theorems. of approximately spherical layers of equal density. Ulti-Clairaut's mately Stokes* pointed out that, only provided the surfaces of equilibrium relative to gravitation alone, and relative to the resultant of gravitation and centrifugal force, are approximately spherical; whether the surfaces of equal density are approximately spherical or not, the same expression (19) holds. A most important practical deduction from this conclusion is that, Figure of irrespectively of any supposition regarding the distribution of determinthe earth's density, the true figure of the sea level can be measuredetermined from pendulum observations alone, without any gravity: hypothesis as to the interior condition of the solid. Let, for brevity, g{1 + 5m (} - cos30)} =ƒ (0, $) ..(21) where m (§ 801) is, and g is known by observation in differ- ƒ(0, 4) =ƒ。+ƒ, (0, $) +ƒ1⁄2 (0, $) + etc..........(22). We have, by (19), F.(0, 4)= i 1 f. (0, $) .(23), and therefore the equation (5) of the level surface becomes 1 r = }ƒ3 - a {1 + } [1ƒ, (0, ¢) + 3ƒ. (0, $) + etc.]} fo .....(24). Confining our attention for a moment to the first two terms we have for f,, by App. B. (38), explicitly the sea level able from ments of ƒ1⁄2 (0, 4) = A。 (cos20 − }) + (A ̧ cos + B1 sin ø) sin @ cos @ + (42 cos 24+ B, sin 24) sin20.....(25). (fo+40-42)2+(fo +} Ao+A2) y2+(fo-340) 22 - B1yz - A1zx-2Bxy=fa3..(26). 0 * "On the Variation of Gravity at the surface of the Earth."-Trans. of the Camb. Phil. Soc., 1849. if ellipsoid with three unequal axes must have one of them coincident with axis of revolution. = Now from $539, 534, we see that, if OX, OY, OZ are principal axes of inertia, the terms of f, which, expressed in rectangular co-ordinates, involve the products yz, zx, xy must disappear: that is to say, we must have B, 0, 4,= 0, B,= 0. But whether B, vanishes or not, if OZ is a principal axis we must have both A1 = 0 and B1 = 0; which therefore is the case, to a very minute accuracy, if we choose for OZ the average axis of the earth's rotation, as will be proved in Vol. II., on the assumption rendered probable by the reasons adduced below, that the earth experiences little or no sensible disturbance in its motion from want of perfect rigidity. Hence the expansion (22) is reduced to ƒ(0, ¢)=ƒ ̧+4 ̧ (cos3 0 −3) + (4 ̧cos 24+ B ̧sin 2p) sin3 0+ƒ ̧(0,$)+etc.... (27). Ο If f(0, 4) and higher terms are neglected the sea level is an ellipsoid, of which one axis must coincide with the axis of the earth's rotation. And, denoting by e the mean ellipticity of meridional sections, e' the ellipticity of the equatorial section, and I the inclination of one of its axes to OX, we have Figure of the sea level determinable from measure ments of gravity; In general, the constants of the expansion (22); ƒ (being the mean force of gravity), A, A, B,, the seven coefficients in f(0, 4), the nine in ƒ(0, 4), and so on; are to be determined from sufficiently numerous and wide-spread observations of the amount of gravity. 796. A first approximate result thus derived from pendulum observations and confirmed by direct geodetic measurements is that the figure of the sea level approximates to an oblate spheroid of revolution of ellipticity about. Both methods are largely affected by local irregularities of the solid surface and underground density, to the elimination of which a vast amount of labour and mathematical ability have been applied, with as yet but partial success. Considering the general disposition of the great tracts of land and ocean, we can scarcely doubt that a careful reduction of the numerous accurate pendulum observations that have been made in locali the sea level able from ments of rendered local irregu ties widely spread over the earth* will lead to the determina- Figure of tion of an ellipsoid with three unequal axes coinciding more determin nearly on the whole with the true figure of the sea level than measuredoes any spheroid of revolution. Until this has been either ac- gravity; complished or proved impracticable it would be vain to specu- difficult by late as to the possibility of obtaining, from attainable data, a larities. yet closer approximation by introducing a harmonic of the third order [f(0, 4) in (27)]. But there is little probability that harmonics of the fourth or higher orders will ever be found useful and local quadratures, after the example first set by Maskelyne in his investigation of the disturbance produced by Schehallien, must be resorted to in order to interpret irregularities in particular districts; whether of the amount of gravity shown by the pendulum; or of its direction, by geodetic observation. We would only remark here, that the problems presented by such local quadratures with reference to the amount of gravity seem about as much easier and simpler than those with reference to its direction as pendulum observations are than geodetic measurements: and that we expect much more knowledge regarding the true figure of the sea level from the former than from the latter, although it is to the reduction of the latter that the most laborious efforts have been hitherto applied. We intend to return to this subject in Vol. II. in explaining, under Properties of Matter, the practical foundation of our knowledge of gravity. geodesy. 797. Since 1860 geodetic work of extreme importance has Results of been in progress, through the co-operation of the Governments of Prussia, Russia, Belgium, France, and England, in connecting the triangulation of France, Belgium, Russia, and Prussia, which were sufficiently advanced for the purpose in 1860, with the principal triangulation of Great Britain and * In 1672, a pendulum conveyed by Richer from Paris to Cayenne first proved variation of gravity. Captain Kater and Dr Thomas Young, Trans. R. S., 1819. Biot, Arago, Mathieu, Bouvard, and Chaix; Base du Système Métrique, Vol. II., Paris, 1821. Captain Edward Sabine, R.E., "Experiments to determine the Figure of the Earth by means of the Pendulum;" published for the Board of Longitude, London, 1825. Stokes "On the Variation of Gravity at the Surface of the Earth."-Camb. Phil. Trans., 1849. geodesy. Results of Ireland, which had been finished in 1851. With reference to this work, General Sir Henry James made the following remarks:-"Before the connexion of the triangulation of the "several countries into one great network of triangles extend"ing across the entire breadth of Europe, and before the dis"covery of the Electric Telegraph, and its extension from "Valentia (Ireland) to the Ural mountains, it was not possible "to execute so vast an undertaking as that which is now in "progress. It is, in fact, a work which could not possibly "have been executed at any earlier period in the history "of the world. The exact determination of the Figure and "Dimensions of the Earth has been one great aim of astrono"mers for upwards of two thousand years; and it is fortunate "that we live in a time when men are so enlightened as to " combine their labours to effect an object which is desired by all, "and at the first moment when it was possible to execute it." For yet a short time, however, we must be contented with the results derived from the recent British Triangulation, with the separate measurements of arcs of meridians in Peru, France, Prussia, Russia, Cape of Good Hope, and India. The investigation of the ellipsoid of revolution agreeing most nearly with the sea level for the whole Earth, has been carried out with remarkable skill by Captain (now Colonel) A. R. Clarke, R.E., and published in 1858, by order of the Master General and Board of Ordnance (in a volume of 780 pages, quarto, almost every page of which is a record of a vast amount of skilled labour). The following account of conclusions subsequently worked out regarding the ellipsoid of three unequal axes most nearly agreeing with the sea level, is extracted from the preface to another volume recently published as one item of the great work of comparison with the recent triangulations of other countries* "In computing the figures of the meridians and of the * "Comparisons of the Standards of Length of England, France, Belgium, Prussia, Russia, India, Australia, made at the Ordnance Survey Office, Southampton, by Captain A. R. Clarke, R.E., under the direction of Colonel Sir Henry James, R.E., F.R.S." Published by order of the Secretary of State for War, 1866. equator for the several measured arcs of meridian, it is found Results of geodesy. 'that the equator is slightly elliptical, having the longer "diameter of the ellipse in 15° 34' east longitude. In the eastern hemisphere the meridian of 15° 34' passes through 'Spitzbergen, a little to the west of Vienna, through the Straits 'of Messina, through Lake Chad in North Africa, and along "the west coast of South Africa, nearly corresponding to the "meridian which passes over the greatest quantity of land in 'that hemisphere. In the western hemisphere this meridian passes through Behring's Straits and through the centre of the "Pacific Ocean, nearly corresponding to the meridian which passes over the greatest quantity of water of that hemisphere. "The meridian of 105° 34' passes near North-East Cape, in "the Arctic Sea, through Tonquin and the Straits of Sunda, and corresponds nearly to the meridian which passes over the 'greatest quantity of land in Asia; and in the western hemi"sphere it passes through Smith's Sound in Behring's Straits, near Montreal. near New York, between Cuba and St Do"mingo, and close along the western coast of South America, "corresponding nearly to the meridian passing over the greatest "amount of land in the western hemisphere. "These meridians, therefore, correspond with the most remarkable physical features of the globe. Feet. "The longest semi-diameter of the equatorial ellipse is 20926350 66 And the shortest 20919972 Giving an ellipticity of the equator equal to ....... 3269'5 'The polar semi-diameter is equal to "The maximum and minimum polar compressions ..... "Or a mean compression of very closely 20853429 Fourteen years later Colonel Clarke corrected this result in the following statement*: "But these are affected by the error Extracted from pages 308, 309 of "Geodesy," by Col. A. R. Clarke, C.B. Oxford, 1880. |