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other; in other words, the figure must be symmetrical, but the mass not so. This may be accomplished by making the weights of equal size and alike in form, and placing them symmetrically in regard to the knife-edges, but having one hollow and the other solid. By this construction, the influence of the air will be eliminated, and the height of the meteorological instruments will be only important, as concerns those variations which may take place between the experiments on the one edge, and on the other. This is the method suggested by M. Bessel. The second method is, to make the experiments altogether in a vacuum in this case, symmetry of construction ceases to be a condition; and the resistance of the air to the motion of the pendulum being removed, the vibration will continue until brought to a close by the mutual action of the plane and knife-edge, enabling the time of vibration to be determined with an increased accuracy, proportioned to the duration of period in which the vibration is continued without fresh impulse. It is probable, that a combination of the two methods may prove, on trial, more satisfactory than either separately; the pendulum being constructed symmetrically, and the experiments made with it both in air and in a vacuum; when the results of both experiments should be the same, if
the conditions are in both cases fulfilled.
Experiments have recently been made in this country, by which the reduction to a vacuum of an invariable pendulum, of the same external form as those which have been lately employed by British officers in determining the relative length of the pendulum in different latitudes, has been obtained directly by its alternate vibration in air of full pressure, and in rarefied air approaching to a vacuum. It appears, from these experiments, and from an examination of the apparatus contrived for the purpose, that, by its means, a convertible pendulum can also be experimented with in a vacuum, or in a rarefied medium extremely near a vacuum, with scarcely less facility than in air, and with increased accuracy in several other respects, besides those which are connected with the influence of a medium. It has resulted, from the experiments with an invariable pendulum, that the true reduction of its vibration to a vacuum is to the correction for buoyancy as
previously computed, in the proportion of about 10.36 to 6.26. The co-efficient of the additional part of the reduction is, in this case, therefore, 0.655.
In considering the new view of the reduction to a vacuum, in reference to the method of obtaining the absolute length of the seconds pendulum by the apparatus of MM. Borda and Biot, in which the experimental pendulum consists of a platina sphere and metallic thread, we might not be far wrong in taking the same co-efficient for the addition to the inertia of the pendulum as applicable to the platina sphere, that M. Bessel derived from his experiments with the brass and ivory spheres. Taking also M. Biot's estimate of the relative densities of platina and air at 15910 to 1 at 32° of temperature and 30 inches of atmospheric pressure, the reduction to a vacuum of the vibration of his experimental pendulum would exceed the amount introduced in the reduction of the experiments with that apparatus at Paris and elsewhere, by about 2 vibrations a day. The length of the simple seconds pendulum derived will consequently be too short by between two and three thousandths of an inch. The published results with that apparatus are, however, capable of being rigorously corrected, whenever the reduction to a vacuum of the identical platina sphere and its suspension thread shall have been experimentally determined. This may be accomplished, either by a proceeding similar to M. Bessel's-viz., by employing an ivory sphere in comparison with the platina sphere; in which case the difference of the specific gravities of platina and ivory much exceeding the difference between those of brass and ivory, the circumstances of the experiment will be more favourable than in M. Bessel's experiment; or by vibrating the platina pendulum in the vacuum apparatus, alternately in air and in the highly rarefied medium which can be readily produced and maintained in that apparatus, equivalent to a pressure of less than one-tenth of an inch, or th part of the pressure of the ordinary atmosphere. In making this experiment, no measurement of the length of the pendulum, which is by far the most difficult and uncertain part of the operations in M. Biot's apparatus, would be necessary, as nothing more
is required than the relative vibrations of the same pendulum in media differing widely in density-a question altogether independent of absolute length. The circumstances influencing the density of the medium and the temperature of the pendulum would be the principal and almost the only objects requiring particular attention.
M. Bessel's view of the incorrectness of the theory by which pendulum experiments have been hitherto reduced to a vacuum, and of the necessity of seeking the true reduction by experiment, must now be considered as fully established; and we are obliged to conclude, that the determinations of the length of the pendulum in England and France, to which the standard measures of the two countries have been more or less formally referred, are in correct; and that, not merely to an amount scarcely surpassing in magnitude the accidental errors to which different parts of the respective operations are liable in single determinations, the correction of which might be deemed by many an attempt to obtain an accuracy more apparent than real; but in magnitude, probably, not less than two or three inches in, a thousand yards of British measure. The admission of this fact does in no degree disparage the great mechanical ingenuity and admirable patience in experimenting that has been manifested by the distinguished persons to whom we owe the methods and the results hitherto obtained. Nor does any thing that has yet appeared in regard to the influence of a medium, prejudice the eventual application of either of the methods in obtaining a true result. But the evidence of facts too strong to be passed by is now afforded, in support of reasoning that had been previously urged, to shew that a conclusion of so much national and scientific importance ought not to be allowed to rest on less than the best evidence that the ingenuity and labour of the age can supply; and that any conclusion ought only to be considered as premature, that is not the concurrent result, not only of all the methods of inquiry which are capable of yielding a just result on competent trial, but also of the repetition of each method, with such variation of circumstances as might serve to detect. errors: from unsuspected sources. Had, for example, the expe-,
riments of Biot been varied by the substitution, in but a single experiment, of a sphere of brass or of other metal, or of any other substance whatsoever, for the sphere of platina ;—or had the experiments of Kater been repeated in air with a reciprocal pendulum, differing in external figure from the one constructed by him, so as to affect the reductions to a vacuum ;—or had they been repeated even with the same pendulum in atmospheric circumstances widely different ;-discrepancies would have appeared, which might have led to an earlier discovery of what M. Bessel is understood to have arrived at, by reflecting on apparent discrepancies occurring in his preliminary experiments, which have now received their explanation.
A very near approximation to the true length of the seconds pendulum in London may be expected from the experiments now in progress, with the original pendulum constructed by Captain Kater, used in the vacuum apparatus. By shifting the place of the weight, this pendulum will be made convertible in a vacuum, and the number of its equal vibrations, free from the influence of a medium, directly ascertained. Presuming the weights to sustain no change of place by the reversal of the pendulum, and the distance between the knife-edges as measured by Captain Kater to be the equivalent length, we should have by this means the true length of the seconds pendulum.
A second approximation will also be shortly obtained by other experiments in progress, for transferring M. Bessel's measurement at Königsberg, to London, by means of invariable pendulums. An invariable pendulum of French construction, the rate of which was ascertained in the Salle de la Méridienne at Paris by M. Arago, has been sent to Königsberg, and comparative observations made with it by M. Bessel, in his observatory. Another invariable pendulum has been sent from London, the rate of which had previously been ascertained by myself, both at the Salle de la Méridienne in Paris, and in Mr. Browne's house in London. This pendulum has also been used at Königsberg; and both pendulums are now on their way to London, returned by M. Bessel for verification; when
JAN. MARCH, 1829.
the French invariable pendulum, which differs in external figure from the British, may also have its true reduction to a vacuum ascertained by means of the apparatus for that purpose; whereby that necessary element will be furnished for the correction of the results obtained with similar pendulums, in the voyages of the Captains Freycinet and Duperrey.
Very delicate and important questions, affecting the measurement of the seconds pendulum by other methods than M. Bessel's, are those of the influence of the cylindrical figure of the knife-edge, and of the elasticity of the supporting planes on the position of the axis of motion in the knife-edge suspension. The influence of the cylindrical figure of the edge was first brought into notice by M. Laplace, whose investigation was grounded on the supposition, that during the motion of the pendulum the knife-edge rolls on the planes on its general cylindrical termination; and that no gliding motion whatsoever takes place, either in the direction in which it would be opposed by friction, or in the contrary direction. The breadth of the cylindrical termination is not, however, sufficient to make known the influence of the departure from a perfect edge on the times of vibration. This depends, as pointed out by M. Biot, not on the general cylindrical curvature, but on the curve formed by the termination of those projecting asperities of the edge, which are successively brought in contact with the planes, by the vibration of the pendulum, and on which the pendulum is actually supported. When the arc is small, the series of these points or minute surfaces, thus brought into action during the vibration, must be confined to a narrow limit in respect to breadth. For example, if the radius of the cylinder be 0.1 L. and the arc of vibration 210, the maximum usually in pendulum observations, the limit corresponding to an arc of 210 of a cylinder of 0.1 L. radius, is a breadth of 0.0043 L.; which breadth would be therefore sufficient to produce the whole influence of a cylinder of 0.1 L. on the length of the pendulum. But, supposing a breadth of 0.0043 L. to be shewn by microscopic observation, optical means are still insufficient to furnish the half diameter