704 B1RKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, igO2 1903. 



The 2nd equation may be transformed to 



d*y 



nl ,"- - 



dt 2 



From (2) we obtain 



dx 



dy _ dx _ 

 X ell ~ y ~dl~ 



whence 



(4) 



Further we have 



dy 

 ~d~t 



dx (Sr 2 3jy 2 ) x dy 



dt 3(^ y z ) y ~dt ' 



dr _ dx . dy . dz . dx . dy 



dt dt dt dt dt dt 



whence 



f dr _ (2r 2 3y 2 ) dy 



'~dt = 3y dt^^'dt 



^ 

 dt 



or 



(5) 



^ dy_ j^dr 

 ~ydi'~2rdt' 



Hence we obtain by integration 



(6) 

 Hence it follows, that 



(c = constant). 



or 



(7) 



x = 



rye* r 



The 3rd equation in (1) may be written in the form 



dy /.ix 



~~ 



~dP ~ (1 + k z ) r 3 dt 

 From this equation in connection with (3) we obtain 



IM 



dfl ' dt* r* \ dt \+k z dt) (1 



Hence we obtain by integration 





Now, however, we also have 



dy dx IM 1 



- == - 



dy dx 

 v ^ .. 



x ji y^7~^ 



3y dr 



and as 



dt J dt 3 ( 1 + 2 ) x 2r dt 2(1 + 

 y VT+7 5 |/r 



dt ' 



r 



