686 



and when 



then 



BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, IQO2 1903. 



Thus is then always less than 4. 



Values of n ^> 4 can thus only be obtained for elliptical velocities, i. e. when 



Then, moreover, / > 3 , whence it follows that 



. Jl 



Hence it will be seen that great values of n can only be obtained for elliptical velocities that ar< 

 very near the parabolic, i. e. when v'l is only a little less than - - . 



133. It might now be interesting to find out whether a negative particle could approach a boundary- 

 circle with positive direction of revolution, if we were to assume that there was a resistance in the 

 medium. We have seen that if there were no resistance, such a motion was impossible. 



When an electrically charged particle moves in the plane of the magnetic equator of a magnetic 

 globe, subject to the magnetism and gravitation from the globe, and moreover a resistance in the medium, 

 we have the following equations of motion : 



d 2 x 



f.M dy 

 ~^~ dt 



(.1 dx 



x ~ m ~ 



where m is the resistance. 

 From this we obtain 



d z y _ IM dx . ^ dy_ 



* = ' *> * y ' m > 



dxd*x . dyd*y ft dr dt((dx\- . fdv\-\ 



*7P+dtWf**- m d S (( l ll) +UJ J' 



f * < fs 



or, if we put . = v , 



(I) 

 We obtain moreover 



dv /LI dr ds 



d 2 x _ A.M dr m ( dy dx 



(II) 



d ( 



- I 



dt\ 



d(D\ 2.M dr m 



_Z. I ^ ----- 



^- _Z. I ^ 



dt 



dt 



m 

 v 



dip 



2 ' 



dt 



Now it is clear that whatever the nature of the resistance may be, it can at any rate be under- 

 stood as a continually positive function of r (possibly multiform, but if the particle were constantly 

 retreating from the globe, it would be uniform). If the particle is able to move in such a manner as 



dr 



to be always retiring from the globe (and approaching a boundary-circle), -. is moreover a continually 



(IS 



positive function of r. For a path such as this then, it should be allowable to put 



