FIGURE OF THE EARTH There are several astronomical methods of determining the flattening of the Earth. For the most part they determine the flattening by means of the observed mechanical effects produced by the Earth's equatorial protuberance. These effects are most noticeable in connection with the Earth's nearest neighbour, the Moon. The equatorial bulge produces periodic perturbations in the Moon's celestial longitude and latitude, and secular changes in the motion of the Moon's perigee and of the node of its orbit on the ecliptic. The Moon in turn acting on the equatorial bulge of the Earth produces the greater part of the slow displacement of the equinoxes known as precession; the Sun contributes a fairly large part of the observed precession and the planets a small remainder. From any one of the effects mentioned above the flattening of the Earth may be deduced. There are theoretical difficulties in all of the methods. Perhaps the flattening deduced from the precession is as satisfactory as any; it agrees substan tially with the flattening of the International Ellipsoid of Refer ence, namely 1/297. The tendency of the flattening deduced by the other lunar methods is to come out a trifle greater than this.
These methods all deal with the average flattening without reference to local irregularities. The flattening or the equatorial radius—one or the other—may also be deduced by a calculation essentially similar in principle to that used by Newton to show that terrestrial gravitation and the force controlling the Moon's orbital motion are one and the same. The result, however, is affected by the fact that the geoid is not a perfect ellipsoid of revolution.
If we think of the elevation of the geoid above the terrestrial ellipsoid, or depression below it, as the case may be, as a mathe matical function of the geographical coordinates (latitude and longitude) of a point on the ellipsoid, then this mathematical function itself may in theory be found by the application of Stokes' formula to gravity observations. Observations of the deflections of the vertical obviously give us the first derivatives of this function and enable us to build up the function itself by a process of integration. The next stage in this line of thought is obviously a consideration of the second derivatives, quantities intimately connected with the curvature of the geoid. It is possi ble to determine certain quantities connected with the curvature by means of the Eotvos torsion balance devised by the late Baron Roland Eotvos of Budapest.
The Eotvos balance is simple in principle, being merely a rod suspended at the middle by a very delicate fiber. In one form of the balance the principal masses are concentrated at the ends of the rod. In another form, more used in practice, the mass at one end is at a different level from the mass at the other.
If we consider only the general conformation of the geoid, it seems to differ in vertical elevation from an exact ellipsoid only by small and slowly changing amounts. If we look, however, at the directions of the tangents to this surface as disclosed by a study of the deflections of the vertical, we find considerable irregularity; if we look to the curvatures as disclosed by the Eotvos balance, we find apparently wild irregularity. This irregularity is due to the fact that the geoid (or any equipotential surface studied in practice by the balance) cuts into discontinuities of density, such as in the sides of hills, walls of buildings, irregularities in subsurface geologic structure, etc. We know that the second derivatives, about which the balance gives us information and on which the curvature of the equipotential surfaces depends, is discontinuous at such discontinuities in density. Near them the second derivative and the curvature, although not actually dis continuous, appear extremely irregular.
The surface of the geoid has been compared to that of a withered apple. This is something of an exaggeration, for actual concavities in the geoid, such as would be found in the apple, though possible are probably quite rare. Moreover, the wrinkles, or more properly irregular undulations, in the geoid are so fine and change character so quickly that in the aggregate they mean comparatively little in the way of actual rise or fall of the geoid as compared with the terrestrial ellipsoid; nevertheless the com parison is suggestive. The details of all the wrinkles in the geoid, as the torsion balance gives them, are really too fine for the pur pose of the geodesist. They simply confuse him.
(9.) NUMERICAL DATA CONNECTED WITH THE FIGURE OF THE EARTH The following table gives the principal determinations of the mean figure of the Earth, beginning with the work of Mechain and Delambre undertaken to establish a basis for the metric system. The table was taken chiefly from "La figure de la Terre" (Revue de geographie annuelle—Tome 1908) by Capt. (now Gen.) Georges Perrier. Other, somewhat different, numerical values may sometimes be found if other sources of information are used. The discrepancies will usually be due to the use of relations between the foot, toise and meter different from those used in this table: As has been remarked, the mean figure for the Earth as a whole is not necessarily the figure best adapted to a particular region. The following table gives some of the ellipsoids actually in use.
A spheroid with a flattening equal to that of the Bessel Spheroid, but with its major axis greater than the major axis of the Bessel Spheroid by i part in io,000 has been extensively used by the Central Bureau of the International Geodetic Association for geodetic calculations relating to Europe. This choice was made because many tables to facilitate computation have been based on the Bessel Spheroid, and these can be adapted to the increased major axis with comparative ease, whereas a change in the flatten ing would require a recomputation of the whole set of tables.
The following table may be useful for reference : Fundamental Elements of the International Ellipsoid of Reference.
a = semi-major axis (equatorial radius) = 6,3 78,388 meters f =ellipticity (flattening) = a b = 297 = 0.003, 367, 0034 Derived Quantities b = semi-minor axis (polar radius) =6,356, 911.946 meters a 2 ez =square of eccentricity— e = o.006, 722, 6700 Length of quadrant of the equator =10,019,148.4 meters Length of quadrant of the meridian = 1°,002,288.3 meters Area of the ellipsoid = 510, Ioo,934 sq. km.
Radius of sphere having same area as ellipsoid = 6,3 71, 2 2 7.7 meters Radius of sphere having same volume as ellipsoid = 6,3 71, 2 21.3 meters Mass of the ellipsoid*= 5.988X metric tonnes *Mean density taken as 5.527, the value found by both Boys, Phil. Trans. A. vol. 186 (1895) p. 1, and Braun, "Denkschriften der Akademie der Wissenschaften zu Wien," Mathematisch-naturwissenschaftliche Klasse, 64, 1896, p. 187.
Geoid contours were constructed by Hayford in connection with his earlier investigation of his figures of the Earth. Within the United States he found a variation in the elevation of the geoid above the Clarke spheroid of 1866 amounting to 38 meters. Similar investigations by others give results of the same order of magnitude. This is somewhat greater than would be inferred from known differences of elevation and perfect isostasy with a depth of compensation of loo km. but is by no means more than might be expected in view of ignorance of the real depth of com pensation and the known imperfectness of isostatic adjustment.
The following are recent formulas for theoretical gravity at the surface of the earth, and the flattening derived from them. In these formulas 4 is the geographic latitude and X the East longi tude reckoned from Greenwich. The unit is dynes per gram, or cm/sec'. The date of publication of the formula is given in parentheses.
Helmert (1915) go= 978.052 sine 24) cos 2(X+17°)], 1/296.7 Bowie (1917) 978.039 [1+0.005294 sin' —0.000007 sin' 20], Zleiskanen (1928) go= 978•o49 [1+0.005293 —o•000007 sinl 4 +o.00o019rnsZcos 2(X-0°)], The fact that the coefficient of sin'2 4 is 0.000007 rather than 0.000006 means that the spheroid on which these formulas are based is depressed a very little in middle latitudes below an exact ellipsoid having the same axes. The longitude term has the effect of turning this spheroid of revolution into a spheroid (approxi mately an ellipsoid) of three unequal axes. The longest semi axis of the equator, according to Heiskanen, lies in the meridian plane of Greenwich and is 121 meters longer than the mean. The least equatorial semi-axis is in longitude and is 121 meters shorter than the mean. The validity of these longitude terms is closely connected with the apparent tendency of gravity at sea to be greater than that on land. Geodesists are not yet agreed as to their validity or their geophysical interpretation, if real. If they are accepted, the conception of isostasy must be modified and the geoid instead of being in general below the mean ellipsoid over the ocean would be above it.
If we take the International Ellipsoid as the basis of a gravity formula and in addition adopt 978.050 cm/sec' for gravity at the equator, we get go= 978.050 [I +0.005288 sin' sin' 20]. INTERNATIONAL GEODETIC ORGANIZATIONS Geodesy is essentially an international science. This was real ized when in 1862 a Central European Geodetic Association (Mit teleuropaische Gradmessung) was organized on the initiative of Lieutenant-General Baeyer of Prussia. The first general con ference of the organization was held in 1864 with representatives of 13 States or countries, many of them being German States later united into the German Empire. General conferences at intervals of three years were arranged for with a permanent committee directing the affairs of the organization between con ferences. At the next conference, in 1867, in recognition of wid ening scope the name was changed to European Geodetic Associa tion (Europaische Gradmessung).
At the general conference in 1883 representatives of England and the United States were present. This conference discussed matters of world-wide interest, such as a common prime meridian and an international time system. At the next conference in 1886 the name International Geodetic Association (Internationale Erdmessung) was adopted to indicate a still wider scope, and a definite international convention was adopted providing for con tributions from the member nations. At the beginning of the same year F. R. Helmert became director of the Prussian Geo detic Institute. He reorganized it in the years that followed, and exerted a powerful and beneficent influence on the work of the International Association, the headquarters of which remained associated with the Prussian Geodetic Institute. The general conference of 1912 at Hamburg was the last held under this organization. The outbreak of the World War prevented the holding of the general conference planned for 1915.
After the World War the International Geodetic and Geo physical Union was organized in connection with the newly created International Research Council. The International Geo detic and Geophysical Union consists of various semi-independent sections, one of which, the section of Geodesy, took over the work of the former International Geodetic Association. The new type of organization emphasizes the fact that geodesy is really a branch of geophysics. The work of the International Latitude Service was taken over jointly by the Section of Geodesy and the newly formed International Astronomical Union, since the subject was of interest to both organizations. Germany and Austria have so far remained outside the new organization, but there has been organized a Baltic Geodetic Commission, which includes representatives of Germany and of other nations border ing on the Baltic and deals with geodetic problems of common interest to them. Some of the Baltic nations are members both of the Baltic Commission and of the International Geodetic and Geophysical Union.
For further information see 1. Bibliographies of Geodesy.—Borsch, Otto, Geoddtisthe Literatur auf Wunsch der Permanenten Commis sion im Centralbureau zusammengestell1 (Berlin, 1889) ; Gore, James Howard, A Bibliography of Geodesy; see also Appendix No. 8 to Report of the U.S. Coast and Geodetic Survey for 1902 (Washing ton, 1903). 2. History of Geodesy.—Beazley, C. R., The Dawn of Modern Geography, 3 vol. (London, 1897-1906) ; Butterfield, A. D., A History of the Determination of the Figure of the Earth from Arc Measurements (Worcester, Mass., 1906) ; Delambre, J. B. J., Grandeur et figure de la Terre (posthumous work ed. by G. Bigourdan, Paris, 1912) ; Delambre, J. B. J., Histoire de l'astronomie ancienne, 2 vol. (Paris, 1817) ; Quodvultdeus, Geschichte der Breitgradmessun gen bis zur peruanischen Gradmessung (Doctoral thesis, Rostock, 1871) ; Todhunter, Isaac, A History of the Mathematical Theories of Attraction and the Figure of the Earth from the time of Newton to that of Laplace, 2 vol. (London, 1873) . 3. General Works.—Clarke, A. R., Geodesy (Oxford, 1880) ; Helmert, F. R., Die mathematischen and physikalischen Theorieen der hoheren Geodasie, 2 vol. (Leipzig, 1880-84) ; Hosmer, George L., Geodesy, including Astronomical Ob servations, Gravity Measurements and the Method of Least Squares (New York, 1919) ; Jordan, Wilhelm, Handbuch der Vermessungs kunde, vol. 3, 4th ed. 1897 ; Perrier, Georges, "La Figure de la Terre," being vol. ii. (1908) of the Revue de Geographie annuelle. 4. Books on special subjects.—Bowie, William, Isostasy (New York, 1927) ; Messerschmidt, J. B., Die Schwerebestimm.ung an der Erdoberfidche (Brunswick, 19o8). 5. Serial Publications.—Verhandlungen der Inter nationalen Erdmessung (Comptes-Rendus de l'Association geodesique internationals), Berlin, G. Reimer, various dates to 1914 ; Travaux de la Section de Geodesic de l'Union geodesique et geophysique inferno tionale (Paris, 1924– ) ; Bulletin Geodesique (Paris, 1924– ) ; Ger lands Beitriige zur Geophysik. Stuttgart (Vols. I. and II.), and Leipzig (Vol. III. and following) ; Monthly Notices of the Royal Astronomi cal Society; Geophysical Supplement (London, 1922- ) ; Zeitschrift fur Geophysik (Brunswick, 1925– ). (W. D. LA.)