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EARTH, FIGURE OF THE

to Greenland and Spitsbergen; and in 1824-1831, Captain Henry Foster (who met his death by drowning in Central America) experimented at sixteen stations; his observations were completed by Francis Baily in London. Of other workers in this field mention may be made of F. B. Lütke (1826-1829), a Russian rear-admiral, and Captains J. B. Basevi and W. T. Heaviside, who observed during 1865 to 1873 at Kew and at 29 Indian stations, particularly at Moré in the Himalayas at a height of 4696 metres. Of the earlier absolute determinations we may mention those of Biot, Kater, and Bessel at Paris, London and Königsberg respectively. The measurements were particularly difficult by reason of the length of the pendulums employed, these generally being second-pendulums over 1 metre long. In about 1880, Colonel Robert von Sterneck of Austria introduced the half-second pendulum, which permitted far quicker and more accurate work. The use of these pendulums spread in all countries, and the number of gravity stations consequently increased: in 1880 there were about 120, in 1900 there were about 1600, of which the greater number were in Europe. Sir E. Sabine[1] calculated the ellipticity to be 1/288.5, a value shown to be too high by Helmert, who in 1884, with the aid of 120 stations, gave the value 1/299.26,[2] and in 1901, with about 1400 stations, derived the value 1/298.3.[3] The reason for the excessive estimate of Sabine is that he did not take into account the systematic difference between the values of $\mathrm{G}\!$ for continents and islands; it was found that in consequence of the constitution of the earth’s crust (Pratt) $\mathrm{G}\!$ is greater on small islands of the ocean than on continents by an amount which may approach to 0.3 cm. Moreover, stations in the neighbourhood of coasts shelving to deep seas have a surplus, but a little smaller. Consequently, Helmert conducted his calculations of 1901 for continents and coasts separately, and obtained $\mathrm{G}\!$ for the coasts 0.036 cm. greater than for the continents, while the value of $\beta\!$ remained the same. The mean value, reduced to continents, is

$\mathrm{G} {{=}} 978.03 (1 + 0.005302 \sin^2\phi - 0.000007 \sin^22\phi )\text{ cm/sec}^2.\!$

The small term involving $\sin^2 2\phi\!$ could not be calculated with sufficient exactness from the observations, and is therefore taken from the theoretical views of Sir G. H. Darwin and E. Wiechert. For the constant $g {{=}} 978.03\!$ cm. another correction has been suggested (1906) by the absolute determinations made by F. Kühnen and Ph. Furtwängler at Potsdam.[4]

A report on the pendulum measurements of the 19th century has been given by Helmert in the Comptes rendus des séances de la 13e conférence générale de l’Association Géod. Internationale à Paris (1900), ii. 139-385.

A difficulty presents itself in the case of the application of measurements of gravity to the determination of the figure of the earth by reason of the extrusion or standing out of the land-masses (continents, &c.) above the sea-level. The potential of gravity has a different mathematical expression outside the masses than inside. The difficulty is removed by assuming (with Sir G. G. Stokes) the vertical condensation of the masses on the sea-level, without its form being considerably altered (scarcely 1 metre radially). Further, the value of gravity (g) measured at the height $\mathrm{H}\!$ is corrected to sea-level by $+ 2g\mathrm{H}/\mathrm{R}\!$, where $\mathrm{R}\!$ is the radius of the earth. Another correction, due to P. Bouguer, is $- \tfrac{3}{2}g\delta \mathrm{H}/\rho \mathrm{R}\!$, where $\delta\!$ is the density of the strata of height $\mathrm{H}\!$, and $\rho\!$ the mean density of the earth. These two corrections are represented in “Bouguer’s Rule”: $g_{\mathrm{H}} {{=}} g_{s} (1 - 2\mathrm{H}/\mathrm{R} + 3\delta \mathrm{H} / 2\rho \mathrm{R})\!$, where $g_{\mathrm{H}}\!$ is the gravity at height $\mathrm{H}\!$, and $g_{s}\!$ the value at sea-level. This is supposed to take into account the attraction of the elevated strata or plateau; but, from the analytical method, this is not correct; it is also disadvantageous since, in general, the land-masses are compensated subterraneously, by reason of the isostasis of the earth’s crust.

In 1849 Stokes showed that the normal elevations $\mathrm{N}\!$ of the geoid towards the ellipsoid are calculable from the deviations $\Delta g\!$ of the acceleration of gravity, i.e. the differences between the observed $g\!$ and the value calculated from the normal $\mathrm{G}\!$ formula. The method assumes that gravity is measured on the earth’s surface at a sufficient number of points, and that it is conformably reduced. In order to secure the convergence of the expansions in spherical harmonics, it is necessary to assume all masses outside a surface parallel to the surface of the sea at a depth of 21 km. (= $\mathrm{R}\!$ × ellipticity) to be condensed on this surface (Helmert, Geod. ii. 172). In addition to the reduction with $2g\mathrm{H}/\mathrm{R}\!$, there still result small reductions with mountain chains and coasts, and somewhat larger ones for islands. The sea-surface generally varies but very little by this condensation. The elevation ($\mathrm{N}\!$) of the geoid is then equal to

$\mathrm{N} {{=}} \mathrm{R} \int^{\pi} \mathrm{FG}^{-1}\Delta g_{\psi}\psi ,\!$

where $\psi\!$ is the spherical distance from the point $\mathrm{N}\!$, and $\Delta g_{\psi}\!$ denotes the mean value of $\Delta g\!$ for all points in the same distance $\psi\!$ around; $\mathrm{F}\!$ is a function of $\psi\!$, and has the following values:—

 Ψ = 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° F = 1 1.22 0.94 0.47 -0.06 -0.54 -0.90 -1.08 -1.08 -0.91 -0.62 -0.27 +0.08 0.36 0.53 0.56 0.46 0.26 0

H. Poincaré (Bull. Astr., 1901, p. 5) has exhibited $\mathrm{N}\!$ by means of Lamé’s functions; in this case the condensation is effected on an ellipsoidal surface, which approximates to the geoid. This condensation is, in practice, the same as to the geoid itself.

If we imagine the outer land-masses to be condensed on the sea-level, and the inner masses (which, together with the outer masses, causes the deviation of the geoid from the ellipsoid) to be compensated in the sea-level by a disturbing stratum (which, according to Gauss, is possible), and if these masses of both kinds correspond at the point $\mathrm{N}\!$ to a stratum of thickness $\mathrm{D}\!$ and density $\delta\!$, then, according to Helmert (Geod. ii. 260) we have approximately

$\Delta g {{=}} \frac{3}{2} \cdot \frac{g}{\mathrm{R}} \left( \frac{\delta \mathrm{D}}{\rho} - \mathrm{N}\right).$

Since $\mathrm{N}\!$ slowly varies empirically, it follows that in restricted regions (of a few 100 km. in diameter) $\Delta g\!$ is a measure of the variation of $\mathrm{D}\!$. By applying the reduction of Bouguer to $g\!$, $\mathrm{D}\!$ is diminished by $\mathrm{H}\!$ and only gives the thickness of the ideal disturbing mass which corresponds to the perturbations due to subterranean masses. $\Delta g\!$ has positive values on coasts, small islands, and high and medium mountain chains, and occasionally in plains; while in valleys and at the foot of mountain ranges it is negative (up to 0.2 cm.). We conclude from this that the masses of smaller density existing under high mountain chains lie not only vertically underneath but also spread out sideways.

The European Arc of Parallel in 52° Lat.

Many measurements of degrees of longitudes along central parallels in Europe were projected and partly carried out as early as the first half of the 19th century; these, however, only became of importance after the introduction of the electric telegraph, through which calculations of astronomical longitudes obtained a much higher degree of accuracy. Of the greatest moment is the measurement near the parallel of 52° lat., which extended from Valentia in Ireland to Orsk in the southern Ural mountains over 69° long, (about 6750 km.). F. G. W. Struve, who is to be regarded as the father of the Russo-Scandinavian latitude-degree measurements, was the originator of this investigation. Having made the requisite arrangements with the

1. Account of Experiments to Determine the Figure of the Earth by means of a Pendulum vibrating Seconds in Different Latitudes (1825).
2. Helmert, Theorien d. höheren Geod. ii., Leipzig, 1884.
3. Helmert, Sitzber. d. kgl. preuss. Ak. d. Wiss. zu Berlin (1901), p. 336.
4. “Bestimmung der absoluten Grösse der Schwerkraft zu Potsdam mit Reversionspendeln” (Veröffentlichung des kgl. preuss. Geod. Inst., N.F., No. 27).