SEC. II.] OF NATURAL PHILOSOPHY. Ill COR. 3. If the orbit is cither a circle, or touches or cuts a circle c< ncentrically, that is, contains with a circle the least angle of contact or sec tion, having the same curvature rnd the same radius of curvature at the point P : and if PV be a chord of this circle, drawn from the body through the centre of force ; the centripetal force will be reciprocally as the solid QP2 SY2 X PV. For PV is - . COR. 4. The same things being supposed, the centripetal force is as the square of the velocity directly, and that chord inversely. For the velocity is reciprocally as the perpendicular SY, by Cor. 1. Prop. I. COR. 5. Hence if any curvilinear figure APQ, is given, and therein a point S is also given, to which a centripetal force is perpetually directed. that law of centripetal force may be found, by which the body P will bcj continually drawn back from a rectilinear course, and. being detained in the perimeter of that figure, will describe the same by a perpetual revoluSP2 x QT2 tion. That is, we are to find, by computation, either the solid ---- or the solid SY2 X PV, reciprocally proportional to this force. Example: of this we shall give in the following Problems. PROPOSITION VII. PROBLEM II. Tf a body revolves in the circumference of a circle; it is proposed to finii the law of centripetal force directed to any given, point. Let VQPA be the circumference of the circle ; S the given point to which as to a centre the force tends : P the body mov ing in the circumference ; Q the next place into which it is to move; and PRZ the tangent of the circle at the preceding place. Through the point S draw the v chord PV, and the diameter VA of the circle : join AP, and draw Q,T perpen dicular to SP, which produced, may meet the tangent PR in Z ; and lastly, through the point Q, draw LR parallel to SP, meeting the circle" in L, and the tangent PZ in R. And, because of the similar triangles ZQR, ZTP. VPA, we shall have RP2, that is. QRL to QT2 as AV2 to PV2. And QRlj x PV2 SI3-' therefore '- —TS -- is equal to QT2. Multiply those equals by -'. and the points P and Q, coinciding, for RL write PV ; then we shall have SP-' X PV5 SP2 x QT2 — • And therefore fl»r Cor 1 and 5. Prop. VI.)