381 



PENDULUM. 



PENDULUM. 



333 



Set the clock going, and when its arc is steady, give the pendulum a 

 small motion, 1 or 1J at the utmost. The white disc, which is at 

 first wholly visible, is gradually invaded by the bar, till at last it is 

 entirely covered, first one side and then the other disappearing.* These 

 times of disappearance are to be noted. Almost immediately after, 

 the white edges reappear, first on one side and then on the other, and 

 these reappearance* are noted. The mean of the four gives the true 

 time of coincidence. It has been found convenient in practice not to ' 

 make the adjustments with the highest degree of precision, for you are 

 then in doubt on which side the first disappearance and reappearance 

 will take place ; whereas if there is a very little displacement, attention 

 need only be fixed on every alternate beat, which is less strain on the 

 eye and on the mind. This displacement however must be trifling, 

 otherwise, when the arc becomes small, the disappearances and reap- 

 pearances will not take place. The reductions for the rate of the 

 clock, for the arc of vibration, the temperature, the buoyancy of the 

 air and its inertia, and the height of the place of observation above 

 the sea-level, are all to be made as in Borda's pendulum. For the 

 details we refer to the standard memoirs which have been published 

 on the subject. But there are two or three points which deserve more 

 attention than they have generally met with. The clock is only to be 

 considered as a counter of the vibrations, and to avoid any error arising 

 from its irregular rate, the observations should be continued without 

 intermission between two epochs, at each of which the error of the 

 clock is ascertained by astronomical observation. If this be done, the 

 only reliance upon the clock is for the short tim which elapses between 

 the last coincidence of one series and first of the next. It is true that 

 this mode of observing requires artificial illumination, for it must be 

 continued day and night. This is no disadvantage, but the contrary, 

 for as the other great practical difficulty is that of ascertaining the 

 exact temperature and keeping it steady, the best place for observation 

 in a deep roomy vault or cellar. 



The correction for the buoyancy of the air depends on the 

 grarity of the pendulum ; that arising from the dragged air, which we 

 have called inertia, on its form. It is very desirable that the experi- 

 ments should be made in racuo, and where that is difficult or impos- 

 sible, 38 in a travelling apparatus, the law and quantity of the correc- 

 tion for each pendulum should have been previously experimentally 

 determined in racuo. The variation for temperature should be got 

 from a comparison of summer and winter experiments at the same 

 place. In addition to the height of the place of observation above the 

 sea-level, a rough survey should be taken of the form of the surround- 

 ing country, and of the kind of strata, with the specific gravity of each. 

 - particularly necessary where the heights are considerable. (See 

 Young's Remarks, ' PhiL Trans.,' 1819, p. 93.) 



Still, when every known precaution has been taken, there are 

 anomalies in the results of this experiment which are not yet clearly 

 understood. It is very difficult to make the edge of the knife per- 

 fectly straight, and to keep it so when it is fastened to the bar. There 

 is considerable uncertainty in measures of distance between the reci- 

 procal knife-edges; and lastly there is no authorised standard of length 

 at present existing in England in terms of which the distance can be 

 expressed. The length of the simple pendulum which vibrates seconds 

 in the latitude of London, is not at present known with the degree of 

 certainty t which the importance of the problem demands, and which 

 is attainable. 



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. Baily 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- 



from the end c, these may be made to agree after two or three trials, or if the 

 pendulum should be too short, a weight may be added at this end, such as is 

 pointed out by the black dot near c. 



* Mr. Baily uses two slips of gold leaf pasted on a tall to bis clock pendu- 

 lum for a disc, which answers very well. In the experiments at Dolcoath 

 mine, Messrs. Airy and Whewell had a disc of enamel at the end of a small rod, 

 which was screwed into tho bob and projected from it. The disc could be 

 inclined so as to appear of the proper breadth and to receive an oblique illumina- 

 tion. This part of the experiment is almost irreproachable, but perhaps in 

 future it would be safer to place the clock and pendulum at greater distances 

 from each other, and to get sharp edges by interposing a lens. 



t Captain Kater's measure was doubtful in several respects, and though It has 

 been partially corrected, it is still uncertain. He expressed the length of the 

 penilulura in terms of Sir George Shuckburgti's scale (which has probably 

 Buffered injury since that time), and this he found to be identical with the par. 

 liamentary measure declared to be standard by the 5 Ceo. IV., e. 74. The Act 

 of Parliament standard was destroyed when the houses of parliament were 

 burned. Fortunately a very beautiful scale of 5 feet had been previously made 

 for the Astronomical Society, and three feet in this scale had been compared 

 many hundred times with the parliamentary standard. As almost all the recent 

 smlis and comparisons which hare been made in England have been compared 

 with this last-mentioned scale, it must be considered to be the scientific 

 English standard, and the best evidence of the parliamentary standard which 

 exuU. The definition for restoring the yard, if lost, according to the Act of 

 Parliament, that is, by the length of the seconds pendulum, as it depends upon 

 an erroneous determination, would give u new standard, differing sensibly from 

 the previous one which it was meant to preserve. (Daily's Report on a Standard 

 Scale, Astron. Soc. Mem.,' vol. ix. p. 35.) 



I 



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 a 

 I 



tainly a great advantage in having a check 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 jform 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- 

 dulums, only partially [understood 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 in 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 kuif e 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 bo 

 controlled by being fground on a narrow slip let into a brass wedge, or 

 the attaching knees should also be of steel. The uncertainty wlu'ch 

 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 Kater'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, ' Phil. Trans.,' 1829, p. 207; 

 Baily, On the Reduction to a Vacuum, ' Phil. Trans.,' 1832, p. 399; 

 Foster's Pendulum Experiments, edited by Mr. Baily, ' Mem. Ast. 

 Soo.,' vol. vii. (at page 96 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 tiber die Lange des einfacheu 

 Secuudenpendels,' Berlin, 1828, which forms ]>art of the ' Transactions 

 of the Academy of Berlin.' The principle of his measure is the same 

 as Hattou's. Bessel procured from Paris a verified copy of the toise of 

 Peru ; 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. 

 Tho final result is that the length of the seconds pendulum at th 

 sea-level in the latitude of Konigsberg, 64 43' N., is 440-8179 lines of 

 the toise of Peru ; this we consider to be a measurement of the actual 



