and the Aurora Borealis. 367 



rents. From this it follows that, the less complete the dis- 

 ruptive discharge in the first-mentioned regions, the more 

 numerous and intense will be the currents in the second. 



We assumed above that the terrestrial magnet coincided 

 with the rotation-axis of the earth, and that the earth was 

 everywhere homogeneous and endowed with the same electric 

 conductivity. Now the magnetic poles of the earth are not 

 situated on the axis of rotation ; or, in other words, the right 

 line connecting the magnetic poles does not coincide with that 

 axis, but makes with it an angle determined by observations 

 to be about 17°. Moreover the conductivity of the air varies 

 with the time and place. Yet these circumstances necessitate 

 only nonessential modifications in what has been said. The 

 circle abed (fig. 11) represents a section passing through the 

 rotation-axis of the earth and through the right line which 

 joins the magnetic poles. This line makes with the axis the 

 angle a (= about 17°). The distance of the magnetic poles 

 from the axis will therefore be p sin a — an expression in which, 

 as already said, p cannot exceed the half of the earth's radius. 



We will now suppose another plane passing through the 

 axis, and forming the angle v with the preceding plane; and 

 we will consider the action of the magnetic poles upon an 

 electric molecule m situated in this plane. During the rota- 

 tion of the earth the magnetic poles describe circles the radius 

 of which is p sin a. The radius of the circle described in the 

 same time by the molecule m will be r cos I, r denoting the 

 distance of the molecule from the centre of the earth, and I its 

 latitude. The relative velocity of the molecule m with respect 

 to the magnetic pole s will be obtained, according to what 

 precedes, by giving the same velocity to m and to s, but in the 

 opposite direction to that of the already existing velocity of 

 the magnetic pole. If the time of rotation of the earth be 

 taken as unit, that velocity will be denoted by 2irp sin a. The 

 magnetic pole will in this way be brought to rest, and the 

 molecule m will move with the velocity 



2<7r V r 2 cos 2 / -f- p 2 sin 2 u — 2rp cos / sin « cos v. 



The relative velocity of the molecule with respect to the other 

 magnetic pole will be 



2-7T Vr 2 cos 2 / + p 2 sin 2 a + 2rp cos / sin a cos v. 



Now it is obvious that these square roots denote the distance 

 of the molecule m from the right line drawn through each pole 

 parallel to the earth's rotation-axis. Hence it follows that the 

 magnetic pole acts upon an electric molecule in the same way 

 as if the pole were at rest and the molecule in rotation about 



