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of its curve with certainty; and no other means present themselves of beiny assured that it may not have an important influence on the length of the pendulum.
M. Bessel has examined the influence of the imperfection of the edge, more closely than 'had been previously done, Concerning the nature of the figure by which it is terminated, he assumes that the polished planes of the prism forming the knife-edge are tangents to the curve of the figure; or, that if the angle formed by these planes is 2 i, tangents drawn to the terminations of the curve on either side will include the angle 2 i; and that the distance (6) between the planes where they meet the curve is capable of microscopic measurement: but in regard to the particular curve of the surface, it remains unknown in his general investigation; and thus much only is assumed in respect to it, that its section perpendicular to the axis is a conical section, of which one axis coincides with that of the pendulum ; its figure is then given by i, b, and an arbitrary eccentricity . By the alteration of e, the curve passes from an hyberbola coinciding with the planes of the prism (that is, from the summit of an angle), through all intermediate steps, to a plane perpendicular to the axis of the pendulum. The results of the investigation are, that if l' is the length of the simple pendulum, oscillating equal times with a pendulum moving round the curved termination of a knife-edge, and I the length of another simple pendulum, oscillating equal times with a pendulum moving on the summit of a knife-edge, then
9 s being the distance of the centre of gravity from the knife-edge, and q a co-efficient dependent on the figure of the curve and the angle of oscillation. For infinitely small arcs the value of
9 = ✓ (1 – e Cos i');
2 Cos i when the edge is the summit of an angle, q equals 0; in the passage of e, through all the intermediate steps, to a plane perpendicular to the axis of the pendulum, q continually increases, and for the last value it is infinitely great, and the oscillation stops. For very great values of ε, q decreases quickly as the arcs of oscillation increase ; for lesser values of ε it is the same or nearly so as for infinitely small arcs; at least with such angles as are usually employed in pendulum experiments.
M. Bessel has given in tables, the value of q for several values of , and for the angles of oscillation 09, 19 and 2°; one of the tables is computed for 2 i = 90°, and the other for 2i = 120°. From these tables it is evident, that, supposing the knife-edge to roll on the planes, its figure may have a perceptible influence on the length of the pendulum, even though the termination b should be but a few thousandths of a line. Unless, therefore, we have sufficient reason for assuming that there is no danger of great negative values of 8 (i. e. much flattened ellipses, with the larger axis horizontal), arising either in the operation of grinding the knife-edge, or from its wearing away by use, we cannot trust to determinations of the length of the pendulum made with knife-edges, unless means are found either of ascertaining the amount of the influence, or of eliminating it from the result.
It has been said, that the influence of the curvature of the knife-edge disappears in the result with the reciprocal pendulum; but M. Bessel remarks, that this very elegant property of that apparatus is only strictly true, when both knife-edges are terminated by equal values of bq. But there is an easy mode of eliminating the effect altogether, which is by so constructing the pendulum, that the knife-edges can be interchanged; when the mean of the experiments before and after changing the knife edges will be a result independent of their figure.
For the purpose of examining practically the influence of different values of b, i. e. of different breadths of termination of the knife-edge, M. Bessel made the following experiments with a pendulum with reciprocal axes, having the knife-edges of steel, alike in figure and weight, and so arranged as to be easily taken out and replaced, after altering their figure. The inclination of the planes of each prism was nearly a right angle. The pendulum consisted of a cylindrical bar of brass, having the great weight fixed, and the small one moveable; and weighed altogether 34260 grains. When the times of vibration on the two knife-edges were made nearly equal, the distance of one of the knife-edges from the centre of gravity was 305.32 L. and of the other 135.7 l. or nearly as 9 to 4; taking
the length of the simple seconds pendulum at 440.8147 L. as found by M. Bessel, the length corresponding to the vibration of the convertible pendulum, when the great weight was above, was 440.8185 (omitting a fraction expressing the unknown part of the reduction to a vacuum, which, as the fraction was almost identical in value in each of the comparative experiments, may be safely neglected).
1. The knife-edge was then removed, ground very fine, and replaced, by which operation it was brought 0.006 L. nearer the centre of gravity; whence l' should be increased 0.0075 L., making 440.8260 L. The length corresponding to the time of vibration was now, however, 440.8668 L.; the difference being 0.0408 L, and showing that bq was lessened 0.0125 L. by the alteration of the knife-edge.
After several experiments had been made on this edge, it was observed to have been injured by use, in consequence of having been ground too fine. The worn edge appeared in a microscope magnifying 200 times to have a flattened edge.
0.005 L.; and was 0.006 L. more distant from the centre of gravity in the last than in the first experiment. The difference in the length corresponding to the vibration between the last and first experiment was -0.0253 L.; whence an increase of bq was manifested, arising from the wearing away of the knifeedge 0.0173 L.
The same knife-edge was then flattened designedly until b= 0.0216 L., increasing the distance from the centre of gravity 0.021 L., whence the length of the pendulum, which had been 440.8572 L., should now have been 440.8689 L. from the vibration, if the effect of the knife-edge had not been altered by the flattening. With the knife-edge in this state, the formula of reduction to infinitely small arcs no longer reduced all the oscillations in different arcs to the same time; the smaller arcs producing less times. The lengths corresponding to the vibration were from 440.5589 L. to 440.5185 L., instead of being 440.8689 L. The influence of the figure of the knife-edge is here fb q = 3 b q; and as b = 13 0.0216 £. by measurement, q, for different angles of oscillation from 84 to 15', varied from 10.1 to 11.2; corresponding to the tabular value of q when the curve is an ellipse, and the pro
portion of the axes as about 1:21. The kuife-edge was then ground again till b was diminished to 0.0135 L., when the length of the simple pendulum corresponding to the vibration was found to be diminished in an arc of 800.2203 L., progressively to 0.2548 l. in an arc of 22', instead of from 0.3150 L. to 0.3504 L., as when b = 0.0216. The value of q varied in this instance from 11.2 in the large arcs to 13 in the small arcs, corresponding to an ellipse having the axes in the proportion of 1:25.
These experiments apparently do not accord with those of M. Biot at Leith and Unst, who employed, in successive observations with his apparatus, two knife-edges, in one of which b= 0.0166 L., and in the other 0.0023 L., and found no difference in the time of vibration produced by the difference of figuré. M. Bessel is inclined to prefer in M. Biot's experiments those results in which the finer knife-edge was employed for the absolute pendulum, and to consider the results with the broader edge as valuable only for the relative pendulum.
In the experiments made by Borda to ascertain the length of the seconds pendulum at Paris, the experimental pendulum consisted of a platina sphere and a metallic thread, four times the length of the seconds pendulum. The influence of the imperfection of the knife-edge would consequently be reduced to one-fourth of what it would have been with the same knifeedge in an experimental pendulum of equal length with the seconds pendulum. I believe that in the experiments that were at one time intended to have been made at Greenwich with the French apparatus, it was designed to have made this property available in eliminating the influence of the figure of the knife-edge employed, by using it successively with pendulums differing greatly in length. The correction of Borda's experiments on account of the figure of the knife-edge would probably be very small; but unless the identical knife-edge used by him were still in existence, the amount of the correction cannot now be ascertained.
M. Bessel remarks of Captain Kater's experiments, that by the great care given by that gentleman to the construction of his knife-edges, it is probable that b did not exceed, in either
edge, 0.001 L. Supposing its value to have been the same in both knife-edges, the influence on the vibration would disappear; if not of equal value, still the influence would only in part affect the ultimate result. By making the knife-edges interchangeable, the influence would in either case disappear.
In all discussions hitherto on the subject of the vibration on knife-edges, it has been assumed that the knife-edge rolls on its cylindrical or curved termination, without having any gliding or sliding motion whatsoever. This strictly supposes, on the one hand, that the friction is sufficient to keep the axis of rotation fast, and on the other, that the supporting planes are planes of perfect hardness. M. Bessel was led by the remarkable circumstance, which occurred in the course of my experiments with invariable pendulums,-of a difference manifesting itself in the time of vibration of an invariable pendulum oscillating on different agate planes,-to institute experiments on the relative vibrations of a pendulum upon supports of different materials and form. On agate planes, glass planes, glass cylinders, and very hard steel planes, no very decided difference was obtained, nor was the arc of vibration affected. On brass planes the time of oscillation was lessened considerably, but still the formula for the reduction to infinitely small arcs applied both to the vibrations in large and small arcs; but on cylinders of brass of 1.63 L. diameter, the diminution in the time of oscillation was still greater, and with the smaller arcs the length of pendulum deduced was considerably less than with the larger arcs. Į
The cylinders were marked by the oscillation more in breadth than in depth. The brass planes were vibrated upon a considerable time before any mark was produced upon them; but during the whole time, the vibration upon them differed greatly from the vibration upon harder planes. It was, therefore, to be inferred that the cause which produced the difference, and which was so sensible in its effect in the one case, might not be altogether insensible in the other. To examine this point more closely, M. Repsold, at M. Bessel's request, contrived an apparatus, capable of measuring any sliding motion that the knife-edge might have upon the planes, even should it amount to no more than the forty-thousandth of a line: the pendulum