Shielding of Concentric Spherical Shells. 125 



But 1 = e+°* 



a 2 pa v ' 



p 2 (a — a ± ) _ p<x x 7]a s 



(P + «i) 2 P + <*l~ ?' 



£p— v*3 

 whence «i is known. 



In the case when /a is so great that £ may be taken =?;, 

 this becomes 



(/? + a 1 ) 2 (p-« 3 )=P 2 («o + />)- 



If the given volume is very nearly =f7r(a 3 3 — a 3 ), 1/p is 

 small. 



In this case the solution reduces to 



. -!-=— 2|— • 



If yu, is large, this becomes 



Hence, if the external and internal radii are given, the 

 best position for a narrow crack within the shell is such 

 that the volume it encloses is the harmonic mean of the 

 volumes enclosed by the inner and outer surfaces of the 

 shell respectively. 



Case of Two contiguous Shells of (liferent Permeabilities. 



It is evident that the question as to the best arrangement 

 of two shells of different permeabilities cannot be very defi- 

 nitely answered. 



If the dimensions of the shells are variable, the more of 

 the more permeable material we use, the more complete will 

 be the shielding. 



If, however, the dimensions of the shells are fixed, under 

 certain circumstances (such as when its thickness is much 

 the greater) it will be better to make the outer shell of the 

 more permeable material, while under other conditions the 

 reverse arrangement may be best. 



Though the problem is not of any great practical interest, 

 it may be worth while to indicate the mode of solution. 



Let /,t =^3=l, and let the permeabilities of the two shells 

 be (a and //, 2 . Then we get 



