120 Prof. A. W. Riicker on the Magnetic 



or i \ t \ «o«i«2 a 3 

 (a — « 3 ) a^ — (« x — a 2 )<a a 3 = g 



•\ by (27), . / x, 



o^-fao— « 3 — («j— « 3 )} = 



R 



... «o-«i=^- 3 (28) 



Hence 



«i = «o(l-§). 

 By combining this with (27), 



« 2 {«o(l-S) 2 + a i}=^a(l-g); • (29) 

 also from (26) and (28), 



and substituting for a 2 from (29) , and dividing by 1 — ^, 

 this becomes 



{?-,+!} {«o<l-l) 8 + ¥(l-l)} 



-i^{ 2 -s}-#=°- • • • ( 3 °) 



The coefficient of a 3 4 in this expression is negative when 

 a 3 = — oo . 



It is positive when a 3 = 0, and negative when a 3 =co . 



Hence the equation has a positive root. In order that the 

 solution may be applicable, this root must be such that 



111 

 > T? ; 



«3 a "> 



i. e. the space enclosed between the inner boundary of the 

 inner shell and the outer boundary of the outer one must be 

 greater than the volume of the material used. The change 

 in the sign of the expression from positive to negative as 

 a 3 increases must therefore take place when 



1 ^ 1 l 



Ct s (Xq SX 



or the expression must be negative when 



1 = ± + i. 



a 3 olq Iv 



