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



, 2Bm . , Bm 



sin a, -j 3— cos a ; 



a^ ' ' a^ 



and the resolved parts in the direction of the axis of c are 



2Bm . . Bm 



-I -5-sinacos(a — ^), + —5- cos asm (a — f). 



Making the sums of these resolved parts equal to nothing, and performing the 

 same reductions as before, the condition of equilibrium of the forces exerted 

 upon the magnet c, in the direction of its axis, is expressed by 



P/ {3cos (2^ - f) + cos f} + Qq' {3 sin (2a -^ + sinf} = 0. (15) 



For the conditions of equilibrium of the disturbing forces exerted upon the 

 three magnets, a, b, c, by their mutual action, we must combine equation (15) 

 with the four equations (10, 11, 12, 13) already given; and, as there are but four 

 arbitrary angles, it follows that complete equilibrium is not attainable, except 

 for determinate values of the relative forces of the magnets. 



It fortunately happens that, for the special purposes which we have here in 

 view, we may, without inconvenience, dispense with one of the conditions of equili- 

 brium, — that, namely, of the forces exerted upon the magnet b resolved in the 

 direction of the magnetic meridian. This condition, (which is expressed by 

 equation (12)) being left unfulfilled, it follows from (13) that the resultant 

 force exerted upon the magnet b by the other two, will be directed in the mag- 

 netic meridian itself, and will therefore conspire with, or directly oppose, the 

 force exerted by the earth on the same magnet. Consequently the changes of 

 position of the magnet bar, (which, in this instrument, are proportional to the 

 changes of force divided by the total force,) are thereby only diminished or in- 

 creased in a constant ratio, — namely, the ratio of the force of the earth to the 

 sum or difference of that force and the resultant force of the two magnets. 

 The changes sought are therefore obtained simply by multiplying by a constant 

 coefficient. Accordingly, the four equations (10, 11, 13, 15) being fulfilled, 

 the disturbing action exerted upon the magnets a and c will be completely 

 balanced ; and, with respect to that exerted upon the magnet b, its effect may 

 be at once eliminated from the results, by altering in a suitable manner the 

 constant in the formula of reduction. 



