4 2 4 



HIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, 1902 1903. 



We will further, in the case of the positive directions, employ the system of coordinates used by 

 HERTZ in his Inaugural Dissertation, "Ueber die Induktion in rotierenden Kugeln'^ 1 ), as in a subsequent 



chapter we shall go into the subject of induction currents, and shall then 

 have occasion to use the developments we here work out, and it is therefore 

 best to introduce these signs at once. The positive directions of ^Y, Y, Z, 0, 

 and 10, are shown by arrows in the figure. 



^ x We will, then, determine the force-components along the radius vector, 



the meridian and the parallel circle in a fixed point upon a sphere with an 

 arbitrary radius Q, (Q supposed < /.). One of the vertical pieces of current 

 produced will intersect the surface of this sphere in a point p, 1", 11. 



The total effect due to a piece of current such as this (see p. roi, 

 Part 1) is 



Fig. 177. 



sn 



I 



sin 2 /J 



sin [i 



L Q cos j 



2 L cos 



L z 



(i) 



Fig. 178. 



/- c cos ft 



where we have put R -\- h = L, and /i? is the arc of the great 

 circle between the place under consideration and the point of 

 intersection of the produced path of the current with the surface 

 of the sphere. We shall, moreover, when not otherwise stated, 

 always make use of the C. G. S. system, and the electro-magnetic 

 system of measurement. 



The three components are thus 



P ? =O, P 9 = Psm v, P,,, = .Pcos v, (2) 



where v is the angle between the direction of the magnetic force 

 and the parallel circle, reckoned positive, as shown in the figure. 

 In the case in which the positive current is flowing away from the 

 sphere, i. e. in the direction of increasing g, we will call the direc- 

 tion of the current positive. 



What we have to do is to find an expression for fi and < r . 



This is given directly by the spheric triangle drawn in the figure- 

 cos /? = cos (o |) sin sin -\- cos C cos 6, 



sin sin (w /u) 

 "sin/? ' 



cos cos 9 cos (i 



sin v = 



(3) 

 (4) 



and 



cos v = - - 



sin |? sin 6 

 By simple combination, the effect of the vertical portions of the current may be found by these formulae. 



We shall then consider the magnetic effect of the curved portion of the current. 



We will call the direction of the current positive when it coincides with the direction of in- 

 creasing io. 



The coordinates of the current-elements we will call L, t and ft, ft thus answering to w. What 

 we have to do, then, is to determine the effect of this element in a point g, 6, >, on the sphere. 



According to Biot & Savart's law, we then have 



, . Lsin'Cdfi . 

 aP = / ,.," sin a, b) 



(') H. HERTZ, "Gesammelte Werke", Hand I. 



