Shielding of Concentric Spherical Shells. 107 



thickness, in which case a 2 =a 3; and then allow a 2 to increase, 

 i. e. the inner radius of the outer shell (a 2 ) to diminish. 

 If a 2 = cc s the coefficient of da% is negative, for a z <a x and 



Hence the addition of a thin external shell reduces M*, i. e. 

 improves the shielding. 



This improvement will reach a maximum when 



a 2_ a^3(£^o— V<*i ) . 



but in order that this may correspond to a physical reality we 

 must have 



The first of these inequalities when combined with the above 

 equation is equivalent to 



«if(«o— #i)>«3(f a o— V^l)' 



As a x is >a 3 and J(a — a 1 )<fa — 7;a 1? this condition may 

 be, but is not necessarily, fulfilled. 

 The second is equivalent to 



which is always true. 



Hence we conclude that if the external radius of the outer 

 shell is so chosen that 



there will be a maximum value of the shielding, while an air- 

 space intervenes between the two shields. There will not be a 

 maximum if a 3 is greater than this limit. In the case of the 

 first harmonic the condition for a maximum becomes 



a d >ai 



l?Oi 3 -«o 3 )J ' 



where a , a 1? and a z are the radii of the corresponding surfaces. 

 If there is such a maximum it is evident that for some 

 thickness of the external shell less than that which gives the 

 best result the shielding must be the same as if the whole of 

 the air-gap were filled. 



To find this value of a 2 we have to put a 2 = u 1 in the 

 expression for M/ 1 , and equate the result to the value of ^ 

 when « 2 is not=«!. 



