694 BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, 19021903. 



or 



drp __ 

 ~dr^ 



v a sin F(r) 



t" 



whence 



rp </) - 





If the particle is to approach a boundary-circle, then of necessity, when r, is the radius of this circle 

 (a) (; + y\ r\ - 2r, - (r a v sin or. - />,))* = 



(b) 



, - f t + (r sin - 



= , 



u 1 



where A, is the value A gets for r ?-, . As the charge is assumed to diminish towards , 



/, = 0, 

 whence, according to (b), 



(c) r t = -- - . 



From (a) we then obtain 



- /'i = (^o sin o ^K)) 2 . 



or 



(d) 



= r a v sin 



noting that F(r t ) r v a sin must be <^0, if the motion is supposed to take place in 

 direction. 



From (d) we obtain 



F(r t ) + V^" 7 , < Vo , 



and as F(ri) is certainly > , we obtain a fortiori 



and by the aid of (c), 



noting, from (c), that 



Then 



or 



H 1 





which is absurd. 



Since the particle cannot approach a boundary-circle in a positive direction, it is clear that if 

 does not change to a negative direction, it must either continue to travel, out indefinitely, or with positi 

 direction change from an out-going to an in-going motion. 



