Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/117

 will be as the vered line directly and the quare of the time inverely. Q. E. D.

And the ame thing may alo be eaily demontrated by corol. 4. lem. 10.

If a body P revolving about the centre S, (Pl. 3. Fig. 2.) decribes a curve line APQ which a right line ZPR touches in any point P; and from any other point Q of the curve. QR is drawn parallel to the ditance SP, meeting the tangent in R; and QT is drawn perpendicular to the ditance SP: the centripetal force will be reciprocally as the olid $$\textstyle \frac {SP^2 \times QT^2}{QR}$$, if the olid be taken of that magnitude which it ultimately acquires when the points P and Q coincide. For QR is equal to the vered ine of double the arc QP, whoe middle is P: and double the triangle SQP, or SP x QT is proportional to the time, in which that double arc is decribed; and therefore may be ued for the exponent of the time.

By a like reaoning, the centripetal force is reciprocally as the olid $$\textstyle \frac {ST^2 \times QP^2}{QR}$$ if ST is a perpendicular from the centre of force on PR the tangent of the orbit. For the rectangles ST x QP and SP x QT are equal.

If the orbit is either a circle, or touches or cuts a circle concentrically, that is contains with a circle the leat angle of contact or ection, having the ame curvature and the ame radius of curvature at the point P; and if P, V be a chord of this circle, drawn from the body through the centre of force; the centripetal force will be reciprocally as the olid $$\scriptstyle ST^2 \times PV$$. For PV is $$\textstyle \frac {QP^2}{QR}$$