Shielding of Concentric Spherical Shells. 101 



the external space are the same, the ratios of the shielded to 

 the unshielded fields are the same for each harmonic term 

 whether the shielded field be external or internal. 



The most interesting application of this proposition is to the 

 first harmonic term. 



In this case the cf> () and <& are coefficients of terms of the 

 form rP lt that is the field is uniform; and yjr and ^ are 

 coefficients of terms of the form VJr 2 , that is, the field is pro- 

 duced by a small magnet placed at the centre of the shells. 



Hence the shielding effect on external space when a small 

 magnet is placed at the centre of the shells is the same as the 

 shielding effect on the enclosed space when the shells are 

 placed in a uniform field. 



In what follows I shall for the most part suppose that the 

 magnetic forces are produced within the shells ; but the 

 above result enables the conclusions arrived at to be applied 

 to the case of shielding against external forces. 



(1) Case of a Single Shell when the Permeabilities of the 

 Internal and External Space are Unity. 



This case is well known, but may be included for the sake 

 of completeness. 



Since /4 = M = 1, the equations (1) reduce to 



^-%^^ + N^ = 0, 



<£o + ^o«o — <t>i — ^i«o = 0, 

 <f) — Na ^ — fi^ + JST^ao^! = 0. 



Hence ^A/^ is the ratio of the shielded to the unshielded 

 potential or, since this ratio is everywhere constant, that of 

 the shielded to the unshielded field, in so far as it depends on 

 the term under consideration, is 



«oMN + l) 2 ;\ m , 2 x 



(N Ml + l)(N + ^0-^(^-1)*' ' " 



If the unshielded field is due to a small magnet placed at 

 the centre of the sphere, the only term in the expression for 

 the potential is A ^ . „ 



Hence 



n = 1, N = (n+ l)/n = 2, cc = a-<*H-» = a~ s , &c. 



