42 J. A. BROUN’S NOTE ON THE BIFILAR MAGNETOMETER. 
the bifilar torsion (that is, the angle made by the upper and lower horizontal 
lines) will be v, and the torsion of each wire will also be v. If p be the torsion 
coefficient of a single wire for unity of arc, then the force exerted by the uni- 
filar torsions will be equal to 2pv; and as the force due to the bifilar torsion is 
G sin 2, the equation of equilibrium will be 
m&=Gsinv+2pv. : (1) 
where m is the magnetic moment of the magnet, X is the horizontal component 
of the earth’s magnetism, and G is a constant depending on the mass suspended, 
and length and intervals of the wires. 
If now we suppose the magnet to be in the magnetic meridian, and that 
each wire receives a torsion v, so that the magnet is moved from the meridian 
by an angle 6 (the bifilar torsion being zero), then 
2pv = mX sin 6 ; (2) 
by (1) and (2) 
G sin v 
mx = as (3) 
Differentiating (1), X and v only being variable, substituting the value of p from 
(2), and dividing by (3) 
AX : 1 1 
x = { cot o+sin 6 G eer <1 AG... (4) 
: 
Since 5 > qny> the coefficient of Av is always greater than cot v, as has been 

found in the experiments for this coefficient by deflections and by weights. 
Experiments for the value of sin 6 have shown that equation (4) will give a 
near approximation to the coefficients by other methods; but the determina- 
tion may always be vitiated through torsion introduced accidentally into the 
wires during adjustments. Thus, at Makerstoun, in two early experiments, 
the unit coefficient, with the same length and interval of wires, was found to be 
k = 0:0001185,* 
when the north end of the magnet was turned towards the east ; and 
k = 00001522, 
when the north end was turned towards the west: this difference was found 
due to torsion existing in the wires. As the methods by deflections, and by 
varying weights, include all the forces acting, it is much more satisfactory to 
employ one of these for the determination of the unit coefficient. 
* By a typographical error this was given 0°000185 in the Makerstoun Observations (Edinburgh 
Royal Society Transactions, xvii. part i. p. 37). 
