692 BIRKELAN'l). THE NORWKGIAN AURORA POLARIS EXPEDITION, 1902 IQO3. 



We then obtain 



dz 



for ;- = r. 2 -j- i , 



and it is thus certain that there are values of r ~> >\_ , for instance r t , such that 



dz 



(IX) 

 and thus also 

 or for an arbitrary r, we have 



-, - 



, when 



s for r t\<~. z for r = r, , 



r**- M ^.1 , r-A^ ., 



(*y\ r 2l _L_ I C" (IF ^" f l ' t" I f9^ r ' 



- 1 1-3 - 1 *-^ J 



Jr '" Jr ^ r 



If we then put r r, , we obtain 



f 

 J 



'i M 



We can then, however, find an angle , such that 

 (X) W>-f f'^jWrfr < 



(// 

 Jr, 



We further put 



9n 9i. 



(XI) 



Since r, may be chosen as little greater than >\ as desired, it may certainly be so chosen that 

 both sin a, and v t can be found as positive quantities. 



It is then clear that if we imagine the negative particle placed at a distance r, from the centre 01 

 the magnetic globe, and possessing a velocity v s , forming an angle a, with the radius vector (a, mint 



be chosen between and re), it will then, from without and the positive way, approach asymptotically 



the circle with radius r., . For since v for r = r, has the positive value sin or, , we can prove, as in 

 Case i, that y must remain positive along the entire in-going path. The particle cannot therefore change 

 to out-going motion again for a value r 3 of r, unless v -f- 1 for ; = r 3 ; that is to say 



sn 



but according to (X) we obtain therefrom 



z for r = r 3 equal to s for ; = r, ; 



and that, on account of (IX), cannot be, if r., > r., . 



For the value r = r, , the velocity v will be determined by the fact that 



2" 2 f r - 



v* -.= ,, 2 + -J- - 2 



r * r i Jr, 



f(r}dr; 



but if we compare this with (XI) we obtain 

 ; and v^ were so chosen, however, that 



