ON A NEW MAGNETICAL INSTRUMENT. 223 



being the inclination, we have 



2p 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 deter- 

 mine it, we have only to place the iron bar horizontally in the 

 magnetic meridian, its acting pole remaining in the same place as 

 before, but pointing alternately to the north and south. The in- 

 ducing force is, in this case, the horizontal 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 



2p = tan + tan if ; (4) 



and dividing (3) by this, 



tan u + tan u' /K \ 



tan 9 = x - - -- r . (5) 



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 inclination, it is not 

 necessary to observe the deflections in the horizontal position 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 : 



A F 2Aw 2tanw AX 



_ 

 Y ~ COS*M (tan u + tan u} tan u + tan u'' X ' 



But from the relation Y = X tan 0, we have 

 AF AX A0 



Y~~^~ + sin 0cos0 ; 



and substituting 



A0 cos?/ At/ + v BJn(i*-tQ AX ^ 



snT20 " cos M sin (u + ') sin (w + n) X ' 



The second term' of the right-hand member of this equation con- 

 tains a correction required for th- sii.iuli:inoous changes of the 

 horizontal intensity ; but this correction will be generally small, 



