108 Prof. A. W. Rucker on the Magnetic 



Remembering that fi(N+ l) 2 =f — v we g©t 



«0 ^Q^1^2 



This leads to a quadratic in a 2 > of which one root must 

 evidently be a 2 ~a 1 . 

 The other root is 



If then we write I = ((?#o~~' , 7 a i)/£( a o~' a i)> where I is a 

 function only of the dimensions of the inner shell, of the 

 permeability of the material and of the order of the harmonic 

 term considered, we have reached the following conclusions. 



If a 3 l>a 1} the shielding improves continuously as the 

 thickness of the external shell increases from without inwards. 



If a 3 I<« 1 , the shielding at first improves and then 

 deteriorates as the thickness increases. 



When a 2 = a 3 I, the shielding is the same as when the 

 whole of the space between the shells is filled, i. e. as when 



When a 2 2 = a 1 oi 3 l ) the shielding is a maximum, after which 

 it diminishes until a 2 = a i* 



In the particular case when the maximum shielding is 

 equal to that when the hollow space is just filled, the con- 

 ditions a 2 = a 5 l and « 2 2==a i a 3^ must be simultaneously 

 fulfilled. Hence a 2 = a, ly an d there is only one maximum, 

 which occurs when the hollow space is just filled. 



This can only be the case when 



„ 2 2_ «l« 3 (&*0 — V*l) 



"2 — a l w \ > 



£(«<) — a l) 



i. e. y when 



g^! 2 — «i(f «o + 7)a s ) + £ a a 3 = 0. 



It may be worth while to give a numerical example to 

 illustrate these results. 



Let the small magnet be placed in the centre. Then the 

 only harmonic term is that for which n—1 and N=2. 



Let o o 



i.e. a 3 =2a and a 1 = l*260a « 



Let /a=501. 



Then 



S = (Njx + 1) (N + p) = 1003 x 503, 



7 ? = N(^~l) 2 =2x500 2 = f nearly ; 



