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the ordinary manner, will give for each the additional part of the reduction to a vacuum, which has hitherto been neglected. Now, as the reduction is in the inverse proportion to the mass; and as the specific gravities of the brass and ivory are respectively as 8.18955 and 1.79443 to that of water at its greatest density, considered as unity; the difference in the oscillations of the two pendulums, when of the same length, and vibrated in air, is so considerable, as to enable the reduction of the required co-efficient; and the consequent elimination, from the two sets of experiments, of the whole reduction to a vacuum for each pendulum. The co-efficient thus derived from M. Bessel's experiments with the brass and ivory spheres, is 0.9459, shewing that the reduction to a vacuum for a pendulum, such as he employed, is nearly double the amount of the correction for buoyancy, which would have been previously supposed to have been the whole reduction.

The number of distinct determinations obtained by M. Bessel, from the mean of which the length of the seconds pendulum at Königsberg is derived, were in all fifteen. Of these, eleven were with the brass sphere, and four with the ivory sphere. Each determination consisted of four experiments with the long pendulum, and two with the short; the number of experiments with the long pendulum being double, on account of the greater difficulty and uncertainty of the observation of coincidences. Of the four experiments with the long pendulum, a was with the lever of the micrometer screw turned to the right; b, with the lever turned to the left, and the thread reversed; e, with the lever turned to the right, and the thread as in b; f, with the lever turned to the left, and the thread as in a and of the two experiments with the short pendulum, c was with the lever turned to the right, and the thread as in a; and d, with the lever turned to the left, and the thread as in b: c and d were made between the pairs a and b, and e and f. During the six experiments belonging to one determination, the clamp fixing the thread to the suspension apparatus was not touched, so that the same part of the brass leaf passed over the cylinder of unrolling.

In the three last determinations with the brass sphere, three different modes of suspension were employed in comparison

with each other, for the purpose of examining the influence of the different modes of suspension, or the situation of the axis of motion. In the first of the three determinations the pendulum was suspended by the brass leaf passing over the cylinder of unrolling as before; in the second, by a knife-edge; and in the third, by the brass leaf clamped, instead of its being passed over a cylinder. The clamp, knife-edge, and cylinder were all laid accurately on the same horizontal plane, and the half-diameter of the cylinder being allowed for, the pendulum in each of the three cases was apparently suspended from the same height. The result of this comparison was, that the cylinder and clamp gave very nearly the same length between the zero of the micrometer-screw and the axis of motion, but both a different length from the knife-edge; the axis of motion with the knife-edge appearing to have been 0.031 L. higher than with the cylinder corrected for its half-diameter, and 0.036 L. higher than with the clamp: whence it may be inferred that in the knife-edge suspension the centre of motion of the axis is higher than in either of the other methods. The three modes serve equally the purpose of determining the length of the seconds pendulum in M. Bessel's apparatus, which requires only that in each distinct determination the same suspension should be employed for the long and short pendulums. Of the four determinations with the ivory sphere, two were with the knife-edge suspension, and two with the cylinder of unrolling the results were in accord with those above noticed with the brass sphere, in shewing that the centre of motion is higher in the knife-edge suspension than in the cylinder: but the experiments with the ivory sphere are not so proper for giving the exact difference between the two modes of suspension as those with the brass-sphere, because the brass leaf must be pressed more strongly against the cylinder with the brass sphere than with the ivory sphere.

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In the experiments with the ivory sphere, the oscillations of the long and short pendulum were made to succeed each other alternately, on account of the liability of the ivory to be affected by changes in the moisture of the atmosphere. Owing to the ivory sphere weighing so much less than the brass, the influence of the resistance of the air in diminishing the arc of

oscillation was much greater, and the results were necessarily obtained from a mean of fewer consecutive coincidences.

The mean result of the fifteen independent determinations gives the length of the simple seconds pendulum in the observatory at Königsberg 440.8147 Parisian lines. The extreme differences of single determinations from the mean are

Line.

Line.

+0.0037 and -0.0038 with the brass sphere, and
+0.0060 and 0.0044 with the ivory sphere.

The ball of the pendulum was 11.2 toises above the mean height of the water of the Pregel, which, as the stream has no perceptible fall, may be regarded as the height above the surface of the Baltic: the diminution of length corresponding to this elevation is 0.0032 L.; whence the length of the simple seconds pendulum, reduced to the surface of the Baltic, is 440.8179.

As the incorrectness of the assumed expansion of the toise (0.0000114 of its length for each centesimal degree) would be productive of a constant error, and is the only source in M. Bessel's opinion of probable constant error, he shews that if 0.00001167 be substituted for 0.0000114, whereby the several partial determinations would be brought into a nearer accord with each other, the final result would be reduced only 0.0003 L. a quantity which may be justly regarded as too small to require consideration.

In further exemplification of the correctness of the new view which M. Bessel has taken of the retardation of a pendulum by the fluid in which it vibrates, he instituted the following experiments. In those which have been related, pendulums of the same figure, but of different specific gravities, had been vibrated in one and the same medium, and the influence of the medium manifested by the different effects it produced on different pendulums. In the experiments now under consideration, the same pendulum was employed in different media. The long pendulum of the apparatus, consisting of the brass sphere and metallic thread, was vibrated in air and in water; its time of vibration in air was 1".7217, and in water 1".9085. By the old formula of reduction, which took into account only the equal bulk of the fluid displaced by the pendulum, the

vibration in air being 1".7217, that in water ought to have been 1.8373, instead of 1′′.9085. In like manner the brass sphere and short metallic thread made in air its time of vibration 1.0020, and in water l'. 1078; whereas, by the old formula, its time of vibration in water ought to have been 1'.0693, corresponding to 1".0020 in air. The co-efficient of the additional retardation derived respectively from the two experiments is 0.648 and 0.602.

Instead of the sphere of brass, a hollow cylinder of the same metal, 36 lines in length and 32 lines in diameter, of the same weight as the brass sphere, but of specific gravity diminished by the included air to 2.0788, was suspended by the thread. The long pendulum formed with this cylinder made its time of vibration in air 1".7244, and in water 2".7892: the short pendulum, so composed, in air, 1".0104, and in water 1". 6385. The times of vibration in water deduced from those in air, by the old formula, ought to have been, respectively, 2".3828 instead of 2.7892, and 1".4021 instead of 1".6385. The co-efficient is here 0.747 and 0.761; compared with the results obtained with the sphere, a perceptible effect of figure is manifest.

To render this effect still more obvious, the bottom plate was then taken away from the hollow cylinder, when its specific gravity became again that of the metal itself, i. e. about 8.3. If the figure had been without influence the cylinder should then have oscillated nearly as the sphere. The long pendulum made its time of vibration in air, 1".7199, and in water, 2".5675: the short pendulum, in air, 1".0019, and in water, 1". 5042: whereas, the computed times in water are 1".8339 instead of 2.5675, and 1.0683 instead of 1". 5042. The co-efficient here derived is 7.99 and 8.21, whence there appears a very great influence of figure. In this case a more than ordinary motion of the particles of the fluid took place, on account of a portion of the fluid in the interior of the cylinder flowing out to supply the vacancy created by the motion of the cylinder; and the con

verse.

The value of the co-efficient found by the experiments in air and water differed from its value obtained by the brass and ivory spheres in air alone, nearly as 2 to 3; whence it is

evident that the co-efficient cannot be derived for ordinary pendulum experiments, from experiments in a fluid. In fact, the reasoning from whence M. Bessel inferred that the part of the reduction hitherto neglected might be in effect represented by an addition to the motion of the pendulum, is expressly limited by him to vibration in media of very small density.

Finally, the time of vibration of a pendulum with reciprocal axis, rendered convertible in air, was 1".0002; when vibrated in water, the time of vibration became, with the great weight, below 1".1177, and with the great weight above, 1". 1450. Thus the equality of vibration was lost, and the pendulum, reciprocal in air, ceased to be so when vibrated in water.

It follows from this experiment, and is, indeed, a necessary consequence of the principles brought forward and established by M. Bessel, that a pendulum designed to be reciprocal, and having its time of vibration made the same when suspended by either axis in air, is not a reciprocal pendulum in a vacuum, unless the external figure be symmetrical on whichsoever of the axes the pendulum be suspended. Now, in the experiments that have been hitherto made, to obtain the length of the seconds' pendulum by the application of the principle of the reciprocity of the axis and centre of oscillation, the pendulums experimented with have been far from possessing that symmetry of form, which would render the reduction to a vacuum the same for the vibrations on each of the axes. There are thus two sources of error, by which the results hitherto obtained by this method are affected, arising from the medium in which the experiments have been made ;-1st, the "correction for buoyancy," hitherto applied as the whole of the reduction to a vacuum, is but a portion of that reduction; and 2d, the condition of reciprocity has not as yet been fulfilled.

To accomplish, then, what is yet a desideratum, viz. to obtain the true length of the seconds pendulum by means of a convertible pendulum, two modes of proceeding present themselves:-1. To make the external figure of the pendulum of experiment symmetrical; so that, being suspended on either of its axes, the motion given to the air shall be precisely the same. This must also be effected in such manner as to leave the centre of gravity nearer one of the knife-edges than the

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