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employed with it was the one already described with reciprocal axes. It was found that, on all the supports, small glidings of the edge took place, always in the direction of the motion of the pendulum; to the right, when the pendulum moved to the right, and the converse: the amount was the same with both knife-edges, and proportional to the arcs of vibration. For all the hard planes, of whatever materials constructed, and for the glass cylinders, it appeared to be the same; namely for each degree of arc 0.06004 L.; for the brass planes, 0.0033 L.; and for the brass cylinders, 0.00049 L. The maximum of motion took place as the pendulum passed the vertical.
M. Bessel is of opinion that these motions cannot be ascribed to an insufficiency of friction for keeping the knife-edge fast: the pendulum requires a force to keep its axis of rotation fast; this force is the friction on the supports; but should the friction be not quite sufficient, so that it should not entirely counteract the motion of the edge, but only reduce it to a small quantity, the remaining motion would be always opposed to the motion of oscillation; whereas the sliding was in the direction of the oscillation. Further, as the distance of the knife-edge from the centre of gravity would enter into the expression for the motion occasioned by the insufficiency of friction, the sliding would not only be in a contrary direction to what it is found, but it would be also of unequal amount in the two knife-edges, which is contrary to experiment. Jointly, then, from the experiments on the times of oscillation and on the motion of the knife-edges, M. Bessel infers that the planes themselves oppose a force to the motion of the pendulum. Now, a force acting only on the edge would not affect the time of oscillation; the action of the force requires, therefore, for its explanation, either that an alteration of the figure of the knife-edge be supposed, or that the knife-edge makes an impression on the support, so that points of the knife-edge, situated higher than its absolute edge, are brought in contact with the planes. If the action of the force produced an alteration in the figure of the knife-edge, the effect would be alike on hard and on soft supports; whereas it is very different. It may, therefore, be inferred with confidence from the experiments, that the second explanation is the true one;—namely, that the knife-edge makes an impression on the
planes, causing a motion of their particles to take place during the motion of the pendulum; and that the opposition to the motion of the pendulum, which lessens the time of vibration, is occasioned by the endeavour of the compressed particles of the planes to recover their natural position in consequence of their elasticity; whence it is presumable that the effect may be greater with rough than with highly polished planes of the same material.
In order to examine the manner in which this impression operates in affecting the time of oscillation, M. Bessel assumes, in conformity with the experiments, that the sliding of the knife-edge is proportioned to the angles of oscillation. Hence he observes that there is a point in the axis of the pendulum, above the knife-edge, which, during the vibration, remains always in the same perpendicular plane which is occupied by the axis of the pendulum and the knife-edge when in repose. The distance of this point from the knife-edge may be determined by the extent of the sliding of the knife-edge and the angle of oscillation. It is found from the experiments---
The conclusion drawn by M. Bessel from these experiments, in regard to the convertible pendulum, is, that the result obtained with that pendulum is only so far influenced by the planes as their effect may be to raise somewhat the point on which the pendulum moves. By interchanging the knife-edges, not only does the influence of their terminating figure entirely disappear, but so also does the influence of the change of figure of the planes during oscillation, should it not be the same for the two knife-edges, which is possible. The mean of the experiments, before and after changing the knife-edges, gives the time of vibration corresponding to the simple distance between the surfaces terminating the knife-edges, increased by two very small quantities, one for each knife-edge; these require to be found, by measuring the amount of the sliding motion of each knife-edge, and deducing therefrom that higher point in the axis of the pendulum, which remains, during the vibration,
always in the perpendicular plane occupied by the axis when the pendulum is at rest.
The convertible pendulum, then, with the knife-edges capable of being interchanged-with the weights fixed to ensure that all the experiments are with an unaltered pendulum,—made symmetrical in external figure but not in mass-or, omitting the last condition, experimented with in a vacuum,—or which is still better, made symmetrical, and experimented with both in vacuum and in the air;-a convertible pendulum, so arranged, will possess the same advantages which M. Bessel has obtained, by a totally different method;-namely, the result will be free from uncertainties arising from the imperfect figure of the knife-edge, and the influence of the medium in which the experimental vibrations are made.
At the direction of the President and Council of the Royal Society, a reciprocal pendulum is now in process of construction, which will fulfil all the necessary conditions. It will also admit of having the distance between the knife-edges referred to a standard scale, whilst the pendulum is suspended on the planes, and the reference repeated as frequently during the experiments as may be desired. The mode of reference will also be free from the uncertainty which has attended the employment of microscopes.
By experiments with this pendulum, varied and repeated as may be seen expedient, the true length of the seconds pendulum will be arrived at, unless there should still exist some yet unknown defect in any part of the process, by which the principle of the reciprocity of the axis and centre of oscillation is applied. The invention by M. Bessel of a method of obtaining the same object by a proceeding totally distinct in all its details, places in our power a comparison of results, which will be a far more efficient means of detecting any remaining error, than could be afforded by the utmost additional consideration given to the details of either method separately.
To accomplish this comparison in a suitable and satisfactory manner, it is necessary that the two methods should be tried at the same spot; and that we should possess, for that purpose, an apparatus upon M. Bessel's plan, adapted to the British measures. By the substitution of a standard bar of the length