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



both magnets are merely diminished or increased in a constant ratio, — namely, 

 in the ratio of the force of the earth to the sum or difference of that force and 

 the resultant force of the magnets. Lastly, it appears from what has been 

 already said, that the mean position of the magnet c is likewise unchanged by 

 the disturbing action, and that its variations of position are only altered is a con- 

 stant ratio. The effect of the disturbing forces, therefore, is in every case 

 readily allowed for. 



As an example of this case of the general problem, let it be required that 

 the three magnets shall be in the same right line, that line being no longer ar- 

 bitrary, as before, but determined. The two equations (11) and (13) are in 

 this case reduced to (2 1 ) and (22). Dividing the former by the latter, we have 



PP' _ i-C0s2a p_^ Q/ ^-co s2ax 



This equation, in which the second member is known, determines the place of 

 the centre of the intermediate magnet. Denoting the second member, for 

 abridgment, by r, we have p = qr, p -\- q ■= 1; whence 



It is manifest from (27) that we cannot have cos 2a = ^, or sin 2a z: 0, and 

 accordingly that the angle a cannot have any of the values 0°, 90°, or 35° 16', 

 otherwise the intermediate magnet would be infinitely near one of the ex- 

 tremes.* 



To determine the azimuth, f, of the plane of the intermediate magnet, we 

 divide either of the original equations (21) or (22) by sin 2 a, and substitute for 



* In order that the intermediate magnet should be equally distant from the other two, the angle 

 must have one of the values determined by the equation 



i— cos2« PA S A / 9 A^ 1 



- =: — ^: — , or tan a : 



sin2« ~ Q~ B' ~4B— \6 B' ^ 2' 



When A^B, or the forces of the extreme magnets equal, this becomes 



tan . = ^-^^^ (r= 1.781, or = - 0.28l); 

 and the corresponding values of a are -}- 60° 41', and — 15o 41'. 



