In all that has been hitherto said, the power of gravity has been supposed con stantly the same. But, if the said power varies, the lengths of pendulums must vary in the same proportion, in order that they may vibrate in equal times ; for we have shewn, that the ratio of the times of vibration and descent through half the lengths is given, and conse quently the times of vibration and de scent through the whole length is given ; but the times of vibration are supposed equal, therefore the times of descent through the lengths of the pendulum are equal. But bodies descending through unequal spaces, in equal times, are im pelled by powers that are as the spaces described, that is, the powers of gravity are as the lengths of the pendulums.
Pendulums' length in latitude of Lon don, to swing Inches.
Seconds 39.2 Seconds 9.84 Seconds 2 45 Length of Pendulums to vibrate Seconds at every Fifth Degree of Latitude.
Mile. To find the length of a pendu lum to make any number of vibrations, and vice versa. Call the pendulum, mak ing 60 vibrations the standard length ; then say, as the square of the given num ber of vibrations is to the square of 60, so is the length of the standard to the length sought. If the length of the pendulum be given, and the number of vibrations it makes in a minute be required ; say, as the given length is to the standard length, so is the square of 60, its vibrations in a minute, to the square of the number re quired. The square root of which will be the number of vibrations made in a minute.
The greatest inconvenience attending this most useful instrument is, that it is constantly liable to an alteration of its length, from the effects of heat and cold, which very sensibly expand and con tract all metalline bodies. See BRAT, PYROMETER, &c.
To remedy this inconvenience, the common method is by applying the bob of the pendulum with a screw ; so that it may be at any time made longer or short er, according as the bob is screwed down wards or upwards, and thereby the time of its vibrations kept always the same. Again, if a glass or metalline tube, uniform throughout, filled with quicksilver, and 58.8 inches long, were applied to a clock, it would vibrate seconds for 39.2=- 4 of 58.8, and such a pendulum admits of a twofold expansion and contraction, viz. one of the metal, and the other of the mercury, and these will be at the same time contrary, and therefore will correct each other. For by what we have shewn, the metal will extend in length with heat, and so the pendulum will vibrate slower on that account. The mercury also will expand with heat, and since by this ex pansion it must extend the length of the column upward, and consequently raise the centre of oscillation; so that by this means its distance from the point of sus pension will be shortened, and therefbre the pendulum on this account will vi brate quicker ; wherefore, it' the circum stances of the tube and mercury are skil fully adjusted, the time of the clock might, by this means, fbr a long course of time, continue the same, without any sensible gain or loss. This was the in vention of Mr. Graham, in the year 1721, who made a clock of this sort, and com pared it with one of the best of the com mon sort for three years together, and found the errors of the fbrmer but about one-eighth part of the latter.
Mr. Graham also made a pendulum consisting of three bars, one of steel be tween two of brass, and the steel bar act ed upon a lever, so as to raise the pendu lum, when lengthened by heat, and to let it down, when shortened by cold ; but he found this clock liable to sudden starts and jerks in its motion.
The ingenious Mr. Ellicott, in the Transactions of the Royal Society, de scribes a pendulum of his invention, composed of brass andiron, with the me thod of applying it, so as to avoid the many jerks to which the machine might be liable.
But besides the irregularities arising from heat and cold, pendulum clocks are liable to others from friction and foulness; to obviate which, Mr. Harrison has seve ral excellent contrivances, whereby his clocks are almost entirely free from fric tion, and never need to be cleaned. See LONGITUDE.
The gridiron pendulum is a contrivance for the same purpose. Instead of one rod, this pendulum is composed of any conve nient odd number of rods, as five, seven or nine ; being so connected, that the ef fect of one set of them counteracts that of the other set; and therefore, if they are properly adjusted to each other, the centres of suspension and oscillation will always be equidistant. Fig. 11 represents a gridiron pendulum composed of nine rods, steel and brass alternately. The two outer rods, A B, CD, which are of steel, are fastened to the cross pieces A C, B D by means of pins. The next two rods, E F, G H, are of brass, and are fastened to the lower bar B D, and to the second upper bar E G. The two follow ing rods are of steel, and are fastened to the cross bars EG and III. The two rods adjacent to the central rod being of brass, are fastened to the cross pieces I K and L and the central rod, to which the ball of the pendulum is attached, is sus pended from the cross piece L M, and passes freely through a perforation in each of the cross bars I K, B D. From this disposition of the rods, it is evident that, by the expansion of the extreme rods, the cross piece B D, and the two rods attached to it, will descend : but since these rods are expanded by the same heat, the cross piece E G will con sequently be raised, and therefore also the two next rods ; but because these rods are also expanded, the cross bar I K will descend : and by the expansion of the two next rods, the piece L SI will be raised a quantity sufficient to counter act the expansion of the central rod. Whence it is obvious, that the effect of the steel rods is to increase the length of the pendulum in hot weather, and to diminish it in cold weather, and that the brass rods have a contrary effect upon the pendulum. The effect of the brass rods must, however, be equivalent not only to that of the steel rods, but also to the part above the frame and spring, which connects it with the cock, and to that part between the lower part of the frame and the centre of the ball.