6i6 



BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, IQOS 1903. 



This relation is the same as that which determines the fields of an elementary magnet. We can 

 then apply the results of STORMER'S mathematical analysis of the orbits of corpuscles in the field of an 

 elementary magnet. 



According to STORMER, the orbits are determined by the following equation: 



(I) 



is the length of the orbit, y is a constant of integration. 



c = 



M 

 H a o t 





H O Q O is a quantity which depends on the stiffness of the rays. Q O is the radius of curvature of 

 the corpuscular orbit, when the magnetic force perpendicular to the orbit is H - 



Introducing the angle 6 which the direction of the orbit forms with the radius vector we get 



' R + 3T 



From the condition that sin 6 must have values between i and + i, STORMER finds that for 

 each value of y the orbits must be restricted to certain regions of space. 

 Suppose at first y is negative and numerically greater than i, or 



y t = y, where 



In this case we shall have an interior and an exterior region for the orbits. 

 The inner region is limited by the two circles, 



The exterior region goes from infinity to the circle 



^3 = c (y, + 1/yl 



( 2 c) 



If y, is less than unity, the exterior circles R% and R 3 cease to exist. 



The rays issuing from points on the equator circle can have any direction inside the two quad- 



rants < 6 < '- and > f. 



*- Ci 



It is of special importance for us to examine the range of those rays which reach the greatest 

 distance. 



It will be those going out in a direction corresponding to sin 6 = + 1 or for these rays: 

 R = a when sin 6 = + 1, which give 



R! = a = c 

 If the rays shall not go towards infinity 



- y,). 



y, > 1 or 



(3) 



