46. THE way of representing the moon’s motion in this case is thus. T denoting the earth (in fig. 105, 106.) suppose the moon in the point A, its apogeon, or greatest distance from the earth, moving in the direction A F perpendicular to A B, and acted upon from the earth by two such forces as have been named. By that power alone, which is reciprocally in the duplicate proportion of the distance, if the moon let out from the point A with a proper degree of velocity, the ellipsis A M B may be described. But if the moon be acted upon by the sum of the forementioned powers, and the velocity of the moon in the point A be augmented in a certain proportion[193]; or if that velocity be diminished in a certain proportion, and the moon be acted upon by the difference of those powers; in both these cases the line A E, which shall be described by the moon, is thus to be determined. Let the point M be that, into which the moon would have arrived in any given space of time, had it moved in the ellipsis A M B. Draw M T, and likewise C T D in such sort, that the angle under A T M shall bear the same proportion to the angle under A T C, as the velocity, with which the ellipsis A M B must have been described, bears to the difference between this velocity, and the velocity, with which the moon must set out from the point A in order to describe the path A E. Let the angle A T C be taken toward the moon (as in fig. 105.) if the moon be attracted by the sum of the powers; but the contrary way (as in fig. 106.) if by their difference. Then let the line A B be moved into the position C D, and the ellipsis A M B into the situation C N D, so that the point M be translated to L: then the point L shall fall upon the path of the moon A E.