Mr. Horner, is still fundamentally alike. At the same time, the methods of investigation, and even the mechanical parts of the operations, are so entirely dissimilar, that no idea can be formed of the two solutions being derived from each other, or from any common original. We are, therefore, necessarily led to the conclusion that these two gentlemen, by following routes altogether different, have arrived very nearly at the same time at the same point of destination; and it is remarkable that it should be one which has been sought in vain by all the most eminent algebraists of the last two centuries. We shall take an early opportunity of noticing in a more particular manner the work of Mr. Nicholson above mentioned.

PART III.

An Account of Experiments for determining the Variation in the Length of the Pendulum vibrating Seconds at the principal Stations of the Trigonometrical Survey of Great Britain. By Captain H. Kater, F. R. S.-The subject of weights and measures having been at various times before the British parliament, an address was presented to the Prince Regent, in pursuance of a resolution of the House of Commons of the 15th of March, 1816, to the following effect:

"Resolved, that an humble address be presented to His Royal Highness the Prince Regent, that he will be graciously pleased to give directions for ascertaining the length of the pendulum vibrating seconds of time in the latitude of London, as compared with the standard measure in the possession of this House, and for determining the variations in length of the said pendulum, at the principal stations of the Trigonometrical Survey extended through Great Britain; and also for comparing the said standard measures, with the ten-millionth part of the quadrant of the meridian, now used as the basis of linear measure on (a part of) the continent of Europe." "

His Royal Highness having been pleased to comply with the prayer of this address,' the proper steps were taken to carry the proposed measures into effect; and Captain Kater, whose experiments relative to the length of a pendulum vibrating seconds in London we have before reported, was appointed to undertake the operations mentioned in the title of this article.

Provided with every necessary apparatus, Captain K. left London on the 24th of June, 1818, and began his operations

• Mr. Nicholson's publication appeared, we believe, in May or June, 1819, and Mr. Horner's paper was read before the Royal Society on the 1st of July.

at Unst, on the 10th of July, his instruments being all fixed and ready for observations by the 22d of the same month. Our readers are aware that Unst is one of the Shetland isles, where, during the preceding summer, M. Biot and Captain Mudge had undertaken a similar, course of observations, while Captain Colby and Dr. Gregory were carrying on a like series with the Ordnance-clock and zenith-sector, in the island of Balta; the distance between the two stations being only about two miles and a half. An account of the latter operations, as far as they relate to the clock, has been since published by Dr. Gregory, to which it may be necessary for us to refer, in one or two instances, in the course of our subsequent remarks on the present memoir.

The principal objects which Captain Kater had in view were, the determination of the latitudes and longitudes of his several stations, and the rates of his clock and chronometer, in order thence to deduce the number of vibrations made by his invariable pendulum in a mean solar day; which would of course indicate the change of intensity in the power of gravity at the several stations, from which the figure of the earth, or at least that of the British meridian, was to be inferred. The operations at each station were precisely of the same kind, the only difference being what might be caused by certain local circumstances. We shall, therefore, content ourselves with slightly indicating the mode of proceeding at the first station, and then confine our remarks wholly to results, with such general observations as they may suggest.

The place selected for the pendulum-experiments was an unfinished cottage belonging to Mr. Edmonstone, very near to M. Biot's former station. One of the walls of this cottage was three feet thick, and to this the iron frame for the invariable pendulum was firmly fixed, and, at a proper distance below it, was placed the clock. The telescope for observing the coincidences was fixed on its proper stand at a convenient distance in front, exactly as described in Captain Kater's former paper. (See M. R. vol. lxxxvii. p. 50.)

The next object was a proper support for the transit-instrument; which being accomplished, and the bell-tent erected over the spot, it only remained to bring the telescope into the plane of the meridian, which was thus effected:

The interval of time between the transits of the same star being all that is required for the present purpose, it is not necessary that the transit-instrument should be accurately in the meridian; it is sufficient that it should always describe the same vertical eircle; it was however brought very near the meridian, at all the stations, by the following method:

• The

The error of the chronometer was determined by altitudes of the sun, and the times were computed when the first and last limb would be on the meridian.

The axis of the transit was carefully levelled, and a little before the time of the sun's first limb coming to the meridian, the middle wire of the transit was brought in contact with it, and kept so by the horizontal adjustment till the calculated time of its arrival on the meridian. The position of the instrument was afterwards farther corrected if necessary by the transit of the second limb. At other of the stations, when the weather permitted, the instrument was brought extremely near the meridian by the transit of the pole-star, the telescope being sufficiently powerful to command this star with ease, at any time of the day.

A mark (generally a flat board sharpened at one end to penetrate the ground) was sent to as great a distance as convenient, and so placed by signal, that it was bisected by the middle wire of the transit; and to this the instrument was carefully adjusted previously to every observation. The preceding detail may serve, with very little difference, for each of the stations, and I have been thus minute in my description of the various adjustments necessary, in order that no difficulty may be experienced by any who may use the pendulum after me.

In observing the time of the transits, the chronometer was used, and was found to be particularly convenient from its beating half seconds. As soon as possible after the passage of the star, the chronometer was carefully compared with the clock, and the difference being applied to the time of the transit shown by the chronometer, and also the computed gain or loss of the clock during the interval between the observation and the comparison; the time shown by the clock at the instant of the transit was obtained.

These comparisons, as well as the whole of the data necessary for the examination of the results given in this paper, will be found in the Appendix.

Every thing being now ready, the transit-observations were begun on the 22d of July, and continued to the 28th. The pendulum-experiments were commenced on the 23d; and at the same time the zenith-distances of the sun's upper limb were taken for the latitude. Of the experiments and observations we cannot in course give any details; and it will be sufficient to indicate the nature of the corrections and reductions afterward made in order to reduce all the results to those that would have taken place at the same temperature and level, and in vacuo.

As the pendulum is vibrating in a fluid, it is obvious that a part of its gravity will be counteracted by the buoyancy of that fluid, or of the atmosphere; and, consequently, the number of vibrations observed will be less than they would be in the same time in vacuo, the difference being greater or less

under

under different degrees of temperature and of barometrical pressure. After all, however, this correction is so minute that we consider the introduction of it as exhibiting rather the appearance of accuracy than the real attainment of it; if it had been entirely omitted, it would not have made a difference amounting to one-fifth of a vibration in twenty-four hours; the greatest correction arising out of this considerThe correction for ation being 6.09, and the least, 5'91. level is a little different from the one that is usually employed, which rests on the erroneous assumption that the mass of any hill, mountain, or table-land, producing the elevation, has no effect in accelerating the pendulum; whereas it is evident that, although the diminution of gravity, as it regards the general mass of the earth, is less as we ascend a mountain, we have the additional attraction of the mountain itself, which has a tendency to diminish the error as it regards distance only. The value of the additive part of this correction is in course almost a matter of assumption: but still, by a little attention to the strata and to the form of the ground, an approximation may be obtained. The reductions for temperature were of the usual kind.

We must now leave the detail of the operations, and confine ourselves to results only; the first series of which are exhibited in the following table:

It is to be observed that the scale of inches, in which the measures in the last column are given, is that of Sir George Shuckburgh, at the temperature of 62°. If, now, the earth were of uniform density, or if the density varied according to any law from the surface to the centre, and if its figure were any one of revolution, the ratio of its axes might be immediately inferred from the preceding experimental data: but

we

we know that neither of these conditions actually obtains; and it is therefore by no means extraordinary that the fractions expressing the ratio in question, or that of the compression, are subject to such irregularities as those that are exhibited in the following table; in which the compression is computed by comparing the result obtained at each station with all the others:

We have made the above extract in order to draw from it the following important inference, viz. that we never can expect to arrive at greater accuracy than we now possess relative to the figure of the earth from pendulum-experiments; these results including degrees of compression which exceed the extreme limits due to the two hypotheses of Newton and Huygens; namely, 3 and 5. It is true that the anomalies due to a particular stratification will become the less apparent as the stations are the more distant: but still we shall never, in any case, be able to eradicate them entirely, nor, perhaps, approach so nearly towards it as Captain K. seems to imagine. He observes:

It must be evident that nothing very decisive respecting the general ellipticity of the Meridian can be deduced from the present experiments. For this purpose it is requisite that the extreme stations should comprise an arc of sufficient length to render the effect of irregular attraction insensible; and this effect might be diminished, if not wholly prevented, by selecting stations of

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