One of the great practical difficulties in this species of pendulum experiments seems to be the extreme and sudden sensibility of the plumb-line to temperature. The whole apparatus is enclosed in a glass cage to exclude currents of air, and the observer is particularly careful, after bringing the lower plane into contact with the bottom of the ball, to absent himself until the temperature is steady, and then to make the contact complete. It is another objection that the different manipulations required are of great delicacy, and therefore not suited to every observer and every place; but when the utmost care and skill are employed, the results appear satisfactory.
The knife-edge, A B, by which the line and ball are suspended, is an ingenious contrivance, exactly similar to Whiteburet's synchronous crutch above described. By the upper screw and weight, the oscilla tions of the knife-edge alone can be made synchronous with the oscilla tions of the whole pendulum, so that the oscillations are just the same as if the knife-edge were immaterial, and the point of suspension exactly in the line of its edge. The platinum ball can be suspended from any side, and if two positions diametrically opposite be taken, the effect of any irregularity of shape or density disappears from the mean of the two results. There arc several corrections to be applied to the quantities immediately given by observation, before the length of the simple pendulum can be concluded. The oscillations are made in an arc of remade extent. Now the time of oscillation in an arc of A° on each side the lowest point, is greater than the time in an infinitely small arc (which is the arc required), in the proportion of I+ A° 16 to I. Au expression which depends upon the first and last arose of each series, gives the correction which is to be added to the number of oscillations observed. This is taken from a table. Between each coincidence, the plumb-line has made two oscillations leas than the clock, therefore subtraoting twice the number of coincidences from the number of seconds elapsed between the first and het coincidences, you will have the number of oscillations of the pendulum during a certain time shown by the clock. Each of these numbers requires a correc tion : to the number of oscillations of the pendulum, must he added the correction for aro just mentioned ; and to the time as shown by the clock, the proportional part of its rate during the experiment. A
simple proportion will now give the number of infinitely small vibra Hone in 24 hours.
Further corrections are to be applied. The length of the wire and ball during the observations must be reduced to the length they would have had at the temperature when the contact with the plena and the measurement was made, and this again must be converted into the equivalent length when the thermometer is at the freezing-point, which is the French standard temperature. The theoretical pendulum is supposed to swing in ratuo, and as the density of the air affects the time of oscillation two ways, both by diminishing the moving force of the pendulum and by • to its inertia by the air carried along with it, a correction is required on this account, which depends upon the barometer and thermometer, on the specific gravity of the materials of the pendulum, and also on its form.
From the dimensions and specific gravities of the parts of the appa ratus, the distance between the centre of oscillation and the bottom of the ball can be computed. Hence the length of the simple pendulum, which oscillates in an infinitely email are a certain number of times in a day and in vacua, can be assigned, from which the length of the seconds pendulum is deduced by simple proportion.
If the place of observation be above the level of the sea, it is usual to reduce the length to what it would have been at the sea-lova: This is a very uncertain quantity, as it depends upon the configuration and density of the strata in and near the spot. In the earlier experi ments gravity was supposed to vary inversely as the square of the dis tance from the centre of the earth, and thus the attractionof the matter between the observer and the sea-level was wholly neglected. I)r. Young showed that in a table-land of average density the correc tion thus obtained was too Large, and should be multiplied by For a full description of Borda's method, with instances, sea du SystZme 3letrique Decimal,' voi. p. 337 (llorda's original memoir); and again, voL iv., p. 441 (` Observations from Formentera to Unst').
Many modifications and improvements leave been introduced into Captain Kater's method of determining the length of the pendulum, and we must refer to his paper (‘ Phil. Trans.,' 1818, p. 33) for a minute description and for a plate of his apparatus.