214 



and the magnet will rest when these moments are equal. 

 Hence the equation of equilibrium is 



pY + g = X tan u. (1) 



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

 and permanent magnetisms are of contrary names, there is 



px — q — si tan u' ; (2) 



in which ii! is the new angle of deflection when the bar is 

 inverted. And adding these equations together, and ob- 

 serving that Y = X tan 0, being the inclination, we have 



2/3 tan = tan u + tan u'. (3) 



This equation would furnish at once the inclination sought, 

 provided we knew the value of the constant k. In order to 

 determine it, we have only to place the iron bar horizon- 

 tally in the magnetic meridian, its acting pole remaining in 

 the same place as before, but pointing alternately to the north 

 and south. The inducing force is, in this case, the hori- 

 zontal component of the earth's magnetic force ; and it will 

 be readily seen that the equations of equilibrium are similar 

 to (1) and (2), substituting x for Y. If therefore v and v' 

 denote the angles of deflection in these positions, we have 

 2 p =: tan f + tan vf \ (4) 



and dividing (3) by this, 



tan u + tan u' ,„. 



tan = ■ J. (o) 



tan V + tan v 



Thus, from the deflections produced in these four positions 

 of the bar, we obtain the inclination. 



In order to determine the changes of the incHnation, it is 

 not necessary to observe the deflections in the horizontal po- 

 sition of the bar. Let equation (1) be differentiated, x, Y, 

 and u being all variable, and let the resulting equation be 

 divided by (3). We thus obtain the following equation, from 

 which p and q are both eliminated : 



