Resuming the inquiry into the **invariability** of mean motions, Poisson carried the approximation, with Lagrange's formulae, as far as the squares of the disturbing forces, hitherto neglected, with the same result as to the stability of the system.

Vii., 1776), Laplace announced his celebrated conclusion of the **invariability** of planetary mean motions, carrying the proof as far as the cubes of the eccentricities and inclinations.

Since the four co-ordinates (Cartesian or other) of these two points are connected by the relation which expresses the **invariability** of the length AB, it is plain that virtually three inde pendent elements are re quired and suffice to specify the position of the lamina.

~ Three-dimensional Kinematics of a Rigid Body.The position of a rigid body is determined when we know the positions of three points A, B, C of it which are not colljnear, for the position of any other point P is then determined by the three distances PA, PB, PC. The nine co-ordinates (Cartesian or other) of A, B, C are subject to the three relations which express the **invariability** of the distances BC, CA, AB, and are therefote equivalent to six independent quantities.

The proper share of each in bringing about this memorable result is not easy to apportion, since they freely imparted and profited by one another's advances and improvements; it need only be said that the fundamental proposition of the **invariability** of the planetary major axes laid down with restrictions by Laplace in 1773, was finally established by Lagrange in 1776; while Laplace in 1784 proved the subsistence of such a relation between the eccentricities of the planetary orbits on the one hand, and their inclinations on the other, that an increase of either element could, in any single case, proceed only to a very small extent.