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itself moves, revolving round the centre of forces, and 

 we are required to ascertain the line in which the body 

 moves in this moving orbit, as related to the line described 

 by a body moving in a fixed orbit ; or conversely to 

 ascertain the motion in the two orbits. This subject 

 divides itself into two branches, according as the planes 

 in which the motions are performed pass through the 

 centre of forces or not. Motions in the planes of the 

 centre form the subject of the Ninth Section ; the Tenth 

 treats of motions in eccentric planes. Under the former 

 division, a principal object of investigation is that which 

 indeed measures the orbit s motion, and is identical with 

 it, the motion of the apsides ; in other words, the positions 

 successively taken by the two points of the revolving 

 orbit, where the tangents are perpendicular to the axis, 

 and where, consequently, the moving body begins to 

 come back towards the centre from its greatest distance 

 in that direction of the axis ; while, under the latter 

 division of the subject, a main point of discussion is the 

 vibration of pendulums. 



i. If a body, revolving round a centre of forces, is acted 

 upon laterally by any other force beside the centripetal and 

 the centrifugal (or tangential), though the centre may re 

 main fixed, the orbit will not remain so. The axis of the 

 curve described will move forward or backward, according 

 to the direction of the disturbing force. This motion of the 

 axis is considered as a revolving motion of the orbit, and is 

 the subject of our present consideration. The great prac 

 tical importance of the inquiry will presently be shown. 

 Suppose a body moves in an ellipse that is very nearly a 

 circle, the centripetal force being inversely as the square of 

 the distance; the centrifugal force is in the direct proportion 

 of the square of the velocity and the inverse proportion 

 of the distance, jointly ; that is, ( being the distance, 



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