60 



by observing the angles of deflection, 6 - »/, at different dis- 

 tances ; it is probable, however, that their values may be 

 inferred, a priori, from the lengths of the needles, with as 

 much accuracy as is attainable in observations of this nature. 

 When the plane of motion is perpendicular to the magnetic 

 meridian, or a = 90°, 



Ycosj)' = mU ; (6) 



which gives, in like manner, the ratio of the vertical compo- 

 nent to the magnetic moment of the needle. 



The total force is determined, absolutely, by means of the 

 two observations in the plane of the meridian : for, multiply- 

 ing the equations (2) (5), m disappears, and we have 



WrU 

 R'- „ (7) 



sin u sm u 



in which the angles of deflection, 0- »j, - »j', are denoted for 

 abridgement by u and u' . Again, dividing the former of these 

 equations by the latter, 



2 Wr %mu' 

 U sin u 



The equations (3) (6) furnish, in like manner, a similar 

 value of the vertical component of the force. 



In order to determine the probable error in the resulting 



value of the force, arising from the errors of the observed 



angles, u and u', we have to observe that the moveable needle 



is acted on, in each case, by two forces, one of which is the 



moment of the earth's magnetic force, niR sin tf, while the 



other is constant. Hence, in any position, the directive force 



is 



F = mR sinM - G. 



Let Uo denote the value of u, corresponding to F= 0, or to the 

 case of equilibrium ; then 7nR sinuo - G, and 



F = jnR{sin u - sin Mo)- 



Let u = ^l^) + A?<oj ^"o being a small angle, — or, in other words, 



