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BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, igO2 1903. 



particles, under certain conditions, could draw nearer and nearer, this giving rise to the formation of 

 planets. It still remains for us to investigate the conditions outside the plane of the equator whether 

 the formation of planets is also possible there, when the particles are flung out anywhere on the sphere 

 or not. This investigation has been carried out as follows. 

 The equations of motion for the particle are 



y 



From these it is easily found that 



ds if" 2iti 



v = -.-= V C C = constant 



at i r 



and if the magnetic field originates in a potential V, 



(3) 



a = constant, 



when ip is a certain function of /? = ^x* -\- y* and z. 



If we assume that the sphere acts as an elementary magnet, i. e. 



then 



(4) 



- 

 <P='/.M~ 



Moreover 



3x 



rj 



whence 



dx 



dy _ 3Mz ( dy dx\ _ ZMz drp 



~ ~ ~ K df 



_ 



y dt~ 



By substitution in the third equation of motion (i), we obtain 



d*z _ 3'j.Ms ^ dtp . z 



TIo" " *^ '* o 



> ' 3 







eii 



'- -~ 



and by substitution of the value of R'- -~ from (3) and (4) we obtain 



As moreover we have 



dt- ~ r* r 3 ' 



t-dtf>- -\- dz~ , 



