166 NOTES ON MAGNETISM. 



through the centre of the needle, as it is when the line of the 

 axis of the needle passes through the centre of the magnet. 



In neither case has the transverse force (2) any tendency to 

 produce rotation. Its effect is destroyed by the fixed centre 

 of the needle. 



Let y be the distance of any element of the needle from the 

 centre, mdy its magnetism. Then, in the first case, we shall 



have 6 = ^ approximately, (R being the distance between the 



centre of the magnet and that of the needle). Consequently the 

 force (1) may be expressed by the formula 



f 2 - 3 } since cos 2 = I - 6* nearly ; 



\ W 



which, neglecting ?/ 2 , becomes 



2 Mmd 



the moment of this round the centre of the needle is - f ^ , 

 and the total moment is, therefore, 



2MM' , 



where M = jmydy. 



In the second case, 6 for every element of the needle, and 

 the moment sought is, therefore, 



MM' . Mm 

 P3 , since (1) then becomes 5- , 



JLl T 



and the sign is immaterial. The moment in this case is, there- 

 fore, one half of what it was before, which was to be proved. 



This result is included in the following general investiga- 

 tion, in which we shall ascertain the moment of rotation due 

 to the action of one magnet or needle upon another, whatever be 

 their relative positions. 



The data by which we shall suppose the position of the 

 magnets to be determined, are the distance between their 

 centres, the angles which the axis of each respectively makes 

 with the line joining their centres, and the angle between the 

 axes themselves. The last element may be readily replaced 

 by the dihedral angle between two planes which intersect in the 

 line joining the centres of the magnets, and one of which passes 



