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 EARTH,

FIGURE

to be made among these data, a part of which only was admitted into the final equations for the determination of the best spheroid (84 data out of 144). The resulting , dimensions of an osculating spheroid were found to be Equatorial radius . Compression (ellipticity)

. 6,378,157+90 metres. . 1/304-5+1-9.

The equatorial radius differs but little from Clarke’s value of 1866, whereas the compression is very different, and comes more nearly to Bessel’s value. The measurement of a great meridian arc, in long. 98° W., has also begun; it has a range of latitude of 23°, and will extend over 50° when produced southwards and northwards by Mexico and Canada. It may afterwards be connected with the new arc of Quito. Geodetical surveys of some extent have also been undertaken in New South Wales and in Queensland. Differences of longitude were measured along the circuit Greenwich-Pulkowa-Vladivostok-Shanghai-MadrasTehran-Potsdam-Greenwich. We transcribe from Mr Schott a comparative table of the dimensions of several spheroids which of late have been often quoted :— Ellipsoid of Bessel, 1841. From ten meridional arcs, total amplitude 90°'S Clarke, 1858. Special spheroid for surface of Great Britain and Ireland; range of lat. 12°, the same in long. ; 75 astron. stations Clarke, 1866. From five merid. arcs, total amplitude 76°'6 Clarke, 1880. From five merid. arcs and longitudinal measures, total amplitude 89°'0 U.S. Coast and Geod. Survey, 1900. Eastern oblique arc; total length 23°‘5; 84 astronomic stations. Bonsdorff, 1899. Russian merid. arc, amplitude 25°‘3 Shdanov, 1899. Russian parallels of 47° and 52°, and three merid. arcs Harkness, 1891. From a variety of sources {Solar Parallax, Washington, 1891) . . ..

Equatorial Polar Semi- Compression Radius a. Axis 6. (a - b)/a.

6,377,397 6,356,079

1/299-15

6,378,494 6,355,746

1/280-4

6,378,206 6,356,584

1/295-0

6,378,249 6,356,515

1/293-5

6,378,157 6,357,210

1/304-5

6,378,344 6,356,983

1/298-6

6,377,717 6,356,437

1/299-7

6,377,972 6,356,727

1/300-2

It is thus evident that great efforts are now being made by all civilized nations to complete the network of triangles that will ere long cover the whole surface of the continents. In due course we shall be in possession of the results of some of the most important undertakings, now in process of execution. Meantime it may be admitted as highly probable that the dimensions of the geoid are nearly those adopted by Clarke (a > 6,378,000 metres), with an ellipticity approaching to Bessel’s value (1/299). The ellipsoid of three unequal axes has been abandoned, mainly for theoretical reasons1 that lead us to prefer an ellipsoid of revolution as a reference spheroid, in relation to which may be stated the prominences and depressions of the true geoid. In Europe, according to Helmert, the deviations of level are scarcely over 100 metres. 1 F. R. Helmert, Die mathematischen und physikalischen Theorieen der holieren Geoddsie, 2 vols., 1880-1884. See t. ii. p. 135.

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But it is necessary to agree precisely on the definition of the geoid. This word is generally understood to mean a level surface coinciding with the mathematical surface of the sea, produced below the continents, which it is imagined to percolate by canals or open cuttings. In that case the reduction of gravity to the sea-level ought to be done simply by the correction for height ^ + 2y^, just as if the superposed strata were condensed upon the surface of the geoid. A somewhat different geoid would be obtained by supposing the surface of the oceans continued below the continents by means of tunnels, or by a levelling in mine galleries, for then the attraction of the upper strata diminishes the gravity at the surface of the geoid, and the latter will be a little below the surface of the former geoid; it has discontinuities of curvature where it enters the continents. If now, according to what is often done, we add to the reduction of gravity Bouguer’s correction ( -^<7—— which goes to subtract the attraction of

A Jti / the continent, 8 being the density of the ground, and A the mean density of the earth, we thereby suppose the upper layers erased; but the process is inadmissible, since we know that mountain masses are generally compensated by a deficiency of matter in the underlying strata. In Europe the compensation is often found to be incomplete, whereas in the Himalaya range the attracting masses are sometimes over-compensated. Professor Helmert, who has devoted much labour and time to the calculations of measures of gravity by means of pendulum observations, has proposed another mode of reduction, which is known by the name of “ condensation method.” He imagines the upper strata condensed upon the surface of a geoid situated at a depth of 21 kilometres (its equatorial radius being, therefore, equal to the polar semi-axis of the earth). This process, conceived with a view to facilitate the development of the potential, presents the inconvenience of introducing many very uncertain data, and may seem somewhat artificial. Professor Helmert has found that it brings into fair agreement the numerous measures of gravity already obtained. The number of stations is about 1400, out of which more than 50 have been connected with the central bureau in Potsdam. A thorough examination of the deflections of the vertical, or local attractions, observed in all parts of Europe has also been undertaken by the central bureau. According to Helmert, the ellipticity of the earth, as derived from pendulum observations, is 1/299-3 (in his latest paper the corrected result is 1/298-3), and from the lunar theory 1/297-8; Hansen’s value is 1/296-2. Lastly, M. Radau has shown2 that by the introduction of the variable rj =

where c, e are the mean radius and ellip-

ticity of a spheroid, it becomes possible to integrate approximately the equations of Clairaut, and to establish a relation which affords the means for an independent evaluation of the ellipticity of the earth. The subject is very fully gone into in two important memoirs, one by M. Callandreau,3 the other by Professor Darwin,4 whose results point to an ellipticity between 1/296 and 1/298. Professor Darwin further concludes that the level surface of the earth is depressed below the true ellipsoid by 3 metres in latitude 45°. It is highly probable that a value 2 Comptes Rendus, t. c., 1885. See also Tisserand, Traite de Mecanique celeste, ii. p. 221. ° O. Callandreau, “ Memoire sur la TMorie de la figure des planetes,” Annates Obs. Paris, vol. xix., 1889. See also Bullet. Astron., 1897. 4 G. H. Darwin, The Theory of the Figure of the Earth, carried to the Second Order of Small Quantities. (Monthly Notices R.A.S., vol. lx., December 1899.) S. III.— 76