166 Attempt to facilitate Observations of 
Such then should be the inclinations of the needle, in order that any constant error 
of position, dé, may have the smallest influence on the calculated dip and inten- 
sity. Of such constant errors the most obvious is the error to which the observer is 
subject in reading the angle on the limb, arising from the smallness or imperfection of 
the divisions. But the error of reading is not the only, or even the most important 
error, to which we are liable in determining the position of the needle. It has been 
already stated that, owing to the friction of the axle, the needle is often brought to 
rest out of its true direction: now the error of position arising from this cause is, 
in struments of the usual size, of greater magnitude than the error of reading, and 
that magnitude is different in the different positions of the needle. 
In order to determine the amount of this error, it will be necessary to consider the 
directive force, by which the needle is urged to its position of equilibrium. This 
force is obviously the difference between the magnetic moment, ¢o sin (6—4), and 
the moment of the counterpoise, » cos 6; so that, if its magnitude be denoted by F, 
F= $s sin (8—@) —p cos 9. 
But, if @, be the position -of equilibrium, there: is ¢ «sin (8—0)—y cos 6=0; and 
substituting the value of «, obtained from this equation, in the preceding formula, 
and observing that 
sin (S—@) cos @ —sin (6-6) cos @=cos é (sin @, cos @—cos @ sin #) =cos 6 sin (0,—6), 
we have finally 
cos } 
F=$¢o6 cos 0, sin (0 —6). 
Hence the directive force varies as the sine of the angular distance from the position 
of equilibrium. Accordingly, when that angular distance is reduced to a certain limit, 
the force becomes equal to the friction, and is balanced by it. Let « be the magnitude 
of the angle, #, — 8, when the directive force becomes equal to the friction, f; then 
cosé.. cos} 
S=¢ooe eee sm <=po0 ——€ 3 
since < is small; and therefore 
cos 0 
feos 0 
E Tema : (6) 
The angle «, thus found, is obviously the limit of error to which we are liable in 
determining the position of the needle, arising from friction. Its value depends, as 
we see, upon the force of friction, the intensity of the terrestrial magnetic force, the 
magnetic moment of the needle, and its position with relation to the dip. In order to 
determine the resulting error which it will produce, in the determination of the dip 
and intensity, we have only to substitute its value for 0, in the equations (5). We 
find, in this manner, 
