PART II. POLAR MAGNETIC PHKNOMENA AND TKRRKLLA EXPKRIMENTS. CHAP. VI. 685 



It will be seen, however, that this is equivalent to 



As we further have, in this case, 

 e can suitably put 



nil then obtain 



(d) HI > . 



The condition (c) assumes the form 



(in 1 ) 2 //- + (2 nft 4 m) n + 3 m 2 ^ . 

 Hence it follows that 





This, in connection with (d) then gives 



K 2 4- 2n + n^2n+ 1 



w z + 2 + 3 

 ,-hence 



;/ 4 + 2 3 + 3 2 <0, 



hile at the same time ]> 1 , which is absurd. 



Tin' particle itinsf thus change to negative direction of revolution before it approaches the boundary-circle. 



Let us return for a little to the equations 



It follows from these that 



If 



icn 



or 



lat is to say, a velocity which, if gravitation acted alone, would remove the particle infinitely, thus a 

 vperbolic velocity. If on the other hand, the initial velocity is hyperbolic, / cannot have other values 

 lan between 2 and 3 ; for if / ^> 3, then 



<" l 



id consequently 



g _ l _ < ,-- g 



It will further be seen that for these hyperbolic velocities, n cannot be greater than 4; for it 

 Hows from (c') that 



/-I - V2(7^T) <; ^ / 1 + V'2(/- 1) ; 



Birkcland. The Norwegian Aurora Polaris Expedition, 19021903. 87 



