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IV. An Account of experiments for determining the length of the
Pendulum vibrating seconds in the latitude of London. By Capt. Henry Kater, F. R. S.
Read January 29, 1818. T.
o determine the distance between the point of suspension and centre of oscillation of a pendulum vibrating seconds in a given latitude, has long been a desideratum in science. Many experiments have been made for this purpose, but the attention of all who have hitherto engaged in the enquiry (excepting WHITEHURST) appears to have been directed to the discovery of the centre of oscillation. The solution of this problem depending, however, on the uniform density and known figure of the body employed, (requisites difficult if not impossible to be ensured in practice,) it is not surprising that the experiments made by different persons should have been productive of various results.
When I had the honour of being appointed one of the committee of the Royal Society for the investigation of this interesting subject, I imagined that the least objectionable mode of proceeding would be to employ a rod drawn as a wire, in which, supposing it to be of equal density and diameter throughout, the centre of oscillation, as it is well known, would be very nearly at the distance of two-thirds of the length of the rod from the point of suspension ; and I purposed by inverting the rod, and taking a mean of the results in each position, to obviate any error which might
arise from a want of uniformity in density or figure. After numerous trials however, and as frequent disappointments, I was at length convinced of the impracticability of obtaining a rod sufficiently uniform, and I was besides aware, that under certain circumstances errors might arise from this cause which it would be impossible by any method to detect.
Not feeling at all satisfied with the prospect which the use of a rod presented, I endeavoured to discover some property of the pendulum of which I might avail myself with greater probability of success; and I was so fortunate as to perceive one, which promised an unexceptionable result. It is known that the centres of suspension and oscillation are reciprocal; or in other words, that if a body be suspended by its centre of oscillation, its former point of suspension becomes the centre of oscillation, and the vibrations in both positions will be performied in equal times. Now the distance of the centre of oscillation from the point of suspension, depending on the figure of the body employed, if the arrangement of its particles be changed, the place of the centre of oscillation will also suffer a change. Suppose then a body to be furnished with a point of suspension, and another point on which it may vibrate, to be fixed as nearly as can be estimated in the centre of oscillation, and in a line with the point of suspension and centre of gravity. If the vibrations in each position should not be equal in equal times, they may readily be made so, by shifting a moveable weight, with which the body is to be furnished, in a line between the centres of suspension and oscillation; when the distance between the two points about which the vibrations were performed being measured, the length of a simple pendulum, and the time of its vibration will at once be known, uninfluenced by any irregularity of density or figure.*
An umexceptionable principle being thus adopted for the construction of the pendulum, it became of considerable inportance to select a mode of suspension equally free from objection. Diamond points, spheres, and the knife edge, were each considered ; but as it was found difficult to procure diamond points sufficiently well executed, the knife edge was preferred, after many experiments had been made with spheres, the result of which it may not be useless for a moment to dwell upon.
In the Connoissance des Temps for 1820, is an article by M. de Prony on a new method of regulating clocks. At the conclusion of this article is a short note, in which the author adds, “ J'ai proposé en 1790 à l'Academie des Sciences un moyen
2 “ de déterminer la longueur du pendule en faisant osciller un pendule composé sur “ deux ou trois axes attachés à ce corps. (voyez mes Leçons de Mécanique, art. “ 1107 et suivans) Il paroit qu'on a fait ou qu'on va faire usage de ce moyen en “ Angleterre.” On referring to the Leçons de Mécanique, as directed, I can perceive no hint whatever of the possibility of determining the length of the seconds pendulum by means of a compound pendulum vibrating on two axes, but it appears that the method of M. de Prony consists in employing a compound pendulum having three fixed axes of suspension, the distances between which, and the time of vibration upon each, being known, the length of three simple equivalent pendulums may thence be calculated by means of formulæ given for that purpose. M. de PRONY indeed proposes employing the theorem of Huygens, of which I have availed myself, of the reciprocity of the axis of suspension and that of oscillation, as one amongst other means of simplifying his formulæ, and says, “ J'ai indiqué les moyens “ de concilier avec la condition à laquelle se rapportent ces formules, celle de rendre “ l'axe moyen le reciproque de l'un des axes extrêmes ; J'emploie pour les ajustemens “ qu’exigent ces diverses conditions un poids curseur dont j'ai exposé les propriétés “ dans un mémoire publié avec la Connoissance des Temps de 1817.” Now it appears evident from this passage, that M. de Prony viewed the theorem of HUYGENS solely with reference to the simplification of his formulæ ; for had he perceived that he might thence have obtained at once the length of the pendulum without further calculation, the inevitable conclusion must instantly have followed that his third axis and his formula were wholly unnecessary.
It is known, that if two curved surfaces be ground together in every possible direction, they will become portions of spheres; and thus a perfect sphere may be formed by grinding a ball in a hemispherical cup. If a penduluin vibrate on such a sphere, working in a conical aperture, it is evident that the centre of the sphere will be accurately in the axis of vibration. In trying this method, however, it was found, that the friction was so considerable, as to bring the pendulum to a state of rest after a few vibrations; and when the friction was sufficiently diminished, by a contrivance which it is unnecessary to describe, the lateral force of the pendulum in an arc of two degrees and a half, was sufficiently powerful to carry
the ball entirely out of the socket; and it was consequently evident, that though the arc of vibration might not be large enough to effect this, it must necessarily cause the ball in some degree to ascend the inclined plane of the
aperture; and this consideration induced me to abandon at once a mode of suspension which I should otherwise have esteemed the best that could have been employed.
The principal objections to the use of a knife edge, appeared to be, the difficulty of forming it perfectly straight, and the possibility that it might suffer a change of figure from wear, during the experiments, which might introduce an error not to be detected. The first of these objections I found to be perfectly groundless, as a knife edge can be made so as not to deviate sensibly from a right line. The second objection would indeed be of weight, were the usual method of determining the time of vibration resorted to, by com