222 ON A NEW MAGNETICAL -INSTRUMENT. 



duced magnetism in producing the deflection ; and it becomes 

 necessary to institute the observations in such a manner as to be 

 able to eliminate the effects of this extraneous cause. For this 

 purpose we have only to invert the bar, so that the acting pole, 

 which was uppermost in one part of the observation, shall be lower- 

 most in the other. The induced polarity will, under these circum- 

 stances, be opposite in the two cases ; and the acting force will in 

 one case be the sum of the induced and permanent forces, and in the 

 other their difference. 



Let X and T denote the horizontal and vertical components of 

 the earth's magnetic force, M the intensity of the permanent mag- 

 netism in the acting pole, and m the magnetic moment of the sus- 

 pended magnet. The intensity of the induced magnetism is, by 

 hypothesis, equal to 



*T, 



k being an unknown constant ; and when this is of the same name 

 as the permanent magnetism, the intensity of the acting force, at 

 the unit of distance, is 



kY+ M. 



Accordingly, the moment of this force to turn the suspended mag- 

 net is (kY+ M) mr cos u, u being the angle of deflection, and r a 

 constant depending on the distance ; or, making, for abridgment, 

 kr = p,Mr = q, 



(p Y + q] m cos u. 



But this deflecting force is resisted by the earth's horizontal force, 

 the moment of which to turn the magnet is 



Xm sin u ; 



and the magnet will rest when these moments are equal. Hence 

 the equation of equilibrium is 



(1) 



By the same reasoning it will appear, that when the induced and 

 permanent magnetisms are of contrary names, there is 



pY- q = Xt&nu'; (2) 



in which u' is the new angle of deflection when the bar is inverted. 

 Adding these equations together, and observing that Y = X tan 6, 



