the Intensity of the Earth's Magnetic Force. 539 



To find the probable error of the force corresponding to the error of the 

 observed angle, we must differentiate the equation of equilibrium, MRsm u = F, 

 with respect to R and ii, where u = — i] ; and we have 



A Z2 sin M + i? cos w Am = 0. 

 But 



M=|(»?l-''/2), 



r/i and fj., being the observed angles of inclination under the two opposite ac- 

 tions of the deflecting force. Hence, the probable error of u is 



Au = \ V^rfi + Aif2 = 7^ A»7 ; 



^/2 

 since Arji = A??,. Accordingly, the second term of the preceding equation be- 



^e 



/ 



1 If 



comes y^ E cos u A »; = 79 iiJ ! ^^^ "^^ ^'^^^ 



AR-. 



M a/2 sin u 



We learn, then, that the probable error of the force varies inversely as the 

 sine of the angle of deflection ; and that it is therefore requisite for accuracy 

 that this angle should be considerable. There is no difliculty in augmenting 

 the angle of deflection as much as we please in the first part of the process, in 

 which the magnet is deflected by a weight ; but in the second, the case is 

 different, and with the slender needles to be employed as deflectors, a large 

 deflection can only be obtained by placing the deflecting needle at a very 

 short distance from the moveable one. The most convenient arrangement 

 appears to be to attach the deflecting needle to the moveable arm of the divided 

 circle which carries the verniers, and at right angles to the wires of the micro- 

 scopes.* So attached, it must always be rendered perpendicular to the 



* To obtain the value of A^ by observation, we must substitute for /its value given above. 

 But when ri = 0, or when the needle is unde&ected, f= MBA6 ; wherefore 



V2 sm u 

 In the instrument with which I made trial of this method, the length of the needles was Sj inches, 

 and the angle of deflection produced, in the position of the deflecting needle here described, was 



