364 Permanent Magnetism [OH. xi 



This distribution is of course simply the " Green's Equivalent Stratum " 

 ( 204) which is necessary to produce the observed external field. 



The bar-magnet already considered in 405, provides an obvious illustra- 

 tion of these results. 



415. Uniformly magnetised sphere. A second and interesting example 

 of a uniformly magnetised body is a sphere, magnetised with uniform 

 intensity /. This acquires its interest from the fact that the earth may, to 

 a very rough approximation, be regarded as a uniformly magnetised sphere. 



If we follow the method of 413, we obtain for the value of VQ> defined 

 by equation (344), 



where a is the radius of the sphere. If we suppose the magnetisation to be 

 in the direction of the axis of x, we have 



COS0 



Thus the potential at any external point is the same as that of a magnetic 

 particle of moment |?ra 3 / at the centre of the sphere. 



To treat the problem by the method of 414, we have to calculate the 

 potential of a surface density I cos 6 spread over the surface of the sphere. 

 Regarding cos as the first zonal harmonic P x (cos 0), the result follows at 

 once from 257. 



Poisson's imaginary Magnetic Matter. 



416. If the magnetisation of the body is not uniform, the value of HQ 

 given in equation (342) cannot be transformed into a surface integral, so 

 that the potential of the magnet cannot be represented as being due to a 

 surface charge of magnetic matter. If we apply Green's Theorem to the 

 integral which occurs in equation (342), we obtain 



where I, m, n are the direction-cosines of the outward-drawn normal to the 

 element dS of surface. 



