CHAP, v.] ELECTRICITY AND MAGNETISM. 271 



shell at a point near it is equal to the strength of the 

 shell multiplied by the solid-angle subtended by the shell 

 at that point; the " strength " of a magnetic shell 

 being the product of its thickness into its surface-density 

 of magnetisation. 



If o) represents the solid-angle subtended at the point 

 P, and i the strength of the shell, then 



V P = <oz. 



Proof. To establish this proposition would require an easy 

 application of the integral calculus. But the following geo- 

 metrical demonstration, though incomplete, must here suffice. 



Let us consider the shell as composed, like that drawn, of 

 a series of small elements of 

 thickness t, and having each an 

 area of surface s. The whole 

 solid -angle subtended at P by 

 the shell may likewise be con- 

 ceived as made up of a number 

 of elementary small cones, each 

 of solid -angle tu : Let r^ and r. 2 

 be the distances from P to the F j 



two faces of the element : Let 



a section be made across the small cone orthogonally, or at 

 right angles to r v and call the area of this section a : Let the 

 angle between the surfaces s and a be called angle (3 : then 



s = -2-g. Let i be the " strength" of the shell (i.e. = its 

 surface-density of magnetisation x its thickness) ; then y = 



surface -density of magnetisation, and s = strength of either 



pole of the little magnet = m. 



_ T .., . area of its orthogonal section 

 Now solid angle c6 = g- 5 > 



therefore a =. & r 2 , 



61* 

 and s - 



cos ' 



u , >* 



Hence ui =. m. 



t cos p 



