For this purpose the pendulum was first let in to a solid piece of mahogany, edgewise, to such a depth that the knife-edges approached to within a twentieth of an inch of the surface. This arrangement is represented Fig. 11. K is an upright piece of wood screwed to the end of the mahogany case ; and to this a string is attached, passing through the ring of a common spring steel-yard, the hook of which is fixed upon the pendulum, and exerts a force rather greater than the weight of the pendulum, equal to about ten pounds. To secure the parallelism of the knife edges, which, on account of the flexibility of the brass rod, was apt to be destroyed when the pendulum was placed in a horizontal position, two screws were passed through the mahogany case, in opposite directions, and made to act transversely on the extremity of the bar, next to the steel-yard. The microscope was then passed over the ex tremities of the knife-edges, and the requisite parallelism readily obtained by means of the screws. Alm this ad justment, four rectangular pieces of brass, each about half an' inch square, were brought into contact with the knife edges, at those points which had rested on the agate planes. On each of these pieces of brass a very fine line had been traced, the distances of which from the points in contact with the knife-edges, had previously been ascer tained by means of the micrometer. The brass pieces were then fixer in their place by means of springs attach ed to the mahogany case, and the pendulum being remov ed, the standard scale was placed beneath the micrometer, and the distance between the fine lines on the brass pieces was ascertained on one side of the pendulum. The scale was then transferred to the other side, and the operation repeated. The mean of the two measures will obviously correct any deviation from parallelism in the knife-edges which might remain, even after the precautions already mentioned had been taken.
The expansion of the pendulum and its specific gravity were determined by accurate experiments, in order to ob tain the corrections for the temperature and the buoyancy of the atmosphere. When these corrections were made, the distances between the knife-edges was found to be 39.44085 inches, and the number of oscillations in a mean solar day, was 86061.30; whence the length of the se conds pendulum, in the room in which the experiments were performed, was found by proportion equal to 39.13908 inches of Sir George Shuckburgh's standard scale, at the temperature of of Fahrenheit.
After this measurement had been effected, Captain Ka ter was requested by the Royal Society to repeat his expe riments at some of the principal stations of the trigono metrical survey. Accordingly, in the month of July, 1818, he sailed for Unst, amply furnished, by the libera lity of government, with every thing that could facilitate his operations. The spot he selected for the experiments, was that at which Biot had made his observations the pre ceding summer. To obviate the necessity of being oblig ed to provide a new stand for the apparatus at each sta tion, a frame of cast iron was employed for receiving the bell metal support of the pendulum, so constructed, that it could be screwed very firmly to the wall of a building. For farther security against any lateral shake, brackets were placed below, which at the bottom spread to a dis tance of three feet. The clock, on the regularity of which
so much depended, was by Arnold, having a gridiron pendulum for the compensation of temperature. The other stations at which experiments were made, were Portsoy, Leith Fort, Clifton, Arbury Hill, and Dunnose in the Isle of Wight.
After the explanation which has already been given of the method of making the observations, it is unnecessary to give a detailed account of his proceedings at any of the stations, as they were all exactly similar to those perform ed in London. The distance between the knife-edges, or the length of the pendulum, being a constant quantity, did not require to be measured at each station, as was the case in Blot's experiments; all that was necessary was to determine the number of its oscillations in 24 hours, and to observe the temperature while it was compared with the clock, in order to make a proper allowance for the di latation. The greatest difficulty, and what required the greatest expense of time at each station, was to ascertain the rate of the clock.
Before giving the results of Captain Kater's experi ments, it is proper to remark, that in allowing for the am plitude of the oscillations, lie did not employ the accurate formula of Borda, considering it to be an unnecessary re finement in practice, especially as there was always an uncertainty in observing the extent of the arc, amounting to one or two hundredths of a degree. The consideration he proceeded on was this : The number of oscillations in 24 hours, in an indefinitely small arc, is known to exceed that in an arc of one degree by 1.635 ; and in very small arcs the times are nearly as the squares of the arcs; hence, if the mean of the arcs observed at the commencement and end of each interval be taken, and its square multi plied by the number 1.635, the correction io be added to the observed number of oscillations will be obtained with sufficient accuracy.
The allowance which he made for the height of the sta tion above the level of the sea, is also different front that which is obtained on the supposition that gravity dimt nishes simply in the inverse ratio of the square of the distance from the centre. Following out an idea suggest ed by Dr. Thomas Young, in a paper published in the Philosophical Transactions for 1819, upon the density of the earth as affecting the reduction on the length of the pendulum, he multiplies the correction ob tained from the formula, by a coefficient, the value of tis hick depends on the attraction of the matter accumulat ed between the general level and the place of observation. However sound this principle may be, its application by Captain Kater, who makes the value of the coefficient de pend also on the geological characters of the surrounding country, is extremely arbitrary; for, unless the ratio of the density of any given mass of matter at the surface of the earth, to that of the strata below, be accurately de termined, (which is perhaps impossible when the mass is extensive,) its influence cannot bc assigned. The re lative density of the earth at different places may be pro perly determined by the measurement of the pendulum ; hut it is unsafe to reverse the process, and assign the length of the pendulum from the assumed density of the earth.