susceptible of, and require much greater exactness; and no such immediate determination is possible. It is not in our power so completely to quiet the needle of the magnetometer that it shall have no visible vibration; at least, it cannot be done with certainty, without some expense of time, and not for long; therefore, instead of an immediate observation, we must substitute such indirect modes of determination as do not require the entire absence of vibratory motion.
The most obvious method consists in observing the needle whilst in vibration; noting two successive extreme positions (a maximum and a minimum) on the scale, and taking a mean between them. This, in itself unexceptionable course, requires however some modification, if the vibrations are of considerable magnitude; and if the vibrations are small, is admissible only under limited conditions. In the first case, the progressive decrease of the arc of vibration in successive oscillations will not be insensible; consequently the deviation from the true meridian on the maximum side will be less than it was on the opposite side at the preceding minimum; and the mean of this minimum and its following maximum will be too small. From the same cause the mean of that maximum and the minimum that follows it will give too great a result. As the decrease of arc is nearly uniform for a few vibrations, the mean of two such means may be considered sufficiently exact, and may be taken to correspond to the instant of the second elongation. To express this by a formula, if , , are the readings for three successive elongations (it is indifferent whether the first and third are minima, and the second a maximum, or the reverse) represents the position of the magnetic meridian at the instant of the elongation . With small vibrations this course is only admissible when the declination does not undergo sensible changes in a short time; and, in such case, the mean between two successive elongations may be taken as the true position of the needle corresponding to the middle time: but this proceeding may be wholly unavailable when the declination is subject to sudden and considerable changes.
These methods of determining the position of the magnetic meridian from observed elongations, have always this inconvenience, that the instant to which the result obtained corresponds may not be that for which the position is required; and, although
