Page:Newton's Principia (1846).djvu/211
Sec. XI.] 205
of natural philosophy.
Cor. 1. Hence it may be easily collected, that if several less bodies P, S, R, &c., revolve about a very great body T, the motion of the innermost revolving body P will be least disturbed by the attractions of the others, when the great body is as well attracted and agitated by the rest (according to the ratio of the accelerative forces) as the rest are by each other mutually.
Cor. 2. In a system of three bodies, T, P, S, if the accelerative attractions of any two of them towards a third be to each other reciprocally as the squares of the distances, the body P, by the radius PT, will describe its area about the body T swifter near the conjunction A and the opposition B than it will near the quadratures C and D. For every force with which the body P is acted on and the body T is not, and which does not act in the direction of the line PT, does either accelerate or retard the description of the area, according as it is directed, whether in consequentia or in antecedentia. Such is the force NM. This force in the passage of the body P from C to A is directed in consequentia to its motion, and therefore accelerates it; then as far as D in antecedentia, and retards the motion; then in consequentia as far as B; and lastly in antecedentia as it moves from B to C.
Cor. 3. And from the same reasoning it appears that the body P caeteris paribus, moves more swiftly in the conjunction and opposition than in the quadratures.
Cor. 4. The orbit of the body P, caeteris paribus, is more curve at the quadratures than at the conjunction and opposition. For the swifter bodies move, the less they deflect from a rectilinear path. And besides the force KL, or NM, at the conjunction and opposition, is contrary to the force with which the body T attracts the body P, and therefore diminishes that force; but the body P will deflect the less from a rectilinear path the less it is impelled towards the body T.
Cor. 5. Hence the body P, caeteris paribus, goes farther from the body T at the quadratures than at the conjunction and opposition. This is said,
however, supposing no regard had to the motion of eccentricity. For if the orbit of the body P be eccentrical, its eccentricity (as will be shewn presently by Cor. 9) will be greatest when the apsides are in the syzygies; and thence it may sometimes come to pass that the body P, in its near approach to the farther apsis, may go farther from the body T at the syzygies than at the quadratures.
Cor. 6. Because the centripetal force of the central body T, by which