For this purpose, the weight of the wire, the weight and centre of gravity of the cup, and the radius and weight of the platina ball, were all determined with the most scrupu lous accuracy. From these elements, the centre of oscilla tion of the compound system was calculated, and found to be 51.08 divisions above the centre of the ball. The radius of the ball was 937 divisions, and the sum of these numbers being deducted from 203952.08, the distance between the plane of suspension and plane of contact, there remained 202964 for the length of the isochronous simple pendulum, in divisions of the rule. Hence, by formula third, the length of the seconds pendulum was obtained equal to 50989.55 divisions.
A correction was now necessary to reduce the length of the pendulum to that which would have been observed had the experiments been performed in vacuo. The ef fect of the pressure of the atmosphere upon the apparatus is to diminish the force of gravity in the ratio of the speci fic gravity of the pendulum to that of the air. It occasions no change in the duration of the oscillations ; for if the ball is retarded while it descends from the highest to the lowest point of the arc, it is equally retarded while it ascends from the lowest to the highest : and as the effect of the re tardation in the one case is to increase the time of the os cillation, and in the other to diminish it, the time of an entire oscillation is the same as if the pressure of the at mosphere were removed. Supposing the specific gravity of the pendulum to be to that of the air in the ratio of n to I, if the square of the number of oscillations in air, during a given tone, be augmented in the same ratio, the number of oscillations in vacuo, during the same time, will be ob tained; and thence the addition to be made to the length of the pendulum. The specific gravity of air varies in versely as its expansion (which is about of its bulk for a degree of Fahranheit) and directly as the height of the barometer. Borda found that when the barometer stood at 28 inches, and the centigrade thermometer at 21°, the specific gravities of the air and of the ball were as t to 17044, and that thus the action of gravity in vacuo, would be more powerful by the part nearly. The addition to be made to the length of the pendulum, in consequence, was 3.02 divisions. This being corrected for the height of the barometer and thermometer at the time of ob servation, became 3.10, which added to 50939.55, gave
50992.65, for the length of the seconds pendulum in va Leo.
The next reduction was to obtain the length at the fixed temperature of melting ice. At the time of measurement, the metallic thermometer stood at 181.5. The term of ice was 151, or 30.5 lower. By an increase of tempera ture, such as to occasion a variation of one division of the thermometer, the rule was lengthened a 216000th part ; and, therefore, by the whole difference of temperature, 30.5 16000' or 7.15 divisions. Consequently the length of the pendulum at the temperature of melting ice, was 50999.8 divisions of the rule. The mean of this and five other ex periments, reduced in the same manner, was 50999.35. Taking afterwards the mean of two series of experiments, in which the ball had been suspended from opposite points, he found it necessary to add 0.34 on account of the in equality of the ball ; and, after numerous other experi ments, he finally concluded with the number 50999.6.
It has been stated that the value of a division of the 864 tile was -- 100.000 of a toise ; that is 100000 of a line.
1\lultiplying therefore 50939.6 by length of the 100000, iiendulum was obtained equal to 440.63654 lines, or in parts of the metre, 0".993827. To reduce this to Eng lish inches, we take the length of the metre as assigned by Captain Rater, in the Transactions of the Royal Society for from a very exact comparison of Sir George Shuck burgh Evelyn's standard scale, with two metres of platina which had been constructed at Paris, and previously com pared with the standard metre deposited in the Archives. From this comparison, it appears that the metre, taken at its normal temperature, which is that of melting ice, is equal to 39.37076 inches of Sir George Shuckburgh's standard scale, taken also at its normal temperature, or 62° of Fahrenheit. Any length 1, therefore, expressed in metres, at the temperature of 32° of Fahrenheit, will be reduced to English inches by multiplying it by 39.37076. This multiplication being performed, we obtain 39.127724 inches.
From the experiments of Borda, therefore, it appears that the length of the seconds pendulum, at the Paris Ob servatory, in vacuo, and at the temperature of is 39.127724 inches of Sir George Shuckburgh's standard scale. Sec Base Metriyue, vol. iii. Delambre, 4strono nue, vol. iii.