The Rev. H. Lloyd on the mutual Action 0/ permanent Magnets. 169 



We have hitherto considered the third magnet as fixed, and serving only to 

 complete the equilibrium of the forces arising from the mutual action of the 

 other two. This magnet may, however, be a moveable one, and its movements 

 serve to exhibit the changes of one of the magnetic elements. In fact, three 

 independent variables are required, in order to determine completely the ter- 

 restrial magnetic force, (or its changes,) in direction and intensity ; and, ac- 

 cordingly, whatever elements be taken as the basis of this determination, three 

 separate instruments will be, in general, requisite for their observation. In this 

 case, then, it becomes necessary to consider the action of the first and second 

 magnet on the third. 



The third magnet employed in the Dublin Magnetical Observatory, is in- 

 tended for the determination of the variations of the vertical component of the 

 earth's magnetic intensity. It is a horizontal magnet, supported on knife edges, 

 and capable of motion in a vertical plane. The plane passing through the 

 centres of the three magnets being horizontal, the axes of the magnets neces- 

 sarily lie in the same plane ; and, consequently, the action of the first and 

 second on the third is directed in that plane. Let this force be resolved into 

 two, one in the direction of the axis of the magnet, and the other perpen- 

 dicular to it. It is obvious that the latter component can have no effect on the 

 position of the magnet, being at right angles to the plane in which it is con- 

 strained to move ; we may, therefore, confine our attention to the former, — 

 that is, to the resolved part of the force in the direction of the magnet. 



Using the same notation as before, the forces exerted by the magnet a, 

 upon any element m of the magnet e, in the direction ac, and in the di- 

 rection perpendicular to ac, respectively, are (5) 



,2Am „ , Am . 



+ — ^cos^, 4.__sm/3; 



and the resolved parts of these forces in the direction of the axis of the magnet 



c are 



,2Am „ ,„ ^. , Am . ^ . , 



+ — ^cospcos(f-^), +-^sm^sm(f-/3). 



In like rtianner, the forces exerted by p upon the same element m of c, in the 

 direction bc, and in the direction perpendicular to bc, are 



VOL. XIX. z 



