PART II. POLAR MAGNETIC PHENOMENA AND TERRELLA EXPERIMENTS. CHAP. VI. 693 



We thus have y = 1 and -r- = for r=>: 2 ; but this means that the particle is asymptoti- 



clr 



lly approaching the circle with 'radius r. . 



In order to obtain the fulfilment of the condition 



gr> for r=r t , 



i is only necessary that f(r t ) shall satisfy the relation 



It this, it is evident, can be done in an endless number of ways, if/\r) is always to be negative. 



The only remaining question is, then, whether the particle expelled from the globe can come to 

 i >ve in this manner. We have not yet succeeded in finding a complete solution of this problem ; but 

 \; have found that the resistance must be so great that the velocity must be diminished during the 

 ijoing motion, in spite of gravitation. 



134. We will now see whether a negatively-charged particle with positive direction of revolution 

 en approach a boundary-circle, if we imagine the charge decreasing to 0. 



The equations of motion for an electrically-charged particle in the plane of a magnetic globe's 

 tuator, influenced by gravitation and magnetism, are 



dP r 3 dt~T~ r 3 * 



d^y _ _ IM dx /( 

 W ~^*~dt + r* y - 



We will now imagine the charge to be variable, in such a manner that it diminishes towards 0, 

 if the length of path increases infinitely. We can then make A equal a function of r, but this will of 

 c irse be multiform if the particle should anywhere change from out-going to in-going motion or 

 v e versa. 



We obtain, in the same way as before, 



al 



By putting 



\\ obtain therefrom 



G)" = " + 2 '' C -7) -? (r v sin a - 



a:l by dividing by 



\\ obtain 



(r t v t sm a, /'Wj 1 



Birkeland. The Norwegian Aurora Polaris Expedition, 19021903. Ss 



