AND ULTIMATE.
Section 2. On Centripetal Force. (1) Equal areas are described in equal times. Six corollaries on the comparison of velocities and forces ; the former inversely as the perpendiculars on the tangents, the latter as the sagittie of area described in equal times. (2) If equal areas bo described in equal times about a centre, fixed-Yr moving straightly and uniformly, the force is centripetal. Two corollaries and scholium. (3) In equiereal motion of a point about a moving centre, that point is acted on by a centripetal force, and by all the accelerating forces which act on the centre. Four corollaries and scholium. (4) In different circles uniformly described, force varies as (vel)' rad. Nino corollaries and scholium indicating the deduction in the ease of the planets. (5) Given the velocities in different parts of an orbit, to find the centre, of force. (6) Centripetal force in the middle of a small arc is as sagitta (tinie)'. Five corollaries ; various ways of comparing forces. (7) The orbit circular, centre anywhere within, to find the law of force. 3 Cor. (S) Ditto, ditto, where the forces act in parallel lines. Scholium ; Same considerations apply to other conic sections. (9) Law of force in equiangular spiral. Lemma 12 (the numbering of the lemmas runs on from the first section). Equality of parallelogram about conjugates in conic sections. (10) Law of force in ellipse about the centre. 2 Cor. and Schol.; extension to the parabola.
Section 3. Motion in conic sections about the focus. (11) Law of force in ellipse about focus. (12) Same for hyperbola. Leinme 13, Latus rectum in parabola always four times focal distance. Lemma 14, Per pendicular on tangent of parabola, mean between focal distances of point of contact and vertex. 3 Cor. (13) Law of force in parabola about focus. 2 Cor. (14) In conic sections about saino focal centre, latent recta are in duplicate ratio of areas described. 1 Cor. (15) In ellipses, periodic tunes are in sesquiplicate ratio of major axes. 1 Cor. (16) And velocities are as perpendiculars on tangents inversely, and subduplicato ratio of latera recta directly. 9 Coe ; comparison of
velocity in conic section and circle. (17) Given initial position and velocity, required conic section describer. 4 Cor. and Schol.
Section 4. On finding conic sections from a qiven focus, and Section 5. On finding conic sections of which no focus is given. These sections, which carry on the lemmas from 15 to 27, both inclusive, and the pro positions from (18) to (29), both Inclusive, are entirely geometrical exercises in drawing conic sections through given points, or touching given straight lines, &e. : the results are hardly of use, even in the rest of_the work, and a particular reference would now be of no use whatever, though the sections themselves are highly interesting to the geometrical inquirer.
Section 6. On finding the motion in a given orbit. (30) To find the place of a body in a parabola at the end of a given time. 3 Cor. Lemma 28. There is no oval figure whose area contained being any two radii, can be obtained by an equation in finite terms. [QUADRATURE OF THE CIRCLE.] 1 Cor.; relates to the ellipse. (31) To find the place of a body in an ellipse at the end of a given time. Scholium ; approximate method.
Section 7. On rectilinear ascent and descent. (32) Required the space described in a given time by a body descending towards a Three cases, derived from the three conic sections. (33) Law of the velocity in the preceding, in the cases derived from the ellipse and hyperbola. 2 Cor. (34) The same in the case derived from the parabola. (35) An equable description of certain areas in the conic sections just alluded to takes place during the motion. (36) The time of the whole descent of a body from rest. (37) The time of the whole descent of a projected body. (38) Velocity and time determined in descent to a centre, the force being as the distance. 2 Cor. (39) Granting the quadrature of curves [QUADRATURE], and the law of force being any whatever, to determine the time and velocity at any point of a descent. 3 Cor.