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so much advantage as a measure of time. The same instrument has also been applied with great success in determining the figure of the Earth. The length of the pendulum, which is the distance between the centre of suspension and the centre of oscillation of the vibrating body, is always in proportion to the force of gravity at the place where it is used, when the time of vibration is constant; and hence the pendulum becomes a proper instrument for measuring its intensity; and, consequently, for ascertaining the comparative distances of different places from the centre of attraction. If we denote the length of the pendulum by 1; the intensity of gravity, as measured by the space which a heavy body would describe in a second with the velocity acquired in the first second of its descent uniformly continued, by g; the ratio of the diameter of a circle to its circumference by ; and the time of one vibration by t; then mathematicians prove that, when the arcs of vibration are small, the relation between the length of the pendulum and the time of its vibration is expressed by the equation,

t = T

g

In any other situation, where the length of the pendulum is l', and the intensity of gravity g', the same relation is expressed by

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Now if the times in both instances be equal, as one

second, for example, we necessarily have

T

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but as is common to both sides of the equation, we may divide by it, and then

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and by converting this equation into a proportion, 1: l'g: g'.

When, therefore, the length of the seconds pendulum is found by experiment in different parts of the globe, the ratio of the intensity of gravity in these places is determined.

As the pendulum is caused to vibrate by the force of gravity, the greater the intensity of this force, the greater will be its velocity, and, consequently, the quicker its vibrations, when the length of the pendulum remains constant. The duration of each of these oscillations is ascertained by the number which a pendulum makes in a given interval between two consecutive passages of a star over the same meridian; for instance, when a pendulum is transported to different parts of the earth, experience shows that the velocity of its oscillations increases from the equator to the poles; and the law of this acceleration, which has been determined with great accuracy, is as the square of the sine of the latitude; as the rotation of the globe on its axis requires. Richer, who was sent by the French Academy of Sciences to Cayenne, for the purpose of making astronomical observations, was the first who put this interesting fact to an experimental proof. He found that his clock, which had been regulated to mean time at Paris, lost a sensible quantity every day at Cayenne. This was the first direct proof that was obtained of the diminution of gravity at the equator; and it has since been very carefully repeated in a great number of places, all of which have confirmed the general result; the resistance of the air being also taken into consideration. The following table shows the general results that have been obtained on this subject; the length

of the pendulum at the equator being unity, and the time of vibration in all of them the same: viz.

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The lengths of pendulums oscillating in the same time being proportional to the gravitating forces by which they are caused to vibrate, it is evident from the preceding series that the force of gravity increases from the equator to the poles; since, in order that the pendulum may make each of its vibrations in the same time, in different latitudes, it is necessary to increase its length, as the latitude augments. This increase in the length of the pendulum is nearly proportional to the square of the sine of latitude, as may easily be proved by a comparison of the numbers in the preceding table with each other; and it affords another proof of the diurnal revolution of the Earth on its axis. These numbers also represent the proportional weights of the same mass of matter, if transported respectively into these latitudes. And likewise, as the spaces through which heavy bodies descend, when at full liberty to obey the action of this force, are always proportional to the forces by which they are put in motion, the relation of these spaces will also be expressed by the preceding numbers.

It is, however, only the ratio of the absolute lengths of the pendulum that is given for the respective latitudes in the above table, and which may be found for any other latitude by means of the square of its sine. In order to find the absolute lengths, one of them must be given. Now it has been determined by numerous experiments, that, in the latitude of London, the length of the seconds pendulum is very nearly equal to 39 English inches. Hence by means of

this length, that for any other latitude may be found by proportion.

But as the length of the pendulum will necessarily vary according to the species of time in which its vibrations are estimated, it becomes the first subject of consideration to fix upon the species of time in which these vibrations are to be reckoned. That which corresponds to the length above specified is the second of mean time. By this we are enabled to ascertain the length of the pendulum, acted upon by the same force of gravity, or, which is the same thing, situated on the same parallel of latitude. Under these circumstances, the lengths of simple pendulums are to each other as the squares of their times of vibration; for we have

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and by omitting the quantities that are common to the last two terms of this proportion, we have

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Then by substituting the numbers in this proportion, gives

12: 99727:39.125: 38.9116,

which is the length of the pendulum vibrating sidereal seconds in the latitude of London; and for any other latitude its length may be found by the principles above stated.

In general terms; if we denote the latitude by L, the corresponding length of the seconds pendulum by 7, and that of the seconds pendulum at the equator by l'; it has been found, from a great number of observations, that the value of I is expressed by the following formula: viz.

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Now the value of l' has been found to be 39.0265 English inches, and hence, by substitution,

139.0265 + 08040804 cos 2L =
39-1069 ·0804 cos 2L,

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in which value 39-1069 is the length of the seconds pendulum at 45° of latitude; and it will be observed that when the latitude, or the value of L, is less than 45°, the latter member will be subtractive, but when it exceeds 45° it will be additive; for then the cos 2L becomes negative; and consequently the product of that cos by 0804 must be added to the number 39-1069. If we take the latitude of 51° 30′ as an example, which is nearly that of London, this formula gives the length of the second's pendulum for that latitude 39-124986 inches; agreeing with the length as found by experiments.

This formula is analogous to that for the increase in the degrees of latitude, and the ellipticity of the Earth as deduced from each is very nearly the same; that from the pendulum being, and that given by the measurement of degrees. The length of a pendulum may also be found from the number of vibrations it makes in a given time, as well as from the time of each vibration, for these are evidently in the reciprocal ration of the times: we have, therefore,

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where n and n' denote the numbers of vibrations cor

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