the moon, with a brief account of two other remarkable results, which Laplace derived from his researches in the lunar theory. The spheroidal figure of the earth occasions a sensible perturbation of the moon's motion both in longitude and latitude. The inequality in longitude was discovered by Mayer, who was ignorant of the physical cause of it, but represented it in his tables by an empiric equation. Laplace derived the equation from theory, and found it to depend on the longitude of the moon's node. Burg, by a comparison of numerous observations, was led to estimate the greatest value of the coefficient at 6".7. This result gives for the earth's ellipticity. The inequality in latitude was discovered by Laplace to vary with the sine of the moon's true longitude. Its value was derived by Burg and Burckhardt, from the combined observations of Bradley and Maskelyne, and was fixed by them at 8′′, a quantity which implies an ellipticity equal to 31.0°

The agreement between the results derivable from these two distinct equations is very interesting. If the earth were homogeneous, it is demonstrable that the ellipticity would be equal to ; it follows, then, that the density must increase towards the centre-a fact which we know to be true from other sources.

A comparison of arcs of the meridian, measured in different parts of the world, presents a series of anomalous results, which lead us to conclude that the figure of the earth is not that of an exact spheroid. It is remarkable, however, that when two arcs are compared, the distance between which is so great as to obviate the effects of any minute inequalities in the spheroidal figure, they indicate an ellipticity almost equal to that derived from the lunar inequalities. Thus, the result of a comparison between a meridional arc at the equator and one measured in France, gives for the earth's ellipticity.

Another striking result which Laplace derived from his researches was the value of the solar parallax. Among the equations in longitude, he found one involving that element, and varying with the angular distance between the sun and moon. The coefficient of this equation, when compared with observation, was found to give 8".6 for the mean value of the solar parallax. This result agrees with the mean of those obtained by astronomers from observations on the transit of Venus in 1769. "It is very remarkable," says Laplace, "that an astronomer, without leaving his observatory, by merely comparing his observations with analysis, has been enabled to determine with accuracy the magnitude and figure of the earth, and its distance from the sun and moon, elements, the knowledge of which has been the fruit of long and troublesome voyages in both hemispheres." *

Exposition du Systême du Monde, tome ii. p. 91.

F

CHAPTER VI.

Theory of the Figure of the Earth.-Newton.-Huygens.-Maclaurin.-Clairaut.-Attraction of Spheroids.-D'Alembert.--Legendre.-Theory of Laplace.-Motion of the Earth about its Centre of Gravity.-Nutation.-Bradley.-Investigation of Precession and Nutation, by D'Alembert.-The Tides.-Equilibrium Theory.-Researches of Laplace. Stability of the Ocean.-Libration of the Moon.-Galileo.- Hevelius.Newton.-Cassini.-Newton's Explanation of the Moon's Physical Libration.-Researches of Lagrange. - Combination of the Principle of virtual Velocities with D'Alembert's Principle.—Laplace investigates the Effect of the secular Inequalities of the mean Motion upon the Libration in Longitude.-His Theory of Saturn's Rings.

This

THE Figure of the Earth was the first of the subjects treated of in the Principia, which engaged the attention of geometers. In 1690 Huygens published his treatise "De Causa Gravitatis," in which he investigated the ratio of the earth's axis in accordance with his own views of gravity. Assuming the density to be homogeneous, he imagined, like Newton, two fluid columns, reaching from the centre of the earth to the surface; one in the plane of the equator and the other along the polar axis. The particles of the equatorial column were acted upon by gravity and by the centrifugal force arising from their rotation; those of the polar column were acted upon by gravity alone. The equatorial particles being, therefore, severally lighter than the polar, and the two columns being also in equilibrium, it was necessary that the equatorial column should compensate, by its superior length, for the diminished pressure of its particles. Huygens assumed that gravity urged the particles to the centre of the earth with a force varying according to the inverse square of the distance. supposition was inconsistent with the theory of gravitation, for Newton had found that, in consequence of the attraction of the surrounding particles, the tendency of each particle to the centre would vary in the direct ratio of the distance. We have already remarked, however, that Huygens rejected the mutual attraction of the particles of matter, and admitted only their gravity towards a central point. Having computed the lengths of the two columns on the supposition that they were in equilibrium, he found that the equatorial column would exceed the polar, assumed equal to unity, by half the ratio of the equatorial centrifugal force, to the equatorial gravity, or by × 5. Hence the ratio of the two axes would be as 579 to 578. He also found that the increase of gravity at the surface, from the equator to the pole, would vary in the proportion of the square of the sine of the latitude, and that the total increase, supposing the equatorial gravity equal to unity, would be equal to twice the ratio of the equatorial centrifugal force to the equatorial gravity, or 2 × Thus the fraction expressing the ellipticity was to that expressing the total increase of gravity as to 2. Newton's theory, on the other hand, gave the same fraction in the one case as in the other; both being measured by ths the ratio of the equatorial centrifugal force to the equatorial gravity. It is remarkable, however, that the sum of the fractions is the same in both theories for the same values of the last

The ratio of the excess of the equatorial over the polar axis to the latter axis is termed the ellipticity of the spheroid. Hence, if the polar axis be assumed equal to unity, the ellipticity will be represented simply by the excess of the equatorial axis over it.

=

mentioned ratio. Thus, in Newton's theory, the two fractions being both equal to their sum is equal to x; in Huygens' theory, the same sum is equal to xa to + 2 × a fo These are only particular cases of a general theorem discovered by Clairaut, connecting the ellipticity of spheroids with the variation of gravity at their sur faces. This theorem, indeed, supposes the mutual gravitation of the particles of matter, which Huygens refused to admit; but the investigation of that philosopher may be considered as founded on the same principle, by imagining the spheroid to be composed of strata of different densities; the exterior stratum being infinitely rare, and the density thence increasing to the centre, where it is infinite.

The theories of Newton and Huygens involve the two extreme cases of density, and therefore assign the limits of ellipticity for a heterogeneous spheroid revolving round a fixed axis. Hence, since there is strong reason to believe that the density of the earth increases towards the centre, it might naturally be expected that the ellipticity would be comprised between these limits. This conclusion has been verified in the most satisfactory manner by the researches of astronomers, who have found that the ellipticity, whether as determined by the measurement of me ridional arcs, by experiments with the pendulum, or by observations on the motion of the moon, lies between and, the values assigned by the two extreme cases of the problem.

Neither Newton nor Huygens demonstrated à priori that the earth might possibly assume the form of an oblate spheroid. This important step was reserved for Maclaurin*. In his prize memoir on the Tides, which appeared in 1740, this distinguished mathematician proved, by a beautiful application of the ancient geometry, that an oblate spheroid would satisfy the conditions of equilibrium of a homogeneous fluid mass, differing little from a sphere, and endued with a rotatory motion round a fixed axis. He also demonstrated that the increase of gravity from the equator to the pole would vary as the square of the sine of the latitude, and that the ratio of the total increase to the gravity at the equator would be expressed by the fraction representing the ellipticity, or, in other words, by ths the ratio of the equatorial centrifugal force to the equatorial gravity. These results confirmed the assumptions of Newton; but, as they were founded on the supposition of a homogeneous fluid, they were not applicable to the earth, which evidently increased in density towards the centre. They formed, however, an important advance towards a more correct theory of the earth's figure, and on this account deserve to be considered as a valuable contribution to Physical Astronomy. The investigations by means of which Maclaurin arrived at these results have been universally admired for their ingenuity and elegance, and are justly considered as rivalling, in these respects, the most finished models of the ancient geometry.

In 1743 Clairaut published his valuable treatise on the Figure of the Earth. In this work the general equations of the equilibrium of fluids, independently of any hypothesis with respect to the density or the law of the attraction, are for the first time given. By means of these equations Clairaut investigated the figure of the earth on the supposition of the density being heterogeneous; and he found, that in this case also an elliptic spheroid would satisfy the conditions of equilibrium, provided the mass was

Born in 1698, at Kilmoddan, in Argyllshire; died at Edinburgh, in 1746.

disposed in concentric strata of similar forms and homogeneous density. The ellipticities of the successive strata will obviously depend on the law of the density, and the other conditions of the problem; but Clairaut discovered that the following theorem is generally true:-the sum of the fractions expressing the ellipticity and the increase of gravity at the pole is equal to two and a half times the fraction expressing the centrifugal force at the equator. This theorem, combined with that relating to the variation of gravity at the surface, enables us to determine the ellipticity of the earth, by means of observations on the force of gravity, in two different latitudes. Its peculiar value consists in being independent of any hypothesis with respect to the internal constitution of the earth. We have seen that the results obtained by Newton and Huygens offer particular illustrations of this important theorem, which is generally designated by the name of its inventor. Little real progress has been made in the theory of the Figure of the Earth beyond the results to which Clairaut was conducted by his admirable researches on this occasion.

The actual ellipticity of the earth may be determined by three distinct methods. The simplest of these in principle depends on the measurement of two arcs of the meridian lying in different latitudes. The other two methods are derived from the theory of gravitation. One of these is suggested by Clairaut's theorem, and requires a knowledge of the force of gravity in two different latitudes. These data may be found by means of experiments with the pendulum. The other method, assigned by theory, depends on the effect of the earth's ellipticity in disturbing the moon's motion. It may not be uninteresting to compare the results obtained by these three methods; and for this purpose we shall select the examples given by Mr. Airy in his treatise on the Figure of the Earth †.

With respect to the first method, Lambton measured an arc of the meridian in India, comprised between lat. 8° 9′ 38′′ .4, and lat. 10° 59′ 48" .9, and found its length to be 1029100.5 feet. Swanberg, on the other hand, measured a similar arc in Sweden from lat. 65° 31' 32" .2, to lat. 67° 8′ 49′′ .8, and found its length 593277.5 feet. These measures assign as the earth's ellipticity.

Again, at Madras, in latitude 13° 4' 9", the length of the seconds pendulum has been found to be equal to 39.0234 inches; at Melville Island, in latitude 74° 47′ 12", the corresponding length is 39.2070 inches. These results give for the ellipticity.

Lastly, the coefficient of the inequality in the moon's latitude, depending on the spheroidal figure of the earth, is found by observation to be equal to 8". This result indicates an ellipticity equal to 71.

The near agreement of these values of the ellipticity, determined by methods so very dissimilar, constitutes a powerful argument in favour of the theory of gravitation.

It is obvious that the question relative to the figure of the earth, and the variation of gravity at the surface, is intimately connected with the theory of the attraction of spheroids. In 1773 Lagrange demonstrated by analysis the results to which Maclaurin was conducted by his researches on this subject, and extended them to the general form of the

In this enunciation of Clairaut's theorem, the unit of force is represented by the equatorial gravity.

Mathematical Tracts on the Lunar and Planetary Perturbations.

This result does not exactly coincide with Laplace's, on account of a slight difference in the data.

ellipsoid. Maclaurin had limited his investigation to the attraction of particles either contiguous to the surface of the spheroid, or situated in its interior. It was desirable, however, to complete the theory of the subject, by determining the attraction of a point situated anywhere without the spheroid. D'Alembert first gave a theorem, by means of which the attraction in this case might be found, when the particle was situated in the prolongation of one of the axes, and Legendre afterwards discovered a similar theorem applicable to an exterior point, situated anywhere whatever when the attracting body was an ellipsoid of revolution. The problem for the general case of the ellipsoid presented analytical difficulties, which continued for some time to elude the researches of the most profound analysts. In 1784 Laplace finally succeeded in effecting its solution, but his method was embarrassed with series, and did not by any means possess the elegance and perfection which distinguished the other parts of the theory.

By

In 1782 Laplace explained a general theory of the attraction of ellipsoids. His researches were based wholly upon a partial differential equation of the second order of a very remarkable character, which has been subsequently employed with great success in many important investigations connected with the Physico-mathematical sciences. simple differentiation, he determined the figure assumed by a heterogeneous mass of fluid, differing only in a small degree from a sphere, and by a similar process he also obtained the law of attraction at the external stratum. His results coincided with those to which Clairaut had been already conducted by a less direct analysis. The investigation. of this great geometer is indeed more remarkable, for the method by which he derives the theorems of his predecessors, than for any new light he throws on the difficult subject to which it relates. The calculus he employs in it is described by one of the most eminent mathematicians of the present age, as the most singular in its character, and the most powerful in its application, which has ever been devised *.

The motion of the earth about its centre of gravity was one of those great problems of the system of the world, which demanded for its solution the most advanced principles of mechanical science. Newton's researches on this subject have been admired as one of the most remarkable triumphs of his genius, but a more complete and systematic investigation was rendered desirable by the improved state of analytical mechanics. Further researches were also called for by Bradley's discovery of Nutation. It did not escape the sagacity of Newton, that besides the motion which occasions the Precession of the Equinoxes, the earth's axis would be affected by an oscillatory motion, arising from the variable position of the plane of the equator with respect to the direction of the sun's disturbing force t. In fact, if we suppose the earth to be situated in the vernal equinox, the sun's disturbing force will pass through the plane of the ring formed by the redundant matter at the equator, and, therefore, it can produce no effect on the position of that plane. As the earth proceeds in her course, the sun's force becoming inclined to the ring, will tend to disturb its position, and this disturbance will continually increase to the solstice, where the inclination reaches its maximum. From this point the tendency of the sun to disturb the ring continually

Airy, Encycl. Metrop. Art. Figure of the Earth. + Principia, book i. prop. 66, cor. 20.