110 Prof. A. W. Riicker on the Magnetic 



As regards roots >a 3 , it is evident, since the whole ex- 

 pression is positive when « 2 = go ,that there is at least one real 

 positive root >a 3 . Also, as %a 2 — rja z is positive, that there 

 can only be one positive root >a 3 . 



This root must be <a l9 i. e. since a 2 a i = a o a 3j a 2 2 < a o a 3J ^ e > 

 the expression must be positive when cc 2 2 = a a 3 . 



This is equivalent to the condition 



f V«o( V«o— v / «3)> V«3(f V«o— *? V^)- 



As a >« 3 and J V*o — £ v ' a 3<? s/^q—V Va 3 , 



this equality may be, but is not necessarily, satisfied. 



Before discussing these equations further, it is convenient 

 to modify their form. 



Let 



- = -=X; i? = f(l-e); - = L. 



a l #3 a S 



Then from (6), 



X 3 (X-l)-L(X-l-fe)=0, .... (7) 



and from (4), 



W„ - f*'(N + l)«X'L . 



/To ~ f{(X-l + e) 2 L-(l-e)\ 2 (X-l) 2 }' {la) 



or from (7), 



/r ° f (X-l) 2 {X 4 -L(l-e)} ~ (X-l) 2 {X 4 -L(l-e) ' [) 

 since jx(N + l) 2 = f — rj = e^. 



In equation (8) SI* becomes infinite when X=l and when 

 X 4 =L(l-e). 



The first of these assumptions is inconsistent with equa- 

 tion (7), and by substituting from \ 4 =L(1 — e) in (7) we get 



(X~l + e) 2 =-e(l-e), 



which is only possible if either e or 1 — e is negative, which 

 they can never be, as £ is always >?/, and r) and £ are always 

 positive. 



In the case when the permeability is very great, f = ^=Njx 2 

 nearly and e may be neglected in equation (7), so that X 3 = L. 



If, however, X— 1 is very small, i.e. if the shells are very 

 thin, the term in e may be comparable with the others, and 

 the approximation is not legitimate. In cases where it may 



