116 Prof. A. W. Riicker on the Magnetic 



Again, since a 2 2=:a 3 a i> 



But 



«3 a 5( ?«0 — l?«l) = f a / ( a — a l) > 



or 



?«o ( a 3 a 5 — a 4 2 ) = «i(^a 3 a 5 — £a 4 2 ) . 



Multiplying this by (17) and putting a 2 2 = a 3«i> 



«3(f«4""^ a 6) ( a 3 a 5 — «4 2 ) =a 4 ( a 4 — *fi) (W*fi — f a 4 2 )j 



.'. (a 3 ~ a 4 ){a 3 a 5 (?«4 — W>) — fa 4 2 (« 4 — « 5 )}=0. 

 If a 3 =a 4 the problem is reduced to the case of two shells. 



(18) 



If « 3 t£« 4 , 



£« 4 — 77« 5 « 4 



f(« 4 — « 6 ) « 3 a 5 



Hence the equations (13), (15), (16), and (18) finally 

 reduce to 



|«o—^ai _a 3 a _«2 a o_ a 4 2 _«o«4 / 1Q \ 



^(«0 — «lj «i« 2 a l a 3 a 5 a 2 



It must be remembered that in obtaining these equations 

 we have thrown out the alternative cases where any two con- 

 secutive values of a become identical ; i. e. cases in which 

 the three shells become two, either by fusion or by the dis- 

 pearance of one. 



Thus the solution for the case of two shells is contained 

 in the general equations for three shells, but the equality of 

 the above groups of ratios holds good only when there are 

 three. 



It remains to assure ourselves that in the case of three 

 shells all the variables give minima simultaneously. 



Thus, considering the coefficient of da, x the coefficient 

 of «! 2 is 



— ^ 4 (f« 4 — f« 5 ) (v^—v^s)- 



Comparing these term by term, we see that the terms are 

 either equal or are less in the second expression. Hence 

 the coefficient of a ± 2 is positive. If, then, a x is a little less 

 than the critical value, the coefficient of da 1 must be negative. 

 Hence, as a, 1 increases up to the critical value ^ diminishes, 

 i. e. the critical value of a x corresponds to a minimum. 



The other coefficients if treated in the same way lead to 

 the same conclusion. We may now proceed to put equation 

 (19) into the simplest form for calculation. 



