124 Prof. A. W. Riicker on the Magnetic 



It will be convenient to throw equation (4) into a form 

 suitable for calculation when the shells are thin and the 

 permeability is great. 



With the same notation as that just employed it becomes 



_ '^ a tt 1 « 2 (N + l) 4 



NV 2 



which, in the case under consideration, leads to the result 



¥/^ = 0-064. 



By (2), if the same amount of material were concentrated 

 in a single shell, we should have 



^/^ = 0*089. 



In this case therefore the advantage of the best arrange- 

 ment is much less than if a larger amount of material were 

 used. 



No very great disadvantage is entailed in such a case if 

 the outer shell be outside its best position, as, if we take 

 l/« 3 = 3, we get 



¥/^ = 0-069, 



which does not differ very much from the minimum value. 



Case of Tivo Shells when the largest and smallest radii and the 

 volume of the material used are given. 



In this case da =dx 3 =0 y and as before 



ill -I-i 



a s a 2 «i «o R 



1 I-i 



where, since a % and a are given, p is a known quantity. 

 Hence substituting from 



Jl = J|in(5) 

 we get 



[?*?{*i*s(«] +«2) — a o a 2 2 }— rf*\*l\ («2~ «s) =0, 



or « 2 2 ( a o ~~ a i) '— a 2 a \ = ~~ ~z a \ a %* 



