690 



BIKKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, igO2 1903. 



Let us now compare the value of v for a point on the in-going orbit, with the same value of r 

 as for a point on the out-going. 



If there had been no resistance, we should have had 



but, owing to the resistance a diminution of the kinetic energy, or of the velocity, has in the mean time- 

 taken place, so that 



Consequently 

 and then also 



If now 

 then 



that is to say, 



and as 



it follows that 



a <^ 



drl , 



V' 



dy dy 

 drin dr m t ' 



From this it evidently follows, that the inequality 



J'in 



in \rV /out 



in == J'out i 



must take place for lesser values of r than the one in question. Now for the value of r for which the 

 change to in-going motion takes place, is of course 



and thus the inequality 



occurs for all smaller values of r. As, moreover, y- m always remains < 0, this means that the particle 

 returns to the globe again along a steeper path than the out-going. 



This proof holds good, if -j- is always > 0. But we may prove that the result is true even il 

 dy " r ont 



were < for certain values of r, at any rate if the outgoing orbit has no point of inflexion. We 



will not here, however, go farther in the discussion of this problem. 



Case 2. The direction of the path of the particle is negative at the moment when the change ti 

 in-going motion takes place. 



In this case it is certain that the resistance might be of such a nature, that the direction of the 

 path of the particle during its in-going motion, changed to positive. Let us suppose, for instance, that 

 for a moment it is suddenly subjected to a very great resistance at the point at which it changes 

 in-going motion. The path will then at first very nearly coincide with the radius vector inwards, but 

 then, as the velocity increases, the magnetism will deflect it in a positive direction. 



It might therefore possibly happen that the particle in this case would approach a boundary-circle 

 the positive way. 



