Page:Scientific Memoirs, Vol. 2 (1841).djvu/56

44 in the majority of cases this may not be very important, yet a preference is clearly due to a method which is free from this objection, and combines convenience, uniformity, and all desirable accuracy; it is accordingly the method adopted by those who take part in the term observations.

This method is founded on the principle, that the mean between two positions of the needle, which correspond exactly to two instants separated from each other by the time of one vibration, coincides with that position of the magnetic meridian which existed at the mean of these instants, in whatever parts of the vibrating period the instants might have fallen. This principle would be mathematically true, if, on the one hand, no external causes (such as the resistance of the air, etc.) occasioned the successive diminution of the arc of vibration; and if, on the other, any possible change in the situation of the magnetic meridian might be regarded as uniform for that short interval. The first circumstance has, however, no perceptible influence, if the method is applied when the vibrations are very small; and, in regard to the second, the changes of declination during so short an interval are generally of themselves hardly perceptible, and therefore we are the more justified in regarding such changes as at least uniform for the short intervals in question. Thus, therefore, the question is solved. In order to learn the position of the needle corresponding to the declination for the time $$T$$, it is only necessary, after the vibrations have been reduced by suitable means, to observe the actual positions for the times $$T-\fract$$, and $$T + \fract$$, and to take the mean; $$t$$ signifying the time of a vibration. For greater accuracy and certainty, however, similar determinations, and of equal number, should be made, at equal intervals, a few moments before, and a few moments after $$T$$: this being done, in as far as the alteration of the declination may be considered uniform for the time in question, the mean of all these results will be the final result corresponding to the time $$T$$, and will deserve more confidence than the single determination for $$T$$ itself.

The mode of performing this is very simple: if, for instance,