Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/270

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in a greater than a triplicate ratio of the altitude, the body at its parting from the apſis, will either deſcend to the centre or aſcend in infinitum, according as it deſcends or aſcends at the beginning of its motion. But if the force in its receſs from the centre either decreaſes in a leſs than a triplicate ratio of the altitude, or increaſes in any ratio of the altitude whatſoever; the body will never deſcend to the centre, but will at ſome time arrive at the lower apſis; and on the contrary, if the body alternately aſcending and deſcending from one apſis to another never comes to the centre, then either the force increaſes in the receſs from the centre, or it decreaſes in a leſs than a triplicate ratio of the altitude; and the ſooner the body returns from one apſis to another, the farther is the ratio of the forces from the triplicate ratio. As if the body ſhould return to and from the upper apſis by an alternate deſcent and aſcent in 8 revolutions, or in 4, or 2. or 1; that is if an ſhould be to n as 8 or 4 or or 2 or 1, and therefore be , or , or , or , then the force will be as , or , or , or ; that is, it will be reciprocally as , or , or , or , If the body after each revolution returns to the ſame apſis, and the apſis remains unmoved, then m will be to n as 1 to 1, and therefore will be equal to or and therefore the decreaſe of the forces will be in a duplicate ratio of the altitude; as was demonſtrateds above. If the body in three fourth parts