PENDULUM. 



PENDULUM. 



ai at rttH, a talesoope having a vertical lino in the focus U placet! 

 a frw fart distant, mo that the cross, the plumb-line. And the fixed 

 wire .-f the telescope are all in the same right line, and all seen dit- 

 limfily.' Such a length i* given to the plumb-line as that it* oscil- 

 lations ahall be a little aluwer than thoe of the clock. 



The clock U now wt a-going, and when it in quite cUvuly , the platinum 

 ball if drawn a little on one aide, and made to swing through a very 

 amall are, MM or two degree* at the most. The observer then phrc* 

 himnlf at the telescope, and note* the exact beat of the clock when 

 the plumb-Una bisect* the crow and coincide* with the fixed wire at 

 the aun* time : thi U called a rainriileitn. As thin observation seems 

 much lea* definite than in Katvr'n method, it is better to note those 

 beats before and after the coincidence, when the eye cannot distinguish 

 whether it ia a coincidence or not, and to take the mean of the two 

 for the trui' time ! coincidence. Immediately after, the extent of the 

 arc through which the plumb-line swings is to be read off on a scale 

 placed just behind it ; the thi-nnometeni which are near the pendulum 

 must also be read off, and also the height of the K-iromeU-r. The 

 obnrver now waiU until this appearance present* itself a second time 

 (which may be in about ten minutes, though the interval is quite <i<l 

 libitum), and having thus established the time which elapse* between 

 the observations, he may leave the apparatus for three, four, or five 

 hours, when on his return the same observations must be repeated of 

 coincidence, arc, temperature, and barometer, and the series is finished. 

 We should recommend closing a* well as commencing a series with 

 two coincidence* in majumn cauleiam, but except as a check, no ob- 

 servations are required except the first and the last. The plumb-line 

 is again set in motion exactly as before, and the second, third, and 

 following series observed. The rate of the clock is established by 

 astronomical observation. 



After a satisfactory number of series has been taken, the next object 

 i* to measure the length of the pendulum. The plane 1 11 is raised by 

 a fine screw until it just touches the bottom of the platinum ball. This 

 is an operation of great nicety, and must be effected by two or three 

 trials, as the exact temperature of the wire at the exact moment of 

 contact is required ; and the presence of the observer, even for a short 

 time, in sufficient to affect a slight wire very sensibly. Suppose this 

 done, the apparatus is removed or pushed on one side, and a bar, which 

 has a knife-edge above and a sliding tongue below, is suspended in its 

 place. The tongue is slid down until it touches the plane 1 11, and the 

 distance between the knife-edge and the bottom of the tongue measures 

 the distance between the top and bottom of the pendulum. This dis- 

 tance may be measured in several ways ; the French astronomers use 

 a contrivance which need not be described here. 



One of the great practical difficulties in this species of pendulum 

 experiments seems to be the extreme and sudden sensibility of the 

 plumb-line to temperature. The whole apparatus is enclosed in a glass 

 cage to exclude currents of air, and the observer is particularly careful, 

 after bringing the lower plane into contact with the bottom of the 

 ball, to absent himself until the temperature is steady, and then to 

 make the contact complete. It is another objection that the different 

 manipulations required are of great delicacy, and therefore not suited 

 to every observer and every place ; but when the utmost care and skill 

 arc employed, the results appear satisfactory. 



The knife-edge, A B, by which the line and ball are suspended, is an 

 ingenious contrivance, exactly similar to White-hurst's synchronous 

 crutch above described. By the upper screw and weight, the oscilla- 

 tions of the knife-edge atone can be made synchronous with the oscilla- 

 tions of the whole pendulum, so that the oscillations are just the same 

 as if the knife-edge were immaterial, and the point of suspension 

 exactly in the line of its edge. The platinum ball can be suspended 

 from any aide, and if two positions diametrically opposite be taken, 

 the effect of any irregularity of shape or density disappears from the 

 mean of the two results. There are several corrections to be applied to 

 the quantities immediately given by observation, before the length of 

 the simple pendulum can be concluded. The oscillations ore made in 

 an arc of sensible extent. Now the time of oscillation in an arc of A 

 on each aide the lowest point, is greater than the time in an infinitely 



mall arc (which is the arc required), in the proportion of 1 + *'"' 



16 



to 1. An expression which depends upon the first and last arcs of 

 each series, give* the correction which is to be added to the number of 

 oscillations observed. This is taken from a table. Between each 

 coincidence, the plumb-line has made two oscillations less than the 

 clock, therefore subtracting twice the number of coincidences from the 

 number of second* elapsed between the first and lost coincidences, you 

 will have the number of oscillations of the pendulum during a certain 

 time shown by the clock. Each of these numbers requires a correc- 

 tion : to the number of oscillations of the pendulum, must be added 

 the correction for arc just mentioned ; and to the time as shown by 

 the clock, the proportional part of its rate during the experiment. A 

 simple proportion will now give the number of infinitely small vibra- 

 tions in 24 hours. 



This U not alwsjm practicable, and It would, generally ipraklag, be better 

 to place a leu between the crow on the clock-bob and the plumb-line, and to 

 form an Imafe of the CTOM upon the plumb-line. It U denlrable on many 

 scout* that the two peodnlunu about d not be very near euh other. 



Further corrections are to be applied. The length of the wire and 

 ball during the observations must be reduced to the length they would 

 have had at the temperature when the contact with the plane an.! the 

 measurement was made, and this again must be converted into the 

 equivalent length whnn the thermometer is at the freezing-jxiint, 

 which is the French standard temperature. The theoretical pendulum 

 is supposed to swing IN raruo, and as the density of the air affects the 

 time of oscillation two ways, both by diminishing the moving force of 

 the pendulum and by adding * to its inertia by the air carried along 

 with it. ,-i correction U required on this account, which de| ml* up >n 

 the barometer and thermometer, on the specific gravity of the m 

 of the pendulum, and also on its form. 



From the dimensions and specific gravities of the parts of the appa- 

 ratus, the distance between the centre of oscillation and the bott 

 the ball can be computed. Hence the length of the simple pendulum, 

 which oscillates in on infinitely small arc a certain number ..f t 

 a day and I'M racuo, can be assigned, from which the length ft the 

 seconds pendulum is deduced by simple proportion. 



If the place of observation be above the level of the sea, it is usual 

 to reduce the length to what it would have been at the sea levcT. 

 This is a very uncertain quantity, as it depend upon the configuration 

 and density of the strata in and near the spot. In the earlier 

 meuU gravity was supposed to vary inversely as the square of the dis- 

 tance from the centre of the earth, and thus the attraction (if tin- 

 matter between the observer and the sea-level was wholly neglected. 

 Dr. Toung showed that in a table-land of average density the correc- 

 tion thus obtained was too large, and should be multiplied by 0-66. 



I ' i a full description of Borda's method, with instances, tus., Be* 

 ' Base du Systcme Mcirique Decimal,' vol. iii., p. 337 (Borda's original 

 memoir) ; and again, vol. iv., p. 441 (' Observations from Formentera 

 to Unst '). 



Many modifications and improvements have been introduced into 

 Captain Rater's method of determining the length of the pendulum, 

 and we must refer to his paper (' Phil. Trans.,' 1818, p. 33) for a minute 

 description and for a plate of his apparatus. 



The principle of his method is founded on the beautiful dynamical 

 theorem due to Huyghens, which we have already mentioned, namely, 

 that the axis of suspension and centre of oscillation are reciprocal in 

 the same body. Now conceive two wedge-formed pieces of steel (knife- 

 edges), with the sharp edges towards each other and parallel, to be 

 fixed at right angles to a bar of any form whatsoever, and in a line 

 passing through its centre of gravity; then if the time of oscillation 

 on either edge be the same, the distance between the knife-edges U 

 the length of the simple pendulum which swings in the same time. 

 Captain Kater, in his experiments, used a pendulum of on incon\ 

 form, the same nearly as in his invariable pendulum (see cut of Kater's 

 invariable pendulum), with a second knife-edge below the centre of tho 

 bob, and there are several doubtful points in the reductions which he 

 employed and the subsequent measurement between the edge.- 

 his memoir must always be carefully perused by any future experi- 

 menter who wishes to make himself thoroughly acquainted with the 

 subject. The method of observing the coincidences is remarkably 

 ingenious and accurate. 



We conceive that the best form of pendulum for the object of 

 measuring the actual length according to Kater's method is 

 that adopted by Mr. Baily. A c is a rectangular bar, with 

 one knife-edge at the end A, and the other at B, about } of 

 the length from the same point. A should be as near the 

 end as sound workmanship will allow. The necessary veri- 

 fications, namely, the parallelism of the knife-edges, their 

 perpendicularity to the bar, and that the line which joins 

 the knife-edges passes also through the centre of gravity, 

 are all very simple, and there is little difficulty in adjusting 

 the oscillations on each knife-edge to synchronism-t This 

 bar pendulum is suspended on agate planes in front of the 

 clock, great core being token that the support is perfectly 

 firm and the agate planes horizontal. A white disc of a 

 spindle shape and exactly the width of the bar is pasted on 

 the clock bob, and the viewing telescope fixed in a line with 

 the disc and bar, so that, both being at rest, the white disc 

 is just covered by the bar. Captain Kater used a diaphragm 

 in his telescope, the sides of which were parallel to the bar 

 and touched its edges; but it has been found more con- 

 venient to place a pair of parallel plates between the bar and 

 the disc, the sides of which ore so adjusted as just to allow 

 the disc on the clock bob to be seen between them. There is 

 a divided scale to mark the arcjof vibration, two or three ther- 

 mometers are hung at different heights about the pendulum 

 bar to register the temperature, and there is a barometer for 

 the atmospheric pressure, all which are to) be read at the beginning 

 and ,end of each series. The observation itself is of the simplest kiud. 



Thin part of the correction due to the air was omitted in the French 

 experiment* and In Captain Hater's: wo Mr. Daily's memoir referred to above, 

 and AIK. 



t ThU may be done two ways. Place the knife edges A anil B at the 

 dlitanco which will give the proper interval for cobudslnga*, 30-4 inches for 

 Instance, A being at one end, and make the bar rather more than one-third 

 longer ; then the otcillaUont on B will be ilowcr than tboie on A. By filing 



