| G = | Mk² | {1 − | 3 | m +( | 5 | m − e) sin² φ}, |
| ac | 2 | 2 |
an expression which contains the theorems we have referred to as discovered by Clairault.
The theory of the figure of the earth as a rotating ellipsoid has been especially investigated by Laplace in his Mécanique celeste. The principal English works are:—Sir George Airy, Mathematical Tracts, a lucid treatment without the use of Laplace’s coefficients; Archdeacon Pratt’s Attractions and Figure of the Earth; and O’Brien’s Mathematical Tracts; in the last two Laplace’s coefficients are used.
In 1845 Sir G.G. Stokes (Camb. Trans. viii.; see also Camb. Dub. Math. Journ., 1849, iv.) proved that if the external form of the sea—imagined to percolate the land by canals—be a spheroid with small ellipticity, then the law of gravity is that which we have shown above; his proof required no assumption as to the ellipticity of the internal strata, or as to the past or present fluidity of the earth. This investigation admits of being regarded conversely, viz. as determining the elliptical form of the earth from measurements of gravity; if G, the observed value of gravity in latitude φ, be expressed in the form G = g(1 + β sin² φ), where g is the value at the equator and β a coefficient. In this investigation, the square and higher powers of the ellipticity are neglected; the solution was completed by F.R. Helmert with regard to the square of the ellipticity, who showed that a term with sin² 2φ appeared (see Helmert, Geodäsie, ii. 83). For the coefficient of this term, the gravity measurements give a small but not sufficiently certain value; we therefore assume a value which agrees best with the hypothesis of the fluid state of the entire earth; this assumption is well supported, since even at a depth of only 50 km. the pressure of the superincumbent crust is so great that rocks become plastic, and behave approximately as fluids, and consequently the crust of the earth floats, to some extent, on the interior (even though this may not be fluid in the usual sense of the word). This is the geological theory of “Isostasis” (cf. [Geology]); it agrees with the results of measurements of gravity (vide infra), and was brought forward in the middle of the 19th century by J.H. Pratt, who deduced it from observations made in India.
The sin² 2φ term in the expression for G, and the corresponding deviation of the meridian from an ellipse, have been analytically established by Sir G.H. Darwin and E. Wiechert; earlier and less complete investigations were made by Sir G.B. Airy and O. Callandreau. In consequence of the sin² 2φ term, two parameters of the level surfaces in the interior of the earth are to be determined; for this purpose, Darwin develops two differential equations in the place of the one by Clairault. By assuming Roche’s law for the variation of the density in the interior of the Earth, viz. ρ = ρ1 − k(c/c1)², k being a coefficient, it is shown that in latitude 45°, the meridian is depressed about 3¼ metres from the ellipse, and the coefficient of the term sin²φ cos²φ (= ¼ sin²2φ) is −0.0000295. According to Wiechert the earth is composed of a kernel and a shell, the kernel being composed of material, chiefly metallic iron, of density near 8.2, and the shell, about 900 miles thick, of silicates, &c., of density about 3.2. On this assumption the depression in latitude 45° is 2¾ metres, and the coefficient of sin²φ cos²φ is, in round numbers, −0.0000280.[2] To this additional term in the formula for G, there corresponds an extension of Clairault’s formula for the calculation of the flattening from β with terms of the higher orders; this was first accomplished by Helmert.
For a long time the assumption of an ellipsoid with three unequal axes has been held possible for the figure of the earth, in consequence of an important theorem due to K.G. Jacobi, who proved that for a homogeneous fluid in rotation a spheroid is not the only form of equilibrium; an ellipsoid rotating round its least axis may with certain proportions of the axes and a certain time of revolution be a form of equilibrium.[3] It has been objected to the figure of three unequal axes that it does not satisfy, in the proportions of the axes, the conditions brought out in Jacobi’s theorem (c : a < 1/√2). Admitting this, it has to be noted, on the other hand, that Jacobi’s theorem contemplates a homogeneous fluid, and this is certainly far from the actual condition of our globe; indeed the irregular distribution of continents and oceans suggests the possibility of a sensible divergence from a perfect surface of revolution. We may, however, assume the ellipsoid with three unequal axes to be an interpolation form. More plausible forms are little adapted for computation.[4] Consequently we now generally take the ellipsoid of rotation as a basis, especially so because measurements of gravity have shown that the deviation from it is but trifling.
Local Attraction.
In speaking of the figure of the earth, we mean the surface of the sea imagined to percolate the continents by canals. That this surface should turn out, after precise measurements, to be exactly an ellipsoid of revolution is a priori improbable. Although it may be highly probable that originally the earth was a fluid mass, yet in the cooling whereby the present crust has resulted, the actual solid surface has been left most irregular in form. It is clear that these irregularities of the visible surface must be accompanied by irregularities in the mathematical figure of the earth, and when we consider the general surface of our globe, its irregular distribution of mountain masses, continents, with oceans and islands, we are prepared to admit that the earth may not be precisely any surface of revolution. Nevertheless, there must exist some spheroid which agrees very closely with the mathematical figure of the earth, and has the same axis of rotation. We must conceive this figure as exhibiting slight departures from the spheroid, the two surfaces cutting one another in various lines; thus a point of the surface is defined by its latitude, longitude, and its height above the “spheroid of reference.” Calling this height N, then of the actual magnitude of this quantity we can generally have no information, it only obtrudes itself on our notice by its variations. In the vicinity of mountains it may change sign in the space of a few miles; N being regarded as a function of the latitude and longitude, if its differential coefficient with respect to the former be zero at a certain point, the normals to the two surfaces then will lie in the prime vertical; if the differential coefficient of N with respect to the longitude be zero, the two normals will lie in the meridian; if both coefficients are zero, the normals will coincide. The comparisons of terrestrial measurements with the corresponding astronomical observations have always been accompanied with discrepancies. Suppose A and B to be two trigonometrical stations, and that at A there is a disturbing force drawing the vertical through an angle δ, then it is evident that the apparent zenith of A will be really that of some other place A′, whose distance from A is rδ, when r is the earth’s radius; and similarly if there be a disturbance at B of the amount δ′, the apparent zenith of B will be really that of some other place B′, whose distance from B is rδ′. Hence we have the discrepancy that, while the geodetic measurements deal with the points A and B, the astronomical observations belong to the points A′, B′. Should δ, δ′ be equal and parallel, the displacements AA′, BB′ will be equal and parallel, and no discrepancy will appear. The non-recognition of this circumstance often led to much perplexity in the early history of geodesy. Suppose that, through the unknown variations of N, the probable error of an observed latitude (that is, the angle between the normal to the mathematical surface of the earth at the given point and that of the corresponding point on the spheroid of reference) be ε, then if we compare two arcs of a degree each in mean latitudes, and near each other, say about five degrees of latitude apart, the probable error of the resulting value of the ellipticity will be approximately ±1⁄500ε, ε being expressed in seconds, so that if ε be so great as 2″ the probable error of the resulting ellipticity will be greater than the ellipticity itself.
It is necessary at times to calculate the attraction of a mountain, and the consequent disturbance of the astronomical zenith, at any point within its influence. The deflection of the plumb-line, caused by a local attraction whose amount is k²Aδ, is measured by the ratio of k²Aδ to the force of gravity at the station. Expressed in seconds, the deflection Λ is
Λ = 12″.447Aδ / ρ,