UNIFORM MAGNETS. 305 



The products ly, I/*, Iv are the components of the magnetisation, 

 and adS, /3dS, ydS, the projections of the element of surface on the 

 co-ordinate planes. We have then 



Ids cos 6 = Kdydz + Edzdx + Cdxdy, 

 and the expression for the potential becomes 



i/DA 3B 3C\ 

 (4) V= -</S- - + + \dxdydz. 



ty ^ / 



The formulae (i) and (4) represent the same potential ; if they 

 are identified it will be seen that the surface density, and the volume 

 density of magnetism, may be expressed as a function of the intensity 

 of the magnetisation in the following manner. 



(5) 



(6) . 



Hence, the surface density is the resolved part of the intensity 

 of magnetisation in the direction of the perpendicular to the surface 

 drawn outward. 



The volume density is equal and of opposite sign to the sum of the 

 partial derivatives of the components of magnetisation referred to three 

 axes. 



The quantities p and o-, which represent the densities of a fluid 

 on Coulomb's hypothesis, may be regarded as purely mathematical 

 quantities. They are two symbols defined by the equations (5) and 

 (6) ; for the sake of brevity the name of densities will be retained, 

 without attaching to this word a literal meaning. 



We may observe that Poisson's equation, relative to secondary 

 differentials, becomes in the present case 



For an external point, the second member is identical with zero 

 since the strength of magnetisation is constant and equal to zero. 



320. UNIFORM MAGNETS. Let us consider the particular case 

 in which the magnetisation is uniform, that is to say, in which the 



x 



