Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/129

Section III.

Of the motion of bodies in eccentric conic ſections.

---

Proposition XI. Problem VI.

If a body revolves in a ellipſis: it is required to find the law of centripetal force tending to the focus of the ellipſis. Pl. 4. Fig. 2.

Plate 4, Figure 2

Let S be the focus of the ellipſis. Draw SP cutting the diameter DK of the ellipſis in E, and the ordinate Qv in x; and compleat the parallelogram QxPR. It is evident than EP, is equal to the greater ſemi-axis AC: for drawing HI from the other focus H of the ellipſis parallel to EC, becauſe CS, CH are equal ES, EI will be alſo equal, ſo that EP is the half ſum of PS, PI that is, (becauſe of the parallels HI, PR, and the equal angles IPR, HPZ) of PS, PH, which taken together are equal to the whole axis 2AC. Draw QT perpendicular to SP, and putting L for the principal latus rectum of the ellipſis (or for $\scriptstyle {\frac {2BC^{2}}{AC}}$) we ſhall have L × QR to