98 Prof. A. W. Riicker on the Magnetic 



more convenient to make the subscript numbers refer to the 

 shell than to the order of the harmonic. 



If, therefore, there are m shells, let the radii of the surfaces 

 of separation be a , %, a g , &c. a m . Then a is the radius of 

 the central space, a 1} a 2) &c. are the external radii of con- 

 secutive shells (see figure). 



Let /ul , fjb 1} . . . . /jb m be the permeabilities of the enclosed 

 space and of the shells, and for generality let M be the per- 

 meability of external space. 



Let the coefficient of P n in the expansion of the potential 

 in the pih shell be 



Then at the boundary between the pth and the (p -f l)th 

 shells (of which the radius is a) the following conditions 

 must be fulfilled : — 



and 



=F f+ H +I «?- 1 -(" i H)t H /<? +8 }; 



or if we write 



N=(n+l)/n, <*,=*;<*+», 



we get ^ + ^^-^+1-^+1^ = 0, 



and ^A" N ^A~^+i^+i + N ^+i a A+i = - 



If we suppose, for the sake of symmetry, and without refer- 

 ence for the moment to the expression of physical facts, that 

 the coefficients of P n in the potentials in the central enclosed 

 space and in external space are 



<$>^ + ^r /r n +1 and ®r n + W/r n+1 



respectively, we have in all 2m + 4 quantities to deal with, 

 and since the m shells have m+1 bounding surfaces, there 



