The rule which was employed in the measurement was of platina, and exceeded 12 feet in length. It was cover ed by another rule, or sheath of copper, 111 feet in length, and firmly attached by means of three pressure screws to the upper extremity MN, (Fig. 6.) of the platina rule, but at liberty to slide along it at the lower. By this means, the difference of the expansion of the two rules was indi cated ; for their divisions being made to ccincide exactly ,at a certain temperature, and the dilatation of copper be ing greater than that of platina, upon any increase or di minution of temperature the divisions would no longer coincide. The distance, therefore, between the corre sponding divisions, indicated by a vernier ST fixed to the platina rule, and observed through a rectangular slit, PR, in the copper, gave the ratio of the dilatation of the two metals, whence, the dilatation of copper having been pre viously determined by experiment, the absolute dilatation of the platina, at any temperature, could easily be dedu ced. The lower extremity of the rule was armed with a graduated tongue of platina, which could he protruded by pressing on the button E. Its divisions were equal to the TO of a toise, a vernier indicated the tenths, or the 1200th of a line nearly. The upper extremity of the rule was terminated by a transverse A BCD of tempered steel, by means of which it was suspended in the plane of the knife-edge, (Fig. 3.) The lower surface of this trans verse had been carefully polished, and the end of the rule was brought into very close contact with it, so that, when the rule was suspended in the plane of the pendulum, its upper extremity was exactly on a level with the plane of suspension.
When a sufficient number of coincidences had been observed, and it was required to measure elle' length of the pendulum, the first thing necessary was to bring it to a state of perfect rest. The horizontal plane IH, (Fig. 3.) of which the support was firmly fixed to a stone project ing from the wall, (Fig. 2.) was then elevated by means of its screw, till it was brought into contact with the lowest point of the ball, and as the threads of the screw were extremely fine, the contact could be observed with the ut most precision. The knife-edge of the pendulum was then removed from the middle of the plane of suspension OP, (Fig. 4.) and the rule which had been previously sus pended at QR was substituted in its place. The tongue of the rule was next protruded, till it touched the plane IH, and the contact having been carefully observed, the divisions of the vernier were read off. Accurate experi ments had previously been made, to determine how..much the rule was lengthened by its own weight when sus "tended ; and the mean of these experiments gave of a division for the part of platina, and for the part of copper.
After this general exposition, an example of the me thod of obtaining the numerical results will serve to eluci date the whole process. In one of the observations, the interval between the first and second coincidences of the pendulums was 4394", as indicated by the clock. The
half, or 2197-1=2196, was the number of oscillations made by the trial pendulum during the sante time, for the clock gained two oscillations on the trial pendulum in the interval between the two coincidences. The clock gained 13".4 per day on the fixed stars, and, therefore, made 86413.4 oscillations during a sidereal day, or 86650 during a mean solar day. Hence, by proportion, the number of oscillations made by the trial pendulum in a mean solar day was 43305.28. The mean of this, and of four other coincidences, was found to be 43305.30.
.
This was the number of oscillations in an arc of definite extent, and varying from the commencement to the end of the interval. The duration of the oscillations is, there fore, greater than it would have been had they been made in an indefinitely small arc, and must be reduced to this case, in order that the results of different experiments may be comparable with each other. lithe arcs are small and 2 a be the amplitude at the commencement of an interval, 2 a' at the end, the mean amplitude may be considered a + a', and the reduction obtained by formula 5th, which in this case would become N— N' 1 + Sin. 2(a +) a' Borda, however not satisfied with this approximation, gave a formula for the correction, which was afterwards demon strated by Biot in the third volume of his Astronome Phy sique, grounded on the more accurate hypothesis, that while the times increase in an arithmetical progression, the amplitudes decrease in a geometrical. Calling the required correction az, the formula is sin. (a a,) sin. (a— a') — 32 M (log, sin. a — log. sin. a') M being the modulus of the common logarithmic tables, or 2.30258509. This formula becomes the same as the former, when the arcs a and a' are so small that their first powers only need to be regarded in the development of their logarithms and sines. Borda calculated the cor rections for five coincidences, and the mean obtained was 0.18. Hence the number of oscillations of the pendulum, in a mean solar day, and in an arc indefinitely small, was 43305.413.
The distance between the plane of suspension and the plane of contact, in divisions of the rule, was 203952.2, but two corrections were required ; one on account of the elongation of the rule by its own weight when suspended, and the other on account of the difference of temperature during the oscillations and at the time of measurement. We have already stated the first of these to be the of a division, which was to be added ; the other was found, by experiment, to be 0.42, and required to be subtracted, the temperature being higher at the time of measurement than it had been during the oscillations. The corrected distance between the two planes was therefore 203932.08 i but it remained to determine the centre of oscillation, in order to obtain the length of the isochronous simplesuendulum.