By means of this formula the following Table is con structed.
Instead of proceeding in this manner, Biot submitted his measurements to a mode of comparison, which shows, in a very perspicuous manner, the relative intensity of gravity at the different stations. From a comparison of the terrestrial degrees, observations on the pendulum, and the values of the lunar inequalities depending on the com pression of the earth, La Place found that the compression indicated by all the phenomena taken in connexion, was 00326, or 3 1 Biot, combining this determination of theory with the measurement at Unst, which he consider ed as the best of the whole, both on account of the obser vations having been made with all the precautions which his former experience could suggest, and on account of the great number of series from which the final result was obtained, formed a theoretic expression for the length of the pendulum, with which he compared the lengths measured at the other stations. Substituting the value of a given by La Place in the formula E=00865— it A' becomes 00865 whence B=A .00539. This value of B being substituted in the general expression /=A+B sin.' L, gives 1=A (1 +00539 sin. 2 L). But at the station of Unst, 1=39.171776 inches, and L=60° 45' This Table agrees with Captain Kater's, in indicating a greater diminution of gravity, on advancing towards the equator, than is given by theory ; but the variation is by no means uniform. Captain Kater's Table shows it to be greater between Unst and Leith Fort than between Unst and Portsoy, greater still at Clifton, suddenly dimi nished at Arbury Hill, increased again at London, and di minished at Dunnose. It increases progressively through France to Bourdeaux, where its effects are most sensible; is diminished somewhat at Figeac, that station being more in the interior, and the country in which it is situated composed of denser materials. At Formentera it appears with a contrary sign, indicating a local excess in the in tensity of gravity. NVhether this variation from theory is uniformly in excess or defect to the south of Formentera, we want farther experiments to determine. The only ex periment indeed which has been made to the south of that station which can be relied on, is that of Mr. Golding ham ; for those of Bouguer, Le Gentil, and others, were not performed with an apparatus.sufficiently delicate to entitle their results to be employed in deciding the ques tion. The formula applied to the latitude of Madras, gives l=39.022436 inches, and Mr. Goldingham's mea surement is 39.02338 inches ; so that the intensity of the gravitating force is, as at Formentera, greater than it is given by theory. On the whole it is sufficiently obvious, that no formula can be obtained to represent, with perfect rigour, the lengths of the pendulum over the globe ; and that the local variations in the density of the strata which compose the crust of the earth prevent us from determin ing the true nature of the meridional curves.* From Biot's formula we have calculated the following Table, showing the length of the pendulum at every fifth degree of latitude from the equator to the pole ; and con sidering the data from which the numerical coefficients are obtained, we presume that it contains much nearer ap proximations to the absolute lengths than any that has hi therto been given.
tiler, being only 39.127724 inches. In the same manner, the length of the seconds pendulum at London, at a height of 83 feet above the level of the sea, is, Comparison of the Seconds Pendulum at Paris and wich.
After Captain Kater had accomplished his measurement in London, the Board of Longitudes, anxious to compare the results of Borda's method with those of one so en tirely different, commissioned M. Arago to make this comparison by a direct experiment. Arago was joined by the celebrated Humboldt, and the first part of their ope rations was, to determine the number of oscillations made during a sidereal day in the Royal Observatory at Paris, by two invariable pendulums of copper constructed by Fortin. They then proceeded to London, and being met by Biot on his return from Shetland, they made the same observations on the two pendulums at the Greenwich Observatory. After their return to Paris, MM. Arago and Humboldt determined anew the number of oscilla tions of their pendulums, in order to assure themselves that they had undergone no derangement by the trans portation.
The mean number of oscillations made by the pendu lums during a sidereal day, in an indefinitely small arc, reduced to the temperature of 10 centigrade, or 50' of Fahrenheit, was observed to be as follows: The acceleration of the first, reduced to seconds, and corrected for the effects of the density of the air, becomes 11".50; and of the second, 10".08: the mean of the two, therefore, or 10" 79, will be the acceleration of a clock at Greenwich in twenty-four hours, regulated at Paris ac cording to sidereal time. Hence it results, that the dif ference of length of two simple pendulums, making re spectively 86,400 oscillations in a mean solar day at I'aris and Greenwich, is .0098033 inches. According to Cap tain Kater, the length of the seconds pendulum, in the apartment at Portland Place, is 39.13908 inches. Sup posing its length at Greenwich to be the same, and de ducting .0098033, the length at Paris is equal to 39.12928 inches. This is precisely the length assigned in the table by Biot and Mathieu, when reduced to the level of the sea. But the allowance they made for the height of the station was .000645 inches; hence the length of the se conds pendulum, as determined by them, is less than that obtained from Captain Kater's experiments, by this quan tity, or 39.128660 inches. Borda's result is less than ei It would be extremely difficult to determine which of the three results approaches nearest to absolute accuracy. The mean, which differs from Biot's determination only by the T00000 of an inch, may perhaps be regarded as nearest the truth. It is rather a remarkable coincidence, that this mean is the precise number deduced by Mr. Watts, in a paper published in the third volume of the Edinburgh Philosophical Journal, from Captain Kater's measurement, by making a more correct allowance for the amplitude of the arcs.