Shielding of Concentric Spherical Shells. 127 



If now we put /^ =yLfc 2 =^4=l, tliis reduces to the case of 

 two concentric shells of permeabilities ft, and ft 3 respectively 

 in air and separated by an air-gap. The equation then becomes 



-*/r a a ia2 . /^ 3 (N + 1 ) 4 /^ 



= - a ia2 a 3 N 0*3 - 1) (ft { - 1) (N + ft 3 ) (Nft, + 1) 



+ ai * 2 2 N (N + /*,) Oh - 1) (^ - 1) (N^ + 1) 



+ ai 2 « 3 N>3-l) 2 0"i--l) 2 



- ai 2 a 2 N(N+^ 3 ) (N^s + 1) (/^i-l) 2 



+ « « 2 a 3 N (N + /* 3 ) (N/^ + 1) fag - 1) (ft! - 1) 



-^/^(N+^CN^ + l)^-!)^!-!) 



-« « ia3 N(N+^) (N^ + l) (/^ 3 -l) 2 

 + « a 1 a s (N/^ + 1) (N + /*i) (N/x 3 + l) (N+/* 3 ). 

 If we write 



ft = (Nft + 1) (N + ft ), ft = (Nft + 1) (N + ft), 

 „=Nfo-l)«, % =N(ft-l)^, 



P=N(ft-l)(ft-l)(N+ft)(Nft + l), 

 the equation reduces to 



^ a a 1 a 2 ^ 1 /i 3 (N + 1)*V 



= «1 (?3 a 2 — W3) (?1«0 — % a l) — P«2 (<*0 — «l) («2 — « 3 ) > 



which, if we proceed to put fti = ft s , further reduces to the 

 expression previously obtained. 

 Differentiating, 



DW 



^ ^ 3 (N + 1) 4 



= {P« 2 a 1 (a 3 — « 3 ) — ^i 2 (| 3 «2— V^s) }oe>ia 2 da 



+ { — P«2« («2 — a s ) + ^i«i 2 (f 3 «2 — V&3) }ot a s da 1 



+ {Pa 2 2 (a — «i)— ^ 3 aia 3 (?i a o — *7i«i) }«o a l^ a s 



+ { — Pa 2 ( a fl — «i) +V a i(Si a o—Vi a O}o^iot2da 3 . 



If the inner shell and the external radius are given, the 

 best value of « 2 2 is deduced from the equation 



2 *?3 «1«3 /£ v 



JT OJq """" £*]^ 



Comparing this with equation (6) we see that the new 

 value of a 2 is greater than that obtained when the permea- 

 bility of the external shell is the same as that of the inner 

 one if 



M32) 



