Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/162

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Section V.


How the orbits are to be found when neither focus is given.




Lemma XVII

If from any point P of a given conic ſection, to the four produced ſides AB, CD, AC, DB of any trapezium ABDC inſcribde in that ſection, as many right lines PQ, PR, PS, PT are drawn in given angles, each line to each ſide; the rectangle PQ x PR of thoſe on the oppoſite ſides AB, CD, will be to the rectangle PS x PT of thoſe on the other two oppoſite ſides AC, BD, in a given ratio.


Plate 8, Figure 4
Plate 8, Figure 4

Case 1. Let us ſuppose firſt that the lines drawn to one pair of oppoſite ſides are parallel to either of the other ſides; as PQ and PR (Pl. 8. Fig. 4) to the ſide AC, and PS and PT to the ſide AB. And farther, that one pair of the oppoſite ſides, as AC and BD, are parallel betwixt themſelves; then