That is, the force of gravity varies as the square of the number of vibrations of a given pendulum in the same time, which is usually taken to be a mean solar day. If, therefore, the number of vibrations of a pendulum in one day at a given place, London for instance, be known, and it is then transported to different places, and the number of vibrations in a day counted, a simple proportion will connect the forces of gravity at London and :every place at which the observation has been made.
If the length l' of the simple pendulum at any station be required from these observations, since 1 : : : g : : : n2 : n'2, 1 n'2 1' = -73-, which gives the length of the pendulum at any place in terms of the length at London, and the number of vibrations per diem at that place and London.
Though it scarcely belongs to our subject, we will give the expres sion by which the ellipticity of the earth is determined from pendulum _ observations. The length of the seconds pendulum at any latitude, X, may be supposed /=.14-13. ain.2 X, where A and n are constant quantities. Now from all the good observations, either of tho actual length or the number of vibrations per day of the same pendulum, determine the values of A and a; then, by Clairaut's theorem, the ellipticity of the earth = *4/08668 — 13 whence the ellipticity is found. By ellipticity is meant the excess of the equatorial over the polar radius of the earth, divided by the polar radius.
The apparatus of Borda will be generally from the follow ing description and figures.
The plumb-lino is suspended from a knife-edge piece, A B, and is attached below to a cup, E, which is ground to fit very exactly the platinum ball below. A little grease is rubbed on the inside of, the cup, making the contact perfect enough to exclude the air and to sus pend the ball. The knife-edge rests on agate planes, a, b, which are carefully levelled, and the frame op, which carries the planes, is fixed immoveably in a horizontal position. The plumb-line is in front of the comparing clock, which has a small cross drawn on the bob. When both are at rest, a telescope having a vertical line in the focus Is placed • few feet distant, so that the cross, the plumb-line, and the fixed wire of the telescope are all in the same right line, and all seen dis theetly.• Such a length is given to the plutob-line as that its oscil lations shall be a little slower than those of the clock.
The clack is now set a-going, and when it is quite steady, the platinum Nall is drawn a little on one aide, and made to swing through a very small Are. one or two degrees at the most. The observer then places
himself at the telescope, and notes the exact beat of the clock when the plumb-line bisects the cross and coincides with the faxed wire at the same time : this is called a coincitknee. As this observation seems much less definite than in Kater'a method, it is better to note those beats before and after the coincidence, when the eye cannot distinguish whether it is a coincidence or not, and to take the mean of the two for the true time of coincidence. Immediately after, the extent of the arc through which the plumb-lino swings is to be read off on a scale placed just behind it • the thermometers which are near the pendulum must &leo be read ell', and also the height of the barometer. The observer new waits until this appearance presents itself a second time (which may be in about ten minutes, though the interval is quite ad libitum), and having thus established the time which elapses between the observations, he may leave the apparatus for three, four, or five hours, when on his return the acme observations must be repeated of coincidence, arc, temperature, and barometer, and the series is finished. We should recommend closing as well as commencing a series with two coincidences in majorent eantelam, but except as n cheek, 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 obeerved. The rate of the clock is established by astronomical observation.
After a satisfactory number of scrim has been taken, the next object is to measure the length of the pendulum. The plane s It is raised by a fine screw until it just touches the bottom of the platimuu ball. This is an operation of great nicety, and must be effected by two or three trials, as the exact temperature of the wire rat the exact moment of contact is required ; and the presence of the observer, even for a short time, is 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 eliding tongue below, is suspended in its place. The tongue is slid down until it touches the plane t n, 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.