—————————eeeeer TTT 
THE OBSERVATIONS OF DEFLECTION. 147 
and as, moreover, in practice the alternations can neither be 
effected instantaneously, or precisely at the prescribed intervals, 
it will seldom happen that the needle will be perfectly at rest 
after the operation is completed; however, the object will be 
sufficiently attained, if the remaining movement be so small that 
the observation of the true position of equilibrium can be at 
once begun in the usual manner. 
Practically, therefore, the above-mentioned suppositions will 
deviate but very little from the truth. The applicability of the 
magnetometer implies that the additional force shall produce 
only a moderate deflection, in which case (with an exception to 
be hereafter noticed) the law contained in the first supposition 
holds good with sufficient exactness. The alteration of the time 
of vibration by the deflecting force is quite insensible when that 
force acts perpendicularly to the magnetic meridian, as is almost 
always the case; but even if it were to act obliquely, inasmuch 
as it is but asmall fraction of the magnetic force of the earth, its 
effect in altering the time of vibration during the short time 
occupied by the operation, would still be quite immaterial. 
There is only one case in which an exception is to be made, 
namely, when the bar oscillates under the influence of a damper, 
by which the arc of vibration is considerably lessened,—in such 
case the above law no longer holds good, and the object would 
not be attained by the method described. On the other hand, 
the inconvenience mentioned in Art. 1 is much less, as a power- 
ful damper does of itself bring the bar to rest within a moderate 
time. As, however, in this case a modification only of the de- 
scribed method is required to obtain the same result, and as it 
must always be desirable to avoid all unnecessary loss of time, it 
is worth while in practice, as well as in theory, to consider the 
question in all its generality. 
4, We have first to solve the following general problems. 
A magnetic bar oscillates under the influence of repeated al- 
terations of the acting forces, the time of vibration and the loga- 
rithmic decrement* remaining, however, unaltered, and the are 
of vibration continuing sufficiently small to allow of quantities 
of the third order being neglected. We are required to deduce, 
from the circumstances of the initial movement, those which take 
place after the last change. 
Let T be the time of vibration, < the logarithmic decrement, 
* Resullate, 1837, p. 68. 
L2 
