128 Prof. A. W. Rttcker on the Magnetic 



L e. } if (to - 1)(N + k) > O^i - 1) (N + ft) , 



i, e., if to > to* 



In like manner, if the outer shell and the internal radius 

 of the inner shell are given, the best arrangement requires 

 that 



2 _ P<* 2 g (« 2 — * B ) 



«1 



%(?3«2 — VdPs)' 



which is greater than the corresponding value when the 

 permeabilities of the shells are the same as that of the inner 

 shell if 



P > * 



^7l(03 a 2 — V&3) Sl^ — Vl a S } 



i. e. } if a 2 ?i(P— V1Z2) >«3*h(P- fi^s)- 

 But 



P-^i?3=N(/, 1 -l)(N + / ,3)(N + l)( / , 3 -^ 1 ), 



P-fi^=N(^ 3 -l) (Njh + 1)(N+ 1) (to-^z) ; 



and therefore the two sides of the inequality are always of 

 opposite signs. 



Hence, if the permeability of the outer shell is increased, 

 the values of oc 2 and a x corresponding to the best arrangement 

 under the circumstances supposed increase, i. e., the outer 

 shell increases and the inner shell diminishes in thickness. 



In the case, therefore, of material such as iron, if the field 

 within the outer shell is such that the permeability of the 

 outer shell is the greater (which will generally be the case 

 when the enclosed space is protected) , it will be better to 

 make the external shell relatively thicker than the above 

 calculation would indicate. Conversely, if the inner shell 

 has the higher permeability its thickness should be increased. 



There is no difficulty in eliminating a x and a 2 in turn from 

 the relations obtained by equating the coefficients of dcc x and 

 da 2 in (32) to zero, and thus finding the best values in any 

 given case for which the equations can be solved by trial. 



Summary. 



In conclusion it is perhaps desirable to summarize the principal 

 results of the foregoing discussion. In doing so I shall con- 

 fine myself exclusively to the corresponding cases of a small 

 magnet placed in the centre of the shells, and of shells placed 

 in a uniform field, and thus the interpretation of the symbols 

 is valid for these cases only. 



The results depend upon two quantities, viz., the ratio of 

 the outermost to the innermost radius of the series of shells. 



