BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, IQO2 1903. 



. ir 2 - f s *>* 

 * ***} 



/> s / J 



? 



We have hereby succeeded, in this case, in making all serial development superfluous. 



(2) k | . R is very great. 



If we look at the conditions near the surface, we find that there, too, k ' . Q is 'very great. But 

 from the last expression for % n in equation (4) it appears that for great values of the argument we 

 may put approximately 



r \ 

 J 



= ( I)" I . 3 . 5 . . . (2 



1+1 



From this we find, since 



k = + 27t(l 



when p s is positive, and 



- / p 



k = + 2/r(i + /) r * when / A . is negative, 



after some reduction, that 



~\ e 



//' - \l ?(R - ?) 



v r * 



when / is positive, and 



I 2 7. \ ( 



when ^, is negative. 



The expressions may also be written in the following form, as 



(91 



2 



i, l ~ i**fij(y*V.J _ Ju, i,' ^-**-/ 



r j r^ ^ l /--(A^I/^ ./?-?)+;) 



-in "J' 



where the upper signs are to be employed when p, is positive, and the lower when p s is negative. 

 this case therefore, in order to find the currents at the surface, we need, only make a single serial 

 development of the potential. 



