SJSG. IL] OF NATURAL PHILOSOPHY. 113 PROPOSITION VIII. PROBLEM III. If a body mi ues in the semi-circuwferencePQA: it is proposed to find the law of the centripetal force tending to a point S, so remote, that all the lines PS. RS drawn thereto, may be taken for parallels. From C, the centre of the semi-circle, let the semi-diameter CA he drawn, cutting the parallels at right angles in M and N, and join CP. Because of the similar triangles CPM, PZT, and RZQ, we shall have CP2 to PM2 as PR2 to QT2; and, from the na ture of the circle, PR2 is equal to the rect angle QR X RN + QN, or, the points P, Q coinciding, to the rectangle QR x 2PM. Therefore CP2 is to PM2 as QR X 2PM to QT2; and QT2 2PM3 QT2 X SP2 2PM3 X SP2 therefore (by QR Corol. 8PM3 X SP2 , and QR And 1 and 5, Prop. VI.), the centripetal force is reciprocally as 2SP2 . that is (neglecting the given ratio -ppr)> reciprocally as PM3. Q.E.L And the same thing is likewise easily inferred from the preceding Pro position. SCHOLIUM. And by a like reasoning, a body will be moved in an ellipsis, or even ia an hyperbola, or parabola, by a centripetal force which is reciprocally ae the cube of the ordinate directed to an infinitely remote centre of force. PROPOSITION IX. PROBLEM IV. If a body revolves in a spiral PQS, cutting all the radii SP, SQ, fyc., in a given angle; it is proposed to find thelaio of the centripetal force tending to tJie centre of that spiral. Suppose the inde finitely small angle AY PSQ to be given ; be cause, then, all the angles are given, the figure SPRQT will , _ be given in specie. v QT Q,T2 Therefore the ratio -7^- is also given, and „„ is as QT, that is (be lot IX QK cause the figure is given in specie), as SP. But if the angle PSQ is any way changed, the right line QR, subtending the angle of contact QPU