XXIV INTRODUCTION. 
26, If uv be the excess of angular motion of the arms of the torsion circle or 
upper extremity of the wires over u, that of the lower extremities or magnetic bar 
in moving the latter from the meridian, the equation of equilibrium is 
9 
m X sin u= W = sin v, 
m, X, W, a, and J, being respectively the magnetic moment of the bar, the horizontal 
component of the earth’s magnetic force, the weight suspended, the interval and 
length of the wires. 
By differentiation and division, the following equation is obtained, u = 90°. 
Aan «a cotv+t(Q+2e—e’), 
n being the number of scale divisions from the zero, or reading when u=90°; « the 
are value in parts of radius of one seale division; ¢ the number of degrees above 
Am : 
the zero of temperature ; Q the value of read for 1°; e and ¢ the coefficients of ex- 
pansion for the brass of the grooved wheel, and silver of the wires. 
27. The tables of abstracts, in parts of the whole horizontal force, are computed 
by this formula. The values of 
K =a cot v, and 
q=Q+2e-¢, 
are given No. 32. 
28. During considerable disturbances, the collimator scale, which contains too 
small an angle, goes out of the field of the reading telescope. In this case it has been 
found necessary to turn the arms of the torsion circle until it again appears ; without 
this it has happened that the greater part of a disturbance would have been lost. As 
there was some doubt that turning the torsion circle after adjustment might affect 
the instrument injuriously, experiments were made in 1842, during periods of slight 
change, which shewed, after turning the torsion circle a few degrees in different di- 
rections, that on recurring to the original value of v, the scale readings were unaltered. 
In altering v, the value of the scale divisions, and the unit of force are also 
changed ; it is therefore necessary to reduce the observations to a common unit. 
Let 8 be the small angle through which the torsion circle is turned, then v becomes 
v=v=s. IfmX=F,W > =G, the equations of equilibrium for the two posi- 
tions are 
F=Gsinv “= 90% (1.) 
FY = Gisin @ =H Aw) (2.) 
cos Av. 
Subtracting (1) from (2), and dividing by (1), 
F—F_ aF_ sinv—sinv , cosv 
EG) oe aetery sin v ~~ sine 
