B1RKF.LAND. THE NORWEGIAN AURORA POLARIS EXPKU1TION, 1902-1903. 



and as 



we obtain 



dR _ 



dt '~'i 



1/1 + 



" h 



dz 



dt 



dR 



+ 



h 



* 



s 

 and finally by substitution of the expressions found for , and 



P dz 



2r 



or 



* 



= 



The coefficients in this equation are irrational functions of R and z. If r is introduced instead of 

 2 as a dependent variable, we obtain a differential equation in r and R with rational coefficients. Thi-; 

 differential equation is 



It will be seen from the second equation in (5) that the surface P=0 is a boundary-surface through 

 which the particle can never penetrate during its movement. The line of intersection of this boundary- 

 surface with the rotation-surface upon which the particle is found, then becomes a boundary-curve, which 

 the particle can never cross. This boundary-curve is always a circle parallel with the plane of the 

 equator. 



It will be seen that for a given point (z, R) upon the surface of rotation upon which the particle 



lies, there are 2 values of j- which are equally great, but have contrary signs, and similarly for 



dt 



- 



. j 



while there is only one value of -J-. From this it will be seen that if the particle moves in such a 



manner that, after a limited time, it reaches the boundary-surface, it will thence turn inwards to the 

 sphere again along a path that is symmetrical with the outward-going path, with reference to the meridian 

 plane -through the point upon the boundary-surface at which the reversal took place. 



We will next see whether the particle could approach this boundary-circle asymptotically. It is 

 clear that no matter how the particle moves, we may put 



and consequently 



= /(*) 



dz 



..V/M' 



As / must increase infinitely when z approaches the value that answers to the boundary-circle, 

 we must have for this value of z 



s = and '(z) = 



