ON THE MAGNETIC INFLUENCE OF THE SUN OR MOON. 307 



mutual distance. Then, if p denote the quantity of free magnet- 

 ism contained in the element ds of the magnet at the point (z,y, r), 

 the force exerted by n on m is 



and its resolved portions in the directions of the three axes of coor- 

 dinates are 



m (a - x) fids m (b - y) pds m (c - s) fads 

 -# ' # 



Let the magnitudes of the lines connecting the points (a, b, c) 

 and (a-, y, c) with the origin be denoted by u and s, and let the 

 angle contained by their directions be o>. Then 



e* = u- - 2us cos a + s 2 ; 



and if s be so small in comparison with u that the squares and 

 higher powers of - may be neglected, 



T 3 = M~ 3 ( 1 + COS (i) ). 



V w / 



Again, if a, j3, 7 denote the angles contained by the axis of the 

 magnet with the three axes of coordinates, 



x - s cos a, y = s cos /3, s = s cos 7. 



Substituting these values in the expressions for the components of 

 the force above given, integrating, and observing that J /ucfe = 0, 

 we have, for the components of the total force exerted by the mag- 

 net on the magnetic element, 



Mmf 

 u 3 \ 



Mmf n o 



-3- cosw - co"s/3 



in which we have put, for abridgement, M = J /i&. The angle w is 

 connected with a, |3, 7 by the relation 



u cos <i> = a cos a f b cos /3 + c cos 7. 



Now let the point (a, b, c) be on the earth's surface, and let us 

 x2 



