on the surface element 

 </S, whose distance from the point P is equal to r. The second 

 integral must be extended to the whole volume of the magnet, p 

 denoting the volume density of magnetism in the element dv at a 

 distance rv from the point P. These densities p and a- may be 

 considered as those of a particular fluid. 



