In the first volume of the Transactions of the Society of Arts,' p. 238, Mr. Hatton proposed, as a mode of fixing a permanent standard of length, to suspend a weight from a fine hair to a clip in an upright bar, sliding up and down in a vertical frame. The hair passed through a fixed clip. The weight was to be swung, and the vibrations counted, in two positions of the bar, and from the difference in the times of vibration and the space through which the bar was moved, the length of the seconds pendulum was to be computed.
In 1787 Mr. John Whitehurst published An attempt towards obtaining invariable Measures of Length,' &c., which is remarkable for its ingenuity. He suspended a leaden ball with a flat steel wire in front of a straight upright frame, the wire being long enough to make forty-two oscillations in a minute. A clock with dead-beat escapement and a clip to hold the wire was slid up and down the frame, and secured and adjustable at two points where the clip made the free oscillations respectively forty-two and eighty-four in a minute. The crutch of the clock, being continued upwards in a screw, carried a weight, by moving which the oscillations of the crutch alone could be regulated to forty-two and eighty-four oscillations, and therefore would not interfere with the free oscillation of the ball and wire, but only keep up their motion. The going weight of the clock was in each case such as sustained an oscillation of 3°. It is clear that if all were properly executed, the clock-frame with its clip must have been shifted between the two positions through a space equal to the difference between the simple pendulums which correspond to forty-two and eighty-four oscillations per minute. A line was drawn in each position along the upper edge of the clock frame upon a brass rule fixed to the upright support, and this space was afterwards accurately measured, and the length of the simple seconds pendulumitheuce computed. Whitehurst's length of the seconds pendulum is 391196 inches of Troughten's standard, but the corrections for the buoyancy of the air and for temperature are not Introduced. It is probable that he introduced greater errors than those he wished to get rid of in Hatton's method, for the real difficulty is not that of counting the vibrations, but of measuring the length between the two clips, in avoiding the errors of temperature, and the uncertainty as to the effective point of suspen sion. The principle of Hatton's method, that of measuring the
difference between two pendulums, has been adopted, as we shall see, by BeaseL The foregoing account is merely a sketch of the history of this mechanical problem, which in the hands of Borda, and more recently of Kater and Bessel. has received a more accurate solution. There are still anomalies and imperfections in some parts of tho processes which require clearing up, but the errors have been reduced within compara tively moderate limits. Before describing these experiments we shall give a brief account of the formulm which they require.
The expression which connects the time of one oscillation of a simple pendulum in an infinitesmal arc, with its length 1, at a place where the /i force of gravity is represented by g, is t = w r being 3141596, or gt circumference to diameter 1; the measure of gravity g, being twice the space through which a body would fall freely in I', or, what is the same thing, the space through which a body would move in 1', with the velocity which it in falling freely for P.
Hence if 1 be the length of the simple seconds pendulum, therefore g is known when 1 can be measured. The process therefore of finding the effective force of gravity at any place is reduced to finding the length of the simple pendulum which vibrates seconds.
The French astronomers, in their great survey of the arc of the meridian, determined the absolute length of the pendulum at different stations between Dunkerque and Fonnentera, and also in the con tinuation to Unat in the Shetland Isles, which is included in the English trigonometrical survey. It is, however, an operation of great delicacy, and when only the variation of gravity between different places is required, as is the case in researches into the figure of the earth, the observation may be more easily performed by swinging the same pendulum in different places, and ascertaining the number of vibrations which it makes in a day. Thus if n and a' be the number of vibrations made in a day by the same pendulum 1, at two different places at which the forces of gravity are fl and 2', and tho duration of one vibration at each place be t and t', then since the time of one vibration = a day divided by the number of vibrations, we shall have 1 / l f l\li I 1 a' g) kg'l ' vl* or g : g' : : n2 : n12.