greater axis of the Moon, causes its motion of rotation to participate in the secular inequalities of its motion of revolution, and the same hemisphere to be constantly directed towards the Earth.

The same theory ought to be extended to all the satellites, in which an equality in their motion of rotation and of revolution round their planet has been observed.

The singular phenomenon of the coineidence of the nodes of the equator of the Moon, with those of its orbit, is another consequence of the terrestrial attraction. This was first demonstrated by Lagrange, who by a beautiful analysis was conducted to a complete explanation of all the observed phenomena of the lunar spheroid. The planes of the equator and of the orbit of the Moon, and the plane passing through its centre parallel to the ecliptic, have always very nearly the same intersection; the secular motions of the ecliptic, neither alter the coincidence of the nodes of these three planes, nor their mean inclination,

which the attraction of the Earth constantly maintains the same.

We may observe here, that the preceding phenomena cannot subsist with the hypothesis in which the Moon, originally fluid and formed of strata of different densities, should have taken the figure suited to their equilibrium. They indicate be tween the axes of the Moon, a greater inequality than would take place in this hypothesis. The great inequalities which we observe at the surface of the Moon, have without doubt a sensible influence on these phenomena.

Whenever nature subjects the mean motions of the celestial bodies to determinate conditions, they are always accompanied by oscillations, whose extent is arbitrary. Thus, the equality of the mean motions of revolution and rotation produce a real·libration in this satellite. In like manner the coincidence of the mean nodes of the equator, and lunar orbit, is accompanied by a libration of the nodes of this equator

round those of the orbit, a libration so small as hitherto to have escaped observation. We have seen that the real libration of the greater lunar axis is insensible, and we have observed, (Chap. VI.) that the libration of the three satellites of Jupiter is also insensible. It is remarkable, that these librations, whose extent is arbitrary and might have been considerable, should nevertheless be so very small; we must attribute this to the same causes as originally established the conditions on which they depend.

But relatively to the arbitrary quantities, which relate to the initial motion of rotation of the celestial bodies, it is natural to think that without foreign attractions, all their parts, in consequence of the friction and resistance which is opposed to their reciprocal motion, would in process of time, acquire a permanent state of equilibrium, which cannot exist but with an uniform motion of rotation, round an invariable axis; so that observation should no longer indicate in this motion any other

inequalities than those derived from these attractions. The most exact observations shew that this is the case with the Earth, the same result extends to the Moon, and probably to the other celestial bodies.

CHAP. XV.

Reflections on the Law of Universal Gravitation.

IN considering the whole of the phenomena of the solar system, we may arrange them in the three following classes :

The first embraces the motions of the centre of gravity about the foci of the principal forces which animate them.

The second includes all that relates to the figure and oscillations of the fluids that surround them.

And the third comprehends the motions of these bodies round their centres of gravity. It is in this order that we have explained the different phenomena, and we have seen that they are necessary consequences of the principle of gravitation. This principle has made us acquainted with a great number of inequalities, which