450 Rey. Mr. Luoyp on a New Method of Observation 
be observed in the usual manner, and if the ratio of the moments u and v be pre- 
viously ascertained, these equations will give the dip, and the relative force, at the 
several places of observation. 
The degree of accuracy with which these elements are thus determined is, however, 
not independent of the moments of the added weights ; and, for a given amount of 
friction of the axle on its supports, the errors of the final results will vary with the 
position of equilibrium of the needle. It has been already shown* from theoretical 
considerations that the probable error in the determination of the dip, arising from 
the friction of the axle, will be least when the needle is entirely unloaded, and of 
course in the line of the dip ;—while the probable error in the determination of the 
force is least, when the needle is at right angles to the same line. Hence the most 
advantageous mode of applying the preceding method consists in observing the po- 
sition of the needle,—first, when unloaded,—and, secondly, when loaded with a weight 
sufficient to bring it into a position nearly perpendicular to the line of the dip. 
It is obvious that if »= 0,¢=6; or the first of the observed inclinations becomes equal 
to the dip, when there is no weight whatever acting with or against the directive force. 
This condition, however, is never perfectly attained in practice. Owing to the want of 
perfect coincidence of the centre of gravity of the needle with the axle, the weight of 
the needle itself has a certain moment, which must deflect it from the true line of the 
dip. But, as this deflexion is, in all cases, small, it will be convenient to consider the 
angle Z as the approximate value of the dip, and to seek the correction necessary to 
reduce it to its exact value. For this purpose, let equation (1) be divided by (2), and 
let the ratio of the moments, “, be denoted by p; then 
v 
ceeGte omnis 4) 
P cos8 ~ sin (S—6) * 
Now making 
d= Ute, . (3) 
the 2d member of the preceding equation becomes aa 9) UP since «is avery 
small quantity, and we haye 
COS 
cos St (¢—6). (4) 
The dip is therefore determined by means of the two equations (3) and (4); and the 
correction due to the want of perfect balance of the needle is inferred from the two 
observed angles, without the reversal of the poles. 
The value of the constant coefficient, p, in equation (4), will be given by the 
formula 
sin s= p 
_ cos § sin (6—Z) 
~ cos Z sin (8—0) 
* Page 167. 
