DETERMINATION OF THE FICTIVE LAYER. 649 



When the magnet reduces to two masses m, on either side of 

 the centre of the sphere, at the distance 2L, the distance from the 

 point to the mass - m being r', the density of the fictive layer is 



m 



(4) 



If this value of o- be developed as a function of the quantity 



2 LR 



we find 



4- 6 4-6. 8.10 



1 



This expression for the density is very different from that of the 

 perpendicular component of the action 



/ 



( 



V 



R - L cos to R + L cos co 



the expansion of which, as a function of the same value of z, has 

 been given above. 



For a magnet of any constitution comprised within the sphere, 

 the superficial layer will be determined by the superposition of the 

 layers o^ relative to all the masses, and, if the magnetisation is 

 symmetrical in respect of the centre, by the superposition of the 

 layers or relative to the magnetic filaments into which it can be 

 decomposed. 



1214. The general problem of the distribution of the fictive 

 layer on the surface S, when we know the internal distribution of 

 the magnetism, is the same as that which consists in calculating 

 the electric layer whose actions are equivalent to those of masses 

 comprised within a surface; it presents great difficulties, and the 

 inverse problem of determining the fictive layer by external actions 

 is of the same kind. 



We observe, in the first place, that the field of a magnet outside 

 a closed surface S, which surrounds it, is completely defined when 

 we know the potential and its perpendicular components at all 

 points of this surface. 



