THE MOON’S MOTIONS.

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body to remain proportionately longer under the in-fluence of the body at S while describing any givensmall arc at p than when describing a correspondingone at P; and it also causes the deflecting influenceof the body at S to be proportionately more effective.Thus the actual curvature of the path at p is exactlyequal to the curvature at P.

At a , then, the curvature is the same as at A, orthe path lies within a circular arc, as la h, about S ascentre. The distance of the body from S begins nowtherefore to diminish. Nor is it difficult to see thatthe course now pursued by the body must be theexact counterpart of that already traversed, only pur-sued in a reverse order ; for all the circumstances aresymmetrically reversed, so to speak. The distance ofthe body diminishing, the course of the body must beinclined at an acute angle to the line from S, and theinfluence of S must therefore act to accelerate themotion of the body. Thus when the body is at p',(as far from S as p is), its course lies for the momentin > the direction p't', and the pull of the orb at S,acting in direction p' S, must needs accelerate thebody’s motion. Also, as the body starts from a withthe same velocity that it had when it reached a, andmoving at the same angle with S a, it is clear that thereduction of its velocity and distance from S, after ithas passed a, must be affected in a manner preciselycorresponding (point for point of its course) to theincrease of its velocity and distance from S before itreached a. Up to the point h (corresponding to the