Captain Kater proposed the following form of pendulum for deter mining the number of oscillations which the same pendulum describes in different places, which is nearly the same as Graham's, above de scribed. Mr. Bally prefers the bar with two knife-edges, the same as that used for determining the absolute length, and he directs observa tions to be made at each place with both knife-edges. There is cer tainly a great advantage in having a cheek upon the knife-edges, as a discrepancy between the two results (supposing the pendulum to have been previously regulated) will show that one of the knife edges has been injured. To ascertain which of the two it is, requires a second pendulum. Upon the whole, though we prefer Mr. Baily's pendulum to Captain Kater's, yet we think a form suggested by Mr. Airy is, for ordinary pur poses, still better—it is, to have the knife-edge placed in that part of the bar where the oscillations take place in the shortest time. In this case it seems probable that any ordinary injury of the knife-edge will not alter the time of vibration. The observer must be very careful in these stiff bars to see that the pendulum is truly vertical, a pre caution which in Kater's flexible bar is less necessary. If an observer chooses to have a second knife-edge opposite the former in the pendulum of minimum vibration, there is no objection except the increased resistance of the air, the increased correction for inertia, and the chance of con founding the two positions. The mere observation of the coincidences, which in its present form was first introduced by Kater, is so simple that it will be successfully performed by any person the first time of attempting it, and an error of a second or two in noting the moment is a matter of little importance. There is a far greater latitude for error in noting the temperature and ascertaining the clock-rate, but there is another anomaly attending the knife-edge pen &Limns, only partially junderstood at present, which is much more vexatious than temperature or clock-rate. As the pendulum swings about those points of the knife-edge which are in contact with the agate planes, it is clear that, supposing any imperfection iu both, these points may be altered, when the axis of suspension, and con sequently the time of oscillation, will be altered too. Hence it is advisable to place the knife-edge as exactly as possible always in the same position on the agate plane, both as to front and back, and also as to right and left. It will generally be found that turning the pendulum half round on the same knife-edge will alter its time of vibration sensibly, so that the face in front must alWays be registered. •But even when these precautions are taken, anomalies are still found, which we can only attribute to the following cause : In most of the pendulums hitherto made the steel knife-edge has been fixed upon a bar of different metal (brass and copper) with stout brass knees, and the two screws which fix the steel knife-edge to the brass knees are at some distance from each other. Now if, from the difference of expansion between the brass knees and the knife-edge, the latter should be curved, not upwards or downwards, but in a horizontal plane, the consequent alteration of time in the oscillation may be very considerable. The steel edge therefore should either be controlled by being !ground on a narrow slip let into a brass wedge, or the attaching knees should also be of steel. The uncertainty which appears due to some such cause as that above described may amount to one, two, or even more seconds a day in the invariable pendulum, either of liater's or Baily's construction. It is evident that in measuring the actual length of the pendulum this error may be further increased, for the problem assumes the perfect parallelism and truth of a pair of knife-edges. For an account of knife-edge bar pendulums, see Kater's Convertible Pendulum, with Young's Appendix, Phil. Trans.,' 1818, p. 33; Kater's Invariable Pendulum, Phil. Trans.,' 1819, p. 337 ; Sabine's Experiments to determine the Figure of the Earth, London, 1825; Sabine's Comparison of the Length of Pendulum at London and Greenwich, Phil. Trans.,' 1829, p. 83 ; Sabine, On the Reduction to a Vacuum, Trans.,' 1829, p. 207; Bally, On the Reduction to a Vacuum, 6 Phil. Trans.,' 1832, p. 399; Foster's Pendulum Experiments, edited by Mr. Mem. Ast. Soc.,' vol. vii. (at page 95 is given a table of the results of the most accurate modern pendulum experiments). See also WEIGHTS AND MEASURES.
The last important measurement of the length of the pendulum, which is undoubtedly the most satisfactory, is due to Bessel, and is described by him in his Untersuchungen uber die LiMge des einfachen Secuudenpendels; Berlin, 1828, which forma part of the Transactions of the Academy of Berlin.' The principle of his measure is the same as Ilatton's. Bessel procured from Paris a verified copy of the toise of I'eru ; this he placed upright, and suspended his ball and wire first from the top and then from the bottom of the toise, the ball being in both cases at the same point below : the time of oscillation in each position was compared with a clock in front. From the difference of the times of oscillation of these two pendulums, and the difference in their length, which is a toise, the length of the seconds pendulum was computed in parts of the toise. Bessel suspended his pendulum by passing a flat and very slender spring over a small cylinder, the spring being continued downwards to the ball in a fine wire. We must refer to the original memoir for a description of the different elaborate con trivances which were introduced by Bessel and Repsold for securing accuracy in every part of the process. All conceivable causes of error are considered, and the effects have been computed and allowed for. The final result is that the length of the seconds pendulum at the sea-level in the latitude of Konigsberg, 54° 43' N., is 44(r8179 lines of the toise of Peru ; this we consider to be a measurement of the actual length of the pendulum, and perhaps the only one, which is deserving of full confidence.
We have already pointed out the manner in which experiments with the pendulum aro used to detorinine the figure of the earth, and stated that for thia purpose the simple bar with a knife-edge at the point of minimum oscillation seems the most fit.. If the present imperfections can be overcome, which we see no reason to despair of, there is another research, that of the mean density of the earth, for which such an instrument is peculiarly adapted. The density which has been hitherto acquiesced in, depends on the attraction of Soho hallien upon a plumb-lino, observed by Maskelyne, and upon the attraction of leaden balls measured by a torsion balance, according to Cavendish, and more recently by Reich. Now in the Schehallien
experiment the whole deflection caused by the mountain was only which, under the circumstances, might very easily be 2" or even 3' wrong either way. The Cavendish experiment has been repeated in this country by Mr. Bally, who from upwards of 2000 experiments has arrived at the result as the density of our globe. Still however it was desirable to adopt other modes of experiment, which we now proceed to point out..
If an invariable pendulum can be made which is only affected by known causes of which the effects can be calculated, the simplicity and perfection of the observation by coincidences, and the unlimited time for which it can be carried on, will supply a much broader and more secure base for the solution of this important problem, and one more readily and universally applicable. Two attempts were made by Mew's. Airy and Whewell, first by themselves, in 1826,and afterwards, in 1828, with the aaaistauce of three Cambridge friends, to deduce the mean density of the earth from observing the oscillations of detached pendulums at the top and bottom of Dolcoath Mine, one of the deepest in Cornwall. As the first operation was less carefully planned, we shall confine our account to the second, which was altogether the best arranged pendulum experiment which had come to our knowledge up to that time, and only failed from a cause then first discovered, namely, that invariable pendulums of Kater'e construction are very uncertain and inaccurate instruments. We will call the two pendulums I and 2. These were placed on their stands opposite each other, and put into motion at the same arc (a board being interposed to prevent their reciprocal action through pulses of the air), and the times of coinci dence observed. A few series were thus made, in which the errors of observation scarcely amounted to a hundredth of a second per day.° Pendulum 1 was then sent to the bottom of the mine, and and act up with its clock and proper accompaniments. Pendulum 2 was established with its clock in a hut at the surface. We will call the observers a, B. C, D, E. A set off at six in the morning, commenced a series with pendulum 2, and compared eight pocket chronometers with the clock by coincident beats through a sidereal hall-seconds chrono meter.t He then descended the mine and commenced a series, compared his chronometers below exactly as he had done above, and then mint them back to the upper station. By this time E had arrived at the upper station to finish the series which a had begun, and to compare the chronometers on their return. In this way the clocks were compared, without a possible error of nth of a second. After finishing his first series and commencinga second, a returned to day (in the miner's language, Co grass:). This eerier was completed by n, who set off at 2 P.M., made a third shorter series, and commenced a fourth. The night-work was taken by c or D, one of whom descended at 10 P.M., finished B's series, and commenced a fresh one; • on his return to day he finished the upper series which a commenced before going to bed, and started another, which was finished by a before his next descent. In tide way observations were Continued from Monday morning, July 11, to Saturday afternoon (127 hours). On comparing the results of each day, it was found that the three first days agreed extremely well, showing an acceleration of about 2' per day in the lower pendulum ; but the two next days, this acceleration was about double. The only mode of accounting for this unexpected and very unwelcome result was, that the knife-edges and agate planes were imperfect, and that the time of oscillation depended on the position of the pendulum on its plane. This had never been suspected before; for confidence in the accuracy of the pendulum was at that time almost an article of faith among experimentalists. To ascertain this, a second series of com parisons was made between pendulum 1 and 2, taking care to place the knife-edges exactly in the same position on their planes, and that the faces were turned the same way. These comparisons were necessarily hurried, as a slip had taken place in the mine near the engine-shaft, which deranged the action of the pumps, and forced the experimenters to repeat their operations below without delay. Pendultnn 1 was again sent below, and the experiment repeated exactly as described above, for three days (79 hours), until the underground observers were driven out by the water. Notwithstanding all the precautions which had been taken," the observations of the last day gave a different ,result from the two first. " The conclusion therefore on which tho experimenters were unavoidably forced was, that even with the care and attention they had used, the pendulum could not be trusted." The pendulums were a third time compared with each other above ground, and the intervals of coincidence were found to vary, without any assignable cause, from 20- 24' to 20" 17', and back again from 20" 20' to 20" 29'; and finally, during one series of oscillations, without being touched, or any one entering the room, the time of coincidence changed from 20" 25* to 20' 13', and at last to 20" 3'. It was there fore clear that while the mode of observing was perhaps sufficient to detect a difference of a hundredth of a second a day (except from the effects of temperature, which might amount to a tenth of a second), the pendulums themselves, though used with the utmost skill and precaution, and under the same circumstances, were liable to errors of two seconds or more per day.
In 1854 Mr. Airy repeated this experiment under more favourable circumstances on the banks of the Tyne, and at the bottom of a pit of Harton colliery, one of the deepest coal-pita in this country (1260 feet below the surface). Advantage was taken of electro-telegraphy to make the comparisons between the upper and lower pendulums, which were found to differ in rate 21 seconds per day, from which it followed that the gravity for that depth was increased by the part. The density of the earth as deduced from this result is between six and seven times that of water, but Mr. Airy imagines that he has not taken full account of the hollow of the Tyne, of the basin named Jarrow slake, of the scoop indicated by the sea, and of the real and observed specific gravity of the rocks which cover the mines of Harton. Further details on this subject will be found under EARTH, MEAN DENSITY OP TR& [See also GYROSCOPE.]