Shielding of Concentric Spherical Shells, 115 



Hence the whole coefficient is negative, or M* diminishes as 

 a Q increases, i. e. as the radius a diminishes. Thus, as we 

 should expect, we obtain the same result as before, viz., that 

 the reduction of the radius of the innermost surface always 

 improves the shielding. In like manner if a 5 increases the 

 shielding improves. 



Let us now suppose, as before, that the outermost and 

 innermost radii are given. The best arrangement is then 

 found by equating the coefficients of dot ly dot^, and du Zj and 

 e?a 4 to zero. 



By eliminating the terms in a 4 and a 5 from the first pair, 

 and also from the second pair of these equations, we get 



a 2 a 3( a l~«2){? a o( a O — «l)<*2 — a l 2 (? a — 17«l)}=0 . (11) 



and 



a 3 (« 2 — ct 3 ) { tjr) (oq — aj) 2 a 2 3 + (f Oq — ^ai) 2 «i 2 « 3 



— ?(?«0— ^«i)aia 2 ( a 2 + «3)K — «!)=0 . (12) 



In the cross multiplications by which these equations are 

 obtained, we assume a 4 — a 5 ^0. 



If a 1 =a 2 or a 2 =a s two of the shells are fused into one. 

 Putting this case aside and substituting for (f«o— V oc i)/(. a o'~ a \) 

 in the second equation from the first we get 



a l«2(?«o — rjoc-i) — £ ot a z (a — a x ) = . 



Hence and from (11), 



jfoo— q«i ___ a 3 a 2 



; r — - — o , • • • • {*■&) 



cc (cc — «i) «ia 2 «i 



a 3 a 1 = a 2 2 (14) 



Again equating the coefficient of da 3 to zero, substituting 

 for (S a o~~V a i)/{ a o mm ' h)) rearranging the terms, and sub- 

 stituting again, we get 



fr-Va. W . = ^ by(14 ),. . (15) 



Next, equating the coefficient of da^ to zero, we get 



£a± 2 —r)a 3 <x 5 _ «2 ifro— *ft*i _ f^p- 

 a 4 2 — a 3 a 5 a 3 a — «j a x 



This again reduces to 



^riK^Tv by(lo) '- • a6) 



12 



