BiFILAR OR HOEIZONTAL FoRCE MAGNETOMETER. XXvii 



30. If v be the excess of the angular motion of the arms of the torsion circle, 

 or upper extremities of the wire, over u, that of the lower extremity or magnetic bar 

 in moving the latter from the meridian, the equation of equilibrium will be 



m X sin m = W — sin v 

 I 



m, X, W, a, and I being respectively the magnetic moment of the bar, the hori- 

 zontal component of the earth's magnetic force, the weight suspended, the interval, 

 and the length of the wires. The differential of this equation (w = 90°) divided by 

 it, gives 



A ^ 



= n a cot V + ; (Q + 2 e — e') 



n being the number of scale divisions from the zero, or scale reading when ^=90°, 

 a the arc value in parts of radius of one scale division, t the number of degrees 

 Fahrenheit which the temperature of the magnet is above the adopted zero, Q the 

 coefficient of the temperature correction for the varying magnetic moment of the 



bar or the value of - — for 1° Fahr., e and e the coefficients of expansion for the 

 m 



brass of the grooved wheel and silver of the wires. 



31. It is assumed, in the previous investigation, that the suspending wire does 

 not act by any inherent elastic force ; that the torsion force depends wholly on the 

 length and interval of the two portions of the wire and the angle of twist : it seems 

 extremely probable that this condition will not be rigorously sustained, and it is 

 very possible that there may be considerable twist in the suspending wire or thread ; 

 for this reason, the following methods, which are independent of the angle of torsion, 

 were employed to determine the coefficient : — 



32. If the equation of equilibrium for the bifilar magnet at right angles to the 

 magnetic meridian be 



mX=F, (1.) 



and if a magnet whose magnetic moment is M be placed with its axis in the mag- 

 netic meridian passing through the centre of the bifilar bar, the centres of the two 

 bars being at a distance r, and the resulting angle of deflection be n scale divisions 

 = A 'y, the equation of equilibrium will be 



m 



{^-i?(in^-^)}-"=^'- 



For a value of the earth's horizontal force X + aX, which would alone have pro- 

 duced the deviation a v, we have 



m 



f X + A X ) cos A ?;=F' ; 



