702 



whence 



BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, IQOa 1903. 



Cr 2/i = - , 



or 



r = 



3C" 



If this is compared with the expressions already found for r 2 , we obtain 



or 



or 



- 16/j 8 = 



Let us next consider a value-system R-\-R lt r-\-r\, R and r indicating the boundary-circle. 

 According to Taylor's development, we then have 



a P 3 P 



In an infinitely small region surrounding the point (R, r), in which P0, -^ ,, = 0, =0, 



5/\ c ; 



obtain then 



For rj = we obtain in particular 



Now 



A 2 AT 2 3a 2 



2 S^ a 



Hence it follows that P has negative values as near the point under consideration as might be 

 desired. 



If the discriminant of the quadratic form 



a/? 2 ' v ' n 



is negative, it follows that P must have negative values all over the area surrounding the point under 

 consideration. Then a particle cannot move towards the circle under consideration. The discriminant 

 must therefore be positive, that is to say, 



or 



r 5 



-" 



or, since IMR 2 = ar 3 , 



..- , 2/ 12aiM 21 1.,,.., , 

 T~ ( ^3 ^ 1& >Q 



