BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, 1902 1903. 



and by multiplication of (i) by (2), we obtain 



The result thus attained is that the necessary and sufficient condition for the approach of a particle 

 to a boundary-circle with radius n r a , when n > I -j- j/2 , is that the last three equations take place for 

 a value of 0- between 1 and -j- 1 . 



If we were to imagine r a , v a and M maintained, we can find those values of /.t and A which give 



ft 



T V 



the boundary-circles. It is at once seen that for great values of n, - - will keep very near 2. Those 



r v 

 values of ft that give rise to great values of will thus approximately be ~-^ . It will be seen, 



r v 

 however, that the greater n is, the nearer to 2 will - - be, and then u must be a little less than 



for smaller values of n. Under otherwise similar circumstances therefore, boundary-circles approach 

 negative particles will be of greater radius than those approached by positive particles. Moreover it will 

 be seen from the last equation that for great values of n, A 2 will be approximately proportional to w; 

 i. e. thai particles with small mass in proportion to their charge will give rise to boundary-circles 

 greater radii than particles with great mass in proportion to their charge. 



The particles that approach a boundary-circle may continue to move there for all time. It is con- 

 ceivable, however, that the number of particles will gradually become so great that they will be capable 

 of collecting into large globules, which in their turn at last unite to form a planet, as the electric charge 

 in the original particles may conceivably be supposed to have been lost. 



In the case of the sudden loss of the charge, the mathematical investigation has shown that the 

 particles will afterwards move in ellipses about the central body with perihelion in the boundary-circle 



and with eccentricity 



1 

 g / o ' 



f 



where/ - k.n, = , and r~v.. sin or- 1.M (\-\-k). 

 >' 



That / is great and thus the eccentricity e small, when r is great in relation to the radius ;- of the 

 central body, will be seen from the following relation, which must be satisfied: 



n + 2 V2w+l <; / <: n -f 2 + \/2 + 1. 



If the electric charge of the particles is gradually lost, it seems evident that the finite orbit com- 

 pletely circumscribes the boundary-circle and very nearly becomes a circle, if the boundary-circle of t 

 particles has a large radius in proportion to r a . 



Let us now, on the supposition that the entire mass of particles near a boundary-circle revolvi 

 directly about the central body (like the .planets), consider the question as to how a planet, originating 

 in the massing together of the globules here assumed, can acquire a direct rotation about its axis, and 

 not a retrograde, as at first sight one would imagine. 



As regards this question, I subscribe to the explanation given by POINCARE in his "Hypothese 

 Cosmogoniques" (p. 51), in which he shows how planets, in the event of their having originated 

 LAPLACE'S ring-formations, can acquire a direct rotation. This explanation is based upon G. H. DARW 

 important investigations on the effect of tidal reaction between a central mass and a body revolving aboul 



