PART I. ON MAGNETIC STORMS. CHAP. IV. 



305 



p;p- 



Fig. 138. 



We will use the following signs: 



The distance from point 5^ to the current is designated r t , 

 the distance from point S 2 to the current, r 2 . The angles these lines 

 make with the ground-line we will call cp t and (jp, and the height of 

 the current above the earth's surface, /. The portions into which 

 the height-line divides the ground-line of this triangle, we will call a l 

 and a . The distance between Kaafjord (/) and Axeleen (2) is 



designated D ; hence the projection of this distance on the above plane is d = D sin i//. For the 

 perturbing forces we will use the signs P', /Y and P,' respectively for the total, the horizontal and 

 the vertical forces at Kaafjord, and correspondingly P", P\" and P," for those at Axeleen. 



If the magnitude and direction of the perturbing forces are given, the problem will be not only 

 determined, but over-determined, so that it affords a test of the correctness of our assumption. 



The direction of the forces, for instance, is sufficient to determine the situation of the current. The 

 strength of the current can then be determined by that of the perturbing force at the one station. The 

 strength of the perturbing force at the other station may then serve as a check. 



The calculation can be made according to the following formulae: 



_/Y' 



*Y 



tan o>j = ^7 , 



sin cp j 



sn 



+ r/> 2 ) 

 sin 



sin (99 j 



Two values will be obtained for the strength of the current, according as the force at Axeleen 

 or that at Kaafjord is employed: 



5 P f 



sn 



n^j -f 9? a ) 



. ^ 5 P" sin (pi d 

 2 sin (9?, +gt> 2 ) 



In these and the succeeding formulae, P, and P, are always to be regarded only as the numerical 

 values of the respective perturbing forces. 



As it occasionally happens that one of the vertical components is wanting, we shall also solve the 

 droblem under that assumption. If the other vertical component is there, it may be used as a check. 



If we introduce: 



and 



P/ = 



_/y_ 3 



we obtain the following equations: 



t -|-a 2 = , (i + d) = D sin (/; = </, 



h 



- = tan gr>, = p , 

 "i 



PI = = '-p cos 2 0>, , 



5 rf 5i 



Birkeland. The Norwegian Aurora Polaris Expedition, 10031003. 



(I) 

 (2) 



(3) 



