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Different means have accordingly been used for avoiding the above inconveniences; and of those 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 experiments of BORDA and CASSINI, 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 a 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, 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 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 certain time, will again approach each other, when the same pheno-menon will take place. The interval between the two coincidences
* Base du Syst. Metrique, tom. iii. p. 343.
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 rass 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 132}"; and four of these, that is, five successive coincidences, gave an interval of 530" or 8 minutes 50 seconds; after which, the are 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 vibratio:s 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 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 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 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 penduluin 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 from 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 = 39.87076 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. The temp. of the scale 62° of Fahr, YOL. XXX. NO. 60.
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
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 kıfe 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. I'he theory of experiments made with the inverted pendulum, is there-, fore 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.
Captain Kater's paper is dated in July 1817, the experiments described in it having been made previously to that time. The same apparatus that was thus perfected has been employed since, for the purpose of ascertaining the length of the seconds pendulum in different latitudes, with a view to the questions about the figure of the earth. That the precise object of the experiments may be the better understood, it may be proper to go back to the summer 1816.
After the bill for the equalization of weights and measures was thrown out, the attention of those who promoted the scheme of equalization, was naturally turned to the determination of the lengths of the pendulum; so that one of the good effects arising from the disappointment of the premature plan of equalization, was probably that of directing the ingenuity of the author of this paper to a subject in which it has been so successfully exerted. This other good effect also resulted from it. The French academicians were known to have directed a great deal of attention to this subject; the experiments of BordA and CASSINI, SO often mentioned, were the most accurate that had yet been made; and the speculations of LAPLACE had deduced, from a collection of the best experiments that he could find, some very important conclusions concerning the figure of the Earth. On this subject, however, more information was still to be