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

 and by conequence by how much the proportion of the econd force LM to the firt force is greater, cæteris paribus. But now the third force SM, attracting the body P in a direction parallel to ST; compoes with the other forces a new force which is no longer directed from P to T; and which varies o much more from this direction, by how much the proportion of this third force to the other forces is greater cæteris paribus; and therefore caues the body P to decribe, by the radius TP, area's no longer proportional to the times; and therefore makes the variation from that proportionality o much greater by how much the proportion of this force to the others is greater. But this third force will increae the variation of the orbit PAB from the elliptical figure before mentioned upon two accounts; firt becaue that force is not directed from P to T; and econdly becaue it is not reciprocally proportional to the quare of the ditance PT. Thee things being premied, it is manifet, that the area's are then mot nearly proportional to the times, when that third force is the leat poible, the ret preerving their former quantity; and that the orbit PAB does then approach nearet to the elliptical figure above-mentioned, when both the econd and third, but epecially the third force, is the leat poible; the firt force remaining in its former quantity.

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