leſs, and is to its greateſt quantity, as the ſine of double the diſtance of the Moon's apogee from the neareſt ſyzygy, or quadrature to the radius.
By the ſame theory of gravity, the action of the Sun upon the Moon is ſomething greater, when the line of the Moon's nodes paſſes through the Sun, than when it is at right angles with the line which joins the Sun and the Earth. And hence ariſes another equation of the Moon's mean motion, which I ſhall call the ſecond ſemi-annual, and this is greateſt when the nodes are in the octants of the Sun, and vaniſhes when they are in the ſyzygies or quadratures; and in other poſitions of the nodes is proportional to the ſine of double the diſtance of either node from the neareſt ſyzygy or quadrature. And it is added to the mean motion of the Moon, if the Sun is in antecedentiâ to the node which is neareſt to him, and ſubducted if in conſequentiâ; and in the octants, where it is of the greateſt magnitude, it ariſes to 47" in the mean diſtance of the Sun from the Earth, as I find from the theory of gravity. In other diſtances of the Sun this equation, greateſt in the octants of the nodes, is reciprocally as the cube of the Sun's diſtance from the Earth, and therefore in the Sun's perigee it comes to about 49", and in its apogee to about 45".
By the ſame theory of gravity, the Moon's apogee goes forward at the greateſt rate, when it is either in conjunction with or in oppoſition to the Sun, but in its quadratures with the Sun it goes backward. And the eccentricity comes, in the former caſe, to its greateſt quantity, in the latter to its leaſt, by cor. 7. 8. and 9. prop. 66, book I. And thoſe inequalities by the corollaries we have name'd, are very great, and generate the principal, which I call the ſemi-annual, equation of the apogee. And this ſemi-annual equation in its greateſt quantity comes to about 12°. 18". as nearly
