Shielding of Concentric Spherical Shells. 119 



Best arrangement in the form of Tico Shells separated by an 

 Air-gap when the innermost radius and the volume of the 

 material are given. 



So far I have supposed that both the innermost and outer- 

 most radii are given. 



It is evident that we may reduce the number of the inde- 

 pendent variables, and I now proceed to discuss two problems 

 which might occur in practice. The enunciation of the first 

 of these is given above. 



In neither case is the solution very simple, and I therefore 

 confine myself to the case of two shells when the magnet is in 

 the centre. In this case a x — l/a* &c, so that the condition 

 that the volume shall be constant is 



Hence 



« 3 <X 2 a l a 



■- + --- =r say. 



da 3 da 2 doc 1 



«3 «2 a l 



Substituting for da 2 in equation (5) , and remembering that 

 da =0, we get as the conditions for a critical value 



?«2(«o— «i){«2 2 — a 3 2 }— «ia 3 0*2— «3)(?«o— V*i) = . (24) 

 and 



{ — ^«ia («2 — a s) + y a i(&2 ~ V a s) }«i 



+ {£ r ) a 2 2 { (X o— a i)— V a i a s(.^ a o — ^«i)}a 2 = 0. . • (25) 

 From the first of these we get 



(?«<> — 17«l)«l«3 = ?«2(«0-«l)( a 2 + a 3)- • • • (26) 



Substituting for f«o— ^ai in (25), 



(f«2 — V a z) a l = ?«2{«0«l( a 2 — «3)+ a 2«3(«0 — a l)}> 



multiply by « x and subtract, 



.'. a 1 2 (a cx 3 — cc 1 a 2 ) = « 2 { — a o a l( a 2"~ a 3) 



+ (« ~ a i) ( a i a 2 + «ia3-"«2«3) } > 

 .*. a^ccQUs — a Y cc^) = a 2 a 3 (a — a x ), 



or ao a 3( a i""«2) = ai«2( a i— ^3) (27) 



££3 #2 Ctj #0 -tv 



