Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/337

Book III. les, and is to its greatet quantity, as the ine of double the ditance of the Moon's apogee from the nearet , or quadrature to the radius.

By the ame theory of gravity, the action of the Sun upon the Moon is omething greater, when the line of the Moon's nodes paes through the Sun, than when it is at right angles with the line which joins the Sun and the Earth. And hence aries another equation of the Moon's mean motion, which I hall call the econd emi-annual, and this is greatet when the nodes are in the octants of the Sun, and vanihes when they are in the yzygies or quadratures; and in other poitions of the nodes is proportional to the ine of double the ditance of either node from the nearet yzygy or quadrature. And it is added to the mean motion of the Moon, if the Sun is in antecedentiâ to the node which is nearet to him, and ubducted if in conequentiâ; and in the octants, where it is of the greatet magnitude, it aries to 47" in the mean ditance of the Sun from the Earth, as I find from the theory of gravity. In other ditances of the Sun this equation, greatet in the octants of the nodes, is reciprocally as the cube of the Sun's ditance from the Earth, and therefore in the Sun's perigee it comes to about 49", and in its apogee to about 45".

By the ame theory of gravity, the Moon's apogee goes forward at the greatet rate, when it is either in conjunction with or in oppoition to the Sun, but in its quadratures with the Sun it goes backward. And the eccentricity comes, in the former cae, to its greatet quantity, in the latter to its leat, by cor. 7. 8. and 9. prop. 66, book I. And thoe inequalities by the corollaries we have name'd, are very great, and generate the principal, which I call the emi-annual, equation of the apogee. And this emi-annual equation in its greatet quantity comes to about 12°. 18". as nearly