The Rev. H. Lloyd on the mutual Action of permanent Magnets. 163 

 Hence the components of the force are 



X= ?|- cos {m, + ^X5 cos^ - 3)1 



[ (7) 

 F=^sIn0|il/,+ |-^X5cos^0-l)}; 



the integral Involving the first dimension of r being denoted, for distinction 

 hy M,. 



When n 0, these values become 



2m r,^ . 21/, 



r=o, X=^^(^.+^-^.); 



and the resultant force is, consequently, directed in the connecting line. 

 When = 90", we find 



and the force is altogether perpendicular to the joining line. , 



Returning to the approximate formulae (5), it is easy to deduce the directive 

 force, or the moment of the action exerted by one magnet on another, the length 

 of each being supposed small in comparison with the distance between them. 

 In this, and other similar applications of the formulae, we may consider the 

 distance a, and the angle 0, as the same for all the elements of the magnet acted 

 upon ; the variations of these quantities being of the order of those which we have 

 already neglected in this approximation. 



Let us assume that the two magnets ns and n's' (Fig. 2) are in the same 

 horizontal plane, and that the magnet acted on, n's', is capable of motion in 

 that plane round an axis passing through its centre of gravity. Let J^ and Y 

 denote, as before, the components of the force exerted by the former upon any 

 element of free magnetism, q', situated at the point q' of the latter. These forces 

 being directed in the line oq', and in the line perpendicular to oq', respectively, 

 their moment to turn the magnet n's' round its centre of motion o', is 



o'a' (Xsin n'q'o — Fcog n'q'o). 



y2 



