Suppose equal to 1", and that the arcs described on each side of the vertical are 5°. The versed sine of 5° is 0,0038053,

The error in each oscillation will, therefore, be

Thus, if a body descend by the action of gravity along a circular curve, and describe arcs infinitely small on each side of the lowest point in a second of time, the duration of each oscillation, no allowance being made for friction or the resistance of the air, would differ only 0,0004757 from that of an oscillation through an arc of 5o on each side of the lowest point, so that in a day or during 24 × 60 × 60 86400" vibrations, the difference would amount to 86400 x 0,0004757 or 41". Thus a pendulum of the length required to vibrate seconds, and performing its oscillations through arcs of 5° on each side of a vertical, would lose only 41′′ a day, when compared with one vibrating in arcs infinitely small.

If the arcs described on each side of the vertical were only 1°, the versed sine of which is 0,0001523, the daily loss would be only 1",64, that is, 12 nearly, and for half a degree, the loss would be 0,41 or of a second daily.

Of Pendulums.

344. What we have said is particularly applicable to pendulums. By a pendulum, is to be understood a rod or thread suspended at one extremity from a fixed point, and supporting at the other extremity one or several bodies. It is called a simple pendulum when it is supposed to consist merely of a mass or weight sustained by a thread or rod without gravity, and when at the same time this mass is of a diameter very small relative to the length of the pendulum. We shall speak for the present only of the simple pendulum.

When the pendulum CB is drawn from its vertical position, the force of gravity acting according to the vertical line AM is not wholly employed in moving the body; a part is exerted against the point C. Let therefore the whole force of gravity, represented by AM, be decomposed into two others, represented the one by AN, directed according to CAN, which will be destroyed, and the other by AP which urges the body along the arc AB. Now as the radius CA is perpendicular to the arc, it will be seen that the motion is here decomposed in the same manner as in the case above considered, where the body is supposed, without any material connection with C, to descend along the arc AB, which has for its radius the length AC of the pendulum. Accordingly every thing which we have said is applicable to pendulums. The following are some of the consequences which are derived from the preceding investigation.

345. We have found for the duration t of an oscillation, the following expression, namely,

Hence, for another pendulum whose length is a', and which is urged by a different force of gravity, or one that is capable of giving the velocity g in a second, we shall have, by calling the duration of an oscillation in this second case,

that is, if two pendulums of different lengths are urged by different gravities, the durations of the oscillations are as the square roots of the lengths of the pendulums, divided by the square roots of the quantities which denote these gravities.

346. As gravity is the same in the same place, we shall have for pendulums of different lengths vibrating in the same place or same part of the earth, g=g, and consequently in this case the proportion becomes

that is, in the same piace the durations of the oscillations are as the square roots of the lengths of the pendulums.

347. But if the same pendulum be successively exposed to the action of two different gravities, a being equal to a', the pro portion

in other words, the durations of the oscillations of the same pendulum. in different places are inversely as the square roots of gravity.

348. Let n be the number of oscillations or vibrations made by the pendulum a in a given time, as one hour or 3600", we shall have t = For the same reason, if we represent

3600"

by n' the number of vibrations made by the pendulum ɗ in the

that is, the number of vibrations made in the same time by two pendulums of different lengths are inversely as the durations of their respective vibrations. Consequently, since

that is, the number of vibrations made in the same time by two pendulums of different lengths, and which are urged by different gravities, are in the inverse ratio of the square roots of the lengths of the pen

dulums divided by the square roots of the gravities; so that if the gravities are the same, the number of vibrations will be reciprocally as the square roots of the lengths of the pendulums; and if the lengths are the same, the number of vibrations will be directly as the square roots of the gravities.

349. Hence if the same pendulum, carried to different parts of the earth, does not make the same number of vibrations in the same interval of time, it is to be inferred that gravity is not the same in these places, and the number of vibrations actually made in the same time by the same pendulum in two different places, will furnish the means of ascertaining the relative intensities of gravity at these places. It is by experiments of this kind, taken in connection with the foregoing proposition, that we are now assured of the diminution of gravity as we approach toward the equator, and on the other hand of its augmentation as we proceed from the equator toward either pole.

350. If we call t the time employed by a heavy body, falling freely, in describing the diameter BD or 2 a, we shall have

Geom.

292

that is, the duration of the descent through any small arc AB is to the time of falling through the diameter, as the fourth of the circumference of a circle is to its diameter. But the fourth of the circumference is less than the diameter; consequently a

body employs less time in descending along a small arc of a circle of which the inferior tangent is horizontal, than it would employ in falling through the diameter; and since the time required to pass through the diameter is the same with that required to describe any chord AB, it will be seen, that a body 332. would pass sooner from A to B, by descending along the arc AB, than by moving through the straight line AB. Therefore, although the straight line is indeed the shortest way from one point to another, it is not that which requires the least time for the pas sage of a heavy body.

Of the Line of swiftest Descent.

351. Not only is not a straight line that along which a heavy body would proceed in the shortest time from one point to another, out of the same vertical, but it is not the arc of a circle which answers to this description; it is the arc of another curve which may be found in the following manner.

Suppose AMR to be the curve sought, or that through which a heavy body would pass in the least time from a given point A, to a given point B. If we take in this curve two points M, m', infinitely near to each other, the arc M m' must also be described in less time than any other arc passing through these same points M, m', since these two points may be taken as the very points in question. Having taken the point N infinitely nearer to Mm' than M is to m', suppose infinitely small straight lines MN, N m2 to be drawn; since the time of describing M m m' must be a minimum, it follows that the difference between the time of passing through Mm m' and the time through MN m', which is the differential of the time, must be zero.

=

Through the points M, N, m', draw the horizontal lines MP, m P', m' P", and through A, the vertical line AC. Call AP, x, PM, y; AM, s, and suppose M m = m m', or that, d s is constant. Then mr = dx, r M dy, mr = dx+d dx, r' m' = dy+ddy. Let u be the velocity with which the body describes Mm; it will be the velocity with which MN is described; and u + du will be that with which mm' and Nm' will be described. Therefore Mech.

29

Fig.166.