3£ w PFaste /Space rowni the Needle of a Galvanometer." 



uniform over the element of length ds, the couple on the element 

 will be mFds, which is Professor Holman's mf cos 6 . ds. But 

 it is to be carefully noticed that the total couple on the magnet 

 is not obtained by integrating this expression from end to end 

 of the magnet ; for the value of F varies along the element, 

 and this variation renders necessary the addition of a term to 

 the quantity to be integrated. 



The necessary correction for this variation is the moment 

 round the axis of suspension of the force tending to produce a 

 motion of translation of the element as a whole. 



The potential energy of the element ds of the magnet in the 

 field is 



— m$ds, 



if S be the magnetic force in the direction of the length of 

 the magnet at the point distant s from the centre. The force 

 producing motion of translation of the element in the horizontal 

 direction perpendicular to the magnet's length (that is, let us 

 say, in the direction of x) is therefore 



dS , 



m — ■ ds. 

 dx 



But clearly d$/dx = a r F /ds; and this force is 



m —-as; 

 ds 



while its moment round the axis of suspension is 



ms — - ds. 

 ds 



The whole expression to be integrated along the magnet is 

 thus 



m 



(»+.g)*«,!*(ft)* 



Hence integrating from s= — I to s=+l (21 being the 

 length of the magnet) we get for the total couple the equation 



where F + j, F_j denote the component forces perpendicular to 

 the magnet at the ends. If these forces be equal we have 



L = 2mZF„ 



or the magnetic moment of the whole magnet multiplied by 

 the component perpendicular to the magnet of horizontal 

 magnetic force at either end. 



This result is independent of the distribution of magnetism 



