684 BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, 19021903. 



The discriminant for the function of the second order of / in (c'| is 



(, t 4. 2)* W 2 _ -2n 3 = 2 + 1 . 



When, in accordance with (a), n ~^> 1 , the discriminant will become > , and consequently there 

 are real values of /, which satisfy (c'). These values are determined by 



Hence it is seen that these values of / are positive. We see moreover that there are always 

 values of / > 2 which satisfy (c'), and then (b') is also satisfied. Hence it follows that by a sir 

 choice of the amount of magnetism and gravitation, initial velocity and angle of expulsion, ,vr can //. 

 obtain an annular formation at any desired distance from the globe, 



It is further seen that for sufficiently large values of n there are permissible values of /, both< 

 and ]> n. 



For l<^n, we obtain 



that is to say 



For / > n , we obtain 



that is to say 



>0. 



The particle can therefore approach a boundary-circle both when the direction of expulsion is 

 positive and when it is negative. 



As this applies to negative particles, it of course also applies to positive particles. 



It might be interesting to see, however, what direction an expelled negative particle will finally take 

 when the globe is so magnetised that ). 



If we assume that 



0, 



din . 



I 10 



the angular velocity *- is negative at the initial moment, and it will then always continue to be s 

 for the change from a negative to a positive revolution-direction, or vice versa, can only take place when 



that is to say when 

 or 



ar + IM = , 



).M r>,, sin a' 



but as this value of r is <^ ;- , no such reversal can take place. 



A positive direction of revolution can thus only take place when a > 0. 



Now we have seen that (c) cannot be satisfied with other than negative values of k, and thus 



we have 



r> sin < IM . 



In order that the particle shall not change from a positive to a negative direction of revolution, it 

 is necessary that the double root r, which is the radius of the boundary-circle, shall be less than 



iAfr. 



Ufr*v a sin cr/ 



