tU THE MATHEMATICAL PRINCIPLES [BOOK J (by Lemma XI) will be changed in the duplicate ratio of PR or QT QT2 Therefore the ratio ~TVD~ remains the same as before, that is, as SP. And QT2 x SP2 — -^ is as SP3, and therefore (by Corol. 1 and 5, Prop. YI) the centripetal force is reciprocally as the cube of the distance SP. Q.E.I. The same otherwise. The perpendicular SY let fall upon the tangent, and the chord PY of the circle concentrically cutting the spiral, are in given ratios to the height SP ; and therefore SP3 is as SY2 X PY, that is (by Corol. 3 and 5, Prop. YI) reciprocally as the centripetal force. LEMMA XII. All parallelograms circumscribed about any conjugate diameters of a given ellipsis or hyperbola are equal among themselves. This is demonstrated by the writers on the conic sections. PROPOSITION X. PROBLEM Y. If a body revolves in an ellipsis ; it is proposed to find the law of thi centripetal force tending to the centre of the ellipsis. Suppose CA, CB to be semi-axes of the ellipsis; GP, DK, con jugate diameters ; PF, Q,T perpendiculars to those diameters; Qvan ^rdinate to the diame ter GP ; and if the parallelogram QvPR be completed, then (by the properties of the jonic sections) the reclangle PvG will be to Qv2 as PC2 to CD2; and (because of the similar triangles Q^T, PCF), Qi>2 to QT2 as PC2 to PF2 ; and, by com position, the ratio of PvG to QT2 is compounded of the ratio of PC2 1< QT2 CD2, and of the ratio of PC2 to PF2, that is, vG to -p— as PC; to_92L^_P_]^_. Put QR for Pr, and (by Lem. XII) BC X CA for CD K PF ; also (the points P and Q coinciding) 2PC for rG; and multiply