yo6 



BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, igO2 1903. 



Hereby it is proved, that trajectories do not exist in any plane passing through the centre of the 

 sphere except in the equatorial plane. 



Hence we may conclude, that formation of planets will hardly be possible outside the equatorial 

 plane. If after all a multitude of trajectories could approach asymptotically a common curve outside the 

 equatorial plane, this curve as we have shown could not lie in a plane passing through the centre of 

 the sphere, and as further the particles certainly very soon will lose their charge, they will come to 

 move in the most different directions. The only possibility for formation of planets must be, that the 

 particles approached a common curve lying in a plane through the centre of the magnetic sphere, and 

 this we have proved to be impossible. 



136. Our mathematical investigations have shown as their result that if boundary-circles exist for 

 all the velocities with which material corpuscles are expelled from the central body, the corpuscles will 

 either return to the central body (this being what will happen in the great majority of cases), or the 

 particles will continue to approach nearer and nearer to the boundary-circles. Possibly some velocities 

 may also be sufficiently great to cause the particles in question to leave the system and retire indefinitely. 



Concerning the charge of the particles, we may imagine three cases: 



I. When the particles are not charged. They will then either retire indefinitely, or fall down again. 



II. When the particles are so highly charged that the electrostatic influence dominates that of 

 gravitation. 



III. When the particles carry a charge of medium strength, so that the electrostatic influence plays 

 an important part side by side with gravitation, which, however, is the dominating force. 



If we consider negative particles in case II, we shall easily be able to prove that they can in 

 approach boundary-circles, but the radius of these circles must be < (1 + V2)r . 



The necessary and sufficient condition for the approach of a particle to a boundary-circle in this 

 case is that the following relations shall be satisfied: 



(a) >1 



(b) (/ 1) (/ 2)< 0, or, otherwise expressed, 1 < /< 2 



(c) ]/2n -\- 1 <^ / 



i i 



<0 



M"" (/) 



(2) 



From (b) and (c) we find that 



n 

 that is to say, we obtain in connection with (a) 



(e) !< 



Necessarily, moreover, 



, or 2 2 



