METHODS OF DEFLECTION. 575 



As the couple is proportional to the sine of the displacement, 

 the needle behaves exactly as if it were situated in a uniform field, 

 parallel to the direction of equilibrium, and of intensity 



sin W 



Let us suppose that the displacement 8 had been obtained by 

 adding to the terrestrial field, a field F parallel and in the same 

 direction, the deflection of the needle is 8. If the couple pro- 

 duced by the fresh displacement is made equal to the couple P, 

 the condition of equilibrium is 



(9) FMsin(0-8) = Csin8 = HM . sin 8. 



sin sin (w - 0) 



This equation gives the ratio of the fields F and H, if the angle 

 w is known, and defects of adjustment are eliminated by an ex- 

 periment made on the other side. 



If the original equilibrium is exactly in the transverse position, 

 the angle 6 being 90, we have 



C 



F = sin w tan 8= - H tan w tan 8. 

 M 



Another field F', parallel to the terrestrial field, will give in like 

 manner a displacement 8', and we deduce 



_sinS sin(0-S') 

 F = sinlF' sin 0-8 ' 



the deflections 8 and 8' being of the same or different signs, according 

 as the fields F and F' are of the same or contrary directions. 



For a first equilibrium in the tranverse position, we have simply 



F tan 8 



It is important to observe, first, that the torsion o> does not enter 

 into the equations (10) ; and then that the deflections 8 and 8' are 



