the line PK, and is thence given. But if the body goes from its place P with a yet greater velocity, the length PH is to be taken on the other ſide the tangent; and ſo the tangent paſſing between the foci, the figure will be an hyperbola having its principal axe equal to the difference of the lines SP and PH, and thence is given. For if the body, in theſe caſes, revolves in a conic ſection ſo found, it is demonſtrated in prop. 11, 12, and 13, that the centripetal force will be reciprocally as the ſquare of the diſtance of the body from the centre of force S; and therefore we have rightly determined the line PQ, which a body let go from a given place P with a given velocity, and in the direction of the right line PR given by poſition, would deſcribe with ſuch a force. Q. E. F.
Cor. 1. Hence in every conic ſection, from the principal vertex D, the latus rectum L, and the focus S given, the other focus H is given. by taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS. For the proportion, SP + PH to PH as 2PS + 2KP to L, becomes, in the caſe of this corollary, DS + DH to DH as 4DS to L, and by diviſion DS to DH as 4DS - L to L.
Cor. 2. Whence if the velocity of a body in the principal vertex D is given, the orbit may be readily found; to wit, by taking its latus rectum to twice the diſtance DS, in the duplicate ratio of this given velocity to the velocity of a body revolving in a circle at the diſtance DS (by cor. 3. prop. 16.) and then taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS.
Cor. 3. Hence alſo if a body move in any conic ſection, and is forced out of its orbit by an impulſe; you may diſcover the orbit in
