J1S THE MATHEMATICAL PRINCIPLES [Book I .of PS, PI; that is (be cause of the parallels IH, PR, and the equal angles IPR, HPZ), of PS, PH, the difference of which is equal to the whole axis 2AC. Draw Q,T perpen dicular to SP; and put ting L for the principal latus rectum of the hy perbola (that is, for 2BC2\ .... -Tp- ) 7 we shall have L X QR to L X Pv as QR to Pv, or Px to Pv, that is (because of the similar tri angles Pxv, PEC), as PE to PC, or AC to PC. And L X Pv will be to Gv X Pv as L to Gv; and (by the properties of the conic sections) the rec tangle G?'P is to Q,v2 as PC2 to CD2 ; and by (Cor. 2, Lem. VII.), Qv2 to Qa* the points Q and P coinciding, becomes a ratio of equality ; and Q,.r2 or Qv2 is to Q,T2 as EP2 to PF2, that is, as CA2 to PF2, or (by Lem. XII.) as CD2 to CB2 : and, compounding all those ratios together, we shall have L X Q,R to Q,T2 as AC X L X PC2 X CD2, or 2CB2 X PC2 X CD2 to PC X Gv X CD2 X CB2, or as 2PC to Gv. But the points P and Q, coinciding. 2PC and Gv are equal. And therefore the quantities L X Q,R arid Q.T2, propor tional to them, will be also equal. Let those equals be drawn into SP2 sp2 x o/r2 ^, and we shall have L X SP2 equal to ^^ . And therefore (by Cor. 1. and 5, Prop. VI.) the centripetal force is reciprocally as L X SP'2. 'hat is, reciprocally in the duplicate ratio of the distance SP. Q,.E.I. TJie same otherwise. Find out the force tending from the centre C of the hype rbola. This will be proportional to the distance CP. But from thence (by Cor. 3, Prop. PE3 VII.) the force tending to the focus S will be as -^-^ th; (t is, because PE is given reciprocally as SP-. Q,.E.I.