Page:Newton's Principia (1846).djvu/117

. 3. If the orbit is either a circle, or touches or cuts a circle concentrically, that is, contains with a circle the least angle of contact or section, having the same curvature and 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 $$\scriptstyle SY^{2}\times PV$$. For PV is $$\scriptstyle \frac{QP^{2}}{QR}$$.

. 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.

. 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 be continually drawn back from a rectilinear course, and being detained in the perimeter of that figure, will describe the same by a perpetual revolution. That is, we are to find, by computation, either the solid $$\scriptstyle \frac{SP^{2}\times QT^{2}}{QR}$$ or the solid $$\scriptstyle SY^{2}\times PV$$, reciprocally proportional to this force. Examples of this we shall give in the following Problems.

Let VQPA be the circumference of the circle; S the given point to which as to a centre the force tends; P the body moving 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 chord PV, and the diameter VA of the circle: join AP, and draw QT perpendicular 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 therefore $$\scriptstyle \frac{QRL\times PV^{2}}{AV^{2}}$$ is equal to QT². Multiply those equals by $$\scriptstyle \frac{SP^{2}}{QR}$$, and the points P and Q coinciding, for RL write PV; then we shall have $$\scriptstyle \frac{SP^{2}\times PV^{3}}{AV^{2}}=\frac{SP^{2}\times QT^{2}}{QR}$$. And therefore (by Cor 1 and 5, Prop. VI.)