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will give the ratio of the time of its own vibrations to the time of those of the pendulum. This experiment must be often repeated, and a mean taken, that if there are any accidental
errors, there may be a probability of their balancing one another.
The method of numbering the vibrations in the experiments of BORDA and CASSINI, was similar, in many repects, to the preceding, and may have been suggested by the same to which BosCovich refers, that of their ingenious countryman, MAIRAN.
The pendulum was placed, as in the former example, right before the clock with which it was to be compared, so that the wire by which the platina ball was suspended, bisected the ball of the clock pendulum when at rest ; the middle point of this last being marked by the intersection of two white lines drawn on a black ground. The two pendulums were viewed through a small telescope, fixed on a stand on the opposite side of the room, and a screen was also placed before the pendulums, the edge of which just covered the wire of the platina pendulum, and therefore concealed behind it one half of each of the balls. The platina pendulum was nearly 12 feet long; so that it made about one vibration while the pendulum of the clock made two.
Suppose, now, that when the pendulums were put in motion, the wire disappeared behind the screen, before the cross; as the times of the vibrations are not supposed accurately as 2 to 1, it would happen that the interval between the disappearances would decrease, till at length both objects came to pass behind the screen at the same instant. The instant of this first coincidence was observed ; the oscillations then began to disagree, afterwards to approach, till at length a second coincidence took place. In the interval between the coincidences, the clock had gained two seconds on the pendulum ; so that the ratio of the times of the vibrations of the two pendulums was given.
Captain KATER's pendulum was compared with two clocks, the property of H. Browne, Esq., in whose house the experiments were made. One of these, a time-piece by Cumming, is of such excellence, that the greatest variation of its daily rate, from the 22d of February to the 31st of July, did not exceed three tenths of a second. The clock, however, with which the immediate comparison was made, and in front of which the pendulum was placed, was one of Arnold's, also of excellent construction. The pendulum was securely suspended in front of this last, and close to it, so that it appeared to pass over the centre of the dial-plate, with its extremity reaching a little below the ball of the pendulum. A circular white disk was painted on a piece of black per, which was attached to the ball of the pendulum clock, and was of such a size, that, when all was at rest, it was just hid from
a Base du Syst. Mètrique, tom. iii. p. 343.
an observer on the opposite side of the room, by one of the slips of deal which form the extremities of the brass pendulum. On the opposite side of the room was fixed a wooden stand, as high as the ball of the pendulum of the clock, serving to support a small telescope, magnifying about four times. A diaphragm in the focus was so adjusted as exactly to take in the white disk, and the diameter of the slip of deal which covered it. "Suppo
sing now both pendulums set in motion, the brass pendulum a • little preceding the clock, the slip of deal will first pass through
the field of view at each vibration, and will be followed by the • white disk. But the brass pendulum being rather the longer,
the pendulum of the clock will gain upon it; the white disk • will gradually approach the slip of deal, and at length, at a cer‘tain vibration, will be wholly concealed by it. The instant of - this total disappearance must be noted. The pendulums will
now appear to separate ; and, after a certain time, will again approach each other, when the same phenomenon will take place. The interval between the two coincidences will give the number of vibrations made by the pendulum of the clock; the number of vibrations of the brass pendulum is greater by two.'
Thus was determined the number of vibrations made by the brass pendulum in a given interval of time: and so, by proportion, the number for a whole day. The interval between the two nearest coincidences was about 1323"; and four of these, that is, five successive coincidences, gave an interval of 530", or 8 minutes 50 seconds; after which, the arc described by the brass pendulum became too small. The pendulum was then stopped, and put in motion anew as oft as it was judged proper to repeat the observations.
Being now in possession of the means of determining, with great accuracy, the number of vibrations performed by his pendulum in a given time, Captain KATER proceeded, by reversing it, to make the vibrations equal in its two opposite positions. The sliding weight mentioned above was used for producing this equality; which, after a series of most accurate and careful experiments, was brought about with a degree of precision that could hardly have been anticipated. By the mean of 12 sets of experiments, each consisting of a great number of individual trials, with the end of the pendulum, which we shall call A, uppermost, the number of vibrations in twenty-four hours was 86058.71 ; and, with the same end, A lowest, the mean of as many others gave 86058.72, differing from the former only by a hundredth part of a vibration. The greatest difference was .43, or less than a half. Such exactness, we believe, has never been exceeded ; and would hardly be thought possible, if the data from which so satisfactory a result was deduced were not given in full detail in the paper before us.
Thus, for the first time, after having been an occasional object of research for more than 150 years, has the centre of oscillation of a compound pendulum been found by experiment alone, according to a inethod also of universal application, and admitting of mathematical precision. The ingenious author has therefore the honour of giving the first solution of a problem, extremely curious and interesting in itself, independently of its immediate connexion with one of the greatest and most important questions in the natural history of the Earth.
The next thing to be done, was to measure the length of the pendulum, or the distance between the knife edges, which had alternately served as the centres of suspension and oscillation, and from thence to deduce the length of the pendulum vibrating seconds in the latitude of London, which, at the spot (Mr. Browne's house in Portland Place) where the observations were made, is 51° 31' 8".4. It is sufficient here to state, that no expedient has been neglected that practical or theoretical science is at present in possession of; for giving precision to this measurement, and that it was in all respects such as to correspond to the accuracy of which we have just seen so striking an example. Including the effects of temperature, of the buoyancy of the atmosphere, of the shortening of the arcs of vibration from the beginning to the end of each trial, and reducing the actual vibrations to those in arcs infinitely small, the length of the seconds pendulum, from a mean of the 12 sets of experiments above mentioned, comes out 39.13829 inches, or 39.1386, reducing it to the level of the sea.a The greatest difference between this result and any one of the 12 of which it is a mean, is .00028 of an inch; that is, less than three of the ten thousandth parts. The mean difference among these results, adding the positive and negative together, as if they had all one sign, or were all on the same side, is little more than one ten thousandth of an inch; and as the above is obviously a supposition more unfavourable than ought
a The scale on which this pendulum is measured, is Sir George Shuckburgh's, the work of Troughton, and of the highest authority. It is described by Sir George in the Phil. Trans. for 1798. Gen. Roy's scale, which is very important, as being that from which are derived all the measurements in the trigonometric survey, was compared with the preceding by Captain Kater. So also was the yard on what is called the parliamentary standard, which was laid off by Bird, but it would seem not so carefully as might have been expected. The scales in the order in which they are now named, appear froni these measures to be as the numbers 1 ;.99963464; 1.00000444.
In another communication from Captain Kater, in the same volume of the Phil. Trans. the length of the French metre is compared with the yard on Sir G. Shuckburgh's scale. He found the metre as marked by two very fine lines on a bar of platina 39.37076 inches on his scale; as marked by the ends of a metal rod in the usual way, the metre 39.37081. Supposing the two of equal authority, the mean length of the metre is 84.87074 inches. The temp. of the scale 62" of Fahr.
to be made, we think the probability is very great that the preceding result does not err so much as a unit in the last decimal place, or in that which denotes ten thousandths of an inch.
The determination given above is considerably different from that which had been received on the authority of the older experiments. The length given to the seconds pendulum, in the bill for the equalization of weights and measures, is 39.13047, differing from that just assigned by .00813; a considerable quantity, in a matter where it appears that a ten thousandth of an inch is a distinguishable magnitude.
To the paper which ends with the measures just given is added, in an Appendix, a letter from Dr. THOMAS YOUNG, containing a demonstration of a very remarkable property of the pendulum recently discovered by M. LAPLACE. The property is, that if the supports of a pendulum, inverted as above described, be two cylindric surfaces, the length of the pendulum is truly measured by the distance of those surfaces. This applies immediately to the experiments we have been considering; because the knife edges, supposing them somewhat blunted, may be regarded as cylindric surfaces of very great curvature, or of very small diameter; and in this way, as Dr. Young very justly remarks, is removed the only doubt that can reasonably be entertained of the extreme accuracy of the conclusions. The theory of experiments made with the inverted pendulum, is therefore much indebted to the calculus of the profound mathematician above named. We have not seen his analysis ; but a demonstration is sketched by Dr. Young, that seems sufficiently concise and simple, considering the recondite nature of the truth to be demonstrated.
Art. III. An Account of Experiments for determining the Variation
in the Length of the Penduluin vibrating Seconds at the principal Stations of the Trigonometrical Survey of Great Britain. By Captain H. KATER, F.R. S. From Pbil. Transactions. London, 1819. Part. III. (Review-Nov. 1820.]
We have now to direct the attention of our readers to a more extended investigation of the same careful observer, by which he has ascertained the length of a Seconds Pendulum, at the principal stations of the great survey of this Island.
It may be recollected, that this inquiry originated in a bill submitted to Parliament, for the general regulation of Weights and Measures, and fortunately thrown out in the House of Lords. We say fortunately,-because those who most readily admit the expediency of adopting some uniform system, will naturally be the first to reject a plan so crude and so ill calculated to attain that desirable object. One good, however, resulted from the discussion; an address was presented to the Crown, praying
416 Edinburgh Review on the Length of the Pendulum.
that instructions might be given for determining the length of a Seconds Pendulum in the latitude of London, as compared with the standard made for the House of Commons in 1758, known by the name of Bird's Parliamentary Standard—for ascertaining the variations in the length of the Pendulum at the different stations, and for comparing the standard measures with the ten millionth part of the quadrant of the meridian, the basis of linear measure in France. In order to carry this purpose into effect, a Committee was appointed by the Royal Society; and Captain Kater, a Member of it, was desired to conduct the inquiry. The choice has been amply justified by the success which has attended his labours.
[We regret that we have not room for the detail given of the experiments made at one of the Shetland Isles—the first station.]
On the 23d of July, 1818, he began to observe the coincidences of his two pendulums; and he found from two series of experiments, each of ten intervals, taken on each day, the mean number of vibrations in 24 hours, the temperature being corrected for 62°. The number of vibrations for each day of the intervals, was deduced from the rate of the clock, during the observed interval; 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. Before employing those seven results to obtain a mean, he attended to the correction of errors.... The chief source of error arises from the position of the transit instrument with respect to the meridian mark : &c.... The next correction 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. [Here a mistake of Capt. Kater is referred to, in drawing an erroneous conclusion from the statement of Dr. Young, in the Phil. Trans., relative to the effect produced by the attraction of the elevated part lying between the general surface and the place of observation.]....Another equation of error is for the buoyancy of the atmosphere.
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.
Length of the Pendulum vibrating Seconds, in parts
of Sir G. Shuckburgh's scale. Unst
60° 45' 28".01 86096.90 39.17146 inches Portsoy
39.16159 Leith Fort
55 58 40 .80 86079.40 39,15554 Clifton
53 27 43.12 86068.90 39.14600 Arbury Hill
52 12 55 .32 86065.05 S9.14250 London
51 31 8.40 86061.52 39.13929 Shanklin Farm
Place of Observation.
57 40 58.65
50 87 23.94