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0.50".63. dded the brations any other
during the observed interval ; consequently, for any other lesser interval and rate, the mean of the vibrations during such interval is taken, and to this is added the difference between the corresponding rate and 50".63, which corrected is positive or negative, according as the rate of the clock has diminished or increased. Proceeding in this way, results were obtained for seven different intervals, the greatest of which was from the 22d to the 28th of July—the least from the 26th to the 28th. In four of these intervals, the rate of the clock was deduced from observations upon stars; and in the other three, from observations upon the sun. But before employing those seven results to obtain a mean, it is necessary to attend to the errors which are likely to accompany each. In observations on the stars, the chief source of error will arise from the position of the transit instrument with respect to the meridian mark, a flat board fixed in the ground at a distance from the transit, and so adjusted, that when the middle wire of the transit bisected it, the instrument was nearly in the meridian. This board was erected for the convenience of more readily placing the transit in the same position previously to every observation; and so much depended on the accuracy of the position, and on the levelling of the instrument's axis, that a deyiation only equal to, the diameter of the silk-worm's thread in the focus of the eye. glass, was found to occasion an error, in the time of transit, amounting to three-tenths of a second. The greater the number of days between the two transits, the less will this error affect the daily rate of the clock; because the whole amount of the error is divided by the number of days which compose the in, terval between the two transits. The accuracy also in a great measure depends on the number of stars observed. It thus appears, that a correct mean will be obtained by multiplying the result for each interval by the product of the number of stars into the interval, and then dividing the sum of the final products by the sum of the factors. In this way the ultimate mean obtained was 86090.77 vibrations in 24 hours, by observations of the stars; and in like manner, by observations of the sun, considering the transits of both limbs as equal to the transits of two stars, we find the vibrations amount to 86090.79. Now in the case of the stars, the sum of the multipliers is 50; in that of the sun, 16; and as the accuracy of the results is in the ratio of those sums, that is, as 8 to 1 nearly, we are entitled to take the final mean equal to 86090.77 vibrations in a mean solar
The next correction to be applied, is the allowance for the height of the station above the level of the sea. This is readily
obtained from the consideration that the force of gravity varies inversely as the squares of the distance from the Earth's centre; and this force is represented by the square of the number of vibrations of the pendulum. Therefore, if we divide the height of the station by the radius of the Earth, and multiply the number of vibrations in 21 hours by the quotient, the correction will be obtained. Now, in á valuable paper published by Dr Young in the Phil. Trans. for 1819, Part I., upon the density of the Earth as affecting the reduction of experiments on the pendulum, some conjectures are hazarded as to the effect which inay be produced by the attraction of the elevated part that lies between the general surface and the place of observation; and as to the allowance to be made for this, in reducing an elevated place of observation to the level of the sea, the meaning of which appears to be merely this, that if we make an observation upon the motion of a pendulum at the height of 100 feet, for example, above the level of the sea, then, in order to bring our observation to the level of the sea, not only is a correction necessary for the elevation of 100 feet, at which the observation was actually made, but a further correction is required, to compensate the attraction produced by the matter accumulated between the level of the sea and the higher position. Putting for the present out of view the accuracy of Dr Young's estimate of the probable amount of this equation, we miay observe that Captain Kater seems to have mistaken the import of Dr Y.'s statement, when he uses this correction for the attraction of matter surrounding the elevated situation. That statement applies only to the attraction of the elevated part interposed be
tween the general surface and the place of observation,' (Phil. Trans. 1819, Pt. I. p. 93), nothing being said of lateral attraction caused by surrounding matter. But Captain Kater applies the correction for the error produced by hills lying round the point of observation; and says, the height of the station at
Unst was found to be 28 feet above low water; whence we • have 0.12 for the correction, as deduced from the squares of " the distances from the Earth's centre; and as the station at • Unst was surrounded by hills composed of serpentine, I shall « take 0.12 X = 0.06 for the correction to be applied in or
der to obtain the number of vibrations which would be made
at the level of the sea.' (Phil. Trans. 1819. Pt. III. p. 354.) It may be said, that the smallness of the quantity makes it immaterial; but in this investigation, extreme accuracy is the only object, and no quantity, however minute, can be disregarded. But suppose Captain Kater's application of the correction was according to Dr Young's true meaning, which we conceive a reference to his own words has disproved, still we think the amount of the correction, as given by that author, should have been adopted with caution in an inquiry like the present, both because the method of arriving at it is somewhat too conjectural, and also because, admitting its general accuracy, we can hardly allow it to be the precise equation for Captain Kater's case. • It is obvious,' says Dr Young, that if we were raised on a sphere of earth a mile in diameter, its attraction would be about sooo of that of the whole globe, and instead of a 'reduction of doo in the force of gravity, we should obtain only gooo, or three-fourths as much ; nor is it at all probable, that the attraction of any hill a mile in height would be so little as this, even supposing its density to be only two-thirds of the medium density of the Earth; that of a hemispherical hill would be more than half as much more, and in the proportion of 1.586 to 1; and it may easily be shown that the attraction of a large tract of table land, considered as an extensive flat surface a mile in thickness, would be three times as great as that of a sphere a mile in diameter, or about twice as great as that of such a sphere of the mean density of the Earth; so that, for a plane so situated, the allowance for elevation would be reduced to one-half; and in almost any country chosen for the experiment, it must remain less than three-fourths of the whole correction, deduced immediately from the duplicate proportion of the distance from the Earth's centre. Supposing the mean density of the Earth 5.5, and that of the surface 2.5 only, the correction for a tract of table land of a mile in thick
3 2.5 - 66 of the whole.' ness, will of course be reduced to 1-3
4.5.5 100 (Phil. Trans. 1819, p. 93.) If then be the correction for an elevation of one mile, on the supposition of its being filled by a solid ring of earth, we cannot perceive the grounds on which Captain Kater takes , only a little less than 16, for the correction applicable to an elevation of 28 feet in the actual state of the superficial inequalities. We may have overlooked some step in his reasoning, or in Dr Young's, but we feel bound to state our difficulty as it occurs.
One other equation of error remains, and that is for the buoyancy of the Atmosphere. The specific gravity of the pendulum was taken at 8.61(); and it was found to be, at the time of making the experiments, to the specific gravity of the Air, as 7.099 to l. This ratio expresses the diminution of the force of gravity arising from the buoyancy of the atmosphere; but the force of gravity varies directly as the length of the pendulum, or inversely as the square of the number of vibrations. Hence, if the square of the number of&vibrations in 24 hours be increased in the ratio of 7.099 to 1, that is, if 6.07 be added to the number of vibrations, the number in vacuo in the same time will be obtained. We have already stated the mean number of vibrations to be 86090.77, as determined by observations of the coincidences of the clock and pendulum : We have, therefore, 86096.84 for the number made by the pendulum in a mean solar day in vacuo; to which must be added the correction for elevation above the level of the sea, or 0.12. Captain Kater deducts from this 0.06, to allow for attraction. We have given our reasons for holding this to be too large an allowance, and we should think 0.12 sufficiently near the truth, without any allowance, for so small a height as 28 feet. According to Captain Kater, however, the corrected number of vibrations in vacuo, and at the level of the sea, is 86096.90..
On the 29th July, having finished his experiments at Unst, Captain K. proceeded to Portsoy, the next station of the Survey, where he arrived on the ist of August. By a process of precisely the same kind with the former, he ascertained the number of vibrations there to be 86086.01 in vacuo and at the level of the sea. The following Table exhibits the results of his observations at all the stations, the experiments being the same at each. They were concluded at the Isle of Wight on the 16th of May 1819.
The instrument used in-determining the latitudes of these stations, was the repeating circle of one foot diameter, made by Troughton; and we cannot omit recording the close agreement which appears between those observations and the latitudes as determined by Colonel Mudge with the zenith sector in the Trigonometrical Survey. This is, in justice, due to the accuracy of that skilful observer; because in a paper of Don Joseph Rodriguez, in Phil. Trans. for 1812, some doubts were expressed upon the subject. In the measuremerit of an arc of the meridian extending nearly three degrees, from Clifton in