analysis; it is, therefore, necessary to estimate the disturbing action of one planet at a time, whence the celebrated problem of the three bodies, originally applied to the moon, the earth, and the sun—namely, the masses being given of three bodies projected from three given points, with velocities given both in quantity and direction; and, supposing the bodies to gravitate to one another with forces that are directly as their masses and inversely as the squares of the distances, to find the lines described by these bodies, and their positions at any given instant.
By this problem the motions of translation of all the celestial bodies are determined. It is an extremely difficult one, and would be infinitely more so, if the disturbing action were not very small when compared with the central force. As the disturbing influence of each body may be found separately, it is assumed that the action of the whole system, in disturbing any one planet, is equal to the sum of all the particular disturbances it experiences, on the general mechanical principle, that the sum of any number of small oscillations is nearly equal to their simultaneous and joint effect.
On account of the reciprocal action of matter, the stability of the system depends upon the intensity of the primitive momentum of the planets, and the ratio of their masses to that of the sun—for the
