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



by suitably determining the positions of the three magnetic bars, whatever 

 (within certain limits) be their relative intensities. 



In the case which we have at present in view, — that is, when the third 

 magnet is merely used as a counteracting power, — its intensity may be taken at 

 pleasure ; and accordingly one of the ratios, P or Q, is disposable, as well as 

 the four angles. It follows from this, as there are but four conditions to be 

 fulfilled, that one of the five quantities abovementioned remains arbitrary ; and 

 the nature of the problem obviously suggests that this should be the angle 7, 

 which determines the position of the line connecting the two principal magnets, 

 and that the conditions of equilibrium should be fulfilled by means of the other 

 variables, which determine the position and force of the subsidiary magnet. 



Let us suppose, for example, that it has been chosen to take the line con- 

 necting the magnets a and b coincident with the magnetic meridian ; or that 



7 = 0. 

 The equations (10, 11, 12, 13) thus become 



3 cos (2 j3 — f ) + cos f = 0, 



3sin(2^-f) + sin^ = 2gy^ 



3 cos (2 a - f ) + cos f = — 4 Pp\ 



3sin(2a — f)+sinf = 0. 



From the first and fourth we have, at once, 



i + cos 2 i3 ^ ^ sin 2 a 



^-X ^=:— tanC=-i ?r-- 



sm 2 /3 ^ ^ — cos 2 a 



Another relation between the angles a and § may be inferred from the second 

 and third of the foregoing equations, from which we obtain, by division and sub- 

 stitution, 



^ — cos2a_ T^ Q q^ ^ B sin' a 



^r2|3 ■" ^ Yf ~ ^'A' sin'jS ' 



From this and the preceding equation, the values of a and j8 may be obtained 

 by elimination. These angles being known, f is given by means of either of the 

 expressions for tan ^ above written ; and one of the ratios, Q or P, by the second 

 or third equation, the other remaining arbitrary. 



