308 ON THE DIRECT MAGNETIC INFLUENCE OF THE SUN 



suppose, for simplicity, that the acting magnet is in the plane of 

 the equator. Let that plane be taken as the plane of (#, y\ and 

 the line connecting the centre of the magnet and that of the earth 

 as the axis of x. Then, if the distance of the acting magnet be 

 considerable, relatively to the earth's radius, b and c are small in 

 comparison with a, and we may neglect the small quantities of the 



b z <? be 



second order, ,, . Wherefore, substituting for cos w its 

 a a a' 



value, the components of the acting force become 

 36 



n 

 H 2 cos a + 



36 3c \ 



cos p + cos y J, 



mf , 36 



-COS/3 + - 



Mmf_ 

 3 V 



cos 7 + cos a J. 



Now, if D denote the distance of the centre of the magnet from 

 the centre of the earth, r the earth's radius, A the latitude of the 

 point (a, b, c) on its surface, and the angle contained by the 

 meridian passing through it with that containing the acting 

 magnet, 



a = D - r cos A cos B, b = r cos A sin B, c = r sin A. 



Hence the maximum values of - and - are equal to - ; and if we 

 a a a 



disregard the terms containing them in comparison with the rest, 

 the preceding values are reduced to 



Mm Mm Mm 



.5-^-cosa, - cos0, - cosy. 



Now, in place of a single magnet, let there be an indefinite 

 number distributed in any manner throughout the entire magnetic 

 body ; and let us make, for abridgement, 



S ( M cos a) = P, s (M cos /3) = Q, S (M cos 7) = R. 

 Then, if the radius of this body be small in comparison with its 

 distance, we may neglect the variations of D y and we shall have, 

 for the three components of the acting forces, 



-^ - mQ ~ mR 



