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of 120 degrees. Every precaution was used to render the edges true, or straight, and to give the hardest temper to the steel; and a long series of experiments proves fully that they have been successful. Every precaution was also taken to give stability to the axes of suspension, when the experiments were made: But for the details of these, we find it necessary to refer to the paper itself.
We come now to the very ingenious method which Captain Kater adopted for determining the number of vibrations made by his pendulum in twenty-four hours. It is no doubt sufficiently understood, from what has been already said, that the pendulum was not to be applied to a clock, nor to receive its motion from any thing but its own weight. When experiments of this kind were attempted, it was for a long time supposed that the pendulum might safely be permitted to receive the continuance of its motion from machinery; and that, as it was then in no danger of coming to rest, the results were more to be depended on. This conclusion, however, proceeded on a great mistake as to the part which the machinery of the clock performs on such occasions. That machinery is hardly ever, we believe, so nicely adjusted as accurately to restore to the pendulum the motion it loses in each vibration, (from friction about the centre, and from the resistance of the air), without either allowing any defect, or producing any excess. A clock, in general, accelerates the natural motion of the pendulum, and forces it to vibrate faster than it would do if impelled only by its own gravity. In experiments, therefore, where the relation of the length of the pendulum to the time of vibration is to be determined, the clock can only be used to measure out a given portion of time, or to assist in numbering the vibrations.
The manner in which this last can be done, is not so obvious as may be imagined. The mere counting of the vibrations one by one, and marking the number at stated intervals of time, would be a very inconvenient and imperfect way of going to work. As the experiment must be long continued, and frequently resumed, the tedium and irksomeness of counting the vibrations would become great, and, like every labour that is tedious and irksome, must be in danger of being inaccurately performed, more especially by mathematicians, the persons into whose hands the operation is most likely to fall. Even if no error were committed, there would still be an insecurity which nothing could remove. It is, indeed, the business of every experimenter to throw as great a share of the responsibility as he can on his apparatus, or on the physical agents he employs; and as little as possible on himself and his living assistants. Different means have accordingly been used for avoiding the above inconveniences; and oftho.se that we are acquainted with, we think Captain Kater's is the best, the least tedious, and the most infallible.
Boscovich, in the 5th volume of his Opera Opt. et Astr. gives an account of a method which he had employed, and which he ascribes to Mairan.
A clock being well regulated, according to mean time, and having its case open, the experimental pendulum was placed right before it at a little distance, with its point of suspension firmly supported. The position of both was such, that, in their state of rest, the pendulums were seen by a person placed in front of them, coinciding with one another, and with a vertical line drawn on the clock-case behind the pendulum. That this coincidence might be more distinctly seen, when it happened to the moving bodies, it was viewed through a hole in a piece of paper fixed to the back of a chair on the opposite side of the room. The two pendulums having been put in motion, and not vibrating exactly in the same time, one would gain upon the other, and after a while they would be seen through the hole in the paper to coincide with another, and with the fixt line on the body of the clock. The instant of this coincidence must be noted. When they next coincide, the difference of the times of their vibrations must have amounted to one entire vibration. This is also to be noted; and thus the information of the clock 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 experiment!? of Borda and CASsixi,'was similar, in many respects, 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 H small telescope, fixed on a stand on the opposite side of the room, and a skreen was also placed before the pendulums, the edge of which just covered the wire of the platina pendulum,nn<l 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 paper, 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 an observer on the opposite side of the room, by one of the slips of deal which form the extremities of the brasspendulum. 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.
'Supposing 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 certain 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 aj-, certain time, will again approach each other, when the same phenomenon will take place. The interval between the two coincidences
* Base du Syst. Metrigue, tom. iii. p. 34-3*
will give the number of vibrations made by the pendulum of the clock; the number of vibrations of the brass pendulum is greater by
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 132J"; 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. Thc 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 method 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 observation^ 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. * 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 to be made, we think the probability is very great that the preceding result does not err so much as a unite 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,
* The scale on which this pendulum is measured, is Sir George Shuckburgh's, the work of Trouguton, 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 from these measures to be as the numbers 1 ; .999634-64; 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 34.37074 inches. ^'he temp. of the scale 62° of Fahr. Vol. xxx. No. 60. E e