The Rev. H. Lloyd on the mutual Action of permanent Magnets. 251 



3 cos (2)3 — f ) + cos f = 0, sin 27 = ; (1) 



3 sin (2/3 — f ) + sin f = 0, 1 — 3 cos 27 = ; (2) 



3 cos (2a — f ) + cos f = 0, 1 + 3 cos 27 = ; (3) 



3 sin (2a — f ) + sin f = 0, sin 27 = ; (4) 



3 cos (2^ — ^) + cos ^ =z 0, 3sin(2a-^) + sinf = 0. (5) 



Now it will be seen, on a little consideration, that of these five pairs of 

 equations, the equations (2) and (3) exclude, each, the other four ; so 

 that if we fulfil the condition expressed by (2), or that expressed by (3), in 

 this way, we cannot at the same time satisfy any other. On the other hand, 

 each pair of the remaining conditions, expressed by the equations (1, 4, 5), 

 has one equation in common ; so that for the fulfilment of these three con- 

 ditions, three equations only are to be satisfied ; and these three equations 

 are not only not inconsistent, but even leave one of the angles still un- 

 determined. 



These equations are 



sin 27 = 0, (6) 



3cos(2/3 — + cosf = 0, (7) 



3 sin (2a — f ) -f sin f =: 0. (8) 



The first of them determines the angle 7 ; and as the other two contain three 

 arbitrary angles, they maybe fulfilled in an infinite variety of ways. Accordingly 

 we must have 



7 = 0, or 7 = 90°; (9) 



that is, the line connecting the magnets a and b must be parallel or perpen- 

 dicular to the magnetic meridian. And the angles, a, /3, f, which determine 

 the place and azimuth of the third magnet, are connected by the relations, 



^ + cos2^ _ sin2a , 



sin2i3 - ^'^^^-i- cos 2a' ^^"^ 



so that when one of these angles is assumed or given, the other two are deter- 

 mined. 



2 k2 



