Terrestrial Magnetism. 165 
meter. The observation of the angles of position of the same needle, when counter- 
poised, was taken immediately before or after, and, as nearly as possible, at the same 
temperature. But unfortunately the moment of inertia of the needle was unguard- 
edly altered during the observations, so that no reliance could be placed on the results. 
Since the preceding trials were made, it appeared to me desirable to examine theo- 
retically the accuracy of the counterpoised needle in its different positions, and to as- 
certain, if possible, the most eligible. It will appear from the foregoing observations, 
that between two distant places, the range, or change of position of the counter- 
poised needle, is considerably greater than the corresponding change in the dip. This 
I had been previously led to expect from theory, and had found that the change of 
position, corresponding to a given change in the dip, would be greatest when 
tan20=1tand. Whend=70", this formula gives, for the angle of maximum range, 
6=17° 15’; and the ratio of the range itself to the corresponding variation in the dip 
was found, in this case, to amount to 3.275. I was thus led to expect that changes 
of dip, which would be inappreciable by the common needle, might be detected with 
this. It appeared, however, on trial, that the limits of error were likewise increased 
in this needle ; so that it was necessary to examine it in another point of view. 
Resuming the original equation 
4 cos 0=0 (6—8), 
and differentiating and dividing by the equation itself, considering 0, 3 and ¢ as all va- 
riable, we find 
cos 9 sin (6—6) “ +cos @ cos (6—6@) dé—cos 8 dd=0. 
This equation will give the error in the dip, or in the force, corresponding to a 
given error in the position of the counterpoised needle ; for, making dg and dé, suc- 
cessively, equal to nothing, there is 
ae cos 6 dé dp _ cos 6 dé 
me = cos 0 cos (8—0)° @ cos @sin (6—@) * 
(5) 
Hence, supposing the error in the position of the needle, d0, to be given, we can find 
the directions of the needle, in which the resulting error in the determination of the 
dip, or of the force, shall be a minimum. It is easily seen that dé is a minimum, when 
Ip . 5 3 
6=49; and that pis is least, when 9=48—45°. Hence, in our latitudes, for the de- 
termination of the dip and of the force, the counterpoises should be such as to bring 
the needle into the positions forming the angles +35° and —10°, respectively, with 
the horizon. 
o 
