The Rev. H. Lloyd on the mutual 'Action of permanent Magnets. 171 



It follows at once from the equations (10, 13, and 15) that 



sJn27 = 0; (16) 



and therefore that 7 = 0, or 7 = 90°. The line connecting the magnets a and 

 B must therefore be parallel or perpendicular to the magnetic meridian. Sub- 

 stituting the former of these values, equations (10, 11, 13) become 



3cos(2j8 — f) + cosf = 0, (17) 



3sin(2,3 — f)'-|-sinf = 2(?9', (18) 



. 3sin(2a — f)-|-sinf = 0; (19) 



in which a = -r—-, r. Equation (15) is rendered identical. When we make 



^ sm(a — /3) ^ ^ 



7 = 90°, the only difference is, that the second member of (18) becomes 



40cos^a . , J f. 2Qsin^a • 1 ^ 1. u 



, mstead 01 . 3- -r. It is easy to see m what manner we should 



sin'(a— i8)' sin^(a — /3) 



proceed for the purpose of eliminating among these equations ; the final equa- 

 tion, however, will be one of much complexity. 



In the application of the original formula it will often occur that we are not at 

 liberty to consider the four angles, a, )3, 7, f, as all arbitrary, some circumstance 

 connected with the locality determining one or more of these quantities, or 

 establishing one or more relations among them. 



Let us suppose, in the first place, that there are but three arbitrary quan- 

 tities, so that we can satisfy but three of the equations of condition. We shall 

 select for that purpose the equations (10, 11, 13), leaving (15) unfulfilled, as 

 well as (12). This being done, the disturbing action exerted upon the magnet 

 c remains unbalanced ; but, as the effective part of this action is directed in the 

 axis of the magnet itself in its mean position, it does not alter that position, 

 but merely diminishes or increases the deviations from it in a given ratio. In the 

 case of this magnet therefore, as in that of the magnet b, the effect of the dis- 

 turbing action may be allowed for, by a suitable alteration in the coefficient by 

 which the changes of angle are multiplied. 



In order to illustrate this, and at the same time to apply the formulae in a 

 very important case, let it be required that the centres of the three magnets 



z2 



