Shielding of Concentric Spherical Shells. 117 



Since 



02 _ «1 

 «3 a 2 



and since 



a 2 a _ a^ ^ °^__ C H. 



«1 2 «3 a 5 ? ' «l""«3' 



so that we have three independent equations, viz., 



— = — = \say, 

 and }>.... (20) 



«1 «2 a 3 



together with 

 also 



= — = — = V say, 



«2 «3 a 4 



(jX-^)V = f(X-l)\; (2D 



'° =X 2 X /3 = L say as before. 



« 



So that equation (21) becomes 



(f\-*)L*=f(X-l)X* 



(\-l + e )L*=(\-l)X* (22) 



If \'=1 all the shells fuse into one (without passing 

 through the intermediate stage of two shells) . In this case 

 X 2 = L. 



Substituting in (22) we get 



( VL-l) 2 =e, 



which is the same as (9) . 



Thus the ratio of the outermost to the innermost radius 

 below is the same for two and for three shells. 



The maxima of advantage in the two cases are independent, 

 and to determine between them their magnitudes must be 

 compared. 



Substituting in (3) from equation (20) and simplifying by 

 means of (22), we finally get 



^ e 3 L 



3T 4 



■6 



^ " (\-l) 4 {\# — Li(l-e)}' 

 where \ is given by equation (22). 



(23) 



