Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/321

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and by conſequence by how much the proportion of the ſecond force LM to the firſt force is greater, cæteris paribus. But now the third force SM, attracting the body P in a direction parallel to ST; compoſes with the other forces a new force which is no longer directed from P to T; and which varies ſo much more from this direction, by how much the proportion of this third force to the other forces is greater cæteris paribus; and therefore cauſes the body P to deſcribe, by the radius TP, area's no longer proportional to the times; and therefore makes the variation from that proportionality ſo much greater by how much the proportion of this force to the others is greater. But this third force will increaſe the variation of the orbit PAB from the elliptical figure before mentioned upon two accounts; firſt becauſe that force is not directed from P to T; and ſecondly becauſe it is not reciprocally proportional to the ſquare of the diſtance PT. Theſe things being premiſed, it is manifeſt, that the area's are then moſt nearly proportional to the times, when that third force is the leaſt poſſible, the reſt preſerving their former quantity; and that the orbit PAB does then approach neareſt to the elliptical figure above-mentioned, when both the ſecond and third, but eſpecially the third force, is the leaſt poſſible; the firſt force remaining in its former quantity.

Let the accelerative attraction of the towards S be expreſſed by the line SN; then if the accelerative attractions SM and SN were equal, theſe, attracting the bodies T and P equally and in parallel directions, would not at all change their ſituation with reſpect to each other. The motions of the bodies between themſelves would be the