Harrison's pendnlum has been greatly improved by Troughton, who substituted for the two parallel brass rods two cylin ders of brass, sliding the one within the other, to which the steel rods are at tached. For a description of this, and various other modes of compensation, we refer the reader to an excellent chap ter on the subject by the late Captain Kater, in the volume of Mee/tanks in the Cabinet Cyclopedia.
Application of the Pendulum to the De termination of the relative Force of Gravi ty at different Places.—There are two methods of determining the relative in tensity of gravity by means of the pen dulum. According to the first, the abso lute length of the simple pendulum which makes a certain number of oscilla tions in a given time is accurately ascer tained at each of the places, and the com parative force of gravity is then given by the formula g' = — g. According to the other method, an invariable pendulum is swung at the different places, and the number of its oscillations noted at each, when the relative gravity is given by the formula g' = -F. Each of these meth ods have been followed in the delicate experiments which have been made for the purpose of determining the figure of the earth ; but though the results of both appear to be nearly equal in point of ac curacy, the latter method, on account of its affording greater facilities in practice, is now generally adopted.
It will readily be conceived that a pen dulum would be altogether unfit for the purpose of determining the minute va riations of gravity if it were attached to a clock, or any machinery by which its motions could be influenced. It must be suspended from a very firm support, to which it can communicate no vibratory motion ; and the most delicate precau tions are necessary to avoid the effects of friction, and other disturbing causes, by which the experiment may be influenced. The method followed by the French as tronomers, in their operations connected with the measurement of the meridian, was this : The pendulum was composed of a sphere of platinum, suspended by a slender iron wire from a knife edge of hardened steel resting on plane surfaces of polished agate. It was placed in front of a well-regulated astronomical clock, with which its oscillations were com pared, and the distance between its cen tres of suspension and oscillation deter mined by calculation from the length of the wire and the diameter of the sphere, ascertained by actual measurement. A different, and in many respects preferable mode of measuring the lengths of the seconds' pendulum, was adopted by Cap tain Kater, grounded on the property of oscillating bodies discovered by Huy gens ; namely that the centres of sus pension and oscillation are convertible.
From this property it follows that if two knife edges, turned in opposite direc tions, are inserted in the same pendu lum, and the mass be so adjusted, by means of a movable weight sliding on the rod, that the oscillations are per formed in exactly equal times when the pendulum is suspended from either knife edge, then the distance between the knife edges is the true length of the iso chronous simple pendulum. In this manner the measurement is effected more directly, and no calculation is re quired for finding the centre of oscilla tion. A third method, lately put in practice by the celebrated astronomer Bessel, consists in suspending a ball and wire from the upper end and then from the lower end of a rod of a given length, the ball being in both cases at the same distance below the rod. From the dif ference of the times of oscillation of the two pendulums thus formed, the length of the simple pendulum can be compu ted in terms of the rod, which is the difference of their lengths. The French
method is described, with all the requi site details, in the third volume of Base lifetrique, in Delambrc's Astronomic, tome iii. ; and in the Recueil d' Observationa Geodes-Epee, 41c., by Biot and Arago, Paris, 1821.
Captain Kater's pendulum was formed of a very thin bar of plate brass, with a heavy bob and movable weight, by means of which the isochron isni was obtained when the pension was made from the oppo site knife edges. But a much simpler modification has been adopted in the recent experi ments. The experimental pendu lums of the Royal Astronomical Society consist merely of a plain straight bar of iron or copper, 2 inches wide, half an inch thick, and about 621 inches long. At the distance of 5 inches from one end of the bar is placed the apex of one of the knife edges, A • and at the distance of 394 inches therefrom the apex of the other knife edge, B ; and the required adjustment to synchronism is produced by filling away one of the ends of the pendulum until the vibra tions are found by trial to be equal in both positions of the pendulum. It is obvious that, for the purpose of merely ascertaining the variations of gravity, a bar of this form with a single knife edge would equally answer the purpose ; but the advantage of the double suspension is this, that besides having two distinct and independent pendulums, each of which is a check upon the other, it fur nishes the means of ascertaining whether the pendulum has sustained any acciden tal injury, which would be immediately discoverable from the inequality of the number of vibrations between the two knife edges.
Corrections.—In order that the results of different sets of experiments may be exactly comparable with each other, seve ral corrections must be applied. The first of these is on account of the length of the arc of vibration, which, being of a finite and variable extent, the duration of the os cillations is consequently unequal, but always greater than in the case of an infinitely small arc. The number of os cillations is reduced to the case of an in finitely small arc by the formula.
M sin. (A + a) sin (A — a) N X 82 (log. sin. A—log. sin. a) where N is the number observed, M the logarithmic modulus = •4342945, A the initial, and a the final arc of vibration ; and as the arcs are always small, the computation may be shortened by using the arcs instead of sines.
In the second place, all the experi ments must be reduced to a common standard of temperature, which, in this country, is assumed at 62° of Fahrenheit. Let e denote the rate of expansion of the metal, t the mean height of the ther mometer at the time of the experiment ; then the correction of the number of vi brations on account of the temperature is N x I e (t-62°.) A third correction is required on ac count of the atmospheric pressure. The effect of the pressure of the atmosphere on the pendulum is to diminish the force of gravity in the ratio of the specific gravity of the pendulum to that of the air ; and on this principle the correction was formerly applied, regard being had to the height of the barometer. But it was recently remarked by Besse] that the pendulum with it a certain portion of air, the amount of which depends on the form of the pendulum ; and, conse quently, the specific gravity of the actu ally moving mass cannot be previously computed, but must be ascertained for each pendulum by actual experiments in air and in a vacuum.