Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/622

600 GOO EARTH the actual instrument P and Q are finely engraved dots at the distance of 10 feet apart. In the measurement the bars when aligned do not come into contact ; an interval of six inches is left between each bar and its neighbour. This small space is measured by an ingenious micrometrical arrangement constructed on exactly the same principle as the bars themselves. The triang illation was computed by least squares. The total number of equations of condition for the triangulation is 920 : if therefore the whole had been reduced in one mass, as it should have been, the solu tion of an equation of 920 unknown quantities would have occurred as a part of the work. To avoid this an approxi mation was resorted to ; the triangulation was divided into twenty-one parts or figures ; four of these, not adjacent, were first adjusted by the method explained, and the cor rections thus determined in these figures carried into the equations of condition of the adjacent figures. The average number of equations in a figure is 44 ; the largest equation is one of 77 unknown quantities. 1 Airy s Zenith Sector is too well known to need descrip tion. The vertical limb is read by four microscopes; altogether, in the complete observation of a star there are 10 micrometer readings and 12 level readings. In some recent observations in Scotland for latitude the Zenith Telescope has been used with very great success ; it is very portable ; and a complete determination of latitude, affected with the mean of the declination errors of two stars, is effected by two micrometer readings and four level read ings. The observation consists in measuring with the telescope micrometer the difference of zenith distances of two stars which cross the meridian, one to the north and the other to the south of the observer at zenith distances which differ by not much more than 10 or 15, the interval of the times of transit being not less than one nor more than twenty minutes. The advantages are that, with simplicity in the construction of the instrument and facility in the manipulation, refraction is eliminated (or nearly so, as the stars are generally selected within 25 of the zenith), and there is no large divided circle. The telescope, which is counterpoised on one side of the vertical axis, has a small circle for finding, and there is also a small horizontal circle. This instrument is universally used in American geodesy. The United States Coast Survey has a principal triangulation extending for about 9 30 along the coast, but the final results are not yet published. In 1860 was published F. G. Struve s Arc du Meridien de 25 20 entre le Danube et la Mtr Glacials mesure depuis 1816 jusqu en 1855. This work is the record of a vast amount of scientific labour and is the greatest con tribution yet made to the question of the figure of the earth. The latitudes of the thirteen astronomical stations of this arc were determined partly with vertical circles and partly by means of the transit instrument in the prime vertical. The triangulation, a great part of which, however, is a simple chain of triangles, is reduced by the method of least squares, and the probable errors of the resulting distances of parallels is given ; the probable error of the whole arc in length is 6 2 toises. Ten base lines were measured. The sum of the lengths of the ten measured bases is 29,863 toises, so that the average length of a base line is 19,100 feet. The azimuths were observed at fourteen stations. In high latitudes the determination of the meridian is a matter of great difficulty ; nevertheless the azimuths at all the northern stations were successfully determined, the probable error of the result at Fuglenoes being 0&quot; 53. 1 See the volume of the Ordnance Survey, entitled Account of the Principal Triangulation of Great Britain and Ireland, by Captaiii A. R. Clarke, R.E., F.R.S., 1858. Mechanical Theory. Newton appears to have been the first to apply his own newly-discovered doctrine of gravitation, combined with the so-called centrifugal force, to the question of the figure of the earth. Assuming that an oblate ellipsoid of rotation is a form of equilibrium for a homogeneous fluid rotating with uniform angular velocity, he obtained the ratio of the axes 229 : 230, and the law of variation of gravity on the surface. A few years later Huyghens published an investigation of the figure of the earth, supposing the attraction of every particle to be towards the centre of the earth, obtaining as a result that the proportion of the axes should be 578:579. In 1740 Maclaurin wrote his celebrated essay on the tides, one of the most elegant geo metrical investigations ever made. He demonstrated that the oblate ellipsoid of revolution is a figure which satisfies the conditions of equilibrium in the case of a revolving homogeneous fluid mass whose particles attract one another according to the law of the inverse square of the distance ; he gave the equation connecting the ellipticity with the proportion of the centrifugal force at the equator to gravity, and he determined the attraction on a particle situated any where on the surface of such a body. Some few years afterwards Clairaut published (1743) his Theorie de la Figure de, la Terre, which contains, among other results, demonstrated with singular elegance, a very remarkable theorem which establishes a relation between the ellipticity of the earth and the variations of gravity at different points of ite surface. Assuming that the earth is composed of concentric ellipsoidal strata having a common axis of rotation, each stratum homogeneous in itself, but the ellipticities and densities of the successive strata varying according to any law, and that the superficial stratum has the same form as if it were fluid, he proves the very important theorem contained in the equation 9 -9 5 V + e=r 2 m Where g, g are the amounts of gravity at the equator and at the pole respectively, e the ellipticity of the meridian, and m the ratio of the centrifugal force at the equator to g. Clairaut also proved that the increase of gravity in pro ceeding from the equator to the poles is as the square of the sine of the latitude. This, taken with the former theorem, gives the means of determining the earth s ellipticity from observation of the comparative force of gravity at any two places. Clairaut would seem almost to have exhausted the subject, for although much has been written since by mathematicians of the greatest eminence, yet, practically, very little of importance has been added. Laplace, himself a prince of mathematicians, who had devoted much of his own time to the same subject, remarks on Clairaut s work that &quot; the importance of all his results and the elegance with which they are presented place this work amongst the most beautiful of mathematical produc tions &quot; (Todhunter s History of the MatJiematical Thtomea of Attraction and the Figure of the Earth, vol. i. p. 229). The problem of the figure of the earth treated as a ques tion of mechanics or hydrostatics is one of great difficulty, and it would be quite impracticable but for the circumstance that the surface differs but little from a sphere. In order to express the forces at any point of the body arising from the attraction of its particles, the form of the surface is required, but this form is the very one which it is the object of the investigation to discover ; hence the com plexity of the subject, and even with all the present resources of mathematicians only a partial and imperfect solution can be obtained, and that not without some labour. We may, however, here briefly indicate the line of reason ing by which some of the most important of the results v/a