Two series of observations were made. From the first, the mean number of oscillations in twenty-four hours, re duced to an indefinitely small arc, and to a temperature of 70° of Fahrenheit, was 86166.108; and from the second 86166.04. The height of the station above the level of the sea was twenty-seven feet; whence the correction given by the formula was 0.095; but Mr. Goldingham, after the example of Captain Kater, multiplied this num ber by 66, in consideration of the attraction of the inter posed mass. The correction allowed was therefore 0.06, to be added to the number of oscillations in twenty-four hours. The correction for the buoyancy of the atmos phere, deduced in the manner already explained, was 6.2075 oscillations for the first series, and 6.220 for the second. These corrections being applied, the number of oscillations in a mean solar day, in vacuo, at the level of the sea, and at the temperature of 70° of Fahrenheit's scale, was obtained as follows : Before the pendulum was sent from London, Captain Kater had found, that the number of its oscillations dur ing a mean solar day, in vacuo, at the level of the sea, and at the temperature of 70°, was 86300.226. In his own experiments, Captain Kater had found the expansion of his pendulum to be .000009959 in parts of its length, for a change of temperature indicated by a variation of one degree of the thermometer. The length of the se conds pendulum in London, at the temperature of 62°, is 39.13929 inches ; at the temperature of 70°, therefore, its length is 39.142408 inches at the same rate of expansion. Hence the length of Mr. Goldingham's trial pendulum is obtained from this proportion, : : 39.142408: 39.232956.
Another proportion gives the length of the seconds pendulum, : : 39.232956: 39.02649.
We have been particular in stating these numbers, be cause the length of the seconds pendulum is stated in Mr. Goldingham's paper to be 39.026302 inches. The differ ence arises from his having assumed the length of the se conds pendulum in London to be 39.1386 inches when re duced to the level of the sea, as it is given in Captain Kater's first Memoir; but it was afterwards found, that a slight oversight had been committed in estimating the specific gravity, and that the true length was 39.13929 inches. On performing the whole calculation with this corrected number, the result is found to be what we have just stated.
It appears, therefore, that the length of the seconds pendulum at Madras, latitude 13° 4' 9" N. at the level of the sea, in vacuo, and temperature is 39.02649 inches. Assuming the expansion of Mr. Goldingham's pendulum to be the same with that of Captain Kater's, the length of the seconds pendulum at Madras, at the temperature of is 39.02338 inches.
Captain John Warren had previously made a set of ex periments, though with a very inferior apparatus, to de termine the length of the seconds pendulum at the same place. His determination agrees very well with Mr. Goldingham's, being 39.026273 inches. An account of his experiments is published in the eleventh volume of the Asiatic Researches.
Deductions relative to the Figure of the Earth.
The experiments, of which we have now given an ac count, furnish some very interesting information relative to the variation of gravity, and consequently to the figure of the earth. Supposing the figure of the earth to be
elliptical, it is demonstrable, from the theory of attrac tion, that the length of the seconds pendulum varies as the square of the sine of latitude; so that, calling the la titude L, and / the length of the pendulum, the value of I will be expressed by a function of this form, /=A+B L ; A and B being two constants, of which A represents the length of the pendulum at the equator, where L is zero, and B the excess of its length at the pole. As experi ments cannot be made at the pole, it is impossible to de termine the value of B directly, but it may be deduced from observations made at known latitudes : thus, let l' and l" be the observed lengths of the pendulum at two sta tions, L' and L" the corresponding latitudes, we have si milarly, for the first =A+B sin. 2 L', and for the second sin. 2 L"; whence I",-- /'=B ; or B= sin. (L"—L') sin. (L"+L') • B being known, the value of A can be found from that of I' or /".
The A expresses the diminution of gravity from the pole to the equator ; and, according to Clairaut's ce lebrated theorem, whatever hypothesis is made respect ing the variation of density in the strata of the earth, the sum of the two fractions expressing the ellipticity and di minution of gravity from the pole to the equator, is always a constant quantity, and equal to of the fraction express ing the ratio of the centrifugal force to that of gravity at the equator. According to the theory of dynamics, this last fraction is being multiplied by it becomes 289.014' 2 .00865; therefore calling e the ellipticity, we find E=.00865— The hypothesis of the gravitating force increasing as the square of the sines of the latitudes, is grounded on the supposition that the spheriod is homogeneous; and if this were really the case, we should obtain the same value ' and consequently the same compression, by A substituting in the formula the lengths observed at any two latitudes whatever, abstraction being made from the errors of observation. But as we know that the strata, at least near the surface of the earth, differ very considerably in density, we may expect, a priori, considerable discre pancies in the results obtained, by combining the lengths observed at different stations. If we combine the lengths given by Biot at Unst and Formentera, a result may be expected very near the truth ; for, on account of the dis tance between the two stations, the influence of any par tial cause of error is greatly diminished; and as both the stations are situated on small isolated rocky islands, the difference of local density cannot be supposed great. Substituting, therefore, the observed lengths at Unst and Formentera in the above formulae, B is found inches, and A=39.012556, whence —= A .005360536, and consequently the ellipticity, or E= 1 .003289464= 304' Captain hater, combining his results by two and two in the same manner, calculated the following Table : 25"; therefore, by substitution, the value of A will be B tained, and consequently that of B, from the ratio . On performing the calculation, they are found as follows : A= :39.011684 inches, and B= 0.2102729 inches. Hence, for any other latitude, /= 39.011684+.2102729 L.