Shielding of Concentric Spherical Shells. 99 



are 2m + 2 equations between them, all of which are linear 

 and of the types given above. 



We thus get a series of equations as follows : — 



Q-^-^a —d> —yjr a = 0. 



1 m rm T m m 



M«|)-MN?a; —Lb 6 +LL Nty cc = 0. 



m imTm ' r~m T m m 



&c., &c. k 1 ) 



$2 + ^2 a l — $1 — " ^l a l = 0. 



/*i<£i — ^iN^o — ^0</>0 + /*oN^o a o = 0. 



In the case of shielding an enclosed space the potential of 

 the inducing forces can be expanded within and in the neigh- 

 bourhood of the shells in terms of positive powers of r, and 

 since the potential of the induced magnetization must be 

 finite when r=0, this also is included in <j> . Hence 

 yfr = 0. In external space the expansion of the potential 

 contains both positive and negative terms, the former due 

 to the inducing, and the latter to the induced magnetization. 



As the inducing forces are given <I> is known. 



As ^0=0 and <I> is given, the number of unknowns is 

 reduced by 2 and is equal to the number of the equations. 



If, on the other hand, the inducing forces are produced in 

 the central space within the shell, their potential must be 

 expanded in inverse powers of r if the series is to be conver- 

 gent at the boundary of the enclosure and beyond it. Hence 

 ^ is known. In external space the terms due both to the 

 induced and inducing magnetizations are expressed in nega- 

 tive powers, and are included in M*. Thus <E> = 0. 



Now <f> Q l& is the ratio of the shielded to the unshielded 

 field in the first case, in so far as it depends on the potential 

 term under consideration, and M^/tyo is the corresponding- 

 quantity when the forces are produced within the shell. 



The relation between these may now be found. 



Putting ^o = and writing A for the determinant formed 

 by the coefficients of the <£'s and i/r's in the above equations, 

 except those which occur in the first and last columns, we 

 have : — 



H2 



