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the same at different periods. Instructions for the accurate determination of the time of vibration will be given subsequently; but a very accurate knowledge is not requisite for the present purpose; and not only may the small variations to which it is subject be neglected, but it is even allowable to substitute the next full second for the accurate value, in order that the instant at which the observer has to determine the point of the scale under the vertical line of the telescope, may correspond always to full seconds. This happens of itself when the approximate time of vibration is an even number of seconds; when it is an odd number, one of the three following means may be chosen.

I. We may still keep to the nearest even number; and we may adopt this course the more readily if the difference between this number and the true value does not exceed half a second. The greater the time of vibration, the more easily will the needle be kept in a nearly quiescent state. The needle in the magnetical observatory at Göttingen has, for instance, at present, a time of vibration of 20^s·64; now, although the number 21 is here the nearest, yet we may generally employ the more convenient number 20^s, as the arc of vibration seldom exceeds a few divisions of the scale: it can easily be demonstrated, that the error originating thence cannot surpass the twentieth part of the arc in a partial result, or the hundredth in a final result. On the other hand, to an observer whose needle has a time of vibration of 10^s·64, and especially if he has not a like perfect quiescence at his command, it is recommended that the number 11 should be chosen, and one of the following modifications adopted.

II. Choose the odd number; but the instants of observation, which, according to the above formula, would fall on half seconds, must be taken either all half a second later, or half a second sooner; which obviously makes only this difference—that the final results do not correspond to the full minutes of clock-time, but to a half second more or less.

III. If the final result is not, as above, based on an odd, but on an even number of partial results, the times of observation fall of themselves on full seconds, whether the next entire number taken for the true value be odd or even. If, for instance, the final result depend on six partial results, then the times of observation are

$T-3t$, $T-2t$, $T-t$, $T$, $T+t$, $T+2t$, $T+3t$.