Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/134

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L x Pv will be to Gv x Pv as L to Gv; and (by the properties of the conic ſections) the rectangle GvP is to as to ; and (by cor. 2. lem. 7.) to , the points Q and P coinciding, becomes a ratio of equality; and Qx or is to as to , that is, as to , or (by lem. 12.) aſ to : and, compounding all thoſe ratio's together, we ſhall have L x QR to as or to , or as 2PC to Gv. But the points P and Q coinciding, 2PC and Gv are equal. And therefore the quantities L x QR and , proportional to them, will be alſo equal. Q. E. I.

Let thoſe equalſbe drawn into, and we ſhall have to . And therefore (by cor. 1 & 5. prop. 6.) the centripetal force is reciprocally as , that is, reciprocally in the duplicate ratio of the diſtance SP. Q. E. I.


The ſame otherwiſe.

Find out the force tending from the centre C of the hyperbola. This will be proportional to the diſtance CP. But from thence (by cor. 3. prop. 7.) the force tending to the focus S will be as , that is, becauſe PE is given, reciprocally as. Q. E. I.

And the ſame way it may be demonſŧrated, that the body having its centripetal changed into a centrifugal force, will move in the conjugate hyperbola.