xxiv Inteoduction. 



26. If V be the excess of angular motion of tlie 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 



m X sin w=W — sin v, 

 I 



m, X, W, a, and Z, 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°. 



A "V 



—— = n a cot « + ^(Q + 2 e — e'), 



n beinff the number of scale divisions from the zero, or reading when t6=90° : a the 

 arc value in parts of radius of one scale division ; t the number of degrees above 



A. ??/- 



the zero of temperature ; Q the value of —^ for 1° ; e and e' 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 

 5- = Q + 2 e - e', 

 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 |S be the small angle through which the torsion circle is turned, then v becomes 



?/ = tf±^. If mX = F, W -T- =G, the equations of equilibrium for the two posi- 

 tions are 



F = G sin » u = 90°. (1.) 



F = G^\n{v'±£,V) (2.) 



cos A v' . 



Subtracting (1) from (2), and dividing by (1), 



F' — F A F sin v'— sin v cos v' 

 F ~ Y ~ sin V sin v 



