Shielding of Concentric Spherical Shells. Ill 



be applied we get from equation (8) the very simple result 



*m- N y (\-i) 3 -(Vl-i) 3 ' 



In this case also the four quantities a , « 1? a 2 , « 3 are in 

 Geometrical Progression. 



If «! = «£, a /a 3 =\ 2 = L, so that (7) reduces to 



(VL-1) 2 =*> (9) 



of which the root greater than unity is \/L=l + s/e. 



This furnishes a limit below which the equatioDs do not 

 apply. 



Below the limiting case lamination must be disadvan- 

 tageous. 



If we remove a thin slice either from the exterior or from 

 the interior surface of a single shell, the shielding power is 

 impaired; and if the shielding does not attain a maximum 

 when this spherical crack is supposed to traverse the shell 

 from inside to outside, it must always be less than if there is 

 no crack. 



Next, comparing the shielding effect of a single shell with 

 that of a double shell of which the innermost and outermost 

 radii are the same as that of the single shell, if we write in 

 equation (2) a s for a 1} and put ?; = £(1 — e), we find by com- 

 paring with (4) that the single shell will shield best if 



a 6 a^a^e 2 



-a 3 (l — e) — (1 - e)a 2 (a —a 1 ) (a 2 — a 3 ) -\-a 1 \a 2 —a z {l — e)}{a — a 1 {l — e) \ 



which reduces to 



\a 3 / V*! / 



<€. 



Now if we diminish a 1} i. e. increase the outer radius of 

 the inner shell, a / a i is increased ; and in like manner by 

 diminishing the inner radius of the outer shell « 2 /a 3 is in- 

 creased. Hence the largest value of the left-hand side of the 

 inequality is attained when a 1 = a 2 , so that if « 2 /a 3 = X 1 , and 



OCq/<Xi^=\j A. A-i =: ±j. 



Thus 



(X 1 -l)(X-l) = ^-l)(X-l), 



of which the largest value is ( yX — I) 2 - 



If e is greater than ( s/h — l) 2 , lamination is injurious, the 

 limit thus fixed being the same as that previously found. 



