114 Prof. A. W. Rucker on the Magnetic 



Thus the limiting thickness in the case of the first three 

 harmonic terms varies only from 6*9 per cent, to 6'6 percent, 

 of the radius, and the shielding factor from '058 to *055. 



If there are three concentric shells, hy differentiating the 

 expression for M* in equation (3) we get 



T/r 0i L6 3 (N + l) 8 a a^fB s a 8 O4 



+ [^(tfci — *7«s) { (£ a 2 — V a s) a \ — £*tfH(*2 — a z) ) 



+ f^« 4 ( a 4 - «s) ( (*7«2 - f^X** — W( a 2 — a 3) } ] — 



Oil 



+ |fy«4 (<*4 — a &) \ (f «0 — ^l) a l a 3 — ^ a 2 2 («0 — «1 ) ! 

 + [^ a 3 2 (? a 4 + ^ a 5){ a l(? a — ^l) — f«2(«0— «l) } 



+ gqownfa — «s) {^2(^0 — «l) — a l(?«0— ^ a l)}] — ~ 



<z 3 



+ [^ a l(? a — *7«l) {£ «4 2 ( a 2 — «3) — a 3 a 5(f«2 — ^3) } 



+ IWjC^O — a l) { (f «3 — V^)"* + V^^( a 2 — «3) } ] * 



a 4 



+ D>7 a l(?«0 — ^ a l){?«4(«2 — «3) + a d(£< x 2 — V< x 2)} 



+ {fo«a(*0 — «l) ( (^ a 2 ~ ?«3)«4 — ^( a 2 — «3) a 3 } ] ^»5» 



We note that in this, as in the previous case, there can be 

 no maximum or minimum if « or a 5 are independent 

 variables. 



Further, writing the coefficient of da in the form 



1?«ia 4 (f«4 — f«5) {V"l ( a 2 — «3) — «2(*7«2— fe)} 



if we remember that the «'s diminish from « x to a 5 and that 

 f is > 77, it is easy to see that the coefficient is the difference 

 of two positive quantities of which the second is the larger. 



For y _. _. 



and 



{ai(f«2 — 1?a 3 ) — f«2(«2 — «3) — {^«l(«S — «s) — «2(*? a 2 — ? a 3)} 



which is positive. 



