THE MOON’S MOTIONS.

65

explanation removes in reality the real difficultiesexperienced by the learner; for it shows that the equalcurvatures at corresponding points, P, p, p', and P',in the four quadrants AB, B a, a b, and b A, is arelation according with the amount of force exertedby the orb at S on the moving body at these fourpoints. This has been already indicated as respectsthe points P and p, and holds in like manner asrespects the points p' and P'; while it needs no de-monstration to show that at p the curvature must bethe same as at p, since the velocities at these pointsare equal, the forces on the moving body also equal,and the retarding action at p precisejy accordant withthe accelerating action at p', so far as the productionof curvature is concerned; and lastly, it follows inlike manner that the curvature at P' is equal to thecurvature at P.

Before passing from this investigation of ellipticmotion, it may be well to notice in what respect thepoints A and a, B and b are critical points of thebody’s motion:—

(i.) At A the velocity is at a maximum, the dis-tance at a minimum, and the direction of the body’smotion has a mean value, being at right angles to theline from S.

(ii.) At a the velocity is at a minimum, the distanceat a maximum, and the direction of the body’s motion

m its proper place,—and in conjunction with what precedes andfollows,—the demonstration is in reality complete

F