Cor. 8. The velocity of a body revolving in any conic ſection is to the velocity of a body revolving in a circle at the diſtance of half the principal latus rectum of the ſection, as that diſtance to the perpendicular let fall from the focus on the tangent of the ſection. This appears from cor. 5.
Cr. 9. Wherefore ſince (by cor. 6. prop. 4.) the velocity of a body revolving in this circle is to the velocity of another body revolving in any other circle, reciprocally in the ſubduplicate ratio of the diſtances; therefore ex æquo the velocity of a body revolving in a conic ſection will be to the velocity of a body revolving in a circle at the lime diſtance, as a mean proportional between that common diſtance and half the principal latus rectum of the ſection, to the perpendicular let fall from the common focus upon the tangent of the ſection.
Proposition XVII. Problem IX.
Suppoſing the centripetal force to be reciprocally proportional to the ſquares of the diſŧances of places from the centre, and that the abſolute quantity of that force is known; it is required to determine the line, which a body will deſcribe that is let go from a given place with a given velocity in the direction of a given right line.
Let the centripetal force tending to the point S (Pl. 6. Fig. 3 .) be ſuch, as will make the body p revolve in any given orbit pq; and ſuppoſe the velocity of this body in the place p is known. Then from the place P, ſuppoſe the body P to be let go with a given velocity
