79 8 



BIRKELAND. THE NORWEGIAN AURORA POLARIS KXPEDIT1ON, IQO2 1903. 



In order to obtain a general idea of the course of the earth-currents induced by the rotation of 

 the earth in relation to a system such as this, we have made a calculation of this current-system upon 

 the assumption that the equatorial system can be replaced by an infinitely long, rectilinear current 

 situated outside the earth in the plane of the equator. 



For the potential of a current such as this, that flows at right angles to the XZ-plane, and inter- 

 sects it in the points x = x\, z = zi, we have, as is well known, the following expression: 



-i a 3j _ i o cos 6 z\ 



V = i . tan - = i . tan 



x #i o sin 6 cos w , 



(45) 



where the direction of the current is reckoned positive when it coincides with the direction of increasing y. 

 If we say that the rotation-velocity equals w, and further, for the sake of brevity, that 



a ^ i sinft) sin 



b = Xi sin 6 cos w 



c = cos 6 



d = sin w sin B 



cos 6 



(46 



we find, after some reductions, that 



3V 



and 



W .c.d( , 



->T-'V{ ? + 



zac 



log nat 



L* 



ay* - 2 b Q + L* 



(47! 



L- 



+ 



2^-c abzi acL 2 _i Q\aL* - 



tan 



: 2 06 ) 



If we here put z\ = o, we obtain the expression for the current-function answering to the equa- 

 torial position of the current. 



We have calculated the current-system answering to an extra-terrestrial current such as this, the 

 result being given in the table below. 





TABLE CXXVII. 



Values of the current- function ifj answering to an extra-terrestrial current 

 situated in the plane of the equator. 



x\ is here put 20. The multiplicator o is left out. 



v. 10 



For 6 == 90 we have ip = o. Further ip (n 6) = t/; (6). 



