consecutive positions, intersect each other in a diameter perpendicular to the line of its nodes, it follows that the inclination of these two planes to the ecliptic, is constant, and the inclination of the ring does not vary, by the mean action of the Sun.

That which has been explained relatively to a ring, may be demonstrated by analysis, to hold true in the case of a spheroid, differing but little from a sphere. The mean action of the Sun produces in the equinoxes a motion proportional to its mass, divided by the cube of its distance, and multiplied by the cosine of the incli nation to the ecliptic. This motion is retrograde when the spheroid is flattened at the poles; its velocity depends on the compression of the spheroid, but the inclination of the equator to the ecliptic, always remains the same.

The action of the Moon produces likewise a similar retrogradation of the nodes of the terrestrial equator, in the plane of its orbit; but the position of this plane and its inclination to the equator inces

santly varying, by the action of the Sun and the retrograde motion of the nodes produced by the action of the Moon, being proportional to the cosine of this inclination, this motion is variable.

Besides, in supposing it uniform, it would, according to the position of the lunar orbit, cause a variation both in the retrograde motion of the equinoxes, and in the inclination of the equator to the ecliptic. A calculation, by no means difficult, is sufficient to show, that the action of the Moon, combined with the motion of the plane of its orbit, produces 1. A mean motion in the equinoxes, equal to that which it would produce if it moved in the plane of the ecliptic. 2. An inequality subtractive, from this retrograde motion, and proportional to the sine of the longitude of the ascending node of the lunar orbit. 3. A diminution in the obliquity of the ecliptic, proportional to the cosine of this same angle. These two inequalities are represented at once by the motion of the extremity of the terrestrial axis (pro

longed to the heavens) round a small ellipse, conformably to the laws explained in Chap. XI. of Book I.

The greater axis of this ellipse is to the lesser, as the cosine of the obliquity of the ecliptic is to the cosine of double this obliquity. We may comprehend from what has been said, the cause of the precession of the equinoxes, and of the nutation of the Earth's axis, but a rigorous calculation, and a comparison of its results with observation, is the true test of the truth of a theory. That of universal gravitation is indebted to d'Alembert, for the advantage of having been thus verified in the case of the two preceding phenomena. This great mathematician first determined by a beautiful analysis the motions of the axis of the Earth, by supposing the strata of the terrestrial spheroid to be of any density or figure whatever, and he not only found his results exactly conformable to observation, but obtained an accurate determination of the dimensions of the small ellipse described by the pole of the

Earth, as to which the observations of Bradley had left some little doubt.

The influence of a heavenly body, either upon the motion of the terrestrial axis of the Earth, or upon the ocean, is always proportional to the mass of that body, divided by the cube of the distance of that body from the Earth. The nutation of the Earth's axis being due to the action of the Moon alone, while the precession of the equinoxes arises from the combined action of the Sun and Moon, it follows that the observed values of these two phenomena, should give the ratio of their respective actions. If we suppose, with Bradley, the annual precession of the equinoxes to be *154"4, and the whole extent of the nutation +55"6, the action of the Moon is found to be double that of the Sun. But a very small difference in the extent of the nutation, produces a very considerable one in the ratio of the actions of these two bodies, to make it equal three

* 50'.

ODL

+ 17'9.

to one, as indicated by the observations of the tides, it is sufficient to suppose the extent of the nutation *62.2. Dr. Maskelyne, by re-examination of the observations. of Bradley, finds this quantity +58"6, which differs but 3"6 from the result obtained by the phenomena of the flux and reflux of the ocean. So small a quantity being nearly insensible in the observations of the fixed stars, the ratio of the solar and lunar action is better determined by that of the tides; it seems to me, therefore, that we should fix the equation of the nutation at §31"1, that of the precession at 582, and the lunar equation of the tables of the Sun, 127"5. The phenomena of the precession and of the nutation throws a new light on the constitution of the terrestrial spheroid. They gave a limit to the compression of the earth supposed elliptic, hence it appears that this compression does not exceed, which accords with the experiments that have

* 20" 1. + 18" 9.1" 1. § 10′′. ¶ 8" 9.