The principle of his method is founded on the beautiful dynamical theorem duo to Huyghene, which we have already mentioned, namely, that the axis of suspension and centre of oscillation are reciprocal in the same body. Now conceive two wedge-formed pieces of steel (knife edges), with the sharp edges towards each other and parallel, to be fixed at right angles to a bar of any form whatsoever, and in a line passing through its centre of gravity; then if the time of oscillation on either edge be the same, the distance between the knife-edges is the length of the simple pendulum which swings in the same time. Captain hater, in his experiments, used a pendulum of an inconvenient form, the same nearly as in his invariable pendulum (see cut of Kater's invariable pendulum), with a second knife-edge below the centre of the bob, and there are several doubtful points in the reductions which be employed and the eubsequent measurement between the edges; still his memoir must always be carefully perused by any future experi menter who wishes to make himself thoroughly acquainted with the subject. The method of observing the coincidences is remarkably ingenious and accurate.
We conceive that the best form of pendulum for the object of measuring the actual length according to Kater's method is that adopted by Mr. Baily. A o is a rectangular bar, with one knife-edge at the end a, and tho other at n, about 3 of the length from the same point. A should be as near the end as sound workmanship will allow, The necessary veri fications, namely, the parallelism of the knife-edges, their perpendicularity to the bar, and that the line which joins the knife-edges passes also through the centre of gravity, are all very simple, and there is little difficulty in adjusting the oscillations on each knife-edge to synchronism+ This bar pendulum is suspended on agate planes hi front of the clock, great care being taken that the support is perfectly firm and the agate planes horizontal. A white disc of a spindle shape and exactly the width of the bar is pasted on the clock bob, and the viewing telescope fixed in a line with the disc and bar, so that, both being at rest, the white disc is just covered by the bar. Captain Kater used a diaphragm in his telescope, the sides of which were parallel to the bar and touched its edges; but it has been found more con venient to place a pair of parallel plates between the bar and the disc, the sides of which ere so adjusted as just to allow the disc on the clock bob to be seen between them. There is a divided 'teak to mark the arc of vibrntion, two or three ther mometers are hung at different heights about the pendulum bar to register the temperature, and there is a barometer for the atmospheric pressure, all which are to be read at the beginning and ,end of each series. The observation itself is of the simplest kind.
Set the clock going, and when its arc is steady, give the pendulum a small motion, 1° or lr at the utmost. The white disc, which is at wholly visible, is gradually invaded by the bar, at last it is entirely covered, first one side and then the other disappearing.* These times of disappearance are to be noted. Almost immediately after, the white edges reappear, first on one side and then on the other, and these reappearances are noted. The mean of the four gives the true time of coincidence. It has been found convenient in practice not to'
make the adjustments with the highest degree of precision, for you are then in doubt on which side the first disappearance and reappearance will take place ; whereas if there is a very little displacement, attention need only be fixed on every alternate beat, which is less strain on the eye and on the mind. This displacement however must be trifling, otherwise, when the arc becomes small, the disappearances and reap pearances will not take place. The reductions for the rate of the clock, for the are of vibration, the temperature, the buoyancy of the air and its inertia, and the height of the place of observation above the sea-level, are all to be made as in Borda's pendulum. For the details we refer to the standard memoirs which have been published on the subject. But there are two or three points which deserve more attention than they have generally met with. The clock is only to be considered as a counter of the vibrations, and to avoid any error arising from its irregular rate, the observations should be continued without intermission between two epochs, at each of which the error of the clock is ascertained by astronomical observation. If this be done, the only reliance upon the clock is for the short time which elapses between the last coincidence of one series and first of the next. It is true that this mode of observing requires artificial illumination, for it must be continued day and night. This is no disadvantage, but the contrary, for as the other great practical difficulty is that of ascertaining the exact temperature and keeping it steady, the best place for observation is a deep roomy vault or cellar.
The correction for the buoyancy of the air depends on the specific grads!, of the pendulum ; that arising from the dragged air, which we have called inertia, on its form. It is very desirable that the experi ments should be Made in vacua, and where that is difficult or impos sible, a,s in a travelling apparatus, the law and quantity of the correc tion for each pendulum should have been previously experimentally determined in vacua. The variation for temperature should be got from a comparison of summer and winter experiments at the same place. In addition to the height of the place of observation above the scndevel, a rough survey should be taken of the form of the surround ing country, and of the kind of strata, with the specific gravity of each. This is particularly necessary where the heights are considerable. (See Young's Remarks,' Phil. Trans.,' 1819, p. 93.) Still, when every known precaution has been taken, there are anomalies in the results of this experiment which are not yet clearly understood. It is very difficult to make the edge of the knife per fectly straight, and to keep it so when it is fastened to the bar. There is considerable uncertainty in measures of distance between the reci procal knife-edges; and lastly there is no authorised standard of length at present existing in England in terms of which the distance can be expressed. The length of the simple pendulum which vibrates seconds in the latitude of London, is not at present known with the degree of certainty t which the importance of the problem demands, and which is attainable.