OP ARTS AND SCIENCES. 349 



for a boundary. Let, now, the surface density of such a spherical 

 magnetic shell be (p, its radius be a, and let there be a unit naagnetic 

 pole R at distance r from C, the ceutre of the sphere. IMoreover, 

 let dS be any small element at Q, and at a distance h from H. Let 



p = J. Let be the value of the flow, or the surface integral, at the 



point Q. Then the mutual potential of the magnetic shell and a unit 

 pole placed at li will be 



— f'L9 

 J da 



But, since h is the distance between the element and the point R, 



= ff^dS = fpc^dS; 

 whence by substitution we find 



But 7? is a homogeneous function of the degree — 1 of the radius a 

 and of the distance r from JR, which gives the conditions, 



dp, dp 



a a ' dr '■ 



or, 



'^ = _ ^ f » _1_ r '^'1 = — - ('^^'^ 

 da ay~'drj a\di- 



substitutinof, we have 



- = -fft''-t'-'^- 



But if T^is the potential at E due to the small element of the shell dS^ 

 we shall have 



therefore 



V=JJcl>pdS; 



Id(rV) , . 



a dr ^ 



which is the expression for the potential due to a magnetic shell of 

 unit strength. 



