NOTES ON MAGNETISM. 165 



The equation to the resultant of these two forces is, since the 

 line passes through P, 



yT sin 6 _ x r cos 6 

 3sin0cos0~~3cos 2 0-l ' 



For y = 0, or at the point , 



X Q r cos 6 Jr sec d (3 cos 2 6 1), 



or X Q = ^r sec 0. 

 Now cD = r sec 0, therefore 



which proves the first part of the construction. 



Next, to find the magnitude of the whole action, square and 

 add (1) and (2), then 



^ (1 - 6 cos 2 + 9 cos*0 + 9 sin 2 cos 2 0)*, 



or 



T" 



is the magnitude sought. Now 



PQ = r {sin 2 + (cos - J sec 0)"}* 



and c 



Therefore = (1 + 3 cos 2 0) *. 



c ty 



Consequently, if E be the magnitude of the resultant sought, 



p _Mm PQ 

 "cP 3 cP' 

 which was to be proved. 



The preceding formulas enable us to determine all the cir- 

 cumstances of tl^e mutual action of two magnets, which are such 

 as to fulfil the conditions of our hypothesis. 



For instance, in the memoir Intensitas vis Magneticce Terres- 

 triSj Gauss has shown that if a magnet and a needle be placed 

 at right angles to one another, then the moment of rotation of 

 the needle due to the action of the magnet is approximately 

 twice as great when the line of the axis of the magnet passes 



