106 Prof. A. W. Riicker on the Magnetic 



consecutive «'s equal to each other ; and it will be found that 

 in all these cases the expression assumes the same form. 

 If a 4 =-a 5 , remembering that £ — 77=/*(N + l) 2 , 



= ^ « oai « 2/ ^ 2 (N + l) 4 [ . (4) 



=¥D say. 

 Then 



T> 2 d^ ^ 



+ {— ffa*2*o( a 2 — a 3)+^ a l 2 (f«2 — ^3) «0 a 2^i > (5) 



+ 1 f 7 ?^ 2 («0 ~ a l) — ^«l a 3( ?«0 — ^«l) } «0«1^*2 



+ {—£v a 2( a o~- a i) +V a l(^ Oi — V a l)} a Oi l a 2^ a d' J 



As « an( i a z do not occur in the coefficients of da and 

 da B respectively, it is evident that no particular values of 

 either these quantities give a maximum or minimum value of 

 M/* when the other three as are arbitrarily selected. 



Again, the coefficient of da may be written 



rja^a 2 {a % (%a 2 — %a 3 ) — rja 1 (%a 2 — rja d ) } . 

 Now a 2 <a ly and since £ is >rj, 



£<H — f a 3 is < %a 2 — 7]a s . 



Hence this expression is always negative, i. e. W diminishes 

 as ot increases, that is as the radius a diminishes. 



Thus adding the permeable material within the shell so as 

 to reduce the internal cavity always improves the shielding. 



In like manner, by discussing the coefficient of da z , it may 

 be shown that an improvement is always effected by increasing 

 the external diameter. 



If, however, the smallest and largest radii are given, M r 

 may be a maximum or minimum if the coefficients of da 1 and 

 da 2 vanish ; i. e. if 



2 _fa 3 a (a 2 — « 3 ) 



a l £ > 



fa 3 — ija 3 



and 



£K— «i) 



Let us first suppose that a x is also given, i. e. that the dimen- 

 sions of the inner shell are fixed, then the equations enable 

 us to determine the effect of adding another external shell 

 of which the outer radius is also given. 



We may first suppose the external shell to be of vanishing 



