Shielding of Concentric Spherical Shells. 

 or to a closer approximation, 



7? = f(l-O009). 



109 



Hence 



1-^0^ = 1+0009. 



?( a o-«i) 



By substituting in (4) we get the values of ^/^r given in 

 the following table for different values of a 2 . The volumes 

 are expressed in terms of the space enclosed by the innermost 

 surface. 



o 2 = 



a 2 = 



Volume of 

 outer 

 shell. 



Total 

 Volume. 



*Mr 



a 3 = u /8. 

 r009a 3 =0-1261a . 



3a 3 /2=3a /16. 



2a 3 = a /4. 



3of3=3of /8. 



a x =4:a 3 =a /2. 



a 3 =2a . 

 0-997a 3 =l*994a . 

 0-873a 3 = l-746a . 

 0-794a 3 =l-588a . 

 0-693a„ = l-386a . 

 0-630a a =l*260a . 





 0-07 

 2-66 

 4-00 

 5-33 

 6-00 



1-00 

 1-07 

 3-66 

 5-00 

 6-33 

 700 



0018 



00102 



00007 



0-0006 



0-0009 



00102 



Turning next to the case when a, 1 and a 2 both vary, we 

 may determine the maximum shielding for given external 

 and internal radii. 



The conditions to be fulfilled are : 



(6) 



— £a 2 «o(«2 — «3) + a iX? a 2 _ V a s) = 



Adding these we get either 



a 2 = «i or a 2 <x 1 = a <x s . 



The first has reference only to the point at which the 

 maximum value is equal to that produced when all the space 

 between the shells is filled. 



The latter leads to the equation 



£ a 2 B (a 2 — a 3 ) — a 3 2 « (jj a 2 —T)a 3 ) — 0. 



This equation is a biquadratic in a 2 . 



Since the expression is positive when a 2 =— oo and nega- 

 tive when ot 2 = a 3} there is at least one real root <a 3) with 

 which, since a 2 >a 2) we are not concerned. 



