V b x—xy 1- 2 The integral of this differential is not known under finite form ; to obtain it in a series, the factor (1— =-) must be developed according to the powers 2 of x. The binomial theorem gives x --A 1 x 1.3 1.3.5 x3 1-.— 2 2 2 2 . 4 4 2 .4 .6 8 consequently d d x . Vb 2' 2 2 . 4 4 .4 . 6 8 The integrals of the different terms of this expression are all comprehended under the general form x" d x A f n being a whole positiv,e number, or zero, and A a co efficient varying with a. To obtain the time of descent from B to A, the integrals must be taken from x=b to Let, therefore be the value of the integral x" d x x — taken between these limits, the index, of A corresponding to the exponent of x, so that A,', A &c. may be the values of The retardation of the seconds pendulum, on being carried towards the equator, was first observed by Richer, in the year 1672, at Cayenne,where he found it necessary to shorten the pendulum of his clock, which had been re gulated at Paris, according to mean time, by a line and a quarter. This phenomenon, when announced in Europe, appeared so very extraordinary, that it did not obtain im plicit credit till confirmed by the observations of Varin and Deshayes, in the islands of Gorec and Cayenne, and in various places in the West Indies. It was then demon strated, both by Huygens and Sir Isaac Newton, that the cause of the phenomenon was to be referred to the dimi nution of gravity occasioned by a compression of the earth at the poles; and since that time, the measurement of the pendulum has occupied much of the attention of astrono mers, and formed a material part in every inquiry relative to the figure of the earth. The French academicians, Bouguer, Godin, and Condamine, when engaged in mea suring the length of a meridional degree in Peru, made a great number of observations on the pendulum in places near the equator ; but the results of their experiments, although they put the fact of the compression of the earth beyond all doubt, did not agree so well with each other as to give any satisfactory information with regard to its amount. Numerous experiments have since been made to determine the length of the pendulum in various parts of the world, but as the greater part of them appear to have been performed with an apparatus by no means suit ed to the extreme delicacy of the operation, little depen dence can be placed on their results. Those of Mairan at Paris, and of our countrymen Mr. George Graham and Whitehurst, may be mentioned as being amongst the most accurate and ingenious. But the first measure ment obtained by a method perfectly unexceptionable, and with the most scrupulous attention to every' circum stance which could possibly influence the results, was that of Borda and Cassini, after the introduction of the " Systeme Metrique" in France. The immediate object which these distinguished philosophers had in view, was to determine the relation of the seconds pendulum to the metre, or ten millionth part of the quadrant of the me ridian, which had been selected by the academy as the unit or basis of a consistent and philosophical system of measures.
Measurement of the Pendulum The apparatus which Borda employed in his experi ments was constructed in the following manner. The pendulum of experiment was formed of a ball of platina, attached to a metallic wire, and suspended from the ex tremity of a block of stone, which rested on the top of a wall of solid masonry. It was placed at such a height, that the ball was nearly on a level with the centre of the bob of a clock pendulum placed at a little distance be hind. The oscillations of the two pendulums were ob served through a small telescope, placed at the distance of six feet, which, by enlarging the arc of oscillation, and augmenting the apparent velocity, enabled the observer to determine the moment of coincidence with much great er precision. In order that no disturbance might be oc casioned by the influence of the external air, the whole apparatus, including the clock, was placed within a wood en case, having glass windows in front, through which the oscillations were observed. ThiS general disposition
is represented, Plate CCCCLVIII. Fig. 2.
The pendulum was suspended by means of a knife-edge, which oscillated on a plane of steel. AB, (Fig. 3.) re presents the knife-edge, CD an appendage to which the wire was fixed, EF a screw, furnished with a nut GH, by means of which the oscillations of the knife-edge were regulated, and rendered of equal duration with those of the pendulum. When this synchronism was established, the oscillations of the pendulum could not be sensibly in fluenced by those of the knife-edge and its appendages, because the centre of gravity of the separate system which they formed, being extremely near the plane of suspen sion, an infinitely small effort was required to bring the oscillatory movement of the knife-edge into perfect accord ance with that of the whole system. The plane of sus pension was fixed to a plate of copper, IKL, (Fig. 4.) and attached to the block of stone, (Fig. 2.) in such a manner as to be in a position perfectly level. The knife-edge was placed over the opening ST; and when an observation was finished, it was moved aside towards S, and its place taken by a rule with which the length of the pendulum was to be compared. For suspending the ball, metallic wires were chosen, on account of their being more uni form than threads composed of vegetable substances; and the iron wire was selected as being the lightest, and presenting least surface in proportion to its strength. The ball was nearly sixteen one-sixth lines in diameter, and weighed 9911 grains—a little more than seventeen ounces.
The lower extremity of the wire was fixed by means of a pressure screw in the top of a small spherical cup of copper, Fig. 5. of the same radius as the ball; and the ball being covered with a thin layer of tallow, and then put into the cup, adhered to it in virtue of the atmosphe ric pressure, and the perfect contact of the surfaces. By this contrivance an opportunity was given of suspending the ball successively by opposite points of its surface, whereby any error could be corrected which might be oc casioned by its unequal density or imperfect sphericity. The other end of the wire was fixed also by means of a pressure screw to the appendage CD of the knife-edge. The trial pendulum s of such a length that it made a little less than one oscillation, while the pendulum of the clock made two, so that their movements coincided only at intervals, longer in proportion as one oscillation of the trial pendulum was more nearly equal to two of the clock. • A small circle of black paper, crossed with two white lines, was pasted on the bob of the clock pendulum, to serve as an index; and both pendulums being at rest, the glass 0 was placed in the direction so that the wire of the pendulum, when seen through it, fell upon the in tersection of the white lines. A dark screen was also placed at a little distance before the pendulum, the verti cal edge of Which concealed half the wire. This dispo sition being made, the clock was put in motion, and its rate having been accurately determined, the trial pendu lum was also made to oscillate. Suppose, now, that at the outset the wire disappeared behind the screen before the cross ; as the times of oscillation were not accurately as two to one, the interval between the disappearances would decrease, till at length both objects came to pass behind the screen at the same instant. The instant of this first coincidence was observed, the oscillations then began to disagree, afterwards to approach, till at length a se cond coincidence took place. In the interval between the two coincidences the clock had gained two seconds on the pendulum, so that the ratio of the times of oscillation of the two pendulums was given. The coincidences were observed till the oscillations became so small that the ob servations could not longer be made with sufficient accu racy. A scale •as•placed at a little distance behind the pendulum, graduated into minutes of a degree, to mea sure the amplitude.