METHODS OF TORSION. 571 



deflection of the needle, the directive couple M H will be given 

 as a function of the coefficient C, by the equation 



(5) C(w-0) = M Hsin0. 



Some precautions are still necessary to eliminate the defects 

 of adjustment. If the needle in its original position, instead of 

 being strictly parallel to the magnetic meridian, makes with this 

 plane a small angle a, the wire itself having an initial torsion e, 

 we have 



Ce = M H sin a, 



and the equation of equilibrium (5) should be replaced by 



(5)' C(a>-6> + e) = M H sin(<9 + a). 



The same experiment, repeated on the other side, gives 

 C fa -6 l -c) = M H sin (O l - a) . 



When the two positions of equilibrium are opposed that is to 

 say, when the sum of the angles and O l is almost TT, these angles 

 being sensibly equal it is easy to prove, by adding and subtracting 

 two equations of equilibrium, that the error of adjustment e is 



d) CO 



. If this difference is small, the angles <o and may be 



replaced by the means of the readings right and left. 



If o> and are the mean values of the angles observed in a first 

 experiment under the influence of the earth alone, w the torsion 

 necessary to obtain the same deflection 6 in the field H-f F, we 

 have 



C (o> - w ) = M F sin (d + a) 



so that for two different fields F and F', 



(6) = ^. 



b w- <o 



If the direction of the fields F and F', instead of being parallel 

 to the terrestrial field, is arranged so that the needle is not deflected, 

 they themselves make an angle a with the meridian. It is sufficient 

 then to produce deflections on one side ; the errors of adjustment 



