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length, was ascertained, experimentally, by weighting both the long and short pendulums with different experimental weights, and measuring by the micrometer screw their consequent increase of length. A mean is always taken between the results obtained by screwing up and screwing down the plane which is brought in contact with the sphere, the difference being found to be about equal to half the value of the upward pressure mentioned above; and is attributed by M. Bessel to friction at the axis of the lever, and not to the dead going of the screw, for which the workmanship is too perfect.
The standard toise used in M. Bessel's measurement is a bar of iron made by M. Fortin of Paris, and designed to be identical with the standard bar preserved there, known by the name of the Toise of Peru. Being carefully compared with the original, by M. Arago and Captain Zahrtman of the Royal Danish Navy, it was found 0.0008 of a French line short. It is 19 lines broad and 4.2 lines thick. Its expansion is assumed at 0.0000114 of its length for each centesimal degree.
Three different modes of suspension were used in the course. of the experiments, the knife-edge, the clamp, and what perhaps may be fitly rendered in English by the expression of the cylinder of unrolling. The principal part of the results were obtained with the last. It is a cylinder of somewhat less than one line diameter, resting on the horizontal plane of support, i. e. on the horizontal shelf in the one case, and on the extremity of the toise in the other. A leaf of brass at the upper extremity of the thread of the pendulum is clamped above the cylinder to an apparatus, to be presently described, and passes from an oblique direction over the cylinder, keeping it steady in its place by the weight of the sphere and thread. The brass leaf rolls and unrolls on the surface of the cylinder in each alternate vibration an amount proportioned to the arc of vibration. In this mode of suspension, consequently, the axis of motion is not a fixed point; but supposing the brass leaf to be perfectly pliable, the centre of the sphere would describe an arc of the curve, the evolute of which is the section circle of the cylinder of unrolling. The vibration in this curve is shewn by M. Bessel, not to differ perceptibly from that of a simple pendulum of the mean length of the unrolling pendulum at the moment of repose. The brass leaf is 0.008 of a line thick, and
1.4 broad; and is clamped above the cylinder of unrolling, to one arm of a balanced lever, which is exactly counterpoised by a weight at the end of the other arm; this part of the suspension apparatus is supported and plays on y's resting on a highly polished steel cylinder, an inch in diameter, which itself rests on y's at right angles to the vertical bar. The y's are capable of exact horizontal adjustment. By means of the detached support and counterpoise of this part of the suspension apparatus, no other weight than that of the pendulum itself, i. e. of the sphere of brass and metallic thread, presses on the plane of suspension. Two sets of the horizontal y's are fixed to the vertical bar; one set at the level of the horizontal shelf, the other set at that of the upper extremity of the toise: and the suspension apparatus is shifted from one to the other, according as the longer or shorter pendulum is designed to be used.
The metallic thread employed by M. Bessel is of steel; it is clamped at both ends to small screws, perfectly equal in weight, one of which is screwed into a hole in the sphere, and the other into a clamp fixed to the brass leaf, which passes over the cylinder of unrolling. The thread is thus capable of reversion in successive experiments, lest it might not be of uniform thickness throughout. The elasticity of the thread prevents its preserving, during vibration, the straight line from the cylinder of unrolling to the sphere in which it hangs when in repose; and occasions a curve to take place in the neighbourhood of the suspension, which decreases with the increasing distance from it, and soon becomes imperceptible. The time of oscillation is thereby altered either a constant quantity, or a quantity varying with the extent of the arc of vibration; but since the curve is insensible beyond a short distance from the suspension, the quantity is not dependant on the length of the pendulum; and consequently affects both the long and short pendulums an equal amount; presuming the same suspension, and the same arc to be preserved in both cases, it is consequently eliminated in the length of the pendulum derived from their difference in length. M. Bessel is led by his experiments to conclude, that the change in the value of the quantity, dependant on the extent of the arc, is not perceptible, and
consequently, that the preservation of similar arcs may be a superfluous precaution.
To avoid the possibility of reciprocal influence of the clock and pendulum, the clock was placed 8 feet 6 inches in front of the pendulum. Between them was placed a telescope of Fraunhofer of 24.36 inches focal distance, with the eye-piece removed, and the object-glass at such intermediate distance, that the image of the pendulum fell exactly on the pendulum of the clock: they were both distinctly seen through a 30 inch telescope of Fraunhofer, placed at 15 feet distance.
The scale for observing the arc of vibration was fixed on the vertical iron bar, 412.5 lines below the suspension of the short pendulum: it was divided into spaces of half a Parisian line, and had in the middle a clear white space, equal in size to one of the divisions: this space was exactly bisected by the pendulum thread when at repose. A hollow cylinder of brass, painted black, two lines high, and 1.25 line broad, was slid upon the thread, for the purpose of observing coincidences; its weight for the short pendulum was 3.81 grains, and for the long pendulum 3.69 grains. When the pendulum is in repose, the coincidence cylinder covers exactly the white space on the scale for observing the arc. At the lower end of the pendulum of the clock was attached a piece of black paper, with a hole cut in it, about three lines diameter. The two pendulums being at rest, the coincidence cylinder painted black, as already hoticed, filled the hole in the paper, when viewed through the telescope, and concealed the white space on the scale; which also was the case when the two pendulums were in coincidence during their motion. As the white space was concealed for more than a single second at the coincidences;-and as the duration of its concealment is liable to vary, from several accidental causes ;-the second, when it was no longer visible, and again, the second, when it re-appeared, were observed, and the mean regarded as the instant of coincidence. The coincidences, both of the long and short pendulums, were observed by the same clock. The clock was compared with the transit clock of the Observatory, by opening a door, which permitted their beats to be audible together. The coincidence clock being regulated to make one second less than the transit
clock in an hour of sidereal time, M. Bessel estimates the accuracy with which they may be compared, by observing when their beats coincide, at the two hundredth part of a second. Three thermometers were let into the vertical bar, one near each extremity, and one in the middle; and two others. were suspended freely, one at the height of the ball of the pendulum, and the other at its point of suspension. The thermometer at the ball gave the temperature for the reduction to a vacuum; and a mean of the two at the ball and at the suspension, gave the temperature of the pendulum. The graduation of the thermometers was examined and corrected by M. Bessel's well-known process.
Having thus described the apparatus, it would appear the most direct course to proceed at once to its results. It will be necessary, however, previously, to give a brief view of one of the valuable subsidiary dissertations of M. Bessel's memoir; since it bears very importantly on the results obtained with his apparatus, as well as on all results of all methods whatsoever, in which the experiments are made in air.
Since the time of Newton, it has been considered, that if the mass of a body falling through the atmosphere be m, and the mass of the air displaced by it be m', the accelerating force acting on the body is equal to the quotient of the moving force by the mass, or to ; and pendulum experiments have been reduced accordingly.
In this it is supposed that the moving force m-m' acting on the body, is distributed over, and exerted on, the mass m; but it must be exerted not only on m, but also on all the particles of the air set in motion by m: consequently, the denominator in the expression for the accelerating force must necessarily be greater than m, and will require to be augmented by the introduction of a new function. The vibration of a pendulum must therefore be more retarded by the medium in which it vibrates, and the reduction to a vacuum must be greater, than has been hitherto allowed for.
In considering the influence of this expenditure of force, in the case of a pendulum, on each of the members of the analytical expression of the unimpaired force, or of the mo
tion in a vacuum, M. Bessel is led to the conclusion, that when the vibration takes place in media of very small density, such as the elastic fluids, the effect on the vibration, of that part of the influence of the medium which has been hitherto neglected, is equivalent to the effect of a simple addition to the inertia of the pendulum. This addition will either be constant, with the same pendulum and medium, or will vary with the extent of the arc of vibration, according as the portion of the medium moving with the pendulum does or does not lose its motion, at the same instant that the pendulum comes to repose; a question which must be determined by experiment,-in the affirmative, if a series of coincidences, begun in large arcs, and ended in small arcs, can be fully represented, by the customary reduction to infinitely small arcs, otherwise in the negative. M. Bessel's experiments lead him to conclude, that the effect is at least very nearly constant, within the compass of the arcs in which he has hitherto experimented.
The value of the addition to the inertia of a pendulum, arising from the causes which we are now considering; or, in other words, the additional retardation which a pendulum vibrating in a medium experiences, beyond the usual correction for buoyancy,-not being susceptible of calculation, requires, in all cases, to be sought by experiment.
The method which first presents itself, and apparently the most simple and decisive, is, the direct experiment of oscillating the pendulum in a vacuum. M. Bessel was deterred from attempting this experiment by the apprehension of meeting with difficulties which might involve doubts of other kinds. The method which he adopted, was to oscillate in air two pendulums, alike in figure, but very dissimilar in mass. This he accomplished in his apparatus, by substituting for the sphere of brass, a sphere of ivory of equal diameter but 4.6 times lighter, and comparing the oscillations of the two pendulums, the one composed of the ivory sphere and metallic thread, and the other of the brass sphere and metallic thread. These pendulums being alike in figure, the movement given to the air may be presumed to be the same; whence the same co-efficient applied to their respective corrections for buoyancy, computed in