126 Prof. A. W. XUicker on the Magnetic 



V{ -« 1 a 2 N(N/t 1 + /^ 2 ) fe-1) (/*i-l) -^^(N + ^G^-^) fc- 

 + cc a 2 (N/h + 1) N (/* 2 - 1) Oi -Ma) + «o«i(Nf*i + 1) (N + m 2 ) (Nfi 2 



=i|r a a 1 (N + l)Vl/*2- 



Hence, if ^ and M^ be the external potential when the 

 permeability of the external shell is fi 2 and /^ respectively, 

 we get, by exchanging the positions of the symbols ^ and fx 2 

 in the above expression and subtracting, 



^« oai (N + l) 3 MiMa 1 ^r - ^r J 



= (Mi-Ma) [-«i 2 N{2/^ 2 + (N-l)(^ 1 + / x 2 )-2N} 

 -« 1 (« 2 +«o)N(N-l)(^-l)(/* 1 -l) 

 + « a 8 N{2N^ 1 A4 8 -(N-l)(^ 1 + /4 8 )-2}]. 



Hence the exchange of materials causes no difference if either 

 fjbi=jjb 2 or if the second factor on the right-hand side vanishes. 



If both the shells are thin let a 2 =a i + t 1 , and a =a 1 —t 0) 

 where the squares of t x and t 2 may be neglected. 



Then 



a 2 =( % + g-^ +1 > = ajL |l-(2^ + l)|} 



ao =(a 1 -^)- (2ra+1) =ai{l+(2 ? z + l)|}. 

 Substituting these approximate values, the second factor 



- (2n + 1)N(N + 1) (w*- 1) fe- * )/«i, 

 which proves that, as we should expect, if the shells are thin 

 the order of their arrangement is indifferent. 



Case of Three Shells of different Permeabilities. 

 The general relation between ^ and yfr when an internal 

 space of permeability /ul is surrounded by three concentric 

 spherical shells of permeabilities Mi> Ma? /a 3 , and when the 

 permeability of external space is /a 4 , is 



^ Q «o«i a 2 • MoMi^aM 3 (N + 1) 7^ 



= a 1 a 2 a 3 N(/X3— fi 4 ) (Mo" Ml) (^Ma + Ms) PVi + Ma) 

 + a 1 a 2 2 N(N^ 4 + /i 3 ) (m 2 — Ms) (Mo— MiK^Mi + Ma) 

 + ai 2 a 3 N 2 (^ 3 — ^ 4 )(ft 2 -^ 3 )(/^i— ^ 2 )(^ — /^i) 

 + «i 2 a 2 N (N/* 4 + Ms) ( N Ms + Ma) (Ml — Ma) (Mo — Ml) 

 + « a 2 a 3 N(N^ 2 + /^ 3 ) QSfii+fio) (Ms" Md) (Ml — Ma) 

 + a a 2 2 N(N/^-i-/* 3 ) (Nmi + Mo) (^a-^s) (Mi~Ma) 

 +«o«i a 3 N (^a+^i) ( N Mi+Mo) (M3-M4) (Ma-Ms) 

 > +«o«i«a(NMi + Mo) C^Ma+Mi) C^M3 + Ma) (Nm 4 +M3)« 



