In the following Table, therefore, which contains the csults of his experiments, it is to be observed, that the correction for the height of the station above the level of the sea, is less than that given by the ratio of the squares of the distance from the centre of the earth.
Captain Kater's method of determining the length of the pendulum possesses several very decided advantages over that of Borda and Cassini. Of these advantages, one arises from a very curious property of the convertible pen dulum, which was not known when Captain Rater made his first experiments, but was subsequently discovered by the celebrated La Place. The property is, that if the supports of a pendulum of this kind be two cylindrical surfaces, the length of the pendulum is truly measured by the distance between those surfaces. Hence, no in accuracy can arise from the knife-edges being blunted, for in this case they may be supposed cylinders of very great curvature, or very small diameter ; and thus a circumstance which gave great alarm to Biot, becomes quite indifferent when the convertible pendulum is em ployed.
A nothd advantage attending the use of Captain Kater's pendulum is, that it supersedes the necessity of a num ber of measurements and calculations, in order to deter mine the centre of oscillation. In Borda's pendulum, before the centre of oscillation can be found, it is not only necessary to measure the distance from the knife-edge to the plane, which is brought into contact with the ball, but also the radius of the ball, and the distance from the knife edge to the suspension of the wire. Besides all this, the wright of the ball, of the wire, and of the cup, must be determined,—an operation scarcely less delicate than their measurement. These circumstances must render Borda's method, in ordinary hands at least, more liable to error than the other.
As experiments were made at two of the stations, viz. Leith Fort and Unst, both by Biot and Captain Kater, we possess the means of comparing their results directly with each other, independently of any hypothesis regarding the compression of the earth, or correction for the height of the station. The results given in the tables differ ex
tremely little; but to show their agreement more perfect ly, it is necessary to calculate the length of the seconds pendulum from the number of oscillations in twenty-four hours, without making any allowance for the elevation. At London, the number of the oscillations of Captain Kater's pendulum in a mean solar day, was 86061.30 ; at Leith Fort, 86079.22; and at Unst, 86096.84; (see Cap tain Kater's paper in the Phil. Trans. for 1819.) The length of the seconds pendulum, in the apartment at Port land Place, was 39.13908 inches ; hence, from the formula the length at the station of Leith Fort, at an elevation of 69 feet, is 39.15538 inches, and at Unst, at an elevation of 28 feet, 39.17141 inches. Leaving out the correction for the elevation, the lengths observed by Biot were, at Leith Fort, 39.15518 inches; and at Unst, 39.17166 inches. The differences are 0.00020 and 0.00025, and affected by contrary signs; Captain Kater's determi nation being in excess at Leith, and in defect at Unst. The mean difference is less than the part of an inch, a quantity scarcely appreciable in the observa tions.
The first part of the Philosophical Transactions of Lon don for the year 1822, contains an account of a series of experiments made at Madras in 1821, by John Golding ham, Esq. with a pendulum in every respect similar to Captain Kater's. It was constructed indeed under the immediate direction of Captain Kater himself, who also determined the number of its oscillations during a mean solar day in London, before it was sent to India, in order that they might afterwards be compared with the number at Madras. The apparatus was set up in the observatory at Madras, and the rate of the clock obtaitied, by compar ing it with the transit clock of the observatory each day at the commencement and conclusion of the observations. The transit clock itself was regulated by transits of the sun and stars; and indeed every part of the experiments was conducted with so much accuracy, as to render the result extremely valuable.