CHAP. XIII.

Of the Precession of the Equinoxes, and of the Nutation of the Axis of the Earth.

EVERY part of nature is linked together, and its general laws connect phenomena with each other, which, in appearance, have not the most remote analogy. Thus, the rotation of the terrestrial spheroid compresses the poles, and this compression, combined with the action of the Sun and Moon, produces the precession of the equinoxes, which, before the discovery of universal gravitation, did not appear to have any connection with the motion of the Earth.

Let us suppose this planet to be an homogeneous spheroid, protuberant at the equator, it may then be considered as composed of a sphere of a diameter equal to

the axis of the poles, and of a meniscus surrounding the sphere, and whose greatest thickness corresponds with the equator of the spheroid. The particles of this meniscus may be considered as so many small moons adhering together, and which make their revolutions in a period equal to the revolution of the Earth on its axis.

The nodes of all their orbits should therefore have a retrograde motion, arising from the action of the Sun, in the same manner as the nodes of the lunar orbit; and from the connection of these bodies together, there should succeed a retrograde motion of the whole meniscus; but this meniscus divides its retrograde motion, with the sphere to which it is attached, which, for this reason, becomes slower; the intersection of the equator and the ecliptic, that is to say, the equinoctial points, should have a retrograde motion. Let us endeavour to investigate both the law and the cause of this phenomena..

And first we will consider the action of the Sun upon a ring, situated in the plane

of the equator. If we conceive the mass of the Sun to be distributed uniformly over the circumference of its orbit, (supposed circular) it is evident that the action of this solid orbit will represent the mean action of the Sun. This action, upon every one of the points of the ring above the ecliptic, being decomposed into two, one in the plane of the ring, and the other perpendicular to it, it follows that the resulting force, arising from these last actions, on all the particles of the ring, is perpendicular to its plane, and situated on its diameter, which is perpendicular to the line of its nodes. The action of the solar orbit, on the part of the ring below the ecliptic, equally produces a resulting force, perpendicular to the plane of the ring, and situated in the inferior part of the same diameter. These two resulting forces combine to draw the ring towards the ecliptic, by giving it a motion round the line of nodes; its inclination, therefore, to the ecliptic, would be diminished by the mean action of the Sun, the nodes

all the time continuing stationary; and this would be the case but for the motion of the ring, which we now suppose to turn round in the same time as the Earth. By this motion, the ring is enabled to preserve a constant inclination to the ecliptic, and to change the effect of the action of the Sun into a retrograde motion of the nodes. It gives to the nodes a variation, which otherwise would be in the inclination, and it gives to the inclination a permanency, which otherwise would rest with the nodes. To conceive the reason of this singular effect, let us suppose the situation of the ring varied an infinitely small quantity, in such a manner that the planes of its two positions intersect each other, in a line perpendicular to the line of nodes.

At the end of any instant whatever, we may decompose the motion of each of its points into two, one of which should subsist alone in the following instant, the other perpendicular to the plane of the ring, and which should be destroyed. It

is evident that the resulting force of these second motions, relative to all the points of the upper part of the ring, will be perpendicular to its plane, and placed on the diameter which we just now considered, and this is equally true for the lower part of the ring. That this resulting force may be destroyed by the action of the solar orbit, and that the ring, by virtue of these forces, may remain in equilibrio on its centre, it is requisite that these forces should be contrary to each other, and their moments, relatively to this point, equal. The first of these conditions requires that the change of position, supposed to be given to the ring, be retrograde; the second condition determines the quantity of this change, and consequently the velocity of the retrograde motion of the nodes. And it is easily demonstrated, that this velocity is proportional to the mass of the Sun, divided by the cube of the distance from the Earth, and multiplied by the cosine of the obliquity of the ecliptic.

Since the planes of the ring, in its two